Phase Separation in Soft Matter: Concept of Dynamic Asymmetry
—Lecture Notes for les Houches 2012 Summer School on
Phase separation is a fundamental phenomenon that produces spatially heterogeneous patterns in soft matter. In this Lecture Note we show that phase separation in these materials generally belongs to what we call “viscoelastic phase separation”, where the morphology is determined by the mechanical balance of not only the thermodynamic force (interface tension) but also the viscoelastic force. The origin of the viscoelastic force is dynamic asymmetry between the components of a mixture, which can be caused by either a size disparity or a difference in the glass transition temperature between the components. We stress that such dynamic asymmetry generally exists in soft matter. The key is that dynamical asymmetry leads to a non-trivial coupling between the concentration, velocity, and stress fields. Viscoelastic phase separation can be explained by viscoelastic relaxation in pattern evolution and the resulting switching of the relevant order parameter, which are induced by the competition between the deformation rate of phase separation and the slowest mechanical relaxation rate of a system. We also discuss an intimate link of viscoelastic phase separation, where deformation fields are spontaneously generated by phase separation itself, to mechanical instability (or fracture) of glassy material, which is induced by externally imposed strain fields. We propose that all these phenomena can be understood as mechanically-driven inhomogeneization in a unified manner.
Soft matter is characterized by its spatio-temporally hierarchical structure, which is absent in ordinary classical fluids. This feature plays crucial roles in the dynamical behaviour of soft matter. As an example of typical hierarchical structures of soft matter, we show such a structure of an amphiphilic surfactant/water mixture in Fig. 1. Amphiphilic molecules spontaneously form bilayer membranes, which further form higher order organizations such as sponge, lamella, gyroid, and onion structures in water. In this system, the motion of the low-level structure, i.e., hydrodynamic motion of water in this case, can be coupled with the motion of membranes in a dynamical manner: membrane motion causes flow and vice versa. This type of dynamical coupling between the different levels of the structure leads to intriguing dynamical behaviour of soft matter. We stress that the characteristic timescale of the high-level structures, e.g., membranes, is generally much longer than that of the low-level structure such as water, reflecting their size difference. This we call “dynamic asymmetry” between the components of a mixture. This can be easily understood by the following scaling arguments. Both softness and slowness of soft matter are consequences of the large size of a high-level structure: The elasticity is scaled as , where is the characteristic lengthscale of the high-level structure and is the characteristic ordering temperature of the system. On the other hand, the characteristic time scale as , with the diffusion constant , where is the Boltzmann constant, is the absolute temperature, and is the viscosity. Typically, the size disparity between the high- and low-level structure of soft matter is the order of , which results in strong dynamic asymmetry between them. In this Lecture Note, we consider how such dynamic asymmetry affects phase separation dynamics and pattern evolution. Please refer to the Chapter by Prof. Mike Cates on phase separation of classical binary mixtures, which is the basis of our understanding of phase separation in soft matter.
Phase-separation phenomena are commonly observed in various kinds of condensed matter including metals, semiconductors, simple liquids, soft materials such as polymers, surfactants, colloids, biological materials, and food materials. The phenomena play key roles in pattern evolution of immiscible multi-component mixtures of any materials. The resulting patterns are linked to optical, electrical, and mechanical properties of materials. Thus, phase-separation dynamics has been intensively studied from both fundamental and applications viewpoints Gunton et al. (1983); Onuki (2002). For example, it was shown recently that a spatially heterogeneous pattern formed by protein phase separation causes a Bragg reflection of light, which is an origin of a colour of bird feathers Dufresne et al. (2009). We speculate that this phase separation may belong to viscoelastic phase separation, which we shall discuss below.
On the basis of the concept of dynamic universality of critical phenomena Hohenberg and Halperin (1976), phase separation in various condensed matter systems were classified into a few groups. Phase separation in each group is described by a specific set of basic equations describing its dynamic process. For example, phase separation in solids is known as “solid model (model B)”, whereas phase separation in fluids as “fluid model (model H)” Gunton et al. (1983); Hohenberg and Halperin (1976). For the former the local concentration can be changed only by material diffusion, whereas for the latter by both diffusion and flow. The universal nature of critical phenomena in each model and the scaling concept based on the self-similar nature of domain growth have been established Gunton et al. (1983); Onuki (2002); Siggia (1979). In all classical theories of critical phenomena and phase separation, however, the same dynamics for the two components of a binary mixture, which we call “dynamic symmetry” Tanaka (1992) between the components, has been implicitly assumed. This assumption can always be justified very near a critical point, where the order parameter fluctuations are far slower than any other internal modes of a system (see Fig. 2). For a mixture having strong dynamic asymmetry between the components, however, this is not necessarily the case far from a critical point, where most of practical phase separation takes place. As mentioned above, the presence of dynamic asymmetry means that there is also a large separation between the soft matter mode and the microscopic mode of a system. Furthermore, there is another gross variable of a system, the velocity field, whose relevance in dynamics comes from the momentum conservation law. Thus, dynamic asymmetry leads to complex couplings between the slow critical fluctuation mode, the slow soft matter mode, and the velocity field (see Fig. 2).
In this article, we review the basic physics of viscoelastic phase separation Tanaka (2000a); Tanaka and Araki (2006); Tanaka (2009) including fracture phase separation Koyama et al. (2009). We show that with an increase in the ratio of the deformation rate of phase separation to the slowest mechanical relaxation rate the type of phase separation changes from fluid phase separation, viscoelastic phase separation, to fracture phase separation. We point out that there is a physical analogy of this to the transition of the mechanical fracture behaviour of materials under shear from liquid-type, ductile, to brittle fracture. This allows us to discuss phase separation and shear-induced instability of disordered materials Furukawa and Tanaka (2006); Furukawa and Tanaka (2009a) including soft matter Onuki (1997, 1989); Helfand and Fredrickson (1989); Doi and Onuki (1992); Ji and Helfand (1995); Milner (1993); Tanaka (1999), on the same physical ground. Finally it should be noted that what we are going to describe in this article has not necessarily been firmly established and there still remain many open problems to be studied in the future. So this Lecture Note should be regarded as preliminary trials towards the understanding of complex interplay between thermodynamics, hydrodynamics, and mechanics, in nonequilibrium pattern formation dynamics in soft matter and glassy matter.
Ii Critical phenomena of dynamically asymmetric mixtures
Before describing how slow modes of soft matter influence phase-separation behaviour, we consider how they affect dynamical critical phenomena. We first review some fundamental knowledges on dynamic critical phenomena, focusing on the mesoscopic spatial and temporal scales associated with critical fluctuations and then consider how the behaviour is modified by the presence of additional mesoscopic spatial and temporal scales of soft matter.
ii.1 Dynamic critical phenomena
Here we consider dynamic critical phenomena of a binary mixture, i.e., model H in the Hohenberg-Halperin classification Hohenberg and Halperin (1976). The basic equations of model H are given as follows:
where is the composition deviation from the average value, is the velocity field, is the viscosity, and is a part of the pressure, which is determined to satisfy the incompressible condition (3). Here we assume for simplicity that the Hamiltonian is given by the following Ginzburg-Landau form:
where , (: a positive constant), and and are positive constants.
Near the critical point , the order parameter fluctuations become mesoscopic. This can be seen from the fact that the spatial correlation of the order parameter fluctuations can be obtained from the above Hamiltonian under a harmonic approximation as
where is the wave number, , and is the susceptibility. This functional form is known as the Ornstein-Zernike correlation function. Both the susceptibility and correlation length diverge towards , respectively, as and , where is a positive constant and is the bare correlation length, is the reduced temperature. Here we note recent experimental Royall et al. (2007) and theoretical studies Campbell et al. (2006) suggest the more appropriate form for is , which provides in the limit of . The critical exponent for the susceptibility is 1 for the mean-field approximation, but 1.24 for the 3D Ising universality after the renormalization. On the other hand, the critical exponent for the correlation length is 1/2 for the mean-field approximation, but 0.63 for the 3D Ising universality.
Reflecting the growth of the correlation length , the dynamics of the order parameter fluctuations also slows down while approaching , which is known as “critical slowing down”. The lifetime of the fluctuations is estimated as , where is the standard dynamical critical exponent. Here is the diffusion constant of a mixture and given by (see below).
Here we briefly review how the diffusional transport can be described under a coupling to the velocity fields in the above framework of model H Onuki (2002). The transport coefficients of fluids in the linear response regime can be expressed as the time integral of flux time correlation functions. By using the relevant flux , the renormalized transport coefficient for is expressed as
where is the (solid-type) bare transport coefficient and negligible compared to the second term. It can be calculated by decoupling the above four-point correlation function as
Since the timescale of the order parameter fluctuations is much slower than that of the momentum diffusion (or, the velocity field ), we apply the adiabatic approximation in the above and assume . Under the incompressible condition, the time correlation function of is expressed as
By using the Ornstein-Zernike form for the correlation function (see Eq. (5)), we obtain
where the scaling function is called the Kawasaki function and for whereas for (see, e.g., Ref. Onuki (2002)).
The order parameter decay rate is then given by
This provides the interesting non-locality of the diffusional transport originating from the presence of the important mesoscopic lengthscale .
Next we consider the viscous transport in a critical binary mixture. The viscosity is expressed as
where is the non-critical background viscosity of a mixture and . By using a decoupling approximation, we can obtain the following expression:
where is the short cut-off wavelength and . Since , we can approximate the above as
The more exact calculation yields , which indicates a very weak (almost logarithmic) divergence of the viscosity. The effects of dynamic asymmetry on this anomaly will be discussed below.
ii.2 Effects of dynamic asymmetry on dynamic critical phenomena
Now we consider how the dynamic asymmetry between the components of a mixture affects dynamic critical phenomena. Here we note that since the effects are purely dynamical, they do not affect the static critical phenomena. As schematically shown in Fig 2, the effects of dynamic asymmetry originate from the fact that there is an additional slow mode coming from the large size of a component of a mixture, whose relaxation time we denote . As described above, there is critical slowing down towards and thus the order parameter fluctuations become slower and slower while approaching . The characteristic lifetime of the fluctuations always becomes the slowest mode near . This clear separation between microscopic and critical length and time scale is the heart of the concept of dynamic universality Hohenberg and Halperin (1976). There is no exception for this limiting behaviour for , however, practically this limit of cannot be accessed experimentally in a system of strong dynamic asymmetry Tanaka (1994). Then dynamic asymmetry affects dynamic critical phenomena significantly Tanaka and Miura (1993); Tanaka (1994); Doi and Onuki (1992); Furukawa (2003a, b); Tanaka et al. (2002); Kostko et al. (2002). The relationship between key timescales are shown schematically in Fig. 2.
The slow dynamics characterized by the slow rheological relaxation time of soft matter affects the above decay rate of the order parameter fluctuations (see Eq. (10)) through the coupling between the two slow modes, critical and rheological modes Doi and Onuki (1992); Tanaka et al. (2002); Kostko et al. (2002). The dynamic asymmetry gives rise to an important new mesoscopic length , which is of purely rheological origin. We call this the viscoelastic (or, magic) length Brochard and de Gennes (1977); Brochard (1983), which is the characteristic length scale above which dynamics is dominated by diffusion and below which by viscoelastic effects. This length is a rheological length scale intrinsic to entangled polymer solutions Brochard and de Gennes (1977); Brochard (1983) and other dynamically asymmetric systems Tanaka (1999), and nothing to do with critical phenomena. We can also interpret this length scale as the length up to which the shear stress can transmit. The shear stress is dominated by the velocity fluctuations of the distance over and of the time . Furukawa predicted that the dynamical asymmetry coupling between the velocity fluctuations and the viscoelastic stress affects the hydrodynamic relaxation process, resulting in a wavenumber-dependent shear viscosity: the viscosity of a polymer solution has the following wavenumber (-) dependence associated with this length Furukawa (2003a, b):
where is the macroscopic viscosity and is the solvent viscosity. The viscoelastic length is approximately given by , where is the blob size de Gennes (1979). The -dependence of is schematically shown in Fig. 3. We stress that the relation between the thermodynamic correlation length and the rheological viscoelastic length depends upon how close to a system is, as illustrated in Fig. 3. So the viscosity felt by the composition fluctuations depends upon the relation between them. Thus, the relation between these lengthscales plays a crucial role in determining how phase separation proceeds in its early stage. This nonlocal nature of the viscous transport is a manifestation of the temporal hierarchical structure of dynamically asymmetric systems. Interestingly, the similar non-locality of the transport coefficient has recently been found by Furukawa and Tanaka for a supercooled liquid having dynamic heterogeneity over the lengthscale Furukawa and Tanaka (2009b); Furukawa and Tanaka (2011, 2012).
Under the coupling between the viscoelastic and critical modes, the time correlation function of , , should be given by Doi and Onuki (1992)
The temperature dependences of and for a critical solution of polystyrene (PS) in diethyl malonate (DEM), PS-7 (the molecular weight ), and for PS-2 () are shown in Fig. 4. Far above where , the theory Brochard and de Gennes (1977); Brochard (1983); Doi and Onuki (1992) predicts that should be approximated as
In this limit, and . We also estimate from the relation
as 0.1 m for PS-7. It is also found that only weakly depends on , as expected from its rheological (non-critical) nature. The value of estimated from light scattering ( s) is compatible with our rheological measurements Tanaka and Miura (1993). Both and are found to be larger for PS-8 () than for PS-7, as expected. Very near where , on the other hand,
This tells us that the critical mode can be free from the coupling to the viscoelastic mode only very near and at small Tanaka (1994). Figure 4 clearly demonstrates that the behaviour of the critical mode is strongly influenced by its dynamic coupling to the viscoelastic mode for PS-7, which supports the above scenario Tanaka et al. (2002).
Next we consider the critical anomaly of the viscosity in polymer solutions Tanaka et al. (2002). We found that the viscosity anomaly is more strongly suppressed for a system of stronger dynamic asymmetry for polymer solutions. Figure 5 plots the value of the exponent for the viscosity anomaly at the critical concentration against the molecular weight of polymers, . This clearly demonstrates that the value of monotonically decreases with an increase in . Since polymer solutions belong to the same dynamic universality class as classical fluids, this exponent should be equal to the universal value of , irrespective of the molecular weight of the polymer, if we are close enough to (see above). Thus, the above results clearly indicate that the viscosity anomaly of critical polymer solutions (especially, of high molecular weight polymers) cannot be described by “fluid model (model H)” at least in the experimentally accessible temperature range. Note that approaches the universal value of classical fluids with a decrease in , i.e, towards the limit of dynamic asymmetry.
We ascribed this non-universal behaviour to the existence of an additional slow mode intrinsic to polymer solutions and its dynamic coupling to the critical mode, which is a direct consequence of dynamic asymmetry Tanaka and Miura (1993); Tanaka (1994); Tanaka et al. (2002). Note that the dynamic universality concept relies on the fact that there exists only one slow mode in a system, which is a critical mode. However, such a situation cannot be realized in practical experiments for a mixture of strong dynamic asymmetry (see Fig. 2). For polymer solutions, the viscoelastic relaxation mode is dynamically coupled with the concentration fluctuation mode, which results in the viscoelastic suppression of the dynamic critical anomaly: The fluctuation modes whose characteristic length is shorter than the magic (viscoelastic) length is significantly influenced by the viscoelastic effects. The theoretical account for this non-universal behaviour of the viscosity was provided by Furukawa Furukawa (2003a, b).
Iii Phenomenological theory of viscoelastic phase separation
Phase separation of soft matter such as polymer blends, polymer solutions, protein solutions, and emulsions had been believed to be the same as that of classical fluid mixtures (model H) Gunton et al. (1983); Hohenberg and Halperin (1976). However, it was shown about two decades ago that it is not necessarily the case Tanaka and Nishi (1988); Tanaka (1992, 1993, 1994, 1996, 1992). In normal phase separation observed in dynamically symmetric mixtures (model H), the phase separation morphology is determined by the balance between the thermodynamic (interfacial) force and the viscous force while satisfying the momentum conservation. In viscoelastic phase separation, on the other hand, the self-generated mechanical force also plays a crucial role in its pattern selection in addition to the thermodynamic and viscous force . Thus we named this type of phase separation “viscoelastic phase separation (VPS)”. In addition to the solid and fluid model, thus, we need the third model for phase separation in condensed matter, i.e., the “viscoelastic model” Tanaka (2000a, 1997a). This model is actually a general model of phase separation of isotropic systems including the solid and fluid model as its special cases Tanaka (1997a) (see Sec. VIII).
Intuitively, viscoelastic phase separation can be explained as follows. When there is a large difference in the dynamics between the components of a mixture, phase separation tends to proceed in a speed between those of the fast and slow components. Then, the slow component cannot catch up with a deformation rate spontaneously generated by phase separation itself, , and thus starts to behave as an elastic body, which switches on the elastic mode of phase separation. Thus, this phenomenon can be regarded as “viscoelastic relaxation in pattern evolution”, which is the reason why we named it viscoelastic phase separation Tanaka (1993). Unlike ordinary mechanical relaxation experiments, the mechanical perturbation is characterized by the rate of deformation induced by phase separation, , and the relaxation rate is that of the slowest mechanical relaxation, , in a system (see Fig. 2).
Without dynamic asymmetry, the deformation rate is always slower than the relaxation rate. Thus, phase separation in such a mixture can always be described by the fluid model, no matter how slow the dynamics of the components is. This is because is always satisfied in the critical regime for dynamically symmetric systems and there is no -dependence in , i.e., is a microscopic length. For example, this is the case for a mixture of two polymers having similar molecular weights and glass transition temperatures Bates and Wiltzius (1989); Hashimoto (1988). We emphasize that dynamic asymmetry, which is prerequisite to viscoelastic phase separation, usually exists in any materials, particularly, in soft matter. In this sense, we may say that “normal phase separation (NPS)” is a special case of viscoelastic phase separation.
iii.2 Two-fluid model of polymer solution and stress-diffusion coupling
Shear effects on complex fluids have attracted much attention because of its unusual nature known as “Reynolds effect”: For example, shear flow that intuitively helps the mixing of the components actually induces phase separation in polymer solutions Onuki (1997, 2002). This is caused by couplings between the shear velocity fields and the elastic internal degrees of freedom of polymers. To explain this counter-intuitive behaviour of polymer solutions under shear, there have been considerable theoretical efforts Onuki (1997, 1989); Helfand and Fredrickson (1989); Doi and Onuki (1992); Ji and Helfand (1995); Milner (1993). Doi and Onuki Doi and Onuki (1992) established a basic set of coarse-grained equations describing critical polymeric mixtures, based on a two-fluid model whose original form was developed by de Gennes and Brochard de Gennes (1976a, b); Brochard (1983) for polymer solutions and by Tanaka and Filmore Tanaka and Fillmore (1979) for chemical gels.
Later we proposed that an additional inclusion of the strong concentration dependence of the bulk stress and/or the transport coefficient, which are not important in shear-induced instability, is necessary for describing viscoelastic phase separation of dynamically asymmetric mixtures, more specifically, the volume shrinking behaviour of the slow-component-rich phase Tanaka (1997b, a); Tanaka and Araki (1997). We also argued its generality beyond polymer solutions to particle-like systems such as colloidal suspensions, emulsions, and protein solutions Tanaka (1999). That is, we showed that the internal degrees of polymer chains and entanglement effects peculiar to polymer systems are not necessary for viscoelastic phase separation to take place and strong dynamic asymmetry between the components of a mixture is the only necessary condition. A main difference between shear-induced phase separation and viscoelastic phase separation is that the velocity fields are induced by external shear fields in the former whereas they are self-induced by phase separation itself in the latter.
Here and are respectively the coarse-grained average velocities of polymer and solvent at point and time , and then the average velocity of a mixture is given by . is the composition of polymer. is the osmotic stress tensor, which is related to the thermodynamic force as
where is the mechanical stress tensor, is the average density, is the solvent viscosity, and is the friction constant per unit volume. Here is a part of the pressure, which is determined to satisfy the incompressible condition . The free energy is given by the following Flory-Huggins-de Gennes form:
where is the degrees of polymerization of polymer and is the interaction parameter between polymer and solvent. The terms containing the mechanical stress tensor cause couplings between the composition and the stress fields via the velocity fields. The above equation (22) clearly tell us that the relative velocity of polymers to the average velocity is determined not only by the thermodynamic osmotic force but also by the mechanical force. The role of mechanical force can be easily understood by considering a case of gel Tanaka and Fillmore (1979). To close these equations, we need a constitutive equation, which describes the time evolution of .
Here it is worth noting that in Eq. (23) the inertia term is not relevant for the description of viscoelastic phase separation in ordinary situations since viscoelasticity suppresses the development of velocity fields. However, this is not necessarily the case for a shear problem and even a nonlinear velocity term plays an important role for high Reynolds number flow. We do not consider the nonlinearity since it is out of scope of this article.
In the above, we consider a case of polymer solution, where only polymers can support viscoelastic stress, for simplicity. For a more general case, where viscoelastic stress is not supported only by one of the components, we need a more general set of equations Tanaka (1997a). In such a case, the constitutive relation may also become more complex (see Sec. III.3).
Finally, we mention a fundamental remaining problem of the two-fluid description. In the above derivation, the dissipation in a mixture is separated into the two contributions: One is viscous dissipation of the liquid component, and the other comes from the friction between the two components. This intuitively looks reasonable, however, the hydrodynamic couplings between the slow components are not considered in a systematic manner in the coarse-gaining procedure. This makes the validity of the above separation a bit obscure. Thus, we need theoretical justification for the treatment of dissipation, which remains a subject for future investigation.
iii.2.1 Kinetic equations describing the time evolution of stress tensor: Constitutive relation
We note that for mixtures composed of a large particle (or molecule) component (component 1) and a simple fluid (liquid or gas) (component 2) the stress division becomes almost perfect ( and ), reflecting the large size disparity and the resulting large difference in the friction constant. The mechanical stress of a system is selectively supported almost by the large component 1 alone. This is the case for polymer solutions Doi and Onuki (1992), and suspensions of colloids, proteins, and emulsions Tanaka (1999). So the velocity relevant to the description of viscoelastic stress is the average velocity of component 1 ( for a polymer solution).
As an example of this type of mixtures, here we consider how the mechanical stress, , should be expressed in the case of polymer solution. In general, we should incorporate relevant constitutive equations into the above two-fluid model, depending upon the type of material. Doi and Onuki Doi and Onuki (1992) employed the upper-convective Maxwell equation as a constitutive relation describing its time evolution for polymer solution Doi and Edwards (1988); Larson (1999):
where and and are the relaxation time and the modulus of the shear stress, respectively. Note that . To make the shear stress a traceless tensor, was defined as , where is the unit tensor and is the space dimensionality.
We proposed to introduce the bulk stress in addition to the shear stress to describe the volume shrinking behaviour of viscoelastic phase separation Tanaka (1997b); Tanaka and Araki (1997); Tanaka (1997a). This stress expresses the connectivity of a transient gel formed in the early stage of viscoelastic phase separation and the resulting suppression of diffusion (see Fig. 6). Since the bulk stress is isotropic, it can be expressed by a scalar variable, namely, : . Then, the bulk stress obeys the following equations:
where and are the relaxation time and the modulus of the bulk stress, respectively.
Here we discuss the rheological functions in the above constitutive equations. In the case of polymer solutions, was estimated Doi and Onuki (1992); Onuki (1989); Helfand and Fredrickson (1989); Milner (1993) on the basis of rheological theories of polymer solution including the reptation theory de Gennes (1979); Doi and Edwards (1988) for good and solvents. The bulk stress related to was not regarded to be important, since the longitudinal relaxation along a tube is much faster than the shear relaxation by reptation Doi and Edwards (1988). This is true in good or solvents.
However, elastic effects associated with the volume deformation may become important in polymer solutions under a poor solvent condition Tanaka (1997b, a). It should be stressed that phase separation of polymer solutions always occurs in a poor-solvent condition. Thus, we cannot apply theories for polymers in good or solvents to our problem. In a poor solvent, there exists attractive interactions between polymer chains. Thus, we expect that there are temporal crosslinkings of energetic origin between polymer chains. The most natural model for polymer solutions under such a poor-solvent condition may be a transient gel model in which interpolymer attractive interactions produce temporal contact (crosslinking) points between polymer chains (see Fig. 6). Since such a connectivity is lost below a certain concentration , there may be a steep concentration dependence of the bulk modulus on . Thus, we mimic this situation by using the bulk modulus , which is proportional to a step-like function ( for and for ) Tanaka (1997b, a); Tanaka and Araki (1997): , where is the threshold composition. We note that the introduction of a step-like dependence of the relaxation time has a similar effect.
If we assume that the lifetime of temporal contacts between chains is , we expect that the bulk relaxational modulus has the relaxation time of the order of . Then, the deformation described by that accompanies a change in the volume occupied by polymer chains causes bulk stress if the characteristic time of the deformation is shorter than . However, since polymer dynamics in a poor solvent is far from being completely understood, we need further theoretical studies on this problem. We point out that this type of attractive interactions between molecules of the same component commonly exist in the unstable region of a mixture, which may generally result in formation of a transient gel in dynamically asymmetric mixtures.
We argued Tanaka (1999) that the same physics may be applied to particle-like systems such as colloidal suspensions, emulsions, and protein solutions on noting that under the action of attractive interactions particles tend to form a transient network with a help of hydrodynamic interactions Tanaka and Araki (2000, 2006); Furukawa and Tanaka (2010). The introduction of a steep -dependence of allows us to include effects of transient gel formation and the resulting transient elasticity due to the gel-like connectivity Tanaka (1997b, a); Tanaka and Araki (1997), as described above.
Besides the above origin, there is a possibility that for particle suspensions the slow bulk stress relaxation may originate from hydrodynamic interactions under the incompressible condition: hydrodynamic squeezing effects Tanaka (1999). What is the relative importance of the energetic and hydrodynamic origins in the bulk stress relaxation remains a problem for future investigation. This is related to the treatment of dissipation in the two-fluid description (see Sec. III.2).
iii.3 A general rule of stress division between the components of a mixture
The above perfect stress division only applies to a mixture of large size disparity. For example, in polymer blends Doi and Onuki (1992); Onuki (1994) and in a system where glass transition has a very different ’s Tanaka (1997a), mechanical stress is supported by both of the two components. In this case, the dynamical equations (21)-(27) must be generalized Tanaka (1997a).
Here we briefly discuss a general rule of the stress division in such a case. First, we introduce the rheologically relevant velocity , which appear in the constitutive relation. It is defined as , with Onuki (1994); Tanaka (1997a). Here is the relative motion of component having the average velocity of to the mean-field rheological environment having the velocity of and is the stress division parameter. For simplicity, we neglect the transport and rotation of the stress tensor, which do not affect the pattern evolution so much since the transport and rotation are very slow in viscoelastic phase separation. In a linear-response regime, then, the most general expression of is formally written by introducing the time-dependent bulk and shear moduli in the theory of elasticity Landau and Lifshitz (1975) as
Here is the velocity relevant to rheological deformation, and for polymer solutions . and are material functions, which we call the shear and bulk relaxation modulus, respectively. It should be noted that the rheological relaxation functions and are functions of the local composition . We note that is a purely mechanical modulus and different from the bulk osmotic modulus, . We have the relation , where is the viscosity of a material.
The second term of the right-hand side of Eq. (28) was introduced to incorporate the effect of volume change into the stress tensor Tanaka (1997a); Tanaka and Araki (1997). In a two-component mixture, the mode associated with can exist as far as , even if the system is incompressible, . As mentioned above, we proposed that this term plays a crucial role in viscoelastic phase separation Tanaka (1997a); Tanaka and Araki (1997) (see also below), although it is not so important when we consider shear-induce demixing Doi and Onuki (1992); Onuki (1989); Helfand and Fredrickson (1989); Milner (1993).
Now we consider the stress division for the above general case. The local friction force is given by , where is the average local friction of component and the mean-field rheological environment at point , where the volume fraction of component is . Here and is proportional to the friction between an individual molecule or particle of the component and the mean-field rheological environment, which we call the generalized friction parameter. Because of the physical definition of the mean-field rheological environment, the two friction forces should be balanced. This fact guarantees that the rheological properties can be described only by . Thus, we have the following relation in general:
Then, the general expression of the stress division parameter is obtained as
The above relation is consistent with a simple physical picture that the friction is the only origin of the coupling between the motion of the component molecules and the rheological medium. We expect that this relation holds, irrespective of the microscopic details of rheological models, and, thus, we may apply it to a mixture of any material, the motion of both of whose components is described by a common mechanism. However, for theoretical estimation of friction coefficients, we need microscopic rheological theories, which are not available for a general case unfortunately. More importantly, as mentioned in Sec. III.2, there is obscureness associated with the treatment of ‘nonlocal’ hydrodynamic couplings in the coarse-graining procedure of the two-fluid model.
iii.4 Dynamic asymmetry in transport associated with glass transition
Dynamic asymmetry affects not only the constitutive relation of a system, but also the kinetics of diffusion Tanaka (1996, 1997a). This is particularly important in a mixture whose components having very different , since there is drastic slowing down of the dynamics towards , which gives rise to an extremely strong dependence of the diffusion coefficient on the composition near . Furthermore, formation of a transient gel is not expected for a mixture having little size disparity between its components. Formation of a transient gel or a gel should be specific to a mixture of large size disparity between its components. Thus, for a mixture whose components having very different , we do not expect a significant role of the bulk stress in suppressing the diffusion, unlike the case of a mixture of large size disparity (see above), since there may be no strong -dependence of . Even in this case, the strong dependence of can cause an effect similar to the bulk stress, as shown below.
The dependence of near the glass transition point can be expressed by the following empirical Vogel-Fulcher-Tammann (VFT) relation: , where is the VFT volume fraction and is the fragility index. Thus, we have to take into account this -dependence of , or the friction coefficient . Effects of a steep -dependence of were studied by numerical simulations Sappelt and Jäckle (1997). It should be noted that large bulk stress in the slow-component-rich phase (see above) and slow diffusion in the phase rich in the high component play similar roles in phase separation: they both suppress rapid growth of the composition fluctuations and slow down the composition change in the more viscoelastic phase. Accordingly, a rate of the material transport between the two phases is limited or controlled by that in the slower phase. In this manner, a disparity in the diffusion coefficient between the two components of a mixture, i.e., a steep -dependence of , has effects on phase separation, which are similar to those in the bulk relaxation modulus .
iii.5 Roles of the steep dependence of bulk stress and/or diffusion in viscoelastic phase separation
Here we briefly discuss roles of the steep dependence of bulk stress and diffusion. The continuity equation
tells us that it is that causes the composition change. The bulk stress caused by the deformation type of , thus, suppresses growth of composition fluctuations if is shorter than . In this way, the bulk stress is directly coupled with the composition change and the volume shrinking Tanaka (1997b, a). Note that the volume change of the polymer-rich (slow-component-rich) phase is directly associated with the deformation described by . So the volume shrinking behaviour peculiar to viscoelastic phase separation is a consequence of (i) the slow bulk stress relaxation due to the connectivity of a transient gel formed by the large component and/or hydrodynamic squeezing effects or (ii) a steep composition dependence of the diffusion constant .
iii.6 Beyond the linear constitutive relations
As will be discussed later, viscoelastic phase separation accompanies mechanical fracture of the slow-component-rich phase and thus nonlinearity of rheology may also play an important role in the pattern evolution. In the above, we assume a simple Maxwell-type constitutive relation. However, we may need more complicated constitutive relations to describe the rheology of materials, since viscoelastic phase separation accompanies mechanical fracture of a transient gel, which intrinsically involves non-linear phenomenon. Two key deformation types in viscoelastic phase separation are elongation (uniaxial stretching) and extension (biaxial extension). The former is important in network-forming viscoelastic phase separation whereas the latter is in cellular one. The importance of strain hardening in the formation of cellular patterns has been recognized for both synthetic polymers Spitael and Macosko (2004) and food polymers (e.g., breads) Dobraszczyk (2004); Dobraszczyk and Roberts (1994). Qualitatively, strain hardening makes the more viscoelastic phase mechanically more resistive to fracture, or makes the morphological selection due to mechanical force balance more robust (see Eq. (40)). Furthermore, the inclusion of the yielding behaviour to the constitutive relation is also of crucial importance in describing ergodic-to-nonergodic transitions such as gelation during phase separation (see Sec. IV.6). It is highly desirable to incorporate these features into the constitutive relation for characterizing these nonlinear effects on a quantitative level.
Iv Dynamics of viscoelastic phase separation and the resulting pattern evolution
iv.1 State diagram
In normal phase separation of a dynamically symmetric mixture, there are only two types of pattern evolution in the unstable region: droplet spinodal decomposition for an asymmetric composition and bicontinuous spinodal decomposition for a symmetric composition. Thus, the morphological selection depends solely on . Contrary to this, the morphological selection in a dynamically asymmetric mixture depends on not only but also the relation between and . Here we show what kind of phase separation proceeds in a polymer solution, depending upon the quench condition (, ), where is the composition and is the final temperature after a quench from a one-phase region. The classification based on microscopy observation of pattern evolution is summarized in Fig. 7. In region A, droplet-type phase separation takes place. The polymer-rich phase appears as droplets. In region B, polymer-rich droplets are initially formed, but they aggregate and form a percolated network. After the percolation, the morphology is determined by the mechanical force balance in the network structure. In region C, phase separation proceeds by mechanical fracture (fracture phase separation). In region D, typical viscoelastic phase separation is observed. After the formation of a transient gel, the minority polymer-rich phase forms a network structure. In this region, phase inversion is observed: The initially majority phase eventually becomes the minority phase with time due to its volume shrinking. In region E, normal phase separation without accompanying phase inversion takes place. The solvent-rich phase appears as droplets here. Later we explain how phase separation proceeds in regions A-D.
Next we consider how the critical temperature and the temperature (the border of region D and E (the dashed line) in Fig. 7), below which viscoelastic phase separation accompanying transient gelation is observed, depend upon the degree of dynamic asymmetry, which is controlled by the degree of polymerization , for a critical polymer solution (). In Fig. 8(a), we plot and as a function of for polystyrene (PS)/diethyl malonate (DEM) mixtures. According to the standard theory of polymer solutions de Gennes (1979), should be on the straight line in this plot, which is indeed confirmed. Interestingly, we found that is also on the straight line and furthermore the and lines meet at the temperature in the limit of : , where = or and is a constant. Thus, the temperature, where binary interactions apparently disappear, is determined as C. In the limit of , the polymer has an infinite molecular weight, and thus it can be regarded as a gel according to the definition based on the concept of percolation. The fact that the and lines meet at the temperature in the limit of implies a connection of a transient gel and a special gel at , which looks physically very natural. This further suggests a general link between phase separation and transient gelation, both of which are driven by attractive interactions between like species.
The fact that is on the straight line passing through the temperature at is a consequence of that the equilibrium phase diagrams of polymer solutions of various can be mapped on the master curve after scaling by and by . Thus, the above result suggests an interesting possibility that the dynamic phase diagrams concerning the transient gelation can also be scaled together with the equilibrium phase diagrams.
To check this possibility, we plot the dynamic phase diagrams for the three solutions of different ’s (PS-2, PS-3, PS-5) in the scaled form (see Fig. 8(b)). We use as a reduced temperature. We find that the dynamic phase diagrams are indeed scaled on the master one, implying the universal nature of this dynamic phase diagram including NPS and VPS. Note that is a decreasing function of the composition separating VPS from NPS, :
Thus, VPS occurs below the binodal line around . This finding may have an impact on our understanding of transient gelation in polymer solutions. We speculate that may be determined by the dynamic crossover between the characteristic deformation rate and the characteristic rheological relaxation rate as a function of and Tanaka (1994, 2000a); Araki and Tanaka (2001). This remains a problem for future investigation.
iv.2 The early stage of viscoelastic phase separation
First we consider viscoelastic effects on the early stage of phase separation. We note that this theory applies to any regions in Fig. 7, including normal and viscoelastic phase separation, as far as phase separation takes place in the unstable region (spinodal decomposition). For simplicity, here we do not consider a difference in the relaxation time between shear and bulk stress and assume . Using the relation , we obtain the linearised equation for :
Here we use the Ginzburg-Landau-type free energy . This form of the free energy is reasonable as far as we concern only a shallow quench near a critical point. Then (see Sec. II.1), where , is the decay rate in the absence of viscoelastic couplings. , where (: a positive constant). The correlation length is given by in the mean-field theory. For a case when the time scale of change is slower than , we can set in Eq. (34) and, thus, the growth rate of is given by
where is the so-called viscoelastic length de Gennes (1976a, b); Brochard and de Gennes (1977) (see Sec. II.2). Without viscoelastic couplings, the relation should hold as the Cahn’s linear theory Gunton et al. (1983) predicts. It was shown Takenaka et al. (2006) that the early stage of phase separation of a polymer solution is well explained by the above Onuki-Taniguchi theory Onuki and Taniguchi (1997).
We emphasize that the early stage of phase separation in dynamically asymmetric mixtures including soft matter should be analysed by this theory. Applications of the Cahn’s theory without considering viscoelastic effects may not be appropriate in many cases since can easily become mesoscopic in dynamically asymmetric mixtures. In relation to this, we note that the above relation [Eq. (35)] also well explains the unusual -dependence of experimentally observed in colloid phase separation Tanaka (1999) (see Fig. 9). This suggests the relevance of the viscoelastic model to phase separation not only in polymer solutions, but also in colloidal suspensions, emulsions, and protein solutions, which further indicates the importance of viscoelastic effects in any dynamically asymmetric mixtures including oxides and metals Tanaka (2000a).
iv.3 Network-forming viscoelastic phase separation
Here we describe the process of pattern evolution in typical viscoelastic phase separation, which are observed in region D of the state diagram in Fig. 7. The phase separation in this region is characterized by (i) the formation of a network structure of the minority phase (see Fig. 10), which is contrary to the well-accepted belief of normal phase separation that the minority phase always forms droplets to minimize the interfacial energy, and (ii) phase inversion .
In normal phase separation, the late stage phase separation is discussed on the basis of the scaling concept, which relies on the fact that the volume of each phase is conserved in the late stage and thus there is only one characteristic length scale, i.e., the domain size, in a system: self-similarity of pattern evolution. For viscoelastic phase separation, however, such a scaling concept is not valid because of the volume shrinking of the slow-component-rich phase during phase separation. The absence of self-similarity is obvious in Fig. 10. Because of this difficulty, there has been no analytical theory of domain coarsening so far. In the following, thus, we describe pattern evolution in phase separation on a qualitative level.
The process of viscoelastic phase separation is schematically shown in Fig. 11. Just after the temperature quench, a mixture first becomes cloudy due to the growth of composition fluctuations (see Sec. IV.2), then after some incubation time, small solvent holes start to appear (see Fig. 11(b)). We call this incubation period the “frozen period”, which is the stage of transient gel formation. Then the number and the size of solvent holes increase with time. In this process, the slow-component-rich matrix phase shrinks by expelling the fast liquid component to solvent holes, which leads to the growth of holes made of the fast-component-rich phase. In this way, the matrix phase keeps shrinking its volume and becomes network-like or sponge-like, (see Figs. 11(c) and (d)). In this volume-shrinking process, the bulk mechanical stress plays a crucial role Tanaka (1997b); Tanaka and Araki (1997). Thin parts of a network-like structure are elongated and eventually broken and disconnected. This results in the release of mechanical stress and allows local mechanical relaxation. In this network-forming process, the pattern is dominated by the mechanical shear force balance condition and thus the shear stress plays a major role Tanaka and Araki (1997). In the final stage, a network-like structure tends to relax to a structure of rounded shape and the domain shape starts to be dominated by the interface tension as in usual fluid-fluid phase separation (see Figs. 11(e) and (f)). Domains finally become spherical. When the slow-component-rich phase is the minority phase, thus, there is a phase inversion during phase separation. This phase inversion is a characteristic feature of viscoelastic phase separation.
If the concentration of the slow-component-rich phase reaches the glass transition composition and the yield stress of the slow-component-rich phase exceeds the mechanical stress generated, a structure is dynamically arrested and becomes stable. This may be regarded as the general scenario for formation of colloidal gels (see below) Tanaka (1999); Cates et al. (2004); Lu et al. (2008); Royall et al. (2008)
Now we describe the temporal evolution of the structure factor in PS/DEM mixtures. For a shallow quench (Figs. 12(a) and (b)), fluid phase separation affected by wetting to walls was observed. In this case, , where is the peak wavenumber, initially increases, then decreases and increases again, which is characteristic of phase separation of a thin film of a binary mixture under wetting effects Tanaka (2001). The first increase reflects initial bicontinuous phase separation in bulk. Then the decrease reflects the following process. The DEM-rich phase is more wettable to the glass walls. Thus it is transported towards the glass walls by a hydrodynamic pumping mechanism Tanaka (2001) such that a bicontinuous phase-separated structure transforms into a three-layer structure. Accordingly, a phase-separation structure tends to disappear in the observation plane. The final increase reflects the growth of the remaining DEM-rich droplets bridging the wetting layers. During this process, monotonically decreases, since the domains keep growing via a hydrodynamic mechanism Tanaka (2001).
For a deep quench (Figs. 12(c) and (d)), on the other hand, we observed viscoelastic phase separation. In this case, increases initially, then has a plateau, and finally increases again. Dynamic arrest of the coarsening by transient gelation of the PS-rich phase leads to the plateau of both and . Finally, shrinking of the PS-rich transient gel increases the concentration difference between the two phases, which results in the increase in . After the formation of a transient gel, a large-scale structure due to “elastic instability” is superimposed onto the initial phase-separation structure and becomes more and more dominant with time, which leads to a double-peaked shape of . This results in the crossover of from one branch to another (Figs. 12(c) and (d)). Thus, the border between viscoelastic and normal phase separation (between regions E and D in Fig. 7) can be determined as the temperature where and change their behaviour between the above two types. We note that this two-step phase separation has not been captured by numerical simulations on the basis of the viscoelastic model yet. This might be due to a too steep dependence of the bulk modulus and/or the setting of to the average composition of a mixture in our simulations.
According to the common sense of normal phase separation, after the formation of a sharp interface between the coexisting phase (namely, in the so-called late stage) the concentration of each phase almost reaches the final equilibrium one and, thus, there should be no change in the volume and concentration of each phase Gunton et al. (1983); Onuki (2002); Siggia (1979). Thus, the sharp front formation of the interface before the saturation of the order parameter to the equilibrium values is a very interesting feature of viscoelastic phase separation. We believe that it is a consequence of the steep dependence of or the bulk stress, which makes the diffusion in a dilute region much faster than that in a concentrated region. We pointed out Tanaka (2000a) that the volume decrease of the more viscoelastic phase with time after the formation of a sharp interface is essentially the same as the volume shrinking of gels during volume phase transition Matsuo and Tanaka (1988, 1992); Sekimoto et al. (1989). The physical reason of this similarity to gel will be discussed later.
The scaling law established in normal phase separation is a direct consequence of the conservation of the volumes of the two phases after the formation of a sharp interface and the resulting self-similar growth of domains. The volume shrinking of the slow-component-rich phase inevitably leads to the absence of self-similarity during viscoelastic phase separation and thus the absence of an extended scaling regime. Nevertheless, we observe a transient scaling law (the characteristic domain size ) in the intermediate coarsening stage for a few systems Tanaka et al. (2005); Tanaka and Nishikawa (2005); Tanaka and Araki (2007), although its physical mechanism remains elusive.
In sum, the whole pattern evolution process can be clearly divided into three regimes: the initial, intermediate, and late stages. The crossovers between these regimes can be explained by viscoelastic relaxation in pattern evolution and the resulting switching of the primary order parameter, as will be described below (see V.1).
Besides the early stage, we do not have any reliable analytical predictions and thus numerical simulations based on the viscoelastic model play a crucial role in its understanding Taniguchi and Onuki (1996); Tanaka and Araki (1997); Bhattacharya et al. (1998); Araki and Tanaka (2001, 2005); Tanaka and Araki (2006, 2007); Zhang et al. (2001). We showed that a steep composition dependence of the bulk modulus or the diffusion constant is the key to volume shrinking and the resulting phase inversion and a rather smooth -dependence of the shear modulus is responsible for the formation of a network-like structure Tanaka and Araki (1997); Araki and Tanaka (2001); Tanaka and Araki (2006). A typical numerical simulation result of the process of viscoelastic phase separation is shown in Fig. 13. An example of 3D pattern evolution of viscoelastic phase separation obtained by numerical simulations is also shown in Fig. 14 (see Refs. Araki and Tanaka (2001); Tanaka and Araki (2006)).
Finally we discuss the spatial correlation in the formation of solvent holes in the slow-component-rich phase. One-dimensional arrays of droplets are often formed in viscoelastic phase separation (see Fig. 15(a), which was simulated by our disconnectable spring model Araki and Tanaka (2005). A similar pattern is also often observed in experiments. This indicates the presence of spatial elastic coupling in the hole (or, solvent droplet) formation. Figure 15(b) schematically shows such an array of nucleated solvent-rich droplets and the mechanical forces acting on the shrinking transient gel. The formation of a droplet array can be explained as follows. First it is energetically more favourable to have two solvent droplets nearby rather than independently since the former situation costs less elastic energy than the latter one. Then, if two droplets are nucleated close to each other in this way, the deformation field around them, which is induced by the shrinking of the more viscoelastic matrix phase, becomes anisotropic. The stress is more concentrated at the edges of the array of droplets. This helps formation of solvent-rich holes there (see the hatched regions in Fig. 15(b)), and thus leads to a further increase in the number of droplets in the array. The overall domain shape of the array is approximated by an elongated ellipsoidal droplet (see the ellipsoids of the broken curve in Fig. 15(a)), if we neglect the thin bridges between droplets. This elastic coupling between emerging droplets is a characteristic of mechanically-dominated pattern evolution.
iv.3.1 Mechanical nature of the coarsening of a network structure
In normal phase separation, the late-stage coarsening is driven by the thermodynamic force coming from the interface energy. In viscoelatic phase separation, on the other had, it is driven mainly by the mechanical force in the volume shrinking stage, whereas by the thermodynamic force in the final stage where the order parameter reaches its equilibrium values and the mechanical force decays almost completely. We revealed that the mechanical stress accumulated in a network structure leads to its coarsening by repeating the following sequence: stress concentration on a weak part of the network, its break up and the resulting stress relaxation, and structural rearrangements towards a structure of lower interface energy Tanaka and Araki (1997); Araki and Tanaka (2001); Tanaka and Araki (2006). Such examples can be seen in 2D and 3D colloid simulations, as shown in Fig. 16. We stress that this process can proceed without any thermal activation. Actually, the simulations in Fig. 16 were performed without any thermal noises, namely, at . This indicates that the coarsening of network-type viscoelastic phase separation can proceed purely mechanically: mechanically driven coarsening. This is markedly different from a conventional picture based on the activation-type coarsening process. It has been widely believed that no coarsening proceeds if the activation barrier exceeds 10 , but this argument cannot be applied to a network structure formed by viscoelastic phase separation. Stress concentration can provide a way to overcome such a barrier. We emphasize that mechanically driven coarsening cannot be characterized by the strength of attractive interactions measured by the thermal energy alone.
iv.3.2 Universality of viscoelastic phase separation to dynamically asymmetric mixtures
Finally, we emphasize that pattern evolution in viscoelastic phase separation is essentially the same between the two types of dynamically asymmetric mixtures Tanaka (2000a); Tanaka and Araki (2006); Tanaka (2009): one is a system like polymer solutions Tanaka (1992, 1993, 1994); Koyama and Tanaka (2007), colloidal suspensions Tanaka et al. (2005), and protein solutions Tanaka and Nishikawa (2005), where the strong dynamic asymmetry comes from a large difference in the molecular size and topology between the components, and the other is a system whose components have a large difference in the glass transition temperature Tanaka (1996). Here we show viscoelastic phase separation observed in four different systems, polymer solutions (a), colloidal suspensions (b), and protein solutions (c), and surfactant systems (d) and (e) in Fig. 17. We can see the universal nature of the pattern evolution in any dynamically asymmetric mixtures, (a)-(d). A pattern in (e) is different from the others because of an interplay between viscoelastic phase separation and smectic ordering (see Sec. IV.6.3).
iv.4 Viscoelastic droplet phase separation: Moving droplet phase
Here we mention another type of viscoelastic phase separation which takes place in a mixture with a low volume fraction of the slow component. This type of phase separation is observed in region A of the state diagram in Fig. 7. At such a low volume fraction, the slow-component-rich phase immediately forms droplets while shrinking their size by expelling a solvent. This shrinking process is finished rather quickly because of the small sizes of droplets in the time scale of (: the droplet size). Thus, the volume fraction inside droplets may rapidly reach a strongly entangled jammed state. Such situations are realized also if the final volume fraction is higher than the glass transition volume fraction. If the collision timescale is shorter than the structural relaxation time, droplets behave as elastic glassy balls. This may also be expressed as follows. If the contact time during droplet collision due to Brownian motion is shorter than the material transport between droplets, droplets do not coalesce and stay without growing for a fairly long time. We referred to this interesting metastable state due to the elastic or glassy nature of droplets as “moving droplet phase (MDP)” Tanaka and Nishi (1988); Tanaka (1992, 1994, 2000a).
Here we consider the droplet stabilization mechanism using polymer solutions as an example. The two important time scales characterizing the situation may be the characteristic time of the collision between two droplets (or the contact time) and the characteristic rheological time of the polymer-rich phase . Viscoelastic effects should play a role when is shorter than or comparable to . Brownian motion of a droplet with mass is characterized by a randomly varying thermal velocity of magnitude and duration (: the diffusion constant of a droplet with radius ). Thus should satisfy the relation , where is the range of interaction. On the other hand, , where is the segment size. For the typical values of the parameters, could be longer than for a large or for a deep quench, in particular, if attractive interactions between chains are taken into account. This means that a droplet may behave as an elastic body on the collision time scale for . For , on the other hand, droplets can coalesce with each other. This viscoelastic effect is probably responsible for the slow coarsening and the unusual dependence of the coarsening rate on the quench depth. Since is strongly dependent on and of a droplet phase, it is natural that this phase exists only for a polymer solution having a large under a deep quench condition. Figures 18(a) and (b) schematically show the elementally process of droplet collision and the resulting coalescence for MDP and that for usual liquid-like droplet phase, respectively. To consider the coarsening behaviour in more detail, we need the information on the distribution functions of and , which are expressed by and , respectively. The typical situations are schematically shown in Figs. 18(c)-(e). The coarsening rate may be determined by the relation between and . For , the coarsening process is similar to usual binary liquid mixtures and described by the Brownian coagulation mechanism Gunton et al. (1983); Siggia (1979). With an increase in , the coarsening rate should become slower and finally MDP might be kinetically stabilized for . The stabilization of MDP may be almost complete for the case when and .
This phenomenon may be used to make rather monodisperse particles whose size is the order of sub microns to microns. The monodisperse nature is a direct consequence of the formation of droplets due to the growth of concentration fluctuations with a characteristic wavenumber and little coarsening after that. So this phenomenon may provide us with a new very simple and low cost method to make particles with a desired size, which may be useful in both soft materials and foods industries. For example, we speculate that the formation of elastic particle gels of proteins van Riemsdijk et al. (2010) may share a common mechanism with the moving droplet phase. Recently, it was also shown that even random nonionic amphiphilic copolymers can form stable aggregates, a mesoglobular phase between individual collapsed single-chain globules and macroscopic precipitation Wu et al. (2004).
Here we note that if the droplet concentration becomes too high, droplets are no more stable and tend to aggregate to form networks due to inter-droplet attractions Tanaka et al. (2005). After the formation of network, the behaviour is similar to viscoelastic phase separation. This may be regarded as two-step viscoelastic phase separation: the formation of elastic gel balls followed by network formation. Such behaviour is observed in region B. This suggests that viscoelastic phase separation is also observed in colloidal suspensions Tanaka (1999).
iv.5 Pattern evolution in fracture phase separation
Here we show a special case of viscoelastic phase separation, where phase separation proceeds accompanying brittle mechanical fracture of a mixture. This type of phase separation is observed in region C of the state diagram in Fig. 7.
iv.5.1 Physical mechanism
The above-described mechanical nature of viscoelastic phase separation implies its close analogy to the mechanical response of materials. Indeed, we recently found novel phase-separation behaviour accompanying mechanical fracture (“fracture phase separation”) in polymer solutions Koyama et al. (2009) (see Figs. 19 and 20). Surprisingly, mechanical fracture becomes the dominant coarsening process in this phase separation. This type of phase separation is observed when the deformation rate of phase separation becomes much faster than the slowest mechanical relaxation time of a system. In this sense, the transition from viscoelastic to fracture phase separation corresponds to the “liquid-ductile-brittle transition” in fracture of materials under shear deformation Furukawa and Tanaka (2009a) (see Fig. 21(b)). The only difference between fracture phase separation and material fracture is whether the deformation is induced internally by phase separation itself or externally by loading.
We argue that fracture phase separation is the process of mechanical fracture of a transient gel against self-generated shear deformation, which is caused by volume shrinking of the slow-component-rich phase. For slow shear deformation, a transient gel behaves as viscoelastic matter and exhibits liquid fracture behaviour for shear deformation: viscoelastic phase separation. A network is stretched continuously under stress, elongated along the stretching direction, and eventually breaks up. This process resembles the process of liquid fracture of a material under a stretching force (see Fig. 21(c)) Spaepen (1977); Argon (1979); Carsson and Fuller Jr. (1996); Falk and Langer (1998). For fast shear deformation, a transient gel should behave solid-like and exhibit brittle (or ductile) fracture behaviour: fracture phase separation (see Fig. 21(d)) . At this moment, it is not so clear whether crack formation in fracture phase separation belongs to ductile or brittle fracture, since we are not able to visualize the deformation field in the coarse of phase separation. We speculate, however, that cracks are formed perpendicular to the stretching direction (see Fig. 21(b)), which is characteristic of brittle fracture. This fracture behaviour is a manifestation of solid-like (or, elastic) behaviour Carsson and Fuller Jr. (1996); Falk and Langer (1998) of a transient gel.
The physical mechanism of this mechanical instability is basically the same as shear-induced fracture of a viscoelastic matter: self-amplification of density fluctuations under shear Furukawa and Tanaka (2006); Furukawa and Tanaka (2009a). In our view, a steep composition dependence of the bulk stress leads to instability of the interaction network for the volume deformation of type , whereas that of the shear stress leads to its instability for shear-type deformation, which should be the origin of fracture-like behaviour. In fracture phase separation, elastic couplings between cracks also play a crucial role in pattern formation. We studied this problem by using a simple spring model Araki and Tanaka (2005) (see, e.g., Fig. 15), but further detailed studies are necessary to elucidate roles of spatio-temporal elastic coupling.
In fracture phase separation, the breakup of bonds is required not only for volume deformation, but also for shear deformation of the network. To represent such a strongly nonlinear behaviour, we introduce a steep (actually, step-like) composition dependence also for the shear modulus Koyama et al. (2009): , where is the threshold polymer composition for the shear modulus. The simulated pattern evolution in this way is shown in Fig. 22, which captures the characteristic features of fracture phase separation observed in experiments (see Fig. 19). may be material specific, reflecting its constitutive relation. We speculate since the instability occurs for volume deformation before it occurs for shear deformation. This is because only volume deformation can induce a composition change and shear deformation cannot. We confirmed that the introduction of a step-like dependence for the relaxation time has a similar effect Koyama et al. (2009).
Finally we stress that fracture phase separation also provides a mechanism for the formation of shrinkage crack patterns, which are widely observed in both nature (tectonic plates, dried mud layers, and cracks on rocks) and materials (cracks in concretes, coatings and grazes on a ceramic mug and crack formation in dried foods). Thus, we expect that the viscoelastic model may be the basis of the understanding of this wide class of pattern formation linked to mechanical instability.
iv.6 Arrest of viscoelastic phase separation
iv.6.1 Gelation as dynamically arrested viscoelastic phase separation
Gels and glasses are important nonergodic states of condensed matter, both of which are dynamically arrested nonequilibrium states Zaccarelli (2007). Unlike crystals, their static elasticity does not come from translational order. These states are particularly important in soft matter and foods. Here we briefly consider the nature of gel and the mechanism of its formation.
A schematic state diagram for colloidal suspensions shown in Fig. 23 indicates that a transient gel is a consequence of viscoelastic phase separation and a permanent gel is that of viscoelastic phase separation dynamically arrested by glass transition Tanaka (1999). Recently, by combining careful experiments and simulations, Lu et al. Lu et al. (2008) showed evidence that colloidal gelation is spinodal decomposition dynamically arrested by glass transition. Here it is worth pointing out that spinodal decomposition may not be the necessary condition, but phase separation including nucleation-growth type is enough to cause gelation if the slow-component-rich phase is the majority phase Tanaka (1999). In this scenario, there is an intimate relation between gels and glasses, since the source of dynamic arrest for these two nonergodic states is the same. However, there are many distinct differences in both structures and dynamics between them (see, e.g., Ref. Tanaka et al. (2004)). Locally the dynamic arrest is a consequence of glass transition. However, since gelation is a consequence of phase separation, it intrinsically has macroscopic spatial heterogeneity. This is always the case if a gel is formed by ordinary attractive interactions between particles. Furthermore, the glassy state forming a gel network is far from equilibrium because of a rapid density increase induced by phase separation. This is particularly the case of spinodal decomposition. Local structural analysis provided such evidence Royall et al. (2008). Furthermore, a gel formed by viscoelastic phase separation is inevitably under the influence of self-generated mechanical stress. For the gel to be (quasi-)stable, this stress should be below its yield stress. This feature is absent in a glass.
In some cases, however, gelation involves specific interactions such as strong hydrogen or covalent bonding and microcrystallite formation (e.g., gelatin gels and agarose gels). We note that the mechanism of gelation in these cases is different from the above scenario, reflecting the difference in the mechanism of local dynamic arrest. For example, in gelatin and agarose crosslinking points are formed by microcrystallites of polymers. The difference in the physical interactions stabilizing a gel network leads to the difference in the stability and yield stress. Upon viscoelastic phase separation, mechanical stress is always generated in the polymer-rich phase but the formation of crosslinkings leads to the increase in the yield stress, which results in the stabilization of the gel under the mechanical stress. We emphasize that this mechanical stress is induced by many body effects (the sum of attractions between many molecules) and thus can well exceed the interaction strength per bond, which is often measured in the unit of . Thus, even for strong attractions (), coarsening can proceed upon phase separation accompanying gelation (see the discussion in Sec. IV.3) if there is a strong driving force for volume shrinking, although stronger bonds of course tend to increase the yield stress and make a gel more stable. The level of coarsening can also crucially depend upon at which stage gelation takes place upon phase separation and what is the level of the yield stress (see the state diagram and the caption of Fig. 23).
iv.6.2 Ageing of gels and glasses
The scenario that gelation is viscoelastic phase separation dynamically arrested by glass transition immediately tells us a crucial difference in the ageing mechanism between gelation and vitrification. In viscoelastic phase separation, the coarsening is driven by elastic stress associated with volume shrinking and interfacial tension. These features are absent in the ageing of glass transition at least in colloidal suspensions due to the conservation of the composition. In ordinary glass transition, which takes place under a condition of constant pressure, the volume of a sample decreases (or, the density increases) during ageing since the ageing accompanies the densification due to attractive interactions. We can say that the ageing of gels proceeds under the momentum conservation while satisfying Eq. (40) Tanaka and Araki (2007). The intrinsic macroscopic heterogeneity of gels, which comes from its link to viscoelastic phase separation, leads to strong inhomogeneity of particle mobility, i.e., faster dynamics near the interface Ohtsuka et al. (2008).
Whether viscoelastic phase separation is dynamically arrested or not may be determined by whether the connectivity of the slow-component-rich phase remains when the system reaches a nonergodic state or not. Once the volume shrinking is completed, the driving force for domain coarsening becomes only the interfacial tension. If the yield stress of a gel is higher than the force exerted by this interfacial tension, the system is basically frozen and only exhibits slow ageing towards lower free-energy configuration.
iv.6.3 Arrest of viscoelastic phase separation by smectic order
So far we have considered only phase separation of a mixture of isotropic disordered materials. However, there is a possibility that viscoelastic phase separation accompanies other ordering phenomena. The most obvious such example is freezing by crystallization. Here we show an interesting example in which viscoelastic phase separation is arrested by smectic ordering Iwashita and Tanaka (2006). We studied phase separation of an ordered phase (lamella) of a lyotropic liquid crystal (tri-ethyleneglycol mono n-decyl ether (CE)/water mixtures) into the coexistence of an ordered (lamella) and a disordered (sponge) phase upon heating. When phase separation cannot follow the heating rate, usual viscoelastic phase separation is observed. The slow lamella phase, which has internal smectic order and anisotropic elasticity, cannot catch up with the fast domain deformation, and thus it transiently behaves like an elastic body and supports most of the mechanical stress. On the other hand, the less viscous sponge phase, which is an isotropic Newtonian liquid, cannot support any stress. This dynamic asymmetry leads to formation of a well-developed network structure of the lamella phase (see Fig. 17(d)).
If phase separation is slow enough to satisfy the quasi-equilibrium condition, then membranes can homeotropically align along the interface between the two phases while keeping their connectivity, to lower the elastic energy. This leads to the formation of a cellular structure (see Figs. 17(e) and 24). The lamellar films forming closed polyhedra cannot exchange material with layers in neighbouring polyhedra, except by permeation (i.e., diffusion of material normal to the layers). For slow permeation, which should be the case in our system, the lamellar films can exhibit dilational elasticity, which leads to a (quasi-)stable cellular structure. This is markedly different from a soap froth, where a stretching fluid film merely pulls in material from the others without any elastic cost. Thus, a high degree of smectic order in the cell walls and borders stabilises the cellular structure: Any deformation of the smectic order increases elastic energy and thus the structure is selected by the elastic force balance condition.
We also note that this is an interesting example showing that the heating rate can be used to control the type of phase separation covering from droplet phase separation, network phase separation, and to foam-like phase separation Iwashita and Tanaka (2006). This may be relevant in pattern formation in various soft materials, where the change in a physical variable, such as temperature, is not instantaneous.
This phenomenon may be applied to phase separation of systems with smectic order, such as lyotropic and thermotropic smectic liquid crystals and block copolymers&