Percolation of Phase-Separating Polymer Mixtures - ACS Macro

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Percolation of Phase-Separating Polymer Mixtures Di-Yao Hsu, Che-Min Chou,* Ching-Yen Chuang, and Po-Da Hong* Department of Materials Science and Engineering, National Taiwan University of Science and Technology, Taipei, 10607, Taiwan S Supporting Information *

ABSTRACT: We study the percolation problem in a binary phaseseparating polymer mixture. By well-designed experiments, we can delineate the percolation line on the phase diagram with sufficient accuracy. Our experiments show that the percolation thresholds start from the random percolation limit (Φ ∼ 0.15) located near spinodal point at T → Tc and then converge toward the geometric coalescence limit (Φ ∼ 0.36) with an increase in the quench depth. This apparent percolation difficulty comes about largely from the Rayleigh instability accompanied by large-amplitude, short-wavelength fluctuations during the spinodal decomposition at deeper quench depth. As a result, the broken “rigid” domains tend to pack closely, and the so-called droplet spinodal decomposition occurs. On the other hand, we observe that, between the selectively attractive walls, the surfacedrying percolating phase will break up into droplets prematurely, thereby shifting its percolation line rather considerably. To our knowledge, such an effect is not yet predicted by theory or simulation. Accordingly, a geometric coalescence limit (Φ ∼ 0.36),13 which the percolation necessarily occurs, is proposed, meaning that the SD probably tends to form a packed droplet structure except for those of near-critical compositions. In this case, the coarsening is through the droplet collision and coalescence, that is, the so-called droplet spinodal decomposition (DSD).5−7 Thus, from the practical point of view, the percolation line ϕp can be more useful in identifying the mode of the phase separation. How to characterize the percolation line in the phase diagram? Binder proposed that the percolation line (for one of the phases) can be simply estimated by ϕ′p = ϕ′ + (ϕ″ − ϕ′)Φp, where ϕ′ and ϕ″ are the two coexisting compositions.14 Since Φp is a constant, ϕp(T) must converge toward the critical point of unmixing, but the experiment did not corroborate Binder’s argument; only isopleth was observed.11 In this letter, we address the problem of the percolation in the phase separation, which has been touched very often but not explored thoroughly. Taking into account that there are many factors controlling the details of the phase separation, we chose a symmetric binary polymer mixture having a similar chemical structure. It can be very helpful in a reliable analysis of the phase diagram, especially in comparing the percolation line with the mean-field spinodal curve. On the other hand, since the growth exponent is obtained by analyzing an asymptotic scaling regime, in principle, the crossover from the diffusion to the hydrodynamics should be sharp. This gives us a convenient way to determine the percolation line. But not all of the droplet growth are governed by the diffusion. In addition to those t1/3 growth,2−7 Nikolayev et al. proposed a hydrodynamic

T

he coexistence curve provides us with basic information on the equilibrium composition and the relative proportion of the two coexisting phases under a given set of conditions. However, materials scientists are interested in how to quantitatively relate the phase diagram to the developing domain structure. Certainly, this involves the kinetic aspects of the phase separation, so that many experiments have been carried out to determine the spinodal curve, which subdivides the miscibility gap into the thermodynamically metastable (nucleationgrowth, NG) and unstable (spinodal decomposition, SD) regions.1 As is well-known, the NG usually implies the formation of isolated droplets in a metastable background, whereas the most striking feature of the SD is the interconnected domains caused by coherent composition fluctuations. This difference is also reflected in their coarsening kinetics. For the isolated droplets, the growth is diffusioncontrolled R ∼ t1/3, where the R is the droplet size.2−7 For the interconnected domains, or rather interpenetrating fluid tubes (Siggia’s phrase), the hydrodynamic effects result in a much faster coarsening ξ ∼ (γ/η)t, where ξ is the correlation length of the domains, γ is the interfacial tension, and η is the viscosity.8 However, as examined later on, that the factors controlling the SD structure depend not only on how it came into being, but also on the volume fraction of the minority phase Φ.9 So even in the SD case, the connectivity of the domains must disappear at Φp. This is just the typical percolation problem,10 and Φp is the so-called percolation threshold. Inasmuch as whether or not the percolation occurs is so crucial for the SD, where is its precise location? The random percolation limit (Φ ∼ 0.15) has in the past been postulated as a possible boundary.8 However, the experimental evidence has shown that the percolation region is much narrower.11,12 © 2015 American Chemical Society

Received: September 25, 2015 Accepted: November 14, 2015 Published: November 17, 2015 1341

DOI: 10.1021/acsmacrolett.5b00692 ACS Macro Lett. 2015, 4, 1341−1345

Letter

ACS Macro Letters coalescence mechanism in 0.26 ≤ Φ ≤ 0.36,13 where the flow induced by the droplet collision likely pushes the neighboring droplets and thereby accelerates the next coalescence (also, see Tanaka’s collision-induced collision via flow,5 where R ∼ (γ/ η)Φ1/3t1). In view of this possibility, great care has to be taken on the experiments. The poly(ethylene glycol) [PEG; M̅ w = 16830 and M̅ w/M̅ n = 1.28 (GPC), Sigma-Aldrich Co.] and the poly(ethylene glycolran-propylene glycol) random copolymer [RAN; ethylene glycol content of about 73%, M̅ w = 14040 and M̅ w/M̅ n = 1.51 (GPC), Sigma-Aldrich Co.] were used in this study. The samples have been purified before use in the experiments (Supporting Information, SI-1). The phase diagram was measured by the homemade turbidimetry (SI-2, Figure S1). The phase separation kinetics was carried out by the smallangle light scattering (SALS),15 where the scattering angle θ = 1.16−37.1° corresponding to the scattering vector q = 0.3−14.4 μm−1. The other details of the experiments are described in SI3. In the conventional turbidity experiments (see Figure 1, white solid line), the boundary of the miscibility gap can be

Figure 2. (a) Comparison of scattering patterns taken from different ϕ at 394.2 K when the characteristic peak has grown to qm ∼ 3.5 μm−1. For a more visual comparison, the intensities of the patterns are equalized so that their brightness and contrast are equal. (b) Composition effects on the coarsening kinetics for the PEG/RAN mixtures at 394.2 K. The inset shows the corresponding kinetic exponent.

also R), the growth exponent α can be estimated from qm ∼ t−α (see Figure 2b and the inset). Surprisingly, the observed hydrodynamic exponent is much larger than the Siggia’s value.8 (We shall return to this point later.) Another interesting observation is the dynamic percolation-to-droplet (cluster) transition at ϕ = 0.49.16 However, these have no influence on the aim of the analysis. Indeed, the crossover between the diffusion and the hydrodynamics is sharp. But before ruling out the possibility of the hydrodynamic coalescence,13 the crossover is best viewed as a kinetic transition ϕk for the moment. In order to establish the phase diagram, we need to check whether system obeys the mean-field approximation,17 so that the spinodal curve can be well-defined. Near the critical temperature Tc, the order parameter is written as Δϕ = ϕ″ − ϕ′ = Aεβ, where A is the critical amplitude, β is the critical exponent, and ε = |Tc − T|/Tc.18 Our result shows good agreement with the mean-field value βMF = 1/2 (see Figure 3a) and gives Tc = 403.56 K and ϕc = 0.458 (where ϕc is the critical composition). Further, the interaction parameter χ(T) can be evaluated from the order parameter (see Figure 3b). Finally, the phase diagram together with the kinetic transition is shown in Figure 3c. (The details of the parameter estimation are described in SI-4.) As can be seen clearly from the circled numbers that represent structure (see Figure 3d), the kinetic transition is just the percolation transition. The phase diagram shows the three important features. First, the percolation region is much narrower than spinodal region. Second, it is not symmetric with respect to the diameter, d = (ϕ″ + ϕ′)/2.

Figure 1. Turbidity landscape of the PEG/RAN mixtures in the bulk. The coexistence points ϕ′ and ϕ″ at a given temperature were defined as the intersection points of the two auxiliary curves and the baseline.

approximated by the cloud point, that is, the first steep rise of the line. We consider here a more delicate approach. For the visualization, a series of the turbidity curves over different PEG compositions ϕ was converted into the 3D landscape. We then define a pair of the coexistence points ϕ′ and ϕ″ from the τ − ϕ profile (where the prime and double prime denote the coexisting RAN-rich and PEG-rich phases, respectively). There are two advantages compared to the conventional method: first, it allows the use of a narrow temperature sampling interval for a representation of the continuous coexistence curve; second, the order-parameter analysis can be performed. We next consider the determination of the percolation line. Figure 2a shows the SALS patterns taken from different ϕ at 394.2 K. Each pattern has the same peak position qm and the equalized intensity. Contrary to our stereotypes, we see the socalled “spinodal ring” even in the NG regions. This suggests that the droplet arrangement can also have a high spatial correlation. Since 2π/qm is the characteristic domain size (ξ and 1342

DOI: 10.1021/acsmacrolett.5b00692 ACS Macro Lett. 2015, 4, 1341−1345

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ACS Macro Letters

Figure 4. Temperature dependence of the percolation thresholds. The gray, thick lines are the guide to a reasonable trend in the data.

fields. We first consider the former case. If the fluidity of two phases is slightly asymmetric, Onuki proposed a condition to judge the percolation η′/Φ′ ∼ η″/Φ″.21 Accordingly, the less viscous phase prefers the percolation, even though it is the minority phase, whereas for the more viscous phase, the reverse is true. But here η394.2K ′ = 2.14 Pa·s and η394.2K ″ = 4.30 Pa·s. The remaining possibility is then the external fields. Indeed, we see that the PEG-rich phase is more wettable to the quartz surfaces than RAN one (see columns 1 and 4 of Figure 3d). This is reminiscent of the surface-directed spinodal decomposition (SDSD), which results in the thickening of the wetting layer with time.22,23 However, shifting Φ′p is hard to explain in terms of the SDSD, because the surface thickening of a phase would cause the breakup of itself in the bulk rather than the surface drying one. So what happens for the surface-drying percolatingphase? First, the interface of the interconnected domains is demonstrated to have an almost zero mean curvature.24 It is conceivable that the surface drying would give rise to an excess bending energy (a higher local curvature).25 Analogous to the phenomenon of “curvature-induced Rayleigh-like instability” in swollen cylindrical micelles,26 it may be rationalized that the percolating RAN-rich domains near the surface is likely to break up, if its tube radius is greater than the radius of curvature at the surfaces. Our study provides a fairly clear picture in the percolation of the phase-separating systems. We explain that the isoplethic percolation line is actually a consequence of the crossover from the geometric percolation of the “soft” domains to the percolation difficulty of the “rigid” droplets. Since it is still a challenge to develop an appropriate theory, some concluding remarks are in order. Initiation of Breakup of Percolating Fluid Tube. We attribute the narrow percolation region to the Rayleigh instability, while the characteristic time of the instability can be estimated to be τins ≈ (ρr30/γ)1/2, where r0 is the radius of the fluid tube.20 This means that the coarsening process would weaken the Rayleigh instability. We may thus infer that the fastest growing mode of the instability occurs right at the onset of the late-stage coarsening. This explains why the DSD phenomena are commonly observed, rather than the dynamic percolation-todroplet transition for later times (see column 3 in Figure 3d). On the other hand, the growing rate of the SD fluctuations can

Figure 3. (a) Double-logarithmic plot of Δϕ against ε. (b) Temperature-dependence of χ parameters. (c) Phase diagram of the PEG/RAN mixture. (d) Comparison of the developing domain structures observed at the conditions marked on the phase diagram by the circled-number symbols.

Third, except near the critical point, the percolation lines are isoplethic, consistent with the previous finding.11 To capture these behaviors, we calculate the percolation threshold for each phases by Φp = 1/2 − |ϕk − d|Δϕ and plot in Figure 4. Let us first look at Φ″p in detail. It starts from the random percolation limit near the spinodal curve and then bends toward the geometric coalescence limit with the quench depth. We seem to face a correlated-percolation problem,19 but it is not so. The observed “percolation difficulty” is very likely related to the Rayleigh instability of a fluid tube, which the elongated minority phase breaks up into droplets due to the interfacial perturbations.20 Because such an instability is proportional to the interfacial tension but inversely proportional to the tube radius, it would be largely enhanced by the deeper quenches where the SD induces large-amplitude (high interfacial tension), short-wavelength (small tube size) fluctuations. On the contrary, since there is diffuse longwavelength fluctuations for shallower quenches, the percolation is only dependent on the geometrical connectivity. Turning now to Φ′p, it is shifted to near Φ = 0.5; namely, the percolation becomes very difficult for the RAN-rich phase. This highlights some possible differences in the properties of the two phases, like the relative fluidity and the response to external 1343

DOI: 10.1021/acsmacrolett.5b00692 ACS Macro Lett. 2015, 4, 1341−1345

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ACS Macro Letters

R are equivalent. This is also the reason why R ∼ q−1 m holds. Obviously, these two length-scales must be decoupled by the degeneracy of the spatial correlation as the coarsening proceeds. Further, if the hydrodynamic coalescence does happen (despite a theoretical possibility),5,13 taking into account the locality of the flow fields, the two growth laws, that is, slow diffusion t1/3 and fast hydrodynamics t1, must be expected to proceed simultaneously, and as a result, a bimodal size distribution will be observed. One final point is due purely to the extinction of the scattered lights by spheres large compared to the light wavelength, that is, the so-called anomalous diffraction problem.37 All of these suggest a failure of the Furukawa’s scaled structure factor. We are now developing an exact scattering modeling for a quantitative analysis of the DSD structures. The results will be published in the near future. Vanishing of Spinodal Singularity. The spinodal point as a kinetic singularity is known to be satisfied for the mean-field system with long-range interactions, whereas for a system with short-range interactions the singularity is smeared out (i.e., the NG and the SD become indistinguishable).9 Although the present system indeed shows the mean-field behavior, the broad DSD region seems to suggest the vague metastability in the SD. It is worth mentioning the work by Balsara et al.,38 who have shown a nonclassical nucleation behavior, that is, fluctuation rather than abrupt change, in the phase separating polymer mixtures. More importantly, no singularity was observed, when the spinodal curve is crossed. In our view, such non-self-consistent results may be due to the highly cooperative and correlated characteristics of the long-chain molecules. Altogether, the present results thoroughly modify the specious arguments in the past and certainly provide some new insights into the general concepts of the percolation in the phase separation.

be characterized by the apparent diffusivity Dapp = Dcϵ, where Dc is the cooperative diffusivity, and ϵ = (χ − χs)/χs is the thermodynamic driving force and χs is the χ parameter at spinodal point.27 We are here concerned with whether the interfacial perturbations can follow the quick change in the SD fluctuations. The characteristic time of the SD is defined as τsd ≈ ξ2/Dapp. If ξ and r0 are of the same order, then we expect an instability likely to occur at τins ≤ τsd, that is, the interfacial perturbations are much faster than the SD composition fluctuations. Ultrafast Hydrodynamic Coarsening. We have noticed that the power-law analysis for the percolation region yields unexpectedly large (α up to 2.5) and ϕ-dependent exponents. Basically, the hydrodynamic coarsening involves the interfacial-tensiondriven squeeze flow in the tube, while the fluid motions depend on the relative importance of viscous dissipation, fluid inertial, and gravity.8,28,29 If ξ is much smaller than the capillary length, Sc = (γ /g Δρ)1/2 where g is the gravitational acceleration, and Δρ is the density difference between the two phases, gravity can be neglected.20 In this case, the Reynolds number, Re ≈ ρνξ(t)/ η where ν is the fluid velocity, is used to identify the flow behavior. For Re < 1, the well-known Siggia’s mechanism (viscous scaling) is satisfied;8 for 1 < Re < Rec, where Rec is the critical Reynolds number in which the turbulence takes place, Furukawa’s inertial scaling ξ ∼ (γ/ρ)1/3 t2/3 would be expected;28,29 for Re > Rec, taking into account the turbulent remixing of the fluids, Grant and Elder speculated that α is no larger than 1/2,30 and Perlekar et al.’s simulation showed that the coarsening process is arrested as the system attains a steady state.31 Obviously, these decreasing exponents do not suffice as explanation. In fact, our Reynolds number always remains much smaller than unity (estimated from η394.2K = 3.22 Pa·s, ρ394.2K = avg avg 3 1.048 g/cm and ξ ∼ 10 μm after about 30 s), so that the estimation of the capillary length then becomes important. Although the density difference does exist (Δρ394.2K ∼ 0.017), the interfacial tension still await measurement by appropriate techniques (e.g., the pendant drop method32). Before γ data can be available, it is worth mentioning that some of the possible gravitational effects on the hydrodynamic coarsening have been proposed in the literature. Goldberg and Huang showed a continuous change in α (between 0.3 and 2) of which they attributed to gravity.33 Moreover, Chan and Goldburg also reported an anomalously fast dynamics α = 3 ± 0.5 (which was derived indirectly from the analysis of Porod’s law) accompanied by a gravity-driven flow.34 In the more recent experiment, by a series of 3D confocal images, Aarts et al. presented how the interconnected domains start to collapse under gravity.35 Although the gravitational scenarios have some experimental support, a proper theory is not yet available. However, taking into account Re ≪ 1 for the present system, if the gravity-driven flow exists, it will occur well before the inertial regime. Structure Factor of Droplet Spinodal Decomposition. Inasmuch as the percolation region is much narrower than the early estimate, this will inevitably heighten the importance of the DSD. Unfortunately, apart from a customary Furukawa’s scaled structure factor,36 it is almost blank! Although no one denies the plausibility of Furukawa’s dynamical scaling hypothesis, that is, the SD can be scaled by the single time-dependent length, we believe that it does not hold here. There are several reasons. First, the DSD structure forms in a nearly close-packed way at the start, and thus the correlation length ξ and the droplet size

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.5b00692. Sample purification, turbidimetry setup, other details of the experiments, and parameter estimation (PDF).

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Fax: +886-2-27376544. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge support from the MOST of the Taiwan under Contract MOST-101-2221-E-011-031-MY3. REFERENCES

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