Non-equilibrium Theory of Arrested Spinodal Decomposition
The Non-equilibrium Self-consistent Generalized Langevin Equation theory of irreversible relaxation [Phys. Rev. E (2010) 82, 061503; ibid. 061504] is applied to the description of the non-equilibrium processes involved in the spinodal decomposition of suddenly and deeply quenched simple liquids. For model liquids with hard-sphere plus attractive (Yukawa or square well) pair potential, the theory predicts that the spinodal curve, besides being the threshold of the thermodynamic stability of homogeneous states, is also the borderline between the regions of ergodic and non-ergodic homogeneous states. It also predicts that the high-density liquid-glass transition line, whose high-temperature limit corresponds to the well-known hard-sphere glass transition, at lower temperature intersects the spinodal curve and continues inside the spinodal region as a glass-glass transition line. Within the region bounded from below by this low-temperature glass-glass transition and from above by the spinodal dynamic arrest line we can recognize two distinct domains with qualitatively different temperature dependence of various physical properties. We interpret these two domains as corresponding to full gas-liquid phase separation conditions and to the formation of physical gels by arrested spinodal decomposition. The resulting theoretical scenario is consistent with the corresponding experimental observations in a specific colloidal model system.
Quenching a liquid from supercritical temperatures into the liquid/gas coexistence region leads to the separation of the system into the equilibrium coexisting phases (1); (2). This phase separation process is a paradigmatic illustration of non-equilibrium irreversible processes, whose description must be based on the general fundamental principles of molecular non-equilibrium statistical thermodynamics (3); (4). When the final density and temperature of the quench correspond to a state point inside the spinodal region, the separation process starts with the amplification of spatial density fluctuations, eventually leading to the two-phase state in thermodynamic equilibrium (5); (6); (7); (8); (9); (10). However, under some conditions, involving mostly colloidal liquids (11); (12); (13); (14); (15); (16), this process of phase separation is interrupted when the colloidal particles cluster in a percolating network of the denser phase, forming an amorphous sponge-like non-equilibrium bicontinuous structure, typical of physical gels (17).
In understanding these experimental observations, computer simulation experiments in well-defined model systems have also played an essential role. This includes Brownian dynamics (BD) (18) and molecular dynamics (MD) (19) investigations of the density- and temperature-dependence of the liquid-gas phase separation kinetics in suddenly-quenched Lennard-Jones and square-well (20) liquids, the BD simulation of gel formation in the colloidal version of the primitive model of electrolytes (a mixture of equally-sized oppositely charged colloids) (12), and MD simulations of bigel formation driven by the demixing of the two colloidal species of a binary mixture(15). From the theoretical side, on the other hand, a fundamental framework is missing that provides an unified description of both, the irreversible evolution of the structure and dynamics of a liquid suddenly quenched to an unstable homogeneous state, and the possibility that along this irreversible process the system encounters the conditions for its structural and dynamical arrest.
Any theoretical attempt to model this complex phenomenon, however, must be guided by the well-established experimental facts and simulation results above, according to which, arrested spinodal decomposition is indeed a possible mechanism for gelation in attractive colloidal suspensions. Although there is still an ongoing discussion regarding specific details, such as the location of the arrest line (14); (11), one can expect that some of these details are actually system-dependent, and that they will eventually be fully understood. In the meanwhile, it is important to focus on the simplest experimental model systems, where this phenomenology has been characterized as carefully and as thoroughly as possible. The simplest such model system should involve, for example, a perfectly monodisperse suspension of spherical particles with attractive interactions that do not require a third species (as colloid-polymer mixtures do), so that their interactions are state-independent, without the need to translate the real temperature into any form of effective temperature. The solutions of the globular protein lysozyme studied by Schurtenberger and collaborators (13); (14) are, to our knowledge, the closest experimental realization of such an ideal model system. Thus, the experimental scenario established by this group may provide an invaluable guide that might serve as a reference for the initial theoretical advances in this field. For this reason, we now quote the main features of such experimental scenario.
According to Refs. (13); (14), the experimental equilibrium phase diagram of aqueous lysozyme solutions shows at high temperatures a stable gas-crystal equilibrium coexistence, and at lower temperatures, a metastable gas-liquid phase separation. Such equilibrium scenario is typical of systems of particles interacting through hard-sphere–like repulsions plus very short-ranged attractions. The non-equilibrium experiments performed consist of the fast temperature quench at fixed overall protein concentration (or volume fraction ), starting with the system in equilibrium at an initial temperature above the gas-liquid coexistence curve , cooling the system to a final temperature lower than . The region below the coexistence curve “can be separated into three areas differing in their kinetic behavior: a region of complete demixing (I), gel formation through an arrested spinodal decomposition (II), and an homogeneous attractive glass (III)” (13); (14). More specifically, for a shallow quench to a final temperature below the spinodal curve, , the classical sequence of spinodal decomposition leading to complete phase separation is observed (region I). However, for quenches below a threshold (or “tie-line”) temperature (), the spinodal domain structure initially coarsens but then completely arrests, forming gels via an arrested spinodal decomposition (region II). Microscopically, these gels correspond to a coexistence of the dilute fluid with a dense percolated glass phase. At lower temperatures, region II is delimited by the boundary with the glass phase (region III), where homogeneous attractive glasses can be reached by quenches at sufficiently low . Many other observations, which can be consulted in the original references (13); (14), enrich the observed experimental scenario with more detail, but the existence of these three kinetically distinct regions constitute the most relevant feature calling for a fundamental explanation.
Some elements needed to construct the desired unifying theoretical framework may already be contained in the classical theory of spinodal decomposition (5); (6); (7); (8); (9). Unfortunately, it is not clear how to incorporate in these approaches the main non-equilibrium features of the process of dynamic arrest, including its history dependence and aging behavior. Similarly, it does not seem obvious how to incorporate the non-stationary evolution of the structure and dynamics of a liquid during the process of spinodal decomposition in existing theories of glassy behavior (21). For example, although many of the predictions of conventional mode coupling theory (MCT) (22); (23); (24); (25) have found beautiful experimental verification, this theory is unable to describe irreversible non-stationary processes. The reason for this is that, in its present form, MCT is in reality a theory of the dynamics of liquids in their thermodynamic equilibrium states. To overcome this fundamental limitation, in 2000 Latz (26) proposed a formal extension of MCT to situations far away from equilibrium which, however, has not yet found a specific quantitative application. In this context, let us also refer to the mean-field theory of the aging dynamics of glassy spin systems developed by Cugliandolo and Kurchan (27). This theory has also made relevant detailed predictions that have been verified in experiments and simulations. Unfortunately, the models involved lack a geometric structure and hence cannot describe the spatial evolution of real colloidal glass formers.
MCT’s limitation above is shared by the self-consistent generalized Langevin equation (SCGLE) theory of dynamic arrest (28); (29); (30), a theory that predicts, in a manner completely analogous to MCT, the existence of dynamically arrested states. In contrast to MCT, however, the SCGLE theory has been extended to describe the irreversible non-equilibrium evolution of glass-forming liquids, thus resulting in the “non-equilibrium self-consistent generalized Langevin equation” (NE-SCGLE) theory (31); (32); (33). This extended theory provides a conceptually simple picture of the crossover from ergodic equilibration to non-equilibrium aging (34); (35), offering a perspective of the glass transition in which the “waiting”-time becomes a relevant active variable and in which indeed the measured properties of the system may depend on the protocol of preparation and on the duration of the actual measurements. Thus, this non-equilibrium theory may provide the general framework in which to build the missing unified description of the processes of arrested spinodal decomposition.
The main purpose of the present paper is to take the first steps in exploring this possibility. The first of these steps is to discuss the manner in which the NE-SCGLE theory may be adapted to address the description of arrested spinodal decomposition. The second refers to obtaining and presenting the concrete results of its application to a specific model system that permits at least qualitative contact with the observed behavior of an experimental model system such as that described above. The physical interpretation of our theoretical results, however, is not as straightforward as, for example, calculating the equilibrium properties of a system using century-old and well-established statistical thermodynamic methods (such as calculating the partition function (2)). Instead, because of the non-equilibrium and non-linear nature of the phenomenon described, the interpretation of the specific predictions is much more subtle, involving features that might seem at first sight strongly counterintuitive. Thus, the third and most relevant step, is to properly interpret these results.
Regarding the first of these steps, let us mention that the most general version of the NE-SCGLE theory of irreversible processes in colloidal liquids (31); (32); (33) consists of two fundamental time-evolution equations, one for the mean value , and another for the covariance of the fluctuations of the local concentration profile of a colloidal liquid. Thus, in the following section and in the appendix we review this general description, to discuss the form adopted by these equations in the absence of external fields, and the possibility that they allow the description of a fingerprint of spinodal decomposition, namely, the early stage of the amplification of spatial heterogeneities. The resulting equations are thus restricted to allow the discussion of the simplest explicit protocol of thermal and mechanical preparation, namely, an instantaneous isochoric quench, to describe the spontaneous relaxation of the system toward its equilibrium state or toward predictable arrested states. To simplify the reading, Section II is essentially a summary of the specific NE-SCGLE equations that we shall actually solve, while the appendix contains a description of the main derivations and approximations leading to these specific equations.
In Sect. III we apply the resulting approximate theory to the discussion of the possibility that the process of spinodal decomposition may be interrupted by the emergence of dynamic arrest conditions. For the sake of concreteness, there we shall have in mind a very specific model system, namely, the “hard-sphere plus attractive Yukawa” (HSAY) potential, subjected to an instantaneous quench at from an initial temperature to a final temperature , with the volume fraction held fixed. The main focus of this application is the determination of the dynamic arrest diagram in the region of unstable homogeneous states. According to the predicted scenario, the spinodal curve turns out to be the borderline between the regions of ergodic and non-ergodic homogeneous states, and the high-density liquid-glass transition line, whose high-temperature limit corresponds to the well-known hard-sphere glass transition, actually intersects the spinodal curve at lower temperatures and densities, and continues inside the spinodal region as a glass-glass transition line.
Some elements of this predicted scenario, however, may seem strongly counterintuitive, and not consistent with experimental observations. For example, this scenario predicts that any quench inside the spinodal region will meet conditions for dynamic arrest, while we know that at least for shallow quenches the system will definitely phase-separate. A satisfactory understanding of these features requires a detailed discussion of a number of issues whose subtlety derives from the lack of familiarity with the solutions of the full NE-SCGLE non-linear equations. Thus, in Sect. IV we discuss the physical interpretation of this theoretical scenario by analyzing in more detail the predicted long-time non-equilibrium arrested structure factor , particularly the length scale associated with the position of its small- peak. This length represents the size of the spatial heterogeneities typical of the early stage of spinodal decomposition described by the Cahn-Hilliard-Cook (CHC) classical linear theory of spinodal decomposition (5); (6), shown to be a particular limit of the NE-SCGLE theory. This analysis results in the recognition that indeed, shallow quenches lead to full phase separation whereas deeper quenches lead to the formation of gels. Thus, in Sect. V we finally establish a more detailed connection between our theoretical scenario and the experimental observations in the lysozyme real model system, and in Sect. VI we summarize the main results of this work.
Ii Review of the NE-SCGLE theory.
Let us start by considering a Brownian liquid formed by particles that execute Brownian motion (characterized by a short-time self-diffusion coefficient ) in a volume while interacting between them through a generic pair potential that is the sum of a hard-sphere (HS) term with HS diameter , plus an attractive tail . Let us imagine that we subject this system to a prescribed protocol of thermal and/or mechanical treatment, described by the chosen temporal variation of the reservoir temperature , of the total volume , and of the applied external fields . In principle, the ultimate goal of the NE-SCGLE theory is to predict the response of our system to each possible protocol of thermal and mechanical manipulation. For simplicity let us assume that the thermal diffusivity of the system is sufficiently large that any temperature gradient is dissipated almost instantly (compared with the relaxation of chemical potential gradients), so that the temperature field inside the system can be considered uniform, and instantly equal to the temperature of the reservoir, .
Let us now imagine that we use these time-dependent fields and constraints to drive the system to a prescribed (but arbitrary) initial state characterized by a given mean concentration profile and covariance , and we then fix the field, the temperature, and the volume afterward, , , and for . The NE-SCGLE theory will then allow us to predict the most fundamental and primary information that characterizes the intrinsic properties of the system, namely, its spontaneous behavior during the irreversible relaxation towards its new expected equilibrium state. This is described in terms of the time-evolution of the non-equilibrium mean concentration profile and covariance for (see Eqs. (18) and (19) of the appendix).
In Ref. (34) we started the systematic discussion of this topic in the context of a model system involving only purely repulsive interactions. For this, we restricted ourselves further to the simplest explicit protocol of thermal and mechanical preparation of our system, namely, its instantaneous cooling or heating from an initial temperature to a final temperature , so that for and for , under isochoric conditions, (so that ) at all times, and in the absence of applied external fields, . As explained in Ref. (34), the NE-SCGLE theory will then describe the spontaneous response of the system in terms only of the non-stationary but uniform covariance , now written as the non-equilibrium static structure factor , whose time-evolution equation for reads
In this equation is the Fourier transform of the thermodynamic functional derivative , evaluated at the uniform density and temperature profiles and , in which case it can be written as or, in Fourier space, as , where is the so-called direct correlation function.
The non-stationary, non-equilibrium mobility function entering in Eq. (1) above, is the most important kinetic property. Within the NE-SCGLE theory this property is determined by (see Refs. (32); (33))
with the -evolving, -dependent friction coefficient given approximately by
in terms of itself and of the collective and self non-equilibrium intermediate scattering functions and , whose memory-function equations are written, in Laplace space, as
with , , and , being the corresponding Laplace transforms (LT) and with being a phenomenological “interpolating function” (30), given by
where the phenomenological cut-off wave-vector depends on the system considered. Since in this paper we are only interested in semi-quantitative trends, here we adopt the value , determined in a calibration procedure involving simulation data of the hard-spheres system (41).
Iii Dynamic arrest diagram of a simple model system.
In this section we apply the previous general theory to the discussion of the possibility that the process of spinodal decomposition may be interrupted by the emergence of dynamic arrest conditions. For this, let us consider a generic liquid formed by particles in a volume interacting through a simple pair potential , which is the sum of a hard-sphere term plus an attractive tail . Although all the results that we shall discuss here are independent of the specific functional form of , for the sake of concreteness, in what follows we shall have in mind a very specific model system, namely, the “hard-sphere plus attractive Yukawa” (HSAY) potential, defined as
where is the hard sphere diameter, the depth of the attractive Yukawa well at contact, and its decay length (in units of ). For given , , and , the state space of this system is spanned by two macroscopic variables, namely, the number density and the temperature , which we express in dimensionless form, as and . In practice, however, from now on we shall use as the unit of length, and as the unit of temperature, so that and ; most frequently, however, we shall refer to the hard-sphere volume fraction .
iii.1 Equilibrium thermodynamic properties.
The most fundamental thermodynamic property of this system, that we need to specify in order to apply the NE-SCGLE theory described above, is the thermodynamic functional derivative . Evaluated at the uniform density and temperature profiles and , we have that or, in Fourier space, as , with being the direct correlation function. In the present paper we shall rely on the approximate prescription proposed by Sharma and Sharma (42) to determine this thermodynamic property, which for the HSAY potential above reads
in which is the FT of the direct correlation function of the hard-sphere liquid, approximated by the Percus-Yevick approximation with Verlet-Weis correction (PYVW) (43); (44) and is the FT of the attractive potential (in units of the thermal energy ), i.e., . Thus, for the HSAY potential, .
For fixed , the equilibrium phase diagram of the HSAY model system in the state space () contains the gas, liquid and (crystalline) solid phases. The previous approximation for allows us to sketch the most prominent features of the fluid phases. For example, the gas-liquid transition involves a coexistence region, with its associated binodal and spinodal lines. The former can be determined by integration of to get the equation of state, along with Maxwell’s construction. The latter can be determined from the condition for thermodynamic instability of uniform fluid states, namely, . The resulting binodal and spinodal lines are illustrated in Fig. 1 for the HSAY system with . On the other hand, the equilibrium static structure factor of the fluid state is determined by the equilibrium (Ornstein-Zernike) condition . From the freezing line, which is another boundary of stability of the uniform liquid state, can be readily sketched using the phenomenological Hansen-Verlet condition (45) that the height of the main peak of the equilibrium static structure factor reaches the value . Besides the spinodal, binodal and freezing lines, Fig. 1 also exhibits the corresponding gas-liquid critical point, located at , and the triple point at . These equilibrium lines serve as a reference to the introduction, in the following subsection, of the liquid-glass dynamic arrest line, also obtained from .
iii.2 Dynamic arrest diagram from the equilibrium SCGLE theory.
The first relevant task in the description of dynamic arrest phenomena in a given system, such as that in our example, is the determination of the dynamic arrest diagram, i.e., the region containing the state points where the system will be able to reach thermodynamic equilibrium (ergodic region) and the region where the system, prevented from crystallizing, will be trapped in a dynamically arrested state (non-ergodic region). Under normal circumstances this can be approached solving the so-called bifurcation equations of mode coupling theory (22), or the corresponding equations of the equilibrium version of the SCGLE theory (30). The latter consists of the following equation for the equilibrium localization length squared ,
The only input of this equation is the equilibrium static structure factor at a given state point . If the solution is , we conclude that the state point lies in the ergodic region, whereas if is finite, the point lies in the non-ergodic region of state space. In this manner we can determine the dynamic arrest transition line.
This procedure has been performed in the past for several model systems, for which the equilibrium static structure factor , given in terms of the equilibrium condition , is well-defined in the entire state space . In the present case we have carried out this exercise for our HSAY model, and the result is the liquid-glass dynamic arrest transition line represented in Fig. 1 by the dark (blue) solid line, with the region to the left and above this curve corresponding to ergodic states. This dynamic arrest line extends from its high temperature limit, corresponding to the hard-sphere glass transition at , down to its intersection with the spinodal curve, indicated by the empty circle in the figure. Unfortunately, for systems with thermodynamically unstable (spinodal) regions, like our HSAY model, we can only apply this method outside such unstable region, since for the state points inside the spinodal no equilibrium static structure factor exists that corresponds to spatially uniform states. It is at this point that the power of the non-equilibrium version of the SCGLE theory is manifested, since this general theory does provide a proper manner to overcome this limitation of the equilibrium theory thus offering an intriguing and unexpected qualitative scenario of the interplay between dynamic arrest and gas-liquid phase separation. As we shall see below, the alternative method is NOT based on the use of the equilibrium bifurcation equation, Eq. (9), but on its non-equilibrium generalization, Eq. (13) below.
iii.3 Detecting dynamic arrest transitions using the Ne-SCGLE theory.
Let us now use the NE-SCGLE theory to detect dynamic arrest transitions. We start by noticing that from the very structure of Eq. (1) one can infer the existence of two fundamentally different kinds of stationary solutions, representing two fundamentally different kinds of stationary states of matter. The first corresponds to the condition in which stationarity is attained because the factor on the right side of Eq. (1) vanishes, i.e., because is able to reach its thermodynamic equilibrium value , while the mobility attains a finite positive long-time limit . These stationary solutions correspond, of course, to the ordinary thermodynamic equilibrium states, in which the difference decays (according to Ec. (1), and in a gross approximation) exponentially fast, , with the equilibration time estimated as .
The second class of stationary solutions of Eq. (1) emerges from the possibility that the long-time asymptotic limit of the kinetic factor vanishes, so that vanishes at long times without requiring the equilibrium condition to be fulfilled. This mathematical possibility has profound and general implications. For example, under these conditions will now approach a distinct non-equilibrium stationary limit, denoted by , which is definitely different from the expected equilibrium value . Furthermore, the difference is predicted to decay to zero not exponentially, but in a much slower fashion, namely, as , with an exponent near unity (33); (34). Contrary to ordinary equilibrium states, whose properties are determined solely by the fundamental condition of maximum entropy, the properties of these stationary but intrinsically non-equilibrium states, such as , may depend on the preparation protocol (in our example of the instantaneous isochoric quench, on and ).
For a given instantaneous isochoric quench from an initial temperature to a lower final temperature , one can also predict if the system will equilibrate or will be trapped in a non-equilibrium arrested state by analyzing the stationary solutions of Eq. (1). This analysis simplifies greatly with the change of variables from the actual evolution time to the so-called “material” time defined by (34)
This allows us to write, for example, the actual solution of Eq. (1) as , with the function being the solution of
where represents the (arbitrary) initial condition.
This expression interpolates between its initial value and the value of the thermodynamic property . Thus, if nothing impedes the system from reaching equilibrium at the state point , the non-stationary static structure factor will eventually attain its equilibrium value . This, however, is only one of the two possibilities described above. In order to determine which of them will govern the course of the spontaneous response of our system, we actually input in the “bifurcation” equation for the square localization length at evolution time (i.e., the long- asymptotic value of the mean squared displacement). Denoting simply as , such an equation reads
As explained in Ref. (34), if we find that for , we conclude that the system will be able to equilibrate after this quench, and hence, that the point lies in the ergodic region. If, instead, a finite value of the parameter exists, such that remains infinite only within a finite interval , the system will no longer equilibrate, but will become kinetically arrested, with the non-equilibrium asymptotic structure factor given by
The parameter and the corresponding long-time asymptotic square localization length are dynamic order parameters in the sense that if , the system will reach equilibrium, but if and are finite, the system will be dynamically arrested. Both of them, as well as the non-equilibrium static structure factor , will depend on the protocol of the quench (in our case, on , , and the final temperature , which from now on will be denoted simply as ). Thus, if one determines the functions , , and , one can in principle draw the dynamic arrest diagram.
We have implemented this protocol and here we present the results of its application to the concrete illustrative case involving the HSAY model. Let us first consider isochores with , which lie to the right of the intersection indicated by the dark circle in Fig. 1. A representative isochoric quench of this class is schematically indicated in the inset of Fig. 2(a) by the downward arrow along the isochore . In the main panel of this figure we plot the resulting as a function of (solid line), together with the corresponding value of (dashed line).
The first feature to notice is that indeed, for quenches to final temperature above the critical temperature , is always infinite (so that , and hence, do not appear in the logarithmic window of the figure), whereas for we detect a finite value of . This value, however, diverges as approaches from below, and decreases monotonically as the final temperature decreases. According to the results in Fig. 2(a), for also attains finite values, thus reaffirming the existence of an arrested phase in this temperature regime. In fact, these results for reveal a second relevant feature, namely, that this dynamic order parameter changes discontinuously at , from a value to a value . We notice that this jump in the value of corresponds to the jump in the solution of the equilibrium bifurcation equation, Eq. (9), and it occurs at the same critical temperature , thus confirming that both methods refer to the same dynamic arrest transition, and that this is a “type B” dynamic arrest transition in the language of mode coupling theory. To complete this identification, we repeated this procedure at other volume fractions to determine the -dependence of the dynamic arrest transition temperature , thus finding that indeed, outside the spinodal curve (i.e., for volume fractions larger than the volume fraction of the intersection point ) this method leads to exactly the same liquid-glass dynamic arrest line previously obtained using Eq. (9).
Let us notice, however, that the curves describing the dependence of on the final temperature do depend on the initial temperature of the quench. However, the resulting transition temperature, , is independent of . This is illustrated in Fig. 2(b), which compares the function obtained with with the results for obtained with and . This independence of on the initial temperature is also a necessary condition for the present method to be a reliable approach to the determination of the liquid-glass dynamic arrest transition. In Fig. 2(b) we have also included the results for corresponding to the quench processes with different initial temperatures, to show that is much less sensitive to the initial temperature of the quench. Let us also notice that for , the results for and as a function of the final temperature , do not exhibit any discontinuous behavior when crossing the spinodal line, i.e., their dependence on does not detect the spinodal. Thus, for the spinodal condition does not seem to have any significant effect on the dynamics of the system. As discussed below, this rather astonishing result is a consequence of the structure of the expression for in Eq. (14), and is in dramatic contrast with what happens in the complementary regime, i.e., for volume fractions , as we now explain.
iii.4 Dynamic arrest inside the spinodal region.
The main advantage of the NE-SCGLE methodology just explained, for determining the dynamic arrest transition line, is that its use can be extended to the interior of the spinodal region. This subsection continues the description of the use of this methodology, but now to unveil the structure of the dynamic arrest diagram of our model in the region inside the spinodal line. Thus, let us repeat the calculations illustrated in Fig. 2, but now for isochores corresponding to volume fractions smaller than the volume fraction of the intersection between the liquid-glass dynamic arrest transition and the spinodal curve. The corresponding illustrative example is contained in Fig. 3, which plots the functions and as a function of for the HSAY model system subjected to an instantaneous quench from an initial temperature to a final temperature at fixed (downward arrow in the inset of the figure).
In contrast with the results for the isochore , in Fig. 2, we notice that in the present case the functions and remain infinite for all temperatures above a critical temperature , and that this singular temperature coincides precisely with the temperature of the spinodal curve for that isochore. The results in Fig. 3 also indicate that both parameters, and , are finite for , and both diverge as approaches from below. This reveals a rather dramatic and unexpected conclusion, namely, that the spinodal curve, besides being the threshold of thermodynamic instability, now turns out to be also the threshold of non-ergodicity. Furthermore, the fact that diverges as approaches from below determines that this transition from ergodic to non-ergodic states occurs in a continuous fashion, i.e., that it is a “type A” dynamic arrest transition in MCT language.
Examining again the same results in Fig. 3, but now at temperatures well below the spinodal curve, we see that the parameters and exhibit a discontinuity at a lower temperature . This discontinuity reveals the existence of still a second dynamic arrest transition, now corresponding to a glass-glass “type B” transition, in which the dynamic order parameter changes discontinuously by about one order of magnitude, from a value to another finite value . Notice that the value is similar to the corresponding value at the isochore . Repeating these calculations at other isochores, we determine both transition temperatures, and , as a function of volume fraction. The corresponding dynamic arrest transition lines are presented in Fig. 4. This figure reveals a rather unexpected global scenario, in which (a) this low- (type B) glass-glass transition (dark blue dashed curve) turns out to be just the continuation to the interior of the spinodal region, of the liquid-glass transition previously determined outside the spinodal (dark blue solid curve), (b) the spinodal line itself is a continuous (type A) ergodic–nonergodic transition, (c) these two transitions merge at the bifurcation point to continue outside the spinodal as the liquid-glass transition line that terminates at the hard-sphere glass transition point , and (d) for the spinodal line does not have any apparent dynamic significance.
At first glance some of these predictions might appear counterintuitive. Let us remind, however, that they are based only on the analysis of the dependence of the dynamic order parameter on the final temperature of the instantaneous isochoric quenches analyzed. This order parameter is a functional of the long-time asymptotic value of the non-stationary static structure factor . Thus, the analysis of the dependence of on and should provide additional pieces of structural information, while the waiting-time dependence of (and of all the other structural and dynamic properties) will provide valuable pieces of kinetic information. When these pieces are assembled together in a structural and kinetic jigsaw puzzle, one should expect that a sound and convincing scenario will emerge. For this, however, we must first identify and describe the most relevant of these pieces. The dynamic arrest diagram in Fig. 4 constitutes the initial and most basic piece for this discussion, but in the rest of the paper we shall identify others, equally fundamental.
Iv Physical interpretation of the dynamic arrest diagram.
The scenario described by the dynamic arrest diagram in Fig. 4 is highly provoking, but also at first sight difficult to reconcile with experience. First, we all know that simple fluids, when quenched to the inside of the spinodal region, are expected to fully phase-separate into the gas phase and its corresponding condensed (liquid or crystal) phase. On the other hand, as explained in the introduction, there are numerous experimental and numerical evidences that for sufficiently deep quenches, the process of spinodal decomposition may be interrupted by its interference with the dynamic arrest of the condensed phase. Thus, the dynamic arrest diagram in Fig. 4 may contain the fundamental explanation of this experimental double scenario of full vs. arrested spinodal decomposition.
To test this expectation, however, we must first understand what exactly is the detailed physical meaning of these NE-SCGLE predictions and what are their most relevant and verifiable physical consequences. In this section we discuss several pieces of information that derive from the solution of the NE-SCGLE equations, and which will dissipate some of the most subtle puzzles of the interpretation of this dynamic arrest diagram. With these details taken into account, we shall see that the resulting theoretical scenario is actually quite consistent with the main features of the experimental scenario of arrested spinodal decomposition in lysozyme protein solutions, thus establishing a concrete link of our theory with physical reality.
iv.1 , , and below the spinodal line.
In this subsection, for example, we discuss the mechanism that allows the existence of a well-behaved non-equilibrium for quenches to state points inside the spinodal region, where the uniform fluid state has become thermodynamically unstable and the equilibrium structure factor does not exist. According to Eqs. (12) and (14), the calculation of and requires the value of the thermodynamic property for . As we know, the condition is a condition for the stability of uniform thermodynamic states, a condition held for all temperatures above the spinodal temperature . Thus, the spinodal condition defines the threshold of thermodynamic instability of uniform states, so that for all states with temperatures below we must have that not only at , but also at least within a finite interval .
This means that the equilibrium static structure factor will attain unphysical negative values in this interval and will exhibit a singularity at . This unphysical behavior, which is a manifestation of the non-existence of spatially uniform equilibrium states inside the spinodal region, is illustrated in Fig. 5, which plots (dashed curves) at the state points and below the spinodal curve. One might then expect that the non-equilibrium static structure factor and its asymptotic value , which are formally an exponential interpolation between and (see Eqs. (12) and (14)), will inherit these singular features of the function .
In reality, this is not the case, since the appearance of in the exponent of the interpolating function, allows us to show, for example, that near we have that and can be approximated, respectively, by and , which are always positive definite. This is illustrated in Fig. 5 with the results for (solid lines) for the quenches along the isochores 0.2 and 0.4, from the same initial temperature , at which the initial static structure factor is given by (dotted lines), to a final temperature . As we can see in Figs. 5(a) and (b), in both cases the non-equilibrium stationary structure factor does not exhibit any singular or non-physical feature at any wave-vector. Of course, for the same reason, the non-equilibrium static structure factor will evolve smoothly from at , to as , without exhibiting any hint of the singular behavior characteristic of the thermodynamic function below the spinodal curve. This is illustrated in Fig. 5 by the light-gray solid lines, corresponding to evaluated at representative finite values of the waiting time . Notice in particular that for the quench illustrated in Fig. 5(a) (corresponding to ), a mild but noticeable small- peak of develops at . This is to be contrasted with the quench along the isochore illustrated in Fig. 5(b), where this small- peak is imperceptible.
iv.2 Relationship with the theory of Cahn, Hilliard, and Cook.
One of the main general features of the non-equilibrium evolution of after an isochoric quench inside the spinodal region is precisely the development of this low- peak of located at the time-dependent wave-vector . This peak is associated with the growing length scale of the clusters formed in the early stage of the process of spinodal decomposition. According to the classical theory developed by Cahn, Hilliard, and Cook (CHC) (5); (6), and reviewed in detail by Furukawa (7), this small- peak evolves with waiting time , moving to progressively smaller wave-vectors, so that . The correct interpretation, however, is that this scenario only describes the very early stages of the phase separation process, up to a point in which the size reaches mesoscopic dimensions so that additional effects (such as surface tension, convection, etc.), not contained in the theory, drive the system to full phase separation.
As it happens, the NE-SCGLE theory contains this classical CHC theory as its linear and small- limit. To see this more explicitly, let us notice that our fundamental time evolution equation for in Eq.(1) above, becomes Cook’s equation (Eq. (3.4) in Ref. (7)) in the linear regime and the long-wavelength limit. The linear regime is achieved in our theory by neglecting the non-linear dependence of the time-dependent mobility on itself, i.e., by approximating , whereas in the long-wavelength limit (small ’s) we can expand the thermodynamic function in powers of up to quadratic order, , with and being wave-vector independent parameters. These two approximations, with , convert Eq. (1) into
which is Eq. (3.4) of Ref. (7). Notice also that Cook’s approximate solution of this equation for shallow quenches (Eq. (3.9) of Ref. (7)) is identical to the approximation quoted above, provided that .
The fact that the NE-SCGLE theory contains the CHC theory as a particular limit also determines the limitation of the NE-SCGLE theory to describe the full phase separation process. Thus, for shallow quenches, where we know that full phase separation occurs, we understand that the NE-SCGLE theory only describes the early stage of spinodal decomposition. Due to its non-linear nature, however, the NE-SCGLE theory complements this classical scenario (in which ) with the prediction that , i.e., with the possibility that the process of spinodal decomposition is arrested when reaches the length scale associated with the low- peak of the non-equilibrium asymptotic structure factor . Of course, if the predicted value of is sufficiently large for surface tension and convection to take over, the system will phase-separate completely. The possibility exists, however, that the predicted value of is not that large, or to be even of the scale of a few particle diameters. Under such circumstances the dynamic arrest of the phase separation process will occur right at its early stage, freezing the current structure and preventing the system from phase separating completely. These conditions are illustrated by the quench to along the isochore , whose is represented by the solid line of Fig. 5(a), for which , and hence, . Thus, we need a criterion to locate the crossover temperature below which the predicted dynamic arrest will in fact occur, and above which full phase separation will be driven by effects not contained in our equations. In the rest of this section we shall search for such criterion in the temperature dependence of the asymptotic structure represented by .
iv.3 Non-equilibrium small- peak of and diverging length scale .
Let us thus analyze in more detail the dependence of on the depth of the quench, i.e., on the final temperature , as well as on the volume fraction . With this intention, in Fig. 6(a) we present a set of results for that illustrate the -dependence of this asymptotic structural property for quenches along the isochore 0.2. All of these quenches start with the system equilibrated at the same initial temperature , whose corresponding equilibrium static structure factor is represented by the dotted curve. Each quench ends at a different final temperature , and the solid lines represent the resulting . Thus, the dotted line and the solid line corresponding to are the same as the dotted and solid lines of Fig. 5(a), but now in the vertical axis we use logarithmic scale to visualize the strong increase in as the final temperature approaches the spinodal temperature from below. This is clearly the most visible trend exhibited by this set of curves, together with the fact that the position of this growing non-equilibrium peak of moves monotonically to the left, so that the corresponding length scale increases with increasing .
In fact, to further visualize these trends, in Fig. 6(b) we plot as a solid line the height of the low- peak of as a function of the final temperature along the isochore . This curve clearly illustrates that the behavior of is distinctly different in the regime where the final temperature falls below the glass-glass transition temperature , and in the regime where falls in the interval . In the former, for the lowest temperatures starts almost constant, with a value similar to (recall that ), but as approaches from below it increases sharply to a value . In the interval , on the other hand, starts from the value (i.e., is continuous at ), and increases monotonically up to the spinodal temperature , at which it diverges as , with . This divergence of is the non-equilibrium counterpart of the divergence of when approaches from above. For completeness, although it has no direct relevance regarding the possibility of dynamic arrest, the divergence of