Comment on Rate of Polymorphic Transformation Between Phases I1 and 111 of Hexachloroethane BY HORIA METIU Department of Chemistry, University of California, Santa Barbara, California 93 106, U.S.A. JOHN ROSS* Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 021 39, U.S.A. AND Received 27th February, 1978 We comment on an article by Koga and Miura in this journal by providing clarifications concerning the use of kinetic equations for phase transitions and of thermodynamic potentials for metastable conditions. The article by Koga and Miura' prompts us to make some clarifications concerning the thermodynamic meaning and the proper use of the kinetic equations describing the rate of phenomena related to phase transitions, such as nucleation,2 spinodal decomposition and c~ndensation.~ The observations made in the present note are not new? and their purpose is to complement the presentation in ref.(4) and hopefully thereby prevent confusion. The macroscopic theory of the kinetics of phase transitions is based on the requirement that the relaxation of the nonequilibrium system proceeds in a way that causes the decrease of some generalized thermodynamic potential. When implemented mathematically this requirement yields two equations : and We use here the example of a binary system in which the " order parameter " is the concentration C(Y, t ) of the component 1 at site Y and time t. F[T,p, C(Y, t)] is the generalized thermodynamic potential. The first equation is valid for cases (diffusion- like) in which the order parameter is conserved : the change in the amount of the component 1 at a given site is equal to the net flux through a surface surrounding the site.The second equation is valid for cases (chemical reaction-like) in which such conservation is not required. One example is provided by phase transitions in some biopolymers :' the order parameter is the concentration of hydrogen bonds and these bonds are formed at the site, by a " chemical reaction ", not transported from the neighbouring regions. The difference between the two equations is substantial and we feel that they are not interchangeable. Koga and Miura's use of eqn (2) instead of (1) makes, in our t The comments made in this Note are available but dispersed in the literature. 2750H.METIU AND J. ROSS 275 1 opinion, any conclusion about agreement between theory and experiment question- able. To see why this is so let us apply the two equations to a familiar case, such as relaxation of a concentration inhomogeneity in a system which is near equilibrium. We use for the generalized potential the customary * form F = 1 dr{flT, P, c(r, t)l+K(Vc)2). (3) Herefis the “ homogeneous part ” of the generalized potential and K(VC)~ takes into account the existence of the concentration gradients. Inserting eqn (3) into (1) and (2) yields ac ( r , tyat = rv2 - -rm4c C) . . and ?f ac ac ( r , tyat = ri - - riKv2c. (4) If we are interested in concentration relaxation near equilibrium we may assume that c(r, t ) - co is small (co is the homogeneous, equilibrium concentration) and hence write Inserting this in eqn (4) we obtain This is the customary diffusion equation (the second term at the right hand side can be neglected if the system is not very close to the consolution critical point, with the diffusion coefficient D = ra2f/dc& Comparing with the known thermodynamic definition of the diffusion coefficient we infer that df/dco is proportional to (,ul/ml) - (p2/m2), where p i and mi are the chemical potential and the mass of the component i and c is the concentration of component 1.Eqn (5), on the other hand, becomes Choosing the generalized potential so that af/aco = 0 at equilibrium [this amounts to subtracting a constant from af/aco and does not modify eqn (7)], we see that eqn (8) describes a system in which the component I undergoes a first order chemical reaction and diffuses.The “ reaction rate ” term is first order only because we consider the case when c(r, t ) - co is small. The time evolution of the concentration given by the two equations is very different and we hope that the application of eqn (1) and (2) to this familiar example illustrates the problems encountered using eqn (2) for cases in which eqn (1) is required. The second topic we want to comment upon concerns the properties and the meaning of the generalized potential. Our views have been strongly influenced by those of Langer.5 The customary thermodynamic treatment of irreversible processes uses the so- called local equilibrium assumption,’ which stipulates that one can coapute the value of a thermodynamic potential for a non-equilibrium state by inserting the non- equilibrium values of the state parameters [e.g., T, p and c(r, t ) ] into the functional2752 RATE OF POLYMORPHIC TRANSFORMATIONS form of the thermodynamic potential at equilibrium.Eqn (6) is one possible way of carrying out this procedure for states near equilibrium. In general terms this amounts to an analytic continuation of the equilibrium potential to non-equilibrium states. In the case of phase transition this procedure has difficulties. The thermo- dynamic potential function ends abruptly lo at the coexistence line of the phase diagram. The coexistence point is a singularity of the potential and it “ blocks ” lo the analytic continuation to values of c corresponding to metastable or unstable systems.The difficulty can, however, be circumvented in the following manner. Consider for example the Helmholtz free energy Here p = 1 /kT, V is the volume andfis the free energy density. We can compute a free energy f corresponding to the unstable homogeneous system of concentration c, by including in the computation of the trace only those states that correspond to the specified concentration c. By imposing this constraint we define a free energyf(c) that must be higher than the one corresponding to the equilibrium state. We emphasize that this method is not as artificial as it may at first appear. True equilibrium states are such only because we impose constraints, as €or example requiring that all the molecules are confined in a box of volume V.If we remove one wall of the box the true equilibrium state becomes unstable. The free energy of this unstable state (without the wall) is defined by the present procedure to be equal to the free energy of the stable equilibrium system obtained when the wall is present. While this procedure explains the molecular (statistical) meaning of the non- equilibrium thermodynamic potential (at concentrations for which a true equilibrium thermodynamic potential does not exist) it is not yet possible to implement it to obtain expressions for f. In order to establish the properties off (here we return to the general problem andfis again the homogeneous part of the generalized potential, not necessarily the free energy) we use the kinetic equation [e.g., eqn (4)] and require that f has features which describe the observed behaviour of the metastable states.Such a requirement is quite reasonable, in view of the fact that metastability is the result of kinetic constraints and that purely thermodynamic arguments predict that a metastable state does not exist. The observed qualitative behaviour of a metastable system is reproduced if we ask thatfhas two minima, one at the concentration of the metastable state c1 and the other at the concentration of the equilibrium state co (the absolute minimum). Between them we must have a maximum, at the concentra- tion c2, and as a consequence the two states, of concentration c1 and co, are separated by a barrier. This construction gives the kinetic equations the desired properties. The homogeneous equilibrium state is a steady state of the kinetic equation (af/aco = 0 implies aco/dt = 0).A small inhomogeneity imposed on the equilibrium state relaxes by diffusion (for conserved variables). The homogeneous metastable state is also a steady state (8f/acl = 0 implies &,/at = 0). A small inhomogeneity in this state relaxes by diffusion and with the diffusion coefficient Ta2f/dcf (which could be different from that of the equilibrium state, which is r a’fj’ac;). Thus a meta- stable state in which only small inhomogeneities are possible is stable forever. Inhomogeneities that are large enough to overcome the barrier between the metastable and the stable states destroy the metastable state (nucleation). The height of the barrier, which is the factor preventing nucleation, must therefore decrease with the degree of supercooling. When the spinodal line is crossed, the barrier disappears, the diffusion coeffcient becomes negative and the phase is unstable.This qualitative picture has been explored quantitatively in detail. 2-4H. METIU AND J . ROSS 2753 To obtain the equation used and criticized by Koga and Miura we have chosen the form of if/& to be given by a(c-cO)(c-c1)(c-c2), which is the simplest form that has the three extrema discussed above. This allows us to ohtain analytically a solution for the kinetic equation. Koga and Miura have objected to the form used by us, since they interpreted it to mean that the chemical potentials of the stable and metastable states are equal, a conclusion which is unacceptable.First, we note that the use of thermodynamk arguments in matters concerning metastability is not safe. Rigorous use of equilibrium thermodynamics would only tell us that metastable states cannot exist. It is necessary to use kinetic arguments, like those presented above, since metastability is a kinetic phenomenon. Second, it is possible to use mean field thermodynamic arguments to show that the type of choice made by us is consistent with a macroscopic, hence '' thermodynamic 'I analysis. The use of a mean field theory allows an analytic continuation of the thermodynamic potential "A Vi VB V 2 V C V D V o v FIG. 1.-Plot of derivative of Helmholtz free energy, a$/av, against v = V - Vc, where Vis the specific volume and V, the specific volume at the critical point.inside the metastable and unstable region since the mean field approximation does not display the singularity at the coexistence line that the exact potential does. If we denote by i,b the Helmholtz free energy per gram and by Y the specific volume, then we have l1 a@/av = -Azv-Bv3/3 +$(T) where v = Y - V,, z = T-T,, A and B are constants and $(T) is a function of temperature which we do not need to specify. A plot of a$(v)/av against v is shown in fig. 1. The points A and D correspond to the coexisting phases at the chosen temperature. Any specific volume between v A and vB, or vc and vD corresponds to a metastable phase ; those between v, and vc are unstable. Such a system can be prepared by starting at point 1, keeping the temperature constant and lowering the pressure in the direction indicated by the arrow.Consider now the Gibbs free energy g(v) = $(v)+plv, where p1 is the externally imposed pressure. We have ag/av = a$(v)/av+p, and we see from the graph that ag/dv = 0 for v = vl, v = v2 and v = vo. This shows that g has the properties required by kinetic arguments. If we put the metastable and the stable phases in contact, as required in a condensation problem (the boundary between the phases changes due to condensation), we see that ag/av = 0, for v 1 and Consider now a system at temperature 2, pressure p1 and volume vl. Here vo corresponds to the true equilibrium state.2754 RATE OF POLYMORPHIC TRANSFORMATIONS vo, is an expression of the mechanical equilibrium condition which requires that the two phases be at the same externally imposed pressure.It does not mean, as interpreted by Koga and Miura, that the two chemical potentials are forced to be equal. In fact, using dp = (V, +v)dP = - (V, + v) at,b/dv dv, one can show that the chemical potentials are equal only for the coexisting phases (pA = pD) which satisfy the Maxwell construction, and that pE # pF. When other slow variables are introduced, and this must often be done, the order parameter becomes very complex;12 the simple situations considered above are only illustrative. H. M. thanks the Research Corporation, and the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this work. J. R. thanks the National Science Foundation and the Air Force Office of Scientific Research for partial support of this work.Y. Koga and R. hi. Miura, J.C.S. Faraday I, 1978, 74, 191 3. J. S. Langer and L. A. Turski, Phys. Rev. A, 1973, 8, 3230. J. W. Cahn, Trans. Metal. SOC. A.I.M.E., 1968, 242, 166; J. S. Langer, Ann. Phys. ( N . Y.), 1973, 78, 421 ; J. S. Langer, in Fluctuations, Instabilities and Phase Transitions, ed. T. Riste (Plenum Press, N.Y., 1975). H. Metiu, K. Kitahara and J. Ross, J. Chem. Phys., 1976, 64, 292. J. S. Langer, Physica, 1974,73, 61. See, for example, H. Metiu, K. Kitahara and J. Ross, J. Chem. Phys., 1976, 65, 393. J. D. van der Waals, 2. phys. Chem. (Leigzig), 1894,6,657 ; V. L. Ginzburg and L. D. Landau, Zhur. Eksp. Teor. Fiz., 1950,20, 1064; J . W. Cahn and J. E. Hilliard, J. Chem. Phys., 1958, 28,258 ; 1959, 31, 688. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, New York, 197% p. 221. l o M. E. Fisher, Physics, 1967, 3, 255 ; J. S . Langer, Ann. Phys. (N. Y.), 1967, 41, 108. l 1 L. D. Landau and E. M. Lifshitz, Statistical Physics (Addison-Wesley, Reading, Mass., 1969). l2 P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys., 1977, 49,435 ; K. Kawasaki, in Phase Transitions and Critical Phenomena, ed. C . Domb and M. S. Green (Academic Press, N.Y., 1976), vol. 5. ' E. Neumann, Angew. Chem. (Int. Edn.), 1972, 12, 356. (PAPER 8/356)