Abstract
A spacespanning network structure is a basic morphology in phase separation of soft and biomatter, alongside a droplet one. Despite its fundamental and industrial importance, the physical principle underlying such networkforming phase separation remains elusive. Here, we study the network coarsening during gasliquidtype phase separation of colloidal suspensions and pure fluids, by hydrodynamic and molecular dynamics simulations, respectively. For both, the detailed analyses of the pore sizes and strain field reveal the selfsimilar network coarsening and the unconventional powerlaw growth more than a decade according to ℓ ∝ t^{1/2}, where ℓ is the characteristic pore size and t is the elapsed time. We find that phaseseparation dynamics is controlled by mechanical relaxation of the networkforming dense phase, whose limiting process is permeation flow of the solvent for colloidal suspensions and heat transport for pure fluids. This universal coarsening law would contribute to the fundamental physical understanding of networkforming phase separation.
Introduction
Phase separation is one of the most fundamental phase transition phenomena and ubiquitous in nature^{1,2}. Demixing of oil and water in salad dressing is a typical example. Very recently, the phenomena have attracted considerable renewed interest since the discovery of biological phase separation in living cells^{3,4,5,6}. In general, phase separation starts from a molecular length scale, and then the characteristic size of phaseseparated domains grows with time to a macroscopic scale. Thus, it is crucial to understand how phaseseparation morphology is selected and coarsens with time. This problem is significant not only from a fundamental viewpoint^{1,2} but also from an application viewpoint, e.g., the processing of soft materials in electric, medical, cosmetics, paint and food industries^{7,8,9,10}.
The dynamics of phase separation was studied intensively in the 20th century, and the fundamental physical mechanism was well understood. The scaling concept has been established based on the selfsimilar growth of the phaseseparation pattern, which leads to the powerlaw growth of the characteristic domain size^{1,2}: ℓ ∝ t^{ν} (t: time). The exponent ν is called the growth exponent, whose value depends on the physical mechanism controlling the coarsening dynamics. It has been well established that for droplet phase separation, the evaporation–condensation or Browniancoagulation mechanisms (for both, ν = 1/3) are relevant, whereas, for bicontinuous phase separation, hydrodynamic coarsening (ν = 1) is relevant. These physical mechanisms have successfully described the dynamics of phase separation of various materials with fluidity, ranging from onecomponent atomic (or molecular) fluids, binary mixtures of simple liquids^{1,2,11,12,13,14,15,16}, to solutions of macromolecules such as polymers, colloids, proteins and emulsions^{6,17,18,19,20}. These fundamental coarsening laws have tremendously contributed to a broad field of basic science and technology.
Here it should be noted that the above theories were developed for phase separation of fluids taking place near the critical point. In reality, however, phase separation in nature and industrial processes often takes place far from a critical point. Although not widely recognised, this means that the applicability of the above theories can be severely limited. The crucial point is that for deep quench far below the critical point, the difference in the particle density between the two phases becomes significantly large, which may lead to a significant difference in the dynamics between them. Under substantial dynamic difference between the two phases, the slower phase cannot catch up with the speed of domain deformation induced by phase separation, behaving as a viscoelastic body rather than a viscous fluid. Accordingly, the viscoelastic nature of the dense (slowercomponentrich) phase plays a critical role in the phasedemixing process, leading to unconventional pattern formation. We called this phenomenon viscoelastic phase separation^{21,22}. The most remarkable feature is the formation of the network structure of the minority phase, which was first discovered for polymer solutions^{21,22}. In this case, the polymerspecific dynamic effect originating from topological entanglements plays a crucial role and leads to the breakdown of the selfsimilar domain growth. Since the selfsimilarity is a prerequisite for the powerlaw growth, there is no universal coarsening law for the viscoelastic phase separation of polymer solutions.
From this respect, gas–liquid phase separation of singlecomponent atomic (or molecular) fluids may be much simpler than that of polymer solutions because atoms (or molecules) have no (or few) conformational degrees of freedom, unlike polymers, implying a possible selfsimilar domain growth. The same logic may apply to the solutions of macromolecules with few internal degrees of freedom, such as colloidal suspensions, globular protein solutions, and emulsions. Indeed, we have noticed that for networkforming gas–liquidtype phase separation under deep quench, unconventional coarsening behaviour, i.e., the powerlaw growth of ν = 1/2, has been seen in a variety of systems, including a singlecomponent atomic and molecular system^{23,24,25,26,27} and macromolecular systems (colloidal suspensions^{28,29,30,31}, protein solutions^{32} and lyotropic liquid crystals^{33}). These observations imply a universal physical mechanism behind this unusual powerlaw growth with the exponent of ν = 1/2.
In this work, we aim to reveal the physical mechanism responsible for the unconventional powerlaw coarsening law of ℓ ∝ t^{1/2} and how universal it is. To this end, we perform numerical simulations of phase separation in two types of systems. One is gas–liquid phase demixing of a colloidal suspension, and the other is that of a singlecomponent atomic (or molecular) fluid. We find that both systems show the powerlaw coarsening of the network structures with the exponent of 1/2. We successfully uncover the underlying physical mechanism, in which mechanical relaxation of the dense phase plays a critical role, and establish the universal coarsening law, which is valid for gas–liquidtype phase separation in a variety of materials ranging from pure fluids to soft matter such as colloidal suspensions, protein solutions, and emulsions.
Results
Phase separation in dynamically asymmetric systems
First, we explain in more detail how the depth of quench affects coarsening process. For a very shallow quench, ordinary phase separation mechanisms are usually valid (see the phase diagram of Fig. 1a and b). For droplet phase separation, the coarsening is driven by the diffusional transport of molecules among droplets (see the top panel of Fig. 1a) or the diffusional transport of droplets and their resulting collision and coalescence (see the middle panel), for both of which ν = 1/3. These mechanisms are widely known as the evaporation–condensation (i.e., Lifshitz–Slyozov–Wagner) and the Browniancoagulation mechanism, respectively^{1,2}. For bicontinuous phase separation, material transport is governed by hydrodynamic flow from a narrower part of the network tube to nearby thicker parts due to the Laplace pressure gradient (see the bottom panel of Fig. 1a), and then the tube eventually breaks up, leading to thickening of the nearby tubes. This mechanism is known as Siggia’s hydrodynamic pumping mechanism^{34}, whose growth exponent is ν = 1.
For a deep quench far below the critical point, the difference in the particle density between the two phases becomes significantly large (see Fig. 1b), which may lead to a significant difference in the structural relaxation time, τ_{α}, i.e., strong dynamic asymmetry, between them^{21,22}. Such a situation is generally realised for gas–liquidtype phase separation under a deep quench (see Fig. 1b). In such a situation, ordinary phaseseparation mechanisms do not necessarily work. As we mentioned above, we observed unconventional domain coarsening with the growth exponent ν = 1/2. To indicate under what situations this exponent is observed, we show phase diagrams of three different softmatter systems together with the type of phase separation observed (see Supplementary Note 1 for the details). In ordinary binary liquid mixtures, such as a wateroil mixture, the region where bicontinuous structure appears is limited only to a region where the volume fraction of the minority phase is higher than 32 ± 3%^{12}. Contrary to this traditional knowledge, even the minority phase, whose volume fraction is much lower than 30%, forms network structures, instead of droplets, as shown for colloidal suspensions^{31} (a), protein solutions^{32} (b), and charged colloidal suspensions^{28} (c) in Fig. 2. Furthermore, for these systems, the characteristic length of the network, ℓ, commonly grows as ℓ ∝ t^{1/2} in the late stage, while retaining the network connectivity (see Fig. 2d–f). Interestingly, the exponent of ν ~ 1/2 has also been observed in the networkforming gas–liquid phase separation of singlecomponent fluids under a deep quench condition^{23,24,25,26,27}. This suggests that the coarsening law may be general, but the underlying physical mechanism and its universality have remained elusive.
Numerical simulations
To reveal the physical mechanism of the peculiar coarsening behaviour, we numerically study phase separation in two systems: gas–liquid phase demixing of a colloidal suspension and that of a singlecomponent atomic (or molecular) fluid. For the former system, we used a hydrodynamic simulation model, fluid particle dynamics (FPD) method^{30,35}, which is based on the direct computation of the incompressible Navier–Stokes equation (Supplementary Note 2 for details). We have recently confirmed^{31} that this simulation method can reproduce the colloidal phaseseparation kinetics experimentally observed without any adjustable parameters once the interaction potential is matched precisely (see Fig. 2d). In this work, we use a Lennard–Jones (LJ) potential as an interaction potential between colloids since its longrange nature prevents dynamic arrest by gelation. It allows us to access a powerlaw coarsening regime for nearly two decades. For the latter system, we used standard molecular dynamics (MD) simulations of singlecomponent LJ fluids (see ‘Methods’).
In the following, we first show the numerical simulation results of colloidal suspensions, and then, discuss those of pure atomic fluids, including the similarity and difference between the two.
Selfsimilar network coarsening of colloidal phase separation
We show in Fig. 3a the time evolution of phaseseparation structures in a colloidal suspension at the colloid volume fraction of 10% and zero temperature. We can see that a spacespanning network structure of the minority colloidrich phase spontaneously forms in the early stage, and its characteristic length scale continuously grows with time. In order to characterise the coarsening behaviour quantitatively, we compute the temporal change of the characteristic wavenumber, 〈q(t)〉 (see ‘Methods’ for its definition), which is inversely proportional to the characteristic length of the network: ℓ(t) = 2π/〈q(t)〉. As shown in Fig. 3b, we can see a clear powerlaw coarsening behaviour extending nearly two decades: 〈q(t)〉 ∝ t^{−1/2}. We confirm that our results are free from finitesize effects (see Supplementary Note 2, C and Supplementary Fig. 1).
To elucidate which length scale of a real network structure the characteristic length ℓ represents, we perform the structural analysis in real space by using the chord length distribution function, P(ℓ_{out})^{36} (see ‘Methods’ for the details). Organizing the results based on the dynamic scaling concept^{1}, we find that the scaling of the length by ℓ(t) leads to the collapse of the distribution functions P(ℓ_{out}) at various t onto a single master curve (see Fig. 3c): ℓ(t)P(ℓ_{out}) = f(ℓ_{out}/ℓ(t)), where f( ⋅ ) is some function. This result indicates the selfsimilarity of the coarsening: phaseseparation patterns at any time are identical to each other in a statistical sense, once their sizes are scaled by the characteristic lengths ℓ(t). This fact further indicates the presence of a unique selfsimilar coarsening mechanism behind this phaseseparation process. In Fig. 3c, we can also see that the peak position of the distribution is around ℓ_{out}(t)/ℓ(t) ~ 0.8, independent of time t. It means that the characteristic length, ℓ(t), obtained by the structure factor, roughly corresponds to the characteristic pore size of the network structure, ℓ_{out}, at any time. Here we mention that the same scaling law is also valid for the chord length distribution function for the colloidrich region, P(ℓ_{in}), and its peak is located around ℓ_{in}/ℓ(t) ~ 0.2 (Supplementary Note 3 and Supplementary Fig. 2). The fact that the ratio between the characteristic length of the two phases is kept constant with time (i.e., ℓ_{out}(t)/ℓ_{in}(t) ~ 4) is consistent with the selfsimilar nature of the network growth.
In Fig. 3d, we show that the structure factor, S(q, t) (the definition being described in ‘Methods’) can also be scaled as 〈q〉^{d}S(q, t) = g(q/〈q〉) (d: the spatial dimension; g( ⋅ ): some function), which is further support for the selfsimilarity in pattern evolution. We also find that the master curve of S(q, t) can be apparently described by Furukawa’s scaling function^{37}: \(g(x)\propto \frac{{x}^{2}}{\frac{\kappa }{2}+{x}^{2+\kappa }}\), where κ = d + 1 (see the grey curve in Fig. 3d), whose low and high qdependences are constrained by the conservation law of the composition and the Porod law due to the sharp domain interface, respectively. Moreover, denoting the qintegral of S(q, t) as I(t), we expect the relation of I(t) ∝ ℓ^{d−1} = ℓ^{2} for selfsimilar domain growth^{1,2}. In the inset of Fig. 3b, we show that I(t) indeed linearly increases with time t (I(t) ∝ t), which is consistent with the coarsening law of ℓ ∝ t^{1/2} found in the above.
In general, the selfsimilar nature of the domain coarsening during phase separation is a consequence of the fact that the volume fractions of the two phases keep constant with time, after the formation of a sharp domain interface between them, i.e., the saturation of the compositions of both phases to their equilibrium ones. In Fig. 4a, we indeed find that the colloid volume fraction in the colloidrich phase ϕ, is kept almost constant with time at ϕ = 0.54 ± 0.03 in the late stage of phase separation, i.e., in the powerlawgrowth time regime. We also confirm that the pressure of randomly packed colloids at zero temperature is almost zero at ϕ ~ 0.54 (see Supplementary Note 4 and Supplementary Fig. 3). This fact is also consistent with the condition for gas–liquid coexistence (note that since there are few colloids in the colloidpoor phase (see Figs. 3a and 4c), its pressure should be nearly zero).
Coarsening mechanism
Now, we turn our attention to the coarsening mechanism. In ordinary bicontinuous phase separation^{34}, a domain responds as viscous fluid to the mechanical stress generated by the interfacial tension. In other words, both phases behave like viscous fluids. This coarsening mechanism due to hydrodynamic flow leads to the growth exponent of ν = 1 (see the right panel of Fig. 1a)^{34}, and thus, the domain coarsens much faster than in the present case with ν = 1/2. Therefore, it is natural to consider that the viscoelastic nature of the colloidrich phase may play a critical role in the slower domain coarsening (see the right lower panel of Fig. 1b). The crucial point is that the elastic deformation of the network structure must be accompanied by small local composition change of δϕ(r) from its average value ϕ_{0} ~ 0.54 (Fig. 4a), i.e., the local volume deformation ϵ(r) = − δϕ(r)/ϕ_{0} (see Fig. 4b). Then, because of the mass conservation law, the volume deformation of the colloidrich phase must be accompanied by the slow solvent transport, which leads to the permeation of the solvent through small gaps among denselypacked colloids. This situation is precisely that of the poroelastic theory^{38}, which, for example, describes slow water transport in soil. We can directly confirm this transport mechanism by looking at the pressure field of the solvent in the colloidrich domain (see Fig. 4c): The pressure field of the solvent, p, is strongly inhomogeneous inside the colloidrich network, which is coupled with the volume deformation field, ϵ. Note that the pressure gradient in the network is the driving force inducing the permeation flow of the solvent. According to the poroelastic theory^{38}, the local volume deformation, ϵ(r, t), should obey the following diffusiontype equation:
where D_{P} is the socalled poroelastic diffusivity^{38,39}. Here, because the average composition, ϕ_{0}, inside the colloidrich phase is almost constant in the coarsening regime (Fig. 4a), the elasticity and permeability of the colloidrich phase should be more or less constant with time, which allows us to treat D_{P} as a constant with time. This result indicates that the mechanical relaxation of the network deformation, which is the crucial process of the network coarsening, is limited by the slow fluid transport through the dense colloidrich phase (i.e., poroelastic deformation, see Fig. 4b). In other words, the characteristic time of domain deformation is given by the time required for the solvent to transport over the characteristic length ℓ. The selfsimilar nature of the network growth indicates that there is a specific characteristic length for the phaseseparation pattern, which is ℓ. This fact justifies choosing ℓ as the space unit of the Laplacian in front of ϵ in Eq. (1) in our scaling analysis. Thus, we obtain the domain coarsening law of \(\ell \sim {({D}_{{\rm{P}}}t)}^{1/2}\).
Evidence for the above mechanism
To confirm the validity of the above mechanism, we now focus on the elastic nature of the colloidrich domain. To this end, we first analyse the strain field, ϵ_{αβ}, by coarsegraining the local displacements of colloidal particles (see ‘Methods’ and Supplementary Note 5 for details). In Fig. 5a, we show the 3D structures of the colloidrich network together with realspace mapping of local volume strain, ϵ (see below). Here we can see that the locations where compression (ϵ < 0) or dilation (ϵ > 0) takes place are not distributed randomly, but distributed with the characteristic length scale of the network width (i.e., the chord length of the colloidrich network), ℓ_{in}. This situation is very similar to the composition and pressure distributions in Fig. 4b, as it should be. Here we note that because of the selfsimilar nature of the network growth, ℓ(t) ∝ ℓ_{in}(t). This fact allows us to treat the elastic deformation of the colloidrich domain in a coarsegrained manner. We also show in Fig. 5b the time evolution of the distribution function, P(ϵ), of local volume strain with respect to the reference time t_{0} (=62.0), \(\epsilon ={\sum }_{\alpha }{\epsilon }_{\alpha \alpha }({t}_{0}\to {t}_{0}+{t}^{\prime})\). We note that t_{0} = 62.0 corresponds to the right most panel in Fig. 3a. We can see that the peak width broadens and the peak height decreases with the increase of \({t}^{\prime}\). Here we stress that the selfsimilarity and dynamic scalability hold for the phenomena. Thus, if the elastic response of the colloidrich phase plays an important role, we expect that the dynamical scaling holds for the distribution function of \(P(\epsilon ,{t}_{0},{t}^{\prime})\). Indeed, we find that it can be scaled as \(({t}^{\prime}/{t}_{0})P(\epsilon ,{t}_{0},{t}^{\prime})=f({t}_{0}\epsilon /{t}^{\prime})\) (f: some function; see Fig. 5c). From this scaling, we may conclude that ϵ is proportional to \({t}^{\prime}\) for a certain t_{0}: \(\epsilon \propto {t}^{\prime}/{t}_{0}\). This proportionality can be explained by the linear nature of the Stokes regime: In a short time duration (\({t}^{\prime}\)), in which the relative displacements between the centreofmass positions of colloidal particles are negligibly small compared to the particle size of σ, the velocities of colloidal particles should be constant with time. Here, (1) ϵ is an infinitesimally small dimensionless quantity, (2) t_{0} ∝ ℓ^{2}, and (3) the above relation of \(\epsilon \propto {t}^{\prime}/{t}_{0}\) is to hold for arbitrary \({t}^{\prime}\). These facts (1)–(3) tell us that \({t}^{\prime}\) should also be proportional to ℓ^{2}. This finding clearly indicates that the growth exponent, ν = 1/2, reflects the elastic response inside the colloidrich phase, whose characteristic time (τ_{ϵ}) and length scales (ℓ_{ϵ}) satisfy \({\tau }_{\epsilon }\propto {\ell }_{\epsilon }^{2}\) (note that ℓ(t) ∝ ℓ_{in}(t) ~ ℓ_{ϵ}(t)). Here we stress that our hydrodynamic simulation method (FPD) strictly satisfies the momentum conservation for colloids and a solvent as well as the incompressibility condition for a solvent, allowing us to observe the local solvent exchange accompanied by subtle volumetric deformation of the colloidrich domain (see Fig. 4c). This technical feature can generally be attained by simulation methods of colloidal suspensions based on the direct computation of the incompressible Navier–Stokes equations (see, e.g., refs. ^{40,41,42}). In refs. ^{31,43}, we discussed the advantages and disadvantages of our simulation method, including comparison with other simulation methods.
Finally, we mention the elementary process of the topological change of the network structure during its coarsening. In the absence of thermal noise (i.e., at zero temperature), the coarsening cannot be due to thermal activation but should be of purely mechanical nature^{44}. The network structure is under mechanical stress to reduce the interfacial energy cost. The resulting mechanical stress is concentrated on weak parts of the network structure, leading to their eventual rupture. Then the whole network relaxes its shape while retaining the momentum balance (or, mechanical balance) condition. Such topological change of the network and the resulting slow mechanical relaxation are the primary mechanisms of the network coarsening (see also Supplementary Movie). Since the mechanical rupture is a rapid nonlinear process, it is not the limiting process controlling the network coarsening. It is slow mechanical relaxation that controls the network coarsening. This situation is similar to the case of Browniancoagulation mechanism for droplet phase separation (see the middle panel of Fig. 1a): slow Brownian motion of droplets is the limiting process of domain coarsening, but rapid droplet coalescence accompanying the topological change is not^{1,2,16,34}. The slow mechanical relaxation process described in the above is characterised by the successful scaling of P(ϵ) with \({\tau }_{\epsilon }\propto {\ell }_{\epsilon }^{2}\), as shown in Fig. 5c.
Crossover of the limiting transport process
Here we note that the above scenario is valid only when the characteristic time required for poroelastic deformation (i.e., \({\tau }_{\epsilon }={\ell }_{\epsilon }^{2}/{D}_{{\rm{P}}}\)) is sufficiently shorter than the structural relaxation time of the colloidrich phase (τ_{α}). τ_{ϵ} increases monotonically with domain growth, whereas τ_{α} is almost constant since it is determined only by the volume fraction of the colloidrich phase, ϕ_{0}, which is constant with time (see Fig. 4a). Thus, the above condition is eventually violated in a very late stage. In such a situation, the colloidrich domains no longer behave as an elastic body and start to behave as a viscous fluid. Then, the domain growth is driven by fluidlike domain deformation, i.e., hydrodynamic transport. Thus, Siggia’s growth exponent (ν = 1; see the right panel of Fig. 1a) is to be observed as long as the bicontinuous structure is preserved. Indeed, such a crossover of the growth exponent from 1/2 to 1 was observed in a microgravity experiment carried out in the International Space Station^{29}, which successfully followed the phase demixing of colloidal suspensions over five decades. On the other hand, if the connectivity of the network structure is lost, the ordinary mechanisms of droplet growth may start to play a significant role in the domain coarsening.
Networkforming phase separation in pure fluids
Next, we turn our attention on networkforming phase separation in singlecomponent fluids. As mentioned above, the domain growth exponent of ν ~ 1/2 has also been observed in the gas–liquid phase separation of singlecomponent fluids under a deep quench condition^{23,24,25,26,27}. To reveal the underlying mechanism, we study the kinetics of gas–liquid phase separation in a singlecomponent 3D Lennard–Jones (LJ) system (see ‘Methods’ on the simulation details). When the gas and liquid phases have significantly different densities, the two phases exhibit strong dynamic asymmetry (see Fig. 1b). In Fig. 6a, we show the temporal change of the characteristic wavenumber, 〈q(t)〉, for various quench conditions. We find that 〈q(t)〉 indeed decays with an almost constant powerlaw exponent close to 1/2 for a wide range of deep quench conditions (from T = 0.5 to 0.01).
As in the case of colloidal phase separation, the dense phase is expected to respond elastically to deformation because of its slow dynamics. The domain deformation should be accompanied by small local density change δρ around its average density of ρ_{0}, as in the case of colloidal phase separation (see Fig. 4b). This local density change may further be coupled to the local kinetic energy for a singlecomponent fluid: The more (less) dense the local density is, the less (more) the local kinetic energy is. Note that thermal expansion is the only mechanism of the density change in a singlecomponent system^{1}. Then, the relevant transport process should be heat transport. To check whether the coarsening mechanism based on the slow heat transport is relevant or not, we calculate the effective temperature, i.e., the kinetic energy of each particle, \({K}_{i}(t)=\frac{1}{2}m\langle {{\bf{V}}}_{i}^{2}\rangle (t)\), where i is the particle index (see ‘Methods’). Figure 7a shows an example of the realspace distribution of K_{i}, where we can see that particles with high/low kinetic energy are not randomly distributed, but heterogeneously with the characteristic length scale of the network structure ℓ. This observation supports the mechanism we proposed above.
The pattern in Fig. 7a is quite similar to those shown in Fig. 5a, although the physical quantity displayed is fundamentally different. This similarity in the pattern between the two types of systems can be understood from the similarity between poroelasticity and thermoelasticity: the fundamental equation of poroelasticity^{38} is known to be mathematically equivalent to that of thermoelasticity^{45} (see, e.g., ref. ^{39}). This fact indicates that the elastic deformation of the dense phase is limited by the transport of the kinetic energy, i.e., heat transport (see Fig. 4b). Similarly to the case of colloidal phase separation, thus, we have the following diffusion equation for volumetric deformation ϵ = − δρ/ρ_{0} of the network domain:
where D_{T} is the thermal diffusion coefficient and almost constant with time since ρ_{0} is almost constant during the coarsening. On noting that the characteristic length is given by ℓ, Eq. (2) leads to the domain coarsening law of \(\ell \sim {({D}_{{\rm{T}}}t)}^{1/2}\). This relation indicates that the characteristic timescale required for thermal diffusion over the length of ℓ is given by \({\tau }_{{\rm{T}}} \sim {\ell }^{2}{D}_{{\rm{T}}}^{1}\). The thermal diffusion coefficient of the dense glassy system with the similar density ρ_{0} is estimated as D_{T} ~ 3 from the literature data^{46,47,48} (see Supplementary Note 6 for details). In Fig. 7a, the characteristic length of domains, or the inhomogeneity of the kinetic energy, is ℓ ~ 10 at t = 45. Consistently, the above relation provides τ_{T} ~ 30 for ℓ ~ 10. This result supports the validity of our mechanism.
Importance of dynamical asymmetry
In the above, we have shown that the powerlaw growth of the exponent 1/2 in networkforming gas–liquid phase separation of colloidal suspensions and singlecomponent atomic (or molecular) fluids is a consequence of the slow elastic response of the dense phase, whose limiting process is solvent transport (poroelasticity) and heat transport (thermoelasticity) in the dense phase, respectively. Here we examine in more detail under what conditions this coarsening behaviour is to be observed. We need a sufficiently deep quench to induce enough strong dynamic asymmetry between the two phases (see Fig. 1b). However, we note that the growth exponent of 1/2 is not observed for ordinary binary liquid mixtures, where dynamics of the two phases is symmetric. In Fig. 6b, we show the temperature dependence of the coarsening behaviour for a dynamically symmetric binary liquid mixture. We can see that there is no distinct powerlaw growth, and the apparent growth exponent (i.e., the slope of the curve) continuously decreases with a decrease in temperature. In contrast, in a singlecomponent 3D fluid (Fig. 6a), the characteristic domain size grows more slowly at a lower temperature in the range of 0.5 ≤ T ≤ 0.01, yet with the same powerlaw exponent (ν ~ 1/2). The dynamically symmetric binary mixture at the lowest temperature (T = 0.1) exhibits a logarithmiclike slow decay of 〈q(t)〉 (see also ref. ^{49}). It is because both phases equally suffer from dynamic arrest due to the glassiness. For such a case, our mechanism is not relevant due to the lack of dynamic contrast between the two phases. For binary mixtures of equalsize particles with dynamic asymmetry, e.g., systems whose components have very different glasstransition temperatures, network phaseseparation patterns may be formed. However, the slow interspecies diffusion prevents the early establishment of the saturation of the composition field, which is prerequisite for the scale invariance of the pattern evolution. Thus, we do not expect the powerlaw domain growth. The absence of the selfsimilarity and powerlaw growth was confirmed for polymer mixtures, whose components have very different glasstransition temperatures^{50}.
In singlecomponent systems, such a situation never takes place: elastic deformation of the dense phase can proceed without being influenced by the dilute gas phase. It is because the relaxation time of the gas phase is rather insensitive to the temperature and always much faster than the timescale of elastic deformation of the dense liquid phase (see the right panel of Fig. 1b). Thus, a considerable difference in the dynamics (structural relaxation time) between the two phases (i.e., the dilute gas and dense liquid phases) is prerequisite for the powerlaw growth of exponent 1/2. It is also the case for colloidal suspensions: although a colloidal suspension should be regarded as a binary mixture, the significant size difference between colloids and solvent molecules leads to the strong dynamic asymmetry between the two phases. We may safely assume that the structural relaxation time of a gas (or, solventrich) phase is significantly faster than that of liquid (or, colloidrich) phase (see again the right panel of Fig. 1b). Note that the characteristic timescale of a particulate system is roughly proportional to the cube of the particle size.
In short, our coarsening mechanism is operative in the case where one phase has a spacespanning network structure and exhibits slow elastic motion during coarsening, and the other phase does not hinder the mechanical relaxation process. This situation is widely satisfied with gas–liquidtype phase separation of dynamically asymmetric mixtures.
Dependence of the growth exponent on the spatial dimensionality
From the above, we may conclude that the growth exponent of 1/2 observed in networkforming phase separation originates from the slow elastic motion of the dense phase under a condition that the other phase does not hinder this process. This condition requires strong dynamic asymmetry between the two phases. This conclusion is valid for three dimensional (3D) systems. However, it may not be necessarily the case for 2D systems. It is because there is an intrinsic topological difference in the percolated network structure between 2D and 3D: in 2D, a bicontinuous network structure can never be formed, unlike in 3D. In the above, we see that for 3D systems, the limiting process of elastic deformation is a slow transport of the solvent or heat, which obeys the diffusionlike equation (Eqs. (1) and (2), respectively). In thermoelasticity, the limiting process is heat transfer, which takes place at any dimension in the same manner. This is confirmed in Fig. 6c: The domain coarsening exponent is ν ~ 1/2, even for 2D. In poroelasticity, on the other hand, the limiting process is fluid flow through the dense colloidrich phase, which obeys Darcy’s law, in which the gradient of fluid pressure induces the flow of the solvent relative to colloids. For a network structure in 2D, the solventrich phase cannot have connectivity, and instead, is divided into isolated domains with different pressure, as shown in Fig. 7b. In this situation, isolated solventrich domains cannot change their volume easily because of the incompressibility of the solvent, which does not allow the volume deformation of each solventrich domain. The only way to change the domain volume is to exchange the solvent between neighbouring solventrich domains through the colloidrich network. This process is very slow. In Fig. 7b, we show a pressure distribution in the solventrich domains together with the network of the colloidrich phase during phase separation of a 2D colloidal suspension. Here we note that the similar pressure field was reported by Yamamoto et al.^{51}. The pressure difference between neighbouring domains leads to solvent transport. This transport mechanism imposes a strict boundary condition on the elastic deformation of the colloidrich network. In Fig. 6d, we show the volumefraction dependence of the temporal change of the characteristic wavenumber during networkforming colloidal phase separation in 2D. We can see that the growth exponent is much less than 1/2 and strongly depends on the volume fractions, as a consequence of complex nonlocal coupling among solventrich isolated domains through a solvent exchange under the constraint of the incompressibility.
The critical point is that pores (i.e., the less dense phase) are isolated for 2D whereas interconnected for 3D. Thus, for 2D colloidal suspensions, permeation flow (Darcy’s law) is induced not only by the pressure gradient inside the percolated network but also by the pressure difference between isolated liquid pores (or, the colloidpoor phase) (see Fig. 7b for the pressure distribution in pores). On the other hand, thermal conduction (Fick’s law) can take place exclusively inside the network for both 2D and 3D singlecomponent fluids since the kinetic energy is inhomogeneous only in the network. Thus, there is no dependence of the coarsening law on the dimensionality for singlecomponent fluids.
Generality of the coarsening law
Here we discuss for what kinds of systems our coarsening law controlled by mechanical relaxation is relevant.
First, we consider the growth exponent ν = 1/2 observed previously by MD simulations of spinodal decomposition. This exponent was reported for 2D gas–liquid spinodal decomposition of singlecomponent fluids^{23,52,53,54}. For this case, the growth exponent of 1/2 was ascribed to the interfacelimited (or, ballistic) evaporation–condensation mechanism, where the transport of molecules is kinematic (or, interfacelimited) rather than diffusive^{1,52,55,56}. The same exponent was also reported for dynamically symmetric binary mixtures^{57,58,59,60}. In this case, on the other hand, it was ascribed to the Browniancoagulation mechanism for 2D fluids^{61}, in which the PlateauRayleigh instability responsible for Siggia’s mechanism in 3D fluids is absent. For both cases, the minority phase forms only droplets unlike our case. Furthermore, the coarsening is governed by thermodynamicallydriven transports (ballistic or diffusional) in these mechanisms, whereas by mechanicallydriven transport in our mechanism.
The growth exponent suggestive of 1/2 was also reported for the gas–liquidtype spinodal decomposition of 3D pure fluids, based on MD simulations^{23,24,25,26,27}. In ref. ^{23}, this exponent was ascribed to the interfacelimited evaporation–condensation mechanism^{52} for both 2D and 3D. In ref. ^{24}, it was speculated that the difference from the coarsening behaviour of the corresponding symmetric binary fluid mixture might be due to the difference in the density and viscosity between the gas and liquid phases, but its exact mechanism has remained elusive. In ref. ^{25}, the exponent of ν = 1/2 was regarded to be transient before a crossover to ordinary hydrodynamic coarsening with ν = 1. In refs. ^{26,27}, similarly, it was suggested to be transient before a crossover to faster growth. We speculate that our mechanism may be responsible for these phaseseparation behaviours.
For the nucleationgrowthtype phase separation with a very asymmetric composition, the same exponent of 1/2 also appears in the time regime where the composition of the majority phase is supersaturated^{62,63,64}. In such a case, diffusional material transport from the surrounding majority phase to droplets of the minority phase controls coarsening dynamics. After the saturation of the majority phase, the evaporation–condensation (or Browniancoagulation) mechanism starts to play a central role in coarsening (see, e.g., refs. ^{63,65}). This mechanism is also governed by thermodynamicallydriven transports and nothing to do with our coarsening mechanism, where mechanical relaxation plays a central role.
Finally, we stress that our coarsening mechanism is relevant for phaseseparation behaviours observed experimentally for colloidal suspensions and globular protein solutions, as shown in Fig. 2d–f, and also for surfactant solutions^{33}. Furthermore, the exponent ν = 1/2 was observed over two decades by microgravity experiments of colloidal phase separation^{29}. These experimental examples include diverse dynamically asymmetric softmatter systems with various interparticle potentials, from shortrange depletion interaction^{29,31} (Fig. 2d), interprotein interaction^{32} (Fig. 2e), to van der Waals interaction^{28} (Fig. 2f). Moreover, although we consider colloids interacting with the LJ potential in our hydrodynamic simulations, the exponent 1/2 is also numerically reproduced for colloids interacting with shortrange depletion attractions^{30,31} (Fig. 2d). These facts indicate the universality of this powerlaw coarsening with the exponent of 1/2 to a wide variety of networkforming phase separation of dynamically asymmetric mixtures.
Discussion
In summary, we discover a universal coarsening law (ℓ ∝ t^{1/2}) for gas–liquidtype networkforming phase separation in soft matter and pure fluids, which is valid in a practically relevant condition far from the critical point. We have revealed that the growth exponent of 1/2 is a consequence of the fact that elastic deformation of the dense phase forming a network is controlled by slow transport of the solvent (heat) through it for soft matter (pure fluids). The universality of the coarsening law relies on the absence of complex internal degrees of freedom in the system element, i.e., the presence of a specific length characterising a system; for example, the atomic (or molecular) size in pure fluids^{23,26,27}, the colloid size in colloidal suspensions^{28,29,30,31}, the protein size in protein solutions^{32}, and the intermembrane spacing in lyotropic liquid crystals^{33}. The mechanism we found here is relevant to gas–liquidtype phase separation of dynamically asymmetric mixtures, in which the coarsening proceeds via solvent (or heat) transport inside the higherdensity phase with elasticity. This condition is satisfied for a quite broad class of materials: They include pure atomic and molecular fluids as well as dynamically asymmetric mixtures, e.g., macromolecular systems containing a solvent as the component. We also stress that this mechanism is not restricted to a limited region of the phase diagram but relevant in its broad region (see Figs. 1b and 2). We expect that this mechanism may be relevant to biological phase separation in living cells^{3,4,5,6}, where various components with different mobility coexist. In this regard, it is notable that very recently, nonspherical networklike morphology has been reported in diverse biosystems (see, e.g., refs. ^{66,67,68}).
We hope that our finding would contribute to the more profound understanding of phase separation in soft and granular matter, including gel formation^{69,70,71}, as well as in living systems. From an application point of view, our coarsening law would provide a useful guide to design the structures of porous materials, which are used in batteries^{72}, ion exchange^{9}, catalysis^{73}, microelectronics^{7} and medical applications^{74,75}.
Methods
Simulations of colloidal phase separation
To study colloidal phase separation numerically, we consider a suspension of colloids interacting with LJ potential, \(U(r)=4{\epsilon }_{{\rm{LJ}}}\{{(r/{\sigma }_{{\rm{LJ}}})}^{12}{(r/{\sigma }_{{\rm{LJ}}})}^{6}\}\), whose interaction range is much longer than a depletion potential. This longrange nature of the interaction allows us to prevent dynamic arrest due to gelation and thus to follow the coarsening behaviour for a long period. Since our interest is in phaseseparation dynamics under a deep quench, we neglect thermal noise. In the data analysis, we use the length unit σ as σ = σ_{LJ} and the time unit τ_{d} as τ_{d} = 3πησ^{3}/ϵ_{LJ}, where η is the viscosity of the solvent. τ_{d} corresponds to the time during which a free colloid under a constant external force of magnitude, ϵ_{LJ}/σ, moves by its diameter σ. The depth of the LJ potential ϵ_{LJ} is set such that the Reynolds number \(Re=\frac{\rho {\sigma }^{2}}{\eta {\tau }_{{\rm{d}}}}=0.8\) (ρ being the density of the solvent). In Supplementary Note 2, B, we confirm the Stokes behaviour for this Reynolds number. We define the volume fraction of colloids, ϕ, as \(\phi =\frac{\pi {\sigma }^{3}N}{6{L}^{3}}\), where N and L are the number of colloids and the side length of our cubic simulation box and set ϕ = 0.1, which is a volume fraction high enough to form network structures upon phase separation. We perform largescale FPD simulations with the system size of L/σ = 69.2 (the corresponding number of the computational grid being 512^{3}) by utilising multiple GPUs at the same time.
Simulations of phase separation of a singlecomponent fluid
To study gas–liquid phase separation, we use a singlecomponent Lennard–Jones system. We utilise the LAMMPS package to perform molecular dynamics simulation with NVT ensemble and control the temperature by Nose–Hoover thermostat. We employ the standard Lennard–Jones units (i.e., σ_{LJ}, \({\tau }_{{\rm{LJ}}}=\sqrt{{\sigma }_{{\rm{LJ}}}^{2}m/{\epsilon }_{{\rm{LJ}}}}\), ϵ_{LJ} for length, time and energy units, respectively). We apply the same simulation box size as in the above FPD simulation (L/σ_{LJ} = 69.2). To simulate gas–liquid phase separation, we first prepare an equilibrium liquid at ρ = 0.33 and T = 1.8 (the critical number density and temperature being ρ_{c} ~ 0.33 and T_{c} ~ 1.2, respectively). Then, we quench the system into various temperature, T = 1.1, 0.5, 0.1, 0.01 in the unit of ϵ_{LJ}/k_{B}, for which we observe the formation of interconnected network structure during phase separation. In the analysis of the local kinetic energy, we calculate the kinetic energy of each particles, \({K}_{i}(t)=\frac{1}{2}m\langle {{\bf{V}}}_{i}^{2}\rangle (t)\) by taking the time average of the velocity of ith particle, V_{i} over the period \([t\frac{{\tau }_{{\rm{LJ}}}}{2},t+\frac{{\tau }_{{\rm{LJ}}}}{2}]\). For simulations of a dynamically symmetric binary mixture (see Fig. 6), we employ an LJ potential with the potential depths, ϵ_{LJ} and ϵ_{LJ}/2, for identical and dissimilar particle pairs, respectively (σ_{LJ} being common for all pairs of particles).
Analysis of the temporal growth of the scattering function during demixing
We calculate the scattering function S(q, t) from the 3D power spectrum of the density correlation function as S(q, t) = ρ_{q}(t)ρ_{−q}(t)/N. Here the density field is defined as \(\rho ({\bf{r}},t)=\frac{6}{\pi {\sigma }^{3}}{\sum }_{i}{{\Theta }}(\frac{\sigma }{2} {\bf{r}}{{\bf{R}}}_{i}(t) )\), where Θ is the step function and {R_{i}} is the set of the centreofmass positions of colloids. To analyse the temporal change of network patterns during phase separation, we compute the temporal change of the characteristic wavenumber, 〈q(t)〉, defined as the first moment of the structure factor S(q, t): \(\langle q(t)\rangle =\frac{\int dq\,qS(q,t)}{\int dq\,S(q,t)}\), which provides the characteristic wavenumber of a network structure.
Structural analysis based on the Chord length distribution
To characterise the typical length of the network structure in real space, we use an analysis method called as the chord length distribution^{36}. To perform this, we first divide the space into the colloidrich and poor regions. Specifically, we apply the Gaussian filter with the standard deviation Δ on the centreofmass positions of the colloidal particles, and construct a coarsegrained density field, \({\rho }_{{\rm{g}}}({\bf{r}})={\sum }_{i}\exp (\frac{ {\bf{r}}{{\bf{R}}}_{i}{ }^{2}}{2{{{\Delta }}}^{2}})\). With this function, we define the part of the space with ρ_{g} > ρ_{th} (ρ_{th} being a threshold value) as the colloidrich region, and the remaining region as the colloidpoor region. In this analysis, we set as Δ = σ and ρ_{th} = 1/2. The inset of Fig. 3c shows an example of the binary field obtained by the above procedure. The black and white parts represent the colloidrich and poor regions, respectively.
The chord length that we use in this paper (ℓ_{in} and ℓ_{out}) is determined in the following way: We randomly choose a point on the space and draw a straight line from the point until the line hitting to the boundary of the colloidrich and poor regions (see the red arrows in the inset of Fig. 3c). If the chosen point is in the colloidrich region, we regard the length of the line as ℓ_{out}; otherwise, as ℓ_{in}.
Construction of strain fields
We compute the strain field ϵ_{αβ}, following ref. ^{76}. Denoting the displacement of ith particle from time t = 0 to t = t as u_{i}(t) = R_{i}(t) − R_{i}(0), a coarsegrained displacement field u(r, t) can be written as, \({\bf{u}}({\bf{r}},t)=\frac{{\sum }_{i}{{\bf{u}}}_{i}(t)G({\bf{r}}{{\bf{R}}}_{i}(t))}{{\sum }_{i}G({\bf{r}}{{\bf{R}}}_{i}(t))},\) where G(r) is a coarsegrain function and we employ the following Gaussian form: \(G({\bf{r}})=\frac{1}{{(\sqrt{\pi }\sigma )}^{3}}\exp (\frac{{{\bf{r}}}^{2}}{{\sigma }^{2}})\). The strain field ϵ_{αβ} is defined as \({\epsilon }_{\alpha \beta }({\bf{r}},t)=\frac{1}{2}(\frac{\partial {u}_{\alpha }({\bf{r}},t)}{\partial {r}_{\beta }}+\frac{\partial {u}_{\beta }({\bf{r}},t)}{\partial {r}_{\alpha }}).\)
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes that are used to generate results in the paper are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors are grateful to Kyohei Takae and ISSP in the University of Tokyo for supports for hybrid MPI + GPU implementation of FPD method. This study was partially supported by GrantsinAid for Specially Promoted Research (JP25000002 and JP20H05619), Scientific Research (A) (JP18H03675) and EarlyCareer Scientists (JP20K14424) from the Japan Society for the Promotion of Science (JSPS). The numerical calculations were partially performed on SGI ICE XA/UV hybrid system at ISSP in the University of Tokyo.
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H.T. conceived and supervised the project, M.T. performed numerical simulations and data analysis, M.T. and H.T. discussed the results and wrote the manuscript.
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Tateno, M., Tanaka, H. Powerlaw coarsening in networkforming phase separation governed by mechanical relaxation. Nat Commun 12, 912 (2021). https://doi.org/10.1038/s41467020207348
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DOI: https://doi.org/10.1038/s41467020207348
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