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Cahn Hilliard的论文

Free Energy of a Nonuniform System. I. Interfacial Free EnergyJohn W. Cahn and John E. Hilliard Citation: The Journal of Chemical Physics 28, 258 (1958); doi: 10.1063/1.1744102 View online: /10.1063/1.1744102 View Table of Contents: /content/aip/journal/jcp/28/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Lennard-Jones systems near solid walls: Computing interfacial free energies from molecular simulation methods J. Chem. Phys. 139, 084705 (2013); 10.1063/1.4819061 Bcc crystal-fluid interfacial free energy in Yukawa systems J. Chem. Phys. 138, 044705 (2013); 10.1063/1.4775744 Calculation of interfacial properties via free-energy-based molecular simulation: The influence of system size J. Chem. Phys. 132, 224702 (2010); 10.1063/1.3431525 Free Energy of a Nonuniform System. III. Nucleation in a TwoComponent Incompressible Fluid J. Chem. Phys. 31, 688 (1959); 10.1063/1.1730447 Free Energy of a Nonuniform System. II. Thermodynamic Basis J. Chem. Phys. 30, 1121 (1959); 10.1063/1.1730145This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 109.171.137.210 On: Sun, 13 Jul 2014 07:39:08THE JOURNAL OFCHEMICALPHYSICSVOLUME 28.NUMBER2FEBRUARY.1958Free Energy of a Nonuniform System. I. Interfacial Free EnergyJOHN W. CARN A:!>.'D JOHN E. HILLIARD General Electric Research Lab(}ratory, Schenectady, New Y(}fk (Received July 29, 1957)I~ is shown that the free energy of a volume V of an isotropic system of nonuniform composition or density given by: Nvfv [fo(C)+K(V'C)2]dV, where N v is the number of molecules per unit volume V'c the composi~ion or density gradient, fo the free energy per molecule qf a homogeneous system, and K a ~arameter which, III general, may be dependent on c and temperature, but for a regular solution is a constant which can be evaluated. Thi~ expression is used to de~ermine the properties of a flat interface between two coexisting phases. I~ pa~tlcular, we. ~nd that the thickness of the interface increases with increasing temperature and becomes Illfimte at the cntical temperature T e, and that at a temperature T just below T the interfacial free C energy rr is proportional to (T.- T)J. :r~e predict~d interfacial free energ>: ~r;d its terr;peratu:e dependence are found to he in agreement with eXlstmg expenmental data. The possibility of usmg optical measurements of the interface thickness to provide an additional check of our treatment is briefly discussed..IS1. INTRODUCTIONN most previous theoretical treatments of interfacial I energies the interface has been arbitrarily restricted to some predetermined thickness. Thus Youngl(a) and Becker)(b) assumed that two adjoining phases are homogeneous up to their common interface, while others2(a) ,(b) have made calculations based on the existence of a single intermediate layer. Thoug

Cahn Hilliard的论文

h such assumptions may be justifiable in certain instances,3 it is evident that they are incorrect in principle since, once the temperature and pressure of the system are specified, the interfacial thickness is no longer an independent variable. Many years ago, Rayleigh 4 noted that the expression derived by Young1(a) was of such a form that the tension of an interface should be inversely proportional to the number of intermediate layers plus one. However, Rayleigh neglected to take into account the increase in free energy resulting from the introduction of nonequilibrium material in a diffuse interface, and he was therefore unable to estimate the interfacial thickness. The first calculation of the equilibrium thickness was apparently made by Ono· (and later repeated, independently, by Hillert 6). Though the approach used by these two authors is undoubtedly correct, it requires the numerical solution of a set of difference equations for each particular case. This procedure is not only tedious, but also obscures certain properties of the interfacial energy and precludes its expression in an analytical form. In addition, the calculations were based on the1 (a) Miscellaneous Works of the Late Thomas Young, George Peacock, editor (J. Murray, London, 1855), Vol. 1, pp. 462-466' (b) R. Becker, Ann. Physik 32, 128 (1938). ' 2 (a) E. A. Guggenheim, Trans. Faraday Soc. 41, 150 (1945); (b) R. Defay and I. Prigogine, Bull. soc. chim. Belges 59, 255 (1950). 3 Murakima, Ono, Tamura, and Kurata, Phys. Soc. Japan 6, 309 (1951). 4 Lord Rayleigh, Phil. Mag. 16, 309 (1883); ibid. 33, 209 (1892). S. Ono, Mem. Fac. Eng. Kyushu Univ. 10, 195 (1947). 6 M. HilJert, "A theory of nucleation for solid metallic solutions," D.Se. thesis, Massachusetts Institute of Technology Cambridge (1956). 'nearest neighbor regular solution* model and there is thus some doubt as to their general validity. The treatment that we will adopt is analogous in some respects to those used for the evaluation of the energy of magnetic 7 and ferroelectric 8 domain walls and . ' of the mterface between a metal in its normal and 9 s~perconducting states. We will derive a general equatIOn for the free energy of a system having a spatial variation in one of its intensive scalar properties, such as composition or density. We will refer to such a system as being "nonuniform." In a subsequent paper we will use this equation as the starting point for a new theory of three-dimensional nucleation. For the present, however, we will confine its application to determining the free energy of a flat interface between two coexisting phases. This will include both a general treatment (Sec. 2) and an evaluation (Sec. 3) in terms of the regular solution theory. In Sec. 4 we will check the predicted interfacial free energy against existing experimental data. The paper will conclude with a brief discussion of certain optical methods which might provide an additional experimental check on the validity of our tre

Cahn Hilliard的论文

atment.2. GENERAL TREATMENTa. Free Energy of a Nonunifonn SystemThe following analysis is valid for any intensive scalar property of the system other than temperature or pressure, but to simplify the exposition we will suppose that the system is a binary solution and that the nonuniform property is c, the mole fraction of the B component. We would expect that the local free energy per molecule, t j, in a region of nonuniform composition will depend both on the local composition and on the* Several different meanings are associated with the term :'regular solutio~." We will use it to denote a solution having an Ide,:l configurattonal entropy and an enthalpy of mixing which vanes parabolically with composition [see Eq. (3.1)]. 7 F. Bloch, Z. Physik 74, 295 (1932). 8 T. Mitsui and J. Furuichi, Phys. Rev. 90, 193 (1953). 9 J. Bardeen, Phys. Rev. 94, 554 (1954). t Symbols are listed in the appendix.258This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 109.171.137.210 On: Sun, 13 Jul 2014 07:39:08INTERFACIAL FREE ENERGY259composItIon of the immediate environment. We will therefore attempt to express 1 as the sum of two contributions which are functions of the local composition and the local composition derivatives, respectively. We will assume that the composition gradient is small compared with the reciprocal of the intermolecular distance and will take c and its derivatives as independent variables. Providing 1 is a continuous function of these variables, it can be expanded in a Taylor series about jo the free energy per molecule of a solution_of uniform composition c. Employing the subscripts i, j, in the usual manner to denote the successive substitution of the x, y, and z components for the variable Xi, and the subscript zero to indicate the value of the parameter in a solution of uniform composition, leading terms in the expansion for j are:IJIII III IIIr--AC.--:--ACe--;IIIoCac~Ch MOL. FRACTION OF BFIG. 1. fo(e) for T < Te.j(c, V c, 'V'lc,· .. ) = jo(c) Li L i ( iJc/ iJXi) L'j KiP) (iJ2cj OXiOXj) + (1/2)Lo KiPl[(iJCj(JXi)(acjaXJ)]+' .. , (2.1)++where N v is the number of molecules per unit volume. By applying the divergence theorem we obtain:whereLi= [ajla(acjax,)Jo, KiP) = [a j I a(a2cj aX,aXj) Jo, KiP) = [a 2j / a (acl aXi)a (acl aXj) Jo.(2.2)fv(KIV'2C)dV = -fv(dKlldc) (V'c)2dV +fs(K1Vc.n)dS.(2.5)In general, KiP) and KiP) are tensors reflecting the crystal symmetry and the L/s are components of a polarization vector in a polar crystal. For a cubic crystal or an isotropic medium (and these are the only cases that we will consider) the free energy must be invariant to the symmetry operations of reflection (X,-"7-Xi) and of rotation about a fourfold axis (x,-"7Xj). Therefore,Since we are not concerned with effects at the external surface, we can ch

Cahn Hilliard的论文

oose a boundary of integration in Eq. (2.4) in such a manner that V'c·n is zero at the boundary. The surface integral therefore vanishes and we can employ Eq. (2.5) to eliminate the term in V'2c from Eq. (2.4) to obtain:F=NvwhereLi=O, K,/l) =Kl= [iJj/av2 c]o for i= j, KiP) =0 for i¢ j, KiP)=K2= [a 2jl(al VCI)2Jo for i= j,andfv[jO+K(V'C)2+ . . JdV,(2.6)K= -dKI/dc+K2 = - [a 2 11 aca'V'lcJo+[a 2fl (a IV'c I)2JO.(2.7)KiP) =0 fori¢ j.Hence for a cubic lattice, Eq. (2.1) reduces totj(c,Vc,"ii2c,' .. ) = jo(C) +K1'V'lC+K2 (VC)2+ .. '.(2.3)Integrating over a volume V of the solution we obtain for the total free energy F of this volume:Equation (2.6) is the central one of the treatment. It reveals that, to a first approximation, the free energy of a small volume of nonuniform solution can be expressed as the sum of two contributions, one being the free energy that this volume would have in a homogeneous solution and the other a "gradient energy" which is a function of the local composition.F=NVifdV =N vIv [jO(C)+K V'2 + K2 (V'C)2+ ... JdV,1 C(2.4)t This equation can also be derived as follows. If we assume that the local free energy J is a function only of io and the composition derivatives then, since i, a scalar, must be invariant with respect to the direction of the gradient, only terms in even powers of the operator V' can appear. The leading terms of the function must therefore be of the form given in Eq. (2.3).b. Free Energy of a Flat Interface We will consider a flat interface of area A between two coexisting isotropic phases 0: and f3 of composition§ Cex and CfJ. It will be assumed that the free energy of nonequilibrium material of composition intermediate between Ca and CfJ can be represented by a continuous function fo(c) of the form shown in Fig. 1.§ If PL and PV (the densities, respectively, of a pure liquid and its vapor) are substituted for Ca and CfJ, then the equations derived in this section will apply specifically to an interface between the condensed phase and its vapor.This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 109.171.137.210 On: Sun, 13 Jul 2014 07:39:08260J. W. CAHN AND J. g. HItLIARDnot explicitly depend on x, the appropriate form 1o of the Euler equation isApplying Eq. (2.6) to the one-dimensional composition change across the interface, and neglecting terms in derivatives higher than the second, we obtain for the total free energy F of the system:1- (dc/dx) [ilI/ il (dc/dx)J= 0,where I represents the integrand. We thus obtain as the condition for a stationary value:(2.13)F=AN vf+00[joCc)+K(dc/dx)2Jdx.(2.8)-00The specific interfacial free energy, (1', is by definition the difference per unit area of interface between the actual free energy of the system and that which it would have if the properties of the phases were continuous thr

Cahn Hilliard的论文

oughout. Hence:The constant in this equation must be zero since ~fCc) and (dc/dx) both tend to zero as ~± ao. Hence for a minimum value of u:(2.14)u=N vJ+00[jo(c)+K(dc/dx)2 -c,c.!B(e)- (l-c)JtA (e)Jdx,(2.9)Using this expression to eliminate KCdc/dx)2 from Eq. (2.10) we find:-00u=2N vJ+00~f(c)dx.-00where JtA(e) and JtB(e) are the chemical potentials per molecule (referred to the same standard states as fo) of the species A and B in the a or (3 phase. For u to be uniquely defined it is obviously necessary that the chemical potential of a particular species be the same in both phases and therefore that the two phases be in equilibrium-a condition which is not required for calculating the energy of an interface with an abrupt composition change. Equation (2.9) can be rewritten:Changing the variable of integration from x to c by means of Eq. (2.14), we finally obtainll :(2.15)u=iV vwhere~f(c)J+00[~f(C)+KCdc/dx)2Jdx,(2.10)-00is defined byIn the next section we will use the regular solution theory for a numerical evaluation of u. The general treatment can, however, be taken a step further to determine the functional dependence of u on temperature in the immediate vicinity of the critical (or conjugate) temperature To at which the two phases attain the same critical composition c e If fo can be expanded in a Taylor series about Tc and10 H. Margenau and G. M. Murphy, The Mathematics oj Physics and Chemistry (D. Van Nostrand Company, Inc., Princeton, 1943), p. 195. II An interface between two fluids which differ in more than one scalar parameter is considerably more complicated. Consider the case in which two parameters, nand m, determine the free energy Aj(n,m) and let the corresponding gradient energy coefficients be Kn and Km. The interfacial energy is the minimum of:~f(c) = fo(c)- [cJtB(e)+ (l-c)JtA (e)J(2.11)~f(c) may therefore be regarded as the free energy referred to a standard state of an equilibrium mixture of a and {3 [Eq. (2.11)J, or as the free energy per molecule of transferring material from an infinite reservoir of composition Ca or Cfl to material of composition c [Eq. (2.12)]' According to Eq. (2.10) the more diffuse the interface is, the smaller will be the contribution of the gradient energy term, K(dc/dx)2, to U. But this decrease in energy can only be achieved by introducing more material at the interface of nonequilibrium composition and thus at the expense of increasing the integrated value of ~f(c). At equilibrium the composition variation will be such that the integral in Eq. (2.10) is a minimum. (This is equivalent t6 the requirement that the chemical potentials be constant throughout the system.) If we substitute the integrand of Eq. (2.10) in the Euler equation, we will obtain a differential equation whose solutions are the composition profile corresponding to stationary values (i.e., maxima, minima, or saddle points) of the integral. Since the integrand doesN vi:

Cahn Hilliard的论文

[Aj(n,m) +K,. (dn/dx)2+ Km (dm/dx)2Jdxwhere we have neglected the cross term. The Euler equations for this problem yield: andiMj/on = Kn(d:'n/dx2) oAj/om=Km(d:'m/dx2). u=2NvJ.~ (K nAf)i[I+(Km/K n)(dm/dn)2Ji dn,These can be combined to give:which reduces to Eq. (2.15) if m is a constant or Km is small. The way m changes with n through the interface is found by solving the two Euler equations to eliminate x. The solution will correspond to the trajectory of a particle of unit mass having a vanishing total energy (kinetic and potential) on a potential surface given by: -Aj(nVKn,mVKm). The particle starts from one of the coexisting phases and slides to the other; it does not follow potential troughs but banks on curves to reduce gradient energy at the expense of volume energy. There may be several different paths, all of which represent stationary values of the interfacial free energy, but only the path having the lowest free energy will correspond to the actual interface.This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 109.171.137.210 On: Sun, 13 Jul 2014 07:39:08INTERFACIAL FREE ENERGY261Ce, the following expression can be derived for Ai:Ai(T~Tc)= -(3(Te-T)[(AC)2- (AC.)2] +Y[(AC)4- (AC.)4]+...Cp----------)1;~(2.16)011in which AC= (c-c e), Ac e= (CIl-C e) = (cc-c",) and (3 and ')' are inherently positive constants defined by the following derivatives of fo evaluated at C=C e and T=Te: (2.17) (3= (lJ3fo/aTac 2)/2!')'=~ g:~II'I/I~~cQI II'. DISTANCEI<.)(a 4fo/ac4)/4!(2.18)-+------1---r---l----jFIG. 2. Interface profile.O~---------------------JThis expansion gives the following relationships:(ACe)2(T~Tc) =(3(T c -T)/2,)" (AC)2]2.(2.19) (2.20)(Ai)(T~Tc)=')'[(ACe)2-From Eqs. (2.15) and (2.20) we obtain:If in the vicinity of the critical point, K is continuous and nonvanishing, then we may, sufficiently close to Te, neglect any variations in K and assume it constant. This is equivalent to expanding K about the critical point and neglecting higher terms, or to applying the mean value theorem. Thus, we can evaluate the integral of Eq. (2.21) and use (2.19) to obtain:(q)(T~Tc) =systems12 ,13 but the coexistence curves for many gases14 and binary liquid mixtures16 ,16 appear to be cubic; i.e., (Te-T) in such cases is approximately proportional to (IACe 1)·3 So far there appears to be no satisfactory explanation for this anomaly. Since we have used Eq. (2.19) in deriving Eq. (2.22) the latter is strictly valid only for those systems having a parabolic coexistence curve. However, we will later show that it may also be a good approximation for the other systems.c. Composition Profile and Thickness of InterfaceThe composition variation across the interface as determined by Eq. (2.14) is such that:(2V2N v/3,),)KW(T e- T)i.(2.22)dc/dx= (Af/K)!.(2.23)Our an

Cahn Hilliard的论文

alysis, therefore, predicts that near the critical temperature the interfacial free energy should be proportional to (Te - T)!. It is fairly easy to prove that any model which confines the thickness of the interface to a fixed number (say p) of molecular planes leads to an expression for q which is proportional to (T e- T) / (p+ 1) and is thus linearly dependent on temperature. fI According to Eq. (2.19) the coexistence curve should be parabolic in the immediate vicinity of Ce. It can be shownll that this functionality should also apply to the density of a liquid and its saturated vapor near the critical point. This is found to be true for certain, This may be proved as follows: let the composition difference between the ith plane and its neighbor be Yi6c . The energy of the interface between these two planes is therefore proportional to (YiAc.)2. The total interfacial energy will be proportional to thepInspection of the Af function (Fig. 1) indicates that to satisfy Eg. (2.23) the composition profile must be sigmoid in shape as shown in Fig. 2. In the vicinity of the critical point we can make the appropriate substitutions from Egs. (2.19) and (2.20) and integrate Eq. (2.23) (assuming as before that K is constant) to obtain(AC/ Ace)(T~Tc)=tanh{[(3(T- Tc) /2K]!X},(2.24)where the distance x is measured from an origin at AC=O (i.e., c=cc)' Using this equation, the thickness 1 of the interface could be defined as the distance x for a given AC/ Ac. ratio. But for convenience in subsequent calculation we will express 1 in terms of the gradient at Cc as follows:pi=(tsum (6C.)2 ~ (Yi)2, plus the sum, ~ .1.fi, of the volume free energy1'=0terms. Sufficiently close to the critical point the latter contribution can be neglected since it varies as (Tc-T)2whereas (.1.C.)2 varies as(Tc- T). The minimum of (.1.C.)2 ~ (y.)2 subject to the conditioni=OpNear the critical temperature we obtain on substitution from Eqs. (2.23), (2.19), and (2.20): (2.26)A. Miinster and K. Sagel, Z. physik Chern. 7, 297 (1956). 13 Krichevskii, Khazanova, and Linshitz, Doklady Akad. Nauk. S.S.S.R. 100, 737 (1955). 14 E. A. Guggenheim, J. Chern. Phys. 13, 253 (1945). 16 J. D. Cox and E. F. G. Herington, Trans. Faraday Soc. 52, 926 (1956). 16 O. K. Rice, J. Chern. Phys. 23, 164 (1955).12p~Yi=1i=Ooccurs when each Yi=1/(p+1). Hence(q)T~Tc0:: (Tc -T)/(p+l). It is interesting to note that this is the expression derived by Rayleigh.4 11 R. Fowler and E. A. Guggenheim, Statistical M ecltanics (Cambridge University Press, London, 1949), pp. 316-318.This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 109.171.137.210 On: Sun, 13 Jul 2014 07:39:08262].W.CAHNAND ].E.HILLIARDThus we see that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature. Before leaving the ge

Cahn Hilliard的论文

neral treatment it should be emphasized that we have tacitly assumed that K is everywhere positive and nonvanishing. This is evidently true for the single phase region of the system as otherwise, contrary to experience, the homogeneous phase would be unstable with respect to periodic composition fluctuations. We can see no reason why K should change sign or vanish in the unstable region, but we are unable to prove that it could not happen. If for some system there were a range over which K:S; 0, then there would be a corresponding discontinuity in the interface profile and the treatment would have to be modified accordingly.3. APPLICATION OF THE REGULAR SOLUTION THEORYwe can obtain C(S) as a function of C(R) by expanding about R. Thus,C(S) =C(R)+ (r· V')C(R) (1/2!) (r· V')2C(R) (1/3!) (r· V')3C(R)+++ .. '.(3.3)Considering now the Zn molecules in the nth coordination shell at a radius r n from R. The probable number of AB bonds, Zn(P AB)n, between a B molecule at Rand the A molecules in its nth shell is, from Eqs. (3.2) and (3.3):Zn(P AB)n=Zn{C(R)[l-C(R)]-C(R)[L:(r' V')C(R) +(1/2!)L:(r·V')2C(R)]}, (3.4)where the summations are over all the sites in the nth shell, and the third and higher derivatives in Eq. (3.3) are neglected. Expressing Eq. (3.4) in terms of the vector components and performing the indicated summations** we obtain for a cubic lattice:The determination of the absolute value of C1 and its temperature dependence outside the range T~ Tc requires the use of a solution model for the evaluation of K and the free-energy function !:.f. For this purpose we believe it worthwhile to apply the regular solution theory despite its well-known shortcomings. Accordingly, we will assume that the free energy fR of a (uniform) solution is given by:fRee) =we(l-e)+kT[e lne+ (l-e) In(1-c)].(3.1)where the probability C has been replaced by the corresponding mol fraction e of the B component. If II n=E AB - (1/2) (EAA + EBB) where the E's are the intermolecular potentials for the nth coordination shell then, from the previous equation, we find that the total energy per molecule at R, u(R), relative to the pure components is:The enthalpy term in this equation is usually derived17 by considering only the molecular interactions between nearest neighbors. The same result can also be obtained18 from a summation of the pairwise interactions throughout the whole system. However, as we will show, these two approaches do not lead to the same value for the gradient energy in a nonuniform solution.u(R) =e(R)[l-e(R)]L:n Znlln - (1/6)e(R)L:n Znrbn.(3.5)As before, the energy is independent of the direction of the gradient in the lattice. Defining: (3.6) andA2= (L:n Znrbn)1 (3 L:n Znlln) ,(3.7)a. Free Energy of a Nonuniform Regular SolutionWe will first determine the enthalpy for a twocomponent cubic lattice. The following assumptions will be made: (1) the lattice parameter is independent of composition, (2) the intermolecular

Cahn Hilliard的论文

potential is a function only of the corresponding intermolecular distance, (3) the distribution of molecules on the lattice sites is locally random. Let C(R) and C(S) be the probabilities of finding a B molecule at sites Rand S, respectively, in the lattice. The probability, P AB, that an AB bond will be formed by a B molecule at R and an A molecule at S iswe obtain on substitution in Eq. (3.5): u(R) =we(l-e) -wA 2 eV'2c/2.(3.8)For a liquid solution the coordination number Zn is replaced by 47rr2p(r)drIV for the probable number of molecules between rand r+dr, where per) is the reduced radial distribution function which is assumed independent of composition and species involved. Substitution for Zn gives for the equations corresponding to (3.6) and (3.7):w= (47rIV)£00 r2p(r)o11(r)dr(3.9)P AB=C(R)[l-C(S)].(3.2)If r is the radius vector of site S relative to site R, then17 E. A. Guggenheim, Mixtures (Oxford University Press, London, 1952). 18 J. H. Hildebrand and S. E. Wood, J. Chern. Phys. 1, 817 (1933).** The odd and mixed derivatives cancel on summation because of the center of inversion. The remaining derivatives are summed as follows. Let the components of r be h, k, and I. Providing not more than two of the components are equal, all the terms arising from permutations of h, k, 1 can be grouped into sets of three, thus [(h,k,I), (k,l,h), (l,h,k)]. Each group gives on summation rn'c(R)\7'c(R). When_h=k=I" th~ term.? can be grouped into sets of four: [(h,h,h), (h,h,n), (h,lt,h), (h,n,It)] giving a sum of (4j3)rn'c(R)VOc(R). In either case the average contribution per molecule is (1/3)r n'c(R)\7'c(R).This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 109.171.137.210 On: Sun, 13 Jul 2014 07:39:08INTERFACIAL FREE ENERGY263andA2=[~'" r4p (r)v(r)dr]/[3i'"rp(r)v(r)drJ(3.10)According to Eqs. (2.6) and (3.12) the total free energy F R is therefore:FR=N vThe parameter A defined by Eqs. (3.7) and (3.10) has the dimensions of length and represents a rms effective "interaction distance" for the energy in a concentration gradient. If interactions other than those between nearest neighbors are neglected, Eq. (3.10) gives a value of ro/v!' for A, where 1'0 is the intermolecular distance. However, if we assume that v" is proportional to r- n and that the radial distribution function per) is approximated by p=l for 1'>1'0 and p=O for 1'<1'0, we obtain:iv[fR+(wA2j2)('V'C)2]dV.(3.13)b. Interfacial Free Energy of a Regular SolutionIn addition to the preceding results we will utilize the following well known properties17 of a regular solution,(3.14)fJ.A=wc2 - kT In(1-c),(3.15)A2= (n-3)rN3(n-S),which gives A=ro for n=6. If a repulsive term proportional to ,-12 is added to Vn then A is increased to a value of (11/7)!ro. Thus A is very sensitive to the exact nature of the long-range inte

Cahn Hilliard的论文

ractions. We have so far only considered the enthalpy of the solution. The entropy can be derived as follows. Assume the lattice to be composed of p equicomposition layers (not necessarily flat). Let one such layer of composition Cp contain N p molecules. The number of ways W p of arranging the molecules within the layer is:W p=N p!/{ (cpN p) ![(1-cp)N p]!}.w= 2kT c,(3.16)(3.17)In[c./ (1-c.)]= (2c.-l)w/kT.From Eqs. (2.11) and (3.15) we obtain!J.fR= -w(c-c e)2+kT{c In (e/e.) +(1-c) In[(1-e)/(1-e e )]} =fR(c)- fR(e.),(3.18)in which c. can be set equal to either of the equilibrium compositions Cee or C{3. Differentiating Eq. (3.18) and substituting in (2.17) and (2.18) gives:f3R=2k, 'YR=4kT./3.Since the layers are of assigned composition and cannot be interchanged, the total number of ways, W, of arranging all the molecules on the lattice is merely:W=IIW p.pSubstituting for W in the Boltzmann expression:S=k InW,We have now evaluated all the parameters introduced into the general treatment Sec. 2. Making the appropriate substitutions in Eq. (2.15) we obtain for (IR: (3.19) where (IT is a reduced interfacial energy defined by (3.20) This integral has been evaluated numerically and is plotted in Fig. 3 as (IT versus T/T. and in Fig. 4 as log «(IT) versus log(1-T/Tc). For the region T"-'T. Eq. (2.22) gives«(IR)(T~Tc)= 2N v AkT.[(Tc - T)/TcJ~.we obtain for the configurational entropy S,S=kln(IIW p)pkLlnW p ,pwhere N is the number of molecules. Stirling's formula gives: S= -k LP N peep lnep+ (1-cp) In(l-cp)]. The configurational entropy per molecule s(R) at lattice point R is therefore:s(R) = -k[c Inc+(1(3.21)In(l--:-c)],(3.11 )which is identical to the entropy in a uniform solution of composition C; consequently there is no contribution to the entropy from a composition gradient. Comparing coefficients in Eqs. (3.8) and (2.3) and using the SUbscript "R" to denote the value of a parameter for a regular solution, we find: K2R=0 and K1R= -cwA2/2. Thus Eq. (2.7) gives(3.12)An expression can also be derived for the case T ",0. At low temperatures the entropy term in Eq. (3.18) is small compared with that for the enthalpy, and C e is approximately equal to 1 or O. Equation (3.20) can therefore be approximated by«(IR)<T~()=2V1NVAkTcf1 {[c(1-e)]!o+T[c Inc+(l-c) In(1-c)]/4T c[c(1-c)Jt}dc.Evaluating the first term of the integral analytically andThis article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: /termsconditions. Downloaded to IP: 109.171.137.210 On: Sun, 13 Jul 2014 07:39:082641. W. CAHN AND J. E. HILLIARDAt low temperatures where there may be a steep gradient in concentration at the interface, a large error might be expected in the calculated value of 0'. However this is not so, because it may be shown that our treatment is equivalent to a sharp interface in which the number of atoms per unit area of interface is 2N vX2/3

Cahn Hilliard的论文

l which at low temperatures is approximately 3N vX/2 and is therefore in good agreement with the number calculated for a sharp interface.4. COMPARISON WITH EXPERIMENTAL RESULTS~ 0.4.-..z...">~ 0.3 " b0.20.1o0102 0.3 0.4 0.5 0.6 0.7 0.8 0.9T/Te-1.0FIG. 3. Reduced interfacial free energy (O'r) versus T ITc for a regular solution.If, as an approximation, one accepts the simple "hole" theory which treats a liquid as a regular solution of holes and molecules, then the equations derived in the previous section are directly applicable to the surface free energy of a pure liquid in equilibrium with its vapor. Our comparison with experimental measurements will therefore include both surface and interfacial free energies.a. Empirical Expressions for the Surface Free EnergyIt is well known that many of the experimental data for the temperature dependence of the surface energy can be fitted by certain empirical expressions. One of the earliest of these was proposed by van der Waals19 and can be written in the form:the second term numerically, we obtain(O'R)(T~O)= 2N vXkT c[(1l'/4v'2) -O.426(T/Tc)].(3.22)An approximate expression for O'R which is valid over the whole temperature range can be obtained by noting that: 0' r=cjl[.1. f R(max)/ kTcJ~.1.ce, where, as will be seen from Eqs. (3.21) and (3.22), cjl varies from tat T= Tc to 1l'/2 at T=O. Using a linear interpolation for cjl we find:(4.1) Ferguson20 21 tested this equation for a variety of organic- ( 1 - TITelO'R"-'2N vX[kTcJ~[1l'.1.ce(.1.fR(max))j/2J X [1- (1l'/2-t) (T /Tc)].0.40.30.20.1(3.23)0.50 0.40 0.30SLOPE 1.22rhis equation in conjunction with (3.17) and (3.18) provides a convenient means of calculating O'R with an error not exceeding one percent.c. Interface Profile for a Regular SolutionFor a regular solution the interface profile is symmetrical about c=!. From Eq. (2.25) defining the thickness I, we obtain on substitution from Eqs. (2.23), (3.12), (3.17), and (3.18):0.20~"" -<z>-.... '"0.10 0.08 0.07 0.06 0.05 0.04 0.03"bIR/X=v'2{ -1- [T In4c e(1-c e)J/ [T c(1-2c e)2J}-tb(3.24)(3.25)and corresponding to Eq. (2.26) we have for T"-'Tc: The quantity lR/X has been calculated numerically from Eq. (3.24) and is plotted versus T /Tc in Fig. 5. It will be recalled that one of the basic assumptions we made when l).eglecting the higher terms in the expansion for f [Eq. (2.1)J was that the gradient was small compared' with the reciprocal of the intermolecular distance. This assumption is undoubtedly valid in the critical region.OD2~~~L-~-L--~__~o 0.20.40.60.7______ 0.8~0.'TITe - -FIG. 4. Log(O'r) versus log(1- T ITc) for a regular solution.19J. D. van der Waals, Z. physik. Chern. 13, 716 (1894).2021A. Ferguson, Trans. Faraday Soc. 19,407 (1923). A. Ferguson, Proc. Phys. Soc. (London) 52, 759 (1940).This article is copyrighted as indicated in the article. Reuse of AIP content is subject

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