Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 18–26www.elsevier.com/locate/calphad

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Calculation of constrained equilibria by Gibbs energy minimization

Pertti Koukkari∗, Risto Pajarre

VTT Processes, P.O. Box 1602, FIN-02044 VTT, Finland

Received 29 September 2005; received in revised form 28 November 2005; accepted 30 November 2005Available online 27 December 2005

Abstract

The Gibbs energy minimization encompasses active use of the chemical potentials (partial molar Gibbs energies) of the constitusystem. Usually, these appear at their equilibrium values as a result of the minimization calculation, the mass balance constraintsnecessary subsidiary conditions. Yet, there are several such physico-chemical circumstances where the system is also constrained by,such as surface effects, potential fields or even by chemical reaction kinetics. In this paper a particular method is presented by which cchemical potentials can be applied in a multi-phase Gibbs energy minimization. The constrained potentials arise typically from worthermodynamic displacements in the system. When Gibbs energy minimization is performed by the Lagrange method, these constrainadditional Lagrangian multipliers. Examples of the constrained potential method are presented in terms of the electrochemical Donnanin aqueous systems containing semi-permeable interfaces, the phase formation in surface-energy controlled systems and in systems wcontrolled by chemical reaction kinetics. The methods have been applied successfully in calculating distribution coefficients for metal ionherwith pH-values in pulp suspensions, in the calculation of surface tension of alloys, and in thermochemical process modeling involvingreaction rates.c© 2005 Elsevier Ltd. All rights reserved.

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1. Introduction

With the improving numerical capacity of present day coputers, Gibbs energy minimization has gained increasing inest not only in the calculation of complex chemical equiliband phase diagrams, but also in performing complicatedcess simulations. The advantage of the thermodynamic meis that it avails a common basis for complex chemical problein multi-phase systems with various proportions and conditioBoth industrial processes and small scale laboratory systcan be calculated successfully with the Gibbs energy metho

However, the multi-phase Gibbs energy minimizatitechnique has not been applicable to more complicaproblems where the thermodynamic system is subjecteddisplacement caused by a generalized work coefficient or wthe chemical or phase change is constrained by slow reackinetics. Such problems are often encountered in practmaterials science and in the simulation of processes, notincluding such topics as membrane separated electrochemequilibrium systems, complex surface energy equilibria a

∗ Corresponding author. Tel.: +358 20 722 6366; fax: +358 20 722 7026.E-mail address: pertti.koukkari@vtt.fi (P. Koukkari).

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finally time-dependent systems controlled with chemireaction rates. A common feature of all these phenomenthat the total Gibbs energy is affected by an additional physconstraint, which is due to a work-related thermodynamdisplacement, including the affinity of kinetically controllechemical reactions.

In the conventional Gibbs energy minimization calculatiothe system is subjected to the mass balance constraints ware deduced from the input amounts of the componentthe equilibrium system. The Gibbs free energy minimumoften solved by using the Lagrange method of undetermimultipliers with the mass balances of the system componas the necessary subsidiary constraints. The chemical poteof the constituents of the multi-component system can tbe solved in terms of the elements of the mass conserva(stoichiometric) matrix and the Lagrange multipliers. Athe elements of the matrix are dimensionless factors,Lagrange multipliers represent chemical potentials ofsystem components. By extension of the matrix, one mintroduce additional constraints for a desired set of constituand thus take into account additional Gibbs energy termsto the surface tension, electrochemical potential or affinitya kinetically controlled reaction. In what follows, we presen

P. Koukkari, R. Pajarre / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 18–26 19

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Lagrangian method, which allows a number of such phenomto be calculated with Gibbs energy minimization.

2. Theory

2.1. Overview of the Lagrangian method

The Gibbs energy of the multicomponent system is writtin terms of the chemical potentials as follows

G =∑α

∑k

nαk µα

k (1)

where µαk = µα

k (T, p, nαk ) is the chemical potential of the

species(k) in the respective phaseα and nαk is its molar

amount. The Gibbs energy is an extensive state variablethe chemical potentialµα

k is the partial molar Gibbs energy othe constituentk.

For the Lagrangian method, the mass balance equationsneeded as follows [1,2]:

φ j = b j −Ψ∑

α=1

Nα∑k=1

aαkj n

αk = 0 ( j = 1, 2, . . . , l) (2)

where b j is the total input amount of a system componeandaα

kj refers to the stoichiometric number of componentj inconstituentk, in its respective phaseα. The number of phases idenoted byΨ , andNα is used for the number of constituentsphaseα. A system component is typically, but not necessara chemical element. The total number of system componenl. The individual mass balances are denoted for brevity asφ j .The Lagrangian function is then written in terms of the Gibenergy and the mass balance conditions:

L = G −l∑

j=1

λ j φ j (3)

where theλ j s are the undetermined multipliers of the Lagranmethod. The minimum condition of the Gibbs energy is tsame as this condition for the Lagrangian function(L), and isreceived at constant temperature and pressure by findingextremum points for the respective partial derivatives:(

∂L

∂nk

)i �=k

= µk −l∑

j=1

akj λ j = 0 (k = 1, 2, . . . , N) (4)

Conditions(2) and(4) together give a set ofN + l equationswith an equal number of unknowns to be solved (N is thetotal number of constituents). The solution gives the moamounts (nks) at equilibrium for the closed system whetemperature and pressure are held constant. In addition,undetermined multipliers(λ j ) become solved. By definition o(3), they connect the mass balances of each system compoto the Gibbs energy. In fact, the solution of the undeterminmultipliers produces eachλ j as representing the chemicapotential of the respective system componentj . To emphasize,we denote this potential byπ j and get, for the chemicapotentials of any constituentk:

µk =l∑

j=1

akj π j (k = 1, 2, . . . , N) (5)

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Eq. (5) gives the chemical potential of any constituentk as alinear combination of the respective potentials of the systcomponents [1]. The respective(N × l) conservation matrix is

A =

a(1)1,1 · · · a(1)

1,l...

. . ....

a(1)N1,1 · · · a(1)

N1,l

a(2)N1+1,1 · · · a(2)

N1+1,l...

. . ....

a(Ψ )N,1 · · · a(Ψ )

N,l

(6)

In the conventional CALPHAD methods, the system coponents represent stoichiometric building blocks of the cstituents and the matrix elementsakj are the respective stoichiometric coefficients. For example, the chemical potentiacarbon dioxide(CO2) in an equilibrium system with the elements carbon (C) and oxygen (O) as system componentsbe given in terms of their potentials. Carbon dioxide consof one unit of carbon and two units of oxygen, and the checal potential is accordinglyµCO2 = πC + 2πO. The conditionis equivalent to the requirement that the affinity of all possichemical reactions is zero at equilibrium.

The independent components of the system maychosen to represent stoichiometric entities other than chemelements. These include, for example, chemical substanions and electronic charge, which characteristically may ocas independent components of a phase constituent.stoichiometric coefficients given in the transformed matmust be consistent with the conservation of mass insystem, which is defined in terms of the total mass(mtot) as∑l

j=1 b j M j = mtot. Here, M j is the molecular mass of thsystem componentj .

2.2. Setting additional constraints with the conservation matrix

The conservation matrixA has a row for each species andcolumn for each independent conservation equation [1,3]. Theconservation matrix is thus made up of the coefficients ofconservation equations valid in the system. As stated abin chemical reactions, atoms of elements and electric chaare conserved. Sometimes, additional conservation equaare required, for example to denote conserved molecgroups [4]. The new constraint appears as an additional coluin the conservation matrix:

A =

a(1)1,1 · · · a(1)

1,l a(1)1,l+1

.... . .

......

a(1)N1,1 · · · a(1)

N1,l a(1)N1,l+1

a(2)N1+1,1 · · · a(2)

N1+1,l a(2)N1+1,l+1

.... . .

......

a(Ψ )N,1 · · · a(Ψ )

N,l a(Ψ )N,l+1

(7)

Here the matrix elements for the phase constituents remequivalent to those in Eq.(6), but the additional column with

20 P. Koukkari, R. Pajarre / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 18–26

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subscriptl + 1 represents the new conservation equation. Ththe elementaα

k,l+1 = 0 for all those constituentsk whichare not affected by the additional constraint, whereasaα

k,l+1 isnot zero for those constituents which are affected by the sconstraint. Thus, for example, the number of aromatic groto be conserved in each aromatic compound can be attachthe Gibbs energy calculation by the new pseudoelement [4]. It isobvious that the mass balance of the total system is not affeif the molecular mass of the pseudoelementMl+1 is chosen tobe zero.

The additional constraint affects the chemical potenof the phase constituents through Eq.(5). As the elementsof the matrix are dimensionless factors, the Lagrangmultipliers represent additive contributions to the chemipotentials of the constituents. Applying this property, omay introduce additional conditions for a desired setconstituents, thus generalizing the conservation matrixapplicable physical constraints of the system. Such constramay be set for the electroneutrality condition of phases5],or for an affinity related metastable or kinetically conservspecies [6,7]. Further, a constraint set for the surface areathe system is similarly linked to the surface energy of tsystem and it can be used to predict the surface tensionsurface compositions of multi-component mixtures [8]. In whatfollows, three simple examples are presented to detail theof the additional constraint when calculating surface tensiof multi-component alloys, to determine Donnan equilibin membrane-separated multi-phase aqueous systems, aconserve the affinity of a kinetically conserved chemicreaction in a multi-phase system.

3. Calculation examples

3.1. Computation of surface tension

Surface energy can appear as an additional factor inGibbs energy function of a multi-component system. If the (flsurface layer is assumed to be one monolayer thick, the tGibbs energy of the system is:

G =Nb∑

k=1

µbknb

k +Ns∑

k=1

µskns

k + σ

Ns∑k=1

Aknsk (8)

Here superscripts and subscriptsb and s have been used fothe bulk and surface phases, respectively. As the same spcan be assumed to be present both in the bulk and insurface, withN being the total number of species, we haNb = Ns = N/2 and the same subscript denotes the sachemical species in both the bulk and the surface phase.surface tension is a function of temperature and compositσ = σ(T, xk). Each constituent of the surface phase occupa characteristic molar surface areaAk . The total surface area isassumed to be constant at equilibrium:

N∑k=1

Aknsk = A (9)

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Considering the two phases, Eq.(9) may also be written as

A/A0 −∑α

N∑k=1

(Aαk /A0)n

αk = 0 (10)

where the molar surface for any species in the bulk phaszero. The area terms are divided by a normalization consA0 with dimensions ofm2/mol. This equation then shows thconstraint of constant surface area, in analogy to the mbalance conditions of Eq.(2). It is then mathematically possiblto consider Eq.(10) as an additional constraint of a multcomponent system, where the surface layer is introducea separate phase and the surface area as an additional scomponent [6].

Using(8) in (3)and by applying(2) and(10), the Lagrangianfunction of the multi-component surface system becomes

L =Nb∑

k=1

µbknb

k +Ns∑

k=1

µskns

k + σ

Ns∑k=1

Aknsk

−l+1∑j=1

λ j

(b j −

(∑k

abkj nb

k +∑

k

askj n

sk

))(11)

Here, summation of the constraints extend over all syscomponents, that is,j = 1, 2, . . . , l, l + 1, where the lastconstraint is the one deduced from the surface area, withbl+1 =A/A0, andas

k,l+1 = Aαk /A0. From Eqs.(4) and(11), the partial

derivative conditions become:(∂L

∂nbk

)i �=k

= µbk −

l∑j=1

abkj λ j = 0 (12)

(∂L

∂nsk

)i �=k

= µsk + Akσ −

l+1∑j=1

askj λ j = 0 (13)

At equilibrium, the chemical potential of each speciesindependent of phase, that is,µb

k = µsk = µk . From Eqs.(12)

and(13), it follows that the surface tension of the mixtureobtained as the additional Lagrange multiplier:

σ · A0 = πl+1 (14)

The numerical value of the constantA0 can be chosenarbitrarily, but for practical calculation reasons it canadvantageous if the ratioAk/A0 is a value close to unitybeing of the same order of magnitude as the stoichiomecoefficients appearing in the conservation matrix.

To perform the calculations with a Gibbs energy minimiziprogram, such as ChemApp [9], the input data must be arrangein terms of the standard state and excess Gibbs energy dachemical potentials of the system constituents. It is suffichere to state that the chemical potentials of speciesk in the bulkand surface phases can be written in terms of the respeactivities as follows:

µbk = µ0

k + RT ln abk (15)

µsk = µ

0,sk + RT ln as

k − Akσ (16)

where the superscript0 refers to the standard state, andabk and

ask are the activities of the constituentk in the bulk phase and

P. Koukkari, R. Pajarre / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 18–26 21

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Table 1Stoichiometric matrix for the Fe–FeO system

Fe FeO Area

BulkFe 1 0 0FeO 0 1 0

SurfaceFe 1 0 3.7187FeO 0 1 5.8782

The molar surface areas for Fe and FeO at 1973 K are 37 187 m2/mol and58 782 m2/mol, respectively [10].

surface phases, respectively. By applying Eqs.(15) and (16)for the case of a pure one-component system, a relationbetween the standard states of the bulk and surface phasbe derived:

µ0,sk = µ0

k + Akσk (17)

Eqs.(15)–(17)then indicate that the necessary input forGibbsian surface energy model must include not only standstate and activity (excess Gibbs energy) data for the constituof the bulk and surface phases, but also the data for surtensions of the pure substances(σk) as well as their molarsurface areas(Ak). The numerical calculation techniquedescribed in detail elsewhere [6]. In Table 1, the extendedmatrix of an FeO/Fe binary system is given. Respectcalculation results at 1970 K are presented inFig. 1 (the molefraction of FeO in the surface phase vs. that in the bulk (left) asurface tension of the mixture (right)). The molar surface arpure substance surface tension values and Gibbs excess eare taken from Tanaka and Hara [10]. The calculated results fothe FeO/Fe binary also agree with those presented in [10].

The obvious advantage of the Gibbsian method isthe direct calculation of surface tensions in multi-componmixtures. An example of a simple ternary system was givearlier in [6].

3.2. Ion exchange equilibria in aqueous multi-phase systems

When two aqueous solutions at the same temperaturepressure) are separated with a membrane that is permeabsome ions but not to others nor to the solvent, a distributknown as Donnan equilibrium. Donnan equilibrium is formin the system of the two compartments [11,12]. The systemconsists of two aqueous phases with water as the soland mobile and immobile ions as solute species. In a muphase system, gas as well as precipitating solids maypresent. Both compartments containing the aqueous soluremain electrically neutral. The essential feature of the Donequilibrium is that, due to the macroscopic charge balancthe separate compartments, immobility of some of the iowill cause an uneven distribution for the mobile ions toThis distribution strongly depends on the acidity (pH) of tsystem in such cases where dissociating molecules in one ocompartments may release mobile hydrogen ions, while trespective (large or bound) counter anions remain immobileto the separating membrane. By applying the electroneutracondition together with other physical conditions of thmembrane system, the constrained potential method allows

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calculation of the multi-phase Donnan equilibrium with Gibenergy minimization. Thus, the distribution of ions in the twcompartments, together with formation of precipitating phafor example, can be calculated.

In Table 2 an example of the stoichiometric matrix fortwo-compartment Donnan system is presented. For thesolution phases present, notations′ and ′′ have been used. Thconstancy of the amount of water in the second compartmeascertained by setting the respective matrix element to unitythere is no molecular mass assigned for this system compothe mass for the constituent H2O in the second solution volum(solvent′′) is obtained from the respective system componeO (for oxygen) and H (for hydrogen). The electronic chargeaqueous ions is introduced to both aqueous phases in termthe negative charge numbers, and an additional electroneutcondition has been set for the secondary aqueous phdenoted in the matrix asq ′′. The immobile anion(Aniona)

has been positioned as a constituent for the secondary aquphase′′. For clarity, just values different from zero are shown

With the given matrix conditions, by using Eq.(5) for thechemical potentials of charged species at equilibrium, we h

µ′′k = µ0′′

k + RT ln a′′k

= µ′k + zkπq′′ = µ0′

k + RT ln a′k + zkπq′′ (18)

where the chemical potentials of the primary and secondaqueous phases have been written in terms of their respeactivities (a′

k and a′′k ). The additional term (zkπq′′) deduced

from Eq. (5) is due to the supplementary electroneutraconstraint set for the secondary aqueous phase. Eq.(18) iscomparable to the general form of electrochemical equilibriof charged species [11]:

µ0′′k + RT ln a′′

k = µ0′k + RT ln a′

k + zk F ϕ (19)

where F is the Faraday constant and ϕ is the electricalpotential difference between the primary and secondaqueous phases. It follows that the solution of the Gibbsproblem with the additional electroneutrality constraint givthis potential difference as the Lagrange multiplierπq′′:

F ϕ = πq′′ (20)

Similarly, from Eq.(5) one may deduce the chemical potentof water in the two aqueous phases in terms of the Lagranmultipliers. Obviously, these two chemical potentials areequal, but differ by the Lagrange multiplierπsolvent′′ . This isbecause the Gibbs energy model does not include the eof the contractive forces of the membrane system, whprevent the transport of water from the primary compartmto the secondary volume. Yet the model can be applieddetermine the activity difference of the two aqueous phaand thus to define the expected pressure difference in vamembrane systems [12,13]. Assuming incompressibility, i.ethat the partial molar volume of water is constant in tmoderate pressure range of the membrane systems, we ha

−πsolvent′′ = (p′′ − p′)V msolvent= RT ln

(a′

solvent

a′′solvent

)(21)

22 P. Koukkari, R. Pajarre / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 18–26

Fig. 1. Surface composition and surface tension as a function of bulk composition in a Fe–FeO mixture at 1970 K.

Table 2Example of the stoichiometric matrix for a Donnan equilibrium system with two membrane-separated compartments containing ionic solutions

O H Na C Ca Solvent′′ Aniona e− q ′′

1. Solution volume

H2O 1 2H+ 1 −1OH− 1 1 1Na+ 1 −1CO2 2 1HCO−

3 3 1 1 1

CO2−3 3 1 2

Ca2+ 1 −2

2. Solution volume

H2O 1 2 1H+ 1 −1 1OH− 1 1 1 −1Na+ 1 −1 1CO2 2 1HCO−

3 3 1 1 1 −1

CO2−3 3 1 2 −2

Ca2+ 1 −2 2Anion−

a 1 1 −1

CaCO3 CaCO3 3 1 1

e

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ion

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ure.talut.

The partial molar volume of the solvent (water) is denotby V m

solvent. The constraining factorπsolvent′′ emerges as thepotential difference due to the different activities of solvewater in the two different compartments. Thus, the pressdrop across the membrane(p′′ − p′) can be calculated on thebasis of either the activities of the solvent in the two volumor the potentialπsolvent′′ .

An example calculation of the above described Donnequilibrium system is a pulp fiber suspension where the fibabsorb both water and solute ions. Fibers contain carboxacid groups of hemicellulose and phenolic groups oflignines. The anions of these groups are generally bounthe fiber, but the protons of these acidic functional groucan be dissolved and transferred through the fiber toexternal bulk solution. Charge neutrality prevails both withthe fiber structure and in the external solution, and ththe dissociation of the functional groups may lead to iexchange between the cell structure of the fiber and the bsolution. The stoichiometric Donnan equilibrium theory hbeen used already by Neale (1929) and Farrar and N(1952) to characterize electrolyte interactions with cellulo

d

tre

s

nrslicetosn

s

lk

lee

fibers [14,15]. In 1996 Towers and Scallan published theDonnan model, which could be used to calculate the iondistribution of mixtures of mono- and divalent cations anmonovalent anions in pulp suspensions [16]. The solutionmodel was further extended by R¨asanen et al. to includethe presence of multivalent anions as well as the formatof hydroxyl complexes and ligands [17]. The multi-phaseGibbs energy model of the fiber suspension, introducedKoukkari, Pajarre and Pakarinen (2002), enables the calculaof the solution equilibria (including the distribution of thecharged species) while precipitating solids and the dissolutof gaseous constituents are also taken into account [5].

The Gibbs energy model of the fiber suspensionessentially based on the electrochemical theory descriabove, the fibers representing the secondary aqueous pcontaining both mobile and immobile ions. The basic input daof the multi-phase Donnan model is similar to any characterisaqueous solution model, requiring the incoming amounof substances and equilibrium temperature and pressThe water content of pulp fibers is based on experimenwater retention values and given as additional model inp

P. Koukkari, R. Pajarre / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 18–26 23

s a

Fig. 2. Measured and modeled concentrations of Na, Ca, Mg, and Mn ions bound by the fibers (f) and in the surrounding solution (s). Experimental resultre taken from Ref. [16]. Low concentrations in the surrounding solutions at pH values above 10 are due to precipitation of hydroxides.

s

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nteant

ntefibhm

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Furthermore, to characterize the immobile anionic speciethe fiber phase, their amounts and Gibbs energy data mbe specified. Unlike the mobile ions and neutral solutes,bound acidic groups are included only in the fiber phaAs no reactions that would change the total amount of thgroups is assumed to take place, the chemical potential oundissociated forms of these groups can be set to zero, wthe chemical potential of the anionic forms can be calculabased on the thermodynamic relation

Gi = −RT ln Ki (22)

where the acidic dissociation constants(Ki ) and thecorresponding molar amounts are determined experimenby potentiometric or conductometric titration. This datasufficient to perform the equilibrium calculations, and tchemical structure of the acids needs not to be knownhas been found quite customarily that both theKi -values andamount of charge are characteristic to a given form of cellulor pulp with a known treatment history. Consequently, tdata appears comparable with standard Gibbs energy daknown substances [18,19]. In Fig. 2, typical ionic distributionsand phase formation in terms of changing pH in variopulp suspensions have been calculated from thermodynequilibrium data and compared with the experimental resof Towers and Scallan [16]. The input amounts (Table 3) arefrom the same source. At low pH of the external solutiothe acidic groups within the fiber phase remain undissociatheir anionic charge is small and the cations, such as C2+,Mg2+, Na+ and K+ typically, are evenly distributed betweethe fiber and the external solution. With decreasing acidity,pH in the external solution is raised and the acid groupsthe fiber are dissociated. This feature triggers the ion exchabetween the two aqueous phases, the protons are transporthe external solution and the corresponding charge in thephase becomes compensated with the metal cations. Atenough pH, precipitating carbonates and hydroxides are for

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heinged toerighed

Table 3Input amounts used for pulp suspension model calculations

H2O 89.12 kgH2Of 1.4 kgBound acid(pK = 4) 0.085 molCa(OH)2 0.0282 molMg(OH)2 0.0174 molMn(OH)2 0.00166 molNaOH 0.0615 molHCl varied for pH control

H2Of denotes the water inside the fibers, H2O the amount of the rest of theaqueous solution (corresponding to the 1. solution volume inTable 2).

as solid phases. The practical perspective of the multi-phequilibria in pulp suspensions lies in the improved controlpH buffering of paper machines and in metal managementpulp bleaching solutions [18].

3.3. Reaction rate controlled systems

The conventional method in studies of reaction mechanisis to distinguish between the fast (equilibrium) reactionand the slow (rate controlled) reactions. This has beenviable approach when applying the stoichiometric method wexplicit reaction equations, with the equilibrium constant daeasily available for the reactions between thermodynamicawell-defined substances and the reaction rate constants usudeduced from experiment. A similar technique, when appliwith the Gibbsian multi-phase method, would avail a flexibroute to calculate reaction rate controlled systems in terof their thermochemical properties. An early approachcombining reaction rates with multi-phase calculations wasextension of the stoichiometric matrix of a Gibbsian multcomponent system by the inert image component [20]. Theimage method was proven successful in many cases whthe driving force of a single reaction is sufficient for reachin100% conversion from reactants to products. The introduct

24 P. Koukkari, R. Pajarre / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 18–26

Table 4Thermodynamic and kinetic data for TiO(OH)2 calcination reactions

Reaction H (1000◦C) (kJ mol−1) G(1000◦C) (kJ mol−1) Reaction rate equation Ea (kJ mol−1) A (h−1)

TiO(OH)2 ↔ TiO2(An) + H2O ↑ 43.5 −83.6 – – –TiO2(An) ↔ TiO2(Ru) −6.6 −5.9 ξ = 1 − (1 − kt)3 441.99 1.80E+17

sb

hb

tnth

tn

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ree

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of conservable groups as additional system componentthe stoichiometric matrix in the Lagrangian method of Gibenergy minimization was originally used by Alberty [4] topreserve aromatic rings in a benzene flame model. Witfurther matrix extension, this technique was shown toapplicable to kinetically conserved species [6] and could beapplied to several related problems [7]. In what follows, themethod is shown to include the characteristics of the potenconstraint technique, in the sense that it conserves the (zero) affinity of the rate controlled reactions step asadditional Lagrange multiplier.

As a simple example, calcination of titanium oxyhydra(TiO(OH)2) slurry is considered, referring to the formatioof titanium dioxide powder in a calciner. The feed consiof (wet) titanium oxyhydrate slurry, the chemical compositibeing approximated as TiO(OH)2 ∗ nH2O. During calcination,the slurry is dried and finally the hydrate decomposes, leavthe product titanium dioxide in the bed. From the oxyhydraat relatively low temperatures (ca. 200◦C) the crystallineform anatase, TiO2(An), is formed first, and only in the hightemperature zone of the furnace end, the thermodynamicstable rutile form TiO2(Ru) appears as the desired product. Treactions are as follows:

TiO(OH)2 ∗ nH2O ↔ TiO2(An) + (n + 1)H2O ↑ (I)

TiO2(An) ↔ TiO2(Ru). (II)

Rutile is the thermodynamically more stable form of thetwo titanium dioxide species (Table 4). Thus a thermodynamiccalculation, such as Gibbs free energy minimization, wouldall temperatures result in rutile and water. This would lead t100% rutilisation of the titania already at temperatures whthe Ti-oxyhydrate is all but calcined by reaction(I). Yet itis well known from practical experience that the rutilisatioreaction(II) is slow and only takes place with a finite rateelevated temperatures (above 850◦C). The simulation of thecalcination reactions must take this feature into account [21].

In Table 5, the stoichiometry of the calcination systempresented. The three first columns with system componoxygen (O), hydrogen (H) and titanium (Ti) show the elemetal abundance-constrainedstoichiometry of the equilibrium stem. The additional column (R) represents the additional kinconstraint, affecting the conservation of rutile content in eaGibbsian calculation sequence.

The formation of the Lagrangian function from Eq.(3)by using the equilibrium matrix is straightforward, as weas the solution of the (zero) affinities for the stoichiometreactions(I) and (II) from the equilibrium condition in Eq.(4). (For simplicity, the Ti-oxyhydrate has been written withothe bound water molecules.) When the additional constrain

tos

ae

ialon-e

e

tsn

nge,

llye

e

atare

nt

nts-s-tich

lc

tis

Table 5Stoichiometric matrix for TiO(OH)2 calcination with kinetically constrainedrutilisation

Index(k) Species O H Ti R

1 O2-gas 2 0 0 02 H2O-gas 1 2 0 03 TiO(OH)2 3 2 1 04 TiO2(An) 2 0 1 05 TiO2(Ru) 2 0 1 1

Substance index(k) is shown in the left-hand column.

taken into account, the Lagrangian function becomes:

L(n) =2∑

k=1

nk

(µ0

k + lnnk

n1 + n2

)+

N=5∑k=3

nkµk

−l∗=4∑j=1

π j

(b j −

N=5∑k=1

akj nk

)(23)

where the first term on the right is the chemical potentialsthe two gaseous species in terms of their partial pressures,second is formed from the three condensed species, and theterm is deduced from the mass balances (a single ascenvalue has been used for the constituent indexk). The totalnumber of mass balance constraints isl∗ = 4, including theadditional zero-mass ‘rutility’ of the formed titanium dioxideFrom(4) and(23), the following conditions are obtained:

µ4 +(

µ02 + ln

n2

n1 + n2

)− µ3 = 0 (24)

µ4 − µ5 = π4. (25)

Eq.(24) is the equilibrium condition for the fast reaction(I)which forms anatase and water from the oxyhydrate at elevatemperatures. Condition(25) gives the additional Lagrangemultiplier as the affinity of reaction(II) and is dependent onthe value of the constraintb4. In a sequential computation,the kinetic constraint is defined as a function of the exteof such a given reaction [bR = f (ξR)] and is set asthe input of the Gibbsian calculation. The reaction raparameters of the rutilisation example are given inTable 4,where the reaction kinetics are deduced from the experimenstudy of McKenzie [22] (model of contracting spheres). Theintegrated reaction rate is obtained as a dimensionless ‘degof rutilisation’, which equals the extent of reaction(II) . At eachsequence of calculation, the constraintb4 = ξ× (number ofTiO(OH)2 moles in the input). Asξ is dimensionless, the unitof bR is moles.

With the procedure described above, the kineticalconstrained Gibbs energy calculation can be performeda sequential procedure, provided that there is a meansdefine the value of the additional system component (bR) as

P. Koukkari, R. Pajarre / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 18–26 25

Fig. 3. Measured points and model curves for the rutile fractions are presented at four temperatures (left). The respective Gibbs energies of the system at the sametemperatures are calculated as monotonically descending curves (right).

thtiol

ad

0

tt

r

aiuioo

lw

ctisith

nc

ah

rate

ea,bbsntialistioneetive

tingtityosefor

ed inen inringrateo betionineds

ilityewns.n ofcedoreainand

pH-ulp

program input. For the ChemApp program, which appliesLagrange method and is hence suitable for the calculaof the constrained problems, we have developed a mindirect procedure, which allows for the control of a kineticalconstrained reaction either in its forward or reverse mode [7,9,23].

The thermodynamic (Gibbs energy) data for the speciesfrom standard sources [26], yet the estimates for the standarenthalpy, entropy and heat capacity of TiO(OH)2 were usedas in [24]. The kinetic calculation is then performed in 6steps with 10 min intervals to cover the experimental dataMcKenzie, ranging up to 10 h at 995◦C. For each step, theGibbs energy of the system is minimized and, as a result,composition and the Gibbs energy of the system are calculaThe calculation method has been described in more dein [23]. The results are shown inFig. 3. The rutile fraction risesfrom zero to one according to the given reaction rate. It is wonoticing that, with the Gibbsian approach, the thermodynamproperties of the system also become calculated for esequential step. The Gibbs energy of the reactive systemmonotonically descending curve at any constant temperatAs the anatase-rutile transformation is an exothermic reactit is of practical interest to also follow the enthalpy changethe system during the gradual process.

Additional constraints may be set to include a more compreaction mechanism. Each kinetically conserved speciesthen be annexed to a rate constraint [23]. It seems viable thata systematic approach, which embeds the necessary rearates as source terms to the multi-phase Gibbsian calculacan be developed on the basis of the additional constraintis emphasized, however, that the method provides a technto connect the experimentally found reaction rates withthermodynamic calculation, but includes no attempt to predreaction rates from thermochemical theory.

4. Summary

The method of constrained potentials provides an extensof the applicability of the Gibbs energy minimizatiomethod to a variety of problems encountered in chemithermodynamics. A common feature of all these problemsthat there is a conservation factor additional to that of mbalance. Above, we have presented three examples, w

eonre

y

re

of

heed.tail

thicchs are.n,f

exill

tionon,. Itquee

ict

ion

alisssere

Table 6Potential constraints for surface energy, electro-chemical and reactionsystems

System Constraint Potential equation

Surface∑

k Ak nsk = A σ · A0 = πsurface

Electro-chemical∑Nα

k=1 zknk = 0 Fϕα = πq−α

Kinetic∑

k vkr nk = br∑

Reactantsνkr µkr −∑Productsνkr µkr = πR

the additional conservation clause follows from surface arelectroneutrality or due to reaction rate. The respective Gienergy factor is then surface energy, electrochemical poteand affinity of the kinetically constrained reaction. Thpotential becomes solved during the Gibbs energy minimizaas the additional Lagrange multiplier. A summary of the threxamples with the additional constraints and the respecpotential factors is given inTable 6.

For the equilibrium systems inTable 6, i.e. the surfaceenergy and electrochemical systems, the equations relathe Lagrange multiplier and the respective physical quanseem to be unambiguous in the form presented. For thsystems, generally one single constraint is necessary. Assystems constrained by reaction rate, the equation presentTable 6is valid for a single stoichiometric reaction, with thconstraint set for one of the reaction products as was givethe example of anatase-rutile transformation. When considea mechanism with more than one linearly independentcontrolled reaction, an equal number of constraints needs tdefined into the Gibbsian system, and the respective relabetween the affinities of these reactions and the constrapotentials(πr ) in terms of their stoichiometric coefficients ithen applicable [23].

The method of constrained potentials avails the possibfor quantitative Gibbs energy calculations of several nphenomena with a wide scope of practical applicatioFrom the examples presented above, the surface tensiometals and alloys, including that of steel, can be deduin various conditions. As the surface energy becomes mdominant for material properties with decreasing particle/grsize, an interesting application is the phase behaviorproperties of nano-size particles [25]. As for electrochemistry,the approach of constrained potentials has enabled thedependent solubility and ion exchange models for p

26 P. Koukkari, R. Pajarre / Computer Coupling of Phase Diagrams and Thermochemistry 30 (2006) 18–26

cieemeonol

h

:,

2

yf

ng

,

01)

e,

s,s

,

).s

suspensions and other membrane separated systems (e.g. [18]).One may expect similar applications in related electrochemiproblems. The constraining of reaction rate-dependent affinitin the Gibbsian calculation extends the applicability of ththermodynamic method to process simulations in systewhere it is essential to follow both chemical and energy changand their interdependence. Finally, as there are many analogphenomena in chemical thermodynamics where an additiodisplacement factor affects the chemical potential of oneseveral species, we expect that the method presented will afind applications other than those mentioned in this text.

References

[1] G. Eriksson, Chem. Scripta 8 (1975) 100–103.[2] S.M. Walas, Phase Equilibria in Chemical Engineering, Butterwort

Publishers, Stoneham, 1985.[3] W.R. Smith, R.W. Missen, Chemical Reaction Equilibrium Analysis

Theory and Algorithms, Krieger Publishing Company, Malabar, Florida1991.

[4] R.A. Alberty, J. Phys. Chem. 93 (1989) 413–417.[5] P. Koukkari, R. Pajarre, H. Pakarinen, J. Solution Chem. 31 (200

627–638.[6] R. Pajarre, Modelling of equilibrium and non-equilibrium systems b

Gibbs energy minimization, Master’s Thesis, Helsinki University oTechnology, Espoo, 2001.

[7] P. Koukkari, R. Pajarre, K. Hack, Z. Metallkd. 92 (2001) 1151–1157.

als

ssusalrso

)

[8] R. Pajarre, P. Koukkari, T. Tanaka, Y. Lee, CALPHAD (in press).[9] http://www.gtt-technologies.de/.

[10] T. Tanaka, S. Hara, Z. Metallkd. 90 (1999) 348–354.[11] E.A. Guggenheim, Thermodynamics, 6th ed., North Holland Publishi

Company, Amsterdam, New York, Oxford, 1975.[12] F. Helfferich, Ion Exchange, Dover Publications, Mineola, New York

1995.[13] R. Pajarre, P. Koukkari, J. Mol. Liq. (in press).[14] S.M. Neale, J. Textile I. 20 (1929) 373.[15] J. Farrar, S.M. Neale, J. Colloid. Sci. 7 (1952) 186–195.[16] M. Towers, A.M. Scallan, J. Pulp Pap. Sci. 22 (1996) J332–J337.[17] E. Rasanen, P. Stenius, P. Tervola, Nord. Pulp Pap. Res. J. 16 (20

130–139.[18] P. Koukkari, R. Pajarre, E. R¨asanen, in: T.M. Letcher (Ed.), Chemical

Thermodynamics in Industry, Royal Society of Chemistry, Cambridg2004, pp. 23–32.

[19] E. Rasanen, Modelling ion exchange and flow in pulp suspensionDoctoral Thesis, Helsinki University of Technology, VTT Publication495, Espoo, 2003.

[20] P. Koukkari, Comput. Chem. Eng. 17 (12) (1993) 1157–1165.[21] M. Ketonen, P. Koukkari, K. Penttil¨a, The European Control Conference

ECC 97, 1997, Brussels, Belgium.[22] K.J.D. McKenzie, Brit. Ceram. Trans. J. 74 (1975) 77–84.[23] P. Koukkari, R. Pajarre, Comput. Chem. Eng. (submitted for publication[24] K. Penttila, Simulation model of a TiO2 calcination furnace, Master’

Thesis, Helsinki University of Technology, Espoo 1996 (in Finnish).[25] J. Lee, J. Lee, T. Tanaka, H. Mori, K. Penttil¨a, JOM-J Min. Met. Mat. S.

57 (2005) 56–59.[26] FACT database Edition 2001, inFactSage 5.0, CRCT, Montreal, Canada

and GTT Technologies, Herzogenrath, Germany, 2001.

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