WITH PHASE TRANSITION:

Harnideh Bastani-Parizi

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Mechanical and Induslial Engineering University of Toronto

O Copyright by Hamideh 8. Parizi 1999

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ABSTRACT THERMAL DESORPTION OF HYDROGEN WITH PHASE TRANSITION:

STATISTICAL RATE THEORY APPROACH

Harnideh Bas tani-Parizi

A thesis submitted for the degree of Doctor of Philosophy, Graduate Department of Mechanical and Industrial Engineering, University of Toronto,

1999

A method is presented for predicting the values of temperature and surface cover-

age at which a surface phase formed by homogeneously adsorbed atoms become an unsta-

ble phase. It is shown that a necessary condition for the intrinsic stability of such a phase

can be expressed in terms of the partial derivative of the chemical potential of the atoms in

the adsorbed phase. To construct the expression for this function, the canonical partition

function is obtained for an atomically adsorbed system. It is found that the adatoms can be

approximated as three dimensional harmonic oscillators with coverage dependent frequen-

cies and potential. To determine the coverage dependent functions appearing in the chemi-

cal potential expression, the measured isotherms are used. To examine the procedure

quantitativeIy, it is applied to examine H2-Ni(100) and H2-Ni(1 lo). It is predicted that the

former does not undergo phase transitions as a result of adsorption, but that the latter can.

These predictions are in agreement with the available, qualitative experimental evidence.

To examine these predictions, the expression for the chemical potential has been

used in the statistical rate theory (SRT) approach and an expression for the net rate of

adsorption developed. This rate expression was then used in a procedure to predict the

temperature programmed desorption (TPD) spectrua. Certain features of the TPD spec-

trum can be predicted and these may be critically compared with the measurements. ]In

particular, it was found that the phase transition in the H2-Ni(ll0) system produces a

sharp change in dependence of the maximal temperature on initial coverage. Previously

this change had been attributed to a change in the "order" of the kinetics, but in SRT pro-

cedure no such empirical parameter enters theory.

The developed approach is used for the H2-W(100) system. A method is presented

to estimate the coverage dependence of the potential for this system by using one TPD

spectrum. When this function is used in the expression for the chemical potential, a first

order phase transition on the surface at coverages above 1.1 is predicted. The SRT equa-

tions are used to obtain the TPD spectra for this system. Two distinguishable peaks in the

TPD spectra are predicted. The maxima1 temperature for the first peak is found to be inde-

pendent of the initial coverage after a certain value. Both predictions are in agreement with

the experimental results.

iii

I had the opportunity to work with two great supervisors during my Ph.D. program.

Professor Charles A, Ward, who opened a new window to the spectacular world of thermo-

dynamics for me. I appreciate his concerns about this work and his helpful comments. I

am also thankful to Professor Javad Mostaghimi with whom I started my graduate studies

at the University of Toronto and who encouraged me to pursue the Ph.D. program. He was

and will always be a source of inspiration to me.

I would like to thank my family: my parents for their constant encouragements through

years of being in different schools and universities; my son for his patience, and my hus-

band for his support.

I would also like to thank Ian Thomson and Payam Rahimi for helping me throughout

the writing process of this thesis.

I am also grateful to other members of my family and friends, without their help, I

could not finish this work.

This work was supported by the Natural Sciences and Engineering Research Council

of Canada and the University of Toronto.

.. ABSTRACT ...................................................................................................................... II

ACKNOWLEDGMENTS .............................................................................................. iv

................................................................................................. TABLE OF CONTENTS v

... LIST OF TABLES ........................................................................................................ VIII

LIST OF FIGURES ....................................................................................................... ix ... LIST OF SYMBOLS .................................................................................................... x111

.................................................................................... CHAPTER 1: INTRODUCTION 1

1.1 Equilibrium Adsorption Isotherms ......................................................................... 2

1.2 Intrinsic Stability of the Surface Phase ................................................................... 5

1.3 Temperature Programmed Desorption Spectra ....................................................... 7

............................................................................................. 1.4 Scope of This Work 1 2

CHAPTER 2: INTRINSIC STABILITY OF SURFACE PHASES: QUANTUM- STATISTICAL APPROACH ................................................................ 18

2.1 Introduction .......................................................................................................... 18

2.2 Intrinsic Stability of an Adsorbed Phase ............................................................... 20

2.2.1 Conditions for intrinsic stability ................................................................. 22

2.2.2 Condition on the Helmholtz function for intrinsic stability ........................ 25

2.2.3 Canonical partition function ....................................................................... 26

......................................................................... 2.3 Equilibrium Adsorption Isotherm 29

............ 2.4 Coverage Dependence of the Chemical Potential and Intrinsic Stability 31

2.4.1 Dependence of NakT lnv - (Do on coverage ............................................ 31

2.4.2 Intrinsic stability of H2-Ni(1 00) and H2-Ni(1 10) ...................................... 35 2.5 Discussion and Conclusion .................................................................................. 37

CHAPTER 3: THERMAL DESORPTION OF HYDROGEN FROM NI(100) AND NI(l10): STATISTICAL RATE THEORY APPROACH .................. 49

3.1 Introduction ........................................................................................................... 49

................................................................................................. 3.2 System definition 51

3.3 Statistical Rate Theory Expression For the Rate of Hydrogen Adsorption .......... 53

3.3.1 Rate of exchange between the surface and gas phase under equilibrium con- ditions in the isolated system .......................................................................... 53

3.3.2 Instantaneous adsorption rate in the isolated system .................................. 58

.................................... 3.4 H2-Ni(1 00) Temperature-Programed-Desorption Spectra 62

.................................... 3.5 H2-Ni(1 10) Temperature-Programed-Desorption Spectra 64

3.6 Discussion: Application of SRT to a Heterogeneous Surface .............................. 66

CHAPTER 4: THERMAL DESORPTION OF HYDROGEN FROM W(100)- .................. PREDICTION OF A SURFACE PHASE TICANSITION 79

........................................................................................................... 4.1 Introduction 79

.......................................................... 4.2 Chemical Potential of H2-W (1 00) System 81

......................................... 4.2.1 Equilibrium adsorption isobars for H2-W(l 00) 3 2

4.2.2 Method for estimating the P function ........................................................ 84

4.3 Temperature Programmed Desorption Spectra ..................................................... 85

4.3.1 The governing equations ............................................................................ -85

............................................................. 4.3.2 TPD measurements of Hz-W(1OO) 86

4.3.3 Procedure for finding the P function .......................................................... 89

...................................................... 4.3.4 TPD spectra calculated for HTW(lOO) 91

.................................................... 4.4 Intrinsic Stability Examination for H2-W(l 00) 92

4.5 Discussion and Conclusion ................................................................................... 94

CHAPTER 5: SUMMARY AND CONCLUSIONS .................................................. 113

Appendix A: Hydrogen .................................................................................................. 118

............................................................................ Appendix B: Numerical Procedure 120

B . 1 Calculating the Equilibrium Exchange Rate ...................................................... 120

.................................................................. B . 1.1 Limits for H2-Ni(1 10) system 122

B . 1.2 Limits for Hz-W(100) system .................................................................. 123

B.2 Numerical Solution of the PDE-s ....................................................................... 123

Appendix C: Fortran Program for Calculating the TPD Spectra of Hz-W(100) System ............................................................................................................................ 128

Appendix D: Numerical Solver for Solving the PDE-s .............................................. 134

vii

LIST OF TABLES

TABLE 2.1 . Properties of Hz-Ni(1 00) .............................................................................. 39

TABLE 2.2. Properties of H2-Ni(1 10) .............................................................................. 40

TABLE 3.1. TPD experimental information for H2-Ni(100) and Hz-Ni(1 10) ................. 69

TABLE 4.1. Coefficients for the vibrational frequencies of hydrogen on W(100) . The fre- ................................. quencies are assumed to be second order polynomials of coverage 98

TABLE 4.2. TPD experimental information for H2-W(100) system ............................... 99

TABLE 4.3. Properties for hydrogen adsorbed on W(100) ........................................ 100

TABLE A . 1 . Properties of hydrogen molecules in a gas phase ...................................... 119

viii

LIST OF FIGURES

CHAPTER 1: INTRODUCTION

FIGURE 1.1. Illustration of a dissociative adsorption process ................... ... .......... 14

FIGURE 1.2. The TPD spectra measured for Hz-Ni(100) by Christmann et aL2The temperature at the maximum pressure moves to the lower temperatures with increasing initial coverage. The initial coverages are in terms of Langmuir (L) ................................................................................................................. 15

FIGURE 1.3. TPD spectra measured by Christmann et a12 for H2-Ni(1 10). The temperature at the maximum pressure moves to higher values by increasing the initial coverages and stays almost constant beyond a certain initial coverage. The initial coverages are in terms of Langmuir (L) ..................... 16

FIGURE 1.4. The TPD spectra of Hz-W(100) measured by Madey and ~ a t e s ' 'at a variable heating rate.While the P, state appears at higher initial coverages, the p, state can be observed for all initial coverages ........................................ 17

CHAPTER 2: INTRINSIC STABILITY OF SUFWACE PHASES: QUANTUM- STATISTICAL APPROACH

FIGURE 2.1. Isotherms for H2-Ni(100) inferred from measurements2 of the dependence of the work function on coverage and of coverage as a function of pressure (solid dots). The isotherms calculated by the quantum-statistical method are

...................................................................................... shown as solid lines 41

FIGURE 2.2. Isotherms for H2-Ni(1 lo). The data points (solid dots) were inferred from n~easurernents~~ 34 of the dependence of the work function on coverage and of coverage as a function of pressure, and isotherms calculated by the quantum-

...................................................................... statistical method (solid lines) .42

FIGURE 2.3. Examination of the procedure used to obtain the analytical expression for the coverage dependence of the chemical potential of hydrogen adsorbed on Ni(100). The symbols are the values of (b - P ) determined from the measured isotherms and the solid lines are the calculated values ................................ 43

FIGURE 2.4. The chemical potential of hydrogen adsorbed on Ni(100) is shown as a function of coverage for two temperatures of 300K and 400 K.., ................ 44

FIGURE 2.5. Examination of the procedure used to obtain the andytical expression for the coverage dependence of the chemical potential of hydrogen adsorbed on Ni(1 lo). The symbols are the values of (b - P ) determined from the measured isotherms and the solid lines are the calculated values. ................................ 45

FIGURE 2.6. The chemical potential of hydrogen adsorbed on Ni(ll0) is shown as a function of coverage for a temperature of 441K. For coverages in the range from 0, to Bp , a homogeneous surface phase is intrinsically unstable. The

.... instability leads to the predicted hysteresis loop indicated by the arrows 46

FIGURE 2.7. Intrinsic stability of H2-Ni(1 10) isotherms. When the ordinate is negative, it is predicted that the homogeneous surface phase is intrinsically unstable. For H2-Ni(1 lo), it is predicted that there are four types of isotherms. One with no phase transitions (upper left), one with an unstable condition at low coverage (lower left), one with two unstable coverage ranges (upper right) and one with an unstable range at higher coverages (lower right). The chemical potential as a function of coverage for the plot on the lower right is shown in Fig. 2.6 ..47

FIGURE 2.8. Instability "islands" for the H2-Ni(1 10) system. When the coverage and temperature correspond to a position in an island, it is predicted that the homogeneous surface phase is unstable. ...................................................... 48

CHAPTER 3: THERMAL DESORPTION OF HYDROGEN FROM NI(100) AND NI(110):STATISTICAL RATE THEORY APPROACH

FIGURE 3.1. Schematic of TPD apparatus and its associated isolated system. .......... 70

FIGURE 3.2. The region for unstable equilibrium, 8, S 9 S Bp shown on an isotherm. The instability leads to the predicted hysteresis loop indicated by the arrows . ....................................................................................................................... 7 1

FIGURE 3.3. Calculated TPD spectra for H2-Ni(lO0) using SRT for the initial coverages indicated when the heating rate was 15 Ws, the pumping rate was 110 L/s and the gas phase volun~e was 100 L. ................................................................. 72

FIGURE 3.4. The TPD spectrum for Hz-Ni(100) that was used to establish the pumping speed and the pressure scale. The solid dots are measurements reported in Ref. 2 when the initial coverage was 5.2 Langmuirs. The solid curve was calculated when the initial coverage was chosen to be 0.45 .......................................... 73

FIGURE 3.5. For Hz-Ni(100) the maximum pressure versus the temperature, T,,, , at which the maximum occurred. The open circles are the calculated values and the solid dots are the measured values. The initial coverages used in the calculation were 0.49, 0.40,0.35,0.30,0.25 0.2,0.15,0.10,0.05, and the experimental values reported in Ref. 2 were 5.2, 1.2,0.7,O.S, 0.4, 0.3, 0.25, 0.2, 0.15, 0.1 Langmuirs. .............................................................................. 74

FIGURE 3.6. The calculated TPD spectra for H2-Ni(1 10) using SRT for the initial coverages indicated when the heating rate was 15 K/s, the pumping rate was

............................................. 110 U s and the gas phase volume was 100 L. 75

FIGURE 3.7. The solid line is the calculated TPD spectrum using SRT for H2-Ni(1 10)

that had a T,,, of 348 K. The data points are from an experimental TPD spectrum that had the same value of T,,, , but its pressure was reported in "arbitrary unit^".^ The experimental pressure has been scaled so that it agrees with the predicted pressure at T,, ................................................................. 76

FIGURE 3.8. For H2-Ni(1 10) the maximum pressure versus the temperature, T,,, , at which the maximum occurred. The open circles are the calculated values and the solid dots are the measured values. The initial coverages used in the calculation were 1.45, 1.2,1.10, 1.0,0.90,0.8,0.70,0.6,0.56,0.53,0.50,0.45, 0.40,0.35,0.30,0.25,0.20,0.10 and the experimental values reported in Ref. 2 were 1.55, 1.30, 1.05, 0.65,0.55,0.40,0.30,0.17 Langmuirs. Note that for initial coverages above 0.5 the slope of the Maximum Pressure-Maxima Temperature changes slope. This is the initial coverage at which a phase

.................... transition takes place during a TPD experiment (see Fig. 3.9). 77

FIGURE 3.9. The solid lines are the predicted coverages as a function of surface temperature for different initial coverages that were calculated using SRT for Ni(ll0) when the heating rate was 15 Ws, the pumping rate was 1 10 Ws and the gas phase volume was 100 L. Note that in all experiments in which the initial coverage was 0.5 or above, the temperature and coverage are predicted to have values that would result in a phase transition, but that none of those with initial coverages below 0.5 are predicted to undergo phase transitions 78

CHAPTER 4: THERMAL DESORPTION OF HYDROGEN FROM W(100)- PREDICTION OF A SURFACE PHASE TRANSITION

FIGURE 4.1. Equilibrium adsorption isobars for Hz-W(100). Symbols are data from Horlacher et ala3 Solid lines are calculated based on the model presented in the text. ............................................................................................................. 101

FIGURE 4.2. Family of P functions for Hz on W(100) obtained from equilibrium adsorption isobars. The additional data points between coverage of 1 and 2 are the artificial data points that are used to make an approximation for the P function. All curves are fifth order polynomials of coverage. The numbers correspond to the different polynomials that are discussed in the text ........ 102

FIGURE 4.3. Conlparison between the calculated TPD spectrum and the measured1 one (solid dots) at initial coverage of 1.96. The expression for the P function for this calculation corresponds to the curve labeled as (I) in Fig. 4.2. The pumping speed was obtained as 0.25 Us. Error bars on the measured data

................. points show the domain of error in temperature measurement. 103

FIGURE 4.4. TPD spectrum at initial coverage of 1.96. Symbols are data inferred from Ref. 11. Solid line is calculated by SRT. The expression for the function for this calculation corresponds to the curve labeled as (4) in Fig. 4.2. The pumping speed was obtained as 0.25 Us. ................................................... 104

FIGURE 4.5. TPD Spectra for H2-W(100) calculated at initial coverages of O.3,O.4,O.SY 0.6,0.8,0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.96, respectively. The experimental parameters were taken from Ref. 11 ..................................... 105

FIGURE 4.6. Maximum pressures versus temperatures extracted from TPD spectra. Solid dots are experimental resu1t.s from Ref. 1 1. Circles are calculated by SRT model. The initial coverages are the same as those in Fig. 4.5. ......... 106

FIGURE 4.7. TPD spectrum at initial coverage 1.7. Symbols are data inferred from Ref. 1 I. Solid line are calculated by SRT. ......................................................... 107

FIGURE 4.8. TPD spectrum at initial coverage 1.4. Symbols are data inferred from Ref. 11. Solid line are calculated by SRT. ..................................................... 108

FIGURE 4.9. TPD spectrunl at initial coverage 0.6. Symbols are data inferred from Ref. 11. Solid line are calculated by SRT. ...................................................... 109

FIGURE 4.10. The criteria for phase transition at 300 K, Eq. (2.19). The chemical potential is calculated from the P function labeled as (4) in Fig. 4.2.. ...... 1 10

FIGURE 4.1 1. The chemical potential of hydrogen adsorbed on W(100) as a function of coverage for a temperature of 300K and the P function labeled as (4) in Fig. 4.2. For coverages in the range of 8, to Bp , a homopneous surface phase is intrinsically unstable.. .................................................................. .I11

FIGURE 4.12. Coverage versus temperature during a TPD experiments. The initial coverages are the same as those in Fig. 4.5. The island of unstable phase is also shown in this figure. Note that at the starting temperature, i.e., 300 K, for the range of initial coverages 1.1324 I 8(0) 2 1 SO02 the homogenous surface becomes unstable. ....................................................................................... .I12

APPENDIX:B.NUMERICAL PROCEDURE FIGURE B.1. The upper and lower limits of the unstable region for H2-Ni(1 10) system.

In this figure only the unstable island at low temperature and coverage is shown .......................................................................................................... 126

FIGURE B.2. The upper and lower limits for the unstable region for H2-W(100) system. ..................................................................................................................... 127

xii

LIST OF SYMBOLS

A surface area of the sample, adsorption.

As surface area of the vacant sites on the surface

b a function defined that is related to the vibrational frequencies, Eq. (2.50)

Do dissociation energy of hydrogen molecules

4 desorption energy used in Polanyi-Wigner equation

f i coefficients in the expression for vibrational frequencies

F the Helmholtz Function

li plank's constant divided by 2n

H enthalpy

J net adsorption rate

k boltzmann constant

ki j defined in Eq. (2.33)

KGM equilibrium exchange rate between gas phase and surface.

K(hj , A,) probability of transition from the quantum state to quantum state.

m reaction order

l n ~ 2 molecular mass of hydrogen

M maximum number of adsorption sites per unit surface area

Mo number of surface atoms per unit surface area

n number of adsorbed particles per unit surface area

N number of adsorbed particles

P pressure

xiii

defined in Eq. (2.32)

partition function for ortho-hydrogen

partition function for para-hydrogen

rotational and nuclear portions of partition function

canonical partition function for the adsorbed particles

heating rate

pumping speed

entropy

sticking coefficient

time

temperature

critical temperature

temperature corresponding to the maximum pressure in TPD spectra

internal energy

volume

potential energy defined in lattice-gas model

Greek Letters

aij coefficients used in polynomial describing the potential energy

P a defined parameter related to the potential energy, Eq. (2.49)

&oi defined in Eq. (2.38)

&ijk possible energy levels for three dimension harmonic oscillators

r potential defined in Eq. (2.32)

xiv

a function related the chemical potential of the gas phase hydrogen

potential of the adsorbed particles

surface tensions after adsorption

surface tension of the clean surface

defined in Eq. (2.37)

denotes a molecular configuration

chemical potential

collision frequency

frequency factor in Polanyi-Wigner equation

surface coverage

potential of the substrate particles

normal stress

vibrational frequencies of the adsorbed atoms

fundamental frequency of the hydrogen molecule

rotational frequency of the hydrogen molecule

the number of quantum mechanical states associated with the molecular

configuration hi

coefficients defined in the expression for potential energy

function related to the vibrational frequencies of the adsorbed atoms,

defined in Eq. (2.40)

Superscripts:

A denotes adsorbed atoms

I den0 tes surface phase

denotes adsorbed molecules

denotes solid phase

denotes adsorbed particles in general

Subscripts:

172 measured values

L denotes Langmuir approach

a denotes boundary for the unstable state

I3 denotes boundary for the unstable state

e denotes equilibrium property

xvi

CHAPTER 1: INTRODUCTION

The interaction of hydrogen with metals is of the fundamental importance in a

number of areas. One of the motivations for investigating this system is the role of hydro-

gen in the field of heterogeneous catalysis where a high proportion of the reactions

involves hydrogen as a reactant or product molecule. The first step in all of the catalytic

reactions is the adsorption of hydrogen and other involved chemical substances on the sur-

face. Therefore, a thorough understanding of this process, both under equilibrium and

non-equilibrium conditions, is necessary.

In the past five decades, numerous experiments have been performed to investigate

the different aspects of the interaction of hydrogen with metal surfaces. These experimen-

tal investigations include (but are not limited to): low energy electron diffraction (LEED)

studies to identify the structure of the surface during the adsorption process; electron

energy loss spectroscopy (EELS) experiments to determine mainly the different resident

frequencies of the vibrating adsorbed particles; equilibrium adsorption isotherm measure-

ments to determine the relation between temperature, pressure and the number of adsorbed

atoms under equilibrium conditions; and measurements of the flash desorption spectra

which are performed by pre-adsorbing a sample and heating it continuously . When the

partial pressure of the desorbed particles in the vacuum system is plotted against tempera-

ture, the so called thermal desorption spectra are obtained.

All the above mentioned experiments aim to clarify the nature of the interaction

between hydrogen and the substrate and to yield some information which might be used in

the analytical modeling of the adsorption process. In this field, however, not much success

has been obtained. In spite of its single valence electron which makes hydrogen the sim-

plest chemically reacting substance, the hydrogen atom is not a very good adsorbate for

theoretical treatment. As a matter of fact, the interaction of hydrogen atoms with metal

surfaces is not as well understood as that of, for example carbon monoxide. Contrary to

some suggestions', even the equilibrium measurements for hydrogen-single crystal sys-

tems are not complete. For example, to our knowledge, there are no measurements of the

calorimetric heat of adsorption for hydrogen on single crystal surfaces. Also the phase

transition of the adsorbed layer that in many cases is observed with hydrogen as the adsor-

bate is not thoroughly understood. Because of the weak scattering nature of the hydrogen

atoms, even the LEED technique does not detect additional diffraction features for this

gas-solid system.

In the remaining sections of this Introduction, the major aspects of hydrogen

adsorption on single crystals will be discussed and an overview of the present theoretical

models for analyzing these processes will be presented. An outline of the analytical

approach that is adopted in this thesis and the advantages of this approach will be also pre-

sented.

1.1 EQUILIBRIUM ADSORPTION ISOTHERMS An illustration of a dissociative adsorption system is shown in Fig. 1.1. As shown

in this figure, the process of dissociative adsorption can be described as the adsorption of a

gas molecule on a surface followed by the dissociation of the gas molecule into localized

adatoms. For hydrogen, the process of dissociation is usually very rapid and under equilib-

rium conditions it may be assumed that only adsorbed hydrogen atoms exist on the sur-

face. This assumption is supported by many experimental observations.' The relation

between the temperature, gas pressure, and the total number of adsorbed particles is one of

the characteristics of the adsorption system and is shown by the adsorption isotherms or

isobars. This collective information for a gas-solid system is the most important in

describing the behavior of the system under equilibrium conditions, (e.g., the heat of

adsorption can be obtained from these isotherm^'^*'^). For systems which may undergo a

phase transition during the adsorption process, adsorption isotherms also contain a great

deal of information.

Compared to the number of adsorption kinetic measurements, there are not many

adsorption isotherms available for hydrogen on single crystals. Practical problems in the

experimental procedure are usually stated as the reason for the lack of measurements. The

adsorption isotherms for H2-Ni(100) and H2-Ni(1 10) were measured by Christmann et a12

25 years ago, but such measurements have not been repeated since. The equilibrium

adsorption isobars for H2-W(100) were measured more recently by Horlacher Smith et

a13, though they did not measure the isobars to the saturation coveraget.

The simplest and most common model for predicting the adsorption isotherms is

the Langmuir isotherm. For this isotherm, the chemical potential of the adsorbed atom is

presented as1 *:

t Coverage, 0 , is defined as the number of adsorbed particles per number of be surface atoms.

where NO is the number of adsorbed particles, T is the temperature, M is the maximum

number of adsorption sites, A is the surface area of the adsorption, and q ( T ) is the parti-

tion function for each adsorbed particle. In this approach it is assumed that the particles in

the adsorbed phase do not interact with each other and therefore, the partition function

depends only on temperature. Under equilibrium conditions the chemical potential of the

adsorbed atoms and the gas molecules are equal. By equating these two chemical poten-

tials, the expression for the adsorption isotherm can be obtained. It is now well known that

the Langmuir relation is not able to predict many types of adsorption isotherms. To partic-

ular, it cannot be used for those systems that undergo a phase tran~ition.~ This is because

the interactions of the adsorbed particles with the surface atoms and with the other

adsorbed particles are neglected in the expression for the chenlical potential.

A different approach, based on quantum and statistical mechanics, for obtaining

the chemical potential of adsorbed particles has been presented by Ward and ~ lmose lh i . '~

In this approach, the potentials of the adsorbed particles are allowed to dcpend on the posi-

tion of the particles, while their displacements rue limited to smdl displaceinents about

their lattice positions. By neglecting the collective nlotion of the parlicles and by

fornli~lg the resulting potential to nornlal coordinates, the Hamiltonian of the adsorbed

particles was separated and were approximated as multi-din~ensional, quantum mechani-

cal harmonic oscillators. The interaction of the adsorbed particles and substrate was then

taken into account through the valiie of the potential when dl the piuticles were at their

lattice positions or adsorption sites. The chenlical potential obtained from this n~odel was

expressed as17*18:

where Qo is the interaction potential which is allowed to depend on the number of

adsorbed particles and w is a function which depends on the vibrational frequencies. In

Refs.17 and 18, however, the dependence of the fundamental frequencies of the harmonic

oscillators on coverage was neglected. In this thesis, the approach is generalized to allow

the fundamental frequencies to depend on coverage. For some hydrogen-metal surfaces

such a dependence has been observed e ~ ~ e r i m e n t a l l ~ . ~ ~ * ~ ~

We will show that the proposed chemical potential expression is not only consis-

tent with the measured adsorption isotherms for three hydrogen-metal systems, but is also

sufficient to define the necessary conditions for the stability of a surface phase.

1.2 INTRINSIC STABILITY OF THE SURFACE PHASE Many analytical approaches have been presented for examining the stability of the

vapor-liquid systems and predicting the conditions under which a phase transition is possi-

ble. But for gas-solid systems, such predictions are not available. The phase transition for

an adsorbed phase can occur when a stable homogeneous surface phase becomes unstable.

Experimentally, the LEED technique has been used to identify the phase transition on a

surface during the adsorption process. Theoretical attempts have been made to find the

interaction energy between the particles during phase transitions. If such an energy is

obtained, it can be used in other theoretical models, such as Monte Carlo simulation, to

obtain the temperature-dependent order of an adsorbed ensemble (phase diagram).'

A lattice gas model is a simple and frequently used approach to obtain the interac-

tion energy for systems with long range order.' The lattice gas approximation assumes the

bonding of the adatoms in specific sites on a periodic substrate. These sites can be empty

or singly occupied and the interaction between the adatoms is only a function of their

mutual distance. In this model, the local geometry is not affected by the adsorption itself,

and the phase diagram is only determined by the lateral adatom interactions, w .' There are

various models in statistical mechanics which predict the degree of long-range order, that

is the configuration of the adatoms, as a function of temperature, for a fixed set of lateraf

interaction energies. For a strictly pair wise w and for a square lattice, the two dimen-

sional Ising model predicts a relation between the critical temperature, T, of the ordered

phase (at a coverage of 0.5) and the interaction energy w as1

But for hydrogen adsorbed on surfaces, there are some serious complications. It is

comparatively small in size and the adsorbed hydrogen is usually located very close to the

surface atoms and can induce relaxation or even reconstruction on the surfaceS. Then the

phase transition becomes more complicated because lattice geometry becomes a function

of hydrogen coverage. Thus the simple gas model no longer applies to the system. '*lo

To model a phase transition on the surface, however, an approach can be adopted

that considers the change in the average interaction energy from a microscopic point of

view, as in the lattice-gas model. The entropy principle for thermodynamic systems

implies that when a system reaches a stable equilibrium state, the entropy should be a

$ Surface reconstruction is also observed for larger adsorbates like 0, CI, and S .

maximum. This means that at the stable equilibrium condition the following relations

must exist for entropy:5

It is important to notice that even when these conditions are met, the equilibrium system

could be in a stable or metastable state, whereas when d2s > 0, the equilibrium state is

unstczble. The investigation of the stability of the equilibrium leads to some interesting and

significant predictions in the thermodynamics of the systems. One is the prediction of the

conditions at which a homogeneous system becomes unstable. If one can examine the con-

ditions for the stability of equilibrium, one may obtain the range of temperatures and pres-

sures at which a system becomes unstable.

In this thesis a thermodynamic definition for the stability of the surfaces, based on

Eq. (IS), is presented and the conditions under which the homogeneous adsorbed phase

becomes unstable are obtained. This development is similar to the Van der Wads approach

for gas-liquid systems. It will be shown, in Chapter 2, that the definition can be expressed

in terms of the chemical potential of the adsorbed atoms. It will be also shown that the

phase transition in a gas-solid system under equilibrium conditions affects the non-equi-

librium behavior of that system. This can be shown from the measurements of temperature

programmed desorption (TPD) spectra for that system.

1.3 TEMPERATURE PROGRAMMED DESORPTION SPECTRA It is now well known that different gas-solid systems show different characteristics

in their TPD spectra. In this measurement, a pre-adsorbed sample is heated continuously

in a vacuum chamber. The desorbing particles are monitored, (e.g. with a mass spectrome-

ter), and are pumped out from the vacuum system. If the partial pressure of the desorbed

particle is plotted against the temperature of the sample and the procedure is repeated for

different initial coverages, a set of curves is obtained that can give valuable information

about the kinetic behavior of the gas-solid system. Some examples of TPD spectra are

shown in Figs. 1.2, 1.3 and 1.4. In these figures the TPD spectra for hydrogen desorbed

from Ni(100), Ni(llO), and W(100) surfaces are shown respectively. These curves have

different characteristics. For example, while the temperature corresponding to the maxi-

mum pressure, TI, , for H2-Ni(100) decreases with increasing initial coverage, the HZ-

Ni(ll0) behaves differently. In the spectra for the latter system, at low initial coverages,

the maximal temperature moves toward higher temperatures as the coverage is increased

and after certain initial coverage, it remains almost constant. For H2-W(100) system, two

distinguishable peaks in the TPD spectra are observed whereas for the two other systems

only one peak appears.

Thermal desorption of adsorbates from solid surfaces has almost always been

modeled by a Polanyi-Wigner equation8:

where Ed and vo are the activation energy and pre-exponential (frequency) factor for des-

orption respectively, and both may depend on coverage. k is Boltzmann's constant, T, is

the surface temperature, n is the number of adsorbed particles per unit surface area, and

m is the order of the process. When it is assumed that the activation energy and frequency

factor are independent of coverage, they may be obtained from measured maximal temper-

atures, T,,, , by a method presented by ~eadhead~ . More details about the procedure for

this calculation can be found in Refs. 8 and 9. According to dams^, an exact agreement

with the assumption of constant frequency factor and desorption energy is rarely found.

In many cases the value of the activation energy and its coverage dependence are

obtained from the equilibrium adsorption isotherms. For this purpose, the isotherms must

be available at sufficiently high pressures (- Torr), and then the isosteric enthalpy of

adsorption AH, can be determined by the Clausius-Clapeyron relation6

At equilibrium the adsorption rate is equal to the desorption rate, i.e.,

where S ( 8 ) is the sticking coefficient, tn , is the adsorbate mass, M, is the number of sur-

face atoms per unit surface area. The pressure can be found

By this definition, Eq. (1.7) leads to:

A H , = E d ( 8 ) + k T / 2 - k[aInv0(B, T ) / a ( l / T ) ] ,

Usually it is suggested that the last two terms in Eq.(l.l.O) are negligible, therefore one can

assume thal AH, = E d @ ) .* With this assumption, now the values for v,(B, T) can be

found by inserting the measured values of temperature, pressure, and coverage into

Eq. (1.9). Note that in doing so, the values of sticking coefficient must be obtained first

and the order of the reaction is assigned from the measured TPD spectra.

Using the above procedure, Seabauer et a16 calculated the values of frequency fac-

tors in terms of coverage for more than 45 systems and found that the variation of the fre-

quency factor with coverage is different for different cases. For some systems like H2-

Ni(100), the frequency factor remains constant with increasing coverage, whereas for oth-

ers like H2-Ni(1 lo), this factor decreases with increasing coverage. For a few adsorption

systems, they found that increasing coverage will increase the calculated frequency factor.

They also tried to apply the existing kinetic modeIs for predicting the desorption rate and

frequency factor and noticed that there are some discrepancies in these models, especially

at high coverage. For example, they found that the precursor state models predict a fre-

quency factor independent of the desorption system, which rises monotonically with cov-

erage, whereas this factor rarely increases with coverage. Based on their analysis of 45

different systems, they concluded that none of the existing methods satisfactorily predicts

the desorption spec tm6 Part of the problem, as they have stated, is that none of the analyt-

ical models takes into account the interaction of the adsorbed gas and the surface and the

possible modification of the surface by the adsorbate.

* There have been some attempts to use this sumpt ti on along with the lattice-gas approach to model the desorption process. But as is stated by Adam the equality of AH and A E is not consistent with tl~e use of a lattice-gas model.

The problem is more severe for more complicated hydrogen-solid interactions.

Many systems, like H2-Ni(100) and H2-Ni(llO), have only a single peak in their TPD

spectra (besides the appearance of small shoulders that could have resulted from surface

defects). But for example H2-W(100) exhibits a more complex behavior than the two Ni-

H2 systems, see Fig. 1.4; The existence of two peaks in the TPD spectra was suggested to

be related to the surface reconstr~ction.~ Tamm and schmidt7 who studied this system in

great detail, tried to use the Polanyi-Wigner equation to reproduce the TPD spectra for Hz-

W(100). They assigned a first order reaction to the first peak and a second order reaction to

the second peak with two different energies of desorption and frequency factors. They

obtained two separate curves that matched the two peaks. Adams, in his comprehensive

workg, used a lattice-gas model along with the absolute rate theory for desorption rdte to

reproduce the thermal desorption spectra for H2-W(100). He obtained the interaction

energy between hydrogen atoms in two cases. First, for a first order reaction where it was

assumed that an activated complex is formed by an adsorbed particle at a single site and

the square lattice consists of a five-site colony. In the other case he assumed a second order

reaction with an activated complex formed by a pair of adsorbed particles occupying the

nearest-neighbor sites and a square lattice that consists of an eight-site colony. The results

of his calculations with two different reaction orders showed the two peak feature of the

spectra but failed to describe the TPD spectra completely. Two major problems were

observed in his model. First, the ratio between peak pressures in a TPD spectrum was not

in agreement with the experimental results. Secondly, he could not predict the same

dependence of the maximal temperatures for two desorption peaks, T,,, , on the initial cov-

erages, as was observed from the measurements.

A different approach, however, can be adopted for predicting the net desorption

rate from a surface. In a proposed mechanism called statistical rate theory ( S R T ) ' ~ - ~ ~ , the

adsorption rate is obtained from a quantum mechanical formulation. In this approach the

adsorption system is assumed to consist of an interphase, as shown in Fig. 1.1, containing

the adsorbed molecules. Since in a dissociative adsorption the gas molecules would disso-

ciate when they interact with two adjacent empty sites, the mechanism must contain two

steps: molecular adsorption and dissociation. The statistical rate theory is a rate deterrnin-

ing approach which takes advantage of these two different steps to determine the adsorp-

tion rate. In this model the net adsorption rate is presented in terms of the chemical

potential of the gas molecules and adsorbed particles. The advantage of the statistical rate

theory over other theories is that the equation describing the net adsorption rale is in terms

of properties which are all material properties or obtained under equilibrium conditions. In

the SRT procedure the "reaction-order" is not assigned empirically and the coverage

dependence of the net adsorption rate is determined by the theory itself. In this thesis, the

SRT approach is modified to predict the TPD spectra for systems that may undergo a

phase transition during the course of the experiment.

1.4 SCOPE OF THIS WORK In Chapter 2 of this thesis, the chemical potential expression for the adsorbed

atoms when the interaction potential and the fundamental frequencies of the adsorbates

are functions of the number of adsorbed particles will be developed. We will show that the

measured equilibrium adsorption isotherms can be used to obtain the unknown functions

in this expression. The chemical potential expression for H2-Ni(100) and H2-Ni(1 10) will

be obtained with the proposed procedure and the corresponding isotheinls will be repro-

duced and compared to the measured ones2. In this chapter, the thermodynamic criteria for

the stability of an adsorbed phase under equilibrium conditions will be explained. It will

be shown that the necessary condition for the stability of the equilibrium state depends on

the chemical potential of the adsorbed atoms. Since the chemical potential expression is

available, the range of the temperatures and surface coverages at which the above two sys-

tems will undergo a phase transition will be obtained.

In Chapter 3, the equations necessary in describing a TPD experiment will be

developed. It will be shown that the number of adsorbed parlicles on the surface during

this process can be obtained from the SRT approach. The equations then will be solved for

H2-Ni(100) and H2-Ni(1 lo). The experimental parameters are taken from the reported

measured spectm2 It will be shown that except for the unknown experimental apparatus

parameters, no other fitting procedure is necessary when calculating these spectra. The

results are compared with the reported spectra for these two systems.2 The theoretical

approach will be also used to investigate the possibility of the occurrence of a first order

phase transition during the course of the measurements.

In Chapter 4, the developed approach is applied to the more complicated HZ-

W(100) system. Since for this system the equilibrium adsorption isobars are not available

to the saturation coverage, a method is presented to predict the expression for the chemical

potential for the whole range of coverage. At the same time the TPD spectra for this sys-

tem will be calculated with the experimental parameters taken from Ref. 11 and will be

compared with the measured ones.

In Chapter 5 a summary of the results and a brief discussion will be presented . It

will be shown that all predictions are in good agreement with the experimental results.

0 Hydrogen atoms

0 Metal atoms Gas Phase

Adsorbed Molecule

FIGUFE 1.1. Illustration of a dissociative adsorption process.

FIGURE 1.2. The TPD spectra measured for H2-Ni(100) by Christrnann et &The temperature at the maximuin pressure moves to lower temperatures with increasing initial coverage.The initial coverages are in terms of Langmuix (L).

temperature I°C1 -

FIGURE 1.3. TPD spectra measured by Christrnann et n12 for H2-Ni(ll0). The temperature at the maximum pressure moves to higher values by increasing the initial coverages and stays almost constant beyond a certain initial coverage.The initial coverages are in terms of Langmuir (L).

FIGURE 1.4. T ~ ~ T P D spectra of Hz-W(100) measured by Madey and ~ates'' at a variable heatina rate.While the P, state appears at higher initial coverages, the 8, state can be observed for all initial coverages.

CHAPTER 2: INTRINSIC STABILITY OF SURFACE PHASES: QUANTUMSTATISTICAL APPROACH

2.1 ~[NTRODUCTION

The chemical potential of particles (molecules or atoms) adsorbed on a solid sur-

face plays a fundamental role in determining the equilibrium and kinetic properties of the

gas-solid system. A complete expression for this function would include effects due to the

interactions between adsorbed particles and the substrate and interactions between the

adsorbed particles. As will be seen, if this function were available, it would be possible to

predict the conditions under which surface phase transitions were produced by adsorption.

In earlier work12, adsorbed particle interactions and changes in the substrate were

neglected. When an adsorbed particle was approximated as a quantum mechanical har-

monic oscillator and these interactions neglected, the expression for the chemical potential

constructed and equated the chemical potential of the particles in the gas phase, the Lang-

muir isotherm was obtained12. However, many systems exhibit isotherms, heats of adsorp-

tion and phase transitions that are not described by the Langmuir expression. By using

classical statistical mechanics and assuming a potential to exist between adsorbed parti-

cles, Fowler l3 introduced ihe lattice gas approximation that is a generalization of the

Langmuir approximation. For a lattice of two dimensions and certain values of the interac-

tion potential, the lattice gas approximation has the important advantage of indicating that

phase transitions can occur, but a principal difficulty has been that of determining the

interaction energy. In the quasi-chemical approximation (a two dimensional lattice gas

appr~ximation),'~ one accounts for the nearest neighbor interactions and assumes the

interacting pairs are independent. This assumption is understood to limit the validity of the

quasi-chemical approximation.14 By establishing the value of the interaction energy one

can then fit the empirical isotherms. This was the procedure adopted by Gijzeman et al. l5

for the CO-Ni(1 11) system.

A different approach can be adopted by allowing the potential to depend on the

position of the adsorbed particles, but limiting the displacement of the adsorbed particles

to small displacements, neglecting their collective motion and transforming the resulting

potential to normal coordinates. This allows the Harniltonian to be separated and the

adsorbed particles to be approximated as multi-dimensiond, quantum mechanical har-

monic oscillators. The interaction of the adsorbed particles and substrate are then taken

into account through the vahe of the potential when all the particles are at their lattice

positions or adsorption sites, and through the frequencies of the oscillators which can

depend on coverage. This is the approach adopted in Refs.17 and 18, except that the

dependence of the frequencies on coverage was neglected in these previous studies.

This quantum-statistical approach to account for the interaction gave important

distinctions from that of the quasi-chemical approxin~ation when both were used to exam-

ine the CO-Ni(l11) system.17* '* The expression for the chemical potential of CO-Ni(l11)

obtained from the quantum-statistical approach was examined further by incorporating it

in the kinetic equations of statistical rate theory, 19-28 and using the result to examine the

adsorption kinetics and the thermal desorption spectra for CO -Ni(11 1).27* 28 Since the

values of the material properties appearing in the expression for the chemical potential had

been determined from the adsorption isotherms, the predictions of certain desorption spec-

tra were made without any fitting constants, and the predictions were found to be in close

agreement with the rnea~urernents.~~

By applying the thermodynamic conditions for intrinsic stability, the quantum-

statistical approach can be used to predict the conditions under which a homogeneous sur-

face phase becomes thermodynamically unstable. To examine this aspect of the quantum-

statistical approach, we consider the H2-Ni(100) and H2-Ni(l 10) systems. No phase tran-

sitions are predicted for the former in the range of coverage and temperature for which iso-

therms are available, but phase transitions are predicted for the latter system. The

predicted conditions under which a phase transition in the Hz-Ni(1 10) system is to be

expected are found to be in qualitative agreement with previously reported measurements

by others.

In Chapter 3, the expressions for the chemical potential of H2 adsorbed on Ni(100)

and on Ni(ll0) obtained herein will be used with statistical rate theory 19-" to predict the

temperature programed desorption spectra for these systems. These predictions are then

compared with the available experimental measurements, and the agreement is shown to

be very good. The predicted phase transition for H2-Ni(1 10) is found to play a fundarnen-

tal role in the TPD spectra. In Chapter 4, the methods presented herein are applied to

develop the expression for chemical potential for H2 adsorbed on W(100) and then this

expression is used with statistical rate theory to predict the TPD spectra for this third sys-

tem.

2.2 INTRINSIC STABILITY OF AN ADSORBED PHASE

We first establish the conditions that the thermodynamic properties of a homoge-

neous surface phase must be satisfied if the phase is to remain homogeneous. Consider a

single crystal, solid surface that is exposed to a gas phase. We make the usual thermody-

namic approximation, and assume the total entropy of the system, S , may be expressed as

a sum of the entropy of each phase: the entropy of the surface, S f , the gas, SG , and of the

buk substrate, S :

To describe the surface thermodynamically, we approximate the surface phase as two

dimensional. Let the number of substrate atoms per unit area of the surface be denoted as

Mo and the number of adsorption sites per unit area be denoted as M. If the number of

adsorbed particles (either atoms or molecules) is denoted as Nay then the entropy of the

surface phase SI may be written as a function of the surface internal energy, U' , the sur-

face area, A , and No. Suppose that SI may be expressed as the sum of two functions, one

that is the surface entropy in the absence of any adsorption and another that represents the

additional entropy that results from the presence of the adsorbed particles:

Sf( U', A, No) = S"(Uo, A) + SQ(Ua, A, NO) (2.2)

where UO is the internal energy of the surface when there is no adsorption. For the internal

energy U' , we suppose a similar relation exists:

U' = u0 + ua The definitions of the intensive properties allow us to write:

where the adsorbed particles are assumed to have the same temperature, T, as the surface,

and the chemical potential of the adsorbed particles is denoted as pu . After taking the total

differential of Eq. (2.2) and making use of Eqs.(2.4) to (2.6), one finds

p' = pa

From the Euler relation for the surface phase and Eq. (2.7)

(I' - TS' = ylA + Nap=

and when this relation is applied in the absence of adsorption

u0 - TSO = ~ O A (2.9)

Equations (2.8) and (2.9) give the expression for the Helmholtz functions ( = U - TS )

when the adsorbed particles are present on the surface and when they are absent. These

functions will be denoted as FI and respectively. The conditions for the intrinsic sta-

bility of the surface phase when the adatoms are present will be expressed in terms of par-

tial derivatives of these functions.

2.2.1 Conditions for intrinsic stability

Suppose an element of the surface phase that has given values of UI, A and No is

isolated and is in a stable equilibrium state. We consider the three mechanisms by which

the element could break into two phases while the constraint of the element being isolated

is maintained. If the equilibrium state is stable, each mechanism will lead to a decrease in

the entropy of the isolated surface element? If the condition is not met, the system is not

intrinsically stable, and would be expected to undergo a first. order phase transition by that

particular mechanism. The fundamental relation is the same function for both surface

phases. Thus, for each mechanism

As will be seen, the second and third mechanisms are not independent. If the surface ele-

ment is stable againsl one mechanism it is stable against the other.

After expanding the first of these relations to second-order and making use of

Eq. (2.4), one finds that

Basically, this condition indicates that in order for the surface phase to be stable, the tem-

perature must increase when the surface is heated.

After the second relation has been expanded to second-order and multiplied by

one finds the result may be written as:

The first term could be zero for certain values of AU and AA ; thus a necessary condition

for the homogeneous state to be a stable equilibrium state can be obtained by taking the

first term in Eq. (2.13) to be zero. Then, after making use of Eq. (2 .4)~he necessary condi-

tion for intrinsic stability may be written as:

From Eq. (2.14), one finds

In general, we may write

and in view of Eq. (2.15), one may write

(2.17)

From Eq. (2.4) one finds that the right-hand side of Eq. (2.17) vanishes. Thus, a necessary

condition for the homogeneous surface phase to be stable is that

An additional necessary condition for the homogeneous phase to be stable can be

obtained from the third mechanism of Eq. (2.10). After following essentially the same pro-

cedure as that leading to Eq. (2.18), one finds

Although Eqs. (2.1 I), (2.18) and (2.19) constitute three necessary conditions for intrinsic

stability of the surface phase, they are not independent. As will be seen Eq. (2.19) implies

Eq. (2.18).

From Eqs. (2.4), (2.7) and (2.8), one may establish the Gibbs adsorption equation:

0 = SidT+Ady i+ NadpG

from which it follows that

and since these thermodynamic quantities may be viewed as composite functions,

Eq. (2.21) may be written

Also, a Gibbs (type) function, GI, may be defined for a surface phase:

u'- ~ s 1 - y ' ~

and from Eqs. (2.4) and (2.7), one finds

dG1 = - S I ~ T - A d y l + pOdN"

From Eq. (2.24) it follows that as

and after making use of Eq. (2.20), Eq. (2.25) may be written

Then from Eqs. (2.22) and (2.26), one finds

Thus, if Eq. (2.19) is satisfied, then so is Eq. (2.18). Hence, the independent, necessary

conditions for intrinsic stability are Eqs. (2.1 1) and (2.19).

2.2.2 Condition on the Helmholtz function for intrinsic stability

To connect the thermodynamic conditions under which the homogeneous system is

intrinsically stable to the molecular description, it is convenient to express the conditions

in terms of the Helmholtz function for the surface phase. From Eqs. (2.2), (2.3)

F' = FO + F O

From Eqs. (2.8), (2.9) and (2.28)

A (y' - yo) = Fa - NGpG (2.29)

The Helmholtz function of the surface in the absence of any adsorption, Fo , is assumed to

be only a function of temperature. From Eqs. (2.4), (2.5) and (2.6), one finds

d Fo = - Sad T + yadA + pad No Then Eq. (2.30) may be used to write

The condition for intrinsic stability given in Eq. (2.1 1) involves the temperature depen-

dence of yo and this dependence is unknown.

2.2.3 Canonical partition function

Since the conditions for intrinsic stability have been expressed in terms of partial

derivatives of Fo, our next objective is to express this function in terms of T, A, No. For

this purpose, we construct the expression for the canonical partition function. This con-

struction requires that the possible quantum states of the system be established. To define

the quantum mechanical problem, we obtain the approxin~ate expression for the potential

by restricting our attention to small displacements of each particle type from its lattice

position. We limit our attention to the solid being monatomic and the adsorbed particles

being atoms. If the generalized coordinates,q,, are measured relative to the lattice posi-

tions and the possibility of a particle leaving its lattice position is neglected, then the

potential, Y , may be approximated by

where 0, is the potential of the substrate particles,ZVS, when each is at its lattice position

and includes energy associated with the formation of the surface; a, is additional poten-

tial when NG particles are adsorbed and all particles are at their lattice positions, and

where the subscript 0 indicates the quantity is to be evaluated when each particle is at its

lattice position. In general, note that @,, 0, and kij depend on the number of adsorbed

particles, but not on their positions. For the adsorbed gas particles, we neglect the collec-

tive motion, and approximate the potential

The solid substrate can be approximated as a Debye solid with a potential energy term that

depends cn the number of adsorbed particles; however, since the stability of the surface

phase has been written in terms of FO , it is only necessary to construct the expression for

the partition function of the adsorbed particles, QO.

Since Q0 does not depend on where the particles are adsorbed, but only on the

number adsorbed, we assume that each of the M sites is equally available for adsorption,

that at most one atom may be adsorbed at an adsorption site, and that there are a total of

Na adsorbed atoms on an area A . The system is assumed to be maintained a t a tempera-

ture T . In the approximation considered, the possible energies of the adsorbed particles

are seen to depend only on the total number of adatoms present.

We suppose that the potential for the adsorbed particles may be transformed to nor-

mal coordinates, and the expression for the possible energy levels of the three din~ensional

harmonic oscillators constructed:

The coverage dependence of 0, and w j will be determined from the measured equilib-

rium adsorption isotherms. The description adopted herein represents a generalization of

the approach used for CO adsorbing on Ni(l1 l).18 In that case the dependence on cover-

age of the six fundamental frequencies of each adsorbed molecule could be neglected.

Although there are some reports of anharmonic effects in vibrational frequencies65, we

will examine the predictions that follow from approximating the adsorbed particles as har-

monic oscillators. Under these constraints, the canonical partition function for the

adsorbed particles is given by 17* l8

(20 = A M ! (kl K ~ K ~ ) ~ ~ No! (AM - No)!

where

From Eq. (2.36), one finds the Helmholtz function for the adsorbed particles

where

and the Planck constant divided by 2n is denoted its 8 . The expression for the chemical

potential of the adatoms may be obtained from Eq. (2.30) and Eq. (2.39):

The expression for rD, and N0kTlnv are to be determined from the measured equilib-

rium adsorption isotherms. It should be noted that if each of the frequencies oi depends

on coverage but not temperature, nonetheless depends on both coverage and tempera-

ture. The latter dependence comes through the introduction of ensemble averaging.

2.3 EQUILIBRIUM ADSORPTION ISOTHERM

Suppose that hydrogen gas, at a pressure P , is exposed to the solid surface consic

ered and that each phase is maintained at a uniform temperature. We neglect the presence

of molecular hydrogen on the surface. Then a necessary condition for equilibrium is that

where yC is the chemical potential of the bulk gas. The chemical potential of the gas mol-

ecules may be written 3'

where $ is given by

G and mH, q,, o , Do are the mass of the hydrogen molecule, the rotational and nuclear

portion of its partition function, its fundamental vibration frequency, and its dissociation

energy respectively. At the conditions we consider, we shall neglect the transition between

ortho-hydrogen and para-hydrogen?l This allows the partition function q, to be

expressed as a product of the partition functions for ortho and para-hydrogen:

3/4 1/4 Yrn = 40 qp (2.45)

If the rotational frequency of hydrogen n~olecules is denoted as or, hen:

and

The properties of hydrogen in the gas phase are listed inTable A. 1.

To obtain an expression for the equilibrium adsorption isotherm, the expression for

the chemical potential of the adatoms, Eq. (2.41), and for the gas phase

molecules,Eq. (2.43), may be substituted into Eq. (2.42):

The main difference between Eq. (2.48) and the Langmuir isotherm is that

NokTlnv - mo in Eq. (2.48) depends on the surface coverage. In the Langmuir isotherm

it would be constant. For a given temperature, the quasi-chemical approximation also

leads to an expression for the pressure in terms of the coverage. The expression contains

two constants: the lattice constant and the potential energy constant.14

2.4 COVERAGE DEPENDENCE OF THE CHEMICAL POTENTIAL AND INTRINSIC STABILITY

For both Hz-Ni(100) and Hz-Ni(1 lo), Christmann et a! report measurements of

the work function versus the gas phase pressure at several temperatures. At room tempera-

ture, they also report that the work function depends linearly on coverage up to at least

50% of the saturation value for the former and up to 80% for the latter system. Experimen-

tal difficulties prevented them from examining higher coverages. We shall assume that a

linear relation exists between work function and coverage up to full coverage for all tem-

peratures.

To establish this linear function for Hz-Ni(100), we will use the saturation cover-

age reported by Andersson 46 who found a value of 0.5 and the corresponding work func-

tion reported by Christmann et aL2 who gives a value of 0.17 eV. After using these values

to obtain the linear relation between work function and coverage, this linear relation may

be used with the measurements of work function as a function of pressure at different tem-

peratures that were reported by Christn~ann et al. to obtain the equilibrium adsorption iso-

therms shown in Fig. 2.1. A similar procedure was followed to obtain the isotherms for

H2-Ni(1 10) that are shown in Fig. 2.2 except that the linear relation was established from

the saturation coverage and corresponding value of the work function, 1.5 and 0.53 eV,

given by Refs. 2 and 34.

2.4.1 Dependence of NokTln yf - 0, on coverage

For convenience, we introduce p and b :

Then from Eq. (2.48)

( b - p) = m n ( No 1 (2.5 1 ) KP(AM- NO)

To complete the determination of the chemical potential, the dependence of (b - P) on

coverage must be determined. Although the procedure described in Refs. 18 and 25 could

be applied to determine this dependence for H2-Ni(100), a preliminary study indicated

that because of the dependence of the frequencies on coverage for H2-Ni(1 lo), the method

of Refs. 18 and 25 is not adequate for the latter system. Since one of our objectives is to

compare the predicted stability of these two systems, the same method will be used to

determine the coverage dependence of their respective chemical potential expressions.

From the isotherms shown in Figs. 2.1 and 2.2, one may determine the value of

b - P by applying Eq. (2.51) at each measured coverage on an isotherm. This allows a

family of b - p curves to be plotted as a function of coverage with T as the parameter. A

sample of such a family of curves, for tenlperatures of 331, 362, 390, and 417 K, is shown

in Fig. 2.3. From these curves, at one coverage, one may determine b - P as a function of

temperature. For H2-Ni(100), one finds a linear dependence; thus

Hence at coverage Bi , one may determine a value of tO(e i ) and a value of 6 , (9,) . The

values of each of these functions may then be pIotted at nine different coverages, and each

function represented by a fifth order polynomial

5 a..8j; i = 0, 1 h=Cj=o 11 (2.53)

The values of aij were determined by minimizing the least-square error between this

polynomial and the measurements. This procedure leads to the values of aij listed in 2.1.

To assess the accuracy of this procedure for each isotherm, the values of aq were

used in Eq. (2.53) to calculate the values of (b - P) on the isotherm. The measured values

of (b - P),ll, 0, at each measured coverage,O,. , were then used to calculate the parameter

a(Tj) for the isotherm:

where N is the total number of measured coverages on the isotherm. If o ( T j ) were zero it

would indicate exact agreement between the isotherm measurements and the analyticai

expression. The maximum value of o(Tj) was 0.00054 and occurred on the 417K iso-

therm*; thus

o(Tj) S 0.00054 (2.55)

In view of the magnitude of o(Tj), it appears that Eq. (2.52) accurately represents the

temperature dependence of b - P for H2-Ni(100). Thus the covemge dependence of b - P also seems to be well described by Eqs. (2.52) and (2.53).

From Eqs.(2.49), (2.50), (2.52) and (2.53)

If the values of aij listed in Table 2.1 are used in Eq. (2.56), the result may be substituted

in Eq. (2.48) and the isotherms calculated. The solid lines shown in Fig. 2.1 were calcu-

lated by this procedure. The agreement seen there between the measurements and the cal-

culations indicates the consistency of the procedure for determining the analytical

expression for the isotherms. It should be noted that Eq. (2.56) may be substituted into

*.As is seen from Fig. 2.3, for this isothcml a large scatter is observed in the data. But since the order of magnitude of tlle error is extremely small, the mathematical model is still consislent for lhis temperature.

Eq. (2.41) and a complete expression for the chemical potential of the adsorbed hydrogen

obtained. The chemical potentials for adsorbed hydrogen on Ni(100) at two temperatures

of 300 K and 400K are shown in Fig. 2.4

A similar procedure may be used to obtain an analytical expression for the chemi-

cal potential of hydrogen adsorbed on Ni(1 lo). However for this case, one finds that at a

given coverage, b - p depends nonlinearly on temperature. Thus, for H2-Ni(1 lo):

The values of aU for HHTNi(l 10) obtained by following a procedure similar to that out-

lined, are listed in Table 2.2, and in Fig. 2.5 plots of b - P as a function of coverage for

different temperatures are shown. The solid lines in this figure were calculated with the

values of ajj listed in Table 2.2. To assess the error in this procedure, the value of o(Tj)

was also calculated in this case. The maximum value was found to be 0.0063 at the 416K

isotherm; thus

o(Tj) 5 0.0063 (2.58)

Thus, Eq. (2.57) appear to give an accurate description of the values of (b - P) for the

H2-Ni(1 10) isotherms.

From Eqs.(2.49), (2.50) and (2.57), one finds for H2-Ni(1 10)

To calculate the isotherms in this case, the values of aij listed in 2.2 may be used in

Eq. (2.59), and the result substituted into Eq. (2.48). The solid curves shown in Fig. 2.2

were calculated with this procedure. By comparing the solid lines with the data given in

Fig. 2.2, one may observe the consistency of the procedure for determining the coverage

dependence of chemical potential.

The type of chemical potential expression that is calculated for H2-Ni(1 10) by this

procedure is illustrated in Fig. 2.6 where the chemical potential of the adsorbed hydrogen,

for one temperature, is plotted as a function of coverage. Under equilibrium conditions,

this chemical potential would be equal to the chemical potential in the gas phase, and as

seen in Eqs. (2.43) and (2.44), for a given temperature, the chemical potential of hydrogen

in the gas phase only depends on the pressure there. Thus, the ordinate in Fig. 2.6 may be

viewed as being directly related to the gas phase pressure. As seen in this figure, for a

given value of the gas phase pressure or the chemical potential of the adsorbed hydrogen,

there are three values of the coverage that satisfy the necessary conditions for equilibrium,

Eq. (2.42). To determine which of these states the system will occupy, their relative stabil-

ity must be considered.

2.4.2 Intrinsic stability of Hz-Ni(lO0) and H2-Ni(110)

For H2-Ni(100), one finds that the condition for intrinsic stability given in

Eq. (2.31) is always satisfied, and for a given gas phase pressure, there is only one cover-

age that satisfies the necessary conditions for equilibrium, see Fig. 2.4. Thus, no phase

transitions are predicted for the temperatures and coverages for which isotherms are avail-

able.

However, a very different result is obtained for H2-Ni(1 lo). As shown in Fig. 2.7,

four types of isotherms are predicted to exist for this system: 1) For a temperature of

298 K, Eq. (2.31) is satisfied. Thus, a homogeneous surface phase is predicted to be stable

at this temperature for all sub-saturation coverages. 2) If the temperature is increased to

346 K, then for coverages near 0.2, Eq. (2.31) is not satisfied, and it is predicted that a

homogeneous surface phase at this temperature and coverage would be unstable. At higher

coverages and this temperature, no instabilities are predicted. 3) For a temperature of 390

K and coverages less than 0.196, Eq. (2.3 1) is satisfied. Thus, the homogeneous Ha-

Ni(1 lo) surface is intrinsically stable in this coverage range. If, at this temperature, the

coverage is increased into the range from 0.20 5 8 < 0.2 1 , Eq. (2.3 1) is no longer satis-

fied. Thus, in this latter coverage range the homogeneous surface phase is unstable. There

is a second coverage range at this temperature where the homogeneous system is no longer

intrinsically stable: 0.69 1 8 S 0.85 . At higher coverages and this temperature, it is

again possible for the system to remain homogeneous. 4) The fourth isotherm type is the

441 K isotherm. The chemical potential of the adsorbed hydrogen for this temperature is

shown in Fig. 2.6. By comparing Figs. 2.6 and 2.7, it may be seen that the homogeneous

surface phase is unstable when the coverage is in the range

where 0, and BP are coverages corresponding to the local maximum and local minimum

of the chemical potential of the adsorbed hydrogen (see Fig. 2.6). These coverages corre-

spond to the coverages where

(see Fig. 2.7). If the surface coverage were initially 8, and the pressure were increased

slightly and both the pressure and temperature held constant, then as indicated in Fig. 2.6,

the system would arrive in a stable condition only when the coverage had reached By.

Similarly, if the initial coverage were ep, a slight reduction in pressure would result in the

system making a transition to the coverage 06, provided the pressure and temperature

were held constant during the transition.

The range of conditions where the homogeneous surface phase is intrinsically

unstable can be calculated by performing a series of calculation such as those shown in

Fig. 2.7. The "islands" of unstable conditions that are predicted from these calculations

are shown in Fig. 2.8.

2,s DISCUSSION AND CONCLUSION

To predict the conditions under which surface phase transitions can be induced by

adsorption requires that the interactions between the system particles be taken into

account. In the method used here, the potential experienced by each particle (substrate and

adsorbed) is allowed to depend on the number of adsorbed hydrogen atoms. As a result, it

is predicted that the vibrational frequencies (phonons) of the adsorbed particles can

depend on the number of particles adsorbed. This dependence was found from the equilib-

rium adsorption isotherms to be negligible for H2-Ni(100), but essential for H2-Ni(1 lo).

This conclusion for H2-Ni(l 10) is supported by independent experimental evi-

dence obtained from high-resolution electron energy loss spectroscopy for this system.

Nishijima et a1.33 show that at 300 K there is a shift in the spectra with changing surface

concentration. Also qualitative studies with low-energy electron diffraction have found

that above 300 K a "streak" phase develops at "fairly" low coverages 34.

In the present theoretical description, the surface concentration of adsorbed hydro-

gen atoms affects the potential experienced by Ni(ll0) atoms. This provides a mechanisn~

by which the adsorption of hydrogen could induce a first-order phase transition in the sub-

strate. As indicated in Figs 2.6 and 2.7, the proposed method can be used to predict the

conditions under which a homogeneous surface phase becomes unstable.

As shown in Fig. 2.8, the proposed method leads to the prediction of a phase tran-

sition at hydrogen coverages above 0.6 and temperatures above 380°K. Although the

phase transition at smaller hydrogen coverages (i.e. near 0.2), has at least qualitative

experimental support,'* '* 34 the phase transition at hydrogen coverages above 0.6 does not

appear to have been identified previously. A partial explanation for this is provided by the

study of thermal desorption of the H2-Ni(ll0) reported in the next chapter. There it is

shown that in previous studies of thermal desorption that were performed at a heating rate

of 15K per second and pumping speed of 110 Us, the hydrogen coverage would be below

0.6 when the temperature reached 380°K. Thus, the conditions under which the phase

transition at higher coverages are predicted to take place have not been encountered in pre-

vious TPD studies.

The stability analysis presented is based on intrinsic stability.5 This approach

allows one to identify the conditions under which a homogeneous phase becomes unsta-

ble, but it does not allow one to determine the conditions under which the phase becomes

metastable. Thus, a phase transition could occur before the system reaches the temperature

and surface concentration ranges labelled as unstable in Fig. 2.8.

The quantitative predictions of the limiting conditions for a system to remain

homogeneous, such as those for H-Ni(ll0) shown in Fig. 2.8, are different in nature from

the predictions of the quasi-chemical approach. For example, the quasi-chemical approach

can be used to predict that a phase transition is possible if the interaction energy between

adatom pairs, w , has a certain value, but a method by which the value w may be estab-

lished is not presently available.

TABLE 2.1. Properties of Ha-Ni(100)

Material Properties:

Mo (1 ~'~atorns/crn~)

TABLE 2.2. Properties of H2-Ni(1 10)

i ao j a1 j * 012 j

Material properties:

l x ~ ~ - l o 1x10-9 M O - ~ lx1o=] lX1om6 I X I O - ~ lx10m4 Pressure , Tom

FIGURE 2.1. Isotherms for H2-Ni(100) inferred horn measuremend of the dependence of the work function on coverage and of coverage as a function of pressure (solid dots). The isotherms calculated by the quantum-statistical method are shown as solid lines.

1x10-9 M O - ~ 1x10-~ M O - ~ M O - ~ 1 x 1 0 ~ Pressure, Toil-

FIGURE 2.2. Isotherms for Hz-Ni 1 1 . The data points (solid dots) were inferred from measurements'. 39)of the dependence of the work function on coverage and of coverage as a function of pressure, and isotherms calculated by the quantum-statistical method (solid lines).

0.2 0.3 0.4

Coverage, N ~ / ( A M ~ )

FIGURE 2.3. Examination of the procedure used to obtain the analytical expression for the coverage dependence of the chemical potential of hydrogen adsorbed on Ni(100). The symbols are the values of ( b - P ) determined from the measured isotherms and the solid lines are the calcuIated values.

. .

6 Coverage, N /(AM,)

FIGURE 2.4. The chemical potential of hydrogen adsorbed on Ni(100) is shown as a function of coverage for two temperatures of 300K and 400 K.

Coverage, N O/(AM ) 0

FIGURE 2.5. Examination of the procedure used to obtain the analytical expression for the coverage dependence of the chemical potential of hydrogen adsorbed on Ni(ll0). The symbols are the values of ( 6 - P ) determined from the measured isotherms arid the solid lines are the calculated values.

Coverage, N O/(AM~)

FIGURE 2.6. The chemical potential of hydrogen adsorbed on Ni( l l0) is shown as a function of coverage for a temperature of 441K. For coverages in the range from 0, to 8 , a homogeneous surface phase is intrinsically unstable. The instaBility leads to the predicted hysteresis loop indicated by the arrows.

0 0.4 0.8 1.2 0 0.4 0.8 1.2 coverage, N~I(AM~) Coverage, N?(AM~)

FIGURE 2.7. Intrinsic stability of H2-Ni(ll0) isothernu. When the ordinate is negative, it is predicted that the homogeneous surface phase is intrinsically unstable. For H2-Ni(llO), it is predicted that there are four types of isotherms. One with no phase transitions (upper left), one with an unstable condition at low coverage (lower left), one with two unstable coverage ranges (upper right) and one with an unstable range at higher coverages (lower right). The chemical potential as a function of coverage for the plot on the lower right is shown in Fig. 2.6.

I 1 1 I I

0

300 350 400 450 500 Temperature, K

FIGURE 2.8. Instability "islands" for the Hz-Ni(1 10) system. When the coverage and temperature correspond to a position in an island, it is predicted that the homogeneous surface phase i s unstable.

CHAPTER 3: THERMAL DESORPTION OF HYDROGEN FROM NI(100) AND NI(110): STATISTICAL RATE T m O R Y APPROACH

3.1 INTRODUCTION

In the previous Chapter, an analytical procedure has been proposed that allows a

measured set of equilibrium adsorption isotherms to be used to determine the expression

for the chemical potential of dissociatively adsorbed hydrogen. This expression has been

used to predict the temperature and surface coverage at which homogeneously adsorbed

hydrogen undergoes a phase transition. It was shown that for certain isotherms of the H2-

Ni(l l0) system, there is a range of (absolute) coverage where a homogeneously adsorbed

gas would be unstable. When the TPD spectra of this system is studied, we find for certain

initial coverages that the temperatures and coverages during an experiment correspond to

those under which a homogeneously adsorbed phase become unstable; thus at these condi-

tions a phase transition would be expected.

Since the phase transition occurs during the course of a TPD spectrum, the surface

makes a transition from a homogeneously adsorbed phase to a heterogeneous surhce

phase. To predict the effect of this phase transition on the TPD spectrum, one could con-

sider adopting the transition state approach. However, the presence of a new heterogeneity

would introduce a new a set of parameters into the theoretical expression for the adsorp-

tion rate (e.g. see Ref. 40). Because of all the parameters introduced by this procedure, it is

not clear that this approach could be used to predict (as distinguished from fitting) the

TPD spectra of a heterogeneous surface. The discrepancies that arise when the parameters

appearing in the Polanyi-Wigner equation (which has its basis in transition state theory)

are taken as independent of coverage have been described by Seebauer et aL6 Further, it

has been shown by ~ d a r n s ~ that a particular lattice gas model that accounts for adsorbate-

adsorbate interaction does not lead to a quantitatively correct prediction of the TPD spec-

tra for the H2-W(100). An approach that does not use the transition state assumption, sta-

tistical rate theory ( S R T ) ' ~ ~ ~ , was used by Rudzinski 41 to analytically examine the TPD

spectra for heterogeneous systems, but he chose to use a Langmuir type chemical potential

for each of the heterogeneities and to introduce an adsorption energy distribution for the

heterogeneities.

Although a great deal of work has been done in the field of lattice -gas model in the

last 25 years to predict the phase transition on the surface, the model is of questionable

validity when adsorption induces surface reconstr~ction.~~ In lattice gas-model, the sur-

face is usually assumed to have the same two dimensional periodicity as the atomic order-

ing in the bulk.66 But in case of a surface reconstruction, a new symmetry appears on the

surface. However, the chemical potential expression that is used in our approach to find

the range of temperatures and coverages at which the surface phase transition occurs is a

more general model that can predict the phase transition quantitatively from isotherms,

even for systems with surface reconstruction

At present there is only qualitative experimental evidence to support the predicted

phase transition for the H,Ni(llO) 34 However, since the predictions have been

made from the expression for the chemical potential, the basis for their predictions can be

examined further. Herein, this function is used with SRT to obtain an expression for the

hydrogen adsorption rate. This expression is then incorporated in a procedure for predict-

ing both the surface coverage during a TPD experiment and the corresponding pressure

spectrum. This allows the expression for the chemical potential to be examined in two

ways: 1) The predicted TPD spectra for both H2-Ni(100) and H2-Ni(1 10) may be directly

compared with the measurements previously reported by Christmann et a12. 2) The condi-

tions at which TPD spectra have been observed to undergo unexplained changes can be

compared with the conditions where a phase transition is predicted to occur.

3.2 SYSTEM DEFINITION

A schematic of a TPD system is shown in Fig. 3.1 along with an isolated system

that would be formed if there were no heating of the substrate nor pumping of the gas

phase. This isolated system plays a fundamental role in applications of SRT. We take the

known experimental information about the TPD system to be that given in Table 3.1. If the

substrate is heated at a constant rate, Rh , for a period t , we assume that the temperature is

uniform in the system and is given by

T ( t ) = T(0) + Rht (3.1)

The change in the number of molecules in the gas phase of the TPD system at the time t

may be expressed in terms of a pumping speed, R, and the gas phase effective volume, V.

These parameters are apparatus constants, since they depend on the vacuum system used

to perform the experiments. Ideally, they would be stated with the description of the exper-

iments.

Molecules enter the gas phase by desorbing from the surface and are removed from

this phase by the pumping system. If the number of adsorbed atoms and molecules are

denoted as NA and N ~ , and the total number of adsorbed hydrogen atoms by NP:

N~ = 2 N " + N A (3.2)

then the change in the number of molecules in the gas phase may be written

where P(0) is the initial pressure and factor (1 - (P*)3 is added to account for the

pumping efficiency. If the gas phase may be approximated as ideal, Eq. (3.3) may be com-

bined with Eq. (3.2) to obtain

The total number of molecules initially present,N(O), in the system is an experimental

parameter, and may be expressed as

Often the given experimental parameters for each experiment are the values of

N~(O) , P(0) and T(0). Thus the total number of gas component particles in the TPD sys-

tem at the initial time may be viewed as a specified quantity in each experiment.

If the net adsorption rate of the atoms is expressed in terms of T, A, NP, IVG :

and of the molecules

(dN"/dt) = f ,(T, A, N", NA, N ~ ) (3.7)

then Eqs. (3.1), (3.4) (3.6) and (3.7) would constitute a closed system of equations to be

solved simultaneously to predict NC as a function of time in the TPD system. If the value

of NG were known, it could then be used with the gas phase volume, the instantaneous

temperature, and the ideal gas equation of state to predict the pressure as a function of time

in the gas phase (i.e., the TPD spectrum). Statistical rate theory will be applied to deter-

mine the explicit expression for the rates indicated in Eqs. (3.6) and (3.7), and thereby a

closed system of equations for predicting the rate of hydrogen adsorption will be formed.

3.3 STATISTICAL RATE THEORY EXPRESSION FOR THE ]RATE OF HYDROGEN ADSORPTION

At one instant, we assume the net rate of desorption from the surface in the isolated

system (Fig. 3.1) is equal to the rate of desorption from the surface in the TPD system.

This instant is the one in which the uniform temperatures of each system, the number of

atoms and molecules adsorbed, and the number of molecules in the gas phase, the area of

the substrate, and the volume of the gas phase in each system are the same. Thus, at the

instant considered the values of T, NA, NM, NG, A and V are treated as given, and at this

instant we assume the rate of adsorption in the two systems to be the same. The expression

for the rate will be developed for the isolated system. Following the instant under consid-

eration, the behavior of the TPD system would be different than that in the isolated sys-

tem. For example, the isolated system would evolve to a state of thermodynamic

equilibrium, but the TPD system wouId not, since the latter system is being heated.

We take the substrate in the isolated system to be of sufficient size so that when the

isolated system comes to equilibrium, the temperature in the final equilibrium state will

have the same vdue as in the initial state. The independent properties of the bulk substrate

are denoted as T, v B and N B .

3.3.1 Rate of exchange between the surface and gas phase under equilibrium conditions in the isolated system

Statistical rate theory will be used to develop the expression for the rate of adsorp-

tion in the isolated system. One of the important factors in the expression is the rate of

molecular exchange between the surface and the gas phase under equilibrium conditions.

We first develop an expression for that factor.

A necessary condition for the isolated system to be in equilibrium is that the chem-

ical potentials of the molecules in the gas phase, pG , of the adsorbed molecules, yM, and

the adsorbed atoms, pA be equal.

p = PM = 2p4 (3.8)

The expression for the chemical potential of the hydrogen molecules in the gas phase is

presented in Eq. (2.43).

Studies of hydrogen adsorption on surfaces that have been carefully cleaned and

their cleanliness monitored indicate that under equilibrium conditions hydrogen is

adsorbed primarily atomically.2* 46 In a prior section, to formulate the expression for the

chemical potential of the adsorbed hydrogen, the presence of molecularly adsorbed hydro-

gen under equilibrium conditions was neglected, the surface phase was approximated as

two dimensional with an area A , the number of substrate atoms and the number of adsorp-

tion sites per unit area of the surface as Mo and M. The position of the surface was taken

to be such that there was no adsorption of the substrate component in the surface phase.

The potential experienced by the hydrogen atoms was allowed to depend on the number of

uniformly distributed adsorbed atoms over the surface. The expression for the chemical

potential of adsorbed atoms thus obtained is:

where the value of the potential of the hydrogen atoms when each atom is at its adsorption

site is denoted as cPo and the function y which depends the three fundamental frequen-

cies of the adsorbed atoms were given in Eq. (2.40). The expression for cPo - i V A k ~ l n y

has been determined from the measured equilibrium adsorption isotherms.

For H2-Ni(100) it was found that the coverage dependence of the fundamental fre-

quencies could be neglected and the expression for a(NokTlnv - Q o ) / ( a N o ) which was

linearly dependent on temperature was presented in Eq. (2.56). A similar relation was also

found for H2-Ni(ll0) with a quadratic temperature dependence and was shown in

55

Eq. (2.57). From Eqs. (3.8) and (3.9) one finds the expression for the adsorption iso-

therms:

P, =

where the properties existing in the isolated system when the isolated system has evolved

to equilibrium are subscripted with an e . Since at the instant being considered the values of lVA(t), NM(t) and NG(t) are

known, the total number of hydrogen molecules in the isolated system, N , would be given

by

and then when the isolated system evolves to equilibrium, the same number of hydrogen

molecules would be present, although they may be differently distributed between the two

phases

(It should be recalled that the number of adsorbed molecules in the equilibrium state is

neglected). Using a numerical procedure, Eqs. (3.10) and (3.12) may be solved simulta-

neously to determine P, and Nt . However, because of the possibility of a phase transi-

tion in the H2-Ni(1 lo), care must be taken to exclude the possibility of calculating the

equilibrium exchange rate in an unstable equilibrium state. One of the isotherms for the

Hz-Ni(1 10) system is shown in Fig. 3.2. As seen there, for a given temperature and for a

pressure in a certain range, there are three possible values of the surface coverage. How-

ever, for a given coverage and temperature, there is only one pressure a t which the system

could be in equilibrium.

First we consider a situation that N is sufficiently small so that 8,s 8,. Then by

solving Eq. (3.12) for P, and inserting the result in Eq. (3.10) one finds an equation that

may be solved numerically, to determine N$ :

and from Eq. (3.12), the corresponding pressure in the gas phase may be determined. The

values of N$ and P , corresponding to N in this range may then be used to calculate the

equilibrium molecular exchange rate between the gas phase and the surface.

Since the assumption of molecular adsorption being rate limiting is a good

assumption that is also confirmed from the experimental observations2, one can find the

equilibrium exchange rate from classical collision frequency. Under equilibrium condi-

tions in the isolated system, the molecular velocity distribution in the gas phase would be

Maxwellian and the collision rate of the gas phase molecules with the surface, v , would

be given by

To determine the equilibrium exchange rate between the gas phase molecules and the sur-

face phase, we follow the SRT procedure of taking the exchange rate to be the product of

the co1lision rate times the area of the vacant sites on the surface,A, , and we take the area

of a site to be the inverse of the number of sites per unit area: 17 -28

( A M - N,A) A, =

M

Thus the equilibrium exchange rate of the molecuIes of the gas phase with the surface

phase, KGM, would be given by

This procedure may be used to calculate the molecular exchange rate for values of N that

correspond 8 , s 8,

Next, suppose N is increased until the calculated value of 8, is in the range

0, c 0, c €Ip. For values of 8, in this range, the surface phase is unstable; thus, we

assume the isolated system will not evolve to this unstable state. Rather, if N, is the value

of N that gives a value of the coverage equal to B,, and N is increased infinitesimally to

N + E , then the numerical solver should look for a solution for coverages greater than

8, and as indicated in Fig. 3.2, the pressure in the gas phase would decrease correspond-

ingly. Thus, a slight increase in N above N, would result in an increase in the coverage

and a decrease in the gas phase pressure. These changes would take place as a result of the

system undergoing the phase change. As may be seen from Eq. (3.16), the calculated value

of the equilibrium molecular exchange rate KGM would undergo a discontinuous

decrease. Finally, for values of N corresponding to 8,Z Bp , N$ and P, depend continu-

ously on N.

Also, if one supposes the surface coverage to be greater than Bp (see Fig. 3.2), and

then N were slowly reduced, no phase change would occur until the coverage had reached

the value Op. If the value of N were then reduced further, when the isolated system had

evolved to equilibrium, the distribution of hydrogen in the system would be such that the

coverage was less than Bp and the pressure would decease.

Note that since P, and N$ are determined by the instantaneous properties of the

TPD system (N and T), the equilibrium exchange rate between the gas phase and the sur-

face phase of the isolated system may be viewed as an instantaneous property of the TPD

system.

3.3.2 Instantaneous adsorption rate in the isolated system

At the instant of interest in the isolated system, non-equilibrium condition exists

between the surface phase and the gas phase. During equilibration, we suppose the process

limiting the approach to equilibrium is the rate of molecular transport from the gas phase

to the solid surface; thus a state of local equilibrium may be assumed to exist in the gas

phase and on the surface. On the surface, this means

2pA = pM (3.17)

The objective is to predict the rate in terms of the values of gas phase properties T, V, NG

and the values of the surface properties T, A, NA, N ~ . Thus, at the macroscopic level each

of these phases of the isolated system is described in terms of the canonical ensemble vari-

ables; thus for each, when described at the quantum mechanical level, there are many pos-

sible states.

For the isolated system, a particular molecular distribution between the phases, hi,

will denote the values of NG, N~ and NM, and the total number of quantum mechanical

states that the system could occupy when in this particle distribution will be denoted as

!2(hj). These states, uv(hj), are all within the energy uncertainty. If one molecule is

transferred from the gas phase to the surface phase, the particle distribution is changed to

h, or NG - 1, N*, NM + 1 , and the states of this second distribution are denoted as

u&). The probability of the system making a transition from a quantum mechanical

state of molecular distribution hi to a state of hk in the time 6t is, according to first order

perturbation theory32, given by

where 6 is the energy density of the states, I V,,I is the matrix element corresponding to a

transition from a state of kj to a state of h, . The probability of the system being in a state

of molecular distribution hi at the time t would be the inverse of the number of states in

the distribution, and thus the probability of the system making a transition from molecular

distribution hj to a state of h, in the time S t would be

and the probability of a transition to h, independently of the state to which the transition

is made, z(hp hk)St, would be

where 7(lcj, 1,) is the probability of the transfer taking place at any instant in time. If the

Boltzmann definition of entropy is introduced, the probability of the transition from hj to

h, at any instant would be

The Euler relation for the gas and surface phases allows their entropies to be written as:

and for the solid substrate, we assume only nomlal stresses are present:

where o is the average normal compressive stress. As a result of the transition from parti-

cle distribution hj to hk one particle is transferred from the gas phase to the solid surface.

We assume that as a result of the transition the intensive properties undergo no significant

changes. For the isolated system,

uG(hj) + Ur(hj) + u B ( h j ) = uc(h,) + ~ ' ( h , ) + uB(hk)

and its volume is unchanged

VG(kj) + v B ( k j ) = vG(kk) + vB(hk) (3.26)

The entropy of the isolated system would be the sum of the entropies of the three phases:

S(h) = SG+Sr+SB (3.27)

and one finds the change in the entropy of the isolated system that results from the change

in particle distribution hj to h, is given by

The expression for the probability of a transition from hi to h, at any instant in time may

now be written as

Wj7 A,) =

In developing the expression for z ( l p 1 , ) . it was

(3.29)

assumed that the transfer of one mole-

cule did not change the intensive properties. We assume the same assumption applies dur-

ing a small period of time st. If the number of transitions are denoted by GNGM, provided

z(hj, hk) is constant during 6t , this number would be given by

Since local equilibrium has been assumed to exist on the surface, the unidirectional

rate of adsorption, jGM may be written from Eqs. (3.17) and (3.30) as

Following a similar procedure, one may obtain an expression for the unidirectional rate of

desorption,

where hi is the molecular distribution that would result if a molecule from the surface

phase were transferred to the gas phase:

Following the statistical rate theory approach, we assume that K[hj(ri ,) , hi(uc)]

has the same value for molecular distributions within the isolated system, since all these

states of molecular distribution would lie within the energy uncertainty. Thus, we may

evaluate Eqs. (3.31) and (3.32) at thermodynamic equilibrium. Since at equilibrium jMG

and jGM are each equal to the equilibrium exchange rate, one finds from Eq. (3.16) that

Hence the net rate of molecular adsorption, .TGM , is given by

Since the molecularly adsorbed hydrogen can dissociate on the surface. The rate of

change of moIecularly adsorbed hydrogen can be written

where J~~ is the net rate of molecular dissociation on the surface, and the rate of produc-

tion of atomically adsorbed hydrogen is given by

Since we have assumed that local

Eq. (3.17)), the rate of dissociation on

equilibrium always exists on the surface (see

the surface must be rapid compared to the rate of

molecular adsorption.22 We assume further that the molecules adsorbed are immediately

dissociated; thus

jGM = JMA

and from Eqs. (3.35) and (3.37) one finds

Since local equilibrium has been assumed to exist within the gas phase and on the surface,

the expressions for pG and pA may be used in Eq. (3.39) to obtain an expression for the

net adsorption rate in terms of T, NA, NG, A, V . Note then that the net adsorption rate

depends only on the properties that the isolated system and the TPD system are assunled

to have in common.

After neglecting the number of molecules on the surface compared to the number

of atoms, Eq. (3.39) may then be used with Eqs. (3.1) and (3.4) to predict NG and the

pressure in the TPD system as a function of time.

3.4 Hz-NI(IOO) TEMPERATURE-PROGRAMED-DESORP'I'ION SPECTRA

Using the numerical procedure outlined in Appendix B, for each of the seven ini-

tial coverages indicated in Fig. 3.3, a thermal desorption spectrum was calculated by solv-

ing Eqs. (3. I) , (3.4) and (3.39). These calculation were performed with the heating rate

and gas volume stated in Refs. 2 and 39 (15*K/s, 100 L) and, for the reasons described

below the pumping rate, R, , chosen to be 110 Lls.

To choose the value of R, , one of the TPD spectra reported by Christmann et oL2

was used. The experimental pressures reported were in "arbitrary units" and the initial

exposures (coverages) in "Langmuirs". The maximum initial experimental coverage was

5.2 Langmuirs and we assume this exposure in Langmuirs would give an initial coverage

near the saturation coverage of 0.5. However, even if saturation were established initially,

some hydrogen would desorb before the start of the TPD procedure. For the calculation

shown in Fig. 3.4 an initial coverage of 0.45 was used. Experimental points from the spec-

trum with an initial coverage of 5.2 Langmuirs are also shown in Fig. 3.4. Increasing the

value of R , was found to move the calculated value of maximal temperature for this ini-

tial coverage, T,,(0.45), to higher temperatures. Since T,,,(0.45) had been measured to

be 359 K, the value of R, was chosen so that the measured value of T,,,(0.45) corre-

sponded to that calculated. This resulted in a value of 110 U s being chosen as the pump-

ing speed.

Once the value of the pumping speed had been chosen, T,, of the calculated spec-

trum and that of the experimental spectrum coincided. However, the experimental pressure

was in "arbitrary units". The value of these units were chosen so that experimental pres-

sure corresponded to the calculated pressure. Thus, note that the pumping speed and pres-

sure scale were chosen independently. One parameter moves the spectrum horizontally

and the other vertically. Thus, the value of both parameters could be determined from one

spectrum. Once these apparatus constants had been chosen they were used to calculate all

of the other spectra shown in Fig. 3.3.

Three predictions regarding the calculated spectra should be noted: 1) The temper-

ature,T,,, , moves to lower values as the initial coverage is increased. This may be clearly

seen in the results presented in Fig. 3.3 and is in agreement with the reported behavior2. 2)

The TPD spectrum shown in Fig. 3.4, at points other than the maximum pressure, was pre-

dicted. As may be seen there, the predicted spectrum agrees reasonably with the measured

pressures. 3) The pressure scale was chosen from one TPD spectrum, and once this scale

was established, as seen in Fig. 3.5, the predicted maximum pressures for other spectra as

a function of initial concentration corresponds well with that measured, and the maximum

pressure of each spectrum is predicted to occur at temperatures that are in agreement with

the measurements.

3.5 Hz-N1(110) TEMPERATURE-PROGRAMED-DESORPTION SPECTRA

Since the same vacuum system2 was used to study both H2-Ni(ll0) and HZ-

Ni(100), we take the values of the pumping speed, the gas volume and heating rate to be

the same (1 10 Us, 100 L, 15 Ws). Using these values and the calculation procedure

described in Appendix B, the TPD spectra were calculated for H2-Ni(1 10) and are shown

in Fig. 3.6. As may be seen there, T,,, is predicted to move to higher values as the initial

coverage is increased until the initial coverige reaches 0.5, then to move more slowly to

higher values as the coverage is increased further. This is in contrast to the predicted and

observed behavior of H2-Ni(100), but it is in accordance with the experimentally

observed2 behavior of Hz-Ni(ll0). Note that this prediction is made without any fitting

constants. To examine the roIe that phase transitions play in the behavior of T,,, , the value

of the experimental pressure is required.

The pressures for the experimental TPD spectra were reported in terms of "arbi-

trary units" and the initial coverage was in terms of Langmuirs ', but in this case (in con-

trast to H2-Ni(100)) no experiments were reported for which the Ni(ll0) was saturated

with hydrogen before measuring its TPD spectrum. To obtain the pressure scale in this

case, the calculated TPD spectrum that had the same value of T,,, as an experimental spec-

trum was used. A value of 348 K for T , was obtained both for the TPD spectrum calcu-

lated when the initial coverage was 0.35 and for the measured TPD spectrum when the

initial coverage was 0.17 Langmuirs. This calculated spectrum and data points from the

measured spectrum2 are shown in Fig. 3.7. The value of the "arbitrary unitsy' for the pres-

sure were chosen so that the maximum pressure of the experimental spectrum and that cal-

culated had the same value. Using this value for the scaling factor, the maximum pressure

in the other experiments of Ref. 2 were each scaled and plotted against TI,, . The results are

shown as the solid dots in Fig. 3.8. Also shown in this figure are the calculated values

(open circles) of the maximum pressure for different initial coverages. As seen there, close

agreement is found between predicted and measured results.

As seen in Fig. 3.8, the experimental and theoretical maximum-pressure versus

T,, plot makes a change in slope when the initial coverage is approximately 0.5. No such

change in slope is seen in the case of H2-Ni(100), this system undergoes no change of

phase during a TPD spectrum.

To investigate the possibility that a phase transition occurs in the H2-Ni(1 10) sur-

face during a TPD experiment, the surface coverage as a function of temperature (or time)

has been calculated for different initial coverages and are shown in Fig. 3.9. Also shown in

this figure are the "instabiiity islands" where the phase transition is predicted to occur. As

may be seen there, for initial coverages below approximately 0.5, the calculated coverage

does not enter the unstable region, but for initial coverages of 0.5 or above, a phase transi-

tion is predicted to occur during the course of a TPD experiment.

Further, the values of TI,, that are calculated for each initial coverage are greater

than the calculated temperature at the time of the phase transition. The values of T,,, are

also indicated in Fig. 3.9. Thus, if the system undergoes a phase transition, the phase tran-

sition occurs before the system reaches the value of T,,, . In summary then, if the initial coverage is less than 0.5, the system does not

undergo a phase transition in the course of a TPD spectrum, and the maximal temperature

moves to the right with increasing initial coverage. As seen in Fig. 3.8, the value of the TI,,

is accurately predicted with SRT. If the initial coverage is greater than 0.5, it is predicted

that before the system reaches T,, the system undergoes a phase transition in the course of

a TPD spectrum. In these cases, the observed behavior is that the values of TI), become

almost independent of the initial coverage, and as seen in Fig. 3.8, the predicted behavior

is in close agreement with these observations. Thus, it appears that the phase transition

that occurs during the course of the TPD spectrum could be the reason for the values of

T,, to become independent of the initial coverage.

3.6 DISCUSSION: APPLICATION OF SRT TO A HETEROGENEOUS SURFACE

To apply the SRT procedure for predicting the molecular transport rate from the

gas phase to the solid surface requires the expression for the chemical potential of the

adsorbed gas species. For both Hz-Ni(100) and H2-Ni(1 lo), this expression has been pre-

viously determined (Section 2.4), and the values of the parameters have been tabulated in

Chapter 2. Their values are given Tables 2.1 and 2.2. Thus, no fitting parameters appear in

the SRT expression for the molecular adsorption rate that arise from the chemical potential

expressions. There are three apparatus constants. Two of these, the system volume and

heating rate, were reported with the experimental data2 The third one was determined

from one TPD spectrum that had been reported for H2-Ni(100). The same experimental

apparatus was used to obtain the TPD spectra; thus, for HZ-Ni(1 10) the same values of the

apparatus constants could be used. This means that certain characteristics of the TPD

spectra for H2-Ni(1 10) were predicted: these include the dependence of the maximal tem-

perature on the initial coverage. The prediction is that for initial coverages of less than 0.5,

T,,, increases as the initial coverage is increased until T,, reaches the value of 352K, then

with further increases in the initial coverage, the slope of the TI,, curve become much

sharper. These predictions are made without fitting constants, and are in agreement with

the observations. The change in slope of TI,, was shown to result from the phase transition

occurring during a TPD experiment when the initial coverage is greater than 0.5.

In a calculational sense, the reason the SRT procedure predicts a change in the

slope of T, , can be traced to the expression for the chemical potential. As shown in the

previous chapter, a homogeneously adsorbed surface phase is unstable when the

(apA/a NA)T is negative. In the TPD experiments with HTNi (1 lo), for the initial surface

coverages and initial temperature considered, the homogeneously adsorbed surface phase

is stable (see Fig. 3.8), but as the experiment proceeds, the coverage decreases and the

temperature increases. If the initial coverage is above 0.5, the system reaches a state where

( a p A / a ~ A ) T is negative. In this range, the decrease in NA imposed by the experimental

procedure results in an increase in the chemical potential value. According to SRT, an

increase in the chemical potential of the adsorbed hydrogen increases the rate of H2 des-

orption from the surface. This would tend to increase the gas phase pressure and cause T,,,

to be reached at a lower temperature. Since T, , increases with initial coverage until the

unstable range is reached during an experiment, the effect of the phase transition is to stop

the increase in T,,, with an increase in initial coverage.

For some systems, the maximal temperature has been found experimentally to

increase with increasing initial coverage and for others, to decrease. An example of the

former is Hz-Ni(1 00) and of the latter is H2-Ni(1 10) when the initial coverage is less than

0.5. The predicted behavior of these system for the same values of the apparatus constants

is in agreement with the observation^.^ Thus, the only reason for the difference in the pre-

dicted dependence of T,,, for these two system is the difference in their chemical poten-

tials.

It has been suggested6 that none of the other theoretical approaches can explain

thermal desorption at high coverages because these approaches do not correctly account

for interaction effects at high coverages. The SRT approach accounts for the interactions

through the potential acting between the particles. This potential is allowed to depend on

coverage, and as a result the chemical potential contains terms that depend on coverage.

This dependence is determined from the empirical adsorption isotherms.

Since SRT has been shown to provide a method for predicting the TPD spectra and

the dependence of T,,, on the initial coverage, these results provide support for the central

hypothesis of this theoretical approach. This hypothesis is that in a thermodynamically

isolated system, the rate of transition between quantum mechanical states has the same

value provided those states are within the energy uncertainty. It is this hypothesis that

allows one to show that K[hj(u,), h,(u,)], the probability of a transition from molecular

distribution hj to hk at any instant in the TPD system is equal to the equilibrium

exchange rate in the isolated system (see Eq. (3.34)). Once this relation is established, the

SRT expression for the adsorption rate can be written in terms of the chemical potential of

the adsorbed particles and other parameters that can be determined independently of the

measured rate. It is now understood that the existing kinetic theories for predicting the

adsorption/desorption rate cannot completely explain the thermal desorption at high cov-

erages and the reason is suggested to be that none of these models incorporate the effects

of adsorbate-induced changes in the surface? The advantage of the SRT approach over

other existing theories of thermal desorption is that it accounts for the interaction of

adsorbed atoms with surface atoms through chemical potential expression. In the SRT pro-

cedure the bbreaction-orderT' is not assigned empirically and the coverage dependence of

the net desorption rate is determined by the theory itself.

This is particularly important for the Hz-Ni(1 10) system. The previous analysis of

this behavior2 was based on the Polanyi-Wigner equation in which the "reaction order" is

empirically assigned. It was suggested that the initial increase in T,,, with increasing cov-

erage "might" be due either to "zero-order9' kinetics or to the heat of adsorption depending

on coverage at small coverages. When T,,, became almost independent of the initial cover-

age, it was suggested that "first-order*' kinetics was then applicable.2

During a portion of the desorption process the surface is heterogeneous. In the

present analysis it was assumed that the surface was in a state of local equilibrium during

the desorption process. As shown in Chapter 2, the chemical potential has the same value

for three different surface concentrations. Thus, the SRT expression for the desorption rate

(see Eq. (3.39)) would predict the same rate of desorption from each portion of the sur-

face, even though the coverage were different in the different portions of the surface.

TABLE 3.1 TPD experimental inforn~ation for H2-Ni(1 00) and H2-Ni(1 10)

Variable Value Source

Heating Rate, R, 15 K/s Ref. 2

Pumping Rate, R, 110 Ws

Surface Area of Crystal, A 0.5 cm2 Ref. 2

Initial Temperature, T(0) 300 K Ref. 2

Initial Pressure in h e gas phase, P(0) 10-10 Torr Ref. 2

Effective Gas Phase Volume, V 100 L Ref. 39

t Vacuum pump

Gas

Substrate

Gas

Surface

Substrate

Isolated System

FIGURE 3.1. Schematic of TPD apparatus and its associated isolated system.

FIGURE 3.2. The region for unstable equilibrium, 8, S 8 1 OP shown on an isotherm. The instability leads to the predicted hysteresis loop indicated by the arrows.

300 350 400 450 500 Temperature, K

FIGURE 3.3. Calculated TPD spectra for H2-Ni(100) using SRT for the initial coverages indicated when the heating rate was 15 'Ws the pumping rate was 1 10 L/s and the gas phase volume was 100 L**".

300 350 400 450 500 Temperature, K

FIGURE 3.4. The TPD spectrum for H2-Ni(100) that was used to establish the pumping speed and the pressure scale. The solid dots are measurements reported in Ref. 2 when the initial covemge was 5.2 Langmuirs. The solid curve was calculated when the initial coverage was chosen to be 0.45.

I I I I I

300 350 400 450 500

Maximal Temperature,T, (K)

FIGURE 3.5. For H2-Ni(100) the maximum pressure versus the temperature, T,,, , at which the maximum occurred. The open circles are the calculated values and the solid dots are the measured values. The initial coverages used in the calculation were 0.49, 0.40, 0.35, 0.30, 0.25 0.2,0.15, 0.10, 0.05, and the experimental values reported in Ref. 2 were 5.2, 1.2, 0.7, 0.5, 0.4, 0.3, 0.25, 0.2, 0.15, 0.1 Langmuirs, respectively.

300 350 400 450 Temperature , K

FIGURE 3.6. The calculated TPD spectra for H2-Ni(1 10) using SRT for the initial coverages indicated when the heating rate was 15 K/s, the pumping rate was 110 L/s and the gas phase volume was 100 L.

300 320 340 360 380 400 420 Temperature, K

FIGURE 3.7. The solid line is the calculated TPD spectrum using SRT for Hz- Ni(ll0) that had a T,,, of 348 K. The data points are from an experimental TPD spectrum that had the same value of T,), , but its pressure was reported in "arbitrary units".2 The experimental pressure has been scaled so that it agrees with the predicted pressure at =,I,

- 300 320 340 360 380 400

Temperature Maxima,T, (K)

FIGURE 3.8. For H2-Ni(1 10) the maximum pressure versus the temperature, T,,, , at which the maximum occurred. The open circles are the calculated values and the solid dots are the measured values. The initid coverages used in the calculation were 1.45, 1.2,1.10, 1.0, 0.90, 0.8, 0.70,0.6,0.56,0.53, 0.50,0.45,0.40,0.35,0.30,0.25,0.20, 0.10 and the experimental values reported in Ref. 2 were 1.55, 1.30, 1.05,0.65, 0.55, 0.40, 0.30, 0.17 Langmuirs. Note that for initial coverages above 0.5 the slope of the Maximum Pressure-Maxima Temperature changes slope. This is the initial coverage at which a phase transition takes place during a TPD experiment (see Fig. 3.9).

300 350 400 450 Temperature , K

FIGURE 3.9. The solid lines are the predicted coverages as a function of surface temperature for different initial coverages that were calculated using SRT for Ni(ll0) when the heating rate was 15 Ws, the pumping rate was 1 10 L/s and the gas phase volume was 100 L. Note that in all experiments in which the initial coverage was 0.5 or above, the temperature and coverage are predicted to have values that would result in a phase transition, but that none of those with initial coverages below 0.5 are predicted to undergo phase transitions.

CHAPTER 4: THERMAL DESORPTION OF HYDROGEN FROM W(100)- PREDICTION OF A SURFACE PHASE TRANSITION

4.1 INTRODUCTION In Chapter 2 we presented a method by which it was possible to predict the values

of temperatures and surface coverages at which a surface phase formed by homogeneously

adsorbed particles under equilibrium conditions becomes unstable. We showed that in

order to apply this approach, the chemical potential of the adsorbed particles must be

obtained. The explicit form of this expression was then obtained from the partition func-

tion of the adsorbed atoms. It was shown that both the expression for the potential when

the particles are at their lattice positions and their adsorption sites and the dependence of

the chemical potential on the vibrational frequencies of the adsorbed atom can be obtained

from the equilibrium adsorption isotherms. These expression were obtained for hydrogen

adsorbed on Ni(100) and Ni(ll0). It was also shown that the H2-Ni(ll0) system can

undergo two phase transitions in the range of temperatures and pressures at which iso-

therm measurements were available.

In Chapter 3, we introduced a quantum mechanical approach, called statistical rate

theory, that allows the concentration of adsorbed particles on the surface during the des-

orption process to be predicted. In this method the net adsorption rate is presented in terms

of the number of particles on the surface and the material properties of the gas-solid sys-

tem. It was shown that in this approach there is no need to assign a "reaction-order7' as is

required in Polanyi-Wigner equation. When the SRT approach was used to predict the

TPD spectra for H2-Ni(1 lo), it was found that certain characteristics of these spectra cor-

responded to the phase transition on the surface. We showed that the coverage dependence

of the temperature corresponding to the maximum pressure in a thermal desorption exper-

iment, T,, , is related to the phase transition on the surface.

In this chapter we will apply the same approach to Hz-W(100) system. Adsorption

of hydrogen on the (100) face of tungsten has been the subject of many studies. The results

obtained from LEED experiments for this system, have revealed sequences of tungsten

surface reconstructions when the amount of hydrogen adsorbed on the surface is

increased. Also the TPD spectra for this system showed two distinct peaks. The first peak,

with corresponding temperature T,),, , appears at high initial coverages. The peak tempera-

ture varies from 430 K at the saturation coverage to 435 K at the lowest initial coverage at

which the peak is observed." The second peak, with corresponding temperature T,,r2,

exists for all initial coverages and appears at temperatures between 530 to 570 K, see

Fig. 1.4.

For the H2-W(100) system, three fundamental frequencies of the adsorbed atoms

have been measured48Ag which were observed to depend on the surface coveragePg Once

the values of these frequencies are inserted into the expression for the chemical potential,

Eq. (2.40), there is only one coverage dependent function that remains to be determined.

This function is determined from the measured equilibrium adsorption isobars: and after

it is determined, the expression for the chemical potential of the adatoms is complete. This

expression is used to determine the possibility of a phase transition of the tungsten surface.

For W(100), however, no measurements of the isobars exist for hydrogen coverages of

higher than 1 (even the results from LEED experiments at these high coverages are not

very conclusive). Therefore, a method is presented to estimate the potential function in the

range of coverages higher than 1.

The SRT equations for calculating the TPD spectra are used to determine this func-

tion and to examine the validity of the expression for the chemical potential for H2-

W(100). In order to perform such calculations, one spectrum is used to find the best

approximation for the potential function and then it is used for predicting the other spectra.

The calculated spectra are round to have two sets of temperature maxinla. It is important

to notice that the SRT equations along with the other equations describing the temperature

and pressure changes in the vacuum system for H2-Ni(lOO), H2-Ni(1 10) and H2-W(100)

are the same. However, the expressions for the chemical potentials that are used in the net

desorption rate equation are different. The primary reason for this difference is the differ-

ence in the equilibrium properties of the systems. This confirms that the different charac-

teristics in the TPD spectra are a reflection of only the different material properties.

4.2 CHEMICAL POTENTIAL OF Hz-W(100) SYSTEM The expressions for the chemical potential and the equilibrium adsorption iso-

therms are obtained in Chapter 2, Eqs. (2.41) and (2.48). The latter equation may be re-

arranged to give:

where p and b are defined in Eqs. (2.49) and (2.50)

All three vibrational frequencies of hydrogen on W(100) were resolved by Barnes

and willis:' They measured the surface vibrational modes of atomic hydrogen on W(100)

in the specular beam direction and -25' off-specular towards the surface normal. They

found that hydrogen occupies the two-fold sites on the surface at all coverages and that its

vibrational frequencies depend on the coverage. The dependence of the vibrational fre-

quencies on coverage can be observed from the series of EELS measurements that have

been presented for different hydrogen coverages. The resolved frequencies taken from

Ref. 49 were plotted in terms of coverage and it was found that each of them can be

approximated by a polynomial of second degree as:

The coefficients fij, obtained from a least square method, are listed in Table 4-1. The

expressions for frequencies then were substituted in Eq. (2.40) and the values of the b

function were calculated from Eq. (2.50) at different temperatures and coverages.

To obtain the mathematical model for the P function and to complete the expres-

sion for the chemical potential of the adsorbed hydrogen on W(100), the measured equi-

librium isobars that were available for this system are used.

4.2.1 Equilibrium adsorption isobars for Hz-W(100)

The adsorption isobars for Hz-W(100) have been measured by Horlacher Smith et

al. and were reported in Refs.3 and 56. The experiments were performed by heating the

sample to a temperature above which no adsorption occurs and then allowing the sample

to slowly cool down, while monitoring the change in the work function as hydrogen

adsorption proceeds. The relation between work function and coverage, which is almost

linear up to the maximum coverage, is presented by ~ ~ e . ~ ~ The maximum coverage for

HTW(lOO) is 2 (two hydrogen atoms per surface tungsten atom) corresponding to a satu-

ration work function of 950 n ~ e ~ . ~ By using a linear relation between work function and

coverage, the equilibrium isobars were converted to the adsorption isobars in terms of cov-

erage, Fig. 4.1. Then Eq. (4.1) has been used to find the values of P at different coverages

and temperatures. These values are shown in Fig. 4.2.

Since the maximum measured work function in Ref. 3 was 500 meV (correspond-

ing to a coverage of 1.05), in order to find the expression for the P function at all cover-

ages, other information was used. Iwata et have reported a set of isothermal

adsorption measurements h i H2 on the W(100) surface at a temperature of 288 K by using

12 the resonant nuclear reaction ' H ( ' ~ N , ay) C. According to their results, at two different

pressures of 4.5 x 10%nd 9.1 x 10-~Torr the tungsten surface was saturated before 300

seconds of exposure to hydrogen gas. These pressures were used to obtain the correspond-

ing p function at saturation coverage. These values are calculated and added to the other

set of data points which are shown in Fig. 4.2.

Note that the values of the p functions which are presented in this Figure, are cal-

culated at different pressures and temperatures. One major assumption in obtaining the

partition function for adsorbed atoms was that @, depended only on coverage and not

temperaturet. Since the values of the p function at different coverages coincide, within

the experimental errors, this figure shows the validity of this assumption very clearly. This

examination was not possible in case of Hz-Ni(100) or Hz-Ni(1 10) because in those two

cases the vibrational frequencies were unknown.

t At the microscopic level where QO is defined in a quantum mechanical approach, temperature has not been defined yet, please see Chapter 2.

To find a mathematical model for the P function, we assume that this function can

be presented by a polynomial of fifth degree in terms of coverage, i-e.,

Then, one may use a least square minimizing error method to find the coefficients aj that

best fit the measured data points. Since there are no measurements between coverages of 1

and 2, if this method were used without any other considerations, the calculated coeffi-

cients could not represent the behavior of the P function physically. This problem can be

observed from Fig. 4.2. The curve labeled as (7), passes through the available data points,

but there is no basis for assuming that this curve is valid in the range of 1 I 9 5 2. For this

reason, we investigate a method to find an expression for the P function in the coverage

range between 1 and 2.

4.2.2 Method for estimating the P function

The method which is adopted in this work is a simple approach. We add an arbi-

trary data point between coverages of 1 and 2 at the same coverage at which the fifth order

polynon~ial (curve number 7) has a minimum. By adding this point to the other measured

data points, a new set of ai coefficients is calculated. Then, the position of this point is

changed along the vertical axis and the calculations are repeated. By repeating this proce-

dure, each time a new set of coefficients are found. A sample of such a family of curves,

plotted from the calcuIated coefficients, are shown in Fig. 4.2. These curves represent the

p function for each set of coefficients. It is obvious that the values of the coefficients <xi

and therefore the P function, are changed if the horizontal locations of these additional

points are also changed. Since there are infinite number of possible configurations for this

portion of the p function, we limit ourselves to a much simpler solution and assume only

the changes in the vertical direction. As will be seen, one of these functions represents a

good estimate of the P function. Note that for all of the infinite number of possibilities for

the p function, the portion of the mathematical model which describes the behavior of this

function between coverage of 0 and 1, remains almost unchanged and accurate.

In order to determine which set of the coefficients gives a good estimate for the P

function and the chemical potentia1 expression, we use the information from the measured

TPD spectra for H2-W(lO0). In the previous section we showed that in SRT equations for

determining the net adsorption rate, the expression for the chemical potential is used. The

TPD experiments are usually performed over a wide range of temperatures and initial cov-

erages, up to the saturation coverage. Therefore, one spectrum can be calculated at this ini-

tial coverage and be compared with the measured one to obtain the best set of coefficients.

By performing such a comparison, the chemical potential expression can be determined.

4.3.1 The governing equations

In Chapter 3, the governing equations describing the heating rate of the substrate,

the number of hydrogen molecules in the gas phase, and the number of adsorbed atoms on

the surface at each instant of time in a TPD system were presented. These equations con-

stitute a closed system of equations that can be solved simultaneously to predict NC as a

function of time in the TPD system. In this chapter, we consider another case in which the

heating rate of the sample is varied. The substrate temperature at time t , then may be rep-

resented by

The expression for R,,(T) would ideally be given as part of the experimental conditions.

After neglecting the number of adsorbed molecules on the surface, Eq. (3.4) can be simpli-

fied to:

and Eq. (3.39) can be expanded to:

( * M - N * ) ~ N ? T $ exp ( 2- bkGj3) - ( A M - N ~ )

N~ N+$ exp (2 2)

Note that the net adsorption rate depends only on the properties that the isolated system

(defined in Chapter 3) and the TPD system are assumed to have in common. If the value of

NG were known, it could then be used with the ideal gas equation of state to predict the

pressure as a function of time in the gas phase.The numerical procedure that was applied

to perform such calculations is outlined in Appendix B.

4.3.2 TPD measurements of Hz- W(100)

The measurements of the TPD spectra for H2-W(100) have been performed by

several groups: Tamm and ~chrnidt,' Yates and ~ a d e ~ , ' li 52 King and ~ h o m a s ~ l , and

Adam and ~ e r m e r ? The experimental results presented in these works, despite the dif-

ferent experimental conditions, i.e., heating rate, pumping speed, and adsorption tempera-

ture have two common characteristics. The first one is that in the TPD spectra two

distinguishable peaks are observed for which the corresponding temperatures depend on

the initial coverage. The first peak with maximal temperature, T,, ,, is only seen for initial

coverages higher than a certain value."* 52 The second peak with corresponding maximal

temperature T,,,, , appears at higher temperatures than Tll,I and is seen for all initial cover-

ages. The second characteristic is that at the initial coverage equal to the saturation value,

the difference between maximal temperatures of the two peaks, (TI,, - TI,,,) is around

1OOK.

In order to solve the system of equations (4.4), (4.3, and (4.6), the expression for

the chemical potential for hydrogen adsorbed on W(100) must be used. Since this expres-

sion contains the mathematical model for the P function, the different sets of coefficients

which were obtained for this function are expected to give different results for the TPD

spectra. From our preliminary analysis we found that the ratio of the first peak pressure to

the pressure of the second peak in a TPD spectrum and, to some extent, the values of the

two maximal temperatures depend on this function. Therefore, these characteristics can be

used to find the closest possible model for the P function.

The experimental parameters for this analysis were taken from Ref. 11, and are

given in Table 4-2. The other set of results that could be used for comparison, were those

presented in Ref. 5 1. But we found two major discrepancies between the results of Ref. 5 1

and the ones presented in the other references. The first problem was associated with the

initial coverage at which the first peak in the TPD spectra is observed. According to Ref.

5 1, at absolute coverage of 1.12, the TPD spectrum has still only one peak. The second

discrepancy was related to the difference between the two maximal temperatures. The

maximum temperature difference in Ref. 51 was found to be only 50K for the saturation

initial coverage which is not in agreement with the other measurement^.^* 52*58

The experiments by Madey and yatesl1* 52 have been performed in an ultra-high

vacuum apparatus with a base pressure of less than 2 x lo-'' torr and a total volume of

approximately 0.27 liters (the volume was calculated from the sketch of the experimental

apparatus that was presented). Since there are other instruments located in the vacuum

chamber, the effective volume was assumed to be 0.25 liters (approximately 90% of the

total volume). Changing the effective volume by +lo % did not affect the final results. The

heating rate was reported to be variable from 20 Ws at 400 K to 30 K/s at 600 K and a

maximum error in the measured temperatures of +10K was reported. By assuming a lin-

ear relation between the heating rate and the temperature, the variable heating rate was

found to obey the following relation:

Note that at this point all parameters in Eqs.(4.4), (4.5), and (4.6) are known except the P

function and the effective pumping speed, Rp. The latter is an apparatus characteristic and

is assumed to be constant throughout the TPD experiments. The two maximal ternpera-

tures are mostly affected by the pumping speed. Increasing the value of R, was found to

move the calculated values of maximal temperatures to higher temperatures. We men-

tioned earlier that the pressure ratio of the two peaks in a TPD spectrum depends on the P

function. Both these two parameters, then, can be determined from one experimental spec-

trum.

4.3.3 Procedure for finding the P function

The spectrum which was used for this purpose was the one measured at the satura-

tion initial coverage. By using a set of coefficients that would describe the P function in

Eq. (4.6), I?, was changed such that the calculated maximal temperatures for the two

peaks became almost the same as the measured ones. Then the ratio of the calculated pres-

sures at the two peaks was obtained and was compared with the measured ratio. The calcu-

lations were repeated for different sets of coefficients until the maximal temperatures and

calculated ratio agreed with the measured spectrum. Each time the calculations were per-

formed with a new set of coefficients, the pumping speed was also adjusted. When the

agreement between the maximal temperatures and the pressure ratio were obtained, that

set of coefficient and the pumping speed were used to obtain the rest of the spectra and the

compatibility of the mathematical model was further examined.

For example, the set of coefficients that would form the curve labeled (1) in

Fig. 4.2, was used for calculating the TPD spectrum at saturation initial coverage. The cal-

culated spectrum exhibited two peaks, see Fig. 4.3, as was observed in the experiments.

The pumping speed was determined such that both maximal temperatures be the same as

the measured ones. But as one may observe from Fig. 4.3, the ratio of two peaks, was less

than the measured values that could be inferred from the spectrum. It is important to men-

tion that changing the pumping speed by more than one order of magnitude, does not

change this ratio significantly. This confirms that it is the P function that is critical in

determining the pressure ratio and not the pumping speed. The error bars on the experi-

mental data points show the margin of the error in the temperature measurement.

In the next step another set of coefficients with corresponding curve labeled (2)

were used. This function showed the same behavior as the function labeled (I), but the

ratio of two peaks was closer to the measured one. The calculations were repeated until we

found that the set of coefficients which represents the curve labeled (4) gives the best ratio

of two peaks. With this P function we could also find a pumping speed that would predict

both temperature maxima, as were measured. In practice, several sets of coefficients were

examined at the vicinity of the function number (4) and based on the calculated results, the

best set was chosen. The values of a,, chosen by this procedure are given in Table 4-3.

The TPD spectrum calculated with this set of coefficients and the corresponding

measured one, taken from Ref. 11, are shown in Fig. 4.4. Note that it is usually impossible

to reach the maximum coverage of 2, therefore, the saturation coverage was chosen to be

1.96. Since T,,,, had been measured to be 430 K and T,,,* to be around 530 K, the value of

R, was chosen so that the calculated values of these two temperatures became almost the

same as those mentioned. This resulted in a value of 0.25 Us as the pumping speed. How-

ever, the experimental pressures were in "arbitrary units". The value of these units were

chosen so that the maximum experimental pressure at the first peak corresponded to the

calculated pressure for the same peak. The other pressures of the experimental spectrum

were obtained with the same pressure scale. Once the P function and the pumping speed

were chosen from one spectrum, they were used to calculate all the other TPD spectra for

H2-W(100) system, shown in Fig. 4.5.

It is necessary to mention that when the ori coefficients, corresponding to the func-

tions labeled with higher numbers than curve (4), were inserted into the expression for the

net desorption rate, two inconsistencies were observed. First, the ratio of two peaks in the

spectrum for 8(0) = 1.96 became larger than the measured one. This ratio increases as

the depth of the P function is increased. Secondly, the maximal temperature T,,,, started

to increase by increasing the initial coverages. This behavior is also in contrast with the

observations from the TPD measurements1 l.

4.3.4 TPD spectra calculated for Hz-W(l00)

As seen in Fig. 4.5, the calculated spectra for Hi2-W(100) show only one peak for

initial coverages less than 0.8. By increasing the initial coverage more than this value,

however, a second peak is observed. The maximal temperatures corresponding to the first

peaks, T,,,, are almost independent of the initial coverage for the initial coverages greater

than 1.1, whereas T,,,, move towards lower temperatures as the initial coverage is

increased. Both observations are in complete agreement with the experimental results.

In Fig. 4.6, the calculated values of the maximum pressures of the TPD spectra for

different initial coverages are plotted against the maximal temperature T,,, (open circles).

In this Figure also the maximum pressures of the experimental TPD spectra of Ref. 11

(solid circles) are also shown. For plotting these data points the value of the pressure scale

obtained from Fig. 4.4 was used as the scaling factor. Close agreement is noted between

the predicted and the measured results, especially for T,,,l.

Since the calculated maximum pressures corresponding to the initial coverages of

1.7 and 1.4 are seen to be the same as the measured pressures, the calculated TPD spectra

for these two initial coverages were also compared with the experimental ones. The results

are shown in Figs 4.7 and 4.8, respectively. The same agreement between the pressure

ratio of two peaks is also observed for these spectra. The calculated spectrum for initial

coverage of 0.6 was also compared with one of the measured spectra that had the same

temperature maximal. As is seen from Fig. 4.9, the calculated spectrum does not have the

same agreement with the measured one. This could be the result of using a inaccurate

heating rate at these high temperatures.

4.4 INTRINSIC STABILITY EXAMINATION FOR H2- W(1OO) In Chapter 2, three expressions describing the intrinsic stability conditions for a

homogeneous surface phase were found. It was shown that by examining only one of

them, Eq. (2.19), the range of coverages and temperatures at which a homogenous surface

phase becomes unstable can be determined. By using the chemical potential of hydrogen,

the stability of the homogeneous surface phase of Hz-W(100) can be examined for the set

of coefficients listed in Table 4-3. One sample of such calculations for a temperature of

300 K, is shown in Fig. 4.10. As one can see the expression ( a P A ) / ( a ~ * ) ~ , a is negative

in a range of 1.13 I 8 I 1.5. In this range the homogeneous surface phase becomes unsta-

ble at this temperature. This instability can be seen clearly from Fig. 4.1 1. In this Figure

the chemical potential of the adsorbed hydrogen at 300K is plotted in terms of coverage.

The slope of the chemical potential is changed between coverages of 8, and ep which

correspond to the local n~aximum and minimum of the chemical potential, respectively.

These coverages are the same coverages at which the expression (dPA)/ (a~*)T, A is

equal to zero,(see Fig. 4.10). If the surface coverage were initially 0, and the pressure

were increased slightly at the same temperature, the system would arrive in a new stable

condition with a higher coverage, as indicated with an arrow in Fig. 4.1 1. Similarly, if the

initial coverage were Bp , a slight reduction in the pressure would result in the system mak-

ing a transition to a smaller coverage provided the temperature were held constant during

the transition. This is an indication of a first order phase transition on the surface.

To investigate the possibility of the occurrence of a phase transition on the HZ-

W(100) surface during a TPD experiment, the range of unstable coverages at different

temperatures has to be obtained. A series of calculations similar to the one shown in

Fig. 4.10 has been carried out at different temperatures. The results of such calculations

determine the range of coverage at which the homogeneous surface phase becomes unsta-

ble. This range can be shown in the form of an island of unstable phase, Fig. 4.12. As is

shown in this figure, by increasing the temperature, the range of coverage at which a phase

transition occurs becomes smaller until at a temperature slightly higher than 500 K, no

instability is observed on the surface.

Fig. 4.12 also shows the variation of the coverage with temperature during the

TPD experiment, obtained at different initial coverages. As may be seen there, for absolute

coverages below approximately 1. I , the surface phase does not experience any phase tran-

sition. However, for initial coverages of 1.1 or above, a phase transition may occur during

the course of a TPD experiment. This prediction is similar to that one made for H2-

Ni(ll0) system. In that case, it was also observed that the corresponding maximal temper-

atures after a phase transition do not depend on the initial coverages (the prediction that

was supported by the experimental observations). In this case, H2-W(100), the same

behavior is observed, i.e., the calculated maximal temperature, T,,, , , remains almost con-

stant after an initial coverage of 1.1, see Fig. 4.6. This observation is in agreement with the

measurements.

Since the approximated expression for the P function is now available, the calcula-

tions for the equilibrium adsorption isobars can be completed. The coefficients listed in

Table 4-3 were inserted into Eq. (2.48) and the equilibrium adsorption isobars were calcu-

lated. The solid lines in Fig. 4.1, are the ones obtained with these coefficients. As was

expected, the model for the P function can predict the measured data points between cov-

erages of 0 and 1. On each isobar, for coverages higher than 1, there is a range of tempera-

tures at which three possible coverages can exist for a specific temperature. The stable

equilibrium state, as was mentioned before, can be determined from the experimental con-

ditions.

4.5 DISCUSSION AND CONCLUSION In the previous chapters we presented a quantum mechanical approach for deter-

mining the expression for the chemical potential of adsorbed atoms. We showed that in

order to find the mathematical models for the potential energy term, P, and the frequency-

dependent term, b, in this expression, the equilibrium adsorption isotherms/isobars can be

used. For H2-W(100), d l three vibrational frequencies of hydrogen have been resolved.

But the equilibrium adsorption isobars were not available to the saturation coverage.

Therefore, in order to find a mathematical model for the potential function, P , an approxi-

mate method was proposed.

To find the best model for this function, it was used in the chemical potential

expression along with the SRT equations to obtain the TPD spectra. Since the TPD spectra

for Hz-W(100) were available, the spectrum measured at the saturation initial coverage

was used for choosing the best approximated P function and the chemical potential

expression was completed. The latter expression was then used to predict the other TPD

spectra for this system and to examine the possibility of a phase transition occurring on the

surface. Through these calculations, the relation between the double peak feature of the

TPD spectra and the phase transition on the surface was also investigated.

Our calculations of TPD spectra for H2-W(100) by the SRT approach could predict

the spectra for this gas-solid system which were in agreement with the experimental mea-

surements. As one may see from Fig. 4.4 to Fig. 4.9, the SRT approach and the model for

the chemical potential predict the additional peak in the TPD spectra with a good quantita-

tive agreement.

To our knowledge, this is the first time that this type of agreement has been

obtained from an analytical model. Adams, in his comprehensive workg, used a lattice-gas

model along with the absolute rate theory for desorption rate to reproduce the thermal des-

orption spectra for H2-W(l 00). He obtained the interaction energy between the hydrogen

atoms in two cases, one for a first-order and another for a second-order reaction. His cal-

culations resulted in predicting the two-peak spectra for this system but with a poor quan-

titative agreement with the measurements. One problem with his model was that the

calculated ratios of the two maximum pressures in the TPD spectra were much smaller

than the experimental values. The second problem was related to the predicted dependence

of the maximal temperatures to the initial coverage. They were not in agreement with the

experimental observations. In another earlier analysis of the desorption kinetics for this

system, performed by Tamm and schmidt7, it was assumed that hydrogen desorbs with a

second-order reaction for the high temperature peak and a first-order reaction for the low

temperature peak. This would result in calculating two frequency factors and desorption

energies which were not confirmed by independent determinations of these parameters.3

As one may notice from Fig. 4.5 and Fig. 4.6, the SRT approach and the model for

the chemical potential predict an additional peak in the TPD spectra starting at an initial

coverage of around 0.8. It should be noticed that this characteristic of the TPD spectra for

H2-W(100) is not the direct result of a first-order phase transition on the surface (the phase

transition starts at coverages higher than 1.1). The coverage of 0.8 is in the range of cover-

ages at which the adsorption isobars are available. Therefore, even without extending the

p function to the saturation coverage, one would expect to obtain the additional peak at

the initial coverage of 0.8. This important result obtained from our calculations can be

related to the surface reconstruction of H2-W(100) system. From the LEED experimental

results, an unidentified type of a disordered phase has been reported in Refs. 10, 51,

61,and 62, which starts at coverages above 0.7 to 0.8. The appearance of the additional

peak could be the result of this surface reconstruction.

On the other hand, for Hz-W(100) another series of surface atoms displacements

have been observed at low coverages (between coverage of 0.3 to 0.7).1° In Ref. 3, this

surface reconstruction has been suggested to be the reason for the existence of the two

peaks in the TPD spectra. The additional peak was also implied to start at an initial cover-

age of 0 .5 .~ Our calculated TPD spectra, however, show no additional peak for this cover-

age, and to our knowledge neither do the measured experimental spectra.' '* 51* 52

As was shown, our model for the potential function, P , and the chemical potential

of the adsorbed atoms, predicted a first-order phase transition on the surface at coverages

of higher than 1.1 and temperatures lower than 500 K, see Fig. 4.12. For T = 300K, the

unstable coverage range was predicted to be 1.1324 S ei 5 1 .SO02 . This phase transition

has not been addressed in the literature explicitly. However, at coverages greater than one,

another type of a disordered phase was reported to occur on the W(100) surface.51* 62 The

nature of this disordered phase has not been identified. Our calculated range of coverage at

which the homogeneous surface phase becomes unstable, is in agreement with these

observations.

To support this phase transition further, we examined the two maximal tempera-

tures appearing in the TPD spectra. As one may see from Fig. 4.5 and Fig. 4.6, the two

temperatures T,,,, and T,,,, are independent of the coverage, when the initial coverage is

above 1.1. We also showed that at initial coverages higher that 1.1, a first-order phase tran-

sition occurs in the course of the TPD measurements, Fig. 4.12. This behavior was also

noticed for Hz-Ni(1 lo), see Chapter 3, which in that case was related to a first-order phase

transition on the surface. The phase transition of Hz-W(100) could be the reason for the

lack of experimental data at high hydrogen coverages (no isobars were reported for cover-

ages higher than 1).

TABLE 4-1. Coefficients for the vibrational frequencies of hydrogen on W(100). The frequencies are assumed to be second order polynomials of coverage as o, = f O j + f l j 8 + f2je2

Lateral Lateral Normal vibration vibration vibration

TABLE 4-2. TPD experimental information for Hz-W(100) system

Variable Value Source Heating Rate

Pumping Rate, RP

Variable, -25, (Ws) Ref. 1 1

0.25, (Ws) Calculated

Surface Area of Crystal, A 0.2827, (cm2) Ref. 11

Initial Temperature, T(0) 300, (K) Ref. 11 Initial Pressure in the Gas Phase, 10-io, (Torr) Ref. 1 1 P ( 0 )

Effective Gas Phase Volume, V 0.25, (L) Estimated from Ref. 1 1

TABLE 4-3. Properties for hydrogen adsorbed on W(100)

a,, (10""~/atorn) -4.9501 E-0 [Calculated]

a,, ( l ~ - ~ ~ ~ / a t o m ) -5.7570 E-I

a, , (10-lg~/atorn) - 1.8528 E-0

a,, ( l ~ - ' ~ ~ l a t o m ) 5.9845 E-1

Material properties:

Mo (I 015 atoms/cm2) 1.0

M (10'~ sites/cm2) 2.0

[Geometry]

Ref. 3

A Coverage, N /(AMo)

FIGURE 4.1. Equilibrium adsor tion isobars for Hz-W(100). Symbols are data from S Horlacher et al. Solid lines are calculated based on the model presented in the text.

0.8 1.2 Coverage, N~/ (AM o)

FIGURE 4.2. Family of P functions for Hz on W(100) obtained from equilibrium adsorption isobars. The additional data points between coverage of 1 and 2 are the artificial data points that are used to make an approximation for the P function. All curves are fifth order polynomials of coverage. The numbers correspond to the different polynomials that are discussed in the text.

300 400 500 600 700 Temperature, K

FIGURE 4.3. Comparison be tween the calculated TPD spectrum and the measured1 ' one (solid dots) at initial coverage of 1.96. The expression for the P function for this calculation corresponds to the curve labeled as (1) in Fig. 4.2. The pumping speed was obtained as 0.25 Lls. Error bars on the measured data points show the domain of error in temperature measurement.

300 400 500 600 700 Temperature, K

FIGURE 4.4. TPD spectrum at initial coverage of 1.96. Symbols are data inferred from Ref. 11. Solid line is calculated by SRT. The expression for the p function for this calculation corresponds to the curve labeled as (4) in Fig. 4.2. The pumping speed was obtained as 0.25 U s .

300 400 500 600 700 Temperature, K

FIGURE 4.5. TPD Spectra for H2-W(100) calculated at initial coverages of 0.3,0.4, 0.5, 0.6, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.96, respectively. The experimental parameters were taken from Ref. 11.

Double

400 500 600 Maximal temperature (T,), K

FIGURE 4.6. Maximum pressures versus temperatures extracted from TPD spectra. Solid dots are experimental results from Ref. 11. Circles are calculated by SRT model. The initial coverages are the same as those in Fig. 4.5.

300 400 500 600 700 Temperature, K

FIGURE 4.7. TPD spectrum at initial coverage 1.7. Symbols are data inferred from Ref. 1 1. Solid line are calculated by SRT.

400 500 600 Temperature, K

FIGURE 4.8. TPD spectrum at initial coverage 1.4. Symbols are data inferred from Ref. 11. Solid line are calculated by SRT.

300 400 500 600 700 Temperature, K

FIGURE 4.9. TPD spectrum at initial coverage 0.6. Symbols are data inferred from Ref. 11. Solid line are calculated by SRT.

FIGURE 4.10. The criteria for phase transition at 300 K, Eq. (2.19). The chemical potential is calculated from the P function labeled as (4) in Fig. 4.2.

Coverage, N* AM^)

FIGURE 4.1 1. The chemical potential of hydrogen adsorbed on W(100) as a function of coverage for a temperature of 300K and the P function labeled as (4) in Fig. 4.2. For coverages in the range of 8, to ep, a homogeneous surface phase is intrinsically unstable.

300 400 500 600 700 Temperature, K

FIGURE 4.12. Coverage versus temperature during a TPD experiments. The initial coverages are the same as those in Fig. 4.5. The island of unstable phase is also shown in this figure. Note that at the starting temperature, i.e., T=300 K, for the range of initial coverages( 1.1324 5 8(0) I 1 .SO02 the homogenous surface becomes unstable.

CHAPTER 5: SUMMARY AND CONCLUSIONS

We predicted the conditions under which a first-order surface phase transition can

be induced by adsorption and found that the interactions between the system particles

plays an important role in this prediction. In the proposed method the potential experi-

enced by each partide (substrate and adsorbed) is allowed to depend on the number of

adsorbed atoms. As a result, it was determined that the vibrational frequencies (phonons)

of the adsorbed particles can depend on the number of particles adsorbed. The method was

applied to two hydrogen-metal systems, H2-Ni(100) and H2-Ni(ll0). This dependence

was found from the equilibrium adsorption isotherms to be negligible for H2-Ni(100), but

essential for H2-Ni(1 lo). This conclusion for the latter system was supported by indepen-

dent experimental evidence obtained from high-resolution electron energy loss spectros-

copy for this system.

The proposed method can be used to predict the conditions under which a homoge-

neous surface phase becomes unstable. For H2-Ni(100) no phase transition was predicted

for the range of coverages and temperatures at which the adsorption isotherms were avail-

able. Two phase transitions, however, were predicted for H2-Ni(1 lo): the first one at low

coverages, near 0.2, and temperatures lower that 380 K; and the second one at hydrogen

coverages above 0.6 and temperatures above 380 K. Although the phase transition at tem-

peratures higher than 300 K+ and smaller hydrogen coverages has qualitative experimental

t It should be mentioned that a sequence of adsorption induced phase transitions has been also observed for H2-Ni(1 10) at temperatures lower than 200 K. But since no measured isotherms are available at these low temperatures, we didn't attempt to predict the phase transition at temperatures lower tha11300 K.

support the phase transition at hydrogen coverages above 0.6 does not appear to

have been identified previously.

We also presented the statistical rate theory approach for predicting the molecuIar

transport rate from the gas phase to the solid surface and showed that in this procedure the

expression for the chemical potential of the adsorbed gas species is required. This proce-

dure was used for predicting the TPD spectra for H2-Ni(100) and Hz-Ni(1 10) systems. We

showed that the dependence of the maximal temperature on the initial coverage differed in

the TPD spectra for these systems. For Hz-Ni(100) there is only one peak in the TPD spec-

trum and its maximal temperature, T,,, , decreases with increasing the initial coverage. For

H2-Ni(1 10) it was found that for initial coverages of less than 0.5, T,,, increases as the ini-

tial coverage is increased until T,, reaches the value of 352 K, then with further increases

in the initial coverage, the slope of the T,,, curve becomes much sharper. The latter predic-

tions, made without fitting constants, are in agreement with the observations. The change

in slope of T,,, resulted from a first-order phase transition that occurred during a TPD

experiment when the initial coverage was greater than 0.5.

The approaches developed above for predicting the phase transition on the surface

and the TPD spectra were then applied to H2-W(100) system. The adsorption isobars for

this system were available but have not been measured to the saturation coverage. The

LEED experimental observations at these high coverages are not accurate or conclusive.

However, all three frequencies for Hz-W(100) have been resolved and found to be depen-

dent on the coverage.

For this system, several estimated mathematical models for the potential energy

were obtained. The proposed functions were used in the chemical potential expression

along with the SRT approach to obtain the TPD spectrum at an initial coverage equal to

the saturation value. The calculated and the measured spectra were compared and the best

function for the p was found. This function was then used in the SRT equations to predict

the other TPD spectra. The predicted TPD spectra showed an additional peak in each spec-

trum as observed in the experiments. The first peak with higher pressure appears only for

initial coverages higher than 0.8, whereas the peak with lower pressure exists for all initial

coverages. We showed that the predicted maximal temperatures, T,,,l were independent of

coverage above the initial coverage of 1.1. Both predictions were in agreement with the

experimental observation^."^^ Since the latter behavior was also predicted for the Hz-

Ni(l10) system, and in that case it was related to the occurrence of a first-order phase tran-

sition on the surface, the conditions for such a transition were also investigated for H2-

W(100) system. It was found that the expression for the chemical potential that predicts

the TPD spectra, with a quantitative agreement with the measured ones, exhibits a first-

order phase transition for this system at coverages greater than 1.1 and temperatures lower

than 500 K. By increasing the temperature, the range of coverage at which the surface

becomes unstable is changed, (see Fig. 4.12) and above a temperature of approximately

500 K, no phase transition occurs on the surface. The sequence of the phase transilions

that was reported '0956*6062 to occur at coverages lower than 0.7, however, has not been

predicted with our approach. Therefore, it can be concluded that these phase transitions

are not the first-order transitions. This conclusion is further supported by the TPD spectra

and the fact that these transitions are reported to be reversible without any hysteresis

effectdo.

Due to the complexity of the TPD spectra and the sequences of the surface recon-

structions of the H2-W(100) system, a theoretical model that could quantitatively predict

the adsorption kinetics for this system had not been developed previously. A lattice-gas

model, along with the absolute rate theory for predicting the desorption rate, have failed to

reproduce the TPD spectra quantitatively.g

Based on the results obtained from the theoretical analysis applied to the H2-

Ni(100), H2-Ni(1 lo), and H2-W(100) systems and the quantitative agreements with the

experimental measurements, the following general remarks can be made:

1. The chemical potential of the adsorbed particles plays a major role in finding the condi-

tions under which a homogeneous surface phase becomes unstable. The potential expe-

rienced by the surface atoms which appears in the expression for the chemical potential

can be obtained from the measured equilibrium adsorption isotherms. The presented

quantitative predictions of the limiting conditions for a system to remain homogeneous,

are different in nature from the predictions of the quasi-chemical approach.

2. The SRT expression and the other equations describing the pressure and the number of

adsorbed atoms on the surface in the course of a TPD measurement were the same for

the three studied systems. Therefore, the difference in their TPD spectra resulted from

the difference in the chemical potential of the adsorbed hydrogen atoms on each sur-

face. As shown, the expression for the chemical potential of the adsorbed atom was

completed from the equilibrium adsorption isotherms/isobars. This means that the dif-

ferent shapes of the TPD spectra are a reflection of the different material properties of

each gas-surface system.

3. The reason the SRT procedure predicts a change in the slope of T,,, can be traced to the

expression for the chemical potential. As shown, a homogeneously adsorbed surface

phase is unstable when the (apA/&VA), is negative. In the range of coverages where

this function is negative, the decrease in NA imposed by the experimental procedure

results in an increase in the chemical potential value. According to SRT, an increase in

the chemical potential of the adsorbed hydrogen increases the rate of H2 desorption

from the surface. This would tend to increase the gas phase pressure and cause T,,, to

stay almost constant.

4. Since SRT has been shown to provide a method for predicting the TPD spectra and the

dependence of T,, on the initial coverage, these results provide support for the central

hypothesis of this theoretical approach. This hypothesis is that in a thermodynamically

isolated system, the rate of transition between quantum mechanical states has the same

value provided those states are within the energy uncertainty. It is this hypothesis that

allows one to show that the probability of a transition from molecular distribution hi to

h, at any instant in the TPD system is equal to the equilibrium exchange rate in the iso-

lated system. Once this relation is established, the SRT expression for the adsorption

rate can be written in terms of the chemical potential of the adsorbed particles and other

parameters that can be determined independently of the measured rate.

Appendix A: Hydrogen Despite its well known electronic structure, hydrogen atom is not a very good

adsorbate to be treated theoretically. As a matter of fact the interaction of a hydrogen atom

with a metal surface is much less understood than that of other adsorbates. Hydrogen mol-

ecule, owing to its strong covalent intermolecular bond, is rather chemically inert. There

are various properties of the ground state hydrogen moIecule that are important and are

listed in Table A.1. Some of other properties that make this species to be distinguished

are1:

-hydrogen is a very small size molecule (H-H distance of only 0.74 A'),

-hydrogen shape is almost spherical,

-the strong influence of the nuclear spin leads to the two species "ortho-" and "para-"

hydrogen with the total nuclear spin being symmetric and anti-symmetric, respectively,

-hydrogen is able to dissociate into the atoms.

This last property of hydrogen molecule deserve particular attention. On various surfaces,

particularly transition metal surfaces, the hydrogen molecule will spontaneously dissoci-

ate into the atoms which then form independent strong metal-hydrogen bonds. This aspect

of adsorption process is illustrated schematically in Fig. I. 1. The dissociation ( which can

also be observed with CO in some cases, depending on the strength of the metal-carbon

interaction) is a relatively complicated process which is difficult to treat theoretically. Fur-

thermore, the ability of hydrogen atoms to diffuse on the surface leads to abnormal kinetic

features. The diffusion of hydrogen atoms into the interior of various metal such as palla-

dium or titanium is another unique property of this element. The investigation of this pro-

cess, absorption, is beyond the scope of this work.

T B L E A. 1. Properties of hydrogen molecules in a gas phase

Property Value Ref. 1 1

Molecular mass, m, ,

( l ~ - ~ ~ k ~ )

Dissociation energy ,Do,

(eV) G Fundamental frequency, o ,

(loL3 rad/s)

Rotational frequency, or, ( 1014 rad/s)

Appendix B: Numerical Procedure

B.1 CALCULATING THE EQUILIBRIUM EXCHANGE RATE In Chapter 3, the expression for the equilibrium exchange rate was presented as:

All parameters in Eq. (B-1) are the material properties of the gas-solid system, except P, A and N, which correspond to the equilibrium pressure and surface coverage in the isolated

system at each instant of time. To obtain these values we use the condition under which the

total number of particles in the TPD system at each instant of time is equal to the total

number of particles in the isolated system that can evolve to an equilibrium state. The total

number of particles in the TPD system at the beginning of the process is:

By assuming that there are only hydrogen atoms adsorbed on the surface and accounting

for the number of hydrogen molecules that are pumped out from the TPD system, Eq. (B-

2) can be rewritten for the time t + At as:

~ p ' ( t ) d t N ( t + At) = LfQl+ iVG(t) -

2 v

Note that at time t = 0, Eq. (B-3) is the same as Eq. (B-2) and the values of N ( 0 ) ,

N*(o) and ~'(0) are ideally given in the experimental conditions. Therefore, the value

of N( t + At ) can be calculated at each time step equal to At using the values of N~ and

N~ at the previous time t. The time step is chosen to be small enough such that the value

of N at each time step increases by only a small amount equal to A E ~

Meantime under equilibrium conditions in the isolated system, the following rela-

tion exists between the total number of particles in the system at time t + At and the num-

ber of adsorbed particles on the surface and in the gas phase:

~ : ( t + At) N( t + At) =

2 + N , C ( ~ + At)

where subscript e denotes equilibrium values. Equation (B-4) can be further expanded in

terms of the equilibrium pressure in the isolated system, i.e.,

where V is the volume of the isolated system and is equal to the effective volume of the

TPD system. The value of N( t + At) can be calculated from Eq. (B-3) and then be substi-

tuted into Eq. (B-5) along with the expression for the equilibrium pressure,

to give,

Given the values of N(t + At) , volume, and temperature, which are all the same as in the

TPD system, Eq. (B-7) can be solved numerically for the number of adsorbed atoms under

equilibrium conditions. This number(s) can then be used i n Eq. (B-6) to obtain the equilib-

rium pressure. By knowing these two values, Eq. (B-1) can now be evaluated.

Note that for some surfaces, like H2-Ni(l 10) and H2-W(100) there are certain tem-

peratures and coverages at which the homogeneous surface becomes unstable, see Fig. 3.9

and Fig. 4.12. In the numerical calculations, therefore, one must apply certain conditions

under which the calculated coverages and pressures correspond to a stable equilibrium

state. In order to imply such conditions, the range of the unstable equilibrium states which

are shown in the shape of islands in Fig. 3.9 and Fig. 4.12, can be defined by two sets of

polynomials. One function defines the lower limit of the coverage and the other one the

upper Iimit of the coverage at which the system becomes unstable. By defining this range,

at any specific temperature, the numerical algorithm which is used for finding the solution

to Eq. (B-7), will not seek for any roots in this range.

B.l .l. Limits for Hz-Ni(1 10) system

From the performed calculations that were resulted in finding the unstable islands

in Chapter 2, the lower and the upper limits of the unstable region boundaries have been

plotted in terms of temperature, see Fig. B.1. As was shown in Chapter 3, Fig. 3.9, El2-

Ni(ll0) system does not experience any unstable conditions during the TPD measure-

ments at higher temperatures and coverages. Therefore only the island corresponding to

the lower temperatures and coverage is considered. By applying a least squared method, a

polynomial of degree four of temperature was obtained for each set of data points. These

polynomials that describe the boundaries are:

For the lower boundary:

For the upper boundary:

B.1.2. Limits for Hz-W(100) system

For H2-W(100) system, the Iower and the upper limits of coverages defining the

unstable region have been plotted against temperature in Fig. B.2. The lower and the upper

limit boundaries are obtained, respectively, as:

-5 2 0, = 1 . 8 9 3 3 ~ 1 0 - ~ ~ ~ - 1.9948~10 T + 7 . 4 9 7 5 ~ 1 0 - ~ ~ + 0.167 (B- 10)

-5 2 0, = ( - 1 . 9 ~ 1 0 - ~ ) ~ ~ + 2.0048~10 T - 7 . 5 2 0 3 5 ~ 1 0 - ~ ~ + 2.465 (B- 1 1)

B.2 NUMERICAL SOLUTION OF TWE PDE-s

As was shown in Chaptefs 3 and 4, the equations describing the heating rate of the

substrate, the number of hydrogen molecules in the gas phase, and the number of adsorbed

atoms on the surface at each instant of time constitute a closed system of equations that

can be solved simultaneously to predict NG as a function of time in the TPD system. If the

value of NG were known, it could then be used with the gas phase volume, the instanta-

neous temperature, and the ideal gas equation of state to predict the pressure as a function

of time in the gas phase (i.e., the TPD spectrum).

The heating rate of the substrate could be constant or be changes with temperature

itself. In general, the dependence of temperature to time can be shown as:

T = To + R,(T)t (B- 12)

The equation describing the net rate of adsorption was obtained from the statistical

rate theory approach as:

By using the perfect gas law, the equation describing the change in the number of the gas

molecules in the TPD system can be shown by:

(B- 14)

Equations B- 12, B- 13 and B- 14 are the initial value ordinary differential equations

that can be solved numerically. A Fortran program was developed to define the initial val-

ues, the conditions in the TPD, and in the isolated systems, as well as to calculate the equi-

librium exchange rate at any instant of time. Then a solver is called each time the

temperature is changed to solve the ODEs. A sample of such a program written for Hz-

W(100) is shown in Appendix C. The numerical solver which was used in this work was

written by Hindmarsh (LSODE) and was taken from CHEMKIN ~ i b r a r ~ ~ ~ . This code

solves the initial value, stiff or nonstiff systems of first order ODEs in the form of:

where NEQ is the number of ODEs to be solved, This solver is design based on the Gear

method which is a multi-value approach for solving the initial value 0 ~ ~ s ~ ~ . The specifi-

cations of this solver is shown in Appendix D.

The time steps at which the output has been asked from the solver is lom3 second

for all calculations. It was found that if this time step is decreased to second, the

improvement of the results would be as follows: for Hz-Ni(IO0) it was found that at initial

coverage of 0.45, the maximum improvement in the pressure calculations would be 1.2 %

at a temperature of 320 K; for H2-Ni(1 10) system at initial coverage of 1.5 this maximum

improvement was 0.55 % at a temperature of 406 K.

320 340 360 380 400 Temperature, K

FIGURE B.1. The upper and lower limits of the unstable region for H2-Ni(1 10)

system. In this figure only the unstable island at low temperature and coverage i s shown.

350 400 450 500 Temperature, K

FIGURE B.2. The upper and Iower limits for the unstable region for H2-W(100) system.

Appendix C: Fortran Program for Calculating the TPD Spectra of H2- W(1OO) System

TDS PROGRAM FOR H/W(lOO) SYSTEM AUGUST, 6-1998* With variable frequencies, variable heating rate IMPLICIT DOUBLE PRECISION ( A-HI 0 - Z ) EXTERNAL FEX DIMENSION Y(2), ATOL(2), RWORK(44), IWORK(22) REAL*8 MH, MO , MT, KRL COMMON /DAT/ RK,H,D0,MH,M0,W1,W2,W3,WG,WRIMT,AIPI COMMON /CAL/ T,V,KRL, PB,Cl,C2 NEQ = 2 PROPERTIES OF THE HYDROGEN H=6.62517E-34 RK=l.38044E-23 D0=4.454*1.60206E-19 MH=3.3204E-27 MO=1.00E+15 WG=l.293Ei-l4 WR=1.779E+12 MT=2.0000 SAMPLE SURFACE AREA King & thomas A-0.25 Yates et al: A=O.2827 Tamm et al: A=0.3166 TO=3OO. PI=ACOS(-1 . ) t=tO PMAX=l.E-15 EFFECTIVE VOLUME V=0.22*1 .E-3 Cl=O.S*A*MO*RK/V C2 =PUMPING SPEED/VOLUME C2= 1.5 ADSORPTION AREA AS= 1.E-4/(2.*MO) THE INITIAL COVERAGE, Y ( 1 ) Y ( 1 ) = 2.*9.80D-1 THE INITIAL PRESSURE,King et allP PB=3.OE-11*101350. /76O. THE INITIAL PRESSURE, Yates et a1 , P PB=2.OE-lO"101350. /76O. THE INITIAL PRESSURE, Tamm et al, P PB=3. OE-1OX101350. /76O. Y (2) = PB*V/ (RK*T) WRITE(*, *)Y (1) ,V,C2 DEFINING THE PARAMETERS FOR NUMERICAL SOLUTION TO PDE'S INDEPENDENT VARIABLE, TIME = TT TTl=O.

TT2=0. ITOL = 2 RTOL = 1.D-4 ATOL(1) = l.D-10 ATOL(2) = 1.D-10 ITASK = 1 ISTATE = 1 IOPT = 0 LRW = 44 LIW = 22 MF = 22 DTT=1. E-2

C*** STARTING TEMPERATURE CHANGE DO 40 IOUT = 1,4000000 TT2 =TT2+DTT

C*** Total Number of hydrogen atoms; TN=A*MO*Y(1)/2.+Y(2) -C2*Y(2)*DTT* (1. -(PB/ (Y (2) *RK*T/V)) * *2 )

C*** Heating Rate, King et al: C T=lO. *TT2+TO C*** Heating Rate, Tamrn et al: C T=25. *TT2+TO C*** Heating Rate, yates et al:

T=-8.003432E-2*TT2**2+13.9788*TT2+298.8668 C** CALCULATING EQUILIBRIUM COVERAGE ,QE

CALL PNUC (T , QRN ) PHI=(H**2/(2.*PI*MH*RK*T))**(l-5)*1./(RK*T)*l./(QRN)*(l.

1 -EXP (-H*WG/ (RK*T) ) ) *EXP (-DO/ (RK*T) )

C*** BISECTION METHOD FOR ROOT SEARCH Defining the upper and lower limit for unstable ranges: Lower Limit QL1=1.893333E-8*T**3-1.994857E-5*T**2

1 +7.497524E-3*T+1.670114E-I Upper Limit QL2=-1.9000E-8*T**32.004857E-5*T**2

1 -7.520357E-3*T+2.465229E+O

AE=O. 0001 BR=2 -0 c=o. 001 E=l . e-6 DO 30 J=1,100 D-C THETAE=AE CALL FUNC (THETAE, TN, 3, PHI , BETAE, FXl G=FX1 THETAE=THETAE+D IF(THETAE.GT.QLl.AND.THETAE.LT.QL2) THETAE=QL2 IF (THETAE .GT .BR)GO TO 250 CALL FUNC (THETAE, TN, 3 , PHI, BETAE,FXI) IF((G*FXl) .LE.O.O) THEN IF(D.LT.E) THEN QEE=THETAE GO TO 241

ELSE THETAE=THETAE-D IF(THETAE.GT.QLl.AND.THETAE.LT.QL2) THETAE=QL2 D=D/2.

ENDIF ENDIF GO TO 330

241 AE=THETAE+C IF(THETAE.GT.QLl.AND.THETAE.LT.QL2) THETAE=QL2

30 CONTINUE 250 CONTINUE C*** CALCULATING EQUILIBRIUM PRESSURE, PE

CALL FUNC(QEE,TN,B, PHI,BETAE,FX) PE=(QEE**2/ (MT-QEE) **2) / (EXP(2. * (B-BETAE) / (RK*T) ) *PHI)

C*** CALCULATING EQUILIBRIUM EXCHANGE RATE, KRL KRL=PE/SQRT(Z.*PI*MH*RK*T)*(MT-QEE)*AS

C*** TIME STEP DEFINITION IF(T.LT.400.) THEN DTT=l .E-3

ENDIF IF(T.GE.400.AND.T.LE.500.) THEN DTT=l. E-3

ENDIF IF(T.GE.5OO.AND.T.LE.525.) THEN DTT=l . E - 3

ENDIF IF(T.GE.525.AND.T.LE.550.) THEN DTT=l. E-3

ENDIF IF(T.GE.550.) THEN DTT=l .E-3

ENDIF C*** CALL FOR SOLVER

CALL LSODE (FEX, NEQ, Y, TT1, TT2, ITOL, RTOL, ATOL, ITASK, ISTATE, 1 IOPT, RWORK, LRW, IWORK, LIW, JEX, MF)

C*** FINDING THE MAXIMUM PRESSURE AND CORRESPONDING C** TEMPERAORE AND COVERAGE

IF((Y(2)*RK*T/V) .GE.PMAX) THEN PMAX=Y (2) *RK*T/V TMAX=T QMAX=Y (1) ENDIF IF(T.GT.200.) THEN IF(mod(IOUT,lO) .EQ.O) THEN WRITE(* , 20)T1Y(2) *RKRT/V*760 ./lOl350. ,Y(l) ,PE*76O./lOl350.,

1 QEE, KRL ENDIF ENDIF

20 FORMAT(F6.2,2x,S(E10.4,2X)) IF (ISTATE .LT. 0) GO TO 80

40 IF(T.GT.650 .)GO TO 850

STOP WRITE ( * ,9 0 ) ISTATE FORMAT(///22H ERROR HALT.. ISTATE =,I3) CONTINUE Write(*, * )

WRITE(*,20)TMAX,PMAX*76O./I01350. WRITE(*, *) END

SUBROUTINES

SUBROUTINE FEX (NEQ, TT, Y, YDOT) THIS SUBROUTINES DEFINES THE TWO DIFFERENTIAL EQUATIONS DIMENSION Y(2), YDOT(2) IMPLICIT DOUBLE PRECISION (A-H, 0 - 2 ) REAL*8 MT,MO,KRL,MH COMMON /DAT/ RK,H, D0,MH,M0,W1,W2t W3,WGIWRIMT,A,PI COMMON /CAL/ T,V,KRL,PB,Cl,C2

CALCULATING B FUNCTION VARIABLE FREQUENCIES W1=(4.2262*Y(l)**2-22.8793*Y(1)+158.41)*1.60206E+12/6.62717 W2=(-1.74509*Y(1)**2+24.549*Y(1)+120.4)*1.60206E+12/6.62717 W3=(-4.40fY (1) **2+21.5212*Y (l)+54.6561) *I .6O2O6E+l2/6 -62517 DW1=(2. *4.2262*Y(l) -22.8793) *1.60206E+12/6.62717 DW2= (-2. *1.74509*Y (1) +24.549) *I. 60206E+l2/6.62717 DW3=(-2 .*4.4O*Y (1)+21.5212) *1.60206E+12/6.62517

DELTA~=EXP(-H*W~/(~.*RK*T))/(~.-EXP(-H*W~/(RK*T))) DELTA~=EXP(-14*~2/ (2. *RK*T) ) / (1. -EXP(-H*W2/ (RK*T) ) )

DELTA~=EXP(-H*W~/(RK*~.*T))/(~.-EXP(-H*W~/(RK*T))) DELTA = DELTAl*DELTA2*DELTA3

CALCULATION BETA

CALL PNUC ( T , QRN ) PHI= (~**2/ (2. *PI*MH*RK*T) ) ** (1.5) *I. / (RK*T) *I. / (QRN) * (I.. 1 -EXP(-H*WG/(RK*T)))*EXP(-DO/(RK*T))

YDOT(~)=~.*KRL*( (MT-Y (1))**2/Y (1) **2*ExP(2.*(~-BETA) / (RK*T) ) 1 *PHI* (Y (2) *RK*T/V) -1. / (EXP (2. * (B-BETA) / (RK*T) ) 2 *PHI* (Y (2) *RK*T/V) * (Y (1) **2/ (MT-Y (1) ) **2) )

RETURN END

SUBROUTINE PNUC (TI QRP) THIS SUBROUTINE CALCULATES THE NUCLEAR & ROTATION PARTITION FUNCTIONS IMPLICIT DOUBLE PRECISION (A-H, 0-Z) REAL*8 MH,MO,MT COMMON /DAT/ RK,H,D0,MH,M0,W1,W2,W3,WG~WRIMT,A,PI J=1 QR=O . QR=(QR+(2*J+l)*EXP(-J*(J+l)*H*WR/(RK*T))I J=J+2 IF (J.GE.50) GO TO 30 GO TO 20 CONTINUE QR1=3. *QR L=O QP=O . QP=QP+ (2*L+l) *EXP(-L* (L+1) *H*WR/ (RK*T) ) L=L+2 IF (L.GE.3) GO TO 50 GO TO 40 CONTINUE QRP = QRl** (0.75) *QP** (0 - 2 5 ) RETURN END

c*************************************************** SUBROUTINE FUNC (THETA, TN, B, PHI, BETA,FX 1 IMPLICIT DOUBLE PRECISION (A-H,O-Z) REAL*8 KRL,K,MT

COMMON /CAL/ T,VIKRL,PB,C1,C2 K= 1.38044D-23 MT=2.0 H=6.62517E-34 W1=(4.2262*THETA**2-22.8793*THETA+158.41)*1.60206E+12/6.62717 W2=(-1.74509*THETA**2+24.549*THETA+120.4)*1.60206E+12/6.62717 W3=(-4.40*THETA**2+21.5212"THETA+54.6561)*1.60206E+12/6.62517 DWl=(2.*4.2262*THETA-22.8793)*1.60206E+1.2/6.62717 DW2=(-2.*1.74509*THETA+24.549)*1.60206E+12/6~62717 DW3=(-2.*4.4O*THETA+21.5212)*1.60206E+l2/6.62517

DELTAl=EXP(-H*W1/(2.*K*T))/(I.-EXP(-H*Wl/(K*T))) DELTA2=EXP(-H*W2/(2.*K*T)}/(l.-EXP(-H*W2/(K*T))) DELTA3=EXP(-H*W3/(K*2.*T))/(I.-EXP(-H*W3/(K*T))) DELTA = DELTAl*DELTA2*DELTA3

PSI~=DW~*(~.+~.*EXP(-H*W~/(~.*K*T)))/(~.-EXP(-H*W~/(K*T))) PSI2=DW2*(1.+2.*EXP(-H*W2/(2.*K*T)))/(l.-EXP(-H*W2/(K*T))) PSI3=DW3*(1.+2.*EXP(-H*W3/(K*2.*T)))/(l.-EXP(-H*W3/(K*T))) PSI=(PSI1+PSI2+PSI3) B=K*T*LOG(DELTA)-THETA*H/2.*PSI

C*** THESE FUNCTIONS ARE THE ONES CAN BE SEEN IN FIGURE (4.2)

C*** (1)n C BETA=l.E-19*(-2.041588E-l*THETA**5+1.243115E+O*THETA**4 C 1 -2.684893E+O*THETA**3+2.398379E+O*THETA**2 C 2 -6.564357E-l*THETA-4.94658E+O) C*** (2)n C BETA=1.E-19*(-1.633231E-l*THETA**5+1.081949E-O*THETA**4 c 1 -2.476867E+O*THETA**3+ 2.291358E+O*THETA**2 C 2 -6.362524E-l*THETA-4.947456E+O) c*** ( 3 ) n C BETA=1.E-19*(-1.224875E-1*11HETA**5+9.207825E-l*~'HETA**4 C 1 -2.268867E+O*THETA**3+2.184338E+O*THETA**2 C 2 -6.160692E-1"THETA-4.948333E-1-0) c*** (4)y C BETA=l.E-19*(-8.165179E-2*THETA**5+7.596162E-l*THETA**4 C 1 -2.060814E+O*THETA**3+2.077317E+O*THETA**2 C 2 -5.958859E-l*THETA-4.949209E+O) C*** (5)y C BETA=l.E-19*(-4.081614E-2*THETA**5+5.984499E-l*THETA**4 C 1 -1.852788E+O*THETA**3+l1970297E+O*THETA**2 C 2 -5.757026E-l*THETA-4.940085E+O) C***(6) C**This curve was used in the final calculations, 29-09-98 c**

BETA=l.E-19*(-2.039831E-2*TI-1ETA**5+5.178668E-l*THETA**4 1 -1.748775E+O*THETA**3+l19l6787E+O*THETA**2 2 -5.65611E-l*THETA-4.950523E+O)

C***Fifth order C BETA=l.E-19*(1.974502E-1*THETA**5-3.419169E-l*THETA**4 C 1 -6.39004E-IE-l*THETA**3+1.345858E+O*THETA**2 C 2 -4.579381E-1"THETA-4.955198E+O)

FX=(THETA**2/(MT-THETA)**2)/(EXP(2.*(B-BETA)/(K*T)) 1 *PHI)/T+THETA*Cl-(K/V)*TN

RETURN END

Appendix D: Numerical Solver for Solving the PDE-s

SUBROUTINE LSODE ( F , NEQ, Y , T , TOUT, I T O L , RTOL, ATOL, I T A S K , 1 ISTATE, I O P T , RWORK, LRW, IWORK, LIW, J A C , MF)

EXTERNAL FEX, J A C INTEGER NEQ, ITOL, ITASK, I S T A T E , I O P T , LRW, IWORK, LIW, MF DOUBLE PRECISION Y, T , TOUT, RTOL, ATOL, RWORK DIMENSION NEQ ( 1 ) , Y ( 1 ) , RTOL ( 1 ) , ATOL ( 1 1 , RWORK ( LRW) , IWORK ( L I W )

C T H I S IS THE JUNE 17 , 1 9 8 0 VERSION OF C L S O D E . . LIVERMORE SOLVER FOR ORDINARY DIFFERENTIAL EQUATIONS. C T H I S VERSION IS I N DOUBLE P R E C I S I O N . C C LSODE SOLVES THE I N I T I A L VALUE PROBLEM FOR S T I F F OR N O N S T I F F C SYSTEMS O F F I R S T ORDER ODE-S, C DY/DT = F ( T , Y ) , OR, I N COMPONENT FORM, C DY ( I ) / D T = F ( 1 ) = F ( I , T , Y (1) ,Y (21,. . . ,Y ( N E Q ) ) ( I = 1, . . . ,NEQ) . C LSODE IS A PACKAGE BASED ON THE GEAR AND GEARB PACKAGES, AND ON T H E C OCTOBER 2 3 , 1 9 7 8 VERSION O F THE TENTATIVE ODEPACK USER INTERFACE C STANDARD, WITH MINOR MODIFICATIONS. C AUTHOR AND CONTACT.. ALAN C . HINDMARSH, C MATHEMATICS AND STATISTICS SECTION, L - 3 0 0 C LAWRENCE LIVERMORE LABORATORY C LIVERMORE, CA 94550 .

C SUMMARY OF USAGE. C C COMMUNICATION BETWEEN THE USER AND THE LSODE PACKAGE, FOR NORMAL C S I T U A T I O N S , IS SUMMARIZED BELOW. S E E THE FULL DESCRIPTION FOR C D E T A I L S , INCLUDING OPTIONAL COMMUNICATION, NONSTANDARD O P T I O N S , C AND INSTRUCTIONS FOR S P E C I A L S I T U A T I O N S . S E E ALSO THE EXAMPLE C PROBLEM (WITH PROGRAM AND OUTPUT) FOLLOW1 NG T H I S SUMMARY. C C A . F I R S T PROVIDE A SUBROUTINE O F THE FORM.. C SUBROUTINE F (NEQ, T I Y , YDOT) C DIMENSION Y ( N E Q ) , YDOT(NEQ) C WHICH S U P P L I E S THE VECTOR FUNCTION F BY LOADING YDOT ( I ) WITH F ( I ) . C C B. NEXT DETERMINE (OR GUESS) WHETHER OR NOT THE PROBLEM IS S T I F F . C S T I F F N E S S OCCURS WHEN THE JACOBIAN MATRIX DF/DY HAS AN EIGENVALUE C WHOSE REAL PART IS NEGATIVE AND LARGE I N MAGNITUDE, COMPARED TO T H E C RECIPROCAL OF THE T SPAN OF INTEREST. IF THE PROBLEM IS N O N S T I F F , C USE A METHOD FLAG M F = 1 0 . I F I T IS S T I F F , THERE ARE FOUR STANDARD C C H O I C E S FOR M F , AND LSODE REQUIRES THE JACOBIAN MATRIX I N SOME FORM. C T H I S MATRIX IS REGARDED EITHER AS FULL (MF = 2 1 OR 2 2 ) , C OR BANDED (MF = 2 4 OR 25 ) . I N THE BANDED CASE, LSODE REQUIRES TWO C HALF-BANDWIDTH PARAMETERS ML AND MU. THESE ARE, RESPECTIVELY, THE C WIDTHS OF THE LOWER AND UPPER PARTS O F THE BAND, EXCLUDING THE MAIN C DIAGONAL. THUS THE BAND CONSISTS OF THE LOCATIONS ( 1 , J ) WITH C I - M L . L E . J . LE. I+MU, AND THE FULL BANDWIDTH IS ML-t-MU+1. C C C . I F THE PROBLEM IS S T I F F , YOU ARE ENCOURAGED TO SUPPLY THE JACOBIAN C DIRECTLY (MF = 2 1 OR 24), BUT I F T H I S I S NOT F E A S I B L E , LSODE W I L L

COMPUTE I T INTERNALLY BY DIFFERENCE QUOTIENTS (MF = 2 2 OR 2 5 ) . I F YOU ARE SUPPLYING THE JACOBIAN, PROVIDE A SUBROUTINE O F T H E FOFU'4..

SUBROUTINE J A C (NEQ, T , Y, ML, MU, PD, NROWPD) DIMENSION Y (NEQ) , PD (NROWPD, NEQ)

WHICH S U P P L I E S DF/DY BY LOADING PD AS FOLLOWS.. FOR A FULL JACOBIAN ( M F = 2 1 ) , LOAD P D ( 1 , J ) WITH D F ( I ) / D Y ( J ) ,

THE PARTIAL DERIVATIVE OF F ( 1 ) WITH RESPECT TO Y ( J ) . ( IGNORE THE ML AND MU ARGUMENTS I N T H I S C A S E . )

FOR A BANDED JACOBIAN (MF = 2 4 ) , LOAD P D ( 1 - J + M U + l , J ) WITH D F ( I ) / D Y ( J ) , I . E . LOAD THE DIAGONAL L I N E S O F DF/DY INTO THE ROWS O F PD FROM THE TOP DOWN.

I N EITHER CASE, ONLY NONZERO ELEMENTS NEED BE LOADED.

D . WRITE A MAIN PROGRAM WHICH CALLS SUBROUTINE LSODE ONCE FOR EACH P O I N T AT WHICH ANSWERS ARE DESIRED. T H I S SHOULD ALSO PROVIDE FOR P O S S I B L E USE O F LOGICAL U N I T 3 FOR OUTPUT O F ERROR MESSAGES BY LSODE. ON THE F I R S T CALL TO LSODE, SUPPLY ARGUMENTS A S FOLLOWS.. F = NAME O F SUBROUTINE FOR RIGHT-HAND S I D E VECTOR F.

T H I S NAME MUST BE DECLARED EXTERNAL I N CALLING PROGRAM. NEQ = NUMBER O F F I R S T ORDER ODE-S. Y = ARRAY O F I N I T I A L VALUES, O F LENGTH NEQ. T = THE I N I T I A L VALUE O F THE INDEPENDENT VARIABLE. TOUT = F I R S T POINT WHERE OUTPUT IS DESIRED { .NE. T ) . I T O L = 1 OR 2 ACCORDING AS ATOL (BELOW) IS A SCALAR OR ARRAY. RTOL = RELATIVE TOLERANCE PARAMETER ( S C A L A R ) . ATOL = ABSOLUTE TOLERANCE PARAMETER (SCALAR OR ARRAY) .

THE LOCAL ERROR I N Y ( I ) WILL BE CONTROLLED S O A S TO BE ROUGHLY L E S S THAN R T O L * A B S ( Y ( I ) ) + ATOL I F I T O L = 1, OR R T O L * A B S ( Y ( I ) ) + A T O L ( 1 ) I F I T O L = 2 . USE RTOL = 0 . 0 FOR PURE ABSOLUTE ERROR CONTROL, ATOL (OR A T O L ( 1 ) ) = 0 . 0 FOR PURE RELATIVE. CAUTION.. ACTUAL (GLOBAL) ERRORS MAY EXCEED THESE TOLERANCES, S O CHOOSE THEM CONSERVATIVELY.

I T A S K = 1 FOR NORMAL COMPUTATION O F OUTPUT VALUES O F Y AT T = TOUT. I S T A T E = INTEGER FLAG ( I N P U T AND O U T P U T ) . S E T I S T A T E = 1. I O P T = 0 TO INDICATE NO OPTIONAL I N P U T S USED. RWORK = REAL WORK ARRAY O F LENGTH AT L E A S T . .

20 + 16*NEQ FOR MF = 1 0 , 2 2 + 9*NEQ + NEQ**2 FOR MF = 2 1 OR 2 2 , 2 2 + 10*NEQ + (2*ML + MU)*NEQ FOR MF = 2 4 OR 2 5 .

LRW = DECLARED LENGTH OF RWORK ( I N USER-S D I M E N S I O N ) . IWORK = INTEGER WORK ARRAY O F LENGTH AT L E A S T . .

2 0 FOR MF = 1 0 , 2 0 + NEQ FOR MF = 2 1 , 2 2 , 2 4 , OR 2 5 .

IF MF = 2 4 OR 25, INPUT I N IWORK ( 1) , IWORK ( 2 ) THE LOWER AND UPPER HALF-BANDWIDTHS ML, MU.

LIW = DECLARED LENGTH O F IWORK ( I N USER-S D I M E N S I O N ) . J A C = NAME O F SUBROUTINE FOR JACOBIAN MATRIX (MF = 2 1 OR 2 4 ) .

IF USED, T H I S NAME MUST BE DECLARED EXTERNAL I N CALLING PROGRAM. I F NOT USED, PASS A DUMMY NAME.

M F = METHOD FLAG. STANDARD VALUES A R E . . 1 0 FOR NONSTIFF (ADAMS) METHOD, NO JACOBIAN USED. 21 FOR S T I F F (BDF) METHOD, USER-SUPPLIED FULL JACOBIAN. 2 2 FOR S T I F F METHOD, INTERNALLY GENERATED FULL JACOBIAN. 2 4 FOR S T I F F METHOD, USER-SUPPLIED BANDED JACOBIAN.

C 2 5 FOR S T I F F METHOD, INTERNALLY GENERATED BANDED JACOBIAN. C C E . THE OUTPUT FROM THE F I R S T CALL (OR ANY CALL) I S . . C Y = ARRAY OF COMPUTED VALUES O F Y(T) VECTOR. C T = CORRESPONDING VALUE OF INDEPENDENT VARIABLE (NORMALLY T O U T ) . C I S T A T E = 2 I F LSODE WAS SUCCESSFUL, NEGATIVE OTHERWISE. C -1 MEANS EXCESS WORK DONE ON THIS CALL (PERHAPS WRONG M F ) . C -2 MEANS EXCESS ACCURACY REQUESTED (TOLERANCES TOO SMALL) . C -3 MEANS ILLEGAL INPUT DETECTED ( S E E PRINTED MESSAGE). C -4 MEANS REPEATED ERROR TEST FAILURES (CHECK ALL I N P U T S ) . C -5 MEANS REPEATED CONVERGENCE FAILURES (PERHAPS BAD JACOBIAN C SUPPLIED OR WRONG CHOICE O F MF OR TOLERANCES) . C -6 MEANS ERROR WEIGHT BECAME ZERO DURING PROBLEM. (SOLUTION C COMPONENT I VANISHED, AND ATOL OR ATOL ( I ) = 0 . ) C C F. TO CONTINUE THE INTEGRATION AFTER A SUCCESSFUL RETURN, SIMPLY C RESET TOUT AND CALL LSODE AGAIN. NO OTHER PARAMETERS NEED BE RESET. C c-----------------------------------------------------------------------

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