Scattering Modeling of Fibrillar Network

Figure 2 (a) shows the SEM micrograph and the corresponding depolarized scattering

pattern of the s-PS/o-xylene gel. As shown in Figure 2 (b) left sketch, the network can be modeled as the rigid fibrillar skeleton connected by the soft entangled junction.

In the common practice of the d-SALS experiment, the structural identification of the scatterers is accomplished by comparison with theoretical form scattering, like Murakami et al.' anisotropic rod model.

³⁴

Figure 2 (a) The SEM micrograph (left) and the representative half depolarized scattering pattern (right) of the fibrillar network, prepared by 0.5 gdL^-1 s-PS/o-xylene solution at 323K. (b) Left: the sketch of the fibrillar skeleton. Right: the theoretical patterns of the crystalline fibril by Murakami et al.’s model³⁴ for different fibril lengths (L = 1 and 5µm). The parameters used are the anisotropies of the fibril δ = 1, the polar angle of the rod axis ω0 = 0, and the refractive index of the medium m1=1.503. (c) The hierarchical structure of fibrillar network.

As shown in Figure 2(b) right, the theoretical pattern (the rod length L=5µm) is similar to the experimental one, but the length estimation differs markedly from the SEM observation

(ca. 1µm). Undoubtedly, the present pattern should be richer in structure. In early works,³⁵ it is implicitly assumed that the structure scattering can be excluded from d-SALS. Later, we knew that the structural information does exist and can be extracted by our scattering modeling approach.^14,15

For the scattering from an evolving hierarchical structure, giving the explicit analytical form of the scattering pattern is rather difficult or even impossible. Instead, the scattering modeling is a compromising phenomenological approach which can lead to a better understanding of the interrelation/interaction between scatterers at different timing and hierarchy during the process. Before building a realistic model, we would like to define two terms: form factor and structure factor. The form factor is the scattering intensity for a scatterer alone and is related to the shape and scattered ability of it. The structure factor is scattering intensity for the interference of the scatterers and is referred to the spatial arrangement of them.

Besides, the form factor also describes the scatterers’ level in the hierarchical structure. If the density-density correlation functions of two structures have the convolution relationship,^36,37 the sub one names as the form factor. Let us return to our main subject. Considering many crystalline fibrils create a topological connectivity and then show a highly correlated evolution.

The model needs two form factors to define the fibrils and their correlation; one more structure factor is needed to characterize the large-scale inhomogeneities of the network, as shown in Figure 2 (c).

Figure 3 Scattering modeling of the fibrillar network.

Two mechanisms were considered: the fibril aggregation/branching and the fluctuation coupling. The former is not new^21,38 and holds for the following case, as shown in Figure 3 top.

First, either an aggregate or a branching object requires “an” origin,²¹ and the fractal characteristic should be detected and analyzed. Secondly, if the cluster is formed by the aggregation, it will be a two-stage process, i.e., the fibril growth and then aggregation. In the

present study, however, no fractal was observed, and the fibrils grew to join one another as a continuous process. Hence, a new fluctuation coupling mechanism goes as follows: the

“growing” fibrils are spread out and embedded in non-localized concentration waves posed by them. As shown in Figure 3 bottom, in addition to the fibril form factor (Murakami et al.'s model), the Debye-Bueche fluctuation model³⁹ as another form factor provided a suitable formalism to describe the fibrils’ correlation. By its very nature, the Debye-Bueche model is often used to analyze the spatial inhomogeneity and makes no reference to the correlation between the discrete fibrils. Nevertheless, just these fibrils give rise to mesoscopically observable inhomogeneity. Given such a “mixing,” the scattered intensity should be expressed by the sum of these two terms. Further, because of a polymer chain inevitably participating in several fibrils, as the fibril growth, the topological entanglement would join the fibrils up. So we treated the structure factor as a loose assembly of such fibril clusters/domains.

The model is then given by

(

^,

) ( )

^Rod

(

^,

) ( )

^{sin 22}

Hv Hv DB

I qϕ =S q NP⎡⎣ qϕ +P q ϕ⎤⎦

(1)

where IHv

(

q^,

ϕ )

is the depolarized scattered intensity (where q is the scattering vector, and

ϕ

is the azimuthal angle), S q( ) 1= −βe^−(γξsq)2 is the structure factor of the spatial correlation between each cluster (where ξ_s is the correlation length, and β and γ are the amplitude and the range of the correlation, respectively),³⁸ N is the number of the fibrils,

(

^,

) (

^{/ 4}

)

²

Rod

PHv qϕ = k Lπ δ

⎡ ⎣

p2

( cos ω

0

) ⎤ ⎦

²P qL

( , ϕ )

is the form factor of the crystalline fibrilgiven by the Murakami et al.’s model [where k is related to the absolute intensity,

L

is the fibril length, δ is the anisotropies of the fibril, ω₀is the polar angle of the fibril axis,

2( )

p x is the second order Legendre function, and ^{P qL}

(

^,ϕ

)

is the scaled form factor],³⁴

( )

^{4 3 2}

(

^{1 2 2}

)

²

DB DB DB DB

P q = π ξK η +qξ is the Debye-Bueche factor (where

K

is a constant, ξDBis the Debye-Bueche correlation length, and η_DB² is the mean-square fluctuation),³⁹ and sin 22 ϕ is the “optical transmission” property of the analyzer from the scattering matrix theory.^40,41

The model consists of eleven parameters (i.e., constant: ,k K, ω₀, γ and δ ; , time-dependent parameters: ,N L, β, ξ_s, ξDB , and η_DB² ). How to better determine their values is a problem, especially in an absolute intensity fitting. However, for a semi-quantitative analysis, we set K = and 1 δ =1, and combined k and Nto be a floating intensity coefficient in the fitting. 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 ω ;³⁴ thus we

set ω₀= . In structure factor, 1 γ is given by 1 1

γ 2 σ π

= −

where

σ

is the ratio of the cluster size to their average distance (if σ =1, the clusters are closely packed; if σ <1, the clusters are close but not in contact; if σ >1, the clusters overlap).³⁸ Considering the loose assembly of the clusters,σ <1seems to be reasonableness;

thus we set σ =0.67(γ =0.42).

Figure 4 A comparison of the experimental (left half a series of quarter patterns) and the best-fitted theoretical (right half) scattering patterns.

For the remainder time-dependent parameters, formally, there may be not just one solution but a range, corresponding to “one” scattering pattern (two-dimensional, space case). The validity of the modeling is limited to the selection of these parameters, because the results may reflect in part the way in which those were selected. Nevertheless, it is possible with appropriate constraints during analysis, like refer to the real-space SEM micrograph. On the other hand,

one must not forget that we face a spatiotemporal evolution (three-dimensional case); only the physical clarity of these solutions appears to be continuous on the time scale and can thus represent a “right” answer. Figure 4 shows that the modeling results agree well with the experimental patterns—even including the background.

在文檔中 探索軟物質介觀結構複雜性的起源 (頁 11-15)