Analysis of solid—gas surface reactions by nonequilibrium thermodynamics

Analysis of Solid-Gas Surface Reactions by Nonequilibrium Thermodynamics G. BERTRAND, M. LALLEMANT, AND G. WATELLE Laboratoire de R e c h e r c h e s sur la R ( a c t i v i t ( des Solides, associ~ an C N R S , Universitk de Dijon, B P 138, 21004 Dijon Cedex, France

Received July 12, 1978; accepted November 27, 1978 To date, experimental studies and interpretations of the heterogeneous kinetics of solid-gas reactions have been reported independently of the concepts of nonequilibrium thermodynamics. The thermodynamic analysis gives to the study of the evolution of solid-gas systems its natural frame. This paper represents an attempt to provide this analysis and involves the following three points: (i) movement diagram defines the various out-of-equilibrium domains characterized by the conservation of a given kinetic feature of the evolving system; (ii) exploitation of the expression of interface entropy production enables us to consider new couplings between irreversible processes, especially for the interface chemical reaction; and (iii) the analysis of the excess entropy production shows that perturbations of the kinetic working conditions may be able to induce instabilities of the reference steady states. INTRODUCTION

Glansdorff (1), referring to the latest Nobel Prize in Chemistry, recently wrote "beside the classical thermodynamics of equilibrium, which in fact only represents the "static" aspect of that field, arose a typically "dynamic" thermodynamics of physico-chemical phenomena." Looking for the evolution laws of systems proved fruitful for homogeneous chemical reactions (2-5); this search, however, is still in its early stages in the case of solidgas reactions, where the solid undergoes a chemical, morphological, and structural transformation such as a dehydration, oxidation, etc. The purpose of this study is to show that using nonequilibrium thermodynamics can also bring changes in reasoning, contribute to understanding phenomena, and widen the scope of this field of research both for the experimental procedure and for the theoretical interpretation of phenomena. The three points on which our attention will be focused are:

First, whereas the kinetist is usually content with phase diagrams showing the domains of equilibrium structures, nonequilibrium thermodynamics shows the interest of nonequilibrium state diagrams. To complete the evolution representation, we propose to add the kinetic characteristics to these diagrams. Second, whereas for the solid-gas systems considered the interface chemical kinetics is often treated independently of heat and matter transport phenomena, on the contrary, the linear thermodynamics of irreversible processes makes it possible for coupled phenomena to be interpreted under the effect of phenomenological relations. Their real importance will be seen for the interface, especially by showing new possibilities of coupling. Finally, the stability problem of steady state has not been raised yet for the reactions we are concerned with, whereas thermodynamics shows that this stability is not guaranteed far from equilibrium and that a sharp change in the system evolution can

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Journal of Colloid and Interface Science, Vol. 70, No. 2, June 15, 1979

0021-9797/79/080223-08502.00/0 Copyright© 1979by AcademicPress, Inc. All rightsof reproductionin any formreserved.

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be observed. We show how important this kind of research is in solid-gas reactions. We thought it useful here to summarize succintly some basic concepts of nonequilibrium thermodynamics before showing some new applications in the kinetics of solid-gas reactions. The purpose of this paper is, thus, to put forward a new approach in the understanding and correlation of resuits in this field. MOVEMENT DIAGRAMS

The Various, Thermodynamic Diagrams of Chemical Systems As we mentioned, the diagrams usually used in classical thermodynamics are phase or equilibrium diagrams. They give no information concerning evolution. To list the data related to this phenomenon, we plot nonequilibrium state diagrams. The axes of the diagrams show the constraints, i.e., variables, that are controlled (3). The domains defined by the conditions of occurrence of the various states, whether steady or not, are displayed on the diagram (3, 5, 6). By varying the constraints, the boundaries of those domains may be crossed, thus showing the possible transitions between nonequilibrium states. To extend our knowledge of the evolution, we have had to consider a second type of diagram which includes the experimental kinetic data, i.e., the movement diagram.

The kinetics of a reaction is studied versus the variation of a constraint (pressure of a gaseous constituent, reactant or product, reactor temperature, etc.). With respect to the distance to equilibrium the continuous or discontinuous evolution of a kinetic datum is noticed. This datum may be the profile of the progress curve, reaction rate, geometrical shape of that part of the solid which reacted, etc. It is that type of result which is expressed on a movement diagram.

Movement Diagram Examples The first example is given by the dehydration kinetics ofpentahydrate magnesium chromate (8). The final state of equilibrium is the dihydrated salt, termed $2. The temperature constraint is varied for a given value of the water vapor pressure constraint (Fig. 1). For T = T1 a monotonous progress curve develops from the initial to the final state. In another experiment, where a higher temperature T~ is fixed, a different evolution is noted: A step occurs on the progress curve corresponding to the occurrence of a structurally defined phase SA. Its composition and structure are different from that of $2. Its lifetime is shown by the boundaries of the step; it disappears spontaneously without any change in the constraints. The phases thus detected have no domain

$2

Concept of Movement Diagram The term "movement of a thermodynamic system" coined by Fer (7) means, at a given time, the set of generalized rates. If the system state is defined by the variables Xi, its movement is defined by dXJdt. At equilibrium there is no movement. As considered by kinetics, where the concentration variation of intermediates is zero, although the whole system changes, steady states are not characterized by a zero movement. For such states, kinetic data must, therefore, be incorporated into the nonequilibrium diagram. Journal of Colloid and Interface Science, Vol. 70, No. 2, June 15, 1979

SA

J

2

3

4

TIME(h~r)

FIG. 1. Reaction MgCrO4, 5H~O---~ MgCrO4, 2H20 + 3H~O. Variation of hydration extent n during dehydration, versus time: Paso = 31.8 Torr, T1 = 72°C, /'2 = 105°C.

225

EVOLUTION IN SOLID-GAS SYSTEMS

o

10 8

I I

6

il I II 1

11 l/Ill

. .

4

~

2

25 S'

~

-)'0

80

90

" a5

~

IL 45

~re)

FIG. 3. Reaction CuSO~, 5H~O---~ CuSO,, 3H20 + 2H=O. Movement diagram with the isokinetic curves (arbitrary units for rates): dotted domain, slowing down rate domain.

100 ~rc)

FIG. 2. Reaction MgCrO4, 5H20 ~ MgCrO4, 2H~O + 3H20. A movement diagram: hatched domain, occurrence domain of step SA; dotted domain, occurrence domain of step SB; S~ domain, domain of monotonous progress curve.

Let us consider another diagram of sulfate CuSO4, 5H20, but where the datum retained is the shape of the dehydrated domain on a monocrystalline face. From the experiment, a movement diagram can be plotted, where the various (P, T) domains are located,

in the equilibrium diagram, whereas the movement diagram shown in Fig. 2 determines their domain of occurrence in outof-equilibrium conditions. Let us consider a second example, given by the dehydration kinetics of pentahydrate copper sulfate into trihydrated salt (9). In this case, where the progress curve profile does not undergo any notable variation, the kinetic information retained is the reaction rate, measured from the slope of the progress curve. The data related to this feature of evolution are listed on the P, T diagram by isokinetic curves, that is to say, the curves linking the constraints for which the rate is constant (Fig. 3). From the equilibrium curve, which is only a particular curve of this network, of rate zero, are located those curves depending on increasing rates. A particular shape, however, occurs showing a domain where the reaction rate slows down. In kinetics (10) this slowing down also occurs in the plotting of the usual rate-pressure and rate-temperature curves.

FIG. 4. Reaction CuSO4, 5H=O---~CuSO4, 3H=O + 2H~O: 0 experiments where elliptical shapes are obtained on monocrystalline platelets; O experiments with angular shapes. Journal o f Colloid and Interface Science,

Vol.70,No.2,June15,1979

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BERTRAND, LALLEMANT, AND WATELLE

GAS(4-] MOBILE INTERFACECA) INTERFACE

experimental aspect of evolution, nonequilibrium thermodynamics can also clarify interpretation in at least two ways that will be examined successively.

SOLID {--)

INTERFACE COUPLING Fro. 5. Schematic view of interface irreversible processes.

each by the occurrence of a well-defined geometrical form (Fig. 4) (I 1). For the same reaction, different diagrams can thus be considered according to the datum retained. It then seems of interest to establish correlations between those different diagrams, perhaps to find out a basic unity. Those examples, which may be multiplied, show the essential and new character of the movement diagrams, i.e., different given domains of constraints each characterized by the conservation of a given kinetic feature of the evolving system. Beyond the unified representation of the

chemical reactions

surface heat transport

Interface Entropy Production The first way is the description of interface processes and their interaction. This description can be easily made qualitatively, as shown in Fig. 5. But its mathematical expression is difficult. However, an interesting approach was made from the expression of interface entropy production ~'(S). It includes within an overall formula the different irreversible processes of the interface which contribute to the system evolution. The different stages of establishment of the formula will not be developed here (12, 13); only the overall expression given by Eq. [1] will be discussed. For the sake of clarity, viscosity, surface tension, and external forces action have been removed.

surface diffusion

+

discontinuity of chemical potential The various symbols are explained in Table I. Eq. [1] consists of two parts. The first part includes the contributions of the interface irreversible processes: chemical reactions, surface heat transfer, and surface diffusion. The second part includes the contributions resulting from the existence of discontinuities at interface level, which are discontinuities in temperature and in the chemical potential of the conJournal of Colloid and Interface Science, Vol. 70, No. 2, June 15, 1979

E(1 l) ~r

3

(q- + p(v_ - _ V ) h )

]

.~_

[1]

discontinuity of temperature

stituents whose contributions also take into account the interface displacement.

Phenomenological Relations Usually the entropy production expression results in the phenomenological relations establishing couplings between irreversible processes. A generalized flow, indeed, does not depend solely on the gen-

EVOLUTION

IN SOLID-GAS

TABLE I List of Symbols Volume

s¢~ g~ h _J~j q T

Interface

~k ~ _3oj ~ ~ y

..v fi'k

~ ~i'k

Yj

~

p V [A] = A + -

A-

Chemicalaffinityof the kth reaction Mass chemicalpotential of speciej Mass enthalpy Diffusion flow of speciej Heat flow vector Temperature Actual velocity of interface progression Particle velocity Molar rate of the kth chemical reaction Mass fraction of speciej Normal to the interface Mass density Space derivation operator Surface derivation operator A+ and A- are the value of the quantity A on either side of the interface

eralized force which is naturally associated with it, but also on other forces of the same tensorial order occurring in entropy production (5). Examples of these relations applied to matter transport in solids were recently envisaged (14). Two applications specific to the interface can be deduced from Eq. [1] and they can also apply to solid-gas reactions. The first concerns the coupling of two interface fluxes resulting from two scalar generalized forces, which are the jumps in temperature and of chemical potential at the interface (15, 16). Equation [2] explains this coupling:

(

rh = Lll g+ T+

gT--

g+ T+

gT --

(

c~ = L21

) + L12( T+1

)(1

+ Lzz T+

T-

1)

[2]

T-

where the coefficients Llz and Lzl are the coupling coefficients or cross-coefficients between the mass th and energy c~ fluxes. The study on the kinetics of dehydration (17) shows that this coupling is responsible

227

SYSTEMS

for the reversed effect of the isokinetic curves obtained for those reactions (see Fig. 3). The second application results from the fact that chemical affinity is no longer the only scalar generalized force but also the temperature jump and the chemical potential jumps. Therefore the rate of an interface reaction no longer depends on chemical affinity but also, through phenomenological relations, on the thermodynamic forces mentioned above. This new relation expresses a case of active transport across the interface. Finally, entropy production writing is interesting in another respect: It plays a major part in the studies of instabilities. This is the last point to be dealt with. INSTABILITIES

IN SOLID-GAS

REACTIONS

When analyzing mathematically the kinetics of solid-gas reactions, steady-state conditions fixed by one or two limiting processes are frequently assumed (18-22). This assumption, together with that required for the simplification of the calculation, which is made linear, leads to a resolution excluding the search of instabilities. But this does not prove that they do not exist, to the same extent as the numerous cases known and investigated in homogeneous kinetics (2-6) or provided by reacting interface media such as membranes (23, 24), solidification fronts (25 to 28), electrochemical reactions (29, 30), or heterogeneous catalysis (31-34).

An Experimental Aspect Experimenting solid-gas reactions often raises doubts, which should urge kinetists to investigate this field. The following example illustrates this fact. In the results obtained from silver sulfidation (35, 36), two different kinds of steady working conditions were detected. The condition corresponding to the reaction carried out under a pressure of 15 Torr of hydrogen sulfide is characterized by the parabolic profile of the progress curves. The other Journal of Colloid and Interface Science, Vol. 70, No, 2, June 15, 1979

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BERTRAND, LALLEMANT, AND WATELLE

condition, located in a pressure range less than 1 Torr, is characterized by the sigmoidal profile of the progress curves. Unfortunately, the intermediate domain has not been investigated by the authors and, thus, the problem of the nature of the transition cannot be fundamentally dealt with. This illustrates a type of result usual in kinetics where the different types of limit steady states of a reaction are studied independently (18-22, 35, 36). It is to be regretted that no consideration has been given to the transition. The existence of an instability may well be considered as a basis for the so-called "working condition change," as suggested by the curves in Fig. 6.

(5, 37). When some terms constituting this expression are negative, it may be foreseen that the corresponding processes are destabilizing in out-of-equilibrium steadystate conditions. They result in a change in the progress of the reaction beyond a given critical point. This means the occurrence of an instability. In the case of a chemical process whose rate ri, and the chemical affinity are known, entropy production o-(S) is written as: 6-(S) = ri, - - . T

The excess entropy production resulting from fluctuations 8~ and 8(~g/T) will then be written as dr(~S) = ~ W S ( ~ )

A Theoretical Aspect In addition, the following, simple calculation shows that the question of instabilities should always be debated. Let us first recall the principle of such a study. Instabilities may be predicted from the analysis of excess entropy production

(2)

variableT Z(1) !i

1

15

variableT (1) ~

(2) CONSTRAINT

CONSTRAINT (PH2s )

INSIAB)LIIY

(a)

State variableA

(b)

State

(1)

~"°''~)

[4]

Let us now look into the expression of excess entropy production in the case of a thermal decomposition reaction of a solid, whose rate in steady-state conditions depends linearly on the distance to equilibrium (38).

State

State

[3]

(2) ~

variableI (1)

(2)

f CONSTRAINvT INSTABItlTY (c)

CONS~.~T ~

NO INSIABItlTY (d)

FIG. 6. Schematic views on steady working conditions changes: (a) experimental results on the reaction 2Ag + H2S ~ Ag2S + H2; (b) hypothesis of a symmetry-breaking transition (instability); (c) multiple steady state (instability); (d) hypothesis of a continuous variation of state variables (no instability). Journal of Colloid and Interface Science, VoL 70, No. 2, June 15, 1979

EVOLUTION IN SOLID-GAS SYSTEMS

In this case,

229

from equilibrium might be accounted for by the instability of steady-state conditions. CONCLUSION

the chemical affinity is expressed by

A T

-

.R I n -

e Pe

[6]

where k, Pe, and P, respectively, denote a kinetic constant, the gas pressure at decomposition equilibrium, and the gas pressure in the reaction conditions. Now, it should be taken into account that, at reacting interface, there are temperature (ST) and pressure (SP) fluctuations around the steady values; the reaction is thermal and releases a gas. Hence, from Eqs. [5] and [6] and account taken of the dependence of Pe and k with temperature, the reckoning of relation [4] gives the following expression of excess entropy production:

+ R 1-

(RT~)2

kE 8TSP] pR7n

[71

where product 8PST is positive as given by the ideal gas law. Studying the sign of the terms of Eq. [7], the first term is always positive, the second either negative or positive depending on the sign of the heat of reaction AH, and the third is always negative. Thus, in these types of steady-state conditions, where, a priori, no destabilizing process seemed present, harmful contributions, i.e., negative ones, appear explicitly. The possible instability of these working conditions beyond a given threshold and the transition toward other working conditions are not to be excluded. Hence, some of the changes in the kinetic working conditions occurring frequently far

In conclusion, the examples given are sufficient to show that the right use of nonequilibrium thermodynamics will help in the knowledge of those phenomena and will finally propose an overall, coherent analysis of the kinetics of solid-gas reactions by integrating it into the general framework of the evolution of nonequilibrium systems. REFERENCES 1. Glansdorff, P., Entropie 78, 4 (1977). 2. Nicolis, G., and Prigogine, I., "Self Organization in Non-equilibrium Systems." Wiley, New York, 1977. 3. Pacault, A., Hanusse, P., de Kepper, P., Vidal, C., and Boissonade, J., Acc. Chem. Res. 9, 438 (1976). 4. Franck, U., Angew. Chem. lnt. Ed. 17, 1 (1978). 5. Glansdorff, P., and Prigogine, I., "Thermodynamics Theory of Structure, Stability and Fluctuations." Wiley (Interscience), New York, 1971. 6. Tyson, J. J., J. Chem. Phys. 66, 905 (1977). 7. Fer, F., "Thermodynamique Macroscopique," Vol. 1, p. 26. Gordon and Breach, Paris, 1970. 8. Lallemant, M., Thesis, University of Dijon, 1974. 9. Bertrand, G., Thesis, University of Dijon, 1976. 10. Bertrand, G., Lallemant, M., and Watelle-Marion, G., J. lnorg. Nucl. Chem. 36, 1303 (1974). 11. Comperat, M., Lallemant, M., Bertrand, G., and Watelle, G., J. Solid State Chem., submitted. 12. Barrere, M., and Prud'homme, R., "Equations Fondamentales de l'Arrothermochimie." Masson, Paris, 1973. 13. Bertrand, G., and Prud'homme, R., Int. J. Quantum Chem. 13 (in press). 14. Wagner, C., in "Progress in Solid State Chemistry" (J. McCaldin and G. Somorjai, Eds.), Vol. 10, p. 3. Pergamon Press, London, 1975. 15. Bornhorst, W. J., and Hatsopoulos, G. W., J. Appl. Mech. 840 (1967). 16. Waldmann, L., and Rtibsamen, R., J. Naturforsch. 27a, 1025 (1972). 17. Bertrand, G., Lallemant, M., Mokhlisse, A., and Watelle, G., J. Inorg. Nucl. Chem. 40, 819 (1978). 18. Barret, P., "Cinrtique Hrtrrog~ne." GauthierVillars, Paris, 1973. 19. Besson, J., in "Reaction Kinetics in Heterogeneous Chemical Systems" (P. Barret, Ed.), p. 463. Elsevier, Amsterdam, 1975. Journal of Colloidand InterfaceScience, Vol,70. No. 2, June 15. 1979

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20. Barret, P., Dufour, L. C., and Delafosse, D., in "Reactivity of Solids" (R. W. Roberts and P. Cannon, Eds.), p. 507. John Wiley, New York, 1969. 21, Bardel, R., and Soustelle, M., J. Chirn. Phys. 71, 21 (1974). 22. Larpin, J. P., Coison, J. C., and Delafosse, D., Mat. Res. Bull. 10, 641 (1975). 23. Ibanez, J. L., and Velarde, M. G., J. Phys. Paris 38, 1479 (1977). 24. Van Lamsweerde-Gallez, D., Bisch, P. M., and Sanfeld, A., in "Proc. 4th Int. Symp. on Bioelectrochemistry." Woods Hole, (in press). 25. Kirkaldy, J. S., in "Non-Equilibrium Thermodynamics, Variational Techniques and Stability," p. 281. University of Chicago Press, Chicago, 1966. 26. Viaud, P., in "Advances in Chemical Physics" (I. Prigogine and S. A. Rice, Eds.),Vol. 32, p. 163. Wiley, New York, 1975. 27. Wollkind, D. J., and Maurer, R. N., J. Cryst. Growth 42, 24 (1977).

Journal of Colloidand Interface Science, Vol. 70, No. 2, June 15, 1979

28. Langer, J. S.,Acta Met. 25, 1113 (1977); 25, 1121 (1977). 29. Poncet, P., Braizaz, M., Pointu, B., and Rousseau, J., J. Chim. Phys. 74, 452 (1977). 30. Epelboin, I., Ksouri, M., and Wiart, R., in "Proc. 4th I.C.C.T., Montpellier" (J. Rouquerol and R. Sabbah, Eds.), Vol. 7, p. 100. 1975. 31. Wankat, P. C., and Schowalter, W. R., A.I.Ch.E. J. 17, 1346 (1971). 32. Bdzil, J., and Frisch, H. L., Phys. Fluids 14, 475 (1971); 4, 1077 (1971). 33. Luss, D., in "Chemical Reaction Engineering," Proc. Int. Symposium 4th, Vol. 2, p. 487. 1976. 34. Eigenberger, G., in "Chemical Reaction Engineering," Vol. l, p. 290. 1976. 35. Barret, P., C.R. Acad. Sci. Paris 266C, 168 (1968). 36. Dvoraczek, J. P., Delafosse, D., Barret, P., Colson, J. C., and Sorbo, B., Bull. Soc. Chim. France 3421 (1970). 37. Defay, R., Prigogine, I., and Sanfeld, A.,J. Colloid Interface Sci. 58, 498 (1977). 38. Barret, P., C.R. Acad. Sci. Paris 266C, 856 (1968).