Cahn-Hilliard model with Schlögl reactions: interplay of equilibrium and non-equilibrium phase transitions. I. Travelling wave solutions
Cahn-Hilliard model with Schlögl reactions: interplayof equilibrium and non-equilibrium phase transitions.I. Travelling wave solutions
P.O. Mchedlov-Petrosyan ∗ , L.N. Davydov † A.I. Akhiezer Institute for Theoretical Physics, National Science Center “Kharkov Institite ofPhysics & Technology”, 1 Akademicheskaya Str., Kharkiv, Ukraine 61108
Received March 17, 2020, in final form May 6, 2020
The present work is devoted to the modelling which is based on the modified Cahn-Hilliard equa-tion, the interplay of equilibrium and non-equilibrium phase transitions. The non-equilibriumphase transitions are modelled by the Schlögl reactions systems. We consider the advancingfronts which combine these both transitions. The traveling wave solutions are obtained; theconditions of their existence and dependence on the parameters of the models are studied indetail. The possibility of the existance of non-equilibrated phase is discussed.Key words: phase transition, nonequilibrium phase transition, Cahn-Hilliard equation, Schlöglreactions
1. Introduction
The present work is devoted to the modelling based on the modified Cahn-Hilliard equation, theinterplay of equilibrium and non-equilibrium phase transitions. We consider the advancing fronts which“combine”, in some sense, these both transitions. To understand the meaning of our modifications, weneed to give some insight into the history and into the existing modifications of this equation. TheCahn-Hilliard equation [1–4] is now a well-established model in the theory of phase transitions as wellas in several other fields. The basic underlying idea of this model is that for inhomogeneous system,e.g., a system undergoing a phase transition, the thermodynamic potential (e.g., a free energy) shoulddepend not only on the order parameter 𝑢 but also on its gradient. The idea of such dependence wasalready introduced by Van der Waals [5] in his theory of capillarity. For an inhomogeneous system, thelocal chemical potential 𝜇 is defined as variational derivative of thermodynamic potential functional. Ifthermodynamic potential is the simplest symmetric-quadratic-function of the gradient, this leads to thelocal chemical potential 𝜇 which depends on Laplacian, while for the one-dimensional case it depends on the second order derivative of the order parameter. The diffusional flux 𝐽 is proportional to the gradient of chemical potential ∇ 𝜇 ; the coefficient of proportionality is called mobility 𝑀 [6]. With such expressionfor the flux, the diffusion equation instead of the usual second order equation becomes a forth-order PDEfor the order parameter 𝑢 (herein our notations differ from the notations in the original papers): 𝜕𝑢𝜕𝑡 (cid:48) = ∇ [ 𝑀 ∇ 𝜇 ] , (1.1) 𝜇 = − ¯ 𝜀 Δ 𝑢 + 𝑓 ( 𝑢 ) . (1.2)Here, 𝑀 is mobility, ¯ 𝜀 is usually assumed to be proportional to the capillarity length, and 𝑓 ( 𝑢 ) = d Φ ( 𝑢 ) d 𝑢 ,where Φ ( 𝑢 ) is homogeneous part of the thermodynamic potential. In the present communication, we ∗ [email protected] † [email protected] This work is licensed under a Creative Commons Attribution 4.0 International License.Further distribution of this work must maintain attribution to the author(s) and the published article’stitle, journal citation, and DOI. a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t .O. Mchedlov-Petrosyan, L.N. Davydov take 𝑓 ( 𝑢 ) in the form of the cubic polynomial (corresponding to the fourth-order polynomial for thehomogeneous part of thermodynamic potential): 𝑓 ( 𝑢 ) = 𝑢 − 𝛿𝑢 − 𝑠𝑢 ; (1.3)rescaling 𝑢 , the coefficient at 𝑢 could be always scaled to one. In the present paper, 𝑢 is assumedto be non-negative, so, even for a symmetric potential, 𝛿 ≠
0; furthermore, the asymmetric potentialnaturally appears in some modifications of the Cahn-Hilliard equation [4]. In this phenomenologicalmodel, we always consider the isothermal situation, so we do not show the temperature dependence ofthe coefficients in (1.3) explicitly. However, if we want to model the approach to the critical state forsuch a model, the approach to critical temperature will be manifested by merging the stationary statestogether, i.e., by two non-zero roots of the right-hand side of (1.3) approaching the third zero root.The classic Cahn-Hilliard equation was introduced as early as 1958 [1, 2]; the stationary solutionswere considered, the linearized version was treated and the corresponding instability of homogeneousstate was identified. However, an intensive study of the fully nonlinear form of this equation startedmuch later [7]. At present, an impressive amount of work is done on nonlinear Cahn-Hilliard equation , as well as on its numerous modifications, see [3, 4]. An important modification was done by Novick-Cohen [8]. Taking into account the dissipation effects which are neglected in the derivation of the classicCahn-Hilliard equation, she introduced the viscous Cahn-Hilliard (VCH) equation 𝜕𝑢 𝜕𝑡 (cid:48) = ∇ (cid:20) 𝑀 ∇ (cid:18) 𝜇 + ¯ 𝜂 𝜕𝑢 𝜕𝑡 (cid:48) (cid:19) (cid:21) , (1.4)where the coefficient ¯ 𝜂 is called viscosity. It was also noticed that VCH equation could be derived asa certain limit of the classic Phase-Field model [9]. Later on, several authors considered the nonlinear convective Cahn-Hilliard equation (CCH) in one space dimension [10–12] 𝜕𝑢𝜕𝑡 (cid:48) − ¯ 𝛼𝑢 𝜕𝑢𝜕𝑥 (cid:48) = 𝜕𝜕𝑥 (cid:48) (cid:18) 𝜕 𝜇𝜕𝑥 (cid:48) (cid:19) . (1.5)Leung [10] proposed this equation as a continual description of lattice gas phase separation under theaction of an external field. Similarly, Emmott and Bray [12] proposed this equation as a model for thespinodal decomposition of a binary alloy in an external field E . As they noticed, if the mobility 𝑀 [6] isindependent of the order parameter (concentration), the term involving E will drop out of the dynamics. Toget nontrivial results, they assumed the simplest possible symmetric dependence of mobility on the orderparameter, viz. 𝑀 ∼ − 𝑟𝑢 . Then, they obtained the Burgers-type convection term in equation (1.5) withthe coefficient ¯ 𝛼 = 𝑟 𝐸 . Thus, the sign of ¯ 𝛼 depends both on the direction of the field and on the sign of 𝑟 .Witelski [11] introduced the equation (1.5) as a generalization of the classic Cahn-Hilliard equation or asa generalization of the Kuramoto-Sivashinsky equation [13, 14] by including a nonlinear diffusion term.In [10–12], and in [15, 16], several approximate solutions and only two exact static kink and anti-kinksolutions were obtained. The “coarsening” of domains separated by kinks and by anti-kinks was alsodiscussed. To study the joint effects of nonlinear convection and viscosity, Witelski [17] introduced theconvective-viscous-Cahn-Hilliard equation (CVCHE) with a general symmetric double-well potential Φ ( 𝑢 ) : 𝜕𝑢𝜕𝑡 (cid:48) − ¯ 𝛼𝑢 𝜕𝑢𝜕𝑥 (cid:48) = 𝜕𝜕𝑥 (cid:48) (cid:20) 𝑀 𝜕𝜕𝑥 (cid:48) (cid:18) 𝜇 + ¯ 𝜂 𝜕𝑢𝜕𝑡 (cid:48) (cid:19) (cid:21) , (1.6) 𝜇 = − ¯ 𝜀 𝜕 𝑢𝜕𝑥 (cid:48) + d Φ ( 𝑢 ) d 𝑢 . (1.7)It is worth noting that all results, including the stability of solutions, were obtained without specifying aparticular functional form of the potential. Thus, they are valid both for the polynomial and logarithmic[3, 4] potential. Moreover, with a constraint imposed on nonlinearity and viscosity, the approximatetravelling-wave solutions were obtained. In [18], for equation (1.6) with polynomial potential, see (1.3), and the balance between the applied field and viscosity, several exact single- and two-wave solutions were obtained. ahn-Hilliard model with Schlögl reactions Another modification of the nonlinear Cahn-Hilliard equation which attracted much interest is theinsertion of linear or nonlinear sink/source terms, e.g., due to a chemical reaction, into this equation.Such a study was pioneered by Huberman [19]. He introduced Cahn-Hilliard equation with additionalkinetic terms corresponding to the reversible first-order autocatalytic chemical reaction and analyzed thelinear stability of stationary states. Cohen and Murray [20] considered the same equation in the biologicalcontext: they used quadratic nonlinearity to describe the growth and dispersal in the population model;they studied the stability and identified bifurcations to spatial structures. Similar equation (with additionalnonlinear term) was used in [21] to study the segregation dynamics of binary mixtures coupled withthe chemical reaction. The same equation as in [19, 20] was used to describe phase transitions in achemisorbed layer [22] and to model the system of cells that move, proliferate and interact via adhesion[23]. Furthermore, for the latter model, several rigorous mathematical results on the existence andasymptotics of solutions were obtained [24, 25]. General observation is that the presence of chemicalreaction can visibly influence the equilibrium phase transition, e.g., freeze the spinodal decompositionor coarsening, stabilizing some stationary inhomogeneous state.On the other hand, the canonical models for non-equilibrium phase transitions in chemical reactionsystems were introduced by Schlögl [26]; here, the different “phases” correspond to different stationarystates of the system. Schlögl considered two reaction systems: the so-called “First Schlögl Reaction” 𝐴 + 𝑋 (cid:29) 𝑋, (1.8) 𝐵 + 𝑋 (cid:29) 𝐶, (1.9)and the “Second Schlögl Reaction” 𝐴 + 𝑋 (cid:29) 𝑋, (1.10) 𝐵 + 𝑋 (cid:29) 𝐶. (1.11)The concentrations of species 𝐴 , 𝐵 and 𝐶 (which are called the “reservoir reagents”) are assumed to beconstant and only concentration of 𝑋 can vary with time and space. For the first Schlögl reaction in theabsence of diffusion, the evolution of 𝑋 is described byd 𝑋 d 𝑡 = − 𝑘 (cid:48) 𝑋 + 𝑘 𝐴𝑋 − 𝑘 𝐵𝑋 + 𝑘 (cid:48) 𝐶. (1.12)Here, the 𝑘 𝑖 𝑗 , 𝑘 (cid:48) 𝑖 𝑗 are the rate constants for the forward and reverse reactions, respectively; the secondlower index is “1” for the first Schlögl reaction, and “2” for the second one. Correspondingly, for thesecond Schlögl reaction in the absence of diffusion, the evolution of 𝑋 is described byd 𝑋 d 𝑡 = − 𝑘 (cid:48) 𝑋 + 𝑘 𝐴𝑋 − 𝑘 𝐵𝑋 + 𝑘 (cid:48) 𝐶. (1.13)The first reaction exhibits a non-equilibrium phase transition of the second order, the second reactionshows a phase transition of the first order (for details see [26]). If the system simultaneously undergoesan equilibrium phase transition accompanied by a phase separation, it could be of considerable interestto study the interaction of an equilibrium and non-equilibrium phase transitions. Apparently, beingunaware of Schlögl paper, Huberman [19] and Cohen and Murray [20] in fact considered the interplayof equilibrium and (the second-order) non-equilibrium phase transitions.In the present communication, we consider the modified Cahn-Hilliard equation complemented bysource/sink terms corresponding both to the first and the second Schlögl reactions. Let us call these mod-ifications Cahn-Hilliard-Huberman-Cohen-Murray (CHHCM) and Cahn-Hilliard-Schlögl (CHS) equa-tions, respectively. We also consider the influence of some additional modifications of the Cahn-Hilliardequation, such as viscous and convective terms [8, 10–12, 17, 18]. We give exact travelling-wave solutionsfor these modifications. For completeness in appendix we also give an exact travelling-wave solution for Puri-Frish modification [21]. In the second part of this paper, some additional exact solutions and stability study are presented. .O. Mchedlov-Petrosyan, L.N. Davydov
2. Convective viscous Cahn-Hilliard-Huberman-Cohen-Murray equation
In the present section we first give exact travelling-wave solutions for convective viscous Cahn-Hilliardequation with second order reaction terms. So, we first take into account the action of both external fieldand dissipation [8, 10–12, 17, 18]; then, we drop the convective and viscous terms, reducing equation toCHHCM equation. To avoid some unnecessary complications, we assume reaction (1.9) to be irreversible,i.e., in (1.12) 𝑘 (cid:48) =
0. In terms of Schlögl model [26], this corresponds to the “analog of zero magneticfield” case. From (1.6), (1.2), (1.3) and (1.12) we write down the Convective Viscous CHHCM equation,first in terms of the initial variable 𝑋 (concentration): 𝜕 𝑋𝜕𝑡 (cid:48) − ¯ 𝛼𝑋 𝜕 𝑋𝜕𝑥 (cid:48) = 𝑀 𝜕 𝜕𝑥 (cid:48) (cid:18) ¯ 𝜇 + ¯ 𝜂 𝜕 𝑋𝜕𝑡 (cid:48) (cid:19) − 𝑘 (cid:48) 𝑋 + 𝑘 𝐴𝑋 − 𝑘 𝐵𝑋, (2.1)¯ 𝜇 = − ¯ 𝜀 𝜕 𝑋𝜕𝑥 (cid:48) + ¯ 𝑓 ( 𝑋 ) , (2.2)¯ 𝑓 ( 𝑋 ) = 𝑞𝑋 − ¯ 𝛿𝑋 − ¯ 𝑠𝑋. (2.3)The equations (2.1)–(2.3) implicitly assume that in the system 𝐴 − 𝐵 − 𝐶 − 𝑋 , the components 𝐴 and 𝐵 are in large excess, and they are not essentially exhausted during the chemical reaction and did notchange essentially due to the phase transition; we also assume 𝑀 to be a constant. Renormalizing ¯ 𝑋 , 𝑥 (cid:48) and 𝑡 (cid:48) , we introduce 𝑋 = 𝑢𝑋 ; 𝑥 (cid:48) = 𝑥𝑙 ; 𝑡 (cid:48) = 𝑡𝜏. (2.4)Here, 𝑋 = √ 𝑞 , 𝜏 = 𝑘 (cid:48) 𝑋 = √ 𝑞𝑘 (cid:48) and 𝑙 = √ 𝑀𝜏 = √︂ 𝑀 √ 𝑞𝑘 (cid:48) . Denoting 𝛼 = ¯ 𝛼 𝑋 𝜏𝑙 = ¯ 𝛼 √︂ √ 𝑞𝑘 (cid:48) 𝑀 , 𝜀 = ¯ 𝜀 𝑙 , 𝜂 = ¯ 𝜂𝜏 , 𝛿 = ¯ 𝛿𝑋 = ¯ 𝛿 √ 𝑞 and 𝑠 = ¯ 𝑠𝑞 we write down equation (2.1) in the non-dimensional form, 𝜕𝑢𝜕𝑡 − 𝛼𝑢 𝜕𝑢𝜕𝑥 = 𝜕 𝜕𝑥 (cid:18) − 𝜀 𝜕 𝑢𝜕𝑥 + 𝑢 − 𝛿𝑢 − 𝑠𝑢 + 𝜂 𝜕𝑢𝜕𝑡 (cid:19) − 𝑢 ( 𝑢 − 𝑢 ) . (2.5)We also introduce 𝑢 = 𝑘 𝐴 − 𝑘 𝐵𝑘 (cid:48) 𝑋 , (2.6)assuming 𝑢 >
0, i.e., 𝑘 𝐴 > 𝑘 𝐵 . Looking for the travelling wave solutions of (2.5), we introduce thetravelling wave coordinate 𝑧 = 𝑥 − 𝑣𝑡 . This yieldsdd 𝑧 (cid:20) 𝑣𝑢 + 𝛼 𝑢 + dd 𝑧 (cid:18) − 𝜀 d 𝑢 d 𝑧 + 𝑢 − 𝛿𝑢 − 𝑠𝑢 − 𝑣𝜂 d 𝑢 d 𝑧 (cid:19) (cid:21) = 𝑢 ( 𝑢 − 𝑢 ) . (2.7)We look for the solution, which connects the stationary state of the reaction system 𝑢 = 𝑢 at 𝑧 = −∞ with the stationary state 𝑢 = 𝑧 = +∞ . The simplest possible ansatz for the anti-kink solution (asusually we call “kinks” the solutions with d 𝑢 d 𝑧 >
0, and “anti-kinks” — solutions with d 𝑢 d 𝑧 <
0) with thisproperty is as follows: d 𝑢 d 𝑧 = 𝜅𝑢 ( 𝑢 − 𝑢 ) , (2.8)where 𝜅 is presently unknown positive constant. Then, equation (2.7) could be written asdd 𝑧 (cid:20) 𝑣𝑢 + 𝛼 𝑢 − 𝜅 𝑢 + dd 𝑧 (cid:18) − 𝜀 d 𝑢 d 𝑧 + 𝑢 − 𝛿𝑢 − 𝑠𝑢 − 𝑣𝜂 d 𝑢 d 𝑧 (cid:19) (cid:21) = . (2.9)Integrating once, we get 𝑣𝑢 + 𝛼 𝑢 − 𝜅 𝑢 + d d 𝑧 (cid:18) − 𝜀 d 𝑢 d 𝑧 + 𝑢 − 𝛿𝑢 − 𝑠𝑢 − 𝑣𝜂 d 𝑢 d 𝑧 (cid:19) = 𝐶 . (2.10) ahn-Hilliard model with Schlögl reactions Regarding the ansatz (2.8), for the latter equation to be satisfied the expression under the derivative shouldbe proportional to 𝑢 . That is, for (2.8) to give the solution of (2.5), two equations should be satisfied forarbitrary 𝑢 𝑣𝑢 + 𝛼 𝑢 − 𝜅 𝑢 + 𝛽 d 𝑢 d 𝑧 = 𝐶 , (2.11) − 𝜀 d 𝑢 d 𝑧 + 𝑢 − 𝛿𝑢 − 𝑠𝑢 − 𝑣𝜂 d 𝑢 d 𝑧 = 𝛽𝑢 + 𝐶 , (2.12)where 𝛽, 𝐶 and 𝐶 are constants. The expression for the second derivative of 𝑢 is easily written as:d 𝑢 d 𝑧 = 𝜅 (cid:16) 𝑢 − 𝑢 𝑢 + 𝑢 𝑢 (cid:17) . (2.13)Then, equations (2.11), (2.12) take the form (cid:16) 𝛼 + 𝛽𝜅 (cid:17) 𝑢 + (cid:18) 𝑣 − 𝜅 − 𝛽𝜅𝑢 (cid:19) 𝑢 = 𝐶 , (2.14) − 𝜀 𝜅 (cid:2) 𝑢 − 𝑢 𝑢 + 𝑢 𝑢 (cid:3) + 𝑢 − 𝛿𝑢 − ( 𝑠 + 𝛽 ) 𝑢 − 𝑣𝜂𝜅 (cid:16) 𝑢 − 𝑢 𝑢 (cid:17) = 𝐶 . (2.15)Rearranging the terms and equating the coefficients at each power of 𝑢 to zero, we finally obtain fiveconstraints on the parameters: 𝛼 + 𝛽𝜅 = , (2.16) 𝑣 = 𝜅 + 𝛽𝜅𝑢 , (2.17) 𝜅 = 𝜀 , (2.18) 𝑣𝜂𝜅 = 𝑢 − 𝛿, (2.19) 𝑣𝜂𝜅𝑢 = 𝑢 + 𝑠 + 𝛽. (2.20)There are five constraints (2.16)–(2.20) and only three unknowns 𝜅, 𝑣 and 𝛽 . That is, for the constantvelocity transition front to exist, two additional constraints on the values of the stationary states of thereaction system and on the values of the equilibrium states for the phase transition should be imposed.Now, there is some freedom in selecting which parameters are “basic”, those related to the reactionsystem, or those related to the “Cahn-Hilliard part”. Assuming the former to be basic, we write theconstraints as 𝛿 = 𝑢 (cid:18) + 𝛼𝜂 √ 𝜀 (cid:19) − 𝜂, (2.21) 𝑠 = − 𝑢 (cid:18) + 𝛼𝜂 √ 𝜀 (cid:19) + 𝜂𝑢 + 𝛼𝜀 √ . (2.22) If the constraints (2.16)–(2.20) are satisfied, the solution of equation (2.8) is simultaneously thesolution of the travelling-wave equation (2.7). Integrating (2.8) once, we get 𝑢 = 𝑢 exp {− 𝜅𝑢 ( 𝑧 + 𝜙 )} + exp {− 𝜅𝑢 ( 𝑧 + 𝜙 )} , (2.23)where 𝜙 is an arbitrary constant. It is natural to take position of the maximal value of the derivative d 𝑢 d 𝑧 (when d 𝑢 d 𝑧 = 𝑧 =
0; then, 𝜙 =
0. The solution (2.23) could be rewritten in the form 𝑢 = 𝑢 (cid:20) − tanh (cid:18) 𝑢 √ 𝜀 ( 𝑥 − 𝑣𝑡 ) (cid:19) (cid:21) . (2.24) .O. Mchedlov-Petrosyan, L.N. Davydov Here, we used 𝜅 = √ 𝜀 , see (2.18); the velocity 𝑣 of the transition front is given by (2.16) and (2.17), 𝑣 = √ 𝜀 + 𝛽𝜅𝑢 = √ 𝜀 − 𝛼𝑢 . (2.25)The roots of equation ˜ 𝑢 (cid:16) ˜ 𝑢 − 𝛿 ˜ 𝑢 − 𝑠 (cid:17) = 𝑢 , ˜ 𝑢 arestable minima and ˜ 𝑢 is unstable maximum. The root ˜ 𝑢 = 𝛿 and 𝑠 , respectively,yields two remaining roots, i.e., two constraints imposed on the values of ˜ 𝑢 , ˜ 𝑢 and 𝑢 ,˜ 𝑢 , = (cid:20) 𝑢 (cid:18) + 𝛼𝜂 √ 𝜀 (cid:19) − 𝜂 ± √ 𝐺 (cid:21) , (2.27) 𝐺 = (cid:20) 𝑢 (cid:18) + 𝛼𝜂 √ 𝜀 (cid:19) − 𝜂 (cid:21) + (cid:34) − 𝑢 (cid:18) + 𝛼𝜂 √ 𝜀 (cid:19) + 𝜂𝑢 + 𝛼𝜀 √ (cid:35) . (2.28)Here, the discriminator of quadratic equation is denoted by 𝐺 for convenience. To understand the mutualeffect of the equilibrium and non-equilibrium transitions, it is practical to consider several special casesof (2.27)–(2.28). First we consider the CHHCM case, i.e., the absence of the applied field and dissipation. For 𝛼 = 𝜂 =
0, expression (2.28) simplifies drastically, yielding 𝐺 = 𝑢 . Then, (2.27) becomes˜ 𝑢 , = 𝑢 ( ± ) . (2.29)This means that for the constant-velocity-transition front to exist, the values of the order parametercorresponding to the equilibrium phases should coincide exactly with the values corresponding to thestationary states of the chemical reactions system, i.e., ˜ 𝑢 = 𝑢 ; ˜ 𝑢 =
0. The thermodynamic potentialshould be symmetric, ˜ 𝑢 = ˜ 𝑢 /
2, with equal-depth wells. The velocity depends on the 𝜀 only, 𝑣 = √ 𝜀 .Now, let 𝛼 = 𝜂 ≠
0. From (2.28) it follows 𝐺 = (cid:16) 𝑢 + 𝜂 (cid:17) ; ˜ 𝑢 , = (cid:20) 𝑢 − 𝜂 ± (cid:16) 𝑢 + 𝜂 (cid:17) (cid:21) . (2.30)That is, stationary values for the equilibrium transition should again coincide with the stationary valuesfor the reaction system, but the unstable value should be shifted to the lower value. As it was mentionedin the introduction, the derivative of the homogeneous part of thermodynamic potential Φ ( 𝑢 ) is givenby (1.3): d Φ ( 𝑢 ) d 𝑢 = 𝑢 − 𝛿𝑢 − 𝑠𝑢. (2.31)Integrating once and substituting values of 𝛿 and 𝑠 for 𝛼 =
0, we obtain the following expressions for the potential values Φ ( ˜ 𝑢 ) and Φ ( ˜ 𝑢 ) Φ ( ˜ 𝑢 ) = − 𝜂 𝑢 + 𝐶 ; Φ ( ˜ 𝑢 ) = Φ ( ) = 𝐶. (2.32)That is, to compensate the dissipation, the potential well corresponding to ˜ 𝑢 should be deeper. On theother hand, if 𝛼 ≠ 𝜂 = 𝐺 = 𝑢 + 𝛼𝜀 √ 𝑢 , = 𝑢 ± √︄ + 𝛼𝜀 √ 𝑢 . (2.33)This means that for positive 𝛼 , the order parameter value for the final state after transition, 𝑢 = 𝑢 , issomewhat lower than the equilibrium value ˜ 𝑢 . To ensure the positivity of ˜ 𝑢 it should be √ 𝛼𝜀 < 𝑢 ; ahn-Hilliard model with Schlögl reactions however, the parameter 𝜀 is small, so it is not a severe limitation. Now, let both 𝛼 ≠ 𝜂 ≠
0. Theexpression for the velocity (2.25) is independent of 𝜂 ; for the special value 𝛼 = √ 𝑢 𝜀 , the velocity is zero,i.e., for the corresponding value of the applied field, the transition front becomes static. Substitution ofthis value of 𝛼 into (2.27) and (2.28) yields˜ 𝑢 , = 𝑢 (cid:34) ± √︄ + 𝜀 𝑢 (cid:35) . (2.34)Interestingly, the viscosity 𝜂 has dropped out from the latter expression. This is physically reasonable:there is no dissipation for the static transition front; the deviation of the order parameter value 𝑢 = 𝑢 forthe final state after transition from its equilibrium value ˜ 𝑢 is exactly the same as given by (2.33) (i.e.,for 𝜂 = 𝛼 .
3. Convective viscous Cahn-Hilliard-Schlögl equation
In this section we first give exact travelling-wave solutions for a convective viscous Cahn-Hilliardequation with third order reaction terms. Again, we first take into account the effect of both external fieldand dissipation [8, 10–12, 17, 18]; then, we drop the convective and viscous terms, reducing the equationto CHS equation. To make the calculations somewhat more transparent we assume the reaction (1.11) to be irreversible, i.e., in (1.13) 𝑘 (cid:48) =
0. From (1.6), (1.2), (1.3) and (1.13), we write down the ConvectiveViscous CHS equation, first in terms of the initial variable 𝑋 (concentration): 𝜕 𝑋𝜕𝑡 (cid:48) − ¯ 𝛼𝑋 𝜕 𝑋𝜕𝑥 (cid:48) = 𝑀 𝜕 𝜕𝑥 (cid:48) (cid:18) ¯ 𝜇 + ¯ 𝜂 𝜕 𝑋𝜕𝑡 (cid:48) (cid:19) − 𝑘 (cid:48) 𝑋 + 𝑘 𝐴𝑋 − 𝑘 𝐵𝑋 , (3.1)¯ 𝜇 = − ¯ 𝜀 𝜕 𝑋𝜕𝑥 (cid:48) + ¯ 𝑓 ( 𝑋 ) , (3.2)¯ 𝑓 ( 𝑋 ) = 𝑞𝑋 − ¯ 𝛿𝑋 − ¯ 𝑠𝑋. (3.3)Writing down equations (3.1)–(3.3), we again assume implicitly that in the system 𝐴 − 𝐵 − 𝐶 − 𝑋 thecomponents 𝐴 and 𝐵 are in large excess and are not essentially exhausted during the chemical reaction;we also assume 𝑀 to be a constant. Renormalizing 𝑋 , 𝑥 (cid:48) and 𝑡 (cid:48) , we introduce 𝑋 = 𝑢𝑋 ; 𝑥 (cid:48) = 𝑥𝑙 ; 𝑡 (cid:48) = 𝑡𝜏. (3.4)Here, 𝑋 = √ 𝑞 , 𝜏 = 𝑘 (cid:48) 𝑋 = 𝑞𝑘 (cid:48) and 𝑙 = √ 𝑀𝜏 = √︃ 𝑀𝑘 (cid:48) 𝑋 = √︃ 𝑀𝑞𝑘 (cid:48) . Denoting 𝛼 = ¯ 𝛼 𝑋 𝜏𝑙 = ¯ 𝛼 √ 𝑘 (cid:48) 𝑀 ; 𝜀 = ¯ 𝜀 𝑙 ; 𝜂 = ¯ 𝜂𝜏 ; 𝛿 = ¯ 𝛿𝑋 = ¯ 𝛿 √ 𝑞 ; 𝑠 = ¯ 𝑠𝑞 ; 𝑅 = 𝑘 𝐴𝑘 (cid:48) 𝑋 and 𝑄 = 𝑘 𝐵𝑘 (cid:48) 𝑋 , we write down equation (3.1) innon-dimensional form 𝜕𝑢 𝜕𝑡 − 𝛼𝑢 𝜕𝑢 𝜕𝑥 = 𝜕 𝜕𝑥 (cid:18) − 𝜀 𝜕 𝑢𝜕𝑥 + 𝑢 − 𝛿𝑢 − 𝑠𝑢 + 𝜂 𝜕𝑢 𝜕𝑡 (cid:19) − 𝑢 (cid:16) 𝑢 − 𝑅𝑢 + 𝑄 (cid:17) . (3.5)Herein below we assume that the quadratic equation 𝑢 − 𝑅𝑢 + 𝑄 = 𝑢 , 𝑢 , 𝑢 (cid:62) 𝑢 ; i.e. 𝑅 − 𝑄 (cid:62) ( 𝑘 𝐴 ) (cid:62) 𝑘 (cid:48) 𝑘 𝐵 . Looking for the travelling wave solutions of (3.5), we introducethe travelling wave coordinate 𝑧 = 𝑥 − 𝑣𝑡 . This yieldsdd 𝑧 (cid:20) 𝑣𝑢 + 𝛼 𝑢 + d d 𝑧 (cid:18) − 𝜀 d 𝑢 d 𝑧 + 𝑢 − 𝛿𝑢 − 𝑠𝑢 − 𝑣𝜂 d 𝑢 d 𝑧 (cid:19) (cid:21) = 𝑢 ( 𝑢 − 𝑢 ) ( 𝑢 − 𝑢 ) . (3.7) .O. Mchedlov-Petrosyan, L.N. Davydov As in the previous section, we look for the solution, which connects the stationary state of the reactionsystem 𝑢 = 𝑢 at 𝑧 = −∞ with the stationary state 𝑢 = 𝑧 = +∞ . Thus, the proper ansatz for theanti-kink solution is again (2.8) 1 𝜅 d 𝑢 d 𝑧 = 𝑢 ( 𝑢 − 𝑢 ) , (3.8)where 𝜅 is presently an unknown positive constant. Then, equation (3.7) could be written asdd 𝑧 (cid:20) 𝑣𝑢 + 𝛼 𝑢 + dd 𝑧 (cid:18) − 𝜀 d 𝑢 d 𝑧 + 𝑢 − 𝛿𝑢 − 𝑠𝑢 − 𝑣𝜂 d 𝑢 d 𝑧 (cid:19) (cid:21) = dd 𝑧 (cid:18) 𝜅 𝑢 − 𝑢 𝜅 𝑢 (cid:19) . (3.9)Integrating once, we get (cid:16) 𝑣 + 𝑢 𝜅 (cid:17) 𝑢 + (cid:18) 𝛼 − 𝜅 (cid:19) 𝑢 + dd 𝑧 (cid:18) − 𝜀 d 𝑢 d 𝑧 + 𝑢 − 𝛿𝑢 − 𝑠𝑢 − 𝑣𝜂 d 𝑢 d 𝑧 (cid:19) = 𝐶 . (3.10)Regarding the ansatz (3.8), for the latter equation to be satisfied, the expression under the derivativeshould be proportional to 𝑢 . That is, for (3.8) to give the solution of (3.7) two equations should be satisfiedfor arbitrary 𝑢 , (cid:16) 𝑣 + 𝑢 𝜅 (cid:17) 𝑢 + (cid:18) 𝛼 − 𝜅 (cid:19) 𝑢 + 𝛽 d 𝑢 d 𝑧 = 𝐶 , (3.11) − 𝜀 d 𝑢 d 𝑧 + 𝑢 − 𝛿𝑢 − 𝑠𝑢 − 𝑣𝜂 d 𝑢 d 𝑧 = 𝛽𝑢 + 𝐶 , (3.12)where 𝐶 , 𝐶 and 𝛽 are constants. If the above constraints are satisfied for arbitrary 𝑢 , the solutionof (3.5) is again given by (2.24), though with different values of 𝑢 , 𝑣, 𝜀 . The expression for the secondderivative of 𝑢 is given again by (2.13). Then, equations (3.11), (3.12) take the form (cid:16) 𝑣 + 𝑢 𝜅 (cid:17) 𝑢 + (cid:18) 𝛼 − 𝜅 (cid:19) 𝑢 + 𝛽𝜅 (cid:16) 𝑢 − 𝑢 𝑢 (cid:17) = 𝐶 , (3.13) − 𝜀 𝜅 (cid:16) 𝑢 − 𝑢 𝑢 + 𝑢 𝑢 (cid:17) + 𝑢 − 𝛿𝑢 − ( 𝑠 + 𝛽 ) 𝑢 − 𝑣𝜂𝜅 (cid:16) 𝑢 − 𝑢 𝑢 (cid:17) = 𝐶 . (3.14)Rearranging the terms and equating coefficients at each power of 𝑢 to zero, we finally obtain fiveconstraints on the parameters: 𝜅 = 𝜀 , (3.15)32 𝑢 − 𝛿 − 𝑣𝜂𝜅 = , (3.16) − 𝑢 − 𝑠 − 𝛽 + 𝑣𝜂𝜅𝑢 = , (3.17) 𝛽 = 𝜅 − 𝛼 𝜅 , (3.18) 𝑣 = 𝛽𝜅𝑢 − 𝑢 𝜅 (3.19)Similarly to (2.16)–(2.20), there are five constraints (3.15)–(3.19) and only three unknowns 𝜅, 𝑣 and 𝛽 . That is, for the constant velocity transition front to exist, two additional constraints on the values of thestationary states of the reaction system and on the values of the equilibrium states for the phase transitionshould be imposed. Assuming, as in section 2, the parameters related to reaction system to be “basic”,we write the constraints as 𝛿 = 𝑢 − (cid:16) 𝑢 − 𝑢 (cid:17) 𝜂 + 𝛼𝑢 √ 𝜀 𝜂 ; (3.20) 𝑠 = − 𝑢 − 𝜀 + (cid:32) 𝑢 − 𝑢 𝑢 (cid:33) 𝜂 + 𝛼 √ (cid:32) 𝜀 − 𝑢 𝜀 𝜂 (cid:33) . (3.21) ahn-Hilliard model with Schlögl reactions The latter expressions impose evident limitations on the roots of˜ 𝑢 (cid:16) ˜ 𝑢 − 𝛿 ˜ 𝑢 − 𝑠 (cid:17) = , (3.22)i.e., on the extrema of the homogeneous part of the thermodynamic potential (1.2), (1.3), here, ˜ 𝑢 , ˜ 𝑢 correspond to stable minima and ˜ 𝑢 to unstable maximum. The root ˜ 𝑢 = 𝑢 , ˜ 𝑢 and 𝑢 . The velocity of the transition front 𝑣 is 𝑣 = √ 𝜀 (cid:16) 𝑢 − 𝑢 (cid:17) − 𝛼𝑢 𝛼 = 𝜂 =
0, these expressions simplify drastically, yielding 𝛿 = 𝑢 ; 𝑠 = − 𝑢 − 𝜀 (3.24)and, correspondingly ˜ 𝑢 , = 𝑢 (cid:32) ± √︄ − 𝜀 𝑢 (cid:33) (cid:39) 𝑢 (cid:34) ± (cid:32) − 𝜀 𝑢 (cid:33) (cid:35) . (3.25) That is, even in the absence of the applied field and viscosity, the order parameter value for the final stateafter transition, 𝑢 = 𝑢 , is somewhat higher than the equilibrium value ˜ 𝑢 . The velocity is 𝑣 = √ 𝜀 (cid:16) 𝑢 − 𝑢 (cid:17) . (3.26)Remarkably, the dependence of velocity on the stationary values of concentration, 𝑢 , 𝑢 , , isexactly the same as for the well known travelling-wave solution for the diffusion equation with cubicnonlinearity; for 𝑢 = 𝑢 /
2, the velocity is zero, that is the front becomes static. However, the coefficientin (3.26) depends on 𝜀 , i.e., on the “Cahn-Hilliard part”. As it was mentioned in the introduction, thederivative of the homogeneous part of the thermodynamic potential Φ ( 𝑢 ) is given by (1.3):d Φ ( 𝑢 ) d 𝑢 = 𝑢 − 𝛿𝑢 − 𝑠𝑢. (3.27)Integrating once and substituting values of 𝛿 and 𝑠 given by (3.24), we obtain the following expressionfor the potential Φ ( 𝑢 ) Φ ( 𝑢 ) = 𝑢 − 𝑢 𝑢 + (cid:18) 𝑢 + 𝜀 (cid:19) 𝑢 + 𝐶, (3.28)where 𝐶 is a constant. Then, final (after transition) value of the potential is Φ ( 𝑢 ) = / 𝜀 𝑢 + 𝐶 .Taking into account 𝜀 (cid:28)
1, to calculate the equilibrium value Φ ( ˜ 𝑢 ) , we use the approximate expressionfrom (3.25), ˜ 𝑢 (cid:39) 𝑢 − 𝜀 / 𝑢 . Substitution into (3.28) and neglecting higher order in 𝜀 terms, yields Φ ( ˜ 𝑢 ) (cid:39) / 𝜀 𝑢 + 𝐶 , i.e., it is nearly equal to the value after transition. It means that despite the deviation of the concentration in the final state after transition from its equilibrium value, the deviations of thermodynamic potential from its equilibrium value are of the higherorder in 𝜀 . Now, let 𝛼 = 𝜂 ≠ 𝑢 , = (cid:40) 𝑢 − (cid:16) 𝑢 − 𝑢 (cid:17) 𝜂 ± √︂(cid:104) 𝑢 + (cid:16) 𝑢 − 𝑢 (cid:17) 𝜂 (cid:105) − 𝜀 (cid:41) . (3.29)From (3.26) (cid:0) 𝑢 − 𝑢 (cid:1) = 𝑣 √ 𝜀 ; comparing (3.25) and (3.29) we see that the deviation term is of theform 𝑣𝜂 √ 𝜀 , i.e., multiple of velocity and viscosity. On the other hand, if 𝛼 ≠ 𝜂 = 𝑢 , = 𝑢 (cid:34) ± √︄ + 𝜀 𝑢 (cid:18) 𝛼 √ − 𝜀 (cid:19) (cid:35) . (3.30) .O. Mchedlov-Petrosyan, L.N. Davydov Now, let both 𝛼 and 𝜂 be non-zero. The expression for the velocity (3.23) is independent of 𝜂 ; for thespecial value of 𝛼 , 𝛼 = √ 𝜀𝑢 (cid:16) 𝑢 − 𝑢 (cid:17) , (3.31)the velocity is zero, i.e., for the corresponding value of the applied field, the transition front becomesstatic even for 𝑢 ≠ 𝑢 . Substitution of this value of 𝛼 into (3.20) and (3.21) yields 𝑠 = − 𝑢 − 𝜀 𝑢 𝑢 ; 𝛿 = 𝑢 , (3.32)and ˜ 𝑢 , = 𝑢 (cid:32) ± √︄ − 𝜀 𝑢 𝑢 (cid:33) . (3.33)Again, the viscosity 𝜂 has self-consistently dropped out of the latter expression, there is no dissipationfor the static transition front; the deviation of the order parameter value 𝑢 after transition from itsequilibrium value ˜ 𝑢 is exactly the same as given by (3.30) (i.e., for 𝜂 = 𝛼 .
4. Discussion
In the present work we have modelled the interplay of equilibrium and non-equilibrium phase tran-sitions. While the equilibrium phase transitions are described on the basis of modified Cahn-Hilliardequation, the non-equilibrium phase transitions are presented by the canonical chemical models intro-duced by Schlögl [26]. In these models, the different “phases” correspond to different stationary states ofthe chemical reactions system. Schlögl considered two reaction systems: the so-called “First Schlögl Re-action”(1.8)–(1.9), which is an analog of the second order equilibrium phase transition, and the “SecondSchlögl Reaction” (1.10)–(1.11), which is an analog of the first order equilibrium phase transition, fordetails see [26]. Each of these reaction systems has four components, though the concentrations of threereagents ( the so-called “reservoir reagents”) are assumed to be kept constant, and only the concentrationof one reagent changes in time and space. If the system is well mixed (or there is no spatial mass transfer),the time evolution of this reagent is governed by a nonlinear ordinary differential equation. It is quadraticpolynomial nonlinearity for the First Schlögl Reaction (1.12), and the cubic nonlinearity for the SecondSchlögl Reaction (1.13). If the mass transfer should be taken into account, it is usually described by dif-fusion equation. However, if the system is essentially inhomogeneous, e.g., undergoes a phase transition,the proper description of the mass transfer is given by the Cahn-Hilliard equation [1–4], complementedwith nonlinear sink/source terms. For the second-order reaction system, such an approach was pioneeredby Huberman [19] and Cohen and Murray [20]. Apparently, being unaware of Schlögl paper, they in factconsidered the interplay of equilibrium and (second-order) non-equilibrium phase transitions. Hubermanintroduced Cahn-Hilliard equation with additional kinetic terms corresponding to the reversible first-order autocatalytic chemical reaction. He analyzed the linear stability of stationary states and the mutualeffect of spinodal decomposition and reaction. Cohen and Murray considered the same equation in thebiological context; using the nonlinear stability analysis based on a multi-scale perturbation method, theyidentified bifurcations to spatial structures. Similar equation with an additional nonlinear derivative termand an inverted sign of the quadratic nonlinearity was used in [21] to study the segregation dynamics ofbinary mixtures coupled with chemical reaction. The same equation as in [19, 20] was used to describethe phase transitions in chemisorbed layer [22] and to model the system of cells that move, proliferateand interact via adhesion [23]. Furthermore, for the latter model, several rigorous mathematical resultson the existence and asymptotics of solutions were obtained in [24, 25]. On the other hand, to the best ofour knowledge, there is no study of the Cahn-Hilliard equation with the third order reaction terms in theliterature.
Our aim in the present work was to consider the possibly simple situation, where the interplay of the equilibrium and non-equilibrium phase transitions could be observed explicitly. Thus, we considered ahn-Hilliard model with Schlögl reactions the advancing fronts which “combine”, in some sense, these both transitions. We obtained several exacttravelling wave solutions, which exhibit an explicit parametric dependence. Naturally, for both transitionsto proceed simultaneously, some additional constraints should be imposed on the parameters of themodel.To get a more direct insight here, we return to dimensional parameters. Starting from the CHHCMequation supplemented by an additional convective term and viscosity, we see that the coexistenceof equilibrium and “second-order” non-equilibrium transformations in the form of a constant-velocitytransition front imposes quite rigid constraints on the parameters. From (2.25) the dimensional velocity 𝑉 = 𝑣𝑙 / 𝜏 is 𝑉 = √ 𝑘 (cid:48) √ 𝑞 ¯ 𝜀 − ¯ 𝛼𝑋 . (4.1)Here, 𝑋 = 𝑢 𝑋 is the dimensional stationary concentration of the reaction system; from (2.6) we have 𝑋 = 𝑢 𝑋 = 𝑘 𝐴 − 𝑘 𝐵𝑘 (cid:48) . (4.2)Remarkably, in the absence of the field, ¯ 𝛼 =
0, the velocity does not depend on this concentration,but on the parameters of the “Cahn-Hilliard part” 𝑞, 𝜀 and on the reaction constant for the reverse firstreaction (1.8) only 𝑉 = √ 𝑘 (cid:48) √ 𝑞 ¯ 𝜀. (4.3)In this case, the velocity of the anti-kink solution is always positive, while that of the kink-solution isnegative. That is, the stable state 𝑋 of the chemical system always spreads on the cost of the unstablezero state. In the absence of the field and viscosity, the constraints imposed on the stationary values ofpolynomial part of the chemical potential ˜ 𝑋 𝑖 = ˜ 𝑢 𝑖 𝑋 = ˜ 𝑢 𝑖 (cid:14) √ 𝑞 are very rigid indeed˜ 𝑋 = 𝑋 ; ˜ 𝑋 = 𝑋 ; ˜ 𝑋 =
0; (4.4)i.e., the stable stationary states for equilibrium transition should coincide with the stationary states for thereaction system. This also means that the homogeneous part of the thermodynamic potential Φ shouldbe a symmetric function with equal-depth wells.As already mentioned, we consider the isothermal situation only; still, it may be interesting to checkthe limit of “critical state” for the equilibrium phase transition, i.e., for the “Cahn-Hilliard part”. As usuallyfor this model, it is assumed 𝑠 ∼ ( 𝑇 − 𝑇 𝑐 ) [for symmetric potential this also means 𝛿 ∼ ( 𝑇 − 𝑇 𝑐 ) / ],where 𝑇 𝑐 is the critical temperature. Then, in terms of our model, 𝑇 → 𝑇 𝑐 corresponds to ˜ 𝑋 →
0. Thus,the larger equilibrium concentration scales as˜ 𝑋 ∼ ( 𝑇 − 𝑇 𝑐 ) / . (4.5)From (4.2) and (4.4), the compatibility of the transitions yields 𝑘 (cid:48) = 𝑘 𝐴 − 𝑘 𝐵 ˜ 𝑋 . (4.6)Substitution of the latter expression into (4.3) shows that if the equilibrium transition approaches thecritical state, the velocity of the front diverges as (cid:0) ˜ 𝑋 (cid:1) − , i.e., 𝑉 ∼ ( 𝑇 − 𝑇 𝑐 ) − / . (4.7)If the viscosity is non-zero (but still ¯ 𝛼 = ˜ 𝑋 = 𝑋 ; ˜ 𝑋 = 𝑋 − ¯ 𝜂𝑘 (cid:48) 𝑞 ; ˜ 𝑋 = . (4.8) .O. Mchedlov-Petrosyan, L.N. Davydov That is, while the stationary states for the equilibrium transition should again coincide with the stationaryvalues for the reaction system, the unstable state should be shifted to the lower value. Thus, to compensatethe additional dissipation, the homogeneous part of the thermodynamic potential becomes asymmetric,the potential well corresponding to 𝑋 is now deeper, see (2.32); the difference, naturally, disappears forzero viscosity ¯ 𝜂 . On the other hand, if ¯ 𝛼 ≠
0; ¯ 𝜂 = 𝑋 , = 𝑋 (cid:34) ± √︄ +
16 ¯ 𝛼 ¯ 𝜀 √ 𝑀𝑞 𝑋 (cid:35) . (4.9)This means that for positive 𝛼 , the order parameter value for the final state after transition, 𝑋 = 𝑋 ,is somewhat lower than the equilibrium value ˜ 𝑋 ; thus, the presence of the field can prevent finalequilibration. The unstable equilibrium value ˜ 𝑋 should be somewhat lower too, so the potential Φ isagain asymmetric. Now, let both 𝛼 ≠ 𝜂 ≠
0. The expression for the velocity is independent of 𝜂 ; forthe special value ¯ 𝛼 = √ 𝜀𝑘 (cid:48) 𝑋 √ 𝑞 (4.10)the velocity is zero, i.e., for the corresponding value of the applied field, the transition front becomesstatic. The latter expression depends both on the Cahn-Hilliard parameters and on 𝑘 (cid:48) and 𝑋 , so the static front is due to the balance of equilibrium and reactive processes. The viscosity ¯ 𝜂 has dropped outfrom the corrections to the stationary states. This is physically reasonable: there is no dissipation for thestatic transition front; the deviation of the equilibrium value ˜ 𝑋 of the order parameter from the finalstate after transition 𝑋 = 𝑋 , see (2.34), is exactly the same as given by (4.9) (i.e., for ¯ 𝜂 = 𝛼 .In appendix we consider the model introduced by Puri and Frish [21]. While it looks similar toCHHCM equation, for their reaction system the stable state is 𝜓 =
0, and 𝜓 = 𝜓 = 𝜓 should nearly merge with zero, 𝜓 (cid:39) − , 𝑉 = 𝑣𝑙 / 𝜏 is now 𝑉 = √ 𝑘 (cid:48) √ 𝑞 ¯ 𝜀 (cid:18) 𝑋 − 𝑋 (cid:19) − ¯ 𝛼𝑋 𝛼 = 𝑉 = √ 𝑘 (cid:48) √ 𝑞 ¯ 𝜀 (cid:18) 𝑋 − 𝑋 (cid:19) . (4.12)Comparing the latter equation with (4.3) we see that this expression is very similar to the coefficientin (4.12) (to avoid confusion we remind that 𝑘 (cid:48) and 𝑘 (cid:48) have different dimensionality). However, thedependence of velocity on the stationary values of concentration, 𝑋 , 𝑋 , , is exactly the same as forthe well known travelling-wave solution for the diffusion equation with cubic nonlinearity; for 𝑋 = 𝑋 ,the velocity is zero, that is the front becomes static. Moreover, for zero field, the viscosity ¯ 𝜂 enters theconstraints (3.20) and (3.21) always multiplied by (cid:16) 𝑋 − 𝑋 (cid:17) , see (3.29). Particularly, for the static front,the stationary concentrations ˜ 𝑋 , ˜ 𝑋 will not depend on ¯ 𝜂 , which is reasonable physically. If additionallyto ¯ 𝛼 = 𝜂 =
0, that is the CHS-case, the final value after transition 𝑋 will deviate from theequilibrium value, see (3.25). Taking into account ¯ 𝜀 (cid:28)
1, we get 𝑋 (cid:39) ˜ 𝑋 + ¯ 𝜀 𝑘 (cid:48) 𝑋 𝑀𝑞 . (4.13) ahn-Hilliard model with Schlögl reactions -6 -4 -2 0 2 4 60 u =1 C on c en t r a t i on , u Travelling wave coordinate, z s =1 s =2 s =5 Figure 1. (Colour online) The trevelling wave front in the case of the 1st Schlögl reaction for severalwave velocities 𝑣 ∼ 𝜎 and 𝑢 = That is, even in the absence of the applied field and viscosity, the order parameter value for the final stateafter transition, 𝑋 = 𝑋 , is somewhat higher than the equilibrium value ˜ 𝑋 , the phase is oversaturatedwith 𝑋 . However, comparing the values of the Φ ( 𝑋 ) and Φ (cid:0) ˜ 𝑋 (cid:1) we see that the deviation of thethermodynamic potential from its equilibrium value is of the higher order in ¯ 𝜀 . Different from CHHCMcase, for CHS case in the limit of critical state for the equilibrium phase transition, i.e., for the “Cahn- Hilliard part”, these transitions become incompatible. Indeed, for 𝑇 → 𝑇 𝑐 we need to take the limit˜ 𝑋 → 𝑋 . For smallervalues, this expression is physically senseless. Thus, if the equilibrium concentration ˜ 𝑋 of the “Cahn-Hilliard part” approaches this limit, the equilibrium and non-equilibrium transitions could not proceedsimultaneously, at the same front.If ¯ 𝛼 ≠
0; ¯ 𝜂 =
0, see (3.30), similar to convective CHHCM, the final state after transition is slightlyundersaturated by 𝑋 due to the presence of the field. If both 𝛼 and 𝜂 are non-zero, for the special value of¯ 𝛼 , see (3.31), the velocity is zero, i.e., for the corresponding value of the applied field, the transition frontbecomes static even for 𝑋 ≠ 𝑋 . Then, the viscosity 𝜂 self-consistently drops out from the correctionsto the stationary states, see (3.33).For the illustration purpose it is instructive to present the connection between the “observable”parameters, e.g., velocity 𝑉 of the transition front and the dimensional effective width of the transitionfront 𝜎 . If 𝑍 is the dimensional travelling wave coordinate, the argument of tanh is 𝑢 √ 𝜀 𝑧 = 𝑋 𝑙𝑋 √ 𝜀 𝑍𝑙 = 𝑋 √ 𝑞 √ 𝜀 𝑍 . (4.14)Then, 𝜎 = √ 𝜀𝑋 √ 𝑞 . This expression is the same for both reactions, though the expressions for 𝑋 aredifferent, see below.For the first reaction (without field) 𝑉 = 𝑘 (cid:48) √ 𝜀 √ 𝑞 = 𝑘 (cid:48) 𝜎𝑋 . (4.15)From the definition of 𝑋 (4.2) we find 𝑉 = 𝜎 𝑘 𝐴 ( − 𝜌 ) , (4.16)where the parameter 𝜌 = 𝑘 𝐵 / 𝑘 𝐴 is characteristic of the first reaction: if 𝜌 →
1, the differencebetween stationary states disappears, if 𝜌 → 𝑉 = 𝜎𝑘 (cid:48) 𝑋 (cid:18) 𝑋 − 𝑋 (cid:19) . (4.17) .O. Mchedlov-Petrosyan, L.N. DavydovFigure 2. (Colour online) The trevelling wave front velocity in the case of 2nd Schlögl reaction asfunction of the front width 𝜎 and parameter 𝜌 . However, 𝑋 , 𝑋 are now the roots of quadratic equation: 𝑢 − 𝑅𝑢 + 𝑄 = 𝑅 = 𝑘 𝐴𝑘 (cid:48) 𝑋 and 𝑄 = 𝑘 𝐵𝑘 (cid:48) 𝑋 . Introducing a characteristic parameter for the second reaction 𝜌 , 𝜌 = 𝑘 (cid:48) 𝑘 𝐵 ( 𝑘 𝐴 ) ; 0 < 𝜌 < , we get 𝑋 = 𝑘 𝐴 𝑘 (cid:48) (cid:104) + √︁ − 𝜌 (cid:105) ; 𝑋 = 𝑘 𝐴 𝑘 (cid:48) (cid:104) − √︁ − 𝜌 (cid:105) (4.18)and 𝑉 = 𝜎 ( 𝑘 𝐴 ) 𝑘 (cid:48) (cid:110) √︁ − 𝜌 − 𝜌 + (cid:111) . (4.19)The dependence of the wave front velocity on its width 𝜎 and parameter 𝜌 according to the aboveequation with ( 𝑘 𝐴 ) / (cid:0) 𝑘 (cid:48) (cid:1) = / 𝑋 in the final state maydeviate from its equilibrium value and even the transition may be stopped. On the other hand, for the CHSequation, the effect of the non-equilibrium transition, i.e., of the reaction system, is much stronger. Thetransition front may be stopped, or even reversed both by changing the stationary states of the reactionsystem and by the field. The final state may be undersaturated or oversaturated, creating non-equilibrated phases. A. Cahn-Hilliard-Puri-Frish equation
Here, for completeness we give the exact travelling wave solution for one-dimensional version ofequation, introduced by Puri and Frish [21]. To match our consideration with [21] we use their originalnotations and normalizations, which are different from these in the other parts of the present paper.After the correction of the evident misprint and considering the generally asymmetric thermodynamicpotential, the equation (9) of [21] is 𝜕𝜓𝜕𝑡 = − 𝜕 𝜕𝑥 (cid:26) 𝜓 + 𝛿𝜓 − 𝜓 + 𝜕 𝜓 𝜕𝑥 (cid:27) − 𝛼𝜓 ( − 𝜓 ) + 𝛼 𝜎 𝜓 𝜕 𝜓 𝜕𝑥 . (A.1) ahn-Hilliard model with Schlögl reactions Here, 𝛼 is a phenomenological constant, proportional to the ratio of the characteristic times for a spinexchange and reaction, 𝜎 is proportional to the inverse square of lattice spacing. Evidently, dropping thelast term in the right-hand side of (A.1) we obtain CHHCM equation, though with an inverted sign ofthe reaction term, see section 2. This means that for the reaction system the stable state is 𝜓 =
0, while 𝜓 = 𝑧 = 𝑥 − 𝑣𝑡 . This yields − 𝑣 d 𝜓 d 𝑧 = − d d 𝑧 (cid:26) 𝜓 + 𝛿𝜓 − 𝜓 + d 𝜓 d 𝑧 (cid:27) − 𝛼𝜓 ( − 𝜓 ) + 𝛼𝜎 𝜓 d 𝜓 d 𝑧 . (A.2)Introducing the ansatz d 𝜓 d 𝑧 = 𝜅𝜓 ( − 𝜓 ) , (A.3)which for the positive 𝜅 and 0 (cid:54) 𝜓 (cid:54) − 𝛼𝜓 ( − 𝜓 ) + 𝛼𝜎 𝜓 d 𝜓 d 𝑧 = dd 𝑧 (cid:20) − 𝛼𝜅 𝜓 + 𝛼 𝜎 𝜅𝜓 − 𝛼 𝜎 𝜅𝜓 (cid:21) . (A.4)Substituting the latter expression into (A.2), integrating once and rearranging the terms we get (cid:16) 𝑣 − 𝛼𝜅 (cid:17) 𝜓 + 𝛼 𝜎 𝜅𝜓 − 𝛼 𝜎 𝜅𝜓 − dd 𝑧 (cid:18) 𝜓 + 𝛿𝜓 − 𝜓 + d 𝜓 d 𝑧 (cid:19) = 𝐶 . (A.5)For the latter equation to be satisfied, the expression under the derivative should be a quadratic functionof 𝑢 : 𝜓 + 𝛿𝜓 − 𝜓 + d 𝜓 d 𝑧 = 𝛽𝜓 + 𝛾𝜓 + 𝐶 , (A.6) (cid:16) 𝑣 − 𝛼𝜅 (cid:17) 𝜓 + 𝛼 𝜎 𝜅𝜓 − 𝛼 𝜎 𝜅𝜓 − ( 𝛽 + 𝛾𝜓 ) d 𝜓 d 𝑧 = 𝐶 , (A.7)where 𝛽, 𝛾, 𝐶 and 𝐶 are constants. Substitution of the corresponding expressions for the derivativesinto (A.6) and (A.7) yields (cid:16) 𝑣 − 𝛼𝜅 (cid:17) 𝜓 + 𝛼 𝜎 𝜅𝜓 − 𝛼 𝜎 𝜅𝜓 + 𝜅 (cid:2) 𝛾𝜓 + ( 𝛽 − 𝛾 ) 𝜓 − 𝛽𝜓 (cid:3) = 𝐶 , (A.8) 𝜓 + 𝛿𝜓 − 𝜓 + 𝜅 (cid:16) 𝜓 − 𝜓 + 𝜓 (cid:17) = 𝛽𝜓 + 𝛾𝜓 + 𝐶 . (A.9)Rearranging and equating to zero coefficients at all powers of 𝜓 , we obtain the system of constraintsimposed on the parameters. 2 𝜅 = , (A.10) 𝛿 − = 𝛾, (A.11) 𝛽 = , (A.12) 𝛾 = 𝛼 𝜎 , (A.13) 𝛼 𝜎 + 𝛽 − 𝛾 = , (A.14) 𝑣 = 𝛼 𝜅 + 𝜅 𝛽. (A.15) .O. Mchedlov-Petrosyan, L.N. Davydov If these constraints are satisfied, the solutions of (A.3) are simultaneously the solutions of the travelling-wave equation (A.2), which, quite analogously to (2.24) is 𝜓 = (cid:26) + tanh (cid:20) √ ( 𝑥 − 𝑣𝑡 ) (cid:21) (cid:27) . (A.16)From (A.10), (A.12) and (A.15) 𝑣 = √ 𝛼 + √ , the velocity of the kink is always positive ( 𝛼 isper definition positive), i.e., the stable 𝜓 = 𝜓 =
1. Theconstraints imposed on the parameters of the model are 𝛼 = 𝜎 ; 𝛿 = /
2. Then, the roots of equation 𝜓 − 𝛿𝜓 − 𝜓 = 𝜓 (cid:39) , 𝜓 = 𝜓 (cid:39) − , Φ ( 𝜓 ) should be very far from symmetric. Acknowledgements
We are thankful to O.S. Bakai for the attention to the paper and valuable remarks.
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П.О. Мчедлов-Петросян, Л.М. Давидов