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1977

Dielectric Relaxation in the UHF and RF Regions Dielectric Relaxation in the UHF and RF Regions

Theodore Bryant Kingsbury College of William & Mary - Arts & Sciences

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DIELECTRIC RELAXATION IN THE UHF AND RE REGIONS

A Thesis Presented to

The Faculty of the Department of Chemistry The College of William and Mary in Virginia

In Partial Fulfillment Of the Requirements for the Degree of

Master of Arts

byTheodore Bryant Kingsbury IV

1977

APPROVAL SHEET

This thesis is submitted in partial fulfillment the requirements for the degree of

Master of Arts

Author "

Approved June 1977

HiK y l ^ E . Do 1 bow

DEDICATION

To my family and friends, whose unceasing encourage ment, support, assistance, and understanding made this possible.

TABLE OF CONTENTSPage

ACKNOWLEDGMENTS............................ viLIST OF FIGURES......................................... viiLIST OF P LATES.........................................viiiABSTRACT............................................. ixCHAPTER I. STATIC PERMITTIVITY....................... 2CHAPTER II. DIELECTRIC RELAXATION ..................17CHAPTER III. THEORY OF UHF MEASUREMENTS..............31CHAPTER IV. UHF EXPERIMENTAL EQUIPMENT AND PROCEDURE. . 43CHAPTER V. UHF EXPERIMENTAL RESULTS.................. 55CHAPTER VI. SUPERCOOLED LIQUIDS AND THE GLASSY STATE. . 61CHAPTER VII. RF AND Q-METER THEORY................ . 66CHAPTER VIII. RF EXPERIMENTAL EQUIPMENT AND PROCEDURE . 78CHAPTER IX. RF Q-METER EXPERIMENTAL RESULTS .......... 87APPENDIX I. COMPUTER PROGRAM TO CALCULATE Cn , b2-a2,

2ab, AND TAN .............................. 93APPENDIX II. TABLE OF Cn , b2-a2 , 2ab, AND TAN 8 . . . . 97APPENDIX III. COMPONENTS OF UHF MEASUREMENT SYSTEM. . . 109APPENDIX IV. COMPUTER PROGRAM TO FIT UHF D A T A ...... IllAPPENDIX V. PLOTS OF e' AND e" VERSUS CONCENTRATION . . 114APPENDIX VI. RELAXATION TIMES Aim SMYTH PARAMETERS

FOR UHF D A T A ............................ 148

iv

PageAPPENDIX VII. UNCERTAINTY IN e" FOR Q FROM 130 TO 1000. 158 APPENDIX VIII. COMPUTER PROGRAMS FOR RF MEASUREMENTS. . 162 APPENDIX IX. TABLES AND GRAPHS OF RF Q-METER RESULTS. . 167 BIBLIOGRAPHY ............................................. 181

V

ACKNOWLEDGMENTS

The writer wishes to express his appreciation for the invaluable assistance and continual encouragement given him by the faculty members of both the chemistry and physics departments; especially Professor David E. Kranbuehl, under whose guidance this research was con­ducted, and Professors Robert A. Orwoll and Kyle Dolbow for their careful reading and criticism of this thesis. Not least of all, Mr. Stan Hummel of the physics shop deserves special thanks for his help in designing and building much of the experimental apparatus.

vi

LIST OF FIGURES

Figure Page1-1 Parallel plate capacitor in vacuo .............. 3l-2a Parallel plate capacitor with isotropic,

non-polar dielectric .............................. 4l-2b Charge distribution with and without an

applied f i e l d * . . . 41-3 Diagram for the calculation of local field

contributions .................................... 92-1 Polarization versus t/T for a suddenly

applied field .............................. 182-2 Polarization versus t/T for a suddenly

removed field ............................ 192-3 Debye plot of €' and e " versus log( C l) T) . . . .202-4 Complex plane representation of €' and £". . . .222-5a Cole-Cole plot of €' and €" versus log ( OJ T)l . .222-5b Cole-Cole plot of €' versus £ " ................. 232-6 Cole-Davidson plot of 6 ’ versus € ” for

various values of a .............................. 234-1 Admittance meter and auxiliary equipment........ 444-2 UHF dielectric c e l l .............................. 484-3 Block diagram of UHF component interconnection . .497-la Q-meter circuit with empty cell attached........ 677-lb Q-meter circuit with filled cell attached . . . .688-1 RF measurement apparatus - interconnection . . . .798-2 Capacitance versus 1/t for 2-terminal cell . . . .808-3 RF 2-terminal dielectric c e l l ................... 82

vii

LIST OF PLATES

PlatePlatePlatePlatePlate

Plate

PlatePlate

Plate

Plate

. 1602-B Admittance Meter - Front

. Admittance Meter - Rear

. UHF Dielectric Cell - Assembled

. UHF Dielectric Cell - Disassembled

. Interconnection of UHF Components - Dielectric Cell and Admittance Meter

. UHF Components - Oscillators and IF Amplifier

. RF Measurement Equipment

. Detail of Q-Meter with Dielectric Cell Attached

. Two-terminal Dielectric Cell for RF Measurements - Assembled

10 Two-terminal Cell - Disassembled

ABSTRACT

The purpose of this investigation was to gain infor­mation on the relaxation processes of certain small polar molecules, both in the liquid state, and also in the super­cooled liquid and glassy states.

The first part of this study was carried out on dilute solutions of nitroaromatic compounds in benzene at 25 and 60°C, using UHF transmission line techniques to measure the complex dielectric constants.

Relaxation times were found for the various compounds studied. In conjunction with data previously obtained in the microwave frequency range, the data indicated a depen­dence of the relaxation time on the volume swept out by the molecule during relaxation, rather than on the molecular volume.

The second part of this investigation was the develop­ment of an experimental technique to measure the comp ex dielectric constants of supercooled dilute solutions of small polar molecules in the weakly polar solvent, ortho- terphenyl.

The presence of a low frequency, cooperative relaxation process in the supercooled state has been well established by previous experimental work. Johari and Goldstein, among others, have proposed the existence of a second relaxation process, due to hindered reorientation of solute molecules "encaged" by relatively immobile solvent molecules. This /3 -process should require a much lower activation energy than the d , or cooperative process, and thus occur at lower temperatures and/or higher frequencies.

As a result of Q-meter measurements in the RF region, this J3 -process appears to occur in the teraperature-frequency region below -70°C and above 5MHz for the compounds studied.As expected, its magnitude is much lower than that of the Ot peak, previous studies having indicated that the Ot -process accounts for the majority of the total loss.

ix

DIELECTRIC RELAXATION IN THE UHF AND RF REGIONS

CHAPTER ISTATIC PERMITTIVITY

All atoms can be thought of as having a particular spatial distribution of electric charge. That distribution is characteristic of the electronic configuration of the atom under consideration, and is also affected by neigh­boring charge distributions. This inherent charge distri­bution of the atom (or molecule) under study can be affected by an external electric field, and that effect can be measured. The study of such interaction between matter and electromagnetic fields constitutes the field of dielec­tric phenomena.

Since molecules are combinations of atoms, the charge distribution of a molecule depends on the types of atoms involved and their structural arrangement. In an electri­cally neutral molecule, although there are equal amounts of positive and negative charge, that charge may be distri­buted in such a way that the positive and negative centers of charge are physically separated, thus giving rise to an electric dipole. Another way to express this is to say that if a molecule belongs to the point group C^, then it does not have a permanent dipole moment. It may, however, possess quadrupole or higher moments which, in most cases, probably make negligible contributions to the electric

2 .

moment. In this paper we will be concerned with the permitti­

vity (dielectric constant) of certain materials which, in most cases, are polar; i.e., they have permanent dipole moments.

In regard to the derivation of the mathematical rela­tions necessary to analysis of dielectric properties, we will first consider a parallel plate condenser. The plate area is large compared to the separation between the plates and there is vacuum around the whole apparatus. A poten­tial, V, is applied across the plates and they acquire charges +Q and -Q per unit area. The capacitance is defined as

/ 1.1 / C0 = Q / V

+ -

+ -

+

+ Q + - ~Q+ -

+ -

+

V1 L

Figure 1-1

When an isotropic, non-polar dielectric is placed between the plates, (Figure l-2a)^ the electric field due to the charges on the plates will polarize the molecules of

the dielectric; i.e., it will cause a distortion of the

4.normally symmetrical charge distributions of the molecules. (Figure l-2b)^

+ (Q + P) +

+++

++

(Q + P)

No F i e l d<--------

F i e l d Appl i edV

(a) (b)Figure 1-2

The capacitor can now hold a charge (Q+P) at potential V, .and the capacitance is now

The permittivity is thus dependent on the polarizability of the material. For a non-polar molecule, polarizability arises from two sources, displacement of the electrons

relative to the nucleus (electronic polarization), and dis­placement of the atomic nuclei relative to one another (atomic polarization), with electronic polarization usually being predominant. If a polar molecule is used instead of a non-polar one, a third effect contributes to the total

/ 1.2 / C = ( Q + P ) / V.

The (static) relative permittivity is defined as^

/I.3 / = C/CQ = (Q + P)/Q

5.polarization of the molecule. The permanent dipole of a molecule will attempt to align itself parallel to the inci­dent electric field, in opposition to the randomizing effect of thermal agitation.^

Obviously, then, the total polarization of polar mole­cules will be temperature dependent; whereas the only effect of temperature on non-polar molecules is to change the density, giving a small effect on the polarization.The total polarizability of the molecule is designated Q , and is the sum of the electronic, atomic, and orientation polarizations mentioned above; i.e.,

/ 1.4 / a T = a + a + aT a e o

When a dielectric material is subjected to an electric field of intensity and direction E, displacement of the charge distribution occurs such that a bound surface charge P is induced. These two vector quantities are related by the equation,-*-

/ 1.5 / €0 = I + 4 7 T P /€ oE

Assuming that the individual dipole moments within such a volume are additive,-*-

/ 1 . 6 / p = N j < ^ >

where is the number of dipoles per unit volume. Also assuming < ^ > is proportional to the local field strength,F, (not the same as E), "*"

6.

/ 1,7 / < > = a F

where Q-y- is the polarizability, then,

/ 1.8 / p = N a FT I T *This can be divided into two parts:

/ 1.8a / P = N-,F(a + a ) , andu ± a e

/ 1.8b / pQ = N1F a Q , with PT = PD + PQ .

The problem of determining the static permittivity of a dielectric is best considered as a two part problem.First, a 0 r the contribution of the permanent dipoles of the molecules to the polarizability, must be calculated; and second, the local field, F, i.e., the actual effective field at the molecule, must be calculated in terms of the applied field, E.

The first theory proposed, and the first which we shall consider, is that of Debye. He assumed that the structure of the material exerted a negligible effect of the orientation of the permanent dipoles of the molecules. The distribution of permanent dipoles about an applied field is in accordance with Boltzmann's law; thus, a dipole of moment jju and with angle Q to the applied field has a potential energy,

/ 1.9 / U = -jJL Fcos 9 (F ^ E)

and the probability that the dipole axis will lie within

7.the solid angle dw, centered around Q is,

/ 1.10 / exp[(M Fcos0 ) / Kt]dw

/ e x p [ ( / x F COS 0)/kl]dw

If dw is the solid angle between Q and 8+ &8 , then dw = 27Tsin9 d9 , and the average value of the dipole com­ponent parallel to the field is

r n /J-FCOS0/KT. o o J f t cosae sinc/da/ l.ll //^<C0SC7> =

r ^ F“ !8/kTsi„ 8d9uQ

Substituting x = cos Q , and integrating the numerator by parts, we get

/ 1.12 / < c o s9> = c o th ( /x F /k T ) ” * (kT //xF )

< c o s 0 > = L ( /x F /k T ) .

A function of the general form L(y) = coth y - (1/y) is called Langevin1s function, and for field strengths that are normal in permittivity measurements, jJL'P/k T « 1, so that L(y) can be approximated satisfactorily by y/3. Thus the polarization due to the dipoles is

/ 1.13 / Po = N| ^t<cos0> = N ( / i 2 F /3 kT

where is the number of dipoles per unit volume. Thepolarization due to distortion has been found previously in equation 1.8, and on adding these two contributions, the total polarization is found to beJ

8.

/ 1.14 / PT = N |(ad+yu.2/3 K T )F ; ad= a g + a Q .

The first part of the problem has been solved, and it remains now to calculate the local field F in terms of the applied field E.

In order to determine the local field, certain assump­tions must be made. First, the following model is postu­lated: A sphere, large in relation to molecular dimensionsbut small on a macroscopic scale, is envisioned with the point at which the local field, F, is to be determined, at the center of the sphere. The interior of the sphere is considered to be composed of individual molecules and the surrounding dielectric is a continuum.-*-

The local field, F, can be thought of as composed of three components; F- which is the contribution from the free charges on the capacitor plates; F2, from the free ends of the dipoles on the surface of the imaginary cavity; and F r which constitutes the contribution from the other molecules which are, in reality, within the cavity and thus close enough to affect the field at the center.^ By definition:

/ 1.15 / F, = E = 47rQ/<r0 .

The contribution from the free ends of the dipoles lining the surface of the cavity can be calculated by referring to figure 1-3.^

9 .

E T

Figure 1-3

The charge density of the cavity walls is caused by bound charges, and is thus determined by the normal component of the polarization, P. Since the cavity was assumed to be spherically symmetric, the horizontal components of P cancel out and only the vertical components need be evalu­ated. These give a surface charge density of Pcos0 normal to the surface, and a component parallel to E of Pcos^0 .Using Coulomb's law, Pcos 2B for the parallel component,

2 •dividing the cavity walls into rings of area 27rr sin 0 o9 , and integrating over all Q , the field contribution of the cavity wall is~

Since F^ can only be determined if information on the geo­

metry and polarizability of the molecules within the sphere

10is available, and then only after extremely difficult mathematical treatment, the assumption is made that these contributions cancel each other so that

/ 1.17 / F3 = 0,

an assumption first made by Mossotti in 1850.^ Thus, we can now write:

/ l . 1 8 / F = F | + F 2 = E + 477-P/3€o ;

and from equation 1.5,

/ 1 .5 ' / 4 77-P = ( €0 - l ) €oE ,

which, upon combination with equation 1.14 give

/ 1.19 / P = N, (a + /u 2/3 k T ) ( 6 o + 2 ) e /3 ,

which can be rearranged to

/ 1.20 /12— ;---- — 47rN,(a + u /3kT) 3e<€„ + Z \ r-

We would like to remove the dependence of the polarization on the density of the substance; and as long as the mole­cules themselves are the dipoles, then this can be done by utilizing the relation,

/ 1.21 / Nq = = 6.023 x 1023 ,P

where NQ is Avogadro's number, M is the molecular weight in kilograms, and p is the density in kilograms per cubic

11meter.Substituting NQ into equation 1.20, we get the

Clausius-Mossotti relation,

/ 1.22 / £0 - I M _ 47rN0/ 2/€0 + 2 p ~ 3ec /^/3kT) ,

the left hand side being called the molar polarization.The analogous equation for energy in the optical region is the Lorentz-Lorenz equation

/ 1 ' 2 3 / 2 I M A Mn - I M 477-No i , 2/_.— 1--------- = — [ a 0 + f j . /6kT) •n + 2 p 3 e0

For the Clausius-Mossotti equation, is to be a realquantity. The equation can be generalized merely by sub­stituting for 1 It must be kept in mind that these equations are valid only so far as the assumption that

= 0 is a reasonable approximation of the real situation.Debye's two main assumptions were, (1) The molecules

will be distributed according to Langevin's law, and (2)The Lorentz field applies; i.e., F3 = 0. The first assump­tion is probably only reasonable for the case of liquids in which the dipoles experience no strong local interactions due to neighboring molecules. The assumption that F^ = 0 will not apply to crystalline solids and will probably only approximate the true situation for gases at low to moderate pressures, and for dilute solutions of polar molecules in non-polar solvents.^

From equation 1.23, it can be seen that, for a non-

12.polar material, the molar polarizability should be a con­

stant, and the permittivity should be directly proportional to density.^

For a polar dielectric, the molar polarizability should be linearly related to 1/T, and jjl can be found from the slope of a plot of molar polarization versus 1/T.Further analysis of this equation shows that there should be a temperature,

/ 1.24 / T 47T NoPAL1 9 M k€0 '

below which spontaneous polarization occurs and the mate­rial becomes ferroelectric.^ Although a few crystalline ferroelectric materials are known, this phenomenon is not common. As an example, water should, according to this equation, be ferroelectric below 1100°K thus making life on Earth impossible. This is an obvious instance of the non­validity of the assumption that F^ = 0. One attempt to

5improve this situation was made by Onsager.Onsager's calculation of the local field is based on

the following model: "The molecule is treated as apolarizable point dipole at the center of a spherical cavity of molecular dimensions in a continuous medium of

o 1(static) permittivity n ." The cavity size is defined by the equation:

/ 1.25 / 4 7tN|C13 / 3 = I ,

13.which merely says that the sum of the volumes of the spher­ical cavities equals the total volume of the material. As in Debye's theory, the assumptions of a spherical cavity and homogeneity outside the cavity, limit applicability to dielectrics with no strong local forces.^

The detailed derivation of Onsager's equation can be found in any book on the theory of permittivity. The result is:

/ 1.26 / ( e D - nZ)(2€o + nZ) 47t N ,^ .2

€o (n 2 + zf ~ 9 k T e0

9where an "internal refractive index", n , is defined as

/ 1 - 27 / (n2 — l ) / ( n 2 *+■ 2.) = a/ d3 47rN,a/3€0 5and where

/ 1.28 / 1 — r a /d 3 = 3(260 + n2)/(26o+ l)(n2+2}

/ 1.29 / 47rN!a/e0= 3(n2— l)/(n2+ 2 ) ; { r =2(£o-i)/(2£0+|)sr0j \

Comparison of the two theories shows that while Debye's theory predicts the occurrence of ferroelectricity (incorrectly in most cases), Onsager's theory predicts that ferroelectricity will not occur. Both Debye and Onsager use "semi-statistical" methods, in that the calculation of the dipolar polarizability uses statistical arguments, whereas the calculation of the local field are based on macroscopic treatments of specific models.^-

Two theories which attempt to carry out a complete

14.statistical treatment of the problem are those of Kirkwood^ and Frbhlich. Differences arise only on the inclusion of distortion polarization; if non-polarizable dipoles are assumed, the two theories give the same result.^ In Kirk­wood's theory, distortion polarization is treated by assu­ming that the dipole has a polarizability a, and that the local field which that dipole feels is Onsager's cavity field — G = 36oE/(2€o + i) . Kirkwood's equation (including distortion polarization) is then:

/ 1.30 / (£0-|)(2£0+1) _ 4 7T N / , g /X2 ■,3£o _ V€0 a 3 kT

where N/V = N^, and g The correlation parameter, g,measures the degree of interaction of the dipoles with their neighbors; JZ is the average moment of a finite spherical region around one molecule which is fixed, and j_i is the moment of that fixed molecule. In other words, if fixing one dipole has no effect on others, then g = 0; if fixing one tends to align others parallel to that one, g > 1;and if in an anti-parallel direction, g < 1.

7 . . .Frohlich, in including distortion polarization, main­tains the non-polarizability of the dipole in the cavityand assumes that instead, the surrounding continuum has a

2permittivity n . This has the effect of changing the cavity field to

/ l . 3 1 / g = 3 6oE / ( 2 £o + n )

15.which leads to Frohlich!s equation for the static permit­tivity :

This is the same as Onsager's equation except for g, the correlation parameter described above.

The differences between the Kirkwood's and Frohlich's theories disappear if the total local field, rather than

These, then, are the major theories which relate per­mittivity to dipole, moment. They all have been based on an applied field which is constant and are thus theories of static permittivity. In chapter II the effects of a time dependent external field will be discussed.

/ i - 32 / ( € o - n 2)(2€o + n2 )€0{n2 + 2 )2

4 77-N g Mg

Onsager's cavity field, is used in Kirkwood's calculations. rr.K+- •: ~ ^ n {z€o ~ \ ) r one uses F = 3 Co

1

16.

NOTES FOR CHAPTER I

1. Nora E. Hill, et. al., "Dielectric Properties and Mole­cular Behavior," Van Nostrand Reinhold Co., New York, N.Y., 1969.

2. Arthur R. Von Hippie, "Dielectrics and Waves," The M.I.T. Press, Cambridge, Massachusetts, 1966.

3. Peter Debye, "Polar Molecules," The Chemical Catalogue Company, Inc., New York, N.Y., 19 29.

4. O.F. Mosotti, Mem. Sac. Ita1., 14, 49(1850).5. L. Onsager, J.' Am. Chem. Soc. , 58 , 1486 (1939).6. J.G. Kirkwood, J . Chem. Phys. , 7_, 911(1939).7. H. Frohlich, "Theory of Dielectrics," Oxford University

Press, London, 1949.

CHAPTER IIDIELECTRIC RELAXATION

Dielectric relaxation occurs when a dielectric mate­rial is subjected to an external field of varying intensity. The permittivity of the dielectric falls off with increa­sing frequency, due to the fact that the dipolar (orienta­tion) part of the polarization cannot change fast enough to reach equilibrium with the applied field. To treat this phenomenon on a macroscopic basis, we first divide the polarization into two parts; the distortion polarization,P^, which consists of the atomic and electronic polariza­tions and can reach its maximum value effectively instan­taneously; and the orientation polarization, ¥ , which requires a significant time to reach its equilibrium value. We then make the assumption that P^ increases at a rate proportional to its departure from equilibrium valued

(P = total polarization at equilibrium.)The macroscopic relaxation time, T, is defined as the

time required for the experimentally observed dielectric

polarization to decrease from Pmax to 1/e times Pm a x * The microscopic, or molecular relaxation time, T , is not the

/ 2.1 / d P2 dt

P. - R>T

17.

18.same as T; and as will be shown, depends on the form chosen for the local field.

Equation 2.1 can be rearranged to give;

/ 2.2 / d(P - PT - P?) _ -dt

for which the solution is,

/ 2.3 / In (P - - P2) = “t/j + C ; C = constant.

Now if a field, E, is suddenly applied at t = 0, then P2 = 0 and C = ln(P - P- ) . Substituting this solution back into equation 2.3 gives

/ 2.4 / P2 = (P - (1 - e~t/T ) ;

that is to say, approaches equilibrium at an exponential rate as shown in Figure 2-1.^

( t /T )Figure 2-1

Similarly, if a constant field is present and is suddenly removed, P^ = 0 , and

19

/ 2.5 /dP.dt T lnP2 = C - t/T .

At t = 0; P2 = P - P p C as shown in Figure 2-2.^*

ln(P - P^) and P^

(t/T) Figure 2-2

(P - Px)e-t/T

2 3 40

An alternating field of frequency 27T f = 0), can be represented by the equation,

iujt/ 2.6 / E Q Q

The static permittivity and the refractive index are given in terms of P and P^ by^

/ 2.7 / 4 7TP/ 2.8 / 47TP

€c( — I ) E , OJ 0

e0( n2 — l ) E

Plugging these values for P and P _ into equation 2.2 we get

/ 2 . 9 / d P2 dt

£2.T

For the steady state, the solution to this equation should have the form P2 = Ae*'0 , where A is found by plugging this solution back into equation 2.9.^

20

/ 2 . 1 0 / A = €q(€o — n2) E o / 477* ( I - f c c u T )

n2 ) E / 47r ( I + c c u T )/ 2 . 1 1 / p 2 = € 0( 6 o

From this it is seen that the orientation part of the polar­ization is out of phase with the applied field. Thus we can write the polarization as a complex quantity,

/ 2.12 /p. + p2 = p ' + l p " =1 2 47T ( n2 — i ) E + — n. ) vEv 47T (I + oa;T)t >

where P" and P" are both real. It follows that if the polarization is complex, so is the permittivity; and it can be expressed as,^

/ 2.13 / € /_ £ € /'== | + 47r(p' — CP'')/<£.

= n2 + (€0_ n2) / ( l + iwT) ,

where both and are real and have the values:

/ 2.14 / ,€ n2 + —

-f cu I€U = ( go — n2) oj T

2 2 i +oj T

When these are plotted versus log(ojj), the graph shown in Figure 2-3^ results.

-2

Figure 2-3

21The fact that there is a phase difference between the

applied field and the orientation polarization causes energy absorption by the dielectric material. When the polarization of the dielectric changes, a displacement current of density i is generated such that,

/2.15/ i = §§,

and Joule heating, H, occurs where

/ 2.16 / H = Ei = E per unit volume.dt

When the average value of H for one complete cycle of the electric field is determined, this equation is found:^

/ 2.17 / H av = Eo€0€ //cu/87r

That is, is directly proportional to the average rate ofJoule heating created by absorption of electrical energy by the dielectric. Because of this, is often called the dielectric loss factor, and the quantity 1 is calledthe loss tangent.^

The permittivity can be represented graphically by utilizing the fact that the equations for £ 7 and £;77 can be reduced and combined into,

/ 2.18 / 2 2(e' - + n )2+g"2 = ( g-s- .n,, f ,

which is the equation of a circle with center2 1 ■ IIand radius ( £0 - n )/2. Only the semicircle for £ >Q is

22meaningful, and the usual representation is shown in Figure2- 4 .1

€

./€Figure 2-4

This curve does fit the data for many simple liquids, but2many other materials do not fit. Cole and Cole have pro­

posed an empirical equation of the form,

/ 2.19 / 2n = n

(l-h)i + (i out)(h is a constant; 0 < h < l . )This seems to fit much data for polymers and long chain molecules whose plots of experimental data resemble the plots shown in Figure 2-5 a and b."

From Debye equat ion

-2

Figure 2-ba

23.

/I Debye curve€

Figure 2-5b

The plot of € versus £ is symmetrical about a line through the center and parallel to the axis. Cole and Davidson found certain materials best fitted a skewed arc, and suggested the equation:

/ 2.20 / * _ 2— f ------ Q_— — ---------- !----- . 0<a< i .

6 o — n ( I + ccuT )

Shapes, of this equation for various values of q are shown in Figure 2-6.^ Experimental data acquired at low tempera­tures seems to fit this equation well with q approaching

41 as the temperature increases.

a =0.80.4a =0.6

0.2

0 0.2 0.60.4 0.8 1.0" n

Figure 2-6

24.Thus far we have been considering macroscopic theories

of dielectric relaxation. Now we will look at this pheno­menon from a microscopic or molecular viewpoint.

Since the phenomenon of dielectric dispersion is directly related to the physical rotation of the molecules under the influence of an external field, it is necessary to examine the mechanism or mechanisms whereby these mole­cules rotate. It is to be expected that this mechanism will depend on a number of factors, including the state ofthe dielectric material.

5Debye assumed that rotation of a molecule due to an external field was interrupted by collisions with neigh­boring molecules. This assumption is reasonable in the case of liquids but less justifiable in the case of solids, or gases of low to medium pressure. Using the theory of Brownian motion as developed by Einstein, and assuming the Lorentz field, Debye derives the following relation:

where T is the molecular relaxation time as defined pre­viously .

Extension of Onsager's theory of static permittivity

I + LcjT

to the high frequency region^ leads to the expression,

/ 2.22 /A6 2 6n

I 4- L cj ~C 2

where A is frequency independent and is given by,

25

/ 2.23 /( n 4- 2X 2C0 4- I) 87T N t yLL

( 2 £0 + n ) 9 kT£0A

Debye's assumption that the applied field rotates molecules at their terminal velocity is seen to hold rea­sonably well up to frequencies near 1 to 2 gigahertz. At higher frequencies, inertial corrections, which Debyeneglects, cause the actual angular velocity to deviate more

6 1and more from the terminal velocity. '

A number of other theories of microscopic relaxation7 1have been proposed. Eynng ' draws an analogy between

dipole rotation and chemical rate processes. Bauer's 8 1theory ' is similar to Eyring's and both theories make the

assumption of a potential barrier of some type which affects9 1free rotation. Cole ' presents a theory which produces a

result of the same form as Debye:

except that here T is the macroscopic relaxation time for a sphere as opposed to the microscopic time w’hich appears in Debye's theory (equation 2.21). The term in brackets is a result of the choice of a spherical model. Changing the assumed shape (for instance, to ellipsoidal) will change the ratio of the two relaxation times (microscopic to

• X 1macroscopic) .A relation is also derived based on a model of a small

/ 2.24 / g* - n22€0 — n

26.9 10sphere within the overall spherical specimen. ' This

result is:

/ 2.25 / Ts = ( I _ A ) Tm

where Ts is the relaxation time of the spherical specimen and Tm is the time for the small internal sphere. Also if the correlation function of a dipole at the center of the small internal sphere is defined by

/ 2.26 / / ( t ) = < fJL {0) * /l(t)> / < JjL{0) • JjL{0)> ,

the surrounding material is treated as a continuum, and is assumed to be a simple exponential (~y (t) = exp(— t/r) ) ;it is found that^'^

/ 2.21 /

/ 2.28 /

Fatuzzo and MasonA"7" have questioned one of theassumptions used in the above derivation, and Klug, Kran-

12 1buehl and Vaughan ' have derived the following equation based on this modified assumption:

/ 2 • 29 / f e ~ n 2)fe^^~n2) _ (£o— n2)(26o + n2) _ 16Fo 1 + LcoT

13Glarum has proposed a model which can account forthe experimentally observed "skewed arc" behavior found by

3 4 14 ■Davidson and Cole ' ; although, as Mopsik points out,this

3 € 0 ( n2 + 2 )

(260+ n ) ( € o i 2 )T

T - 3£c2 €0 + n

T11.1

27.model "admittedly has many defects."

In the normal case, all molecules feel the same aver­age hindrance to reorientation and thus exhibit a single relaxation time. This hindrance is a result of the envi­ronment of each molecule, and these environments are assumed to be the same for all molecules. In Glarum's model, "defects" are present in the solid or liquid, and are distributed throughout the sample according to a diffu­sive law. The presence of a "defect" in the environment of a molecule greatly decreases the hindrance to reorientation, thus allowing the dipole to relax essentially spontaneously. This process is assumed to be independent of the normal relaxation process. Although the expression for the general case is complicated, three specific cases yield interesting results. If the rate of arrival of defects is small com­pared to the normal relaxation time, then a result corres­ponding to Debye type relaxation is obtained. If the rate of arrival is equal to the normal relaxation time, then a skewed arc with CL = 0.5 (eq. 2.20) is obtained; and if the rate of arrival is much greater than the normal relaxation time, then a semicircle with h = 0.5 (eq. 2.19) is pre­dicted .

A number of a u t h o r s ^ h a v e proposed theories based on the concept of "free volume", defined differently by different authors, but basically the local room which each molecule has available. These theories are related to Glarum's in that all assume more than one time dependent

28.process which affects reorientation.

Thus far we have considered dielectric relaxation only in general microscopic terms. We would like to relate the relaxation to some macroscopic property or properties of the specimen. Debye suggested making the assumption thatthe molecule could be represented by a sphere rotating in ahomogenous medium of viscosity qq equal to the viscosity of the bulk sample according to Stokes law. This assumption, when applied to Debye's theory gives:

/ 2.30 / T = 47T7^a3/ k T ,

where a is the radius of the molecular "sphere". Thisgives results which are 5 to 10 times too high, and nume-

17-20rous attempts at improvement have been made.A basic problem with Debye's assumption is that unless

the polar molecule is large compared to the surroundingnon-polar solvent, the solvent cannot really be treated asa continuous medium of viscosity tj .

20Wirtz and his co-workers have attempted to take into account the discontinuous nature of the surrounding medium. Their results indicate that when the polar molecules are large compared to the surrounding molecules

/ 2.31 / T = 4 7r 77 a3/ k T ,

the same result obtained by Debye. When the molecules are all the same size (e.g., a pure polar liquid)^,

29

/ 2.32 / T = 2 77" 77 CJ3 / 3 k T ,

a relaxation time six times smaller than that predicted by Debye, and in much better agreement with experiment.

30.

NOTES FOR CHAPTER II

1. Nora E. Hill, et.al., "Dielectric Properties and Molecu­lar Behavior," Van Nostrand Reinhold Co., N.Y., N.Y., 1969.

2. K.S. Cole and R.H. Cole, J. Chem. Phys., 9_, 341 (1949).3. D.W. Davidson, R.H. Cole, J. Chem. Phys., 18, 1417(1951).4. D.W. Davidson, R.H. Cole, J . Chem. Phys., 19, 1484(1951).5. Peter Debye, "Polar Molecules," The Chemical Catalogue

Company, Inc., New York, N.Y., 19 29.6 . M.Y. Rocard, J. Phys. Radium, Paris, 4_, 247 (1933).7. S. Glasstone, K.I. Laidler, H. Eyring, "The Theory of

Rate Processes," McGraw-Hill, New York, N.Y., 19 41.8. E. Bauer, Cah. Phys., 20, 1(1944).9. R.H. Cole, J. Chem. Phys., 42, 637(1965).

10. S.H. Glarum, J . Chem. Phys., 33, 1371(1960).11. E. Fatuzzo, P. Mason, Proc. Phys. Soc., 90, 741(1967).12. Dennis D. Klug, David E. Kranbuehl, and Worth E. Vaughan,

J. Chem. Phys., 50_(9_) , 39 0 4 (19 69) .13. S.H. Glarum, J . Chem. Phys., 33, 369(1960).14. F.I. Mopsik, Doctoral Thesis, Brown University, 1964;

University Microfilms, 65-2229, Ann Arbor, MI, 1975.15. J. Anderson, R. Ullman, J . Chem. Phys., 47, 2178(1967).16. M. Cohen, D. Turnbull, J . Chem. Phys., 31, 1164(1959).17. F. Perrin, J . Phys. Rad ium, Paris, 5_, 497 (1934).18. E. Fischer, Phys. Z., 60, 645(1939).19. A. Budo, E. Fischer, S. Miyanoto, Phys. Z., 40, 337(1939).20. A. Gierer and K. Wirtz, Z . Naturf., 8a, 532(1953).

CHAPTER IIITHEORETICAL CONSIDERATIONS OF UHF DIELECTRIC MEASUREMENTS

Transmission Line Theory

The theoretical approach used for a given experiment depends upon the frequency range under consideration. At lower frequencies the sample is small enough compared to the wavelength,that bridge or resonant methods may be used in conjunction with either lumped or distributed circuit analysis. Above approximately 200Mhz, the sample is no longer small compared to the wavelength of the applied field, and transmission line techniques must be used. An upper limit of approximately 7 5GHz is imposed on this tech­nique by the difficulty of fabricating components with the required precision in their physical dimensions. Free space methods can be used in this frequency range.^

In the frequency ranges which require transmission line methods it is necessary to investigate the electric and magnetic fields which are present. This is best done by starting from Maxwell's equations: From any generaltext on electromagnetics, the following pertinent equations can be found.

/ 3.1/ divD = V • D = 47Tp ,

where p is the free charge density and D is the dielectric

31.

32displacement.

}/ 3.2 / divB = V • B = 0

where B is the magnetic induction or magnetic flux density.

/ 3 . 3 / c u r j H = V X H — D + 4 7T J

where H is the magnetic field intensity; J is the current density; and c is the speed of light in vacuo.

/ 3 . 4 / curl E = V x E = c" B ,

where E is the electric field intensity. The following three relations also hold.

/ 3.5/ D = £ 1E

/ 3 . 6 / J = orc E j

7 3 .7/ H = B/f±Q ; B/H = j-L 0 ;

where is the real part of the permittivity and is the ohmic conductivity of the medium. jJLQ±s the permeability.

2The following derivation is based on that of Glarum. We assume that the materials of interest are non-magnetic so that fji0 = 1 . We can now write for a conducting medium:

/ 3.8 / V - D = V • (c 'e ) ;/ 3.9 / V • B = V • H = o ;/ 3.10 / V x E = ----5-H

Substituting into equation 3.3 gives

33.

/ 3.11 / V X H = E + 4 7rcr E

If we assume that the applied field is alternating sinudoi- dially, then,

/ 3.12 / E (t) Eo £ c cut

from which it can be shown that

/ 3.13 / E = “ i E cu

which, when substituted into equation 3.11, gives

/ 3.14 /V x H / • _ L4va-rE

cu= E c

J _ lAircrr _ cu

A normal dielectric has zero free charge density so that V • D = V * ( ;E) — Arrp — 0 . If we now assume that thedielectric constant is complex and of the form

/ 3.15 / € = i £ ii

then

/ 3.16 / V X H =

We see that equation 3.14 is of the same form as that for a

non-conducting dielectric.Consequently, to a solution of the field equa­tions involving a nonconducting dielectric, there exists a formally similar solution in which a complex dielectric constant appears in place of the real dielectric constant provided, of course, that the electric fields involved are of the form given by Equation 3.12 .^

34If the complex dielectric constants given in equations

3.15 and 2.13 are equivalent, is related to oj; as shown in equations 2.11 through 2.17.

To generate the pertinent field equations we take the curl of both sides of equation 3.10:

/ 3.17 / Vx(VxE) = - 4

Due to the interchangeability of space and time derivatives we can rewrite 3.17 as

/ 3.18 /V x (VxE) d(V x H )

“ 11

and substituting from equation 3.16, we get

/ 3.19 / a o f ./at) atFrom standard vector relationships, we obtain the identity

/ 3.20 / V x ( V x E ) = V ( V - E ) - ( V - V ) E •,

and since V'E- 0, we can combine 3.19 and. 3. 20 to give the desired wave equation:

/ 3.21 / V 2 E

In exactly the same manner, we find that

/ 3.22 / V 2 H = -^5- H

35.For a fixed frequency, plane waves travelling in a

waveguide in the z direction will reasonably give

/ 3.23 / E = Eo(x,y) 6 ; and

/ 3.24 / H = Ho(x,y) eL^ ' rz .

The variable y in the above equations is known as the pro­pagation constant. It is complex and can be written in general as

/ 3.25 / / = a + i /3 ,

where a is an attenuation factor and J3 is a measure of2the phase velocity of the wave.

It can be shown that if propagation is to occur in a hollow waveguide, either E0 or H0 must have a non-zero z component. Modes of propagation with z components of the E and H fields equal to zero are designated TE and TM respectively. If a center conductor is present, as in co­axial line, neither E nor H need have a z component, and

2the resulting mode is designated TEM.Substitution of equations 3.2 3 and 3.2 4 into the wave

2equations 3.21 and 3.22 results in a solution for

/ 3.26 / 7 2 = k* - ^ ,

where kQ is a geometric parameter which depends on the mode of propagation and the dimensions of the waveguide. Thereis a cut-off frequency oj such that

36

/ 3.27 / oic — C kc ;

and therefore, if oj is less than cu , and the wave guide is in vacuum, so that = 1 ,

2 2 / 2 2 OJp— OJ _ V O U r - CU/ 3.28 / r = - c - 2 ■» r = - - - 0- -C C

This indicates that above the cut-off frequency, propaga­tion with no attenuation occurs; and below the cut-off frequency, attenuation occurs without propagation. For a waveguide of given dimensions, there are an infinite numberof modes (designated TE and TM , with m and n 0,1,2,etc.) mn m n 'each having a certain cut-off frequency. Below a certainfrequency none of the modes will propagate and above some

2other frequency, more than one mode can propagate. Since one normally wishes to study only the lowest mode possible for a given waveguide, the bandwidth for each waveguide is limited.

For the TEM mode for coaxial line, k is zero. Due tocthe possibility of higher modes, an upper frequency limit

2exists which is given by

/ 3.29 / OJ r= 2 . 1 3 6 c / (a + b) ,

where a is the outside radius of the inner conductor and b is the inside radius of the outer conductor. At this fre­quency another, higher mode will begin to propagate; and since we only want the TEM mode, we must operate between

37.zero and ouc.

If \ Q is the free space wavelength corresponding to OJ and \c corresponds to GJC , then the complex dielectric constant can be written from equation 3.26 as

/ 3.30 / e * = (X0/ X c)2 — ( X a/27r)2y 2 .

Thus £ may be found by experimental determination of the propagation constant. This determination may, in practice, involve a number of problems.

In our work we will use another method which necessi­tates the development of the concept of wave impedance.This is defined as the ratio of the transverse component of the electric field to that of the magnetic field. For a traveling wave in a coaxial line (TEM mode), it can be shown that the characteristic wave impedance Tj0 , is con­stant and given by

/ 3.31 / V ° ~ ~ VA'-o /*= >

where the plus and minus signs refer to the direction of2travel of the waves. For an air filled coaxial line such

as we used, both fJL0 and € are essentially equal to 1 so that TjQ = ± 1. This is not to be confused with the surge impedance, Z0 , which is defined as the ratio of voltage to current and given by

/ 3.32 / Z0 = -^(-^rfln (b/a)

A dielectric filled section of coaxial line will have

38.a wave impedance 7702 and a propagation constant yz . Sup­pose that this section has a termination with a complex reflection coefficient, p 2 , which is defined as the ratio

Assuming that both the incident and reflected waves travel in a positive direction along the z axis, and that the ter­mination is at z= 0; then, at some distance d in front of the termination, the z coordinate will be d and -d for the reflected and incident waves respectively. For sinusoidal waves this gives

of the reflected field to the incident field: p9— E* / E*r c. f tIt can be shown that pz — — we can write the

2following equations which apply at the termination.

/ 3.33 / E f = p2E*

/ 3.34 /

/ 3.35 / ^ Z = EoS1icbt + Yz

/ 3.36 / Et = Eo(S.£'Ct,t_ =Eo(S>ccut - Yz^

d

From the definition of wave impedance

/ 3.39 / Et + E?it+ i li-

and substituting from 3.35-3.38 we obtain

39 .

/ 3.40 / 7

/ 3.41 /

(Eoe Ccouy2(i - PzEoe^ - r z d ) r ,02

Voi> (e7*6 + Pze~Tz6){ e YzA - P z e -yzA)

For the important case when the termination is a short,

The usefullness of this experimental procedure lies in the fact that it allows the use of real rather than complex values. Most other methods using transmission line tech­niques require complicated mathematical procedures toobtain the desired results from the experimentally deter-

2 4mined parameters. 'These complex values are eliminated by using a vari­

able length section of dielectric filled coaxial line, terminated in a short circuit. The distance can be ad­justed so that, at one or more positions of the short, the input impedance Tj is entirely resistive, i.e., real. These distances are functions of the real and imaginary parts of the propagation constant. If we know a distance for whichthe impedance is real, and the corresponding impedance, we

2 , Acan find the propagation constant.For a dielectric filled line with propagation constant

X / and where is the characteristic wave impedance,

2Pz = -1, and equation 3.41 becomes

/ 3.42 /Vd = Voz e 7zd + ' e-Yzd j = t o n h •

r2d -Yz d ie - <g .

40equations 3.25, 3.30 and 3.31 can be combined to give

/ 3.43 / I / TJ0 = ( X0/27r) ( j3 ~ i a )

The wave admittance Y is defined as 1/TJ , so that equation 3.42 now gives

/ 3.44 / Y = 1/770 coth yd ,

which on substituting from equation 3.43, gives

/ 3.45 /Y = { \ 0/27r){/3-La) b sinh2ad—L sin 2/3d

cosh 2ad— cos2/3d

Setting 2 ad = a, and 2/3 d = b, the above equation becomes

/ 3.46 /Y = (X /477-d) (bsinha— a sinb)-i(a sinhg-fb sinb)

cosh a — cos b

When we experimentally adjust the distance so that Y is real, the second term in the bracketed numerator is zero;i.e., (a sinh a + b sin b) = 0, and

/ 3.47 /Y =(X„/47rd) b sinh a -a sin b

_ cosh a — cosb _ (X0/47rd) Cn(a,b)

For any value of Y/(X0/47rd), there are multiple solutions of C (a,b). Using equations 3.25 and 3.30 and remembering a = 2 ad, b = 2/3d, and that 0Jc = 0 for the TEM mode, the complex dielectric constant can be separated and expressed as

41.

= (X0/47rd)2 ( b2 — a2) >

/ 3.49 / //£ = (XQ/477" d) 2ab

^ 2 2 and tan o = 2ab/(b - a ) . A computer program and theo oresultant tables of values for Cn (a,b), b - a , 2ab, and

tan 8 , are presented in Appendices I and II.

42.

NOTES FOR CHAPTER III

1. Worth E. Vaughan, Chapter IT: Experimental Methods, in "Dielectric Properties and Molecular Behavior," Nora E. Hill, Ed., Van Nostrand Reinhold, Co., New York, N.Y., 1969.

2. S.H. Glarum, Doctoral Thesis, Brown University, 1960;University Microfilms, 62-5745, Ann Arbor, Michigan, 1975.

3. W.T. Scott, "The Physics of Electricity and Magnetism,"John Wiley & Sons, Inc., New York, N.Y., 1966.

4. F .I . Mopsik, Doctoral Thesis, Brown University, 19 64;University Microfilms, 65-2229, Ann Arbor, Michigan, 1975.

CHAPTER IVUHF EXPERIMENTAL

The equipment and procedure are based on previous work1 2 done by Mopsik and Glarum .

All electronic equipment was purchased from General Radio Company, West Concord, Massachusetts. A complete list of all components is given in Appendix III.

The central component of the measurement apparatus was a General Radio type 16 02-B admittance meter which is a direct reading instrument and covers the frequency range from 4 0 to 1500 megahertz (to 20MHz with the appropriate correction). Its stated accuracy is ±(3% + 0.2mmhos) from 0 to 20 millimhos below 1000MHz, and with the 1607-P10 mul­tiplier plate provided, the accuracy is improved to

"3±(3% ± 0.04mmhos) from 0 to 2.0mmhos. A schematic diagram is shown in Figure 4-1, and includes the auxiliary equipment -- (generator, detector, standards, and the unknown).

43.

44.

Figure 4-1

The admittance meter is also shown in Plates 1 and 2. Although it is a null reading device, the admittance meter is not a true bridge. It has three coaxial lines which are fed from a common junction, and the current flowing in these lines is measured by three rotatable coupling loops which sample the magnetic component of the field in each of three arms (unknown, conductive, and susceptive). The susceptance loop is rotatable through 180° and the conduc­tance and unknown loops can be rotated through 90° „ The voltages induced in the loops are proportional to the

45.cosines of the angles which the loops make with the coaxial line. These three loops are connected in parallel to a co­axial line which feeds the crystal mixer and heterodyne detector. When the loops are properly oriented, the com­bined output is zero and a null is read on the detector. In

3this sense, the admittance meter balances as does a bridge.The conductance standard is a pure resistance equal to

50 ohms, the characteristic surge impedance of the coaxial line; and the susceptance standard is a short circuited length of coaxial line. In actual practice, the susceptance standard was replaced with a type 8 7 4-WOL open circuit, since our procedure is based on the premise that the imagi­nary (susceptive) part of the admittance is zero.

The admittance meter actually measures the admittance at a point directly under the center of the unknown coupling loop. Thus, if the admittance is desired at some other point on the coaxial line, correction for the effect of the length of the transmission line must be made. If the elec­trical length of the line between the center of the coupling loop and the point at which the admittance is desired is exactly an integral number of half-wavelengths, then the measured and unknown admittances will be equal. Also, if the distance is an odd number of quarter-wavelengths, the admittance meter will read the resistive and reactive (rather than conductive and susceptive) components of the unknown. For example, in our procedure, a short circuit is placed h wavelength from the interface of the cell and

the constant impedance adjustable line; and the adjustable line is set to give a null reading, which occurs when the

cell window is then % wavelength from a point above the center of the coupling loop, and the meter will measure the input admittance of the cell.

If we are to use the equations previously presented, we need to know the relationship between the admittance measured by the meter and the input wave admittance, Y, which appears in equations 3.44 - 3.47.

/ 4.1 / Y = -±- coth 7d ; { p = - \ ) ■ (eg.3.44)

If we treat the dielectric filled section as a termination with complex reflection coefficient, p , equation 4,1 can

be rewritten as

where 7j0t is the characteristic wave impedance of the sec­tion in front of the interface. Since d=0 at the interface, 4.2 becomes

3overall electrical length is wavelength. Therefore, the

/ 4.2 / i e - p e ' e rd + p e ~ rd

Y

/ 4.3 /Y

From the definition of p , it can be shown that

47.where YQ is the characteristic surge admittance of the transmission line (20mmhos), and Y is the characteristicXwave admittance of the unknown. Therefore,

, . ~ , measured admittance/ 4.5 / Y = Yx/Yq = ------ ZOmmhos------

Thus the measured admittance must be divided by 20mmhosbefore being used in equation 3.48 and 3.49.

The cell was constructed by the William and MaryPhysics Department shop using the 50cm stub and BNC adapter,following the description given by Mopsik.^

It (the cell) consists of a modified General Radio type 874-D50 adjustable stub. The outer conductor was cut to the desired length and surrounded with a jacket through which thermostating fluid could be pumped. Also entry was provided for the inser­tion of a thermocouple for the measurement of temperature. The bottom end of the outer conduc­tor butted against a small land in the jacket for alignment purposes and was made the same inner diameter as the outer conductor so that electri­cally it would be continuous. A Kel-F window sealed off the bottom which was made the same physical size as the polystyrene window in a Gene­ral Radio type 8 74 connector so that when the entire cell was assembled, any discontinuities would be no worse than that due to a similar con­nector. Around the outer edge, and the inner one, small rings of Teflon were placed to provide a liquid-tight seal. Below the window is the outer part of a type 8 7 4-QBPA adapter with the 8 74 con­nector removed. A nut threaded into the bottom of the outer jacket tightened down on this piece by means of a small ring inserted in a groove in the adapter. By this means, sufficient pressure could be applied for the Teflon seal to be liquid tight. When the entire assemble was put together, it should be electrically as smooth as the top half of a General Radio connector and therefore, adequate for the purposes of measurement.

The inner conductor of the stub on top of the window was joined to the inner conductor of the adaptor below the window by a short sec­tion of threaded brass rod which also allowed

48.sufficient tightening to be applied to the Teflon seal. The shorting piston assembly was made from the contact fingers and a small length of solid continuation in the original stub. These could be separated into inner and outer finger pieces which were reassembled around a brass ring of the proper dimensions to fill the space between the pieces up to where the fingers began. Then a thin brass tube was affixed to the brass ring, concentric to the center conductor, of sufficient length to reach above the cell when the plunger was at the bottom of the cell. The entire plunger assemble was made so that the fingers were on top and the brass ring was flush with the bottom in order to present a smooth, flat surface to the electrical circuit.

The finished cell is shown in Figure 4-2 and in Plates 3and 4. The only major differences between our cell and thatdescribed by Mopsik are the lack of a third outlet in thethermo jacket, use of a finely scribed line on the plungeras a measuring reference since measurement was by opticalrather than physical means; and the use of stainless steelfor the plunger tube.

Figure 4-3 and Plates 5 and 6 show the interconnectionof the components.

BNC L-J C onnec to r

R e fe renceMark

PlungerKe!-FWindow

Figure 4-2

1 264 -B 1362PWR oscSUPPLY

Pi558 7 4 -p l oooLr

8 7 4 -G I0 L -

11

JJHF CELL

m

1602- R4

874-WOL

A A

8 7 4 - L K 2 0 L

—874 - QB J L

— 8 7 4 - Q B P A L

— 8 7 4 - E L L

.874 - M R ALL 0 1 B

if input

362236F AM P

Figure 4-3

In addition to the interconnection as shown, the cell was mounted vertically and rigidly on a vacuum rack, but in such a way that it could be easily removed for solution changes, cleaning and maintenance. The cathetometer, which was used to measure the extension of the plunger, was placed on a low table close to the cell so that there was an unob­structed view of the plunger tube.

Operation of the equipment was relatively simple. The signal generator was turned on, and the oscillator was set to the desired frequency. According to the instruction manual, drift should be less than 0.1% after fifteen minutes and negligible after one hour. In practice the signal generator and the detector were allowed to warm up for at

50 .least two hours; and if continual use over a several day period was anticipated, they were left on, although turned down as low as possible in between measurement periods.

Once sufficient warm up time had passed, the modula­tion level was set to an arbitrarily chosen level (seven on the scale of ten). The IF oscillator was tuned to a fre­quency 30 Megahertz lower (or higher) than the signal gener­ator, as indicated by a peak reading on the detector meter. The IF amplifier was used in the normal switch position and the calibrated attenuation adjusted as necessary. Once the detector had been tuned, the mixer current was checked by placing the switch on the IF amplifier in the appropriate position. The current should read between 50 and 100 on the scale and was adjusted with the sliding attenuator on the oscillator section of the IF amplifier. A setting of 8 5 was used throughout the measurements.

The multiplier plate type 1607-P10 was installed in the admittance meter so that the conductance scale would read times 0.1. The multiplier arm was set to zero as was the susceptance arm. With the connections and settings made as described, and the cell hooked up; calibration of the equip­ment using carbon tetrachloride and benzene was performed.

To calibrate the cell, the plunger was pushed all the way to the bottom of the cell; and then, using the catheto- meter, the plunger was extended one quarter wave-length (calculated from the nominal generator frequency with the speed of light equal to 299,792.4562^ kilometers per second).

51.With the cell length thus fixed, the constant impedance line was adjusted for a null on the meter. Then the con­ductance arm of the meter was adjusted for a further null.A very slight adjustment of the susceptance arm may be necessary to achieve a perfect null. The value of the con­ductance was found to be approximately 0.1 millimhos when the frequency was 500 megahertz. The constant impedance line was locked in place and not moved unless the frequency was changed; in which case, the procedure in this paragraph was repeated.

The solution to be measured was introduced into the cell, and the plunger and the conductance arm were reset for a null. The plunger height was measured with the cathe- tometer to give the value of d, and the admittance, Y, was calculated by subtracting the zero offset determined above from the new measurement and dividing that difference by 20, the value in millimhos of the standard 50 ohm termina­tion.

The complex dielectric constant could then be calcu­lated from equations 3.4 8 and 3.49:

/ 3.48 / , . \ 0 ,2 2 2€ ( 47Td I ' ( b — a )

47rdThe values of b - a and 2ab for the experimentally deter­mined value of Cn (a,b) were found from the table in Appendix

2II. Glarum has shown that simple interpolation is

52.sufficient to determine these values accurately.

If this calculated value of the dielectric constant did not agree with the accepted value for the reference liquid, the above procedure was repeated using a certain amount of offset in the physical setting of the plunger.The actual amount of offset was dependent on the physical configuration of the plunger since that configuration directly affects the electrical properties of the plunger. The offset was found to be on the order of 0.5 cm. The actual value was determined by choosing a certain arbitrary amount and setting the plunger that much shorter than a quarter wave length. For example, with a h wavelength of 10.0 cm and a trial offset of 0.5 cm, the plunger would be set to a physical distance of 9.5 cm. The constant impe­dance line and the conductance arm were reset for a null with the cell empty, liquid introduced again into the cell, and the plunger moved and the conductance adjusted until a new null is achieved. However, this time, the trial offset was added to the actual measured plunger distance to give d. This procedure was repeated with various trial offsets until the calculated dielectric constant was in agreement with the accepted value. The offset should be frequency independent as pointed out by Mopsik”, but our findings seem to indicate a small variation on the order of 0.0 5 cm per 100 megahertz. Therefore, a determination of the offset for each different frequency is recommended.

Once this calibration has been carried out, the cell

can be used to measure the complex dielectric constants unknown solutions.

54.

NOTES FOR CHAPTER IV

F.I. Mopsik, Doctoral Thesis, Brown University, 19 64 ; University Microfilms, 65-2229, Ann Arbor, Michigan,1975.S.H. Glarum, Doctoral Thesis, Brown University 1960; University Microfilms, 62-574 5, Ann Arbor, Michigan,1975.General Radio Company, Instruction Manual, Type 16 02-B Admittance Meter, General Radio Company, West Concord, Massachusetts, 1968.Robert C. Weast, Ed., "Handbook of Chemistry and Physics, 54th Edition" CRC Press, Cleveland, Ohio, 1973.

CHAPTER VEXPERIMENTAL RESULTS

Measurements on a number of solutions at two frequencies (500 and 700 MHz) and at two temperatures (25° and 60°C.) have been carried out. The following mixtures were mea­sured :

nitrobenzene in benzene,

ortho-nitrobiphenyl in benzene, para-nitrobipheny1 in benzene, and 2 ,2 1-dinitrobiphenyl in benzene.

For each combination, either three or four solutions with a linear change in concentration were made; the most concen­trated solution being on the order of 1.5 mole per cent for the first two compounds above; and, due to limited solu­bility, about 0.4 mole per cent for the second two.

A typical measurement on one solution was made as follows: The frequency generator had been previously turnedon and allowed to warm up for at least two hours. The desired frequency was selected and the IF oscillator was tuned to thirty megahertz below the generator frequency.The mixer current was set to 85. The multiplier arm and the susceptance arm were set to zero. By means of diffe­rence measurements, the cell plunger was extended so that the physical distance was one quarter wavelength, less the

55.

56.offset for the particular frequency. Since this setting did not usually correspond exactly to an electrical quarter wavelength, the constant impedance line was adjusted to give a null. Also, because there are slight conductances present in the system, the conductance scale may not null at exactly zero. This conductance offset, as well as the plunger offset, must be noted and taken into account in the calculations. Once this null had been found with the cell empty, the unknown liquid was poured into the cell and allowed to reach temperature equilibrium. The plunger and the conductance arm were then adjusted to give the new null point. Again by difference measurements, the distance, d, was measured. Before this value could be used, the plunger offset had to be added. Likewise, the conductance offset must be subtracted from the value of Y as read from the scale. This difference must then be divided by 20 millim­hos, the value of the standard termination, to give thevalue of Y to be used in equations 3.48 and 3.49.

Typical values for a nominal 1.0 mole per cent solution of nitrobenzene in benzene at 25°C. and 700MHz are:

distance offset = 0.47 cm., conductance offset = 0.15 mmhos,Y = (measured Y - offset)/20 = (.42 - ,15)/20 = 0.0135,d = measured d + offset = 6.317 + .47 = 6.787 cm.

From these values, the value of C (a,b) can be found as

/ 5.1 / Cn (a ,b) 47rd Y -22.6884 x 10

57.9 9Appendix II then gives values for - a^ and 2ab of 9.8699

and 0.1075 respectively. These, in turn, yield values for (i1 and of 2.489 and 0 .0271.

The solution data is analysed by the method of C. P. Smyth and co-workers. This "provides the 'best1 values for the solute, whilst also checking for the presence of inter- molecular interaction (solute-solute) over the concentrationrange studied. „1

The values for ^ and for each solution - frequency - temperature combination are plotted versus the concentra­tion of the solute ( in mole per cent) and the best straight lines are found by a computer least squares analysis. These

slopes, a 1 and a", and the similarly obtained slopes, aQ and a^ , are the Smyth parameters per unit concentration of the solute. The macroscopic relaxation time was then cal­culated for each slope from the equations,

/ 5.2 /

anda = qQ - a3Q_

+ oj T2 -r2 + a,

/ 5.3 / / /a = (dp - Qcr, )<^TI + oj2 T 2

The data which were collected for the four benzenesolutions were combined with data which Howe had gathered

2at higher frequencies and a computer program was written to find the best fit of all data. This program, and plots

of £ 7 and versus concentration, appear in Appendices IV

and V.

58.The macroscopic relaxation time is calculated from

equations 5.2 and 5.3. These equations were solved bycomputer for a' and a" as functions of T. The relaxationtime is then determined by finding the minimum in the sumof the residuals between the calculated and experimentalvalues of a' and a".

Howe estimates the error in this determination to beabout ±15% due to a combination of experimental error and

2the use of the Debye model.The Debye model applies to rigid, spherical molecules

whose total dipole moment is in one direction. The mole­cules studied are, however, non-spherical, non-rigid, and some have multidirectional components of their dipole moments.

The Debye equation for the microscopic relaxation time (eq. 2.30) (for dilute solutions, macroscopic and micro­scopic times are essentially equal) has three parameters which can affect the theoretical relaxation time. These are temperature, molecular radius (or volume, if the Van der Waals volume is used), and the bulk viscosity of the solu­tion. Howe analyzes the effects of each parameter andconcludes that the Debye model predicts the temperature

2 . .dependence fairly well. The consistent overestimation ofthe relaxation time by equation 2.30 is usually attributed to the use of the macroscopic (bulk) viscosity rather than the internal molecular viscosity. One of the attempts to improve theoretical and experimental agreement by modifying

59.the viscosity parameter can be found in reference 20 of Chapter II. A more complete discussion can be found in Hill.1

Since the ortho and para-nitrobiphenyls have the samemolecular volume, equation 2.30 predicts that the relaxationtimes will be the same, whereas the experimental valuesdiffer by approximately a factor of 2. Likewise, the ratioof molecular volumes for nitrobenzene and p-nitrobipheny1is 1:1.7, whereas the experimental relaxation times have a

2ratio of 1:5.Howe explains these discrepancies by showing that the

"effective" molecular volume is essentially the volume swept out by the molecule during relaxation. An analysis of the geometry of the molecules studied, the components of their dipole moments, and the possible modes of relaxa­tion, leads to logical explanations for the experimentally observed relationships among the relaxation times.

For example, p-nitrobiphenyl sweeps out a volume 4.9 times larger than that swept out by nitrobenzene, and itsrelaxation time is 5 times larger. The ratio of molecular

2volumes is only 1.7:1.The conclusion can then be drawn that it is the volume

swept out during relaxation, rather than the molecular volume, which is important in determining the relaxation time.^

60.

NOTES FOR CHAPTER V

1. Nora E. Hill, et. al., "Dielectric Properties and Mole­cular Behavior," Van Nostrand Reinhold Co., New York, N.Y., 1969.

2. Allen K. Howe, Jr., Honors Thesis in Chemistry, The College of William and Mary in Virginia, 1974.

CHAPTER VISUPERCOOLED LIQUIDS AND THE GLASSY STATE

A liquid can be simply defined as "a fluid which if placed in a closed vessel at once conforms to the shape of the vessel without necessarily filling the whole of its volume."^ A liquid is thus distinguished from a solid by its indefinite shape, and from a gas by the fact that it may not completely fill its container. While this definition is usually adequate, some materials (for example some polymers) do not fit. These materials appear as solids at room temperature, but on heating, begin to flow, usually with high viscosity, and without exhibiting a sharp melting point. These last two properties are characteristic of a supercooled liquid; when the viscosity becomes infinite (or when the flow is imperceptible) the material is termed a glass. A glass, then, has the external appearance of a solid and the internal characteristics of a liquid.^

Most liquids can be made to solidify in the glassy state if they are cooled through the crystallization temper­ature range faster than the time needed for crystal nuclei

Qto form. ' This is fairly easy to do for liquids of a low order of symmetry, or if the rate of rotational isomeriza­tion from the normal liquid configuration to that required for crystallization is slow at temperatures below the

61.

62.2melting point.

The temperature at which the transition to a glassoccurs is characterized by a second order thermodynamictransition; and in the vicinity of the glass transitiontemperature, T , the viscosity may change by several ordersy O *7 Aof magnitude within 10°K. 7 7 This transition is notthermodynamic in nature since the temperature at which athermodynamic transition due to a structure change occursis independent of the thermal history of the material.The faster the rate of cooling, the higher the glass transi-

7 1 4tion temperature will be. 7 7 In a supercooled liquid,the volume decrease which occurs in normal liquids on cooling is delayed, so that the density decreases as the rate of cooling increases.

Above and adjacent to T in the supercooled liquid region there seem to be two regions of dielectric disper­sion, indicative of two independent relaxation processes.One theory proposes that the low frequency or CL process is due to cooperative relaxation of a group of molecules, and the j3 t or higher frequency process is caused by hindered reorientation of individual molecules which are surrounded by other molecules which are unable to reorient coopera- tively. 7

Studies by Williams and Hains^ of dilute solutions of small molecules in supercooled ortho-terphenyl, support the theory that cooperative motion is the mechanism by which the CL process occurs. Comparison of data from some

63.polymers and copolymers suggests that a similar process may

7be present in these materials.The a process has rather strict requirements due to

the necessity of simultaneous interaction among a number of• 5molecules. Johan has found that mtermolecular potential

barriers are significant in producing f3 relaxation. He and Goldstein have proposed that the potential barriers necessary for the f3 process arise when a molecule is "en­caged" by other molecules in such a way that the requirements

5 8for cooperative motion are not met. 'Other studies by Johari indicate that the strength of

the (3 process is related to the thermal history of the liquid. As mentioned previously, the faster the rate of cooling, the lower the density; thus, the more sites avail­able in which the potential barriers are low enough to

3 5allow the "encaged" molecules to reorient. 'Since each "encaged" molecule, depending on its orien­

tation, can be presented with a number of different poten­tial barriers, and since the potential barriers may varyfrom one site to another, the j3 process has a distribution

3 5of relaxation times. 'Johari and Goldstein have found from studies of small

rigid molecules that the (3 process is not due to internalhindered rotation of the molecules themselves, but due

3 8entirely to intermolecular forces. 'Experimental results seem to indicate that the rela­

tive occurrence of the CL and (3 processes is temperature

64.dependent. In normal liquids, both processes occur. Asthe supercooled liquid approaches T , the CL processbecomes less frequent, finally ceasing at T , while the /3y3process continues to be present.

65.

NOTES FOR CHAPTER VI

1. Gilroy Harrison, "The Dynamic Properties of Supercooled Liquids," Academic Press, New York, N.Y., 1976.

2. Arnold A. Bondi, "Physical Properties of Molecular Crystals, Liquids, and Glasses," John Wiley & Sons, Inc., New York, N.Y., 1968.

3. Karen Rae Trimmer, Honors Thesis in Chemistry, TheCollege of William and Mary in Virginia, 1975.

4. G.P. Johari, J. Chem. Ed., 51, 23(1974).5. G.P. Johari, J . Chem. Phy s. , _4, 1766 (1973).6. G. Williams and P.J. Hains, Faraday Society of the

Chemical Symposium, 6_, 14 (1972) .7.' G. Williams, M. Cook, and P.J. Hains, Faraday Trans­

actions II, 68 , 1045 (1972).8 . M. Goldstein, Am. Chem. Soc., 30, 117(1970).

CHAPTER VII RF AND Q-METER THEORY

Bridge measurement methods have a practical upper frequency limit imposed by the increase in the admittance associated with stray capacitance. The use of substitution methods using simple resonant circuits can essentially eliminate this source of error."''

A Q-meter is a commercially available instrument which uses a resonant circuit, and can be used to measure induc­tance, capacitance, resistance, and Q of a circuit or component. Our experiment makes use of a Hewlett-Packard type 4342A Q-meter to experimentally determine the complex dielectric constant of an unknown.

Before proceeding with the derivation of the pertinent equations for £ / and £ //, a brief look at some basic AC circuit relations will be helpful.

In an AC circuit which contains inductance, capaci­tance, and resistance, the inductive reactance, X^ = ojIj, and the capacitive reactance, Xc = 1/cuC, act as effective resistances in addition to the pure resistance present.The voltages due to the two reactances are 180° out of phase and each is 9 0° out of phase with the resistive voltage. The overall effective circuit resistance is called the impedance, and is the vector sum of resistance and

66 .

67.reactance.

/ 7.1 / Z 2

where X = XL - X , by convention. Since X^ is zero when

^ = 0 , and increases with increasing frequency; and X^ isa maximum when 0J = 0 and decreases with increasing fre­quency, there is obviously some frequency at which the two

they cancel and only the pure resistance is effectively in the circuit. This condition when XL = X , is known as

and we see that current, and thus the voltage measured across the capacitors, will be a maximum when Z is a mini­mum; i.e., at resonance.

A generalized Q-meter circuit with an empty dielectric

sample holder in place is shown in Figure 7-la.

are equal in magnitude. Since they are 18 0° out of phase,

resonance. We can write Ohm's law as 2

/ 7.2 / L

C E L L

© Rc

Figure 7-1 (a)

CELL

ii

Figure 7-1 (b)— Q-meter main tuning capacitor

Cv - Q-meter vernier tuning capacitor = Capacitance of empty cell

Cs - Capacitance of sampleCL = Capacitance of leadsRq = Series resistance of Q-meter circuitRc = Series resistance of the empty cellRs = Parallel resistance (loss) of the sample

Figure 7-lb shows the same circuit with the sample in placein the cell. With the sample out, the resistances are inseries, and the total circuit resistance, R , is equal toRq + R . Placing the sample in the cell has the effect ofadding a resistance, R , in parallel with R so that thes ctotal circuit resistance is now R^ = Rq + (RSRC)/(RS + Rc). The difference of the reciprocals can be written

69.and if we assume that R „ « Rg/R_ (Rn £2 1.0 milliohms forQ C wthe 4342-A Q-meter), equation 7.3 reduces to

/ 7.4 / Rs + RnR; R c Rs

The measurement theory is quite simple and is based on the equation for the capacitance of a parallel plate capacitor:

the plates, €0 is the permittivity of free space, A is thearea between the plates in appropriate units, and t is theseparation between the plates.

The measurement method is likewise simple and isbasically a substitution procedure using reference liquidsof accurately known dielectric constant. The technique is

3 4based on work done by several authors ' , but is primarily5that of Kakimoto.

Although the determination of is basically an empirical procedure, the equations fore require some derivation.

The actual procedure, which will be described in detail later, requires that Cg be set equal to , so that the total circuit capacitances, and C0 respectively, are also equal. By definition, the dissipation factor of the sample

is

/ 7.5 / C € 'g „ At

where £ is the dielectric constant of the medium between

70.

/ 7.6 / n = J _ = Is Q S w RsCs

Substituting for 1/RS gives

/ 7.7 / n = — (— !— - ! )s Cs v wR| ojR0 'Multiplying the right hand side by CQ/C^ and rearrangingthe terms gives

/ 7 ‘8 / Do = CCsOjRoCo ~ Cl) R; Ci

_ (o»R0C 0 ) (oj Ri Ci ) -

If the appropriate Qn ’s are substituted from equation 7.6, this becomes

/ 7.9 / n -C®. ( Q° Q ‘ 'is Cs V Q. Qs ;

We would like to convert this to an equation in terms of parameters which are measurable by our particular ex­perimental technique. To do this, we make use of the fact that our measurement procedure sets CQ = C^. Remembering that we use a reference liquid for the sample out measure­ment, we can write from equation 7.5, and by referring to Figure 7-lb

/ 7.10 / „ £rpf ACo = c Q + Cv + c L + ref t2.......

where €. , is the dielectric constant of the reference refliquid at the experimental temperature and t^ is the plate

71.separation (in mm.) at resonance. The capacitance of the sample is

/ 7.11 / Cc = €ref€°A ^ g g°As t , t

where t is the dielectric constant of the unknown, and t^ is the plate separation with the unknown in place, (nominally 3.0mm). Using the relations in 7.10 and 7.11, and letting

= Cq + Cv + CL , equation 7.8 becomes

/ 7 . 1 2 / D , = ^ ( C ° ' % y ; e , g ° A ) ( i ^ >

by definition, D = , so that 7.12 can be rewritten as

/ 7 13 / e " = t ( CQ + - £ r e L ) ( Q °~ Q j _ )7 7 € €c A + t2 M QcQi >

This is the working equation which we use for all our Q-meter measurements.

We can derive the equation which is used by Tay and 4Walker by rewriting equation 7.9 as

/ 7.14 / n = -i- ( Q “ Qis Cs Q0 Q

We now need to evaluate 1/Q . The voltage measured across the circuit capacitance is the circuit current times the capacitive reactance. Using equation 7.2, we can write

72.

/ 7.15 / V = - £r 2 +

1/2 C J C

where C is the total circuit capacitance. Squaring both sides, we obtain

/ 7.16 / V aoi R C + (cu LC — I)

which will have a maximum at

/ 7.17 / C. Lr->2 , 2,2R 4 CU L

CQ is just the total circuit capacitance at resonance Substituting this result into equation 7.16 gives

<ff2( R 2 + cj2 L2) R 2

/ 7.18 / Vr

where Vr is the measured voltage at resonance. If we choose V so that the ratio Vr :V is 1:^/2", then we can divide equation 7.18 by equation 7.16 to obtain

/ 7.19 / V 2/V2 = 2R2uj2C2 +(ci/l_C - I f __ 2 . 2,2 R 4- cu L

R

This can be rewritten as a quadratic equation in C

/ 7.20 / o)2(R2 + cu2L2)C2 - 2 u/LC 2 R2 2 2 R + cu LT 4 I

Solving this equation for C gives two solutions:73.

/ 7.21 / C = 2 2 2 ±L + R______R 2 + ou2|_2 “ cu ( R2 + oi2 L2 )

Multiplying the second term by L/L, and substituting from equation 7.17, this becomes

/ 7.22 / C = C„ ± RC<?cuL

By definition, OJL/R = Q, so we can write the two equations for C as

/ 7.23 / C H = Co + Co/Qo

/ 7.24 / C|_ = Co - Co/Qo

where and are merely the capacitances which are higher and lower than the resonance capacitance, C . These equa­tions indicate that the voltage resonance curve is symmet­rical about C , unlike the current resonance curve which broadens slightly for C > C Q . If A c q is defined as CH - C^, the expression for Aco is obtained:

/ 7.25 / AC. = C. + ^ — C. + = - j £ ~

Rearranging this gives the desired equation for 1/Q^

/ 7.26/ ,/Q. =

74.

A c ,o

Substituting this result into equation 7.14 gives the desired equation for the dissipation factor, and since D = as mentioned, we obtain

7 7*27 7 € " = A C ^ ( Qo Q; )2 C s Qj

4the working equation of Tay and Walker.Broadhurst and Bur derive a similar equation

/ 7.2 8 / ^ c'r2t, Cg CgJ

The term in brackets can be left out since this is an air gap correction, and applies only to solid samples. Like­wise, the third term can be ignored since this accounts for

3lead inductance which is negligible below 30MHz. This then leaves us with

7 7'2’ 7 e" = A(A> > A <-ir> = <757 - 4>Their tg corresponds to our t- , and is the total circuit capacitance which is the same whether the sample is in or out; i.e., CT = CQ = C^. Substituting from equation 7.10 and remembering that Cq = Cq 4- Cv + , we obtain

/ 7. 30 / e" = t, ( i Q ..± ^ .g a A / t ^ ) ( U ------1 )€n A Ul u ©

75.which on rearrangement proves to be the same as our working equation, 7.13:

/ 7.31 / £ // = f ( Cq^oA + ^ref \ (Q0- Qj

QaQi-)

We can determine the effect of Q on the uncertainty in £ as given by equation 7.13. It is

/ /

/ 7.32 / A c " _■//

At, + At;t. + A A

A ( Q o - Q i )Q«- Qi

+

A

A Q

+ A(Cot24-€rpf €oA)(Cq tj? 4- € ref a )

+

Q<+ AQ;

Qi

If we arbitrarily assume the following values and uncer­tainties :

II

CQc reft„

0.01114 Omm^ 200pf 2.4718 2. 0mm 1.8 345mm

A A =

A c q = ^^ref =

A t l =A t 2 =

±1.5mm^± 0 . l p f

±0.01 (T = 25°C) ±0.001mm ±0.00141mm ;

the first four terms become approximately 0.00293. The two ranges on the Q-meter which we use are 0-300 and 0-1000; and for both ranges, the Q-analog output is 0 to 1.000 volts The digital voltmeter can be read to ±0.0001V, so the uncertainties in Q for the two ranges are ±0.03 and ±0.1 at best. In practice A q was probably closer to ±0.06 or 0.2 (or even possibly as high as ±1.0). If, in addition to

76.the above assumptions, we arbitrarily choose a value for Qq ,we can back calculate from equation 7.13. A computerprogram was written to calculate and plot the per cent un-

_ / /certainty m 6: for values of Qq between 130 and 1000.This program, along with the resultant graph and a table ofvalues, is given in Appendix VI. For the 0-300 range,

A s - nassuming a of ±0.06, the uncertainty m c (for c =0.01) decreases from approximately 3% at Qq = 130, to 1.1% at Qq = 300. On the 0-1000 range, for A q = ±0.2, the uncertainty decreases from 2.2% at Qq = 300, to .86% at Qq = 1000.

77.

NOTES FOR CHAPTER VII

1. Nora E. Hill, et. al., "Dielectric Properties and Mole­cular Behavior," Van Nostrand Reinhold Co., New York, N.Y., 1969.

2. W.T. Scott, "The Physics of Electricity and Magnetism," John Wiley & Sons, Inc., New York, N.Y., 19 66.

3. Martin G. Broadhurst and Anthony J. Bur, J. Res. NBS 69C (Eng, and Instr.), No. 3, 165(1965).

4. S.P. Tay and S. Walker, J. Chem. Phys., 63, 1634(1975).5. Akira Kakimoto, Rev. Sci. Instr., 43, 763(19 72).

CHAPTER VIIIEXPERIMENTAL

The experimental equipment for measurement of the complex dielectric constant in the RF region consists of a Hewlett-Packard type 4 34 2A Q-meter, a H-P type 5 381A frequency counter, a Data Technology Model 4 0 digital volt­meter to measure the analog output of the Q-meter, the dielectric cell, and associated temperature control and measurement equipment.

The temperature control and measurement apparatus had1 2been previously developed by Trimmer and Wilkes . The

cooling apparatus consists of a 25 liter dewar containing liquid nitrogen, into which a heating element consisting of a two foot length of resistance wire was placed. Two Variacs in series were used for coarse and fine control of the heating element which boiled off nitrogen, thus genera­ting a variable flow of cold nitrogen gas. The gas, after cooling the cell, exhausted into the plastic bag containing the cell, thus giving a moisture free atmosphere. (Conden­sation on the cell can cause serious errors.) Temperature measurement was accomplished with a Yellow Springs Instru­ments type 44201 thermistor, and a voltage-resistance network, built by the William and Mary Physics Department electronics shop, which provided a linear output voltage

78 .

79over the temperature range -50° to +5 0°C.

The cooling apparatus, the Q-meter, cell, and asso­ciated equipment, were connected as shown in Figure 8-1 and in Plates 7 and 8.

ToDVM

Jk

Standard I nductor.

©

rt. angle /adaptor , ^

'© 7 ///////J

To H-P 5381 A Frequency Counter

ThermistorTktemp. probe* K

t m—

GU4342A Q-Meter

frequency range on Q range AQ□ □□□□□□ □ □ □ □ □ □ off

— i~10i~5—Cot 45C 40

Plastic bag

Figure 8-1

The samples were dilute solutions of the compound of interest, in the slightly polar solvent, o-terphenyl. The o-terphenyl, obtained from Eastman, was purified by recrys­tallization from methanol. After recrystallization, excess methanol was removed by treatment in a vacuum oven for six hours. The proper amounts of the solute and solvent were weighed and mixed together by heating to 60°C, giving a homogeneous solution which was then ready to be poured into the cell.

80.Prior to the actual measurements, a number of calibra­

tions were performed.In order to calculate the area of the cell plates, the

capacitance was measured as a function of plate separation on a General Radio type 1615-A capacitance bridge. A plot of C versus l/t (see equation 7.4) should have a slope, £/£0A.As can be seen from Figure 8-2, the function is not linear.

2Because of this, an area of 1140.1mm was used, calculated from the measured plate diameter of 1.5 inches. Inspection of the plot does seem to indicate that the slope approaches a limiting value corresponding to this calculated area.110

100

90

80

70

C(pf)50

40

30

20

10l/t (mm.)

Figure 8-2

81.The true zero point for the micrometer is easily found

by connecting an ohmmeter across the cell terminals and slowly decreasing the plate separation until a short is indicated. This was done prior to each actual measurement and provides a check on the dimensional stability of the cell.

The other correction which must be considered is thevoltage offset and drift of the Q-meter analog output. This

3voltage should be 0.00 ± 0.01V after three hours. This voltage was measured as a function of time with a chart recorder and was seen to decay exponentially with a fairly short time constant. Drift was negligible after a warm up time of three hours. In practice, the value of this volt­age offset was checked before and after each series of measurements, and the measured values of Q adjusted accor­dingly .

The original dielectric cell was based on the design of Broadhurst and Bur^; but because of problems with leak­age and non-linear response, no satisfactory data could be obtained. The cell did, however, allow the validity of the measurement technique to be verified.

A new cell was designed and built by Mr. Stan Hummel of the William and Mary Physics Department shop. The body was machined from solid brass with tubing for thermostatic control built directly into the body of the cell. The electrodes were also brass, and the stationary high poten­tial electrode was insulated from the grounded case by

82.

mounting it in a 3/4 inch thick piece of Teflon. The piston shaped movable electrode was made in two parts which were joined by a threaded stud on one and a tapped hole in the other. A tapered Teflon o-ring was placed in a circum­ferential groove at the junction of the two halves of the electrode. By tightening the two halves, the pressure on the o-ring could be adjusted to ensure a good seal. A L.S. Starrett type 1-46IMP (2 5mm) micrometer head was mounted on the cell so that the electrode separation could be measured to the nearest 0.001mm. The cell is shown in Figure 8-3 and also in Plates 9 and 10.

Cooling outletOne piece brass body

eflon insulatorCooling coi Compression

spring

Micrometerassembly

ovable electrodePlexiglas backing plate High electrode Cooling inlet

Figure 8-3

83.Measurement of £'

Due to the fact that the high electrode cannot bemoved with the supercooled o-terphenyl solution in the cell,

5a slight modification of Kakimoto’s procedure was neces­sary .

The equipment is allowed to warm up for at least three hours. An appropriate reference inductor is plugged into the Q-meter and the capacitance is fixed at 200.Opf. This is an arbitrary value and is chosen so that, in combination with the reference inductor, resonance will occur in a convenient segment of one of the seven fixed frequency ranges. The reference inductors can be purchased separately from Hewlett-Packard. These are similar to inductors which were obtained from the National Bureau of Standards and which had Q values in the range from 150 to 250. In order to achieve better precision, the Q of the circuit should be as high as possible; (see Appendix VI) and, to this end, "Ultra High Q" inductors made by Rutherford Research were obtained. These inductors had Q values near 800 and would resonate with the 200pf capacitance of the Q-meter at frequencies ranging from approximately 150KHz to 1.5MHz.

The cell is connected, through the plastic bag, to the Q-meter terminals. Since these terminals are vertical banana jacks and the cell terminals are horizontal banana plugs, a right angle adaptor was fabricated from two banana plugs, two banana jacks, and short pieces of \ inch copper tubing. Another adaptor was made by embedding a small

84.piece of printed circuit board with banana plugs and jacks attached, in a block of epoxy. This proved to have too much loss and was not used.

Since the capacitance and Q of the circuit are ex­tremely sensitive to the physical orientation of the cell and the adaptor, we constructed a framework in which to rigidly clamp the cell. This framework was fastened to the top of the Q-meter by removing two cover screws and one of the ground connection posts and replacing these with brass screws of sufficient length to go through the framework.The cell could thus be located reproducibly for each run.

With the cell in place, the micrometer was set to 3.000mm (taking into account the offset previously measured). This is an arbitrarily chosen distance and need only be wide enough to allow the hot o-terphenyl solution to bepoured into the cell without trapping any bubbles betweenthe plates. It must also be small enough so that when thecalibration material with the highest dielectric constantis used, the plates can be opened wide enough to reestablish resonance. After placing the thermistor probe in the cell, the cooling apparatus is connected and turned on.

When the temperature reaches approximately -30°C, the hot o-terphenyl is poured into the cell and temperature equilibrium is established. With the temperature stable at the desired value, the frequency of the Q-meter oscillator is adjusted for resonance. The frequency and corresponding value of Q,(Q^), are measured for each reference inductor

85.at each of the measurement temperatures.

For each temperature-solution combination, a calibra­tion curve must be constructed. This is done by measuring a series of reference liquids whose dielectric constants range above and below 2.7, the nominal value for o-terphenyl.

Using the temperature-inductor-frequency combinations determined for each o-terphenyl solution, each reference liquid is placed in the cell and the circuit brought to resonance by adjusting the micrometer. The maximum value of Q, (Q0)/ is noted, as well as the Q-analog output zero offset. Since this Q value is measured at the peak of the voltage versus spacing curve, accurate determination of the spacing may be difficult if the Q is low; and a better value may be obtained by averaging the spacings measured at the "half power points".

Tha values thus obtained (t2 in the relevant equations) are plotted versus the corresponding dielectric constants (calculated from the measurement temperature, the known dielectric constant of the reference liquid, and its temp­erature coefficient). The resulting curves are fitted to a second order equation:

/ 8.1 / at^ + bt2 + c = 0 ,

and the o-terphenyl spacing can then be plugged into this/ ^ / /equation to give € . Equation 7.13 is used to calculate £

86.

NOTES FOR CHAPTER VIII

1. Karen Rae Trimmer, Honors Thesis in Chemistry, TheCollege of William and Mary in Virginia, 19 75.

2. Charles A. Wilkes, Honors Thesis in Chemistry, TheCollege of William and Mary in Virginia, 197 6.

3. Hewlett Packard, Operating and Service Manual - Model4342A Q Meter, Yokogawa-Hewlett-Packard, Ltd., Tokyo,1973.

4. Martin G. Broadhurst and Anthony J. Bus, J. Res. NBS 69C (Eng, and Instr.) , No. 3 , 165 (1965) .

5. Akira Kakimoto, Rev. Sc I. Instr., 43, 763(1972).

CHAPTER IX RESULTS OF O-METER MEASUREMENTS

Measurements of the loss and dielectric constant for a number of nitroaromatic compounds in supercooled o-ter­phenyl were made in an attempt to observe the (3 -relaxation process discussed in Chapter VI.

Of the two processes proposed to be present, the a , or cooperative process is predominant. It has an activa­tion energy on the order of 3 0 to 7 0 kcal mol’"'*", and appears

1-4 nm the low frequency region (ca. lKHz). The p-process,due to hindered rotation of the small, polar solute mole­cules, has an activation energy on the order of 2 to 10 kcal

— 1 3mol , and occurs at a much higher frequency. In addition,the CL -process should cease to occur at or below T , since,yat this temperature, the structure of the solvent becomes frozen in and cooperative motion is no longer possible. The/3-jporocess, however, should continue to occur.

1 2 3 4Data gathered by Trimmer , Wilkes , and others 'indicates that the a -process accounts for 50 to 90% (orperhaps more) of the total loss in the supercooled regionabove T (ca. -10°C; T_ ~ -30°C). y y

As mentioned, the /3-process is thought to have an activation energy near 2 to 10 kcal mol This is

87.

88.

reasonable if one assumes that this relaxation process is similar to the process which occurs in the liquid phase.For example, Howe found an activation energy for nitroben­zene in benzene from 25 to 60°C of 1.8kcal mol"'*'. It would seem reasonable that the potential energy barriers which the encaged solute molecules would encounter would be greater in the supercooled state due to the decreased mo­bility and more ordered structure of the solvent molecules.

Based on the above range of activation energies, we would like to be able to predict the frequency range in which the f3 -process should occur. If we assume that the relaxation process follows first order kinetics, then the rate constant for the process will be proportional to the frequency of maximum loss. Applying the Arrhenius rate theory allows us to write

/ 9.1 / fmax CC k = A<9"A E */RT

where A e * is the activation energy in kcal mol \ R is the— *3 “ 1 O “ 1gas constant equal to 1.987 x 10 kcal mol °K , and T

is the temperature in degrees Kelvin. We also assume that the activation energy is temperature independent, an assump­tion which seems to hold fairly well for normal liquids, but whose validity for the supercooled and glassy states is not certain.

Using equation 9.1, and a nominal relaxation time forr-

nitrobenzene in benzene at 25°C of 10 picoseconds3, we can

89.construct the following table of frequency and temperature for various activation energies.\e* \ t °c 25

298-25248

-60213

-85188

-115158

2 50,612 25,979 13,858 5,014

5 18,234 3,440 715 56

7 105 9,223 894 99 2.8

10 3,321 118 5 32KHz

15 605 4.1 37KHz 18Hz(All frequencies in MHz unless otherwise noted)

Table 9-1The highest frequency attainable with our experimental apparatus was approximately 13 MHz. Inspection of the above table indicates that for the J peak to ke i-n the experimentally accessible range, the activation energy needs to greater than approximately 7.5 kcal mol~^. This is based on a lower temperature limit of -115°C, with the frequency of maximum loss occurring at -100°C.

Initially, the lowest temperature used was -85°C, and while the tail of the (X peak was clearly discernible, the J3 peak was not obvious. The data seemed to show that the loss was beginning to increase, but that it had not yet peaked. We felt that perhaps the J peak. was indeed being shifted into the observable range, but that at the same time, its magnitude was decreasing, thus making detection difficult.

One possible solution to this problem would be to

90.obtain reference inductors which would resonate at higher frequencies. However, since the maximum frequency at which the Q-meter operates is 70 MHz, this solution may not yield much improvement because Table 9-1 indicates that the acti­vation energy still must be greater than approximately 6 kcal mol“ .

If the magnitude of the J3 peak does decrease withdecreasing temperature, measurement of compounds which havea large absolute magnitude of loss (i.e., a large dipolemoment) which is attributable to the ^-process, might allowdetection of the peak. To this end, measurements weremade on 2,2*-dinitrobiphenyl and 2-bromo-2-methyl propane.While the percentage of the total loss due to the y3~Processfor the former compound is approximately the same as for

1nitrobenzene (ca. 20%) * , the magnitude of the total loss is greater. For the latter compound, the magnitude of the total loss is still relatively large, and the percentage due to the yQ“Process is rauch greater (ca. 50+%)^.

The step which proved to be decisive in allowing the definite observation of they3Pea^ was t Le decrease of the experimental lower temperature limit to -115°C. (See the graphs in Appendix IX.)

In thus lowering the experimental temperature, it was found that the thermistor being used was extremely non-linear below -60°C. A calibration curve was constructed, allowing the thermistor to be used to measure temperatures with rea­sonable accuracy to -90°C. For the measurements at -115°C, a chrome1-alumel thermocouple was used, the reference junction

91.at 0°C. While the accuracy of the thermocouple was not as good as that of the thermistor, the effect on the results was not appreciable.

Although the measurement technique does not allow the precise determination of exact values for the dielectric constant and loss, the relative values are readily measured. This is sufficient to allow observation of the peak in the low temperature-high frequency region. The trends seen are as expected from theoretical predictions. At the lower frequencies (ca. 150 KHz); the peak occurs below -115°C; and as the frequency is increased to 13 MHz, the peak shifts up in temperature to near -85°C. The presence of the high frequency-low temperature tail of the OC peak is also clearly evident. Although the peak cannot be located precisely from the available data, if we assume that it occurs at -85°C and 13 MHz, this is found to correspond to an activation energy of 9 kcal mol™^. This is only a rough estimate, but it is obviously in the range of 2 to 10 kcal mol~^ previously quoted for the /^peak.

92.

NOTES FOR CHAPTER IX

1. Karen Rae Trimmer, Honors Thesis in Chemistry, The Col­lege of William and Mary in Virginia, 1975.

2. Charles A. Wilkes, Honors Thesis in Chemistry, The Col­lege of William and Mary in Virginia, 19 76.

3. G.P. Johari,4. G. Williams and P.J. Hains, Faraday Society of the

Chemical Symposium, 6_, 14 (1972) .5. Allen K. Howe, Jr., Honors Thesis in Chemistry, The Col­

lege of William and Mary in Virginia, 1974.

PLATE I Admittance Meter - Front

PLATE 2 Admittance Meter - Rear

_ . - PLATE 3

UHF Cell - Assembled

PLATE 4UHF Cell - Disassembled

tr

P L A T E 5Interconnection of UHF Components

f; |V*J*s- Iv* \ i

jm *k;*ir^-

i iir#%$V \f?pu |» j

}■ \ -v • •% j

.,v- . ..3F4*V S' W

s r : ' ••;W * -n;: . i • ••• • •#' - .

- O-X •»,:..fS§f * f

ejf J-. . &:*'■'

9 ~ z__JL^Esr— ^ .

;i s?

f m

<v.',;;*rtr. 1tr ?&*:■&% JV•■a :5’it*..-3»t I i i «* & U 't v" *•■ * |5 3 «T\-|| jj $g*.

‘■W' ,’Z-. :: Vv*:..

>y O .

o

T .1 « l

o

f ■

PLATE 6 UHF Signal Generator and IF Amplifier

r - ■ • '» f1"'-' ;L ’' * .i(X.«‘

P L A T E 7 Interconnection of RF Components

PLATE 8Interconnection of RF Components - Detail

r

PLAT E 9RF Cell — Assembled

PLATE 10RF Cell — Disassembled

APPENDIX I

COMPUTER PROGRAM TO CALCULATE Cn , b2-a2, 2ab, AND TANS

93.

94.

V PR 0 C COr i n c f + 1 3 5r 2 i K + 1r 33 SOP+ 1 0 1 3 pO04 ] AB2 + 10 1 3 pOr s i TA F + 10 1 3 p Or e ] AF- + 1 0 1 2 oO0 7 ] CF 0 + 4 0 4 pOr e ] c f l -*- 10 1 4 p 00 9 ] CF2 + 80 4 pOr i o ] CR+ 0 l x i 1 0 1m i LOOP:K+-K + 1r 12 ] C+cor i 3 ] =2 ) / D O U P L Fr i ' H T F C + 0 . 1r i s ] + S K I Pr i 6 ] DOU E L F : T F C + 0 . 2r 17 ] S K I P :c7+0r i s ] I I :A + 0r i 9 ] B + o C E 0 Klr 20 ] PA I F A , Er 2 i i APS Jr 22 ] SORT e/; 5 ]-*-(./ 5 0 5 ; 1 ] *2 ) - ( / 5 r 5 ; 0 ] *2 )r 23 ] A B 2 l J \ K " \ + 2 ' x ( A B [ . J \ 0 ' \ ) x ( A P S J ; l l )r 24 ] TAP 0 J i K l H A32T J i K l i S Q P T J i K l )r 25 ] C + C + I F Cr 2 6 ] -*■ ( 100 >J+tJ + 1 ) / I Tr 27 ] M X <2 ) / L O O P0 2 8 ] CF 0 0 O ] - ( -40 + (750 2 0 ] CFO 0 1 ] «-4 0 + 5 0 7 0 ; O ]r 3 o ] r r o O 2 ]«-4 0 AAR 2 0 ; O ]0 3 1 ] 550 o 3 ] +>4 O + TA F r ; O ]0 3 2 ] OF 1 [ o l + c s0 3 3 ] CF 10 1 1 + s o p t ; i ]r 341 CF 10 2 1 + A P 2T ; 1 ]0 3 5 ] CF 10 3 ] +TA FT ; 1 ]0 36 ] 5 5 2 0 0 1 +0 . 2 x i 8 00 3 7 ] CF 2 0 1 ] +-8 0 A SOP 0 ; 2 ]0 38 ] CF 2 0 2 ] -*-8 0 + / F 2 0 ; 2 ]0 39 ] CF 2 o 3 ] +8 O + TA F 0 ; 2 ]0 40 ] t mvp 7? t » pp 7* p p t t rpn pp t F T 0 I IP T A P I R S

95.

V MY I F Xri] i+- or 2 ] Y*-X[ 3 ] L 1 : Z + - ( F N Y ) W F P Yf 4 ] -+L 2 x \ ( 50<_r -< -J+l )r 5 1 ( C r / I .3 ) <0 . 00 0 0 1 )[6] y^y+z[ 7 ] + L 1[ 8 ] L 2 : ->0 , p r^- ' DTD POT COP VFRGF ’

V

7 R + F N P ; / ; Pr i] /«-yro]r 2] b + xri]r 3] P+(/x5o/)+(Pxio^)[ 4 ] R+-P , ( ( (P x 50/1 )-(/xl o y ) ) v( (6 0 / ) - 2 Or ) ) - r

V

V P+-FP J ; / ; P ; F F F l ; F U F 2 ; FFP'3 ; F£/ / '4r i] /.«-*[ o]r 2 ] f + x t i ][ 3 ] P+- 2 2 p 0[ 4 ] P [ 0 ; 0 ] « - ( / x 6 o / ) + 5 0 / r 5 ] /?f 0 ; l ] « - ( F *2oZ? ) + 1 0 5f 6 ] NUMl+( ( ( 6 0 / ) - 2 OP ) x ( (P X 6 0 / ) - 1 0 5 ) )[ 7 ] N U M 2 + ( ( 5 x 5 0 / ) - / x l o F ) x 5 0 /r R1 RUM 3 ( ( (6 0/) -2oP)x(( 5 0/ ) -/x2C5) )r 9 ] FUFP+( ( B X 5 0 / ) - / X 1 0 5 ) X 10?no] PF7>-( ( 6 0 / ) -2 05 ) *2r i l l p r 1 ; 0 ] -*-( F U F 1 - F I J I *2 ) * 5 FF[ 1 2 ] P [ 1 ; 1 ] ^ ( P UF 3 - F IF i P PA’

V

96.

v? ~PR T / ” T7 * , ' r

r 1] ' w r t o r c r r. ] d o yo ur 2] x + rr 3 ] M K =1 ) / K 0m + ( k = 3 ) / x ir 5 ] X+-2[ 6 ] n 7j rp~p {j m r j 1 9r 7] - i .^yp fr e ] X0 : K*-0i—iCoi_

0 UTPUT+-CN 0r 10 ] -+TYPFr 11] V1 ; p'^-1[ 12] r\ 7’ rr> 7T 7 1 rn . r* ?■> -1I.' ... r U J. •*- L. L. Ir i 3 ] <vypr . r r: p r? rr . i -j_ 2 1 5L 1 ]

t tr i s ] ? r

r i 6 ] t t

t—^ i_i C7 T 7 A p T T 1 T7 r7 r p p ) 1 m p £ j m

r?

APPENDIX II

TABLE OF Cn (11=1 ,2 ,3), b2-a2 , 2ab, AND TAN B

97.

£i0 0

1 020

3 0

'i-Oc n

5 0

70

8 0o 7 'j00

102 03 0

7050

6 070

3 0

Q ft

98

b - a9.80050

9.87360

9.88587

9.90607

9 ,9 3 9 4 2 n n n n o r

10 1 0 1 0 10

1 010.3 5 84

1 0 Li t; f) n

0 15 3 7

0 6 7 9 3

1 2 8 54

10 7 17

2 7 38 1

1 0 10

1 0 10 11 11 11

5 514 1

6 5 9 7 4

7 7 59 1

g q q g p

0 3 15 1

17100

3 13 0 7

2 a b

.00 0

4 0 0

80 0

1 . 19 9

1 . 5 9 31 o nr

2 . 3 9 4

2.790

3 . 18 5

3 . 5 7 9

3.971

4 .351

4.749

5 . 13 6

5 . 5 2 0

5.901

6 . 2 3 0

6 .656 7.029

7 3 9 9

t a n S0 0 0 0 0 0 40 51

0 80 90

12 10 6 16 0 8 7o n

2 3 9 00

2 77 1 2

3 14 46

3 50 9 5

3 8 6 5 0

6 9 10 24 5 4 ( l 5

4 8 6 7 3

5 1781

5 4 7 6 3

5 76 16

6 0 3 3 7

6 2 9 2 2

6 5 3 7 2

99.

£ i b 2 - a 2 2 a b t a n 8

2 . 00 1 1 . 4 7 2 7 4 7 . 7 6 5 . 6 7 6 8 4

2 . 10 1 1 , 6 349 5 8 . 1 2 8 . 6 9 8 59

2 . 20 1 1 . 8 0 4 6 3 8 . 4 8 7 . 7 1 89 7

2 . 30 1 1 . 9 8 1 7 1 8 , 8 4 2 . 7 3 7 9 8

2 . 40 1 2 . 1 6 6 1 2 9 , 1 9 3 . 7 556 5

2 . 5 0 1 2 . 3 5 779 9 , 5 4 0 . 7 7 198

2 . 0 0 12 . 5 5 6 6 2 9 , 8 8 2 . 7 8 7 0 1

2 . 70 1 2 . 7 6 2 54 1 0 . 2 2 0 . 8 0 0 76

2 . 80 12 . 9 7 54 5 1 0 . 5 5 2 . 8 1 3 2 5

2 . 9 0 13 . 19 52 3 1 0 . 8 80 . 8 2 4 5 3

3 . 00 1 3 . 4 2 1 8 4 1 1 . 2 0 2 . 8 346 2

3 . 10 1 3 . 6 5 5 1 1 1 1 . 5 19 . 8 4 3 5 6

3 . 20 1 3 . 89 49 3 1 1 . 8 30 . 8 5 1 4 0

3 . 30 1 4 . 1 4 1 2 0 12 . 136 . 8 58 17

3 . 4 0 1 4 . 3 9 3 7 7 12 , 4 35 . 86 390

3 . 50 14 . 6 52 51 1 2 . 7 2 8 . 8 6 8 6 5

3 . 6 0 1 4 . 9 1 7 2 8 1 3 . 0 1 5 . 8 7 2 4 6

3 . 70 1 5 . 1 8 7 9 2 1 3 . 2 9 5 . 8 7 5 3 5

3 . 80 1 5 . 4 6 4 2 7 1 3 , 5 6 8 . 8 7 7 3 9

3 . 9 0 1 5 . 7 4 6 15 1 3 . 8 35 . 8 7 8 6 0

100.

Co b - a- 3

. 0 0 8 8 . 8 26 44

. 10 8 8 . 8 26 89

. 20 8 8 , 8 2 8 2 4

. 30 8 8 . 8 30 50

. 40 8 8 . 8 336 7

. 50 8 8 . 8 3 7 7 6

. 6 0 8 8 . 8 42 77

. 70 8 8 . 8 4 8 7 3

. 80 8 8 , 8 5 5 6 5

. 90 8 8 . 8 6 3 54

1 . 0 0 88 , 8 7 2 4 3

1 . 10 8 8 . 8 8 2 3 3

1 . 20 8 8 . 8 9 3 2 8

1 . 3 0 8 8 . 9 0 529

1 . 40 8 8 . 9 18 41

1. 50 8 8 , 9 3 2 4 2

l . f i O 8 8 . 9 4 7 7 7

1 . 70 8 8 . 9 6 4 3 2

1 . 8 0 8 8 . 9 8 2 10

1 . 9 0 8 9 . 0 0 1 16

2ab tan 6

. 00 0 . 0 0 0 0 0

. 40 0 . 0 0 4 50

. 80 0 . 0 0 9 0 1

1 . 2 0 0 . 0 13 51

1 . 6 0 1 . 0 1 8 02

2 . 0 0 2 . 0 22 53

2 , 4 0 3 . 0 2 70 5

2 , 8 0 4 . 0 31 56

3 . 2 0 7 . 0 3 6 0 9

3 . 6 1 0 . 0 40 6 2

4 , 0 1 3 . 04 5 16

4 . 4 1 7 . 0 4 9 7 0

4 . 8 2 3 . 0 5 42 5

5 . 2 29 . 0 5 8 8 1

5 . 6 3 6 . 0 6 338

6 . 0 4 4 . 0 6 796

6 . 4 5 4 . 0 7 2 5 5

6 . 8 6 4 . 0 77 16

7 . 2 7 6 . 0 8 1 7 7

7 . 6 9 0 . 0 8 640

101.

c 3 2 a b t a n 8

2 . 00 8 9 . 0 2 156 8 . 1 0 5 . 09 10 4

2 . 1 0 8 9 . 0 4 3 3 2 8 . 5 2 1 . 0 9 570

2 . 20 8 9 . 0 6 6 52 8 . 9 40 . 1 00 37

2 . 30 89 . 09 1 2 2 9 . 360 . 10 506

2 . 40 8 9 . 1 1 7 4 6 9 . 7 8 2 . 10 9 76

2 . 5 0 89 . 14 532 1 0 . 2 0 6 . 1 1 4 4 8

2 . 5 0 89 . 1 7 4 8 7 1 0 . 6 32 . 1 1 9 22

2 . 70 89 . 206 18 1 1 . 0 6 0 . 12 398

2 . 80 89 . 2 39 34 1 1 . 4 9 0 . 1 28 75

2 . 90 8 9 . 2 7 4 4 2 1 1 . 9 2 2 . 1 3 3 55

3 . 00 8 9 . 3 1 1 5 2 1 2 . 3 5 7 . 1 3 8 3 6

3 . 10 8 9 . 3 5 0 7 3 1 2 . 7 9 5 . 1 4 320

3 . 20 8 9 . 3 9 2 1 5 1 3 . 2 35 . 14 80 5

3. 30 8 9 . 4 3 5 8 8 1 3 . 6 7 7 . 1 5 2 9 3

3 . 40 89 . 48 20 5 1 4 , 1 2 2 . 1 5 7 8 2

3 . 50 8 9 . 5 3 0 7 6 14 . 5 70 . 1 6 2 74

3 . 6 0 8 9 . 5 8 2 1 5 1 5 . 0 2 1 . 16 76 8

3 . 70 8 9 . 6 3 6 3 5 1 5 . 4 7 5 . 1 7 2 6 4

3 . 80 8 9 . 6 9 349 1 5 , 9 32 . 17 76 3

3 . 3 0 8 9 . 7 5 3 7 2 1 6 . 3 9 2 . 1 8 26 3

102.

— 3<4.00

b2 - a28 9 . 8 1 7 1 9

4 . 10 8 9 . 8 8 4 0 8

4 . 2 0 8 9 . 9 54 54

4 . 3 0 9 0 . 0 28 76

4 . 40 9 0 . 1 0 6 9 2

4 . 5 0 9 0 . 1 8 9 2 2

4 . 6 0 9 0 . 2 7 5 8 7

4 . 70 9 0 . 36 70 7

4 . 8 0 9 0 . 4 6 3 0 5

4 . 9 0 90 . 5 6 4 0 3

5 . 00 9 0 . 6 70 49

5 . 1 0 9 0 . 78 229

5 . 20 9 0 . 8 9 9 8 5

5 . 30 9 1 . 0 2 3 4 5

5 . 40 9 1 . 1 5 336

5 . 50 9 1 . 2 8 9 8 5

5 . 6 0 9 1 . 4 3 3 2 3

5 . 70 9 1 . 5 83 78

5 . 8 0 9 1 , 7 4 1 8 1

5 . 9 0 9 1 . 9 0 7 6 1

2 a b t a n 8

16 . 8 55 . 1 8 7 6 6

1 7 . 3 21 . 1 92 71

1 7 . 79 1 . 1 9 7 7 8

18 . 2 6 4 . 2 0 2 87

1 8 . 7 4 1 . 2 0 7 9 8

19 . 22 1 . 2 1 3 12

19 . 70 4 . 2 18 27

20 . 19 1 . 22 3 4 4

20 . 68 2 . 2 2 8 6 2

2 1 . 1 76 . 2 3 3 8 3

2 1 . 6 74 . 239 04

2 2 . 1 7 6 . 2 4 4 2 7

2 2 . 68 1 . 2 49 51

2 3 . 19 0 . 2 5 4 7 6

2 3 . 70 2 . 2 6 0 0 2

2 4 . 2 18 . 2 6 528

2 4 . 7 3 7 . 2 7 0 55

2 5 . 2 59 . 2 7 5 8 1

2 5 . 78 5 . 2 8 1 0 6

26 . 3 1 4 . 2 8 6 31

103.

C3 b 2 - a 2

6 . 00 9 2 . 0 8 1 4 9

6 . 10 9 2 . 26 3 77

6 . 20 9 2 . 4 5 4 7 4

6 . 30 9 2 . 6 5 4 7 2

6 . 40 92 . 86 40 1

6 . 5 0 9 3 . 0 8 2 9 0

6 . 60 9 3. 3 1 1 6 8

6 . 70 9 3. 5 5 0 6 4

6 . 30 9 3 . 8 0 0 0 5

6 . 9 0 9 4 . 0 6 0 1 7

7. 00 9 4 . 3 3 1 2 3

7 . 10 9 4 . 6 1 3 4 8

7 . 20 9 4 . 9 0 7 1 1

7. 30 9 5 . 2 1 2 3 2

7 . 40 9 5. 5 2 9 2 7

7 . 5 0 9 5 . 8 5 8 1 1

7 . 6 0 9 6 . 198 96

7 . 70 96 . 5 5 19 1

7 . 80 9 6 . 9 1 7 0 3

7 . 9 0 9 7 . 2 9 4 3 7

2 a b t a n v

26 . 8 46 . 2 9 1 55

2 7 , 3 8 1 . 296 77

2 7 . 9 1 9 . 3 0 1 9 7

2 8 . 4 59 . 30 7 15

2 9 . 0 0 1 . 3 1 2 2 9

2 9 . 5 4 5 . 3 1 7 4 1

3 0 . 0 9 1 . 3 2 2 4 8

30 ..6 39 . 32 7 5 1

3 1 . 1 8 7 . 3 3 2 4 8

3 1 . 7 3 6 . 3 3 7 4 1

3 2 . 2 8 6 . 3 4 2 26

3 2 . 8 36 . 34 706

3 3 . 3 8 6 . 3 5 1 7 7

3 3 . 9 3 5 . 3 5 6 4 1

3 4 . 4 8 2 . 36 0 96

3 5 . 0 2 8 . 36 542

3 5 . 5 7 3 . 36 9 78

36 , 1 1 4 . 3 7 4 0 4

36 , 6 53 . 3 7 8 1 9

3 7 . 1 8 8 . 3 8 2 22

104.

~3 b - a 2ab tan 88 . 0 0 9 7 , 6 8 3 9 4 3 7 . 7 19 . 3 8 6 13

8 . 10 9 8 . 0 8 5 7 3 3 8 . 2 4 5 . 3 8 9 9 2

8 . 20 98 . 49 9 70 3 8 . 7 6 7 . 39 3 58

8 . 30 98 . 9 2 5 80 3 9 . 2 8 3 . 3 9 7 10

8 . 4 0 9 9 . 3 6 39 3 3 9 , 7 9 4 . 4 0 0 49

8 . 50 9 9 . 8 139 7 4 0 , 2 9 8 . 4 0 3 73

8 . 6 0 1 00 . 2 7 5 79 4 0 . 7 9 5 . 4 0 6 8 3

8 . 70 1 0 0 . 749 22 4 1. 28 5 . 4 0 9 78

8 . 8 0 1 0 1 . 2 3 4 0 8 4 1 . 7 6 7 . 4 1 2 58

8 . 9 0 1 0 1 . 7 3 0 1 6 4 2 . 2 4 1 . 4 1 5 2 2

9 . 0 0 1 0 2 . 2 3 7 2 3 4 2 . 7 0 6 . 4 1 7 71

9 . 1 0 1 0 2 . 7 5 505 4 3. 16 2 . 4 2 0 0 5

9 . 20 1 0 3 . 2 8 3 3 5 4 3 . 6 0 9 . 4 2 2 2 2

9 . 3 0 10 3 . 8 2 1 8 5 4 4 . 0 4 5 . 4 2 4 2 4

9 . 40 1 0 4 . 3 7 0 2 5 4 4 . 4 7 2 . 4 2 6 09

9 . 50 10 4 . 9 2 8 2 3 4 4 . 8 8 7 . 4 2 7 79

9 . 6 0 10 5 . 4 9 546 4 5 . 2 9 2 . 4 2 9 33

9 . 70 1 06 . 0 7 1 6 0 4 5 . 6 8 6 . 4 30 71

9 . 8 0 1 06 . 6 56 2 8 4 6 . 0 5 7 . 4 3 1 9 2

9 . 9 0 1 0 7 . 2 4 9 4 4 4 6 . 4 3 7 . 4 3 2 9 8

10 . 00 10 7 . 8 5 0 2 2 4 6 . 7 9 5 . 4 3 3 8 9

105 .

-5 b2 - a2.00 246.74011.20 246.74076.40 246.74271.60 246.74596.80 246.75054

1.00 246.756451 .20 246.763721. 40 246.772381.60 246.782461.80 246.794002.00 246.807042.20 246.821642.40 246.837852 .60 246.855732 .80 2 4 6.875363 .00 246.896663.20 246.919993.40 246.945303 .60 246.972723.80 247.00234

2ab tan S.000 .00000.800 .00324

1. 600 .006492 . 401 .009733 .203 .012984.005 .016234.809 .019495 .614 .022756.421 .026027.230 .029308.041 .032588.855 .035889.672 .03918

10.492 .0425011.315 .0458312.141 .0491812.972 .0525313.807 .0559114.646 .0593015.490 .06271

106.

-5 k 2 _ 2 d - a 2ab tan S4.00 247.03430 16.339 .066144.20 247.06873 17.194 .069594.40 247.10577 18.054 .073064.60 247.14558 18 . 921 .076564.80 247.18833 19.794 .080085 . 00 247.23421 20.674 .083625.20 247.28343 21.561 .087195 .40 247.33620 22.456 .090795 .60 247.39277 23.358 .094425 . 80 247.45340 24.269 .098086.00 247.51838 25.189 .101776.20 247.58800 26.118 . 105496. 40 247.66262 27.057 .109256 .60 247.74260 28.006 .113046.80 247.82834 28.965 .116877 . 00 247.92027 29.935 .120757 .20 248.01888 30.917 .124667.40 248.12467 31.911 .128617.60 248.23820 32.917 .132607.80 248.36011 33.937 .13664

107.

—5 b2 - a2 2ab tan S8.00 248.49104 34.970 .140738.20 248.63175 36.017 .144868.40 248.78301 37.079 .149048 . 60 248.94570 38.156 .153278.80 249.12076 39.249 .157559.00 249.30923 40.358 .161889.20 249.51222 41 .483 .166269.40 249.73095 42.626 .170699.60 249.96674 43.787 .175179.80 250.22103 44.965 .17970

10.00 250.49536 46.162 .1842810.20 250 .7 9141 47.378 .1889110.40 251.11098 48.613 .1935910 . 60 251.45599 49.866 .1983110.80 251.82895 51.139 .2030711 . 00 252.23132 52.430 .2078611.20 252.66576 53.740 .2126911.40 253.13470 55.067 .2175411.60 253.64070 56.412 .2224111.80 254.18638 57.772 .22728

108 .

—5 b2 - a2 2ab tan S12.00 254.77444 59.147 .2321612.20 255.40759 60.536 .2370212.40 256.08855 61.935 .2418512.60 256.81996 63.342 .2466412.80 257.60436 64.756 .2513813.00 258.44415 66.173 .2560413.20 259.34147 67.589 .2606213.40 260.29821 69.000 .2650813.60 261.31592 70.403 .2694213.80 262.39576 71.794 .2736114.00 263.53845 73.167 .2776314.20 264.74456 74.517 .2814714.40 266.01348 75.841 .2851014.60 267.34468 77.133 .2885114. 80 268.73700 78.388 .2916915.00 270.18879 79.602 .2946215.20 271.69795 80.770 .2972815.40 273.26194 81.887 .2996715.60 274.87786 82.950 . 3017715.80 276.54243 83.955 . 30359

TABLE OF CONTENTS

Solute Freq (MHz) T (K) Plot Page500. . 25. . t VS Cone. • 116500. . 25. . . G " VS Cone. • • 117700. . 25. . . g ' VS Cone. • 118

Nitrobenzene 700. . 25. . . g " VS Cone. • 119500. . 60. . . g 1 VS Cone. . • 120500. . 60. . . € " VS Cone. • * 121700. . 60. . . G 1 vs Cone. * • 122700. . 60. . . e " vs Cone. • • 1235 00. . 25. . . g 1 vs Cone. • • 124500. . 25. . . G " vs Cone. • • 125700. . 25. . . e ' vs Cone. • 126

o-Nitrobiphenyl 700. . 25. . . g " vs Cone. • • 127500. . 60. . . e 1 vs Cone. • • 128500. . 60. . . c " vs Cone. • • 129700. . 60. . . G 1 vs Cone. • 130700. . 60. . . € " vs Cone. • • 131500. . 25. . . G 1 vs Cone. 132500. . 25. . . € " vs Cone. • • 133700. . 25. . . g 1 vs Cone. • • 134

p-Nitrobiphenyl 700. . 25. . . g " vs Cone. • • 135500. . 60. . . G * vs Cone. • • 136500. . 60. . . g " vs Cone. • • 137700. . 60. . . G 1 vs Cone. • • 138700. . 60. . . G " vs Cone. • • 139500. . 25. . . g 1 vs Cone. • • 140500. . 25. . . € " vs Cone. * • 141700. . 25. . . 6 ' vs Cone. « • 142

2,2'-Dinitrobiphenyl 700. . 25. . . € " vs Cone. • • 143500. . 60. . . c 1 vs Cone. • • 144500. . 60. . . e " vs Cone. • • 145700. . 60. . . G 1 vs Cone. • • 146700. . 60. . . G " vs Cone. • • 147

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APPENDIX III

COMPONENTS OF UHF MEASUREMENT SYSTEM

109 .

1 1 0.

Signal Generator:

General Radio Type 1264-B Modulating Power Supply and Type 1362 UHF Oscillator (220-920MHz)

Detector:General Radio Type 1362 UHF Oscillator General Radio Type 1236 IF Amplifier General Radio Type 8 7 4-MRAL Mixer

Measuring Equipment:General Radio Type 1602-B UHF Admittance Meter General Radio Type 1602-P4 50 Ohm Termination Gaertner Scientific 0.001 cm. Cathetometer

Connecting Equipment:General Radio TypeGeneral Radio TypeGeneral Radio TypeGeneral Radio TypeGeneral Radio TypeGeneral Radio TypeLine

874-R22LA Patch Cords ( 2 ea.) 874-F1000L Low Pass Filter 874-G10L lOdb Attenuator Pad 8 74-WOL Open Circuit Termination 874-ELL 90° Elbow874-LK20L 20 cm. Constant Impedance

UHF Cell:*General Radio Type 874-D50L 50 cm. Adjustable Stub *General Radio Type 8 7 4-QBPAL BNC Jack to Type 8 74 Adapter

*These parts were used in the construction of the cell as described on pages 47 and 48.

1 1 0.

APPENDIX IV

COMPUTER PROGRAM TO FIND BEST FIT FOR ALL UHF DATA

111.

112.

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APPENDIX V

LEAST SQUARES PLOTS OF £ ’ VS CONCENTRATION

AND £ "

114 .

TABLE OF CONTENTS

Solute Freq(MHz) T (K) Plot Page500. . 25. . .e i VS Cone. . 116500. . 25. . . € VS Cone. . • • 117700. . 25. • . e VS Cone. . 118

Nitrobenzene 700. . 75. . . € VS Cone. . « • 119500. . 60. . . € VS Cone. . • • 120500. . 60. . . € VS Cone. . • • 121700. . 60. . . e vs Cone. . • • 122700. . 60. . . € vs Cone. . . • 123500. . 25. . . € vs Cone. . 124500. . 25. . . € vs Cone. . m , 125700. . 25. . . € vs Cone. . 126

o-Nitrobiphenyl 700. . 25. . . € vs Cone. . • . 127500. . 60. . . € vs Cone. . * • 128500. . 60. . . € vs Cone. . , • 129700. . 60. . e vs Cone. . , , 130700. . 60. . . € vs Cone. . • • 131500. . 25. . . € vs Cone. . 132500. . 25. . . € vs Cone. . « * 133700. . 25. . . € vs Cone. . 134

p-Nitrobiphenyl 700. . 25. . . e vs Cone. . • • 135500. . 60. . . € vs Cone. . • • 136500. . 60. . . € vs Cone. . • • 137700. . 60. . . € vs Cone. . • • 138700. . 60. . . € vs Cone. . « • 139500. . 25. . . € vs Cone. . • # 140500. . 25. . . 6 vs Cone. . • • 141700. . 25. . . € vs Cone. . • « 142

2, 2 1-Dinitrobiphenyl 700. . 25. . . € vs Cone. . • 143500. . 60. . . 6 vs Cone. . 144500. . 60. . . e vs Cone. . m * 145700. . 60. . . € vs Cone. . • • 146700. . 60. . . € vs Cone. . . * 147

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APPENDIX VI

RELAXATION TIMES AND SMYTH PARAMETERS FOR UHF DATA

148.

TABLE OF CONTENTSPage

Nitrobenzene at 2 5 ° C ...................................... 150Nitrobenzene at 6 0 ° C ...................................... 151o-Nitrobiphenyl at 25°C.................................... 152o-Nitrobiphenyl at 60°C.................................... 153p-Nitrobiphenyl at 25°C.................................... 154p-Nitrobiphenyl at 60°C.................................... 1552 , 2 ' -Dinitrobiphenyl at 2 5 ° C ..............................1562,2'-Dinitrobiphenyl at 6 0 ° C ..............................157

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(picoseconds)

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of the

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of the

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s

157.

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CO lo CO CN oo rH <j* CO • • • *

r—I CNJ I I

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1 CO CNi 1— t LO CO rH • * • *

CM CO <t CO

co co co coCO VO CO CM * * # *

CO <J- 1— I CNJ i—! i—I

co cr> vo co r-' r- co vo • • • •I CNl '—I

O CNJ LO CNl CO vO 00 CO • • • •CNi CNJ C7\ I - CO CO

r-'» CO r-i voo c o N S j- • • * *

CNI C7\ <— ! CO CO CNJ '—1

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APPENDIX VII

COMPUTER PROGRAM, TABLE OF VALUES AND PLOT OF UNCERTAINTY IN € " VS. Q

158.

159.

1— I i— i —H CN Hi_i •*> O ’

1_iO' I--J•I- OOo O— ' fe:H-/—- i— i1— 1 CN OK - o?1_1 i_io oOf•I* Ofo oo+■ i— is—\ CN -s—\ • * W1— 1 1_1 -H P -i_: Os UJN POi O -i— i O -H p —t__i ^ VJO (— i <3o ▼H•*

1_iin Of«o ri o H* P O''■'» o CN rH 1—1

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P in O in rHOs zi- P —j P rHP CO co • ID CU 1_1P CO ID ID P O P.« r-> CO o C y— \ P rH 0sH X CN p c o P Cl P•« c o • o O CO P —

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1—1i— i1—11—11— ti— i1—11—1i— ii— iI— i1—1i— i1-11—11— I1-1 o rH CN CO Zi" LO (DrH CN CO Zf in ID r- CO (J) rH rH rH rH rH rH rH rH1_il__iU_i 1_1i__il_1 i— ii_i1_1I_i1_11_11_11_11_1i__;1_;

160.

A e T 1/ € ' ’ (%) QO Q I A e " / e " (%) QO Q I

2 . 9 5 1 3 0 1 2 6 . 1 1 . 1 6 5 7 0 5 0 2 . 32 . 64 1 4 0 1 3 5 . 5 1 . 1 4 5 8 0 5 1 0 . 12 . 4 0 1 5 0 1 4 4 . 9 1 . 1 3 5 9 0 5 1 7 . 82 . 2 0 16 0 1 5 4 . 2 1 . 1 2 6 0 0 5 2 5 . 52 . 0 3 1 7 0 1 6 3 . 4 1 . 1 0 6 1 0 5 3 3 . 11 . 8 9 1 8 0 1 7 2 . 7 1 . 0 9 6 2 0 5 4 0 . 81 . 7 7 1 9 0 1 8 1 . 8 1 . 0 8 6 3 0 5 48 . 41 . 6 7 2 0 0 1 9 1 . 0 1 . 0 7 6 4 0 5 5 5 . 91 . 5 8 2 1 0 2 0 0 . 1 1 . 0 6 ' 6 5 0 5 6 3 . 41 . 5 0 2 2 0 2 0 9 . 1 1 . 0 5 6 6 0 5 7 0 . 91 . 4 3 2 3 0 2 1 8 . 1 1 . 0 4 6 7 0 5 7 8 . 41 . 37 2 4 0 2 2 7 . 1 1 . 0 3 6 8 0 5 85 . 81 . 3 2 2 5 0 2 3 6 . 1 1 . 0 2 6 9 0 5 9 3 . 31 . 2 8 2 6 0 2 4 4 . 9 1 . 01 7 0 0 6 0 0 . 61 . 2 3 2 7 0 2 5 3 . 8 1 . 0 0 7 1 0 6 0 8 . 01 . 2 0 2 8 0 2 6 2 . 6 . 9 9 7 2 0 6 1 5 . 31 . 1 6 2 90 2 7 1 . 4 . 9 9 7 3 0 6 2 2 . 62 . 2 3 3 0 0 2 8 0 . 1 . 9 8 7 4 0 6 2 9 . 82 . 1 3 3 1 0 2 8 8 . 8 . 9 7 7 5 0 6 3 7 . 12 . 0 5 3 2 0 2 9 7 . 5 . 9 7 7 6 0 6 4 4 . 31 . 9 8 3 30 3 0 6 . 1 . 9 6 7 7 0 6 5 1 . 41 . 9 1 34 0 3 1 4 . 7 . 9 5 7 8 0 6 5 8 . 61 . 8 4 35 0 3 2 3 . 3 . 9 5 7 9 0 6 6 5 . 71 . 7 8 3 6 0 3 3 1 . 8 . 94 8 0 0 6 7 2 . 81 . 7 3 3 7 0 3 4 0 . 2 . 9 4 8 1 0 6 7 9 . 81 . 6 8 3 8 0 3 4 8 . 7 . 9 3 8 2 0 6 8 6 . 91 . 6 3 3 9 0 3 5 7 . 1 . 9 3 8 3 0 6 9 3 . 91 . 5 9 4 0 0 3 6 5 . 5 . 9 2 8 4 0 7 0 0 . 91 . 5 5 4 1 0 3 7 3 . 8 . 9 2 8 5 0 7 0 7 . 81 . 5 1 4 2 0 3 8 2 . 1 . 9 1 8 6 0 7 1 4 . 71 . 4 8 4 3 0 3 9 0 . 3 . 9 1 8 7 0 7 2 1 . 61 . 44 4 4 0 3 9 8 . 6 . 9 0 8 8 0 7 2 8 . 51 . 4 1 4 5 0 4 0 6 . 7 . 9 0 8 9 0 7 3 5 . 31 . 3 8 4 6 0 4 1 4 . 9 . 8 9 9 0 0 7 4 2 . 11 . 3 6 4 7 0 4 2 3 . 0 . 8 9 9 1 0 7 4 8 . 91 . 33 4 8 0 4 3 1 . 1 . 8 8 9 2 0 7 55 . 71 . 3 1 4 90 4 3 9 . 1 . 8 8 9 3 0 7 6 2 . 41 . 2 8 5 00 4 4 7 . 2 . 8 8 9 4 0 7 6 9 . 11 . 2 6 5 1 0 4 5 5 . 1 . 8 7 9 5 0 7 7 5 . 81 . 2 4 5 2 0 4 6 3 . 1 . 8 7 9 6 0 7 8 2 . 51 . 2 2 5 3 0 4 7 1 . 0 . 8 7 9 7 0 7 8 9 . 11 . 2 1 54 0 4 7 8 . 9 . 8 6 9 8 0 7 9 5 . 71 . 1 9 5 50 4 8 6 . 7 . 8 6 9 9 0 8 0 2 . 31 . 1 7 5 6 0 4 9 4 . 5 . 8 6 1 0 0 0 8 0 8 . 8

PER

CENT

UNCE

RTAI

NTY

IN e"

FOR

QO FROM

130

TO 10

00

161.

o in O in O •- in O in O in oc r— in CM o in CM o r- h• • • • • W w • • • m •

CO CM CM CM CM < r l H r l r l o

100

200

300

i+oo

500

600

700

BOO

900

1000

APPENDIX VIII

COMPUTER PROGRAMS FOR Q-METER DATA CALCULATIONS

162.

163.

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VV U N K IN P U T

1] FP+-12 ] FPS4 7 F C L 0 ; ]« -FP«-n,0prW- ' F R E Q U E N C I E S ? '3 ] O T S P l L C } * - D , 0 p O ’ 0 -T E R P H E N Y L S P A C I N G ? '4 ] O T T L L C ; ]<-□ , O p O - ( V * F F F T E M P S , Pf/ tf » ) , ▼ £ £5 ] + ( ( p 0 F F [ F 0 ; ] ) * ( p F F ) ) / 46 ] 0 F 5 [ F C , ; ] « - n * O p f > ( T 'U N K Q " S , F£/A7 *7 ] ->( ( p 0 F 2 [ F ( 7 ; ] ) * ( p F P ) ) / 6

VV R E F I N P U T \ I

1 ] F P F F ^ - n , 0 p [ j ^ 'NUMBER OF REFERENCE L I Q U I D S ? '2 ] -K ’ F 1 = ( 1 tP] ) ) / 0 L 0 » 0p [> - ’ NEW REFERENCE D A T A ? '3 ] I *~04 ] ' G I V E A C O E F F ' ' S % THEN a' ' S . '5 ] R E F L O O P : I + T + l6 ] R E F S P Z L C ; I i l + U > 0 p O O ’ F F F » ) , ( ▼ / ) , ( ▼ ’ SPi4 0 r P £ * )7 ] -K ( p F F F F P C F F j J ; ] ) * ( p F F ) ) / 68 ] R EFTL LC ; I i ] - * -0 , 0 pH-*- ( T ' R E F ' ) , ( ? ! ) , O » TEMP ’ )9 ] + ( ( p P F F P r F C , ; J ; ] ) ^ ( p F P ) ) / 81 0 ] R E F Q t L C ; J ; ]-«-□ , Op ["!-«-( ▼ ’ REF ' ) , O F ) , O » Q " S ' )1 1 ] ->( ( p F F F ^ C F F ; J ; ] ) * ( p F P ) ) / 1 01 2 ] ->( I < N R E F ) /R E F L O O P1 3 ] O L D : 'R E F E R E N C E NUMBER FOR RUN ' , J L C1 4 ] OLDREF*- □1 5 ] P F F 5 P C F 0 ; ; l + R E F S P Z O L D R E F ; ; ]1 6 ] P F F F C F P ; ; ]^ -P F F P C O LD REF ; ; ]1 7 ] R E F Q l L C ; ; ]^ F F F < ? [0 L P F F F ; ; ]

V

164.

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f t f t f t E - ^ f t f t f t f t f t f t f t f t f t f t f t +

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i—ii—ii—ii—ii—ii—ii—irl CN CO J- LO to r*>L_i I 11— 11 II— II— II 1

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I— II— II— i rl CN CO I I I I I I

APPENDIX IX

TABLES AND GRAPHS OF RF Q-METER DATA

167.

168.

TABLE OF CONTENTS

FIRST NITROBENZENE .......... TABLE OF € *t 1

SECOND NITROBENZENE . . . . . . TABLE OF € 't 1

SECOND NITROBENZENE ........ . . £ 1 * VS 1/T

SECOND NITROBENZENE . . . . FREQUENCY . . 1

FIRST 2 , 2 * -DINITROBIPHENYL . . .TABLE OF €' t

SECOND 2 , 2 * -DINITROBIPHENYL . . TABLE OF € *t

SECOND 2 , 2 * -DINITROBIPHENYL . . € * * VS 1/T . . 1

SECOND 2 ,2 1-DINITROBIPHENYL . . £ » » VS LOG FREQUENCY . . 1

FIRST T-BUTYL BROMIDE . . . . . TABLE OF £' i

SECOND T-BUTYL BROMIDE . . . . . TABLE OF £ *t

SECOND T-BUTYL BROMIDE . . . . . £ * * VS 1/T

SECOND T-BUTYL BROMIDE . . . . . £ » 1 VS LOG FREQUENCY . . 1

69

70

71

72

73

7475

76

77

78

79

80

" FOR

4 PER

CERT

RITR

OBEN

ZERE

FR

EQUE

RCIE

S AVERAGED

170 .

C o L n t ' - ' C O v H r - L D C N C o m t " ' ' t o o c r > c - ' - T H c o i £ > tH tH tH tH O O tH O O t H O O O O O O O O | • • • • • • • •

O c o i o o m o c n t ^ oi n C N C O O C T i C D t H f O ^ t CO tH tH tH O O tH tH tHI o o o o o o o o

O o u o o o o c n c o c o c oo c n c n c n c n c o c n t H Ol O O O O O O O t H t H I O O O O O O O O

• • • • • • • «

C j > C O ^ J - t H C D t O C O C n t H o t D O r - c ^ c c c n o c n . l O O O O O O O t H O

I O O O O O O O O • • • • • • « «

O CO z}' tH tH CO Zj J" C'- o j - ^ - c ^ c o c n o c N OJ t O O O O O O t H t H I O O O O O O O O

• • • • • • • •

O O C N O C M C N C O O O O O CO CO 00 tH CM CN CO tOCO O O O tH tH tH tH tHI o o o o o o o o

• • • • • • • •

O i o c o o c o o o c o o f "o c o c s i r ' o r - i o c o r -C N t H t H t H t H t H t H t H t HI o o o o o o o o

• • • • • • • •

U) x - s C D C O t H C O l O C O C M C y >C ^ t ^ d - O t H t H i n L O C O H j C N C N C O O cD C O O O w o r o c o c N O i o z r t H O f i n o t ^ o o c - ' C — c^thf c j t H ^ - c o r - ' O O c o m c n Dq tH CO COftj tH

FOR

H PER

CENT

NITR

OBEN

ZENE

FR

EQUE

NCIE

S AV

ERAG

ED

1 6 9.

o CD rH 00 o CNl CO o oUO o cn cn cn On cn o00 rH o o o O o rH x~i1 c G o o o o O O

• * • • • • • «

Co tH rH .n- rH CM t~- UOo o O OO 00 on CNI CM CNICD rH rH o o o rH rH rH1 O O o o o o O O

• • • « • • • •

Co rH 00 r- r- st CMo CD rH 00 co O CO LO 00LO rH rH o o O rl rH rH1 O O o o O O O G

• • • • • • « •

Co o st CNI CO st CO Oo CO co o o CN CO On OnSt rH rH rH rH rH rH rH rH1 O O O O O O O C

• • • • • • • •

Co r- o rH" C"- o f- cn COo CN J" o rH J" 00 rHCO rH rH rH rH rH rH CN CN1 O O O O O O O O• • • • • « • «

Co r-" 00 O cm UO rH CO COo rH rH CO rH CO =t in COCN rH rH rH rH rH rH rH rH1 O O O O O O o O• « • • • • • •

Co rH CN rH CN st CN sto CN O CO C" CO CN CN rHrH st CO CO . CN CN CN CN CN1 O o o O O O O O• • • • • • • •

/-- CO CO CO CN co CO in C"«XJ CO O 00 CN on C-" CO st

to CO r- CN c~' 00 o rHo CO CO CN cn o t*- C-

O LO O C" CO C" E'­ r- rHrH St CO f- CO en m cm

Ct; t-i co t"~ ro

171.

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184.

VITA

Theodore Bryant Kingsbury IV

Born in Richmond, Virginia, October 28, 1947.Graduated as valedictorian of Franklin High School, Franklin, Virginia, in June, 19 65. Attended Dartmouth College in Hanover, New Hampshire, and St. Andrews College in Laurinburg, North Carolina from 1965 to 1969. Served from 1969 to 1973 in the United States Air Force, attaining the rank of Staff Sergeant in the field of airborne electronics. B.S., Morris- Harvey College, Charleston, West Virginia, 1974.

The author entered the graduate program of the Depart­ment of Chemistry of the College of William and Mary in January, 1975.