Morphogenetic Transition in Weak Gelation of Crystallizable Linear

Apr 30, 2012 - Moreover, it comes as a surprise when such a transition exhibits a spinodal singularity that reveals a mean-field-like “mesoscopic”...
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Morphogenetic Transition in Weak Gelation of Crystallizable Linear Polymers Che-Min Chou and Po-Da Hong* Department of Materials Science and Engineering, National Taiwan University of Science and Technology, Taipei, 10607, Taiwan S Supporting Information *

ABSTRACT: Fibrillar networks and spherulite assemblies are the two most frequently observed textures in weak gelation of crystallizable linear polymers. We find such two textures in response to the kinetic distinction between instability/spinodal and metastability/nucleation of the polymer crystallization and prove the morphogenetic transition in between. Moreover, it comes as a surprise when such a transition exhibits a spinodal singularity that reveals a mean-field-like “mesoscopic” phase transition behavior.

he percolation model presents a “physically correct” picture for an understanding of the polymer gelation.1,2 The basic idea is to approach the formation of a topologically disordered network from the randomly assigned two-state property (connected or disconnected between the neighboring polymer chain segments).3 Thus, many experiments have been focused on the microstructures (microcrystal, helical structure, molecular compound, etc.) associated with the cross-links in an imaginary disordered network.4,5 However, the vast observation of diverse, hierarchical gel morphologies shows that the polymer self-organization/assembly seems to be a reasonably realistic description.6−13 As early as 1979, de Gennes has distinguished between strong gelation and weak gelation: a gelation is termed “strong” if it occurs by the percolation way; otherwise, it is a “weak” gelation that corresponds to the progressive freezing of degrees of freedom of the system.14 A typical example would be the localized vitrification of the polymer-rich domain during the liquid−liquid phase separation process, as the temperature is lowered below the polymer glass transition.7,8 However, the glass transition does not take place in most polymer gelation phenomena. In fact, the temperature is not the only variable where the dynamical arrest occurs. The dynamics may also slow down dramatically when density is increased, the so-called jamming transition.15−17 In a very general sense, the jamming transition induced by the self-organization/assembly of the microstructures seem to be of relevance for de Gennes’s viewpoint; moreover, the fragility under external stresses of nonequilibrium “jammed solids” is just what the term “weak gelation” is meant to describe. The hard argument for the weak gelation of crystallizable linear polymers comes from our recent time-resolved depolarized small-angle light scattering (d-SALS) investigation.9−11 We have captured the morphogenetic details of the two most frequently observed textures, that is, fibrillar

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© 2012 American Chemical Society

networks9 and spherulite assemblies.10,11 Which outcome occurs depends on the manner where the packing of polymer chains was achieved in the crystal, that is, the fringed-micelle crystal (interchain packing) and the chain-folded lamellar crystal (intrachain packing), respectively. Generally, expanding a polymer chain to increase the crystallization possible with other chains seems to be a most efficient way to gelation. However, from the crystallographic point of view, the direct interchain packing is believed to be forbidden on kinetic grounds, and the metastability of a fringed-micelle crystal becomes an academic matter.18 Recently, we have identified the instability of the fringed-micelle formation (a process called “coherent evolution”, the nonlocalized, fibril nucleation via the fluctuation instability and fibril growth coupled with a cooperatively coarsening concentration wave); at this point, we further speculated that the fibrillar networks might represent the “observable” spinodal crystallization of polymers.9 In contrast to the fibrillar networks, because the polymer chain folds into a crystalline grain, the self-assembling mechanism provides the only available means to form a “network” without the topological entanglement. The cartoon in Figure 1 shows the process. The crystalline grains self-assemble into compact spherical clusters (spherulites) and further pack these clusters into a jammed solid.10−13 In this letter, we consider whether a morphogenetic transition may occur, if the fringed-micelle crystal and the chain-folded lamellar crystal result from the kinetic distinction between instability and metastability. We start with the spherulite assemblies, which we have made clear during these years and named “nucleation gel” according to its kinetics,10−13 and focus our attention on what may happen in the early stage. Received: January 6, 2012 Accepted: April 11, 2012 Published: April 30, 2012 646

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Figure 1. Image plot of time-resolved depolarized scattering profiles at azimuthal angle φ = 45° for C = 7 gdL−1 solution after the temperature jump from 433 to 293 K. The three scattering patterns show the representatives in nucleation-growth (typical four-leaf-clover characterization), aggregation-coarsening (the four-crescent-moon shape), and ripening-gelation stage, respectively.10.

again, is not so clear. (Note: the freezing point of the solvent is much lower than the experiment temperatures.) A clue may be found in Figure 2b. The exceptional divergence of ac at low temperature evokes the image of the mean-f ield spinodal nucleation predicted by Cahn and Hill-

All results offer valuable insights into the theoretical treatment and dynamic experiments of polymer weak gelation in the future. The polymer used in this study was poly(vinylidene fluoride) (PVDF; M̅ w = 2.75 × 105 and M̅ w/M̅ n = 2.75, Aldrich Chemical Co.). The solvent was a mixture of tetra(ethylene glycol) dimethyl ether (TG) and LiCF3SO3 salt. The salt was added to give an O/Li ratio (oxygen atoms in TG/lithium atoms in the salt) of 12:1. The gels were prepared by quenching homogeneous solution (C = 7 gdL−1) from 433 K to the gelation temperatures. All experiments were carried out by the time-resolved d-SALS, which the optical arrangement has been described previously.13 In the present study, the reliable data are about scattering angle θ = 0.82−25°, corresponding to scattering vector q = 0.2−6.17 μm−1. Figure 1 shows all the details of the gelation process.10−13 Let us focus on the cluster nucleation−growth stage. qm is the peak locus in the intensity map and tn is the nucleation time determined by the time of remarkable in the scattering °−0° 45°−0° = ∫∞ (q)q2dq (see Figure S1). For a invariant, Q45 Hv 0 IHv (category 2) birefringent sphere,11 that is, spherical cluster assembled by crystalline grains, its characteristic size, a, is estimated by a ∼ 4/qm,11,19 and its growth kinetics is identified by the simple power-law relationship a(t) ∼ tα; meanwhile, the critical cluster size, ac, can extrapolate the power-law relationship back to tn. Figure 2a shows the scaled cluster size, ã, versus scaled time, τ, at different temperatures. Unexpectedly, the results indicates temperature dependence of α, which ranges between 0.5 (diffusion-controlled growth) and 1 (interfacecontrolled growth), see inset of Figure 2a. The highly diluted cluster nuclei in which the long-range diffusion of the crystalline grains becomes important may account for the high temperature crossover. But, at low temperature, the crossover, once

Figure 2. Nucleation-growth kinetics of clusters. (a) Power-law behavior of the scaled cluster size ã vs scaled time τ at different temperatures. The inset shows the kinetic exponent, α. (b) Crossover behavior of critical cluster nucleation. The experimental critical cluster size, ac (open circles),20 the best-fitted curve (thick line) calculated by Kalikmanvo’s model, nc (thick dash line), and the corresponding changes in average packing fraction, ϕ, and cluster dimension, D. The shading is the crossover region from CNT domain to spinodal nucleation domain. 647

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iard.21 The qualitative shape of the ac curve may be examine by Kalikmanov’s crossover function.22 Accordingly, the number of the crystalline grains constituting the critical cluster, nc, is given by ⎤ ⎡ 3 nc = B⎢2X −3 + κ(1 − X )−1/4 ⎥ , ⎦ ⎣ 4

0≤X≤1

(1)

where B is the thermodynamic coefficient, κ is the dimensionless parameter, and X is the quench depth within the region of the metastability. In the function, when X → 0+, nc behaves corresponding to the classical nucleation theory (CNT), while when X → 1−, nc shows a singularity predicted by the spinodal nucleation. However, the critical cluster is not always a compact nucleus, because toward the spinodal nucleation, a more or less fractal feature is established.22−24 To quantitatively calculate ac from nc, one must account for its shape. For a fractal cluster, the size is written as ac = a0 (nc/ ϕ)1/3, where a0 is the size of the elementary grains and ϕ is the average packing fraction of grains within a spherical shell of radius ac. From ϕ = A (ac/a0)D−3 and nc = (ac/a0)D,25 we have ϕ = An1−3/D , where A is the packing factor and D is the c dimension. Thus, for a fractal cluster, ϕ decreases as the cluster grows larger. We set A = 0.64 for the close-packed but still remaining complete disorder26 and set a0 = 0.2 μm by our SEM micrograph10 and by Park et al.’s result for the core−shell structure of a PVDF chain in the dilute solution (Rg ∼ 0.22 μm, Rh/Rg ∼ 1.45).27 In regard to the shape of the critical cluster, we assume a Boltzmann sigmoidal form: D(X) = df + (3 − df)/ [1 + e(X−X0)/dX], where df = 1 for a ramified fractal (chain-like) cluster.22 Figure 2b shows nc (thick dash line) with the bestfitted result of ac (thick line) and the corresponding changes in D and ϕ, respectively. Clearly, the diffusion-controlled kinetics at low temperature can be associated with a continuous and collective spinodal nucleation. Having mentioned the cluster spinodal nucleation, we raise further questions to be explored: First, at first glance, the crystalline grains perhaps might seem to be a short-range attractive colloidal system, but it is not so (instead, highlight the long-range interaction nature of the mean-field system). Second, the singularity behavior allows an extrapolation to locate the lower limit of the thermodynamic metastability of cluster nucleation, that is, the so-called “pseudospinodal.”28−30 So what is the specific texture beyond this limit? The answer to these questions may be the evidence of the morphogenetic transition we are looking for. In principle, the scattering pattern is composed of vast independent, (intensity) fluctuating speckles. The speckle is formed by the scattering of coherent light from an inhomogeneous or a perturbational medium, especially by the interference of partial waves scattered from various position of the medium.31 The speckle size is proportional to the distance between the detector and the scatterers, and inversely proportional to the diameter of the collimated incident beam. When the instrumental geometry is determined, the speckle intensity and size are related to the spatial and temporal coherence of scatterers. Figure 3a indicates that, on crossing the metastability limit,32,33 the background speckles gradually mask the patterns. As the left intensity chart of Figure 3b, since the short-time average can “denoise” the data, there are thermal fluctuations in the homogeneous surrounding medium. On the contrary, in the right chart, the speckles with larger size and longer lifetime mean a comparatively immovable and appreciable inhomogeneous “structure” coexisting with the

Figure 3. (a) Speckle patterns and the temporal fluctuations of speckle intensity on crossing the metastability limit. (b) The location of the speckle profiles are shown by the white marks above the 293 and 268 K patterns, respectively.

growing clusters. We have a good reason to think that near the pseudospinodal point, the cluster nucleation may occur in a “flickering” elastic background constituted by such a structure. More significantly, Klein et al. have highlighted that the nucleation in the system with long-range elastic interactions behaves like a near mean-field system.29,30 By the scattering modeling approach,9,10 we can easily discern whether the cluster nucleation is true in a network. A simple model is given by (Supporting Information) I Hv(q , φ) = PHsphere (q , φ)(1 − ϕfibril) + PHfibril (q , φ)ϕfibril v v + P DB(q)sin 2 2φ

(2)

where I Hv (q,φ) is the depolarized scattered intensity, 34 Psphere Hv (q,φ) is the form factor of the birefringent sphere, 35 DB Pfibril P (q) Hv (q,φ) is the form factor of the birefringent fibril, 36 2 is the Debye-Bueche factor, sin 2φ is the optical transmission property of the analyzer from the scattering matrix theory,34 and ϕfibril is the fibril fraction. The last two terms are the structural description of the fibrillar network in our previous work.9 Two simple assumptions underlie the model: First, the crystalline grain and the fibril can coexist and follow the conservation law; Second, for the sake of simplicity, only the spherical clusters were assumed. Of course, there is a difference between the spherical and the fractal cluster, particularly near the pseudospinodal point. However, as shown in Figure 3a, when the background structure becomes visible, the scattering signal of the cluster would be too blurred for an accurate determination of its structure. Therefore, one may still work with the sphere by using an effective radius, and the qualitative cluster structure was represented expediently by manipulating the birefringence and the polydispersity of the sphere (see Table S1 for all modeling parameters). Clearly, Figure 4a shows that the calculated scattering profiles (each color line) agree quite well with the corresponding data and demonstrate that the elastic background structure is the fibrillar network. Because the birefringent sphere is most convenient for a comparison with the conceived model, the black dash line is the scaled form φ=45° 3 factor of it defined by P̃sphere Hv (qa) = IHv (qa,t)qm(t)/Qan, where Qan is the equivalent invariant of the anisotropic term (Supporting Information).11 Naturally, the scaling does not 648

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requirement (i.e., the critical nucleus size smaller than the correlation length of the transient entanglement network). In summary, our d-SALS analyses offer a very clear picture on the morphogenetic transition of the polymer weak gelation. From the spinodal singularity of the cluster nucleation, it may be seen that characteristic kinetics of such a mesoscopic transition closely resemble a thermodynamic phase transition. At the same time, the present results are in line with our previous supposition: forming the fibrillar networks may be consequent on the spinodal crystallization of polymers. There is no doubt that only the fibrillar network can exist beyond the pseudospinodal point. Thus, now we call it “spinodal gel” in contrast with the nucleation gel. A natural question, then, is whether one would expect a universal mesoscopic phase diagram, not a morphological map, for dealing with such phenomena. It will certainly provide some new insights into the general concepts of polymer gelation.

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Figure 4. Scattering modeling of morphogenetic transition. (a) Scaled form factor in nucleation-growth stage: each color lines show the best theoretical fit for the experimental profiles; the black dash line is P̃ sphere Hv (qa); each color dash-dotted lines are the contributions of DB 2 [P̃ fibril Hv (q,φ) + P (q) sin 2φ]. The fraction of fibrils, ϕfibril, and the 2 strength of network, ⟨ηDB⟩, are labeled and shown inset in the figure, respectively. (b) Relationship between four characteristic lengths a0, ac, L, and ξDB.

ASSOCIATED CONTENT

* Supporting Information S

Expanded detail on experimental determination of the nucleation time, scattering modeling of the fibrillar network and birefringent sphere, and modeling parameters. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

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

hold, but does highlight the contribution of the fibrillar network (see each color dash-dotted lines with labels and in the inset). Indeed, the increases of the mean-square Debye-Bueche fluctuation, ⟨η2DB⟩, and ϕfibril indicated the fibrillar network emerged gradually. A more visual description of the modeling result is given in Figure 4b, where the larger, isolated critical clusters, ac, are dispersed and embedded in the fibrillar network with smaller mesh size, ξDB (Debye-Bueche correlation length), thereby providing evidence of the direct elastic effect on the nucleating clusters. Another intriguing observation is that unlike the singularity shown in the mesoscopic scale, a smooth crossover is between the instable fringed-micelle crystal and the metastable chainfolded lamellar crystal, see Figure 4b, the fibril length, L, and a0. Even though we have made this priori assumption, as with many long-standing problems, their coexistence and crossover is deceptively normal. We must remember the historically socalled “δl catastrophe” which causes a singular behavior of the average lamellar thickness at a finite supercooling in Hoffman− Lauritzen secondary nucleation theory.18 Additionally, in terms of the mean-field nature of polymers, the same is true of Cahn− Hilliard spinodal nucleation theory (as a crystal fold surface should be rough/diffuse far from equilibrium).21 One possible explanation of the smeared-out singularity may rest in Ryan et al.’s SAXS experiment37 and Muthukumar’s simulation38,39 on the down turn of the Cahn−Hilliard plot, that is, R(q)/q2 versus q2, at low q. In brief, the down turn amounts to the suppression of large-scale fluctuation/nucleation, and from Muthukumar’s phenomenological model, it may be referred to the intrinsic topological connectivity/entanglement of polymers. That is, when the metastability limit is satisfied (i.e., the crystal nucleation time faster than the relaxation time of the metastable parent phase32 or, rather, the polymer disentanglement), the nucleation must be confined within the obvious “steric”

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge support from the NSC of the Taiwan under Contract NSC-98-2221-E-011-009-MY3. REFERENCES

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