Note pubs.acs.org/Macromolecules
Anomalous Phase Separation Dynamics under Asymmetric Viscoelastic Effect: Where Fluidic and Elastic Properties Meet Weichao Shi, Jingjing Yang, Wei Liu, Lina Zhang, and Charles C. Han* Beijing National Laboratory for Molecular Sciences, Joint Laboratory of Polymer Science and Materials, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100190, China S Supporting Information *
V
In the previous work,15 we found that the crystallization of PEO would be frustrated by introducing even a minor amount of PMMA. The large PMMA chains may maintain well entangled structure in this case. We proposed the slow relaxation of PMMA may play a significant role as additional interface energy to prohibit the crystallization of PEO. In this study, we carried out experiment in PMMA/PEO blend with fixed weight ratio at 1/9, where crystallization is totally screened out on experimental time scale (Supporting Information, part B). Phase separation dynamics is fast for a shallowly quenched sample (120 °C). In scattering experiment, the initial growth mode locates at a small q and grows exponentially before the growth mode shifts to even lower qs (Figure 1a). In the late stage of phase separation, we note that a I ∼ q−4 power law emerges in the large q region (inset in Figure 1a), which indicates the formation of sharp interfaces on small length scales. Here two points should be noted: (1) The optimum growth mode locates at a much smaller q-region than that reported in dynamically symmetric blends. In dynamically symmetric blends, the initial growth mode usually locates on the order of molecular sizes (usually tens of nanometers).16,20 In this study, the initial growth mode appears at about 1.8 μm−1. Hashimoto also reported the initial optimum growth mode at about 3.5 μm−1 in an asymmetric polymer solution of polystyrene (Mw = 5.48 × 106g/mol) and dioctyl phthalate.21 (2) The concentration growth rate was obtained via Cahn− Hilliard relation. The drastic drop of the growth rate curve demonstrates the depression of the concentration growth at large q region (inset in Figure 1b). There is not a linear relationship from the conventional Cahn plot of Γ(q)/q2 versus q2 (Figure 1b). For an intermediate quench depth to a lower temperature (below 80 °C), the scattered intensity shows a stepwise growth. Here we show the time-resolved scattering curves at 40 °C. In the initial period, there is a quick growth of scattered intensity to small amplitude at a relatively large q (Figure 2, parts a and b). This conveys that there are only mild concentration fluctuations even when the blend is highly unstable. In the subsequent long period, the scattered intensity fluctuates around a certain value and the blend seems to be in a “frozen” stage (Figure 2, parts c and d). After that, there is a second-step
iscoelasticity is a distinguished property of polymers. In general, polymers behave like fluids to a slow perturbation and show elastic responses to a fast perturbation.1,2 The viscoelastic property may also play an important role in phase transitions.3−15 In the conventional concept of polymer phase separation kinetics, equivalent viscoelastic properties of component molecules are usually assumed implicitly.16 The early stage of concentration growth usually occurs on molecular dimension, which can be simply described by the well-known Cahn− Hilliard equation with an exponential relation:17 I(q , t ) = I(q , 0) exp( −Γ(q)t )
(1)
Here I is the scattered intensity and q the wave vector. Γ(q) = L(q)q2(r0 + Cq2) is the growth rate, where L(q) is the Onsager coefficient, r0 indicates thermodynamic instability and C the interfacial contribution.16 However, such blends with dynamic symmetry are really minor in practice. Polymer mixtures with different viscoelastic properties, which are called dynamic asymmetry, are the more common cases in nature. Recent experiments6−15 and theoretical considerations3−5 have given interesting results on dynamically asymmetric phase separation. But the mechanism is still far from fully understood. In this study, we revealed the anomalous phase separation dynamics at different quenches and tried to give a unified explanation that bridges the fluidic and elastic properties, when thermodynamic instability competes with extremely large dynamic asymmetry. This work is potentially helpful for understanding and controlling morphology and functional properties in materials’ science. Dynamic asymmetry usually arises from two factors: (1) contrast in molecular sizes (polymer solutions) and (2) friction contrast between elementary units (polymer constituents with different glass transition temperatures Tg). Usually the second factor is much more effective to determine the motion of a polymer chain. In this study, we deliberately enhance dynamic asymmetry combining the above two factors. Experiments were carried out in a binary blends composed of slow poly(methyl methacrylate) (PMMA, Mw ≅ 100 kg/mol and Tg ≅ 116 °C) and fast poly(ethylene oxide) (PEO, Mw ≅ 24 kg/mol and Tg ≅ −60 °C). The polydispersity for PMMA and PEO are 1.78 and 1.52, respectively. The blend has an upper critical solution temperature at 135 °C, corresponding to 20% weight fraction of PMMA.14,15,18 The sample preparation is given in the Supporting Information (Part A). The detail of the scattering facility was reported elsewhere.19 © 2013 American Chemical Society
Received: January 6, 2013 Revised: March 5, 2013 Published: March 15, 2013 2516
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(Figure 2, parts e and f). In the late stage of phase separation, we note that the I ∼ q−4 power law emerges first from the small q region while I ∼ q−2 power law stays in the large q region after the initial growth (inset in Figure 2e). In a normal phase separation, the initial growth mode continuously shifts toward the lower q region with growing intensity. In this study, the initial growth mode stops growing after the initial stage and a new growth mode appears at a smaller q with faster growth rate after a long “frozen” period. The stepwise growth phenomenon emerges at around 80 °C. The initial growth stage and the “frozen” period become longer as temperature decreases (Supporting Information, part C). At 20 °C, the blend shows weak intensity growth in the initial 1000 (± 100) min, and then enters a “frozen” period lasting as long as 9000 (± 1000) min. As crystallization is screened out in the current composition, this stepwise growth should reflect the intrinsic feature of phase separation under large dynamic asymmetry. For a very deep quench (below −30 °C), the phase separation dynamics returns to be fast. The pattern evolution is distinguishably different from the cocontinuous growth at higher temperatures. At −50 °C (Figure 3), small PEO-rich domains tend to nucleate and grow in the PMMA-rich matrix, accompanied by an apparent volume shrinking of the matrix, which is quite similar to the volume phase transition in crosslinked gels.22 The area ratio of the PMMA-rich phase decreases from 65% to 40%. We also found the initial isotropic domains (1 min) may be anisotropically deformed later, which is a direct indication of the strong network stress built up in the system. However, the anisotropic domains relaxed to isotropic network after a long time (10 h). A clear interface boundary can be seen from the very beginning. The interconnected PMMA-rich phase can keep the shape for a long time, even though PMMA is the minor component. By analyzing the volume shrinking with respect to time,7−10 we estimate the characteristic time for phase separation is about 7 (± 1) min. In a usual phase separation, the relaxation does not play an important role on a large length scale (only significant within
Figure 1. (a) Experimental data under time-resolved small angle light scattering at 120 °C. The intensity at 80 min is drawn in a double− logarithmic plot as the inset of part a. (b) Cahn plot. The growth rate is shown as the inset of part b.
of growth which emerges at much lower q region, and the intensity has sharp decreasing dependence with respect to q
Figure 2. Phase separation dynamics monitored by time-resolved small angle light scattering at 40 °C. The inset of part e shows double−logarithmic plot of I versus q at 6000 min. The intensity is plotted in linear form in parts a, c, and e and in semilogarithmic form in parts b, d, and f. 2517
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Figure 3. Morphological evolution monitored by phase-contrast optical microscopy −50 °C.
molecular sizes).16,20 However, if the stress relaxation time of one component is very long, even a small perturbation could produce a large transient stress. The microscopic origin of this stress arises from entanglement of the slow component with finite (but relatively long) time. If the deformation rate (due to phase separation) is slower than relaxation, then the blend behaves like a highly viscous fluid. On the contrary case, the blend shows a soft elastic behavior. Doi and Onuki proposed a two-fluid model to interpret the viscous behavior in dynamically asymmetric systems.4 The twofluid model modified the conventional time-dependent Ginzburg−Landau (TDGL) equation by including the extra contribution from the local stress. In the initial stage, the concentration fluctuation can be linearly described as3 ∂δϕ(q , t ) 4 = −Γ(q)δϕ(q , t ) − L(q)q2α 2 ∂t 3 t ∂δϕ(q , t ′) dt ′ Gs(t − t ′) 0 ∂t ′
∫
plays a dominant role in the formation of network-like structure and leads to dynamic depression in the off-diagonal part of deformation velocity.3,11,12 So Tanaka proposed a growth rate around initial concentration ϕ0 at deep quenches by including a longitudinal modulus:13 ⎡ ⎛ ⎞⎤ 4 R(q) ≅ L(q)q2⎢r0 + Cq2 + ϕ0−2⎜Gb(ϕ0) + Gs(ϕ0)⎟⎥ ⎝ ⎠⎦ ⎣ 3 (5)
Tanaka also pointed out that the bulk modulus builds up earlier and relaxes faster than the shear modulus, but a stepwise growth phenomenon has never been depicted in his simulation.11 In addition, we employ a parameter λ = |ro|/ [ϕ0−2(Gb + 4Gs/3)] to denote dynamical coupling between the real thermodynamic instability and the long-range stress.3 We should note that the unstable blend is “frozen” to show dynamic stability for a long time if λ < 1. A gel-like volume shrinking appears when λ ≫ 1. The viscous fluidic21 and soft elastic effects6−10 on concentration growth were separately reported in dynamically asymmetric blends before. In this study, we additionally report that the competition between thermodynamic perturbation and slow relaxation may lead to a stepwise growth between the two cases. The dominating growth mode in each stage can be originated from different relaxation modulus. We may interpret the anomalous phase separation dynamics in this study as following. Under a shallow quench at a high temperature (120 °C), the blend shows viscous behavior and molecular conformation can quickly adapt to the driving force. The bulk relaxation modulus does not play important role because of weak thermodynamic instability. Only the transient entanglement of PMMA chains in diffusion leads to a dynamically depressed region (ξv in eq 3), which is solely originated from shear relaxation modulus. The fitted viscoelastic length is about 2.91 ± 0.27 μm, which is much larger than the molecular size (Supporting Information, part D). The growth rate is sensitive to the wave vector at small q range (qξv ∼ 1) but approaches to a constant value at large q range (qξv ≫ 1). This is predicted as in the two-fluid model.3,4 Under a very deep quench (−50 °C), it is highly unstable on thermodynamic sense while the entangled PMMA chains behave like a swollen gel. For λ ≫ 1 in this study, the strong instability leads to fast phase separation dynamics via quick cooperative motion of solvent-like PEO molecules.23 The selection of a nucleation−growth type pathway will minimize the total induced network stress as much as possible. Here, the role of bulk relaxation modulus is significantly important and mainly responsible for volume shrinking. The shear relaxation modulus, otherwise, sustains the long-living bicontinuous structure even though PMMA is the minority component. This follows the improved viscoelastic model by Tanaka.12,13 Under an intermediate quench (40 °C), a stepwise growth mode could happen but is unusual and, to our knowledge, has
(2)
Gs is the local plateau shear modulus. α is the dynamic asymmetry parameter, expressed as α = (N1ζo1 − N2ζo2)/ (ϕ1N1ζo1 + ϕ2N2ζo2), where N is polymerization index and ζo friction coefficient for each component. In the symmetric case when α = 0, eq 2 returns to the conventional Cahn−Hilliard theory. Using Maxwell model with Gs(t) = G0 exp(−t/τ), from eq 2, the concentration growth still keeps an exponential relation with time, like Cahn−Hilliard equation, but the growth rate is modified as R(q) ≅ L(q)q2(r0 + Cq2)/(1 + ξv 2q2),
if Γqτ < >1
(4)
ξv = (4/3)Lα η is called viscoelastic length with η = dt′ Gs(t − t′). The molecular diffusion is dynamically depressed within the viscoelastic length. Although the Doi−Onuki model can successfully explain the dynamic depression and the formation of the network structure, however, it cannot explain the phenomenon of volume shrinking at deep quenches.5 Tanaka further developed a viscoelastic model, and pointed out that it was not only a shear modulus but also a bulk relaxation modulus that play key roles in viscoelastic phase separation.7−13 Comparative to the volume phase transition in cross-linked gels, the local thermodynamic instability is partially balanced by the bulk modulus in the concentration gradient. The effective osmotic pressure for diffusion is expressed as πeff = (ϕ((∂f)/(∂ϕ)) − f) − ∫ t−∞ dt′Gb(t − t′)▽ . v(t′),7−13 where v is the diffusion velocity and Gb the bulk stress-relaxation modulus. This is the origin of bulk stress. Volume shrinking cannot take place without bulk modulus.11,12 The shear modulus, however, is not directly related to diffusion. It mainly 2
∫ t0
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growth.25 However, an even earlier growth mode in large-q regions before the “frozen” stage cannot be easily observed, which needs adequate dynamic asymmetry. In most cases, the initial intensity growth by local response is smeared with a background scattering. As we noted, however, a double-mode fluctuation was observed in polymer solutions by sensitive dynamic light scattering.26 A double relaxation was reported in the dissolution process of shear-induced phase separation after cessation of shearing.27 Recently, a double concentration growth was revealed by 2D fast Fourier transform of optical microscopic photographs in the phase separation of polystyrene and diethyl malonate system.9 However, all previous work did not differentiate a stepwise behavior from viscous and elastic behaviors clearly in phase separation. To give a more complete picture on dynamically asymmetric phase separation, another phenomenon appears in the dynamic process when the induced network stress cannot be easily relaxed by concentration variation. The accumulated internal stress may go beyond the mechanical stability of the material and lead to fracture of the gel-like matrix. This is reported as fracture phase separation recently.10 In our opinion, we think the fracture phase separation is more likely to occur in the frozen stage of the stepwise regime, where concentration growth is depressed (Supporting Information, part E). In conclusion, we reported the concentration growth between the viscous and elastic behaviors in phase separation. A more complete picture on phase separation behavior is schematically shown in Figure 5 based on previous study.15
not been reported before. Then, the growth rate of each period is shown in Figure 4. In the initial growth period, the growth
Figure 4. q-dependent growth rates in the initial growth period (a) and the second growth period (b).
rate increases with wave vector and the maximum growth mode seems located in the high q range. The growth rate in the “frozen” stage is near and/or around zero and is not given here. In the second growth period, the growth rate decreases dramatically with wave vector and reaches a bottom in the high q range. The maximum growth mode seems to be in a much smaller q region which is out of the scattering window in our measurement. Phenomenologically, this stepwise growth can be depicted in a form which combines eq 3 and 5: R(q) ≅ L(q)q2[r0 + Cq2 + ϕ0−2(Gb + 4Gs /3)] /(1 + ξv 2q2)
(6)
We note there are three featured lengths (Supporting Information, part D): a viscoelastic length ξv arising from the shear modulus, a network instable length ξn from the bulk modulus when R(q) = 0 in eq 6, and a thermal instable length ξth under dynamic symmetry when Γ(q) = 0 in eq 1. The values of ξv and ξn become much larger than ξth with increasing dynamic asymmetry. We attribute this stepwise growth to the different relaxation modes of bulk stress and shear stress. As quench depth increases, it is more unstable on local scales (r0 more negative). But slow molecular relaxation may depress concentration fluctuation in the concentration gradient. As this bulk modulus relaxes, the concentration growth may be accommodated locally on the instable length ξn. So, the initial small amplitude growth should be controlled by the competition between thermodynamic instability and bulk stress relaxation. Meanwhile, the slow molecular relaxation induces a transient network with a much larger viscoelastic length. This leads to the depression of molecular diffusion for a long time, which is the origin of the “frozen” period. The second growth mode mainly corresponds to the relaxation of the shear modulus on the viscoelastic length ξv. From a microscopic picture, intuitively, the dynamic evolution at an intermediate quench may be still accomplished by the interdiffusion of PMMA and PEO, but the slow PMMA chains may need a long time to fully relax (the “frozen” stage).24 However, at −50 °C, the thermodynamic instability may overwhelm the slow relaxation of entangled PMMA− PMMA pairs, and the dynamic evolution is accomplished by the one-sided fast motion of mobile PEO, similar to a volume phase transition of physically linked PMMA network in solution. Although this stepwise growth should be a general rule for asymmetric polymer blend systems, it may not be easily captured without large dynamic asymmetry. There have been reports on a short transient “frozen” stage before apparent
Figure 5. Phase diagram of the PEO/PMMA blend in this study. The solid curves were directed with the eyes based on the experimental data points. The violet dashed line is the static symmetry line which corresponds to the equal volume fraction of the equilibrium phases. The green dotted line indicates a limiting case that the minor PMMArich phase can transiently form percolating network structure. The red stars correspond to the viscous regime, the yellow ones to the stepwise regime, and the blue ones to the elastic regime.
Here the static symmetry line is directed by glass transition line at the low temperatures, compared with the usual case that is solely directed by the binodal line. We propose the fluidic, viscous and elastic properties may well meet more generally in the phase separation of polymer blends, depending on viscoelasticity and thermodynamic instability (Supporting Information, part F). The roles of shear and bulk stresses intervene into the phase separation as temperature lowers. We 2519
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(24) Watanabe, H.; Chen, Q.; Kawasaki, Y.; Matsumiya, Y.; Inoue, T.; Urakawa, O. Macromolecules 2011, 44, 1570. (25) Clarke, N.; McLeish, T. C. B.; Pavawongsak, S.; Higgins, J. S. Macromolecules 1997, 30, 4459. (26) Hair, D. W.; Hobbie, E. K.; Douglas, J.; Han, C. C. Phys. Rev. Lett. 1992, 68, 2476. (27) Koizumi, S.; Inoue, T. Soft Matter 2011, 7, 9248.
hope our research will attract more attention on further careful examination of phase separation in dynamically asymmetric systems, especially in the initial stage.
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ASSOCIATED CONTENT
S Supporting Information *
Sample preparation, phase behavior at 20 °C, small angle light scattering data, initial growth length, fracture phase separation, and fluidic, viscous, stepwise, and elastic behaviors in phase separation. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*Telephone: +86 10 82618089. Fax: +86 10 62521519. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by the National Basic Research Program of China (973 Program, 2012CB821503) and National Natural Science Foundation of China (No. 50930003).
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REFERENCES
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