Phase Transitions Affected by Molecular Interconversion
AA Mean Field Theory of Phase Transitions Affected by MolecularInterconversion
Thomas J. Longo and Mikhail A. Anisimov
1, 2 Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742,USA Department of Chemical and Biomolecular Engineering, University of Maryland, College Park, MD 20742,USA a) (Dated: 25 February 2021) In this paper, we describe the diffusive and interconversion properties of a symmetric mixture of two different dynamics,characteristic for conserved and non-conserved order parameters, in the presence or absence of an external “source”of interconversion. We show that the interactions of the two competing dynamics of the order parameter (diffusionand interconversion) results in the phenomenon of “phase amplification,” when one phase grows at the expense ofanother one. This phenomenon occurs when the order parameter exhibits even a small probability of non-conserveddynamics, thus breaking the particle conservation law. Also, we show that the addition of a source of interconversiondrives the system away from equilibrium and creates the possibility for arrested phase separation - the existence ofnon-growing (steady-state) mesoscopic phase domains. These steady-state phase domains are an example of a simpledissipative structure. The change of the dynamics from phase amplification to microphase separation can be consideredas a nonequilibrium “phase transition” in the dissipative system. The theoretical description is used to describe phasetransitions affected by interconversion of an equilibrium and nonequilibrium lattice model.Solving the problem of condensed-matter systems with alarge number of strongly interacting degrees of freedom - theproblem of phase transitions and critical phenomena - has rev-olutionized macroscopic physics in the 20th century . Phasetransitions play a crucial role in condensed-matter physics,biophysics, biology, high-energy physics, metallurgy, andastrophysics . Examples include structural, superconduc-tive, and ferromagnetic transitions in solids, superfluidity inhelium, as well as a variety of phase transformations in mul-ticomponent fluids . While equilibrium phase transitionsand criticality in simple systems are well studied and under-stood, the description of phase transformations in the presenceof chemical reactions and in systems far from equilibriumis much less developed . Under certain conditions, thesetransitions can lead to liquid polyamorphism (the existenceof liquid-liquid separation in a single-component fluid ) orto non-equilibrium microphase separation, an observed phe-nomenon that we hypothesize occurs during the formation ofmembraneless organelles . Understanding the formation ofthese microphases is a precursor to understanding more com-plex self-organizing structures in non-equilibrium systems.The onset of non-equilibrium arrested phase separation cantherefore be considered as a simpler version of dissipativestructures described by Prigogine’s theory and a precursorto self-organizing criticality in more complex structures .In this paper, we present a general description of phaseseparation driven by spinodal decomposition in a binary fluidwith molecular interconversion of the components. Through-out the paper, we focus on the two primary findings of thepresent theory: first, without a “source” of interconversion,the most stable state of the system will always be macroscopicphase separation . The addition of a source of interconversiondrives the system away from this equilibrium and creates the a) Electronic mail: [email protected] possibility for arrested phase separation - the existence of non-growing (steady-state) mesoscopic phase domains . Second,a symmetric, equilibrium interconverting system can be mod-eled using a mixed lattice gas and Ising model with the sameorder parameter. This is possible because these models arethermodynamically identical, but they differ in dynamics .As a result, the interactions of the two dynamically competingorder parameters (the conserved lattice gas order parameter,density, and the non-conserved Ising model order parameter,magnetization) results in the phenomenon of “phase ampli-fication,” when one phase grows at the expensive of anotherone.The developed theory is applied to describe phase tran-sitions affected by interconversion in an equilibrium andnonequilibrium lattice model. First, we consider an equi-librium model with mixed lattice-gas (conserved order-parameter) and Ising-lattice (non-conserved order-parameter)dynamics . Second, we consider a nonequilibrium modelwith mixed lattice-gas and Ising-lattice dynamics. This modelwas originally formulated by Glotzer et al. , and was asubject of criticism because the source of dissipation was amystery . We resolve this mystery by showing that thesource of dissipation originates from a disbalance in chemicalpotentials associated with interconversion and phase separa-tion. I. THEORY OF SPINODAL DECOMPOSITIONAFFECTED BY INTERCONVERSION
Consider a symmetric, reversible chemical reaction A k −− (cid:42)(cid:41) −− B (1)where k is the forward and reverse reaction rate. c A and c B aredefined as the concentrations of species A and B respectively. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b This system can be described in the mean field approximationby a Landau-Ginzburg free-energy functional with one orderparameter φ = c A − / . This functional reads as F [ { φ } ] = (cid:90) V (cid:18) f ( φ ) + κ | ∇ φ | (cid:19) d V (2)For an isotropic system, κ is the square of the range of in-termolecular interactions on the order of the molecular size.In general, throughout this paper κ ≈
1. Using the Euler-Lagrange derivative, the chemical potential for this system is µ ≡ δ F δ φ = αφ + β φ − κ ∇ φ (3)where α ∝ ( T − T c ) and β is a positive constant. For a lattice-gas system β = /
3. This equation can be rewritten in thelinearized form µ = f (cid:48)(cid:48) φ − κ ∇ φ (4)where f (cid:48)(cid:48) = ∂ f / ∂ φ ∝ − T / T c . For the case of a systemwith one order parameter and two different types of dynam-ics, the most general form of the time evolution of the orderparameter is given by ∂ φ∂ t = π − L µ + M ∇ µ (5)This equation can be separated into three key terms. First, π is known as the interconversion energy source or sink thatforces the system into a non-equilibrium steady-state. De-pending on the model, this term can be written in a variety offorms, but in this paper it is assumed that π ∝ φ . The secondterm describes the relaxation of the nonconserved dynamicsof the concentration to equilibrium . This As a result, onlyone order parameter (with two different dynamics) is used tocharacterize the behavior of the system. The Ising spin in-terconversion behavior corresponds to the chemical equilib-rium constant being independent of temperature or pressure,such that ln K =
0, meaning that there is no enthalpy of re-action ( ∆ H rxn = A to B mirrors the flipping of spins inthe Ising model. In this case, µ is the deviation of the chem-ical potential difference ( µ = µ A − µ B ) from its equilibriumvalue - µ =
0, and L is a mobility constant that determines therate of interconversion. In equilibrium thermodynamics, theinterconversion and phase separation follows the same equi-librium conditions . Therefore, in equilibrium systems, phaseseparation (even accompanied by interconversion) should al-ways reach complete microscopic phase separation. Lastly,the third term describes the conserved dynamics of the orderparameter . This component mirrors the diffusion of particlesin the lattice gas model where the total number of particlesis conserved. M is the diffusion mobility (an Onsager coef-ficient). The interactions between diffusion, interconversion,and the source of interconversion, given in the general timeevolution of the order parameter can be explained by consid-ering the following three examples: Equilibrium diffusion - When there is no interconversionof any kind ( π = L = : ∂ ˆ c A ∂ t = M ∇ µ (6)This equations describes the dynamics of a system where thetotal number of particles is conserved, and as such, in thespinodal region, phase separation into two distinct phases willoccur. A well-known example is the diffusion of particlesin the lattice gas model. In addition, this equation can besolved to determine the growth rate equation (known as the“Amplification factor”) of the phase domains, and is given as: ω ( q ) = − M ∆ T q ( − ξ q ) , where ξ is the correlation lengthof concentration fluctuations - defined as ξ = (cid:112) κ / f (cid:48)(cid:48) . Thecoefficient M ∆ T is known as the diffusion coefficient, D . Equilibrium interconversion - In the absence of a sourceand a diffusive term ( π = M = : ∂ ˆ c A ∂ t = − L µ (7)As a result, this equation describes the dynamics of a systemwhere the total number of particles is not conserved. Thissystem will relax to equilibrium along with the order param-eter. Thus, this equation describes the dynamics of an “Ising-like” model; if the system starts in a homogeneous symmet-ric mixture, then after infinite time only one phase will sur-vive. The growth rate for this system is determined to be: ω ( q ) = − L ∆ T ( − ξ q ) . This equation is similar to the am-plification factor in the classical Cahn-Hilliard theory, but in-stead, describes the non-conserved dynamics of the order pa-rameter. Nonequilibrium diffusion - If we consider the effect of asource of interconversion on a system with no non-conservedorder-parameter dynamics ( L = et al . . In this model, the reactionsource is constrained to be linearly dependent on concentra-tion. Therefore, as a simple approximation, π = C − k ˆ c A ,where k is the reaction rate and C is a constant. This typeof source randomly converts one species to another, and is in-dependent on the thermodynamic properties of the system. Inthis case, the mass-balance equation can be written as ∂ ˆ c A ∂ t = C − k ˆ c A + M ∇ µ (8)Glotzer et al. modeled the dynamics described by this equa-tion by randomly choosing a spin and forcing it to intercon-vert independently from the dynamics of phase separation.They found that the addition of a source term causes chem-ically arrested microphase separation. This toy model waslater recognized as being a special case only valid far awayfrom equilibrium . The growth rate for this system can bedetermined to be: ω ( q ) = − k − M ∆ T q ( − ξ q ) . Thus, aswill be discussed in more detail later, the inclusion of a reac-tion rate term k in the growth rate results in phase separationto be arrested. Nonequilibrium diffusion with interconversion - To solvethe general mass-balance equation, the same interconversionsource used by Glotzer et al. and the chemical potential fromEq. (4) is used. This gives ∂ ˆ c A ∂ t = C − ( k + L f (cid:48)(cid:48) ) ˆ c A + ( L κ + M f (cid:48)(cid:48) ) ∇ ˆ c A − M κ ∇ ˆ c A (9)This differential equation has a solution ofˆ c A = C ( k + L f (cid:48)(cid:48) ) + ∑ i Ae R ( q i ) t cos ( q i r ) (10)where A is a constant determined by the initial conditions at t =
0, and R ( q ) is the general growth rate amplification factordefined by ω ( q ) = − ( L f (cid:48)(cid:48) + k ) − ( M f (cid:48)(cid:48) + κ L ) q − M κ q (11)This equation, in a slightly different form, was also obtainedby Lefever et al. . This equation is best expressed throughthe correlation length, ξ ; re-writing it in this from gives ω ( q ) = − k − f (cid:48)(cid:48) ( L + Mq )( − ξ q ) (12)This equation describes the characteristics of domain growthin both phase amplification and microphase separation. II. PHASE AMPLIFICATION
VS.
MICROPHASESEPARATION: ORDER PARAMETER CHARACTERISTICS
As shown in Eq. (12), there are two competing rates thatdictate the behavior of the system. The first is found from ω ( q = ) , where the reaction rate term k competes with thediffusion-interconversion coupling term ( L f (cid:48)(cid:48) ). The secondis determined from ∂ ω / ∂ q | q = , where the diffusion ( M f (cid:48)(cid:48) )and interconversion ( L κ ) rates compete with each other. Theresults of these competitions will either produce phase ampli-fication or microphase separation as shown in Fig. (1) anddepicted in Table (I). To achieve arrested phase separation,the source of interconversion must overcome the diffusion-interconversion coupling term, k > L f (cid:48)(cid:48) , while
M f (cid:48)(cid:48) > κ L such that the diffusion rate is faster than the reaction rate.Meanwhile, phase amplification occurs when the diffusion-interconversion coupling term is greater than the reaction rate L f (cid:48)(cid:48) > k , and the rate of amplification is enhanced when M f (cid:48)(cid:48) < L κ . TABLE I. Conditions for phase amplification and arrested phase sep-aration as illustrated in Fig. (1). The left column corresponds to thesolid lines and the right column corresponds to the dashed lines.
M f (cid:48)(cid:48) > L κ M f (cid:48)(cid:48) < L κ f (cid:48)(cid:48) L > k Phase Amplification “Ising-like” Growth f (cid:48)(cid:48) L < k Arrested Phase Sep. No Growth
As will be mentioned in more detail later, Eq. (12) hasthree characteristic wavenumbers: q m , the wave number of the FIG. 1. The growth rate factor for different interactions betweendiffusion, interconversion, and the source of interconversion. L , M ,and k were varied to represent the conditions of Table (1). Thesolid curves corresponds to M | f (cid:48)(cid:48) | > L κ , where phase amplification(purple) and microphase separation (green) are shown, and and thedashed curves correspond to M | f (cid:48)(cid:48) | < L κ , where “Ising-like” ampli-fication (purple) and no growth (green) are shown. The red line rep-resents classical Cahn-Hilliard spinodal decomposition where L = k = fastest growing inhomogeneities; q + c , the upper cut-off wavenumber; and q − c , the lower cut-off wave number. The exis-tence of a non-zero q − c indicates that arrested phase separa-tion will occur; such that, after infinite time, the observabledomain size will not exceed 1 / q − c . As a result, we predict that q − c is the order parameter that characterizes microphase sep-aration. When the system experiences phase amplification,shown by the purple curve in Fig. (1), q − c is imaginary andmicrophase separation doesn’t occur. Only by increasing thestrength of the racemizing source (given by k ) will the ampli-fication factor be shifted down until q − c becomes non-zero asillustrated by the green line in Fig. (1). Meanwhile, we predictthat the Cahn-Hilliard result (when there is only a conservedorder parameter so π = L =
0) is a critical line distin-guishing microphase separation from phase amplification andmarked at the point where q − c =
0. Thus, we predict that asystem going from the equilibrium phase amplification regionto the nonequilibrium microphase separation region will ex-perience a large “jump” in its internal energy. This transitionwill be similar to a first order phase transition, and it indicatesthe formation of a dissipative structure.
III. STRUCTURE FACTOR: DEMONSTRATION OFMICROPHASE SEPARATION
The central prediction that the domain size will be arrestedin the presence of an external source can be proven throughthe structure factor for the system. Cahn-Hilliard theory islimited in that it is only applicable in the early stages of spin-odal decomposition. Thus, to understand how the system tran-sitions from the later stages of spinodal decomposition, wheredomain coarsening occurs, to the nucleation regime, the origi-nal theory is expanded to allow for concentration fluctuations.Then, by approximating the crossover from the later stages
FIG. 2. The structure factor for a conserved order parameter system(where π = L =
0) exhibiting crossover from Cahn-Hilliardspinodal decomposition, t = t = t =
150 (red), the nucleation limit, t → ∞ (black). Themaximum of the structure factor saturates at S m ( q , t ) = / ξ = ∆ T = − . / ξ and the corresponding q value saturates at q − c . of spinodal decomposition to the nucleation regime, it can beshown that the inclusion of an external interconversion sourcecauses the phase domain sizes to be arrested.To characterize the late stages of spinodal decompositionand the transition to the coarsening regime, the theory ofspinodal decomposition is expanded to allow concentrationfluctuations. Defining the fluctuation variable as u ( r , t ) ≡ c ( r , t ) − c , the Structure factor is given by the correlationfunction for the concentration fluctuations of the system. Suchthat S ( q , t ) = (cid:90) d r < u ( r , t ) , u ( r , t ) > e iq · r (13) As suggested by Langer et al. , the equation of motion of S ( q , t ) , is found by multiplying the key differential equation -Eq. (5) - by u ( r ) u ( r ) and integrating over the space of func-tions u . The resulting differential equation was first presentedby H. E. Cook and modified for the case of mixed diffusion-interconversion dynamics asd S ( q ) d t = R ( q ) S ( q ) + M (cid:18) + LM κ (cid:19) q (14)The structure factor immediately after the quench (which hasthe form of the well known Ornstein-Zernike structure factor)is defined as S χ = − ( M + L κ ) q R ( q ) (15)It can be seen that in the case when L = k =
0, this equa-tion reduces to the Ornstein-Zernike - S OZ = ξ / ( + ξ q ) . FIG. 4. The development of the maximum wavenumber for thestructure factor during the crossover from spinodal decomposition, q ∝ t / (red, dashed), to the nucleation regime, q ∝ t / (green,dashed). Using this factor, the above differential equation can be in-tegrated, using the condition that as t → ∞ , the structure factortakes the form of S χ ( q ) , to get S CHC ( q , t ) = S χ ( q ) (cid:16) − e R ( q ) t (cid:17) (16)Where S CHC ( q , t ) (representing the Cahn-Hilliard-Cook struc-ture factor) is valid from the initial quench to the late stages ofspinodal decomposition. To expand this equation to accountfor the coarsening regime after decomposition has finished, atime dependent f (cid:48)(cid:48) is adapted into S χ as suggested by Binder et al. . The exact form of the time dependence of f (cid:48)(cid:48) ( t ) isempirically estimated to have the form f (cid:48)(cid:48) ( t ) = f (cid:48)(cid:48) (cid:32) e − √ t / τ + e − √ t / τ (cid:33) (17) FIG. 5. Domain growth, given by R ( t ) = / q max , at constant L , M ,and f (cid:48)(cid:48) for no source (black), a small source ( k = . k = . M and τ (used for the crossover).The infinite time domain size R ( t → ∞ ) is proportional to the choiceof L and k . Where τ controls the transition rate from spinodal decompo-sition to coarsening.As suggested by Binder et al. , a crossover term is in-troduced to connect the coarsening regime with the nucle-ation regime. As depicted in Lifshitz-Slyozov theory forOswald ripening, the nucleation regime is characterized byan Ornstein-Zernike structure factor that scales with time as S ( q , t ) ∝ t and a domain size that goes as q ∝ t / . Forreference, the domain size grows as q ∝ t / in the classicalCahn-Hilliard theory . To account for the transition from t / to t / , the same scaling relation as used by Binder etal. is adopted. That in the nucleation regime, the exponentialgrowth rate scales with time as tq . Thus, the following Padéapproximant is used in the exponential factor of Eq. (16), suchthat the time, t , is replaced by the crossover time, t × , given by t × = t τ ( + t / τ ) + t / q (18)Where τ is the same crossover parameter used in f (cid:48)(cid:48) ( t ) .Therefore, the crossover between spinodal-decompositiongrowth and nucleation growth is phenomenologically intro-duced as S ( q , t ) = S OZ ( q ) (cid:18) S CHC ( q , t × ) S OZ ( q ) + S CHC ( q , t × ) (cid:19) (19)Where now, S CHC ( q , t ) = S χ ( q , t ) (cid:16) − e R ( q ) t × (cid:17) . Therefore, S ( q , t ) is the complete crossover structure factor. Using thisequation for the crossover, the structure factor for the caseof only diffusive order-parameter dynamics is compared tothe case of the same system when an external interconversionsource is introduced. It can be shown that the behavior ofthese systems through all three regimes: spinodal decomposi-tion, coarsening, and nucleation in Figs (2 & 3). It is observed that the introduction of a small source to the system causes thestructure factor curve to be arrested around a minimum valuegiven on the growth rate curve as q − c . Lastly, the crossover be-havior for a system following the transition of classical Cahn-Hilliard theory to the nucleation limit is shown in Fig. (4).The behavior of the maximum q value goes to zero accordingto the two limits: q ∝ t / (initial time, spinodal decompostionregime) and q ∝ t / (long time, nucleation regime). IV. EQUILIBRIUM MODEL: PHASE AMPLIFICATION
In 1952, Lee and Yang showed that the Ising model for ananisotropic ferromagnet and the lattice gas model for a fluidare mathematically equivalent . It was later proven that allfluids exhibiting phase separation, whether simple or com-plex, belong to the same class of critical-point universalityin thermodynamics as the Ising model . Within the sameuniversality class, systems demonstrate the same critical sin-gularities and the same critical equation of state, provided thatthe appropriately defined order parameter has the same sym-metry. The one-component-vector order parameter (the mag-netization) in the Ising model and the scalar order parameter(the density) in the lattice gas posses the same symmetry, Z up-down symmetry . FIG. 6. (a) The spontaneous equilibrium order parameter ( φ = φ )in the lattice gas / lattice binary mixture along the liquid-vapor phasecoexistence (red domain). One of the two alternative magnetizations( φ > φ <
0) in the Ising ferromagnet in zero field are shownin the red domain with blue arrows. The solid curve is the crossoverfrom mean-field behavior (dashed) to the asymptotic scaling powerlaw φ ∝ ∆ T β with β = . , while the crosses are simulationdata reported in . (b) The amplification factor given by Eq. (12) forthe initial growth of the order parameter for three conditions: con-served L =
0, non-conserved M =
0, and mixed L / ( M + L ) = . ∆ T = − . However, while the lattice gas and Ising model are equiv-alent in thermodynamics, they are fundamentally different indynamics. The order parameter in the lattice gas is conserved,while the order parameter in the Ising model is not. Thus, thetwo order parameters belong to different universality classesin dynamics . As a result, the density relaxes to equilibriumby spatial-dependent diffusion, while the relaxation of magne-tization in the lowest approximation is not spatial-dependent.An important consequence of this contrast in dynamics is inthe difference in the equilibrium states as illustrated in Fig.(6a). In the lattice gas, below the critical temperature, twoequilibrium fluid phases must coexist to conserve the totalnumber of particles (occupied cells), while in the Ising fer-romagnet only one of the alternative magnetizations, positiveor negative, will survive . Since the interface between thetwo alternative magnetic phases is energetically costly, even-tually, one phase will win over the other. This is the strikingphenomenon of “phase amplification”, as originally coined byLatinwo, Stillinger, and Debenedetti . FIG. 7. Phase amplification - the growth of the order parameter fordifferent probabilities of Ising dynamics (simulations performed inreference ). (a) Full-time behavior, T = . p r =
1; (b-d) initial time behavior, T = . p r = .
0, (c) p r = .
1, and (d) p r = . × − . The solid horizontalline, φ =
0, corresponds to lattice-gas dynamics, p r = The growth rate equation for a system with a mixture ofconserved and non-conserved order-parameter dynamics isfound to be the combined growth rate of the Cahn-Hilliardequation and the Cahn-Allen equation , but it can also beshown as the solution to Eq. (12) when there is no intercon-version source ( k = ω ( q ) = − f (cid:48)(cid:48) ( L + Mq )( − ξ q ) (20)As shown in Fig. (6b), when M =
0, the order parametergrows according to Ising dynamics , while when L =
0, theorder parameter grows according to lattice gas dynamics .From Eq. (20), the probability that the system will exhibitIsing-model spin interconversion is defined as p r = L / ( M + L ) . If p r =
1, the system always relaxes to equilibriumthrough fast amplification to one of two alternative phaseswith either positive or negative order parameter. If p r = < p r <
1, the rate of phase amplifica- tion depends on the Ising-dynamics probability, the distanceto the critical temperature, and the system size .The competition between the two types of dynamics of theorder parameter determines the initial growth of the phase do-mains generated in the unstable region. In a mixed lattice-gasand Ising system, this competition eventually results in onephase amplification the other. However, the rate of amplifica-tion depends on the probability for the system to follow Ising(non-conserved) order-parameter dynamics, see Fig. (7b-d)where the phenomenon of phase amplification is shown fordifferent probabilities from pure Ising, p r =
1, to extremelylow probability p r = . × − . For the extremely low proba-bility most realizations just fluctuate around < φ > =
0. How-ever, it is remarkable that even for this probability phase am-plification still occurs . The phenomenon of phase amplifi-cation can also be seen in the more complicated models thatfollow (provided that there is a weak interconversion energysource, k < L f (cid:48)(cid:48) ). V. NONEQUILIBRIUM MODEL: MICROPHASESEPARATION
The diffusion (conserved order-parameter) and interconver-sion (nonconserved order-parameter) dynamics employed inthe previous section, may also be used to describe a nonequi-librium model. By writing f (cid:48)(cid:48) ∝ − T / T C in the equilibriumchemical potential, given by Eq. (4), we may choose a sourceof interconversion that cancels with the temperature and spa-tial dependent components of the equilibrium chemical po-tential. For example, the source of interconversion may bechosen as π = − L (cid:18) TT C φ − κ ∇ φ (cid:19) (21)As a result, this creates a disbalance in the equilibrium chem-ical potential for the nonconseved dynamics of the mixed lat-tice gas /Ising-lattice model. Thereby, the time evolution ofthe order parameter, given by Eq. (5) may be rewritten in theform ∂ φ∂ t = − L φ + M ∇ µ (22)Thereby, we have created a disbalance in the equilibriumchemical potential. This nonequilibrium time evolution equa-tion for the order parameter is the same as the one consideredby Glotzer et al . As a result, the steady-state growth rategiven is given by the solution of this equation as ω ( q ) = − L − M f (cid:48)(cid:48) q ( − ξ q ) (23)As discussed previously, the order parameter for the nonequi-librium system is given by the existence of q − c . If we solvethis equation for the root q − c , we obtain ( q − c ) = ξ − (cid:115) + L ξ M f (cid:48)(cid:48) (24) FIG. 8. The amplification factor, given by Equation 12, at constanttemperature The red-dashed line corresponds to the inverse maxi-mum size of the observable phase domain on the length scale of thesimulation box. The existence of a non-zero q − c > q ∗ indicates kinet-ically arrested microphase separation. This equation can be simplified if we apply the first order bi-nomial expansion under the condition that
M f (cid:48)(cid:48) ξ > L . Doingso gives the simple relation that ( q − c ) = − LM f (cid:48)(cid:48) (25)In this model, due to the disbalance in chemical potentials, q − c is always nonzero (provided L (cid:54) = f (cid:48)(cid:48) < D m = − M f (cid:48)(cid:48) , it is alsoobserved that the order parameter for the system q − c dependson the ratio of L and D m . Such that if q − c >
1, then no growthwill occur (meaning that interconversion is faster than phaseseparation so no phases are able to grow), and if q − c <
1, thenmicrophase separation will occur because the rate of phaseseparation will be faster than interconversion.It should be noted that the growth of the domain size iscontrolled by the fastest growing wavenumber q m = / ξ √ . However,as discussed in Sec. III, due to the crossover from spinodaldecomposition to nucleation, q m → q − c at infinite time. As aresult, the domain size of the system will be limited by theorder-parameter q − c . VI. FINITE SIZE EFFECT ON THE NONEQUILIBRIUMMODEL
Within the nonequilibrium system, a unique situation arisesfor small systems sizes. If the system attempts to develop adomain size that is greater than the characteristic size of thesimulation box, the simulation box will restrict the size of thedomain such that phase separation will occur on the lengthscale of the simulation box. Thus, the observer will find thattwo phase separation (as predicted by Cahn-Hilliard theory)will occur. In this case, the order parameter is restricted bythe inverse size of the box, which we call q ∗ . Thus, it will appear like a phase transition between two phase separation(characterized by q − c < q ∗ ) and microphase separation (char-acterized by q − c > q ∗ ). Since the typical variables that char-acterize the dynamics of phase separation and interconversionare a (where a is related to p r the probability for a reaction tooccur), T , and T C , which are contained within M , L , and f (cid:48)(cid:48) ,then we can characterize the point of this transition by solvingfor the location where q − c = q ∗ , thus from Eq. (25) we wouldobtain q ∗ = L ( a , T ) D m ( T , T C ) (26)Therefore, by solving this equation when T = T ∗ , the tem-perature obtained when q − c = q ∗ and defined as the onset ofmicrophase separation temperature, and a = a ∗ , (likewise theonset of microphase separation reaction probability) a transi-tion line in T , a -space that separates these two regions. FIG. 9. Phase separation onset temperatures T ∗ (solid black) and T ∗∗ (dashed black). For any given value of a and T , the light regionabove T ∗∗ is a homogeneous region (only fluctuations are present),the region in the middle depicts microphase separation, and the darkregion below T ∗ is the two phase region. Example: L ∝ T a - As an example of this effect, let usconsider a hypothetical system where L = MT a . In this case,the line separating the two phase region ( q − c < q ∗ ) and themicrophase region ( q − c > q ∗ ) would be found from evaluating: ( q ∗ ) = ξ − (cid:115) + ( T ∗ ) a ∗ ξ f (cid:48)(cid:48) (27)If we also assume that κ ≈
1, so that ξ ∝ − / ∆ T , then thisequation can be expanded to second order to get ( q ∗ ) = ( T ∗ ) a ∗ ( − ∆ T ∗ ) (cid:20) − ( T ∗ ) a ∗ ( ∆ T ∗ ) (cid:21) (28)which can be solved for the onset temperature T ∗ or (equiva-lently) for the onset probability b ∗ as a ∗ = (cid:18) ∆ T ∗ T ∗ (cid:19) (cid:34) − (cid:114) + ( q ∗ ) ∆ T ∗ (cid:35) (29)The figure below demonstrates how this line would separatea system between the two phase region and the microphaseregion. Scaling Adjustments to the Amplification Factor - Themean field approach does not incorporate the effects of diverg-ing fluctuations near the critical point. To adapt the proposedtheory to account for the dynamics near the critical point, thetheoretical equation for the amplification factor needs to beadjusted. As the system approaches the critical point, themutual diffusion coefficient will diverge in two ways. First,the susceptibility (given as 1 / f (cid:48)(cid:48) ) diverges as ( ∆ T ) − ν (where ν = .
63) and, second, as the system gets much closer to thecritical point, the mobility also diverges proportionally to thecorrelation length ( ξ ∝ ( ∆ T ) − ν ). The net effect is that themutual diffusion coefficient asymptotically close to the crit-ical point becomes D m ∝ ( ∆ T ) ν . It is our assumption,in accordance with Eq. (25), that if the mobility M divergesproportionally to the correlation length, then the interconver-sion Onsager coefficient L must also diverge similarly to keepthe ratio of L / M independent of the distance to the criticalpoint. Therefore, in a first-order approximation to scalingtheory, we can adopt this prediction into the proposed the-ory by keeping M constant and changing the susceptibility inEq. (23) according to scaling theory. Thus, f (cid:48)(cid:48) ∝ ( ∆ T ) ν and ξ ∝ ( ∆ T ) − ν . Repeating the calculation of q ∗ assuming that L = MT a (where M is a constant independent of the correla-tion length), it is found that ( q ∗ ) = ξ − (cid:115) + ( T ∗ ) a ∗ ξ f (cid:48)(cid:48) (30)After expansion to second order using the scaling theory cor-relation function becomes ( q ∗ ) = ( T ∗ ) a ∗ ( − ∆ T ∗ ) ν (cid:20) − ( T ∗ ) a ∗ ( ∆ T ∗ ) ν (cid:21) (31)Solving for the onset probability b ∗ gives a ∗ = ( ∆ T ∗ ) ν ( T ∗ ) (cid:34) − (cid:114) + ( q ∗ ) ∆ T ∗ (cid:35) (32)In the case when ν = /
2, this indeed reduces to the meanfield result as given in Eq. (29). A more detailed descrip-tion of microphase separation would need to account for thecrossover between the mean field approximation far awayfrom the critical point to scaling theory in the critical region.
VII. THE POINT WHERE GROWTH STOPS
In addition to q ∗ , there is another point that affects or-der parameter for the domain growth, q − c . This point occurswhen the racemizing source of interconversion becomes sostrong that the entire amplification factor is shifted down. Inthis case, when the growth rate becomes entirely negative, nogrowth will occur and the system will remain homogeneous. FIG. 10. Amplification factors for different source strengths. Asthe source increases, the amplification factor is shifted down startingfrom the onset of phase separation where q − c = q ∗ (red), to the mi-crophase region (green), to the termination point of domain growth(blue) where q − c = q m = q + c = q ∗∗ , and to the no growth regime (or-ange). This is an indication that system is racemizing to a state of50/50 concentration, phase separation is too slow to occur.Just like with the case of the onset temperature T ∗ and theonset probability a ∗ , we can characterize a system throughthe termination temperature T ∗∗ and the termination proba-bility a ∗∗ . These points can be found at the location where ω ( q m ) =
0. At this location, the three characteristic wavenum-bers merge into a single point ( q m = q + c = q + c = q ∗∗ ). See thefigure below For the case of the nonequilibrium model, thisoccurs where L ≥ D m ξ (33)Therefore, whenever L exceeds this ration, microphase sepa-ration will not occur and a homogeneous solution will remain. Example: L ∝ T a - Returning to our previous example,where L = MT a , we can solve the equation for ω ( q m ) = a ∗∗ = (cid:18) ∆ T ∗∗ T ∗∗ (cid:19) (34)The termination line is shown on Fig. 8 along with the onsetphase separation line. VIII. PHASE TRANSITION - PHASE AMPLIFICATIONTO MICROPHASE SEPARATION
So far, we have only discussed scenarios where the sys-tem could exhibit phase amplification or microphase separa-tion. However, we can also consider a model that allows forthe transition between these two phenomenon. Such a modelwould be characterized by the general time evolution givenby Eq. (12). The nonequilibrium source of interconversionwithin this model could be generated through the interactionsof “massless” particles that react with the two species A and B with a reaction probability, a and energy E . Depending onthe frequency of their interactions and the magnitude of theirenergy, these particles could cause interconversion of species.We note that a similar study was conducted where these par-ticles caused diffusion which resists phase separation in animmiscible alloy .Considering the particles as an interconverting source, wecharacterize this system in accordance with the amplificationfactor given by Eq. (12). For the system, the wave numbercorresponding to the fastest growing inhomogeneties is foundas q m = ξ (cid:18) − L ξ M (cid:19) (35)which (depending on the strength of the nonconserved order-parameter dynamics L ) is slightly smaller than the previouspredicted q m . In addition, the order parameter for microphaseseparation can be solved for and written as ( q − c ) = ξ q m (cid:18) kD m − LM (cid:19) = ξ (cid:18) q k q m (cid:19) (36)where q k is given by k / D m − L / M . q k is the central com-ponent that determines the order of the system as it containsthe two conditions outlined in Table (I) that the source mustbe stronger than the nonconserved order parameter dynamics,given by k > L f (cid:48)(cid:48) , and the diffusion rate must be greater thanthe interconversion rate, given by
M f (cid:48)(cid:48) > κ L . Thus, it canbe shown that if either of these conditions is violated then q k (and consequently q − c ) becomes imaginary, meaning that thesystem will be in the phase amplification regime. In addition,the order parameter is inversely proportional to the correla-tion length. Thus, as the system approaches the critical pointand fluctuations become more pronounced (provided that thesystem doesn’t reach the termination point first), the order pa-rameter will go to zero, and the dissipative structure formedin the microphase separated regime will not be able to form.Lastly, the other solution to the amplification factor given by q + c is calculated as ( q + c ) = q m − ( q − c ) (37)Therefore, the transition from phase amplification to mi-crophase separation will be characterized by the emergence of q − c , the order parameter for the microphase separation region. IX. CONCLUSIONS AND FUTURE WORK
We have developed a mean field theory of phase transitionsaffected by molecular interconversion that describes the ef-fects of interconversion on phase separation. The theory isapplicable to any system where the order parameter possessesboth conserved and non-conserved dynamics, such as a mix-ture of a phase separating symmetric lattice gas model and theIsing model for an anisotropic ferromagnet. We have shown that the introduction of an external interconversion source willcause arrested microphase separation, and the competition be-tween order-parameter dynamics results in the phenomenon ofphase amplification, when one phase grows at the expense ofanother one. These phenomena are all derived from the gen-eral mass-balance equation, Eq. (5), and they can be used tocharacterize the behavior of two different atomistic models.In the future, the present theory could be extended to in-corporate fluctuations of the order parameter near the criticalpoint more rigorously. The theory could also be applied tolarger more complex interconverting structures. One exam-ple is the formation of membraneless organelles. Just likemolecular interconversion in the presence of an external en-ergy source, the interconversion of proteins from a foldedto unfolded state can be treated with a similar approach.Previous studies of the formation of such subcellular struc-tures have used Flory-Huggins’ theory of phase separationin polymer solutions to describe the self-assembled state atequilibrium . However, Flory theory only predicts liquid-liquid macro-phase separation, and recent studies have shownthat the spontaneous self-assembly of these organelles is anexample of non-equilibrium micro-phase separation . Adetailed theoretical understanding of the kinetic and diffusivecharacteristics leading to the formation and growth of thesemicrophase droplets has yet to be developed.In addition, arrested microphase structures caused by theinfluence of interconversion can be considered as a simpletype of dissipative structure, and it would be promising toconnect our approach with the general theory of dissipativestructures of Prigogine et. al. It could also have interest-ing cross-disciplinary applications; for example, it would haveparallels with other nonlinear phenomena like hydrodynamicinstabilities , phase transitions in bifurcation theory, catas-trophe theory, and dissipative cellular structures in conditionsfar from equilibrium. ACKNOWLEDGMENTS
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