Scattering Modeling of the Morphogenetic Transition

Figure 11 (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 marks at above right of the 293 K and 268 K patterns, respectively.

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.⁷⁹ 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 11 (a) indicates that, on crossing the metastability limit,^34,80 the background speckles gradually mask the patterns. As the left intensity chart of Figure 11 (b), 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 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.⁵⁷

By the scattering modeling approach,^14,15 we can easily discern whether the cluster nucleation is true in a network. A simple model is given by (Supporting Information)

(

^,

) (

^,

) (

¹

) ⁽

^,

^{) ( )}

^{sin 22}

the Debye-Bueche factor,³⁹ sin 2² ϕ is the optical transmission property of the analyzer from the scattering matrix theory,⁸⁰ and φ_fibril is the fibril fraction. The last two terms are the structural description of the fibrillar network in our previous work.¹³ Two simple assumptions underlie the model: First, the crystalline grain and the fibril can coexist and follow the conservation law. Secondly, for the sake of simplicity, only the spherical clusters were assumed. Of course, there is difference between the spherical and the fractal cluster, particularly near the pseudospinodal point. However, as shown in Figure 11 (a), 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 1 for all modeling parameters). Clearly, Figure 12 (a) 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. Since the birefringent sphere is most convenient

for a comparison with the conceived model, the black dash line is the scaled form factor of it defined by ^P_H^sphere_v

( )

^{qa =I}_H^ϕ_v⁼⁴⁵

(

^{qa t q t Q,}

) ( )

_m³ _an^{, where} Q is the equivalent invariant of the _an anisotropic term (Supporting Information).¹¹ Naturally, the scaling does not 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, η_DB² , and φ_fibrilindicated the fibrillar network emerged gradually. A more visual description of the modeling result is given in Figure 4b, where the larger, isolated critical clusters, a , are _c dispersed and embedded in the fibrillar network with smaller mesh size, ξ_DB(Debye-Bueche correlation length), thereby providing evidence of the direct elastic effect on the nucleating clusters.

Figure 12 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

v

( )

The fraction of fibrils, φ_fibril, and the strength of network, η_DB² , are labeled and shown inset in the figure, respectively. (b) Relationship between four characteristic lengths a₀, a_c, L, and ξ_DB

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 chain-folded lamellar crystal, see Figure 12 (b), the fibril length, L , and a . Even though ₀ we have made this priori assumption, as with many long-standing problems, their coexistence and crossover is deceptively normal. We must remember the historically so-called “ lδ catastrophe” which causes a singular behavior of the average lamellar thickness at a finite supercooling in Hoffman-Lauritzen secondary nucleation theory.³⁰ 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).⁵² One possible explanation

of the smeared-out singularity may rest in Ryan et al.'s SAXS experiment⁵⁴ and Muthukumar's simulation⁵⁶ on the down turn of the Cahn-Hilliard plot, i.e., R q q vs.

( )

² q , at low q . In ² 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 phase,⁵⁷ or rather the polymer disentanglement), the nucleation must be confined within the obvious

“steric” requirement (i.e., the critical nucleus size smaller than the correlation length of the transient entanglement network).

(1) Because the concentration of the solute decreases as the nuclei grow, m₁ is time dependent. However, in the end of the nucleation and growth stage, a similar environment for the birefringent scatterers was assumed. m₁ was approximated from the Fig. 3 (a) in Ref. 14.

(2) Unlike the category 2 birefringent sphere which ∆ , µ µ, and m₁ are closely linked together and depend on the size of the growing sphere,¹⁴ we set δ= for a uniform microstructure of the fibril and for a semi-quantitative analysis. 1

(3) In the Murakami et al.’s model, if the polymer in the fibril has a helical conformation, the angular dependence of the d-SALS pattern is independent of ω₀.³⁴ In the present system, the PVDF micro-crystal is the -phaseγ with a TTTGTTTG’

conformation;⁸¹ thus, we set ω₀= ° . 0

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.

在文檔中 探索軟物質介觀結構複雜性的起源 (頁 24-28)