Impact of complex topology of porous media on phaseseparation of binary mixturesRyotaro Shimizu and Hajime Tanaka*
Porousmaterials, which are characterized by the large surface area and percolated nature crucial for transport, play animportant role inmany technological applications including battery, ion exchange, catalysis,microelectronics,medicaldiagnosis, and oil recovery. Phase separation of amixture in such a porous structure should be strongly influenced byboth surface wetting and strong geometrical confinement effects. Despite its fundamental and technological impor-tance, however, this problem has remained elusive for a long time because of the difficulty associated with thecomplex geometry of pore structures. We overcome this by developing a novel phase-field model of two coupledorder parameters, the composition field of a binary mixture and the density field of a porous structure. We find thatdemixing behavior in complex pore structures is severely affected by the topological characteristics of porousmaterials, contrary to the conventional belief that it can be inferred from the behavior in a simple cylindrical pore.Our finding not only reveals the physical mechanism of demixing in random porous structures but also has an impacton technological applications.
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INTRODUCTIONWetting effects on phase separation in random porousmaterials of me-soscopic scale (1–18) are of great technological interest because they arerelevant to many important industrial applications, for example, phaseseparation in porousmaterials can be used for the coating of the internalsurface of a porous material (17), and the stability of phase-separateddomain structures is a key to oil recovery processes (16). Thus, thereis a high demand for the basic understanding of phase-separation be-havior in a complex pore geometry. However, there are fundamentalquestions that remain to be answered.
The difficulty originates from the complex dynamical interplay be-tween phase separation, surface wetting, and geometrical confinement.For example, phase separation in porous materials cannot proceed in-definitely and is trapped in ametastable state, unlike phase separation inbulk. The key question is, “Which pore characteristics affect the kineticpathway of phase separation and its arrest to a final metastable state?”The two important characteristics of pores are surface wettability andgeometrical structure. The former, which characterizes the friendlinessof the pore surface to the components of amixture, can be grouped intothree cases: neutral, perfect, and partial wetting (see Fig. 1). On the otherhand, the latter can be characterized by its size and topology. The sizeeffect is rather well understood. It can be characterized by the relation-ship between two key length scales: the characteristic pore size l (here,we define it as the pore diameter) and the domain interface thickness,which is comparable to the correlation length of composition fluctua-tions x (strictly,
ffiffiffi2
px). The correlation length x increases when
approaching a critical point Tc as x = x0(|T − Tc|/Tc)−n, where x0
is the molecular size and n is the critical exponent [≈0.63 for a three-dimensional (3D) system]. Thus, x is large near a critical point or forpolymers and is usually in the range of 1 to 100 nm. Then, the situationcan be classified into two cases in terms of the ratio of l/x: (i) For l≤ x,the randomness in the pore geometry and/or surface chemistry applies aquenched random field (pinning) to the fluid, resulting in separationinto domains larger than the pore size. Thus, the randomness isexpected to result in random field Ising model behavior (19, 20), that
is, the collective behavior of magnetic spins under random magneticfields. (ii) On the other hand, for l ≫ x, a simple pore model is pro-posed to work: Themetastability and the slow kinetics of the coarseningprocess can be explained by the geometric confinement of phase-separated domains in a cylindrical pore (21–24). Now we have a con-sensus that case (ii) is relevant formost practical applications. However,this is not the end of the story, because there is a possibility that not onlythe size but also the complex topology of a porous material matters.However, the impact of complex topology of a porous material onphase-separation behavior has not been addressed to date because ofthe following reasons.
One reason is the experimental difficulty in studying phase-separationbehaviors in a mesoscopic pore, particularly in random porous mate-rials, because not only is the pore radius of Vycor glasses too small forthe direct real-space observation but also extracting detailed real-spaceinformation from scattering experiments is extremely difficult. Therehave been detailed experimental studies on phase separation in rathermacroscopic pores, which allows real-space observation (24). Only re-cently, phase separation in more complex geometries has been studiedexperimentally: Kanamori et al. (17) observed coarsening dynamics ofphase-separating mixtures using the scanning electron microscopytechnique. Synchrotron x-ray tomographymeasurements were also ap-plied to phase separation of oil/watermixtures in porousmaterials (25, 26).However, there have been few experimental studies that can addressthe role of pore topology in the demixing behavior. Another reason isthe difficulty in theoretical and numerical studies because of thecomplex nature of pore structures. Some numerical works revealedphase separation under surface fields for simple geometries such as athin film or a simple cylindrical pore bymeans of the phase-field model(6, 12, 27) and molecular dynamics (MD) simulations (12, 28). Howev-er, actual porous materials have much more complicated labyrinth-likepore structures than a simple cylindrical pore, whichmakes it difficult toreveal how phase-separated domains wet and grow in such complexconfining geometries. There were some numerical studies on phase de-mixing in complicated geometries by the phase-fieldmodel (29, 30), thelattice Boltzmann method (31), and MD simulations (32). However,these studies are limited to phase demixing in 2D porous structuresand to the rather early stage of demixing. Because there are manymetastable states in porous materials under an influence of wetting,
it is crucial to understand a nonequilibrium kinetic pathway in whichphase-separated domains wet walls and coarsen in random porousmaterials. It is expected that the spatial dimensionality and the topologyof porousmedia should have significant impacts on the kinetic pathwayof phase separation and the resulting structural formation, but theirroles have not been understood yet even at a primitive level.
Here, we address this problem by numerically studying phase sepa-ration in random porous materials in both 2D and 3D. To this end, wedevelop a novel phase-field method, which allows us to deal with a va-riety of porous structures. We reveal an intrinsic difference in patternevolution between 2D and 3D porous structures and the crucial role oftopological characteristics of 3D porous materials in the selection of thefinal phase-separated structure.
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RESULTSSimulation methodWe consider a binary mixture immersed in a porous material, whosesurface interacts with the mixture via surface wettability. To study thephase-separation kinetics in porous materials, we use a phase-fieldmodel described by two order parameters, which represent thecomposition field of the binary mixture, y(r), and the density field re-presenting the porous material, r(r), with their coupling strength, thatis, wettability, h1. Here, r is the position vector. Throughout this article,we consider a symmetric (50:50) binary mixture of the averagecomposition �y ¼ 0:5. The porous structure is represented by the den-sity field of r(r) and its surface expressed by a smooth but sharpinterface. The details of this method are explained in the MaterialsandMethods. This method has several merits: We can produce variousporous structures by using either a physical process of demixing or amathematical function that represents various regular pore structures(33). Furthermore, we can introduce hydrodynamic degrees of freedom(27) straightforwardly in the samemanner as our fluid particle dynam-
Shimizu and Tanaka, Sci. Adv. 2017;3 : eaap9570 22 December 2017
ics method to simulate colloidal suspensions (34), although here, weconsider only diffusion as the transport mechanism. Furthermore, wenote that our phase-field model can incorporate charges, additionalcomponent, and liquid-crystalline order in a straightforward manner.
We check the validity of our simulation method by studying a well-known problem of wetting of a liquid droplet on a flat solid substrate,as shown below. We prepare complex porous structures by freezingspinodal decomposition concerning the r(r) field at a certain timing,which is similar to the preparation method of the so-called Vycor glass,which is formed by spinodal decomposition of a borosilicate glass. Theporous structure is characterized by the volume (area) fraction of thesolid porous matrixF (that is, the mean surface curvatureH of the po-rous material) and the average pore size l.
Results for a simple wetting geometryFirst, we check the validity of our method by studying wetting be-haviors of a droplet on a flat wall. As the solid wall, we set the wall po-sition to be y = L1 and y = L2, and the interfacial profile of the wall isgiven by rðrÞ ¼
ffiffi3
p2 ½1þ tanhð�y þ L1Þ� þ
ffiffi3
p2 ½1þ tanhðy � L2Þ�. As
an initial condition, we put a droplet on the bottom wall with the con-tact angle q ¼ 90°. Thus, the initial profile is
yðr ¼ ðx; yÞÞ ¼
12ðtanhððy � L2Þ=xyÞ � 1Þ
ðL2≤ y≤ LyÞtanhððjr� rcj � RdÞ=xyÞ
ðL1≤ y≤ L2Þ0
ð0≤ y≤ L1Þ
8>>>>>>><>>>>>>>:
ð1Þ
Here, the center of droplet rc = (L1, Ly/2), and Rd is the radius of thedroplet. In this simulation, we set Lx = Ly = 128 and L1 = 20, L2 = 108,andRd = 30. Figure 1A shows the stationary wetting behavior of a drop-let for the wetting parameters h1 = 0, −2, and −10, respectively.We notethat h1 = 0 or −10 should correspond to the neutral wetting, that is, theequilibrium contact angle of 90°, and the complete wetting, that is theequilibrium contact angle of 0°, respectively. This is confirmed in Fig. 1A,indicating the validity of our simulation method. We also show theequilibrium spatial profile of the two fields, y and r, in the directionperpendicular to the wall in Fig. 1B.
Neutral wetting caseDemixing in 3D porous materialsFirst, we consider the case of neutral wetting, that is, h1 = 0 (q ¼ 90°), ina 3D symmetric bicontinuous porous structure (H = 0, that is,F = 1/2).Figure 2 shows the time evolution of the concentration field y in theporous structure. Here, the black interface shows the surface of the po-rous structure by the iso-interface of r = 0.8. The blue and green inter-faces represent the iso-surfaces of the A and B phases, y = 0.6 andy =−0.6, respectively. Initially, wetting layers are not formed on the wall. In-stead, small tortuous domains contact the wall with the contact angle of≈90° (see the image at t ≈ 40 of Fig. 2A). The temporal change of thestructure factor S(k) (where k is the wave number) is shown in Fig. 3A.Because our pore size l is much larger than the correlation length xof the y field (l/x ≈ 20), the peak wave number k1 of S(k) is about thesame as that in bulk in the initial stage of phase demixing, and theinitial growth of composition fluctuations is not affected so seriouslyby the spatial confinement. Accordingly, the growth exponent seen inFig. 3B is not so different from the well-known growth exponent in
A
B
Fig. 1. Wetting of a droplet on a flat wall. (A) The wetting behavior of a dropleton a flat wall for different wetting parameter h1. h1 = 0, −2, and −10 from left toright. (B) The cross section of a droplet and a wall along x = Lx/2 for the case ofneutral wetting (h1 = 0).
bulk, 1/3 (35). We note that Bailey et al. (36) performed pressure-quench experiments of binary mixtures confined in dilute silica gelsand reported the similar domain growth exponent of 1/3.
As domains grow, they start to exceed the pore size and, thus, bridgethe pore structure around t≈ 500. However, even after the formation ofthese plug structures, domains keep growing, as can be seen in Fig. 2.Themotion of each interface of disconnected domains is determined by
Shimizu and Tanaka, Sci. Adv. 2017;3 : eaap9570 22 December 2017
the difference of the curvature of neighboring domain interfaces. Thecurvature difference between surfaces of neighboring plugs leads tothe difference in the chemical potential between them, which inducesthe diffusion flux from a domain of high interface curvature to that of alow one. This is basically the same as the evaporation-condensation (orLifshitz-Slyozov-Wagner) mechanism in bulk phase separation (35). Inthe bulk case, the domain interface curvature of disconnected droplets is
A CB
Fig. 2. Time evolution of the concentration field for a 3D random symmetric porous structure in a neutral wetting case (h1 = 0). (A) Two demixed phases [the A(green) and B (blue) phases] together with the porous structure surface (black). See also movie S1. (B) Only the A phase but without the wetting layer formed on thesurface of the porous structure by drawing the iso-interface of y = 0.6 in the pore space where r < 0.173. (C) Only the B phase by drawing the iso-interface of y = −0.6.Here, the green and blue interfaces represent the iso-concentration surface of y = 0.6 for the A phase and that of y = −0.6 for the B phase, respectively.
A B
C D
10
0.1 1 102 104
0.1
0.1
1
110–2
0
10–12
10–6
10–4
10–6
10–8
S(k)
K1(t)
K1(t)
S(k)
k t
0.1 1 102 104 106k t
t –1/3
t –1/3
t = 10,000
t = 10,000
t = 10t = 100t = 1000
t = 10t = 3
t = 100t = 1000
t = 100,000Porous material
t = 100,000Porous material
h1 = 0
h1 = –2
h1 = –5
h1 = –10
h1 = 0
h1 = –2
h1 = –5
h1 = –10
Fig. 3. Structural evolution in k-space for a binary mixture in porous structures. Time evolution of S(k) in a neutral wetting (h1 = 0) case (A) and the first moment ofwave number k1 for various wettability (B) for a 3D symmetric porous structure. Time evolution of S(k) in a neutral wetting (h1 = 0) case (C) and the first moment ofwave number k1 for various wettability (D) for a 2D symmetric porous structure. For both 2D and 3D, the characteristic wave number k1 decreases as t−1/3 in theintermediate stage for h1 = 0 but more slowly for h1 ≠ 0.
inversely proportional to the droplet radius, and the diffusion continu-ously takes place from smaller to larger droplets. Unlike this case, in aporousmedium, the curvature of the domain interface is determined bythe confining pore-wall structure and independent of the domain size(or the plug length). Thus, the coarsening rate, which is significantlyslower than that in bulk due to the confinement, further slows downwith time, and, eventually, the coarsening almost stops around t ≈100,000 when the interface curvatures of all the plugs become almostidentical (see Fig. 3B). We may say that the domain structure in a po-rousmedia self-organizes to attain a configuration of homogeneous do-main interface curvature. This feature can be more clearly seen in the2D case (see Fig. 4A).
Next, we discuss a very interesting phase-separation structureformed under a neutral wetting condition. To see the structure andconnectivity of each of the phase-separated two phases separately, weshow in Fig. 2 (B and C) only the A (green) and B (blue) phases bythe iso-surface of y = 0.6 and y = −0.6, respectively, for the inside ofthe pore region where r < 0.173. From these images, we can clearly seethat a 3D percolated network structure of each phase is formed in thebicontinuous pore structure while the plug state blocks each pore. Thatis, two percolated network structures are simultaneously formed, thanksto both the symmetry provided by neutral wetting and the symmetry ofthe pore structure itself (F = 0.5, orH = 0). Because the curvature of thedomain interface is minimized when it is located in a junction part ofthe porous network structure, the domain structure is self-organizedinto this interesting structure by the evaporation-condensation mecha-nism.We note that this bicontinuous double-network structure in a po-rous material is formed to minimize the domain interface energy.Reflecting that the characteristic length of this network pattern is abouttwice of that of the porous structure, we can see a distinct peak in thestructure factor at the half of the peak wave number of the porous struc-ture (see Fig. 3A).
This unique double-network phase-separation structuremay be use-ful formany applications, for example, for coating its inner pore surfacesuch that the two percolated surfaces have distinct wetting properties.Furthermore, the percolated nature of each phase should provideunique transport properties.Demixing in 2D porous materialsTo study the role of the dimensionality in pattern formation, we alsoperformed numerical simulations under the same condition as theabove 3D case for a 2D symmetric porous structure (F = 1/2).We show
Shimizu and Tanaka, Sci. Adv. 2017;3 : eaap9570 22 December 2017
the time evolution of phase separation in 2D in Fig. 4A, where the redand white domains show phase-separated domains and the black re-gions represent the porous matrix. The temporal changes in S(k) andk1 are also shown in Fig. 3 (C and D, respectively). The basic behaviorsin 2D are similar to that in 3D, but there is a crucial difference in thepattern formed in the late stage between 2D and 3D. In 2D, a networkstructure is immediately destroyed, a plug state is formed, and then thelengths of plugs increase with time by the evaporation-condensationmechanism (see Fig. 4A). In 3D, on the other hand, bicontinuous doublenetworks of the two phases can be formed, as shown in Fig. 2. This showsthat there is an essential effect of the dimensionality of a pore structure onthe connectivity of phase-separated domains. The difference simplycomes from the fact that in 2D, a bicontinuous porous structure canneverbe formed even for the symmetric situation unlike in 3D.
Perfect wetting caseDemixing in 3D porous materialsNext, we consider the case of perfect wetting, that is, h1 = −10 or q ¼ 0°.Figure 5 shows the time evolution of the concentration field in the 3Dsymmetric bicontinuous pore (H=0, orF =1/2), which is similar to thatin Fig. 2A. The black regions show the internal surface of the solid po-rous structure. The green and blue surfaces show the more wettable Aphase and the less wettable B phase, respectively, by the iso-surfaces ofy = 0.6 and y = −0.6. We also show in Fig. 5C only the blue surface ofthe less wettable B phase by the iso-surface ofy = −0.6, whereas in Fig. 5B,we show the more wettable A phase by the iso-interface of y = 0.6 forthe pore region where r < 0.173, as above. Note that the choice of r <0.173 is not to show the surfacewetting layer of the A phase tomake theA phase in the middle part of the pore visible. In the initial stage justafter the quench, the more wettable A phase (y = 1) wets the surface ofthe porous material: In Fig. 5A, we can see that the green surface coversthe black wall around t = 3. Then, the layered tube structure composedof two parts (the surface wetting layer of the A phase and the core tubeof the B phase) is formed around t= 10 (see Fig. 5A). This process of thelayered tube structure formation is a consequence of the surface-directedspinodal decomposition (4, 12). The emergence of a domain structure ismuch faster than for the neutral wetting case (see Fig. 3B). This is becausestrong adsorption of the more wettable phase to the wall accelerates do-main growth.
After t ≈ 10, the internal phase-separated structure is formed andthen domains grow. During this time regime, the temporal change of
Fig. 4. Time evolution of the concentration field in 2D random symmetric porous material. (A) A neutral wetting case (h1 = 0). The black region represents thesymmetric porous structure, and the red and white domains show the A and B phase, respectively. See movie S2. (B) A complete wetting case (h1 = –10). See movie S4.(C) A partial wetting case (h1 = –2).
k1(t) can be described by the relation t−1/3, as seen in Fig. 3B. Thisgrowth exponent is the same as that observed in the growth processof a wetting film on a plain wall. After t ≈ 100, the structure factor ata higher k region does not change. In the late stage, narrow pores arefilled by the more wettable A phase. This process can also be seen as abreakup of the tube configuration of the less wettable B phase, but thepercolated nature of the B phase is still preserved. In this regime, thetemporal change of k1(t) becomes slower and weaker compared to thatin the initial stage, and after t=10,000, the domain coarsening stops (seeFig. 3B). Although the phase-separated domain structure is very differ-ent between the neutral and complete wetting cases, the two phases arepercolated for both cases: the double-network structure for neutralwetting and the double-tube structure for complete wetting.Demixing in 2D porous materialsNow, we turn to the corresponding 2D case under the same completewetting condition (see Fig. 4B). Here, the red and white domains cor-respond to the more and less wettable phases, respectively. The initialdemixing process is similar to that in 3D, as seen in Fig. 4B. After theinitial surface wetting stage, in which the three-layer configuration isformed, the inner tube part becomes unstable and capillary bridgesare formed. In the entire process, the pore surface is always coveredby the wetting layer. Then, the thinning of surface wetting layers andthe growth of capillary bridges take place. Then, a crucial differencefrom the 3D case starts to emerge. Unlike the 3Dporous structure, thereare appendix parts (terminal parts of tubes) in the 2D porous structurebecause there is no bicontinuous porous structure for 2D. The appendixparts of the porous structure gradually get filled with the more wettablered (A) domains in the late stage. This is because the total interfacialarea between the two phases can be reduced by filling appendix regions,which reduces the total interfacial area of the tube parts. Locally, this isdriven by the chemical potential gradient between the interfaces with amore negative curvature of the more wettable A phase filling an appen-dix and the interface of tube parts with a less negative curvature. Thisdiffusive transport process leads to slow but continuous growth of thedomain size, as shown in Fig. 3D. We perform independent simulation
Shimizu and Tanaka, Sci. Adv. 2017;3 : eaap9570 22 December 2017
runs for h1 = 0 and h1 = −10 to see how reproducible the structure for-mation is. In the case of partial wetting, the final domain configurationsare different from run to run, and thus not reproducible. In the case ofcomplete wetting, on the other hand, the final domain configurationsare reproducible. This indicates that the nearly equal domain interfacecurvatures of plug ends can be realized in many ways for neutralwetting, but appendix parts have deterministic roles in the selectionof the domain configuration for complete wetting.
Partial wetting caseDemixing in 3D porous materialsNext, we consider a partial wetting case intermediate between neutraland complete wetting, that is, a case of 0 > h1 > −10. Here, we set h1 = −2(see Figs. 4C and 6, respectively, for 2D and 3D). As in the case ofcomplete wetting, a thin wetting layer and phase-separated stripe do-mains are formed in the initial stage, but the thickness and compositiondifference is smaller compared to the complete wetting case. Then, theless wettable B phase domains are deformed and get contact with thesurrounding pore walls. After this process, there is a crucial differencebetween 3Dand 2D. For 3D, a double-network structure can be formed,although it is not so symmetric compared to the neutral wetting case(see Fig. 6). The temporal change of k1(t) is summarized in Fig. 3Bfor 3D.
As we can see above, the behavior for 0 > h1 > −10 is a complicatedcombination of the behaviors of complete and neutral wetting. The be-havior changes gradually with an increase in the wetting parameter h1from the behavior of complete wetting to that of neutral wetting.Demixing in 2D porous materialsFor 2D, on the other hand, the elongated domains are disconnectedand, thus, the pore structure is divided by disconnected plugs (seeFig. 4C). After this happens, the subsequent growth is similar tothe case of neutral wetting. The temporal change of k1(t) is summarizedin Fig. 3D for 2D. Similar to the above 3D case, the behavior changesgradually with an increase in the wetting parameter h1 from the behav-ior of complete wetting to that of neutral wetting.
A B C
Fig. 5. Time evolution of the concentration field for a 3D random symmetric porous structure in a completewetting case (h1 = –10). (A) Two demixed phases [themore(green) and less wettable (blue) phases] together with the porous structure surface (black). See also movie S3. (B) Only the more wettable phase but without the wetting layerformed on the surface of the porous structure by drawing the iso-interface of y = 0.6 in the pore space where r < 0.173. (C) Only the less wettable phase by drawing the iso-interface ofy = −0.6. Here, the green and blue interfaces represent the iso-concentration surface ofy = 0.6 for themorewettable phase and that of y = −0.6 for the less wettablephase, respectively.
Demixing in off-symmetric porous structuresSo far, we focus on the effects of compositional symmetry on phase de-mixing in a symmetric bicontinuous porous structure (F = 0.5). Next,we turn our attention to the effects of the symmetry of the porous struc-ture. As an example, we study phase separation in an asymmetric po-rous structure (F = 0.65). The results are shown in Fig. 7. We can seethat even for neutralwetting, a double-network structure cannot be formedand theBphase is disconnected. For completewetting, a double-tube struc-ture is formed even in this case. For partial wetting (h1 = −2), the situa-tion is marginal. This indicates that a double-network structure is favoredfor higher wetting symmetry (that is, a more neutral wetting condition)and higher surface curvature symmetry of the solid matrix.
Comparison with scattering experimentsNow, we compare the temporal change of S(k) between our results andthe experimental ones. Lin et al. (37) experimentally studied a water/
Shimizu and Tanaka, Sci. Adv. 2017;3 : eaap9570 22 December 2017
lutidine mixture of a critical composition confined in a Vycor glass. Forsmall-angle neutron scattering experiments, they made contrast-matching between the mixture and the Vycor glass by appropriatelymixing protonated and deuterated water. They interpreted the resultsas a combination of a lutidine-rich wetting layer coating the internalsurfaces of the Vycor glass and random single-phase domains occupy-ing small pores. Our structure factors show the behavior similar to theirresults. However, it is worth mentioning the following possibility: Con-trary to the assumption made in the previous works that the domainsize does not exceed the pore size, we reveal that 3D double-percolatednetwork structures can be formed for a neutral wetting condition.This means that the characteristic size of domains along pores canbe macroscopic. Here, we note that such a situation was experimentallyobserved by Iglauer et al. (25) for sandstones containing a residual non-wetting phase. Their synchrotron x-ray tomography showed that theresidual phase forms large clusters, spanning over many pores. These
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A CB
Fig. 6. Time evolution of the concentration field for a 3D random symmetric porous structure in a partial wetting case (h1 = –2). (A) Two demixed phases [themore (green) and less wettable (blue) phases] together with the porous structure surface (black). (B) Only the more wettable phase but without the wetting layerformed on the surface of the porous structure by drawing the iso-interface of y = 0.6 in the pore space where r < 0.173. (C) Only the less wettable phase by drawing theiso-interface of y = −0.6. Here, the green and blue interfaces represent the iso-concentration surface of y = 0.6 for the more wettable phase and that of y = −0.6 for theless wettable phase, respectively.
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BA C
Fig. 7. Time evolution of the concentration field for an asymmetric bicontinuous porous structure (F = 0.65). (A) Neutral wetting (h1 = 0). (B) Partial wetting (h1 = –2).(C) Complete wetting (h1 = –10).
facts indicate the limitation of the k-space approach and the importanceof real-space observation in revealing the domain structure in acomplete geometry.
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DISCUSSIONIn summary, we have studied phase-separation behavior in mesoscopicrandom porous structures by numerical simulations. This allows us tocompare the time evolution of the structure factor of binary mixturesconfined in a random porous structure between numerical simulationsand experiments for the first time. In the initial process of phase sepa-ration, the domain size grows as et1=3 for both complete and partialwetting conditions. This apparently looks similar to bulk, but thereal-space analysis reveals that the domain orientation is stronglyaffected by the difference in the wettability between the two phases,which cannot be easily inferred from scattering experiments.
We find a significant impact of spatial dimensionality by comparingresults of 3D and 2D. The breakdown of wetting symmetry between thetwo components, that is, a small degree of preferential wetting, leads tothe formation of a wetting layer on the surface of pores. This seriouslyaffects the final domain configuration. Only for a neutral wetting con-dition, the two phases equally wet the pore surface. For 2D, this leads tothe formation of a plug configuration. This is the only way to keep thewetting symmetry in 2D. For 3D, on the other hand, there is a novelconfiguration that allows a system to keep the wetting symmetry, thatis, the formation of double-percolated bicontinuous structures. Thismay provide us with interesting applications. For example, we canput the two distinct surface wetting properties to two channels of per-colated bicontinuous pores by using a coating technique (17). We canalso fill two types of liquids with very different properties so that each ofthem is percolated in a bicontinuous pore structure.
We also reveal a strong impact of complex topology of porous mediaon demixing, whichmakes phase-separation behavior fundamentally dif-ferent from that in a simple cylindrical pore.One crucial difference comesfrom the variationof the pore radius in randomporousmaterials.What isimportant here is that the assumption that the total composition is fixed,which is valid for a simple pore geometry, can be violated locally for ran-domporousmedia. In our system, the overall composition averaged overa system is of course fixed during phase separation.However, because thepore radius varies from place to place, the composition is not conservedlocally and varies spatially even though it is conserved globally. This fea-ture leads to the formation of a much more heterogeneous complexphase-separation structure in random porous media, unlike a rather pe-riodic structure formed in a cylindrical pore (9, 24). Another importantdifference is the tortuous structure of a bicontinuous network pore. In asimple cylindrical pore, domain coarsening stops when the plug sizebecomes comparable to the pore radius. This is because the domaininterface curvatures are simply determined by the unique characteristiclength scale of the tube, that is, the pore diameter, and thus become ho-mogeneous. On the other hand, the random tortuous porous structureallows domains to further coarsen even after exceeding the local poreradius, because inhomogeneous domain interface curvatures due to var-iable local pore radius and the presence of junctions keep the evaporation-condensation mechanism active. Our study shows that these fundamentaldifferences lead to drastic differences in the physical mechanism ofstructural formation between simple and complex pore geometry. Thus,the randomness, topology, and bicontinuity of pore structures are quiteimportant in structural formation and its kinetics, which should be re-levant to many industrial applications.
Shimizu and Tanaka, Sci. Adv. 2017;3 : eaap9570 22 December 2017
Finally, we alsomention a potential application of ourmethod to studyfluid dynamics in a complex porousmedia. For example, the dynamics ofimbibition, by which a wetting liquid is drawn into a porousmedium, is avery important problem in both technological and geophysical viewpoints(38). Because we can easily incorporate the hydrodynamic degrees of free-dom into our method [see the study by Tanaka and Araki (34)], it will beuseful to study fluid transport through a porous media.
MATERIALS AND METHODSA new phase-field simulation method using twoorder parametersWe developed a novel two-order parameter model, which can describethe phase-separation process of binary mixtures in arbitrary porestructures. In this model, we adopted the concentration difference ofbinary mixtures y and the solid-phase composition r as the two rele-vant order parameters. The solidwall was expressed by a smoothly vary-ing field r, and thus, the interface between the solid phase and the emptyphasewas described by the surfacewith a finite thickness. The usage of asmooth interface to represent a solid object was proposed to describe asolid colloidal particle (34). This is a natural extension of the phase-fieldmodel, which is often used for studying crystal growth and melting.However, there was no time evolution of the r field, which just actsas a solid wall for the y field. In this model, the total free energy of asystem is expressed as
Fðy; rÞ ¼ ∫dr f ðy; rÞ þ 12
�Kr þ h1y þ g
2y2
�j∇rj2 þ Ky
2j∇yj2
� �;
ð2Þ
where
f ðy; rÞ ¼ � 12y2 þ 1
4y4 þ 1
2r2y2: ð3Þ
Here, f(y, r) is the free-energy density of the system,which has threeminima: The two of them, (y, r) = (±1, 0), correspond to the two phase-separated phases, whereas ðy; rÞ ¼ ð0; ffiffiffi
3p Þ correspond to the solid
matrix. The second term in the right-hand side of Eq. 2 describes thesurface wetting energy at porous walls. The parameters h1 and g repre-sent the strength of the short-range surface field and the surface en-hancement field, respectively. The third term in the right-hand sideof Eq. 2 is the gradient termof the concentration field.Here, we consideronly a case of short-ranged interactions with the solid wall. In the thininterface limit (xr → 0), the total free energy is expressed as
FðyÞ ¼ ∫dr f ðy; r ¼ 0Þ þ K2j∇yj2
� �þ h1ys þ
g2y2s ; ð4Þ
where ys is the concentration difference at the wall ys = y(0). Then, wedescribed the dynamic evolution of y by the following generalized dif-fusion equation
This is an extension of the so-called model B equation in theHohenberg-Halperin notation (39). We set the transport coefficientof the concentration difference as LyðrÞ ¼ Ly0ð1� r=
ffiffiffi3
p Þ so that itis zero inside the solid matrix.
Using themethod described above, we can deal with any type of wallstructures. Here, we use a randomporouswall structure.We prepared itby numerically solving model B for the bulk spinodal decomposition ofa 50:50 binary mixture and stopping the simulation in the late stage at atime when the structure becomes a desired size (29, 33). This modelporous structure mimics a Vycor glass, which is made by acid leachingof the boron-rich phase of a spinodally decomposed borosilicate glass.The porosity of this system is 50%. The peak wave number of the struc-ture factor of the porousmaterial is kp≈ k1≈ 0.2; thus, the average poresize is about 2p/kp = 31. This is about 30 times of the interfacial thick-ness of domains in bulk.
For simulations, we used the above equations after scaling the timeand space by the interface length, x, and the characteristic diffusion timeof concentration field, t = x2/D. We solved the scaled kinetic equationsby the Euler method with a periodic boundary condition. Inexperiments for a solid pore immersed in a reservoir liquid mixture,exchange with the reservoir can lead to a very slow change in y whenwe vary the temperature, and capillary phase separation can also hap-pen when the pore size is small. In our simulations, on the other hand,the concentration is strictly fixed due to the periodic boundary condi-tion, and the capillary phase separation does not happen. We set thesystem size to be 256 and the grid size to be Dx = Dy = Dz = 1. For sim-plicity, we do not introduce temporal thermal fluctuations. The initialprofile of y is given by y ¼ ðy0 þ DyÞð1� r=
ffiffiffi3
p Þ, where y0 is theaverage concentration difference and Dy is the variance of initial ran-dom composition fluctuations. We set y0 to be 0 and Dy = 0.01. Wedefine the interface between two immiscible fluids as a surfacewhere thevalue of the concentration differencey is equal to 0 and at the same timethe value of the wall profile parameter r is smaller than
ffiffiffi3
p=2.
Calculation of the structure factor S(k) and the peak wavenumber k1To extract the characteristic lengthscale of phase-separated structures,we calculate the power spectrum of y. The structure factor S(k) is ob-tained by spherically averaging the power spectrum of y. Thecharacteristic wave number is calculated as
k1ðtÞ ¼ ∫dkkSðk; tÞ=∫dkSðk; tÞ ð6Þ
Because we set the average concentration difference y0 to be 0, thestructure factor S(k) obtained in this way should correspond to thatexperimentally observed by light scattering or neutron scattering un-der an index-matched condition.
SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/12/eaap9570/DC1movie S1. Movie corresponding to Fig. 2A.movie S2. Movie corresponding to Fig. 4A.movie S3. Movie corresponding to Fig. 5.movie S4. Movie corresponding to Fig. 4B.
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AcknowledgmentsFunding: This work was partially supported by Grants-in-Aid for Specially Promoted Research(25000002) from the Japan Society of the Promotion of Science. Author contributions: H.T.conceived and supervised the project, R.S. performed simulations, and R.S. and H.T. discussed
Shimizu and Tanaka, Sci. Adv. 2017;3 : eaap9570 22 December 2017
and wrote the manuscript. Competing interests: The authors declare that they have nocompeting interests. Data and materials availability: All data needed to evaluate theconclusions in the paper are present in the paper. Additional data related to this paper may berequested from the authors.
Submitted 14 September 2017Accepted 6 November 2017Published 22 December 201710.1126/sciadv.aap9570
Citation: R. Shimizu, H. Tanaka, Impact of complex topology of porous media on phaseseparation of binary mixtures. Sci. Adv. 3, eaap9570 (2017).
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