Hierarchically Self-Organized Dissipative Structures of Filler Particles

Article pubs.acs.org/Macromolecules

Hierarchically Self-Organized Dissipative Structures of Filler Particles in Poly(styrene-ran-butadiene) Rubbers Daisuke Yamaguchi,† Takeshi Yuasa,‡ Takuo Sone,‡ Tetsuo Tominaga,‡ Yohei Noda,†,§ Satoshi Koizumi,†,§ and Takeji Hashimoto*,†,∥,⊥ †

Quantum Beam Science Center, Japan Atomic Energy Agency, Tokai, Naka, Ibaraki 319-1195, Japan Yokkaichi Research Center, JSR Corporation, 100 Kawajiri-cho, Yokkaichi, Mie 510-8552, Japan § Faculty of Engineering, Ibaraki University, Nakanarusawa-cho 4-12-1, Hitachi-shi, Ibaraki 316-8511, Japan ∥ Kyoto University, Kyoto 606-8501, Japan ⊥ National Tsing Hua University, Hsinchu 30013, Taiwan ‡

S Supporting Information *

ABSTRACT: We elucidated the spatial distribution of filler particles in cross-linked poly(styrene-ran-butadiene) rubbers (SBR) developed under a typical fillers/rubbers compounding process as one of dissipative structures formed under a stress field imposed on the given system. Two types of fillers and two types of SBR were used to prepare four kinds of the fillers/rubbers composites to investigate effects of specific polymer/filler interactions on the hierarchically self-organized dissipative structures under a given processing condition. The dispersion structures of the filler particles were explored by using the combined small-angle scattering (CSAS) method, which enables the exploration of the structures existing over the wide length scale ranging from ∼6 nm to ∼20 μm. The measured CSAS profiles were analyzed by using a newly developed scattering theory on fractal structures built up by the “cluster” as their lower cutoff objects. This cluster is composed of a few aggregates, defined as the fused primary filler particles, bound by the SBR chains. The scattering from the cluster having characteristic internal structures built up by the fillers and the SBR chains was theoretically formulated by generalizing the Debye−Bueche fluctuation theory for infinite space to the theory for a “confined space” relevant to the cluster size. The dispersion state of the fillers in SBR was clarified on the basis of hierarchical structures consisting of five structure levels as detailed in the text. More specifically, it has the following characteristics depending on the specific interactions: Small, compact clusters build up compact mass-fractal structure, while large, loose clusters build up open mass-fractal structures. “primary” particles as will be detailed later. The external fields relevant to our systems are (1) a field involved first by mixing a bulk of rubbery polymers and that of powders of filler particles, (2) a field involved subsequently by pressing the mixtures into a sheet, and (3) a field finally involved by chemical cross-linking of the matrix polymer chains. The fields (1) and (2) first develop fluctuations in a spatial dispersion of filler particles. The fluctuations developed are subsequently grown into the dissipative structures, which are eventually locked or trapped in the cross-linked network chains by the field (3). We will investigate the locked, static dissipative structures to gain insight into the self-organization process. The rubber/filler composites, one of the most successful composites exhibiting both the viscoelastic properties of soft rubbers and the hard characteristics of filler particles,2,3 have been improved always to satisfy increasing demands on the

1. INTRODUCTION Elucidation of self-organizing processes and structures of nanoparticles (NPs) in polymer matrices under external fields is an important topic from viewpoints of both fundamental polymer physics and polymer engineering on nanocomposites. From scientific viewpoints, the topic is relevant to universal problems on pattern formation or self-organization of dissipative structures and/or ordered structures through fluctuations in nonequilibrium systems open to external fields.1 Herein, we define the dissipative structures as structures formed in nonequilibrium systems under external fields: the dissipative structures are formed through kinetic pathways in which the systems efficiently dissipate the energy stored under the given external fields. From the engineering viewpoints, it is crucial to control the dissipative structures at will by manipulating the effective interactions between NPs and polymers and the compounding conditions as well to develop materials with advanced properties. In this work, we focus on to investigate nanocomposites of rubber-like polymers and filler particles having NPs as © XXXX American Chemical Society

Received: May 19, 2017 Revised: August 25, 2017

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DOI: 10.1021/acs.macromol.7b01052 Macromolecules XXXX, XXX, XXX−XXX

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Those studies9,13,14 described above investigated the filler dispersion morphology on the model compounds in which some common additives, such as zinc oxide (ZnO), etc., were excluded. Thus, those systems described above could be regarded as simply two-component systems with respect to the scattering contrast (the filler and the matrix). This work aimed to elucidate the effect of the endfunctionalized SBR with bifunctional amino and alkoxysilyl groups on the spatial dispersion morphology of the filler particles15 at a wide range of the reciprocal scale (from 3 × 10−4 to 1 nm−1) using CSAS methods. The compounding components employed in this work resemble those used in industrial products and include various additives such as ZnO, as will be shown later in Table 2. Those additives are the one required to achieve desirable properties for the general use of the rubber/filler composites. This paper focuses solely on structural analysis with the CSAS method; other fundamental data, including rheological properties and real space structural observations by TEM, have been previously published for the same materials studied in this work.5 This study addresses the following two issues: (i) the extraction of the scattering arising from the filler component (silica or carbon black particles, designated hereafter as Si or CB, respectively) or the filler components (Si/CB = 50/4 wt/ wt in the two-component filler systems) dispersed in the complex industrial composite systems which contain ZnO particles, etc., from the net scattering from the composite; (ii) the elucidation of the effect of the end-functionalized SBR on the filler dispersion morphology. We used two kinds of SBR as the matrix rubber, i.e., the SBR with and without the endfunctional group designated hereafter f-SBR and n-SBR, respectively. Thus, we investigated four kinds of the filler/ rubber composites of Si/f-SBR, Si/n-SBR, CB/f-SBR, and CB/ n-SBR plus two kinds of more complex composites of Si/CB/ (= 50/4 wt/wt)/f-SBR and Si/CB/(= 50/4 wt/wt)/n-SBR, all of which have a common formulation of the common additives except for silane coupling agent as will be shown later in Table 2.

performance required for their application to economic and ecological car tires. This requires the inclusion of new additives such as coupling reagents or accelerators into rubbers to enhance their chemical affinity for the fillers.4 A chemical modification of the rubber is anticipated to also promote effectively its affinity for the fillers. In fact, the composites consisting poly(styrene-random-butadiene) rubber (SBR) modified by the amino and alkoxysilyl groups at one of the chain ends exhibited a considerable improvement in the mechanical properties with respect to the loss tangent (tan δ) and the nonlinear Payne effect.5 Structural analyses have been attempted with respect to the filler-dispersion morphology6−9 and to the specific polymer layer built up on the filler surface, i.e., the so-called bound rubber layer,10 in order to find relationships between macroscopic properties and various mesoscopic- to microscopic-structural parameters in order to establish a valid scientific strategy for the development of advanced rubber/filler composites. This work, which was also motivated by a considerable improvement in the rheological properties of the rubber/filler composite brought about by the chemical modification of the matrix rubber,5 focused on to elucidate the chemical-modification-induced changes in the selfassembled dissipative structure of fillers in the composites. Filler powders themselves generally possess the following hierarchical structure in the matrix of air (or toluene):9 (1) primary particles which are the smallest units in the filler particles, (2) “aggregates”, which are dimers, trimers, and/or multimers of the primary particles fused and welded together, and (3) the mass-fractal agglomerates which are higher-order structures of the aggregates formed by physical interactions between the aggregates in air (see for example Figure 12 in ref 9). The aggregates are unbreakable under further compounding processes of the fillers in the polymer matrix. The aggregates themselves act as the building blocks of the agglomerates in the filler powders dispersed in air matrix.9 On the other hand, when the fillers are compounded with polymers, a small number of the aggregates bound together by polymers act as the building blocks for the mass-fractal agglomerates and serve as the dispersible units in the rubber matrix (see for example Figure 13 in ref 9), rather than the aggregates themselves. The size of the dispersible units in the rubber matrix may be controlled by the stress involved during the mixing process of the SBR bulk and filler powders which is coupled with the physical interactions between the aggregates of the primary filler particles and the polymer chains and those between the aggregates themselves. This work aimed to investigate also the universality of the dispersible unit of fillers formed in the rubber matrix for various rubber/filler systems as will be discussed below. Our previous studies9 fully characterized the hierarchical structures of the fillers, based on the unified power-law/Guinier theory proposed by Beaucage,11 at a given volume fraction of 0.2 in the rubber matrices composed of the polymers without the end-functional groups at an extremely wide range of q scale from 3 × 10−4 to 1 or even up to 3 × 101 nm−1 by using the combined small-angle scattering12 (CSAS) and wide-angle Xray diffraction measurements. The effects of the end-functional polymer content13 and filler content14 on the dispersion structure of the fillers were also examined by SAXS and TEM. In these studies, the end-functional polymer employed was the SBR with an −Si(CH3)2OH group at its single chain end [i.e., SBR−Si(CH3)2OH], and in the most of the cases the functional SBR was mixed with the nonfunctional SBR.

2. EXPERIMENTAL METHODS 2.1. Polymers. Two types of solution-polymerized SBR copolymers whose characteristics are shown in Table 1 were supplied by JSR

Table 1. Two Kinds of the Solution Polymerized SBR Employed in This Study code

functional group(s)

vinylb (%)

wPSc (%)

Mwd (kg/mol)

Tge (°C)

f-SBR n-SBR

Si-OR + NH2a none

61 62

20 21

194 199

−36 −36

a

One of the chain ends was substituted with bifunctional amino (−NH2) and alkoxysilyl (−SiOR; R = alkyl group) groups.5 bVinyl content in butadiene sequence. cWeight fraction of styrene sequence. d Weight-average molecular weight. eGlass transition temperature. Corp. (Tokyo, Japan).5 One denoted as f-SBR possesses a functionalized chain end in one of the copolymer chain ends which has the chemical affinity for the filler particles. The other is a conventional (i.e., nonfunctional) SBR (denoted as n-SBR) that has the affinity less than f-SBR. Further characteristics of the polymers are summarized in Table 1. 2.2. Composites. We investigated four types of samples, the difference of which is characterized by the compositions of components described in the first four rows in Table 2. B

DOI: 10.1021/acs.macromol.7b01052 Macromolecules XXXX, XXX, XXX−XXX

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and wavelength of the incident beam, respectively. λ = 0.2 nm for the PNO. The obtained USANS profiles were corrected for the background scattering and the slit-height smearing.20 The SANS-J-II spectrometer covers a wide q-range of 5 × 10−3 ≤ q ≤ 1 (nm−1) by combining MgF2 focusing lens optics and conventional pinhole optics. The focusing lens optics covers the q-range (5 × 10−3 ≤ q ≤ 7 × 10−2 nm−1) for USANS designated hereafter as focusing USANS, and the pinhole optics covers the q-range for SANS (3 × 10−2 ≤ q ≤ 1 nm−1). The wavelength of incident neutron beam was λ = 0.65 nm for the SANS-J-II, and its distribution was (Δλ/λ) = 0.13. Further details of the SANS-J-II instrument were provided elsewhere.18 The data were corrected for the background electric noise, the background scattering from the empty cell, and the detector efficiency. After the implementation of these corrections, the absolute scattering intensity was obtained using a precalibrated secondary standard of an irradiated aluminum.21 The q-independent scattering appeared at a high q-region of the SANS profile due to the incoherent scattering from hydrogen atoms. In this study, the incoherent scattering was subtracted from the profiles. We designate hereafter the combined USANS and SANS measurements as CSANS. The filler-dispersion morphology of as-received Si or CB filler in the state of pelletized powder itself was also investigated, using CSANS, as a reference of the composites containing the fillers. For this purpose, the same filler powder as that used in the composite was simply packed in a quartz cell of 1 mm thickness. The fillers dispersed in air are hereafter denoted as Si powder/air and CB powder/air. The volume fraction of pores (ϕV) in the filler powders in the cell was obtained from density measurements, where (1 − ϕV) corresponds to the ratio of the measured density of the filler in the cell to the reported density value (1.95 g cm−3) for either Si or CB filler. The results were ϕV = 0.85 for Si powder/air and ϕV = 0.82 for CB powder/air. The SAXS measurements were performed on beamline BL19B2 of the SPring-8 synchrotron radiation facility (Hyogo, Japan).22 The incident wavelength of the X-ray was 0.69 Å (18 keV), and the distance from the sample to the detector was 35 m. The obtained twodimensional scattering pattern was circularly averaged into a onedimensional profile and corrected for the background scattering from the sample cell, the air scattering, and the transmittance of the specimen. Although the SAXS intensity measured was not further converted to the absolute intensity, the measurement conditions and the sample thicknesses (1 mm) were made identical for all the sample specimens, and thereby the obtained SAXS profiles can be compared to each other. The accessible q-ranges of the SAXS measurements were 4 × 10−3 ≤ q (nm−1) ≤ 2 × 10−1 for Si/SBR, CB/SBR, Si powder/air, and CB powder/air and 4 × 10−3 ≤ q (nm−1) ≤ 3 for SBR (no filler).

Table 2. Formulation of Composites composition SBRa silane coupling agent (Si75) silica; Nipsil AQ (Si) carbon black; N339 (CB) oil stearic acid zinc oxide (ZnO) ao.; 6Cb acc.; DPGc acc.; CBSd sulfur total

Si/CB/SBR

Si/SBR

CB/SBR

SBR (no filler)

100 4

100 0

100 0

100 0

50 4 10 2 3 1 1.5 1.8 1.5 178.8

50 0 10 2 3 1 1.5 1.8 1.5 170.8

0 54 10 2 3 1 1.5 1.8 1.5 174.8

0 0 10 2 3 1 1.5 1.8 1.5 120.8

a

Both f-SBR and n-SBR were used for SBR in each composite. bN(1,3-Dimethylbutyl)-N′-phenyl-p-phenylenediamine. cDiphenylguanidine. dN-Cyclohexyl-2-benzothiazylsulfenamide. A silane coupling reagent (Si75; Evonik-Degussa GmbH, Germany), which improves the compatibility of the SBR chains and silica fillers, was added only in one specific composite denoted by Si/ CB/SBR, as described below. The amounts of all ingredients are described in Table 2, where the unit is per hundred rubber with respect to weight (phr). Each sample has a common set of the additives having the same formulation as described in the second seven rows in Table 2. These additives are composed of activators (stearic acid and ZnO), a cross-linker (sulfur), and accelerators (acc; DPG and CBS). Oil, acting as the plasticizer of matrix rubber, and an antioxidant (ao.; 6C) were also included in the compounds. The four types of the composites shown in Table 2 are different from each other with respect to the type of the filler, when it is added, and whether or not the silane coupling reagent was added. The composite Si/CB/SBR, which most closely resembles industrial composites, includes the silane coupling reagent together with 4 phr of CB (N339; Tokai Carbon Co., Ltd., Japan) as well as 50 phr of Si (Nipsil AQ; Tosoh silica Corp., Japan). The other three composites (Si/SBR, CB/SBR, and SBR) lacking silane coupling reagent were employed for a control experiment to facilitate the analysis of the filler dispersion morphology in the cross-linked SBR. Si/SBR contained only Si fillers, CB/SBR contained only CB fillers, and SBR was only a rubber matrix without any fillers. Both n-SBR and f-SBR were employed as the matrix polymer in each composite. Thus, the numbers of the specimens prepared and investigated totaled up to eight. 2.3. Compounding Process. The ingredients shown in Table 2 were mixed twice in a mixer (75 mL of Plasto-Mill; Toyo Seiki Co. Ltd.) at 60 rpm at 90 °C for 1.5 min for the first mixing and at 70 °C for 1.5 min for the second mixing. More precisely, in the first-step mixing, the filler(s) (Si for Si/SBR or CB for CB/SBR or both Si and CB for Si/CB/SBR), silane (only for Si/CB/SBR), oil, stearic acid, antioxidant, and ZnO were simultaneously added to the SBR and mixed to prepare the master batch. Then in the second-step mixing the cross-linker and accelerator were added to and mixed into the master batch. The composite Si/CB/SBR was prepared as a reference for the four kinds of the model composites (Si/f-SBR, Si/n-SBR, CB/f-SBR, and CB/n-SBR). Then for the curing process, the temperature was raised to and held constant at 160 °C for 40 min. 2.4. Scattering Experiments. The CSAS method used in this study is based on SANS and SAXS measurements. The SANS measurements were performed on either the PNO spectrometer16,17 or the SANS-J-II spectrometer18 at the research reactor JRR-3 (Tokai, Japan). The PNO spectrometer uses the double crystal, as the original Bonse-Hart camera19 does, and enables ultra-SANS (USANS) measurements with an accessible q-range from 3 × 10−4 to 3 × 10−2 nm−1 where q = (4π/λ) sin(θ/2), θ and λ being the scattering angle

3. EXPERIMENTAL RESULTS 3.1. CSANS Results on Model Nanocomposites. We present here the CSANS profiles from the model nanocomposites composed of only one type of the filler, Si/SBR and CB/SBR. The studies on the model nanocomposites are aimed to serve as a control experiment to elucidate effects of the end-functionality of the SBR chains and the type of the fillers on the filler dispersion morphology in the rubber matrix. Moreover, the results deduced from the control experiment are anticipated to provide very fundamental information to elucidate the filler-dispersion morphology on the Si/CB/SBR composite whose result will be discussed later in section 7. The CSANS profiles for the four kinds of the model composites (Si/f-SBR, Si/n-SBR, CB/f-SBR, and CB/n-SBR) are presented in Figure 1. Figure 2 is also presented, in addition to Figure 1, to facilitate comparisons of the CSANS from these four composites. These figures present the scattering profiles covered over a very wide q-scale, as wide as ∼4 orders of magnitude, and a very large dynamic range of the intensity (as large as 9 orders) thanks to the combined use of the double-crystal USANS, focusing USANS, and the pinhole-SANS as indicated in the figures. In Figure 1, parts a and c enable direct comparisons of the scattering from the given two composites, while parts b and d facilitate to observe detailed features of the individual C

DOI: 10.1021/acs.macromol.7b01052 Macromolecules XXXX, XXX, XXX−XXX

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An overall feature of all the scattering profiles is roughly classified into three regions as shown in the figures. In region I, the profile is sensitive to the filler particles themselves, more specifically to the primary particle and its aggregates. Thereby, the profile hardly depends on the matrix rubber as revealed from Figures 1a and 1c. This fact is evidenced also from the difference in the profiles shown in Figures 2b and 2d where the matrix rubbers are the same, so that the difference is obviously due to that in the fillers. In region II the profile primarily depends on the “clusters” composed of the aggregates of the primary filler particles bound together by the SBR. The size of the clusters appears to depend on the matrix polymer (f-SBR or n-SBR) through the attractive interactions between the SBR and the aggregates as observed in the difference of the profiles in Figures 1a and 1c. These results would provide us an intuition that the clusters of the Si and CB aggregates in the f-SBR matrix may be smaller than those in the n-SBR matrix, respectively. In region III the scattering intensity profile with q, Iexp(q), can be characterized by the power law Iexp(q) ∼ q−α

(1)

in which the exponents α for Si/f-SBR and Si/n-SBR were 2.5 and 2.3, respectively, and α’s for CB/f-SBR and CB/n-SBR were 2.4 and 2.1, respectively. For a given filler, the exponent α for the f-SBR is larger than that for the n-SBR as shown in Figure 1a for Si and Figure 1c for CB. For the given n-SBR matrix, the exponent α for Si is much larger than that for CB (2.3 vs 2.1) (Figure 2b), though the exponent is slightly larger for Si than that for CB (2.5 vs 2.4) for the given f-SBR matrix (Figure 2d). These results shown in Figures 1a, 2b, and 2d are consistent with the value α for CB/f-SBR being larger than that for Si/ n-SBR as shown in Figure 2a and the large disparity in α for the two composites shown in Figures 1c and 2c. More quantitative analyses and discussion on the power-law exponent α will be deferred until sections 5.3 to 5.5 and 6. In these four composites, the values for the exponents α satisfy 1 < α < 3, implying existence of the mass-fractal structures built up by the clusters as the lower cutoff objects. This piece of evidence disclosed herewith reveals the important fact that the building blocks of the mass-fractal structures, i.e., the higher-order agglomerates of the fillers, are the clusters but neither the aggregates nor the primary particles themselves as will be discussed later in sections 3.5, 5, and 6 and schematically presented later in Figure 8. The critical values of q between different regions (I/II and II/III) quantitatively depend on the composites as will be detailed later in sections 5 and 6. Figure 3 presents the CSANS profiles for the Si and CB fillers themselves (Si powder/air and CB powder/air) used for the composites. The results indicate some differences between these fillers with respect to (i) the size of the primary particles and their

Figure 1. CSANS profiles for Si/f-SBR and Si/n-SBR (a) and for CB/ f-SBR and CB/n-SBR (c). The profiles for Si/f-SBR and CB/f-SBR are vertically shifted respectively by a factor of 10−1 relative to those for Si/n-SBR and CB/n-SBR in (b) and (d), respectively.

Figure 2. Comparisons of CSANS profiles (a) between Si/n-SBR and CB/f-SBR, (b) between Si/n-SBR and CB/n-SBR, (c) between Si/fSBR and CB/n-SBR, and (d) between Si/f-SBR and CB/f-SBR.

scattering profiles owing to a vertical shift of the bottom profile relative to the upper one (by a factor of 10−1) which avoids the overlap of the two profiles.

Figure 3. CSANS profiles for Si powder/air and CB powder/air. D

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Figure 4. (a) Comparisons of the CSANS profiles from the f-SBR composite [profile (1) shown by the unfilled circles] and the n-SBR composite [profile (2) shown by the unfilled squares, vertically shifted by 10−2 relative to profile (1)] without the fillers (see Table 2). Note that profiles (1) and (2) completely overlapped each other without the vertical shift as shown by profile (3) drawn with solid red line and green broken line obtained by the vertical shifts of them [the unshifted profiles (1) and (2)] by 10−4. Profile (4) is the partial structure factor SZnO−ZnO(q) for the ZnO particles extracted by using the contrast variation method with DNP for the Si/f-SBR composite. (b) Comparisons of CSANS [profile (1)] and SAXS profile [profile (5)] from the f-SBR composite. The CSANS profile shown by the symbols is displayed in the absolute intensity, the scale of which is referred to the left-hand-side ordinate, while the SAXS intensity profile shown by the symbols is displayed in an arbitrary unit, the scale of which is referred to the right-hand-side ordinate. The CSANS profile (1) and the SAXS profile (5) completely overlap one another as shown by the solid line (1) and the symbols (5) in the bottom profile with the vertical shifts by a factor of 10−4 and 10−2, respectively.

Table 3. Calculated Scattering Length Density of Each Component for Either X-ray or Neutron Beam component a

SBR silane coupling agent (Si75) silica (Si) carbon black; N339 (CB) oil stearic acid zinc oxide (ZnO) A.O.; 6Cb acc.; DPGc acc.; CBSd sulfur

ρM [g cm−3] 0.92 1.03 1.95 1.95 0.92 0.94 5.68 1.09 1.18 1.31 2.01

M [g mol−1] 64.1 486.1 60.1 12.0 13.9 284.5 81.4 268.4 211.3 264.4 32.1

bneutron [cm−2] 6.89 1.58 3.08 6.50 −1.53 −6.75 4.83 1.19 2.22 1.52 1.07

× × × × × × × × × × ×

Δbp,n [cm−2]e

9

10 109 1010 1010 109 108 1010 1010 1010 1010 1010

−5.31 2.39 5.81 −8.42 −7.57 4.14 5.01 1.53 8.31 3.81

× × × × × × × × × ×

109 1010 1010 109 109 1010 109 1010 109 109

bX‑ray [cm−2] 8.52 9.42 1.64 1.64 8.79 8.91 4.47 9.99 1.05 1.17 1.69

× × × × × × × × × × ×

Δbp,x [cm−2]f

10

10 1010 1011 1011 1010 1010 1011 1010 1011 1011 1011

9 7.88 7.88 2.7 3.9 3.62 1.47 1.98 3.18 8.38

× × × × × × × × × ×

109 1010 1010 109 109 1011 1010 1010 1010 1010

a Calculation was carried out by monomer unit. bN-(1,3-Dimethylbutyl)-N′-phenyl-p-phenylenediamine. cDiphenylguanidine. dN-Cyclohexyl-2benzothiazylsulfenamide. eΔbP,n = bP,n − bSBR,n where bP,n and bSBR,n are the scattering length density for P (P = Si, CB, ZnO, and the other ingredients other than SBR) and SBR for neutron, respectively. fΔbP,X = bP,X − bSBR,X where bP,X and bSBR,X are the scattering length density for P (P = Si, CB, or ZnO, and the other ingredients other than SBR) and SBR for X-ray, respectively.

background scattering to obtain the scattering only from the filler particles in the filler/rubber composites containing a small amount of the ZnO particles. Figure 4a presents the CSANS profiles for the f-SBR composite [profile (1) shown by the unfilled circles] and n-SBR composite [profile (2) shown by the unfilled squares], both without the fillers. Profile (2) is presented after a vertical shift by the factor of 10−2 in order to avoid an overlap with the profile (1). The two profiles (1) and (2) are actually overlapped each other as shown in the profile (3) obtained only by the vertical shifts of them by the same factors of 10−4 and 10−2 relative to profiles (1) and (2), respectively, in the doublelogarithmic plots, indicating that the given compounding process yielded an identical characteristic size of the ZnO particles dispersed in both the f-SBR and n-SBR composites without the fillers. Figure 4b presents the CSANS profile [profile (1)] and SAXS profile [profile (5)] for the f-SBR composite. The results reveal that

aggregates, (ii) the clusters composed of the primary particles and the aggregates, and (iii) the power-law exponent α, judging from the disparities of their scattering profiles in region I, II, and III, respectively. The detailed analyses of the characteristic parameters will be done in section 5. 3.2. CSANS and SAXS from the SBR Composites without Fillers. The scattering from the SBR composites without the fillers should primarily arise from the zinc oxide particles (ZnO). This scattering contributes as the background scattering to the scattering from the Si/SBR and CB/SBR composites which contain the same amount of ZnO as the SBR composites without the fillers (see Table 2), provided that the size of ZnO and their spatial distribution in the SBR composites without the fillers are essentially the same as those with the filler/rubber systems. In this section, we aim to investigate this background scattering arising from the dispersion of the ZnO particles and also whether we can justify the method to correct the E

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Macromolecules the CSANS and SAXS profiles are superposable one another only by the vertical shifts as shown in the bottom profiles indicated by the solid line marked (1) and filled symbols marked by (5), despite the differences in the scattering length densities, bneutron and bX‑ray, of the various ingredients for the neutron and X-ray beam, respectively, as summarized in Table 3. This piece of evidence reflects that the SBR composites without the fillers can be essentially treated as a two-phase system in which the ZnO particles are dispersed in the virtually homogeneous matrix of the polymers and the ingredients other than ZnO. Moreover, the ZnO particles must dominantly contribute to the net CSANS and SAXS scattering profiles at least in the q-range covered in the experiment. Similar results are well expected for the cured n-SBR composites without the fillers. Figure 4a also includes the partial structure factor SZnO−ZnO(q) which is shown by the filled symbols marked (4).23 This structure factor was extracted from a set of the CSANS profiles from the Si/fSBR composite which were obtained by varying the scattering length density (SLD) of the matrix SBR phase on the basis of the contrastvariation method with the DNP technique.24−26 This self-structure factor from the ZnO particles represents in principle purely the scattering from the ZnO particles themselves independently of the Si fillers which coexist together with the ZnO particles and are dispersed in the Si/f-SBR composites. Although the q-range covered by this structure factor (4) is relatively narrow compared with the CSANS profile (3), the structure factor [profile (4)] is essentially identical to the CSANS profiles from the f-SBR and n-SBR composites [profile (3)], revealing that the characteristic size of the ZnO particles developed in the Si/f-SBR composite is essentially identical to that developed in the f-SBR and n-SBR composites without the Si filler under the given compounding conditions employed in this work; the sizes of ZnO particles evaluated were the same in these matrices, 55 nm in the average radius and 0.68 nm in the standard deviation from the average size in the context of the log-normal distribution. This piece of evidence satisfies at least the necessary condition for the background correction method. The sufficient condition will be discussed later in section 3.4. 3.3. Comparisons between the Scattering Profiles from the Composites with the Fillers (Si/SBR and CB/SBR) and Those without the Fillers (SBR). Figure 5 presents comparisons between the scattering profiles from Si/f-SBR and those from f-SBR without Si (part a) and comparisons between those from Si/n-SBR and those from n-SBR without Si (part b). On the other hand, Figure 6 presents

Figure 6. CSANS profiles from (a) CB/f-SBR and (b) CB/n-SBR. Profiles (1) to (3) have the same meaning as those specified in Figure 5 with the replacement of Si by CB.

corresponding comparisons between CB/f-SBR and f-SBR and those between CB/n-SBR and n-SBR. More precisely, in each part of each figure the top two profiles numbered (1) and (2) compare the CSANS profiles from the composites with and without the fillers, respectively. The main components which contribute to the CSANS in these systems are Si, CB, and ZnO particles in terms of both the size and composition of the components, while the SBR and other components more or less form a uniform matrix for those particles. The contribution of the scattering from each component to the net scattering is determined by the scattering power ΔbP,n2, i.e., the square of the difference in the SLD of each component bP,n and the SLD of the SBR polymer bSBBR,n as the major component of the matrix for neutron, ΔbP,n2 ΔbP,n = bP,n − bSBR,n

(2)

where the subscript P designates the particle, Si, CB, ZnO, etc. In Figures 5 and 6 the CSANS profiles from the composites with the fillers [profile (1)] have the scattered intensity higher than those without the fillers [profile (2)] in the whole q range covered, which is natural because of the extra contribution of the fillers (Si or CB) to the scattering (see Table 2). The difference becomes small only at q’s near qC as shown by the arrow marked by qC in those figures. We note that the differences in the scattering intensity at qC are still substantially large, though they may appear to be small in the compressed ordinate scale. These differences in the scattering intensity can be accounted for on the basis of the differences in the scattering powers between ΔbZnO,n2 and ΔbSi,n2 or those between ΔbZnO,n2 and ΔbCB,n2 as shown in Table 3 and the differences in the volume fraction of ZnO, Si, and CB (3.3 × 10−3, 1.6 × 10−1, and 1.3 × 10−2, respectively) evaluated from Table 2. 3.4. Extraction of CSANS Profiles Arising from the Spatial Distribution of the Fillers (Si or CB) Only. We treated the scattering from the SBR composites without the fillers (but with ZnO) as the background scattering for the filler/SBR composites (Si/f-SBR, Si/n-SBR, CB/f-SBR, and CB/n-SBR), based on an assumption that the same background scattering exists for the scattering from the corresponding filler/SBR composites. This assumption is validated by the fact that the composites with and without the fillers prepared under the same compounding process and condition have essentially identical size of the ZnO particles as presented by the profiles (3) and (4) in Figure 4a. Thus, we subtracted the contribution of the scattering from the ZnO particles to the net scattering from the filler/SBR systems by subtracting the background scattering (profile 2) from the

Figure 5. CSANS profiles from (a) Si/f-SBR and (b) Si/n-SBR. In each part, profiles (1) and (2) are for the profiles with and without the fillers, respectively. Profile (3) is the profile corrected for the scattering from the ZnO particles obtained by subtracting profile (2) from profile (1). F

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Macromolecules net scattering (profile 1) from the composites with the fillers containing ZnO by the same amount (see Table 2).27 The correction for the contribution of ZnO to the net scattering from the filler/SBR systems is also based upon the assumption that the interparticle interference between ZnO and fillers are immaterial. This assumption is also anticipated to be legitimate from a viewpoint of a very dilute dispersion of ZnO particles in the matrix composed of the filler/rubber dispersion. The volume fractions of the ZnO particles in the set of ZnO and filler(s) in Si/CB/SBR, Si/SBR, and CB/SBR were 0.02, 0.03, and 0.03, respectively. Thus, the ZnO particles give rise to the independent scattering, contributing independently to the net scattering from the composite and thereby satisfying the sufficient condition for the background correction method. Profile (3) shown by the solid line in parts a and b of Figures 5 and 6 presents the CSANS profiles corrected for the contribution of the ZnO particles by the subtraction as described above. The corrected profile (3) shown by the solid lines is compared with the uncorrected net scattering profile (1) shown by the symbols in parts a and b in Figures 5 and 6. The results clearly revealed that the ZnO particles in these systems hardly contributes to the net CSANS profile in the whole q-range covered, which is common for all the four composites with the fillers. However, it should be noted that this conclusion is not self-evident but is obtained only after the laborious, rigorous analyses. We conducted the same analyses for the SAXS profiles measured for the same four filler/SBR composites and SBR composites without the fillers (but with ZnO) as presented in Supporting Information S1 with Figures S1 and S2. The result revealed that the corrected CSANS and SAXS profiles are superposable each other only by the vertical shift in the double-logarithmic plot of the intensity vs q, commonly for all the four composites. This revealed an important conclusion that the corrected (or even uncorrected) CSANS and corrected SAXS profiles are essentially described by the two-phase structure having the dispersed Si and CB fillers in the homogeneous matrix composed of the SBR polymer and the other components, in which each component has a small amount relative to the polymer. 3.5. Summary and Conclusion Deduced from the Experimental Results. Figure 7 summarizes the CSANS profiles for the Si

composites: Si/f-SBR (profile 1), Si/n-SBR (profile 2), and Si powder/air (profile 3) and for the CB composites: CB/f-SBR (profile 4), CB/n-SBR (profile 5), and CB powder/air (profile 6), where the intensity scale for the Si and CB composites should be referred to the left- and right-hand side ordinate scale, respectively. Note that the profiles (1), (2), (4), and (5) are corrected for the scattering from ZnO as described in the preceding section. The crossover border lines between regions I and II and between regions II and III are the same ones commonly set in Figures 1−4 in order to qualitatively indicate the change in the nature of the scattering entity which brings about the change in the q dependence of the scattering intensity. The power-law exponent α in eq 1, for the scattering in region III from Si/f-SBR, Si/n-SBR, CB/f-SBR, and CB/n-SBR satisfied 1 ≤ α ≤ 3, so that the scattering arises from the mass-fractal (MF) objects in 3dimensional (3D) space with the MF dimension Dm given by

Dm = α

(3)

On the other hand, the exponent α for the scattering in region III from Si powder/air and CB powder/air satisfied 3 < α < 4, so that the scattering arises from the surface fractal (SF) objects in 3D space with the SF dimension Ds given by Ds = 6 − α

(4)

This SF structure may be built up on the surfaces of the interstitial space constructed by the closely packed clusters as the building block of this structure. These clusters must be composed of the aggregates bound together by the van der Waals type attractive interaggregate interactions. Close analyses of the scattering profiles elucidated that the crossover values q’s depend on the composites as follows: (1) the crossover borderline between region I and II actually depends on the filler and should be thus indicated by the q values shown by the arrows marked qSi and qCB which are common for all the compounds involving Si and CB, respectively; (2) the crossover borderline between region II and III depends on the individual composite defined (1) to (6) in the figure and should be replaced by the arrows marked qC,1 to qC,6, respectively. The crossover q values defined by qSi and qCB were determined as the values, above which the scattering profiles become independent of the matrix phase, i.e., f-SBR, n-SBR, and air, but depend only on the filler employed, i.e., either Si or CB. The crossover q values defined by qC, i (i = 1−6) were determined as the values, above which the scattering profiles in the smallest q range deviate from the straight line with the slope −α in the doublelogarithmic scale. Figure 7 indicates also the theoretical scattering profiles of isolated spherical primary particles for the Si fillers and for the CB fillers best fitted with the CSANS profiles at q > qSi and q > qCB, respectively, as shown by the profiles (7) and (8) for Si and the profile (9) for CB. The theoretical profile (7) for the Si spheres of the average radius ⟨Rprimary⟩ = 10.4 nm and the standard deviation σR,primary = 0.33 nm for the Gaussian size distribution of the particle size fitted well the CSANS from all the composites with the Si fillers at q ≥ qSi, including the firstorder form factor peak shown the thick solid arrow marked qF. The theoretical profile for the monodisperse sphere with Rprimary = 11 nm and σR,primary = 0 nm is also presented as a reference. The theoretical profiles for the CB spheres with ⟨Rprimary⟩ = 12.0 nm and σR,primary = 0.35 nm also fitted well the CSANS profiles from all the composites with the CB fillers at q ≥ qCB. It is crucial to note here that profiles (7) and (9) are not directly interconnected to the power-law scattering profiles in profiles (1) to (3) and those (4) to (6) in region III, respectively, which arise from the fractal structure. This clearly indicates that the primary particle itself is not a building block of the fractal structures; the clusters of the primary particles and their aggregates bound together by the SBR chains must be the building blocks of the fractal structure. Note that this conclusion is valid also for the surface fractal structure in the Si and CB powders in air, though the previous work9 claimed that the mass-fractal structure in the CB powder dispersed in toluene with a stirrer using ultrasonic energy is built up by the aggregates themselves but not by the clusters of the aggregates. The difference in these two

Figure 7. CSANS profiles from the Si composites (1) Si/f-SBR, (2) Si/n-SBR, and (3) Si powder/air and the CB composites (4) CB/fSBR, (5) CB/n-SBR, and (6) CB powder/air. Profiles (7) to (9) are the theoretical scattering profiles from the single primary particle with ⟨Rprimary⟩ = 10.4 nm, σR,primary = 0.33 nm, Rprimary = 11.0 nm and σR,primary = 0 nm, and ⟨Rprimary⟩ = 12.0 nm and σR,primary = 0.35 nm, respectively. G

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Macromolecules Table 4. Hierarchical Structures of the Composites structure level

relevant q-window

structure entity primary particles of Si, CB fillers aggregates of primary particles fused together into multimers which are unbreakable during the compounding process clusters composed of the primary particles and their aggregates bound together by the cured SBR fractal structures (agglomerates) built up by the clusters and dispersed in the cured SBR matrix bulk compounds composed of an assembly of the fractal structures

1 2 3 4 5

region I (q > qprimary) region I (qagg < q < qprimary) region II region III q < 10−4 nm−1

Figure 8. Hierarchically self-organized dissipative structures for the Si or CB fillers in the cured f-SBR and n-SBR.

the clusters in Vir given by ϕC = NCVC/Vir with NC and VC being the number of the clusters in Vir and the volume of the cluster. SMF(q) is the structure factor of the MF object. In the isotropic system SMF depends only on q = |q| and given by29

results may be due to the difference in the CB powders used and in the treatments of the powders, including the surrounding medium as well (air vs solvent). The scattering from the clusters themselves contributes to the scattering profiles in region I and II at the values q’s higher than those qC,1 to qC,6 for profiles (1) to (6), respectively. The crossovers from region I to II occurs at qSi and qCB. In this stage, we can disclose hierarchical structures of the Si and CB fillers dispersed in the cured SBR being composed of the five structure levels: level 1 to level 5 as summarized in Table 4 and as schematically shown in Figure 8. Table 4 summarizes the hierarchical structures of the composites with respect to the structure level 1 to 5, structural entity, and relevant q-window corresponding to each of the structure level. The structure level 1 and level 2 are characterized by the scattering profiles in the high-q window (at q > qprimary) and the low-q window (at qagg < q < qprimary) in region I, respectively, where qagg = qSi or qCB defined in conjunction with Figure 7. The structure level 3 and level 4 are characterized mainly by the scattering profiles in regions II and III, respectively. The structure level 5 is characterized by the scattering profiles at q < 10−4 nm−1, the q-window of which was unable to be covered by the present USANS apparatus. The scattering entity corresponding to each of the structural level is schematically illustrated in Figure 8. The details including the definition of qprimary will be further discussed later in section 5.

SMF(q) = 1 +

(qR C)Dm [1 + (qξ)−2 ](Dm − 1)/2 (6)

where Γ (x) is the gamma function of argument x, ξ is the upper cutoff length that characterizes the size of the upper bound of the MF object, and RC is the lower cutoff length of the MF object that is equal to the average radius of the sphere when the cluster is assumed to be spherical. SMF(q) has the following asymptotic behaviors ⎧ q−Dm when q < q < q ⎪ RC ξ SMF(q) ∝ ⎨ ⎪1 when q > qR ⎩ C

4. THEORETICAL BACKGROUND FOR ANALYSES OF SCATTERING PROFILES 4.1. Scattering from the Fractal Object. The differential scattering cross-section28 from the MF object [∂Σ(q)/∂Ω]MF per the volume Vir irradiated by the incident beam is given by 2 ⎛ ϕC ⎞ ⎡ ∂Σ(q) ⎤ pcluster (q) ⎢ ⎥ = ⎜ ⎟SMF(q)∼ ⎣ ∂Ω ⎦MF ⎝ VC ⎠

DmΓ (Dm − 1) sin[(Dm − 1) tan−1(qξ)]

(7)

where qRC ≡ (1/RC) and qξ ≡ (1/ξ) are the upper and lower cutoff wavenumbers of the MF object. The differential scattering cross section of the SF object per Vir, [∂Σ(q)/∂Ω]SF, is given by 2 ⎛ ϕC ⎞ ⎡ ∂Σ(q) ⎤ pcluster (q) ⎢ ⎥ = ⎜ ⎟SSF(q)∼ ⎣ ∂Ω ⎦SF ⎝ VC ⎠

(5)

2

p cluster(q) is the structure factor from the cluster to be where ∼ described in the next section, and ϕC is the volume fraction of

(8)

in the case when the system under consideration is isotropic. The structure factor SSF(q) for the SF object is given by30,31 H

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Macromolecules π 2 2 (Ds − 2) Γ (5 − Ds)R C Ds p l2 2 p sin[π (3 − Ds)/2] × q−(6 − 2Ds) (qR C)Ds [π (3 − Ds)/2]

SLD fluctuations within the cluster. In our earlier paper,37 we neglected the term K by setting K = 1, which affects only the evaluation of the absolute value of ⟨η2⟩/(Δp0)2. In the case when the cluster size is infinitely large, Γ (q) = δ(q), so that eq 11 reduces to

SSF(q) = 1 +

(9)

where pp is the SLD of the filler particle and l2 is the upper cutoff length of the SF structure. SSF(q) given by eq 9 has the asymptotic behaviors as follows: ⎧ −(6 − Ds) ⎪q when 1/l 2 < q < 1/R C SSF(q) ∝ ⎨ ⎪ when 1/R C < q ⎩1

2

∼ pcluster (q) → VC⟨η2⟩γ(q)

and thereby to the equation originally given by the DB theory. In this case the newly introduced term Γ(q) does not play roles at all for the scattering at |q| other than zero. In the case when the cluster is a sphere of radius RC, q is replaced by q, and Γ (q) in eq 11 is given from eq A5 by

(10)

Note that eq 9 was modified version of eq 15 originally given by Wong and Bray30 in order to incorporate the asymptotic behavior given by eq 10 which accounts for the crossover of the scattering profile from the scattering due to the SF object,30 2 p S (q), to that due to the clusters, ∼ (q), with increasing q SF

(12)

2

Γ (q) = σ∼(q)/VC = VCΦ2(u)

(13)

where Φ(u) is well-known as the reduced structure amplitude for a sphere and given by

cluster

across the crossover wavenumber qC = 1/RC (see eqs 8−10). 4.2. Scattering from the Clusters: Extension of the Debye−Bueche Theory to Confined Space. The clusters of the fillers were discovered to be the dispersible unit which builds up both the MF and SF structures for all the six composites shown in Figure 7. The clusters are composed of the primary particles, and their aggregates fused together into dimers, trimers, multimers, etc., as indicated schematically in Figure 8c and will be detailed later in section 5.2. However, their internal structures are not well-defined because the spatial distributions of the aggregates of various size and shapes in the clusters with respect to both their positions and orientations are not known. Thus, we will take an advantage of a statistical description of the scattering in terms of the fluctuation of the SLD within the cluster. Debye and Bueche (DB)32 developed the statistical theory of the scattering from the systems having the density fluctuations in infinite space. Here, we shall extend and generalize the DB theory to the system having SLD fluctuations, pcluster(r), in a confined space relevant to the cluster size. In this section, we discuss the scattering equations generally applicable to neutron, X-ray, and light scattering and thereby define the SLD as p(r) instead of b(r) used conventionally for the neutron scattering.33 A. Basic Formula. The structure factor for the cluster, 2 ∼ p (q), is given by

Φ(u) =

3 (sin u − u cos u), u3

u = qR C

(14)

On the other hand, γ(q) in eq 11 is given by the DB theory for the isotropic SLD fluctuations γ(q) =

8πa3 (1 + a 2q2)2

(15)

where a is the correlation distance of the DB correlation function γ(r) = exp(−r/a) for the statistically random fluctuations. B. 3D Convolution. Noting that the convolution product defined by eq A11 in the Appendix for spherically symmetric functions of f(q) and g(q) in 3D space is given by f (q) ∗g (q) =

2π q

∫0

u+q

∞

du[uf (u)]

∫u−q

dy[yg (y)]

(16)

Φ2(q)∗γ(q) involved in eqs 11 and 13−15 is given after the integration of the term related to γ(q) by Φ2(q)∗ γ(q) = 32π 2a3

∫0

∞

Φ2(u)u 2 du [1 + a (u − q2)][1 + a2(u + q)2 ] 2

2

(17)

The limiting value of eq 17 at q → 0 is given by

cluster

⎡ ⎤ ⎛ ⎞ 1 ⟨η2⟩ ⎟ ∼ ⎥ pcluster (q) = KVC(Δp0 )2 ⎢Γ (q) + ⎜⎜ Γ ( q ) ∗ γ ( q ) ⎢⎣ ⎥⎦ K ⎝ (Δp0 )2 ⎟⎠ 2

(11)

Φ2(u)u 2 32π 2 ⎛ a ⎞ du ≅ ⎜ ⎟ 2 2 2 3 ⎝ RC ⎠ (1 + a u ) (18)

where K = (2π) , Δp0 = p0 − pm with p0 and pm being the mean SLD of the clusters and their surrounding medium, respectively, Γ (q) is the Fourier transform of the correlation function of the single cluster, ⟨η2⟩ is the mean-squared fluctuations of the SLD within the cluster which reflects the spatial distribution of the aggregates within the cluster, and γ(q) is the Fourier transform of the correlation function of the SLD fluctuation η(r) within the cluster. Γ (q)∗γ(q) is the convolution product of Γ (q) and γ(q) defined by eq A11 in the Appendix. The detailed calculations leading to eq 11 will be given in the Appendix. The first term of the right-hand side (rhs) of eq 11 is responsible for the scattering from the homogeneous cluster with ⟨η2⟩ = 0, while the second term for the scattering from the

since Φ2(u) ≅ 1 at u ≲ 1/RC and Φ2(u) ≅ 0 at u > 1/RC, and (1 + a2u2)−2 ≅ 1 at u ≲ 1/RC in the case of (a/RC) ≪ 1. C. Structural Interpretation of the Internal Fluctuations of the Clusters with “Equivalent-Sphere” Concept. In this section, we focus on to extract some information on the internal structure of the clusters from the statistical parameters ⟨η2⟩ and a by introducing the following concept of “equivalent spheres” relevant to the aggregates. The concept assumes that an aggregate of an arbitrary shape and size which has statistically random orientation in the clusters is able to be treated statistically as a sphere of radius Reqv having its volume and SLD equivalent to the volume and the SLD of the aggregate, respectively. The equivalent spheres, which have a size distribution corresponding to the size distribution of the

lim Φ2(q)∗ γ(q) = 32π 2a3

q→0

3 34−36

I

∫0

∞

3

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Macromolecules aggregates, are statistically dispersed randomly within the clusters. Debye−Anderson−Brumberger (DAB) derived the correlation distance a given by38 a = 4ϕ(1 − ϕ)V /S

⟨X(R C)⟩ =

(24)

The average structure factor is then given from eqs 11, 13−15, and 24 by ⎡ ⎛ ⎞ 2 1 ⟨η2⟩ ⎟ ⟨∼ pcluster (q; R C)⟩ = (Δp0 )2 K ⎢⟨VC 2 Φ2(u)⟩ + ⎜⎜ ⎢⎣ K ⎝ (Δp0 )2 ⎟⎠

(19)

for the statistically random two-phase system composed of particles of an arbitrary shape and size dispersed randomly in the matrix. Here ϕ is the volume fraction of the particles; V and S are the total volume and interface area of the two-phase system, respectively. In our case the particles, ϕ, V, and S in the DAB theory correspond to the aggregates (of the primary particle as will be clarified in section 5.1 later), the volume fraction of the aggregates within the cluster, the volume of the cluster VC, and the interface area of the aggregates Sagg, within the cluster, respectively. The volume fraction of the aggregate is equal to that of “equivalent spheres”, defined hereafter as ϕeqv. Let the total number of the equivalent spheres in the cluster be Neqv; then VCϕeqv = (4π/3)Reqv3Neqv and Sagg = 4πReqv2Neqv. Thus, a = (4/3)(1 − ϕeqv)Reqv. When the polydispersity in Reqv exists, it should be replaced by the average quantity ⟨Reqv⟩ a = (4/3)(1 − ϕeqv )⟨R eqv⟩

∫ P(R C)X(R C) dR C/∫ P(R C) dR C

× ⟨VC 2 Φ2(u) ∗⟩

⎤ 8πa3 ⎥ (1 + a 2q2)2 ⎥⎦

(25)

Here, a is assumed legitimately to be independent of RC because a depends only on ϕeqv and Reqv but not essentially on RC. Equations 18 and 25 lead to the limiting value at q → 0 2

lim

q→0

p (q; R C)⟩ ⟨∼ K (Δp0 )2 ⟨VC 2⟩ =1+

≅1+

⎛ ⎞ 128π 3 ⟨VC⟩ ⎜ η2 ⎟ 3 a 9K ⟨VC 2⟩ ⎜⎝ (Δp0 )2 ⎟⎠

3 ⎛ ⎞ 32π 2 ⟨R C ⟩ ⎜ η2 ⎟ 3 a 3K ⟨R C6⟩ ⎜⎝ (Δp0 )2 ⎟⎠

(20)

(26)

Thus, a depends on the volume fraction of the equivalent spheres ϕeqv and ⟨Reqv⟩. The mean-squared amplitude of the fluctuation of the SLD in the cluster, ⟨η2⟩, is given by

E. Theoretical Structure Factor for the Cluster. Figure 9 presents an example of the numerically calculated theoretical

⟨η2⟩ = (Δpeqv )2 ϕeqv (1 − ϕeqv )

(21)

where Δpeqv ≡ peqv − pm with peqv and pm being the SLD of the equivalent spheres and the matrix, respectively. peqv is identical to the SLD of the aggregates pagg or the primary particles of the fillers given by bneutron or bX‑ray for the fillers in Table 3. Thus, ⟨η2⟩ depends on ϕeqv for a given Δpeqv. The matrix is either the cured SBR surrounding the clusters in the case of filler/SBR or air in the case of filler powder/air. The mean SLD of the cluster, p0, is given by p0 = Δpeqvϕeqv + pm. Moreover, the reduced mean-squared fluctuation of the SLD with respect to (Δp0)2 is given by 1 − ϕeqv ⟨η2⟩ = , 2 ϕeqv (Δp0 )

Δp0 ≡ p0 − pm

Figure 9. Theoretical scattering profile from the cluster and its various components: (1) the scattering from the internal structures of the cluster, (2) the scattering from the homogeneous cluster, and (3) the net scattering structure factor from the cluster which is given by a sum of components (1) and (2). K = 8π3, ⟨η2⟩/(Δp0)2 = 1.53, ⟨RC2⟩/⟨RC6⟩ = 2.77 × 10−6 nm−3, a = 14.6 nm, and σRC = 0.54 nm.

(22)

and thereby depends only on ϕeqv. Thus, in the context of the equivalent sphere concept, a and ⟨η2⟩/(Δp0)2 give ϕeqv and ⟨Reqv⟩ as a fundamental structural information on the cluster. In the case when most of the aggregates exist within the clusters, the net volume fraction of the filler in the composite ϕfiller,net is approximately given by ϕfiller,net ≅ ϕeqv ϕcluster

2

p cluster(q;RC)⟩ given by eq 25 for a given set of structure factor ⟨ ∼ parameters: K = 8π3, ⟨η2⟩/(Δp0)2 = 1.53, a = 14.6 nm, ⟨RC3⟩ = 2.61 × 104 nm3, and ⟨RC6⟩ = 9.42 × 109 nm6 where P(RC) is given by

(23)

P(R C) ∝ exp[(R − ⟨R C⟩)2 /2σR C]

where ϕcluster is the volume fraction of the clusters in the whole composite. In this case, ϕcluster can be evaluated from ϕfiller,net, known from the composition of the composite (Table 2) and the densities of the components (Table 3), and ϕeqv, evaluated by using eq 21 or 22. D. Polydispersity in the Cluster Size. When there is a polydispersity in the cluster size RC, the RC-dependent terms in eq 14, X(RC), should be average with the distribution function of the cluster size, P(RC). The average quantity ⟨X(RC)⟩ is given by

(27)

with ⟨RC⟩ = 22.2 nm and σRC = 0.54 nm. The structure factor normalized by K(Δp0)2⟨VC2⟩ was plotted as a function of q in the double-logarithmic scale. In the figure, profile (1) (the red broken line) and profile (2) (the blue solid line) present the contribution of the scattering from the DB type fluctuations confined in the cluster (the second term in the rhs of eq 25) and that from the homogeneous cluster (the first term in the rhs of eq 25), respectively, to the net reduced structure factor J

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Macromolecules Table 5. Summary on the Parameters Characterizing the Hierarchical Structures of the Fillers

Estimated by the best-fit of the theoretical and experimental scattering profiles at the high q range in region I (q ≥ qp = 0.15 nm−1). bThe lower cutoff length is specified by ⟨RC⟩, while the upper cutoff length is larger than ∼20 μm [= (2π/3 × 10−4) nm], the largest length scale covered by the present experiments. a

influences the scattering in region III and characterizes the fractal structure and dimension. These parameters evaluated from the best-fit between the theoretical and experimental scattering profiles based on eqs 5, 6, 8, 9, and 25 will be shown later in Table 5. In addition to these parameters described above, Table 5 contains also such parameters as (Δp0)2, ⟨η2⟩/(Δp0)2, ϕeqv, Reqv, and ϕcluster, all of which are not independent parameters for the model. Δp0 defined by eq 22 is known from the SLDs of the fillers pagg and the matrix rubbers (pm) and the estimated value of ϕeqv. The parameters ϕeqv, Reqv, and ϕcluster can be evaluated from ⟨η2⟩ and a. Nevertheless, these parameters give very useful information on the structure level 3 and thereby are included in Table 5. These two sets of the parameters are distinguished in the structure levels 3 and 4 in Table 5 by the rows with or without the gray background. The parameters in structure level 1 in Table 5 were obtained independently of the theory described in section 4, and their analyses will be discussed in the next section. 5.2. Primary Particles and Their Aggregates. Here, we focus on to analyze the primary particles and their aggregates which form the cluster by analyzing the profile in a high q-range

shown by profile (3) drawn by the dash-dot line. The characteristic parameters for the cluster are ⟨η2⟩/(Δp0)2, a, ⟨RC⟩, and σRC. Each set of the parameters, e.g., (⟨η2⟩/(Δp0)2, a) or (⟨RC⟩, σRC), differently affects the reduced structure factor, as demonstrated by curve (1) and curve (2), respectively, which enables to assess each parameter with a reasonable precision through the best fit between the theoretical and experimental scattering profiles as will be detailed later in section 5.3.

5. ANALYSIS 5.1. List of Parameters Needed in the Model. Before conducting the detailed theoretical analysis of the experimental scattering profiles, it will be useful to list up the parameters, together with a summary explanation, which are required for the theoretical analysis of the experimental profiles based on the scattering theory described in section 4. These parameters are (1) ⟨η2⟩ and a which primarily influence the scattering in region I and characterize the internal structure of the cluster, (2) ⟨RC⟩ and σRC which primarily influence the scattering in region II and characterize the cluster as the building block of the fractal structure, and (3) Dm or Ds which primarily K

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Macromolecules

14. The differential scattering cross section was averaged for the polydispersity in Rprimary, the distribution function of which was given by eq 27 with replacements of RC and σRC by Rprimary and σR,primary, respectively. The theoretical profiles numerically calculated for ⟨Rprimary⟩ = 10.4 nm, and σR,primary = 0.20 nm gave the best fit with the experimental profile for Si powder/air at q ≥ qprimary. All the theoretical profiles #1 to #4 merge together and become identical at q ≥ qprimary, as indicated by the arrow marked with qprimary, implying that the scattering in this q range dominantly arises from the primary particle even in the case when the particles form multimers, confirming that the arrow marked qF is related to the q value at the form factor peak or shoulder for the primary particle, which provides the basis that this part of the scattering enables a reliable evaluation on the size of the primary particle, ⟨Rprimary⟩ and σR,primary. The theoretical scattering profiles at q < qprimary show the trend such that the intensity systematically decreases as shown by the downward arrow and increases as shown by the upward arrow with increasing n at q larger and smaller than the crossover wavenumber qx shown by the arrow marked with qx in Figure 10b, respectively, due to the increasing destructive and constructive interprimary-particle interference effects on the scattering from the aggregates, respectively. The best-fit between the experimental and theoretical scattering profiles at qP ≤ q ≤ qX was obtained for n = 3. Thus, the analysis concludes that the aggregates are most likely dimers or trimers of the primary particles, though the monomers (primary particles) and tetramers also may exist as minor components of the building blocks for the cluster as shown in Figure 8c. The deviation of the theoretical scattering profiles for the dimer and trimer from the experimental profile is observed at q ≤ qx. This deviation is expected to be due to the interparticle interference effects between the aggregates within the cluster. The results of those analyses described above are summarized in Table 5 as the structure levels 1 and 2. 5.3. Fractal Structures Built Up by Clusters. The structure analysis in the preceding section identifies the primary particles and their aggregates of the fillers as the basic structural units designated as levels 1 and 2 in Figure 8, respectively, in the hierarchical structures of the composites. This section focuses on the analysis of the higher order structure and its structural unit, i.e., the cluster (the level 3 structural unit) and the fractal structures (the level 4 structural unit) built up by the clusters. Figure 11 summarizes the theoretical analyses of all the experimental CSANS profiles shown in Figure 7 based on eqs 5, 6, 8, 9, and 25. Parts a−c present the results for Si/f-SBR, Si/ n-SBR, and Si powder/air, respectively, while parts d−f present those for CB/f-SBR, CB/n-SBR, and CB powder/air, respectively. In each figure, profile #6 shown by the symbol O is the experimental profile designated as Iexp and best-fitted with the theoretical profile #5 shown by the solid line designated as Itheory. Each theoretical profile in part (a) for Si/f-SBR, (b) for Si/n-SBR, (d) for CB/f-SBR, and (e) for CB/ n-SBR is decomposed into the profile #4 from the mass-fractal structure (designated SMF and given by eq 6) and profile #3 from the cluster (designated Scluster and given by eq 25), while each theoretical profile in part (c) for Si powder/air and in part (f) for CB powder/air is decomposed into profile #4 from the surface-fractal structure (designated SMF and given by eq 9) and profile #3 from the cluster. Commonly to all the six scattering

in region I. Figure 10a presents the scattering profiles for (1) Si/f-SBR, (2) Si/n-SBR, and (3) Si powder/air. It is important

Figure 10. (a) Experimental scattering profiles from (1) Si/f-SBR, (2) Si/n-SBR, and (3) Si powder/air in the high q range and (b) the bestfit of the profile from (3) Si powder/air with the theoretical scattering from the monomer (#1), dimer (#2), trimer (#3), and tetramer (#4) of the primary particles characterized by ⟨Rprimary⟩ = 10.4 nm and σR,primary = 0.20 nm.

to note that the three profiles were identical at q ≥ qSi ≅ 0.15 nm−1; they were different at q < qSi. The latter fact means that the spatial distribution of the Si fillers within the cluster is different in the length scale r of r > 1/qSi, depending on the nature of the matrix (f-SBR, n-SBR, and air), while that in the length scale r < 1/qSi is the same, independent of the matrix but dependent only on the Si filler itself. Thus, in this q range we can precisely assess the primary particles and their aggregates, independently of their matrix phases. The profiles in this q range showed a broad maximum or shoulder at q ≡ qF ≅ 0.58 nm−1 as indicated by the arrow marked by qF due to the firstorder form factor peak or shoulder of the primary particle as will be clarified below. Figure 10b presents the best-fit of the experimental scattering profile (3) for Si powder/air shown by the filled circles and the broken line with the theoretical profile from a monomer (profile #1), a dimer (profile #2), a trimer (profile #3), or a tetramer (profile #4) of the primary particle(s). Here the dimer, trimer, or tetramer are fused together at the contact point(s) to form the aggregate. The trimers and tetramers employed here are assumed to form the equidistant triangle and tetrahedron, respectively, as schematically shown in Figure 8b. The differential scattering cross section per Vir from the randomly oriented, isolated, n-mers composed of the identical spheres of the radius Rprimary is given by the Debye formula39 ∂Σ(q; R primary ) ∂Ω

= ϕprimary Vprimary Δp2 Φ2(qR primary )1(q)

1(q) = 1 +

2 n

n−1

n

∑ ∑ i=1 j=i+1

sin qDij qDij (28)

where ϕprimary is the volume fraction of the primary particles in Vir, Δp is the difference of the SLD’s between the primary particles and their matrix, Dij is the intersphere distance, and n is the number of the primary particles with the volume Vprimary fused together in the aggregates, and Φ(qRprimary) is given by eq L

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cluster in air by the van der Waals interaggregate attractive interactions. The correlation distance a and the reduced meansquared fluctuations of the SLD ⟨η2⟩/(Δp0)2 further characterize statistically the internal structure of the cluster. The detailed discussion on these parameters will be given later in section 6. 5.5. Parameters Characterizing the Fractal Structures. Three parameters as described immediately below were obtained from profile #4 which gave the best fit between Iexp(q) and Itheory(q) in region III and characterize the structure level 4 in the hierarchical structures of the composites. The mass-fractal dimension Dm or the surface fractal dimension Ds was obtained from the power law behavior of profile #4 for SMF(q) given by eq 6 or from that for SSF(q) given by eq 9, respectively. The lower cutoff size or the upper cutoff wavenumber of the fractal structure for the six composites is the size of the cluster ⟨RC⟩ or the crossover wavenumber qC, respectively, which are clearly shown to vary depending on the type of the filler (Si, CB) and the matrix (f-SBR, n-SBR, air). The clusters are building blocks commonly for both the mass and surface fractal structure. The upper cutoff size or the lower cutoff wavenumber of the fractal structure could not be assessed by this experiment because the power-law scattering given by eqs 7 or 10 was observed down to the smallest q value (qmin = 3 × 10−4 nm−1) covered by this experiment, implying that the upper cutoff size is greater than ∼2π/qmin ≅ 20 μm and the lower cutoff wavenumber is smaller than qmin = 3 × 10−4 nm−1. A detailed discussion on the parameters will also be given in section 6.

6. DISCUSSION 6.1. Structure Level 1 and Level 2. The primary particles of Si and CB fillers characterized in section 5.2 are summarized in Table 5 as the structure level 1. In the case of the Si filler used in this work, the primary particles are fused together into the aggregates, most of which are the dimer and trimer of the primary particles, as shown in Table 5 as the structure level 2. Although the aggregates for the CB filler used in this work were not investigated, they are expected to exist.9 6.2. Clusters Formed in Filler Powder/Air. The size of the cluster ⟨RC⟩ for Si powder/air (27.9 nm) is large compared with that for Si/f-SBR (18.1 nm) and Si/n-SBR (22.2 nm) as shown in Table 5 as the structure level 3. Similarly, the size of the cluster for CB powder/air (28.9 nm) is larger than that for CB/f-SBR (21.7 nm) and CB/n-SBR (25.3 nm). These results are interpreted as follows. In the densely packed Si or CB powder in air, the aggregates of Si or CB in the cluster are weakly bound together by the van der Waals force. When the filler powders are mixed together with bulk SBR, the stress involved during the mixing process of the SBR bulk and the filler powder, which is expected to depend on the strength of attractive interactions between the SBR chains and the fillers, will break the weak physical force between the aggregates to result in formation of the clusters of the aggregates bound together by SBR chains. This may suggest the following:

Figure 11. Best-fit of the experimental scattering profiles Iexp (#6 shown by unfilled circles) with the theoretical scattering profiles Itheory (#5 shown by the solid lines) for (a) Si/f-SBR, (b) Si/n-SBR, and (c) Si powder/air, (d) CB/f-SBR, (e) CB/n-SBR, and (f) CB powder/air. Profiles 1 to 3 (designated SDB, Shomog.C, and Scluster, respectively) in each part of this figure present the structure factors related to profiles (1) to (3) in Figure 9, respectively, while profile 4 presents the massfractal structure factor SMF for parts (a), (b), (d), and (e) and the surface-fractal structure SSF for parts (c) and (f), both of which are built up by the cluster as the lower cutoff object.

profiles, profile #3 from the cluster is given by the sum of profile #1 (designated SDB given by the second term on the rhs of eq 25) and profile #2 (designated Shomong.C given by the first term on the rhs of eq 25), the physical meanings of which were already discussed in conjunction with Figure 9. Good agreements were obtained between Iexp and Itheory for all the six composites. Thus, the analyses would elucidate various important parameters characterizing the cluster and fractal structures, as summarized in the rows labeled as the structure level 3 and those labeled as the structure level 4 in Table 5, respectively. 5.4. Parameters Characterizing the Cluster. These parameters in structure level 3 were obtained from profile #3, Scluster(q), which gave the best fit between Iexp(q) and Itheory(q) in regions I and II. The parameters ⟨RC⟩ and σRC which concern the size of the cluster control the crossover q value shown by the arrow marked with qC around which profiles #3 and #4 exhibit the characteristic change with respect to the q dependence. The cluster is composed of the aggregates bound together by SBR in the case of (Si or CB)/(f-SBR or n-SBR). On the other hand, in the case of the (Si or CB) powder/air systems, the aggregates are bound together into the

(AI)Si/SBR > (AI)Si/Si or (AI)SBR/SBR , (AI)CB/SBR > (AI)CB/CB or (AI)SBR/SBR

where (AI)K/SBR is the attractive interaction strength between K and SBR chains (K = Si, CB), and (AI)J/J is the attractive interaction strength between J and J (J = Si, CB, SBR chains). 6.3. Clusters Formed in Filler/SBR: Cluster Size and Internal Structures. Let us first discuss the parameters listed M

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Table 6. Conclusions 1−4 Obtained on the Mass-Fractal Structure Characterized by (RC)K and (Dm)K and Conclusions 5−8 Obtained on the Internal Structures of the Cluster Characterized by the Relative Mean-Squared Fluctuations (RMF) Defined by (RFM)K ≡ [⟨η2⟩/(Δρ0)2]K, aK, and (ϕeqv) conclusion

(RC)Ka

(Dm)Ka

remarkb

1

(R C)Si/ f ‐ SBR < (R C)Si/ n ‐ SBR

(Dm)Si/ f ‐ SBR > (Dm)Si/ n ‐ SBR

(AI)Si/ f ‐ SBR > (AI)Si/ n ‐ SBR

2

(R C)CB/ f ‐ SBR < (R C)CB/ n ‐ SBR

(Dm)CB/ f ‐ SBR > (Dm)CB/ n ‐ SBR

(AI)CB/ f ‐ SBR > (AI)CB/ n ‐ SBR

3

(R C)Si/ f ‐ SBR > (R C)CB/ f ‐ SBR

(Dm)Si/ f ‐ SBR > (Dm)CB/ f ‐ SBR

(AI)Si/ f ‐ SBR > (AI)CB/ f ‐ SBR

4

(R C)Si/ n ‐ SBR < (R C)CB/ n ‐ SBR

(Dm)Si/ n ‐ SBR > (Dm)CB/ n ‐ SBR

(AI)Si/ n ‐ SBR > (AI)CB/ n ‐ SBR

conclusion

(RMF)ka

aKa

remarkc

5

(RMF)Si/ f ‐ SBR < (RMF)Si/ n ‐ SBR

(a)Si/ f ‐ SBR < (a)Si/ n ‐ SBR

(ϕeqv )Si/ f ‐ SBR > (ϕeqv )Si/ n ‐ SBR

6

(RMF)CB/ f ‐ SBR < (RMF)CB/ n ‐ SBR

(a)CB/ f ‐ SBR < (a)CB/ n ‐ SBR

(ϕeqv )CB/ f ‐ SBR > (ϕeqv )CB/ n ‐ SBR

7

(RMF)Si/ f ‐ SBR < (RMF)CB/ f ‐ SBR

(a)Si/ f ‐ SBR < (a)CB/ f ‐ SBR

(ϕeqv )Si/ f ‐ SBR > (ϕeqv )CB/ f ‐ SBR

8

(RMF)Si/ n ‐ SBR < (RMF)CB/ n ‐ SBR

(a)Si/ n ‐ SBR < (a)CB/ n ‐ SBR

(ϕeqv )Si/ n ‐ SBR > (ϕeqv )CB/ n ‐ SBR

(X)K designates parameter X (= RC, Dm, AI, ⟨η2⟩/p02, a or ϕeqv) for the composite K (= Si/f-SBR, Si/n-SBR, CB/f-SBR, or CB/n-SBR). b(AI)K designates attractive interactions in the composite K (e.g., AI between Si and f-SBR if K = Si/f-SBR). cϕeqv is the parameter characterizing the compactness of the packing of the aggregates in the cluster: the larger the value ϕeqv is, the more compact is the packing. a

in the structure level 3 in Table 5. For a given filler particle (Si or CB), the cluster size RC is smaller in f-SBR than in n-SBR as summarized in conclusions 1 and 2 in Table 6. For a given SBR (f-SBR or n-SBR), Si gives rise to the smaller cluster size RC than CB commonly for both f-SBR and n-SBR, as summarized in conclusions 3 and 4 in Table 6. The reduced size distribution of the cluster σRC/⟨RC⟩ is about the same for all the composites (equal to ∼0.02). The parameter ⟨η2⟩ which characterizes the electron density fluctuations within the cluster is larger for CB than for Si for the given matrix, i.e., SBR (f-SBR or n-SBR or air). This is quite rational because the parameter (Δbpn)2 itself is larger for CB than for Si for neutron beam by about 6 times (Table 3). The set of the parameters (⟨η2⟩/(Δp0)2, a) for a given filler particle (Si or CB) reveals the following characteristic feature: ⟨η2⟩/ (Δp0)2 [i.e., (RMF)K ] in f-SBR is smaller than that in n-SBR, and the correlation distance a in f-SBR is smaller than that in nSBR as summarized in Table 6 (conclusion 5 for Si and conclusion 6 for CB). The same set of the parameters for a given rubber (f-SBR or n-SBR) reveals that ⟨η2⟩/(Δp0)2 and a for Si are smaller than ⟨η2⟩/(Δp0)2 and a for CB, respectively, as summarized in conclusion 7 for f-SBR and conclusion 8 for n-SBR in Table 6. The equivalent-sphere concept provides a phenomenally clear interpretation on the conclusions 5−8 obtained above. The values ϕeqv evaluated from ⟨η2⟩/(Δp0)2 (eq 22) for the composites shown in Table 5 consistently elucidate the conclusions as summarized in the remark column for conclusions 5−8. These conclusions together with the conclusion on (ϕeqv)CB/f‑SBR ≳ (ϕeqv)Si/n‑SBR (also obtained in Table 5) lead to the following conclusion:

The trends given by eqs 29 and 30 imply that the system develops a larger cluster with less compact packing of the aggregates in the same order of Si/f-SBR, CB/f-SBR, Si/n-SBR, and CB/n-SBR. This trend also accounts for the volume fraction of the clusters ϕcluster in the whole system increasing in this order, though ϕ cluster for CB/f-SBR is about equal to that for Si/n-SBR. ⟨Reqv⟩ evaluated is 12.5 nm for Si and 16.1 nm for CB. ⟨Reqv⟩ for a given filler (Si or CB) should be naturally identical within the experimental errors and independent of the nature of the matrix rubber. ⟨Reqv⟩ for CB is larger than ⟨Reqv⟩ for Si, which is also rational because Rp for CB is larger than Rp for Si. 6.4. Mass-Fractal Structures. Let us now focus on the mass-fractal structures of the fillers formed in the rubber matrix. For a given filler particle (Si or CB), the mass-fractal dimension Dm is larger for f-SBR than for n-SBR as summarized in conclusion 1 on Dm for Si and conclusion 2 on Dm for CB shown in Table 6. These results together with the results on ⟨RC⟩ and ϕeqv reveal that f-SBR forms a more compact massfractal structure of the fillers with the larger Dm built up by the smaller and more compactly packed clusters of the fillers than n-SBR (conclusions 1 and 5 for Si and conclusions 2 and 6 for CB in Table 6): in other words, n-SBR forms a more open or looser mass-fractal structure of the fillers with the smaller Dm built up by the larger and less compactly packed clusters of the fillers than f-SBR. The same concept is applied to the effect of the filler on the mass-fractal structure in a given rubber matrix (f-SBR or n-SBR): The Si filler forms a more compact massfractal structure built up by the smaller and more compactly packed clusters than the CB filler (conclusions 3 and 7 for fSBR and conclusions 4 and 8 for n-SBR), while the CB filler forms a more open mass-fractal structure built up by the larger and less compactly packed clusters than the Si filler. The results on Dm ≅ 2.5 in Table 5 possibly imply that the mass-fractal structure is formed via the diffusion-limited aggregation40,41 of the clusters involved by the compounding process. 6.5. State of the Filler Dispersion in the SBR Rubbers as Revealed by ⟨RC⟩ and Dm. The results summarized in Table 5 and the conclusions presented in Table 6 and in eqs 29 and 30 characterize precisely and specifically the state of the filler dispersion in the SBR rubber with respect to the

(ϕeqv )Si/ f ‐ SBR > (ϕeqv )CB/ f ‐ SBR ≳ (ϕeqv )Si/ n ‐ SBR > (ϕeqv )CB/ n ‐ SBR (29)

Equation 29 predicts that the compactness of the filler packing in the cluster decreases in this order of Si/f-SBR, CB/f-SBR, Si/ n-SBR, and CB/n-SBR. Moreover, we found that the cluster size ⟨RC⟩K follows the trend opposite to (ϕeqv)K, i.e. ⟨R C⟩Si/ f ‐ SBR < ⟨R C⟩CB/ f ‐ SBR ≲ ⟨R C⟩Si/ n ‐ SBR < ⟨R C⟩CB/ n ‐ SBR (30) N

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of the SBR chains tethered on their surfaces. In this stage, the mechanical energy imposed on the system may be dissipated dominantly through the flow-induced stress relaxation of SBR chains in the matrix. The size of the clusters may depend on the stress level built up and the mechanical energy dissipated in the compounding process which are expected to be intricately coupled with the strength of attractive interactions between the aggregations, that between aggregates and the SBR chains, and that between SBR chains. The compact, small clusters may be formed in the f-SBR matrix, while the loose, large clusters may be formed in the n-SBR matrix, as a dissipative structure. Effective attractive interactions between the stabilized small clusters may further build up the flow-induced diffusion-limited aggregation of the clusters into the mass-fractal structures. The strength of the effective attractive interactions will be determined by a balance of attractive interactions among clusters/clusters, clusters/SBR chains, and SBR chains/SBR chains. From the viewpoint as described above, conclusions 5 and 6 on the effect of the SBR on the compactness of the clusters for Si and CB given in the remark column in Table 6, respectively, may infer that the effective attractive interaction, designated AI in Table 6, between the filler and f-SBR is stronger than that between the filler and n-SBR, giving rise to conclusion 1 for Si and conclusion 2 for CB in the remark column in Table 6, respectively. Similarly, conclusion 7 [(ϕ eqv ) Si/f‑SBR > (ϕ eqv ) CB/f‑SBR ] may infer conclusion 3 [(AI) Si/f‑SBR > (AI)CB/f‑SBR]. Stretching the discussion along the line may lead us to expect that conclusion 8 on (ϕeqv)K infers conclusion 4 on (AI)K. In these systems, the difference in the interactions is attributed to the difference in the chains adhered on the fillers, which may play an important role on the difference in ϕeqv. A systematic decrease of the parameters ϕeqv (given in eq 29) and Dm in the order of Si/f-SBR, CB/f-SBR, Si/n-SBR, and CB/n-SBR may infer the effective filler/SBR attractive interaction strength (AI)filler/rubber satisfying the following relationship:

compactness of the cluster structure and the mass-fractal structure. The small and compact clusters build up the compact mass-fractal structures, while the large and loose clusters build up the open mass-fractal structures as schematically illustrated by Figures 12a and 12b, respectively. The compact cluster and

Figure 12. Schematic illustrations of two kinds of mass-fractal structures: (a) a compact mass-fractal structure having a larger Dm built up by smaller clusters having a higher volume fraction ϕeqv of the aggregates which are more tightly bound by SBR chains as shown in the inset to parts (a) and (b) an open mass-fractal structure having a smaller Dm built up by larger clusters having a smaller volume fraction ϕeqv of the aggregates which are more loosely bound by SBR chains as shown in the inset to part (b).

mass-fractal structure as illustrated in Figure 12a may have been recognized as a well-dispersed filler state, while the loose cluster and mass-fractal structure illustrated in Figure 12b as a poorly dispersed filler state. 6.6. A Plausible Model for Self-Organization of Cluster Structures and Mass-Fractal Structures. What are physical factors which control the compactness of the cluster and mass-fractal structure? During the mixing process of the SBR bulk and the filler powder, the stress is built up in this system. This stress breaks the attractive physical force between the aggregates of the filler comprising the powder and thereby breaks the powder into smaller but a larger number of grains of the powder. This process may serve as a dominant process, through which the mechanical energy imposed on the system is dissipated. The functionalized chain ends of the SBR have a strong attractive interaction with the filler surface, which causes the chain ends to be tethered on to the surface of the grains. The tethered chains may entangle with free SBR chains or SBR chains partly adhered to other grains, which may further raise the stress level of the grains/SBR systems during the given mixing process of the grains and the rubber. The built-up stress will break the grains into smaller ones to dissipate again the imposed mechanical energy, thereby resulting again in the stress relaxation. The smaller grains have new interfacial area, with respect to the SBR matrix, on which new tethered chains are formed: This process will again result in building up the stress level in the system, causing the grains to become even smaller. The process described above which would occur repeatedly during the mixing process will eventually form small clusters composed of the aggregates bound together by the SBR chains. The small clusters will be eventually stabilized against the stress-induced breakup and also against the diffusioncoalescence growth of the clusters due to the steric stabilization

(AI)Si/ f ‐ SBR > (AI)CB/ f ‐ SBR ≳ (AI)Si/ n ‐ SBR > (AI)CB/ n ‐ SBR (31)

This work conducted for the four-different filler/SBR systems confirm the universality about the dispersible unit (equivalent to the cluster in this work) of the filler particles previously proposed for a different filler/rubber system:9 The dispersible unit which builds up the mass-fractal agglomerates is composed of a few aggregates bound together by SBR, rather than the aggregate or the primary particle itself.

7. PERSPECTIVES In this section will apply the basic principles on the scattering method introduced in this work to a more complex filler particle system with some practical interests. The system to be treated here is the two-component filler systems of Si/CB/SBR shown in Table 2 in which the same f-SBR and n-SBR, as shown in Table 1, and the same Si and CB fillers, as shown in Table 2, were employed. This two-component filler system is complex because it may form a two-phase or three-phase system composed of Si, CB, and the homogeneous matrix even after the contributions of the ZnO particles and other additives (shown in Table 2) to the net scattering being successfully subtracted as the background scattering. Figure 13 presents the CSANS profiles (Iexp) from the Si/ CB/f-SBR (red filled circles) and Si/CB/n-SBR composites O

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Figure 14. Schematic presentation of the models for the clusters on the top: (a) Si/CB/SBR = 50/0/100, (b) SI/CB/SBR = 50/4/100, and (c) Si/CB/SBR = 0/54/100 and the models for the mass-fractal structures on the bottom: (d) the fractal built up by the single clusters composed of mixed Si and CB, (e) the fractal built up by the two kinds of clusters, i.e., clusters of Si and those of CB, and (f) two kinds of fractals built up by different type of single clusters; one is composed of only Si, and the other is composed of only CB. In each model, the weight ratio of Si and CB are kept equal to 50/4. SBR is either f-SBR or n-SBR.

Figure 13. Experimental CSANS profiles Iexp from Si/CB/n-SBR (profile #5 shown by the blue filled circles) and Si/CB/f-SBR (profile #5 shown by the red filled circles) and the theoretical scattering profiles best-fitted to the experimental scattering profile for Si/CB/nSBR (the blue solid line #6 and the blue unfilled circles #5, respectively) and that for Si/CB/f-SBR (the red solid line #6 and the red unfilled circles #5, respectively). The best-fitted profiles for Si/CB/ n-SBR and those for Si/CB/f-SBR are presented with the vertical shift of the experimental profiles by a factor 10−2 and 10−3, respectively.

Model B. Si and CB separately form the clusters of Si and the clusters of CB which are essentially identical to those formed for the Si/CB = 50/0 and Si/CB = 0/54 systems, respectively, as shown in Figures 14a and 14c, respectively. The two kinds of the clusters form single mass-fractal structures with their relative fractions being kept in proportion to the composition of Si/CB = 50/4 wt/wt, as shown in Figure 14e. The system forms a three-phase system comprising the Si clusters, the CB clusters, and the SBR matrix. Model C. Si and CB separately form their own clusters as in model B. However, the Si clusters and the CB clusters independently form their own mass-fractal structures with the relative fractions of the Si mass-fractals and the CB massfractals satisfying the composition of Si/CB = 50/4. The system forms a three-phase system comprising the Si mass-fractal structures, CB mass-fractal structures, and the SBR matrix. Let us first consider model C and try to fit Iexp(q) for Si/CB/ K-SBR (K = f and n) with the theoretical scattering profiles, designated hereafter as Iav,K-SBR(q)(K = f and n), obtained from the weighted average of the theoretical scattering profile for the single-filler composites, i.e., Si/K-SBR = 50/100 wt/wt and that for CB/K-SBR = 54/100 wt/wt, designated as Itheor/K-SBR(q) and Itheor(q)(K = f and n), respectively, which gave the best-fit with the corresponding experimental profile for Si/K-SBR = 50/100 wt/wt and that for CB/K-SBR = 54/100 wt/wt (K = f and n), respectively, as shown in Figures 11a and 11d for f-SBR and Figures 11b and 11e for n-SBR, respectively. The profiles Iav,theor(q) given by

(blue filled circles) corrected for the background scattering, where the top two profiles overlapped each other are plotted with the absolute intensity scale (the differential scattering cross section per Vir), while the two profiles marked with (−2) and (−3) shown below the top profiles are the profiles of Si/CB/nSBR and Si/CB/f-SBR vertically shifted downward by the 2 and 3 orders of magnitude, respectively. Note that the top two profiles are identical in the high q range of q ≳ 0.3 nm−1. This is quite reasonable because the profiles in the high q range depend only on the total amount of the fillers incorporated in the composites which are identical. In this work, we consider the following three possible models for the binary filler systems, all of which can be treated by the scattering theory described in section 4. Figure 14 schematically presents the models for the clusters (a), (b), and (c) on the top and the models for the mass-fractal structures (d) and (e) on the bottom. Model A. The small amount of CB is mixed together with the large amount of Si to form single clusters bound by the SBR chains with their relative fractions being statistically kept in proportion to the composition of Si/CB = 50/4 wt/wt as shown in Figure 14b, and the single clusters build up the single mass-fractal structures as shown in Figure 14d. The system forms a two-phase system comprising the single clusters and the SBR matrix. P

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Macromolecules Table 7. Parameters Characterizing the Hierarchical Structure of the Fillers in the Si/CB/SBR Compounds structure level 3 matrix polymer

⟨RC⟩ [nm]

f-SBR n-SBR

20.7 23.3

σ

RC

[nm]

0.53 0.55

structure level 4

(Δp0)2 [×1010 cm−2]2

⟨η2⟩/(Δp0)2

ϕeqv

⟨Reqv⟩ [nm]

ϕcluster

Dm

0.97 0.72

1.68 2.12

0.37 0.32

12.5 12.5

0.47 0.55

2.4 2.3

Table 8. Conclusions Obtained from the Results Shown in Tables 5 and 7a

a

(1)

(R C)Si/ f ‐ SBR < (R C)Si/CB/ f ‐ SBR

(2)

(R C)Si/ n ‐ SBR < (R C)Si/CB/ n ‐ SBR

(3)

(R C)Si/CB/ f ‐ SBR < (R C)CB/ f ‐ SBR

(4)

(R C)Si/CB/ n ‐ SBR < (R C)CB/ n ‐ SBR

(5)

(R C)Si/CB/ f ‐ SBR < (R C)Si/CB/ n ‐ SBR

(RC)K is the average cluster size in the Kth system (K = Si/f-SBR, Si/n-SBR, CB/f-SBR, CB/n-SBR, Si/CB/f-SBR, and Si/CB/n-SBR).

nm) and Si/n-SBR (22.2 nm) (Table 5), respectively, as summarized in Table 8 (conclusions 1 and 2, respectively) and as schematically presented in Figures 14b and 14a, respectively. This is simply because in the two-component filler system the additional amount of CB (4 phr) was simply added to the onecomponent filler system having Si by 50 phr. It is important to note that the cluster size for the twocomponent filler system of Si/CB (= 50/4 wt/wt)/f-SBR is smaller than that for the one-component filler system of CB/fSBR (21.7 nm) as shown in conclusion 3 in Table 8 and as schematically presented in Figures 14b and 14c, respectively. The same conclusion is found to be valid also for n-SBR (23.3 nm vs 25.3 nm) as shown in conclusion 4 in Table 8 and schematically shown also in Figures 14b and 14c. The conclusions 1 and 3 and the conclusions 2 and 4 in Table 8 lead to the following relationships, respectively:

Iav,theor(q) = IK−SBR/Si/K‐SBR (q) + (4/54)Itheor/CB/K‐SBR (q) (K = f or n)

(32)

did not fit well with the experimental scattering profiles both for Si/CB/f-SBR and for Si/CB/n-SBR, as presented in Figure S4 of the Supporting Information. These results shown in Figure S4 provide an important conclusion that the binary filler systems do not form the following two coexisting and independent mass-fractal structures either in the matrix of fSBR or in the matrix of n-SBR: one built up only by the clusters composed of the Si fillers and the other built up only by the clusters composed of the CB fillers. Let us now focus on model A and model B. These two models are commonly based on the single mass-fractal structures. Thus, the two models are different only with respect to the clusters: The clusters in model A are composed of the single clusters in which the Si and CB fillers are mixed together within the clusters, while the clusters in model B are composed of the clusters of Si and those of CB which are mixed together within the fractals. In the context of the present scattering theory, the average SLD of the clusters, i.e., Δp0 in eq 25, for these two models is commonly regarded as the SLD equal to those of the Si fillers and the CB fillers averaged with respect to their weight fractions which are the same and given by Si/CB = 50/4. Thus, the average SLD of the clusters for these two models are the same. Similarly, the quantities such as Δpeqv, peqv, and Reqv are the corresponding quantities averaged for the Si and CB equivalent spheres (aggregates) in the clusters. These quantities are also the same for models A and B. Consequently, the present scattering theory is unable to distinguish these two models. In Figure 13, the experimental profiles Iexp(q) shown by the blue unfilled circles and red unfilled circles for Si/CB/n-SBR and Si/CB/f-SBR, respectively, were well fitted with the theoretical profiles Itheory(q) for model A and model B shown by the blue and red solid lines, respectively. The parameters which gave the best-fittings are summarized in Table 7. The contributions of the scattering from the mass-fractal structure factor SMF(q) and those of the scattering from the cluster structure factor Scluster(q) to Itheory(q) which gives the best-fit with Iexp(q) are also presented by the profiles #4 and #3, respectively. It is quite natural to observe that the average cluster radii for the two-component filler systems Si/CB/f-SBR (20.7 nm) and Si/CB/n-SBR (23.3 nm) (Table 7) are larger than those for the corresponding one-component filler systems Si/f-SBR (18.1

(R C)Si/ f ‐ SBR < (R C)Si/CB/ f ‐ SBR < (R C)CB/ f ‐ SBR

(33)

(R C)Si/ n ‐ SBR < (R C)Si/CB/ n ‐ SBR < (R C)CB/ n ‐ SBR

(34)

as also presented schematically in Figures 14a, 14b, and 14c. The selective attractive interactions between Si and n-SBR anticipated from eq 34 in the two-component filler system may be enhanced by the silane coupling agent introduced to the system. Those relationships given above by eqs 33 and 34 infer such a plausible cluster structure that the simultaneous mixing of both the Si filler powder and the CB filler powder with the fSBR or n-SBR rubber at the weight ratio of Si/CB/ (f-SBR or n-SBR) = 50/4/100 will develop a core−shell cluster as envisioned by model A (shown in Figure 14b) in which the Si aggregates first develops the core cluster bound by f-SBR or nSBR. This core−shell cluster is formed because the Si aggregates have the stronger effective attractive interactions with f-SBR or n-SBR than the CB aggregates and because the filler−filler attractive interactions may be stronger for Si than for CB. The clusters of the Si aggregates first-formed are expected to provide seeds for clustering of the CB aggregates on the surface of the as-grown Si cluster to result in formation of the shell cluster of the CB aggregates also bound by f-SBR or n-SBR, as envisioned by Figure 14b. It seems plausible from a statistical mechanical viewpoint on ensemble-averaged clusters that some CB aggregates are incorporated in the Si core clusters during the clustering process of the Si aggregates. The size of the core cluster of Si may be primarily controlled by the effective attractive interactions between Si and f-SBR or Si and n-SBR relative to the filler−filler interactions in the mixture of Si/CB = 50/4 and f-SBR or n-SBR and thereby the Q

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Macromolecules stress level built up and the dissipated mechanical energy level involved in the system under the given compounding conditions. Since CB is a minor fraction of the two-component filler systems, the size of the Si core cluster is expected to be essentially identical to or slightly larger (in the case of a part of CB being statistically incorporated in the core Si cluster) than the Si cluster formed in the corresponding one-component filler system, Si/f-SBR, as shown in Figures 14a and 14b. In other words, the small amount of the CB particles are anticipated to be virtually unable to affect the core clustering process of the Si component. While model A described above is one extreme model representing the plausible models for Si/CB/SBR, model B is another extreme model. We may further envision the plausible model intermediate between model A and model B by visualizing the case in which the single mass-fractal structures are built up by the clusters composed of Si and CB mixed together with varying fractions under the given net fraction of Si/CB = 50/4. We hope the plausible models may be clearly distinguished by the CSANS experiments combined with the contrast variation methods via either the H−D substitution method or the DNP method, which deserves future works. It is natural to observe that the cluster size of the twocomponent filler system in f-SBR is smaller than that in n-SBR as shown in Table 7 and in conclusion 5 in Table 8. This is because the filler/polymer attractive interactions coupled with the stress level built up and the energy level dissipated for fSBR are larger than those for n-SBR.

This work was devoted to investigating the physical factor (ii) described above under given compounding conditions (i). Specifically, the factor (ii) was explored by using two kinds of the fillers [the silica (Si) and the carbon black (CB)] and two kinds of poly(styrene-ran-butadiene) (SBR) with and without the end-functional group at one of the chain ends (f-SBR and n-SBR, respectively). The effect of factor (ii) on the structure levels 3 to 5 may be symbolically illuminated by a set of the structure parameters (RC, ϕeqv, Dm) characterized by CSANS, where RC is the average cluster size and ϕeqv is the average volume fraction of the aggregates in the cluster in the structure level 3, and Dm is the dimension of the mass-fractal structure in the structure level 4 which is built up by the clusters. We can extract the four set of the parameters (RC, ϕeqv, Dm) for the four sets of the filler/rubber composites from Table 5 and list them below: Si/f ‐SBR: (18 nm, 0.52, 2.5), CB/f ‐SBR: (22 nm, 0.41, 2.4),

Si/n‐SBR: (22 nm, 0.40, 2.3) CB/n‐SBR: (25 nm, 0.33, 2.1)

where the two composites on the first line or the second line compare the effect of the type of the SBR (i.e., f-SBR or n-SBR, respectively) for the given filler on the set of the parameters, while the two composites on the first or the second columns compare the effect of the filler (Si or CB, respectively) for the given SBR. In the former comparisons, both parameters RC and ϕeqv are equally useful because the same filler is involved for the comparison, while in the latter comparisons, the parameter ϕeqv is more suitable than the parameter RC because the parameter RC depends also on the size of the primary particles which are different between Si and CB. The results revealed such a remarkable trend that ϕeqv decreases in the order of Si/f-SBR, CB/f-SBR, Si/n-SBR, and CB/n-SBR. If the SBR having the strong interactions with the aggregates is able to bind them to form small and compact clusters, the trend described above implies that the attractive interactions may decrease in the same order as described above (see eq 31 also). If this is the case, the above list suggests also a remarkable trend that with increasing polymer/filler interactions the composites form more compact mass-fractal agglomerates having larger Dm which are built up by smaller and more compact clusters (Figure 12). The selectively strong interactions of a matrix polymer with a particular filler particle may give a strong influence on the cluster structures in multicomponent filler/rubber systems, as schematically presented in Figure 14b for the composite having two component fillers of Si/CB = 50/4 wt/wt in the f-SBR rubber. The model suggested in the present work would motivate further profound and quantitative analyses of the selforganized filler structures in the multicomponent filler systems based on the contrast-variation methods.

8. CONCLUDING REMARKS We investigated hierarchically self-organized dissipative structures of the filler particles in the polymer matrix. The structures were first developed in the nonequilibrium system open to the external fields involved by the given compounding conditions and subsequently locked by cross-linking of the matrix polymer chains. The locked dissipative structures were investigated by the combined small-angle neutron scattering (CSANS) method covering 4 orders of magnitude in the length scale (several nm to a few 10 μm) based on the scattering theory for fractal structures built up by the clusters where the scattering from clusters is formulated with a newly developed theory based on the Debye−Bueche fluctuations theory extended and generalized to a finite space. The scattering analyses yielded the hierarchical dissipative structure developed through the thermal fluctuations and locked at nonequilibrium, as summarized in Tables 4 and 5 and concluded in Table 6. The structure levels 1 and 2 are independent of the polymers employed but controlled by production processes of the filler particles. On the other hand, the higher order structure levels depend on the polymers and the filler: the structure level 3 is the cluster composed of the aggregates physically bound together by the polymers; the structure level 4 is the fractal structure (agglomerate) built up by the clusters (neither by the aggregates nor by the primary particles). The bulk filler/rubber composites (structure level 5) is composed of the fractal agglomerates dispersed in the crosslinked polymer matrix. The structure levels 3 to 5 depend on (i) the compounding conditions employed and (ii) the fillerparticle/polymer-chain interactions, the cluster/cluster interactions, and the polymer-chain/polymer-chain interactions which are coupled with the stress level built up and the energy level dissipated in the compounding process.

■

APPENDIX 2

p cluster(q), is given by The structure factor for the cluster, ∼ 2

∼ pcluster (q) =

2

∫V ∼pcluster (r) exp[i(q·r)] d r C

(A1)

2

p cluster(r) is the autocorrelation function of pcluster(r) and where ∼ ∫ VC...dr is the volume integral over the volume VC of the cluster. The autocorrelation function of g(r) is defined by R

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Macromolecules 2

∼ g (r) ≡

∫ g (u ) g ( r + u ) d u

T (q) ≡

(A2)

where ∫ ...du is the volume integral. pcluster(r) is given by pcluster (r) = p∞ (r)σ(r)

K = (2π )3

by using the shape function of the cluster defined by

C

(A4)

cluster

■

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b01052. S-1: comparisons between the SAXS and CSANS profiles from the composites with the fillers (Si/SBR and CB/ SBR) and those from the composites without the fillers (SBR), together with Figures S1−S3. S-2: extraction of CSANS and SAXS profiles arising from the spatial distribution of the fillers (Si or CB) only. S-3: comparison of CSANS profiles for Si/CB/K-SBR (K = f and n) composites with the theoretical scattering profiles for model C, together with Figure S4 (PDF)

cluster

2

2

2

∫∞ ∼p∞ (r)σ∼(r) exp[i(q·r)] d r

(A5)

Here, ∫ ∞...dr is the volume integral over the infinite space: 2 2 ∼ p∞(r) and σ∼(r) are the autocorrelation function of p∞(r) and σ(r) defined by eq A2, respectively. The correlation function of the shape function Γ (r) is defined by 2

Γ (r) ≡ σ∼(r)/VC

(A6)

■

where the correlation function Γ (r) decays from 1 to 0 with increasing |r| from zero. p∞(r) is expressed with respect to its fluctuations η(r) around its average value p0, p∞ (r) = p0 + η(r)

*E-mail: [email protected] (T.H.).

(A7)

ORCID

Daisuke Yamaguchi: 0000-0001-9643-4513 Notes

The authors declare no competing financial interest. T.H.: Professor Emeritus, Kyoto University, Kyoto 606-8501, Japan, and Honorary Chair Professor, National Tsing Hua University, Hsinchu, 30013, Taiwan.

∞

■

2

(A8)

ACKNOWLEDGMENTS The synchrotron radiation (SAXS) experiments were performed at the BL19B2 in the SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2009B2021).

where γ(r) is the correlation function of the internal fluctuations defined by γ(r) = ⟨η(r1)η(r + r1)⟩r /⟨η2⟩

(A9)

■

⟨η ⟩ is the mean-squared fluctuations of η(r). Eqs A5, A6, A8 lead to a generalized version of the DB theory applicable to the confined space also 2

2

∼ pcluster (q) = VC

∫∞ Γ(r)[p02 + ⟨η2⟩γ(r)] exp[i(qr)] d r (A10)

where the symbol ∗ designates the convolution product defined by

∫ f (u )g (q − u ) d u

REFERENCES

(1) Nikolis, G.; Prigogine, I. Self-Organization in Nonequilibrium Systems. From Dissipative Structure to Order through Fluctuations; John Wiley & Sons, Inc.: New York, 1977. (2) Kraus, G., Ed.; Reinforcement of Elastomers; Interscience: New York, 1965. (3) Heinrich, G., Ed.; Advanced Rubber Composites; Springer-Verlag: Berlin, 2011; Vol. 239. (4) Tsutsumi, F.; Sakakibara, M.; Oshima, N. Structure and Dynamic Properties of Solution SBR Coupled with Tin Compounds. Rubber Chem. Technol. 1990, 63, 8−22. (5) Yuasa, T.; Tominaga, T.; Sone, T. Analysis of Filler Aggregation in Compounds Using Small-angle X-ray Scattering: Effect of Functional Group Introduced into Polymer-ends of Solutionpolymerized SBR. Nippon Gomu Kyokaishi 2013, 86, 249−255.

= VCΓ (q) ∗[p0 2 T (q) + ⟨η2⟩γ(q)]

f (q) ∗g (q) ≡

AUTHOR INFORMATION

Corresponding Author

Note that p0 in eqs A7 and A8 as well shown below should be replaced by p0 − pm ≡ Δp0 in the case when the clusters are dispersed in the medium having its SLD of pm. Noting that η(r) is positive and negative with equal 2 p (r) is given by32 probability, ∼ ∼ p∞ (r) = Vir[p0 2 + ⟨η2⟩γ(r)]

ASSOCIATED CONTENT

S Supporting Information *

p ∞(r)σ∼(r)/Vir Thus, eqs A1 and A3 lead to =∼ 2 1 ∼ pcluster (q) = Vir

(A13)

for any shapes of the 3D volume irradiated by the incident beam, e.g., the rectangular-shaped or cylindrical-shaped irradiated volume, when the volume is much larger than the volume covered by scattering experiments. Here, Γ (q) and γ(q) are the Fourier transform of Γ (r) and γ(r), respectively; δ(q) is the Dirac’s delta function. Equations A10, A12, and A13 lead to eq 11 in the text.

and p∞(r) is the spatial distribution of the SLD of the scattering elements in infinite space which is statistically replicated by the spatial distribution of the SLD within the cluster. The infinite space is considered to be practically equal to the volume irradiated by the incident beam Vir, when Vir is much larger 2 2 p p than V . From eqs A2 and A3 ∼ (r) is given by42 ∼ (r) 2

(A12)

where K is given by34−36 (A3)

⎧ 1 at r ∈ (cluster) σ(r) = ⎨ ⎩0 otherwise

∫∞ exp[i(q·r)] d r = Kδ(q)

(A11)

and T(q) is defined by S

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Macromolecules (6) Schaefer, D. W.; Suryawanshi, C.; Pakdel, P.; Ilavsky, J.; Jemian, P. R. Challenges and opportunities in complex materials: silicareinforced elastomers. Phys. A 2002, 314, 686−695. (7) Botti, A.; Pyckhout-Hintzen, W.; Richter, D.; Urban, V.; Straube, E. A microscopic look at the reinforcement of silica-filled rubbers. J. Chem. Phys. 2006, 124, 174908−5. (8) Morfin, I.; Ehrburger-Dolle, F.; Grillo, I.; Livet, F.; Bley, F. ASAXS, SAXS and SANS investigations of vulcanized elastomers filled with carbon black. J. Synchrotron Radiat. 2006, 13, 445−452. (9) Koga, T.; Hashimoto, T.; Takenaka, M.; Aizawa, K.; Amino, N.; Nakamura, M.; Yamaguchi, D.; Koizumi, S. New Insight into Hierarchical Structure of Carbon Black Dispersed in Polymer Matrices: A Combined Small-Angle Scattering Study. Macromolecules 2008, 41, 453−464. (10) Wolff, S.; Wang, M. J.; Tan, E. H. Filler-Elastomer Interactions. Part VII. Study on Bound Rubber. Rubber Chem. Technol. 1993, 66, 163−167. (11) Beaucage, G.; Kammler, H. K.; Pratsinis, S. E. Particle Size Distributions from Small-angle Scattering Using Global Scattering Functions. J. Appl. Crystallogr. 2004, 37, 523−535. (12) Hashimoto, T.; Koizumi, S. Combined Small-Angle Scattering for Characterization of Hierarchically Structured Polymer Systems over Nano-to-Micro Meter: Part I Experiments. In Polymer Science: A Comprehensive Reference; Matyjaszewski, K., Möller, M., Eds.; Elsevier: Amsterdam BV, 2012; Vol 2, pp 381−398. (13) Baeza, G. P.; Genix, A.-C.; Degrandcourt, C.; Petitjean, L.; Gummel, J.; Couty, M.; Oberdisse, J. Multiscale Filler Structure in Simplified industrial Nanocomposite Silica/SBR Systems Studied by SAXS and TEM. Macromolecules 2013, 46, 317−329. (14) Baeza, G. P.; Genix, A.-C.; Degrandcourt, C.; Petitjean, L.; Gummel, J.; Schweins, R.; Couty, M.; Oberdisse, J. Effect of Grafting on Rheology and Structure of a Simplified Industrial Nanocomposite Silica/SBR. Macromolecules 2013, 46, 6621−6633. (15) Compounding the fillers with the SBR having functionalized chain ends, such as f-SBR, might seem nonsensical at a glance because the end-functional groups are expected to associate themselves to form micelles in the matrix composed of main chains, which may prevent the groups from having the attractive interactions with the fillers. However, in the practical rubber compounding process, where the SBR chains are highly entangled, and the entangled chains are subjected to an external force under the mechanical mixing with the fillers, the micelles may not be formed or may be dissociated so that the end groups are expected to associate with the surfaces of the filler, thereby promoting the filler dispersion. (16) Aizawa, K.; Tomimitsu, H. Design and use of a double crystal diffractometer for very small angle neutron scattering at JRR-3M. Phys. B 1995, 213-214, 884−886. (17) Yamaguchi, D.; Koizumi, S.; Motokawa, R.; Kumada, T.; Aizawa, K.; Hashimoto, T. Tandem analyzer crystals system doubles counting rate for Bonse-Hart ultra-small-angle neutron-scattering spectrometer. Phys. B 2006, 385−386, 1190−1193. (18) Koizumi, S.; Iwase, H.; Suzuki, J.; Oku, T.; Motokawa, R.; Sasao, H.; Tanaka, H.; Yamaguchi, D.; Shimizu, H. M.; Hashimoto, T. Focusing and polarized neutron small-angle scattering spectrometer (SANS-J-II). The challenge of observation over length scales from an ångström to a micrometre. J. Appl. Crystallogr. 2007, 40, s474−s479. (19) Bonse, U.; Hart, M. Tailless X-ray Single Crystal Reflection Curves Obtained by Multiple Reflection. Appl. Phys. Lett. 1965, 7, 238−240. (20) Guinier, A.; Fournet, G. Small-Angle Scattering of X-rays; John Wiley and Sons, Inc.: New York, 1955. (21) Russell, T. P.; Lin, J. S.; Spooner, S.; Wignall, G. D. Intercalibration of small-angle X-ray and neutron scattering data. J. Appl. Crystallogr. 1988, 21, 629−638. (22) Yagi, N.; Inoue, K. Ultra-small-angle X-ray diffraction and scattering experiments using medium-length beamlines at SPring-8. J. Appl. Crystallogr. 2003, 36, 783−786. (23) Results reported by Noda, Y.; Yamaguchi, D.; Hashimoto, T.; Shamoto, S.; Koizumi, S.; Yuasa, T.; Tominaga, T.; Sone, T. Partly

presented at SAS 2012 Conference, November 2012, Sydney, Australia. Noda, Y.; et al. Phys. Procedia 2013, 42, 52−57. (24) Stuhrmann, H. B.; Schärpf, O.; Krumpolc, M.; Niinikoski, T. O.; Rieubland, M.; Rijllart, A. Dynamic nuclear polarization of biological matter. Eur. Biophys. J. 1986, 14, 1−6. (25) van den Brandt, B.; Bunyatova, E. I.; Hautle, P.; Konter, J. A.; Mango, S. Dynamic nuclear polarization in thin polymer foils and tubes. Nucl. Instrum. Methods Phys. Res., Sect. A 1995, 356, 36−38. (26) Kumada, T.; Noda, Y.; Hashimoto, T.; Koizumi, S. Dynamic nuclear polarization system for the SANS-J-II spectrometer at JAEA. Phys. B 2009, 404, 2637−2639. (27) The slight discrepancy between the profiles (3) and (4) at high q’s of q ≥ 10−1 nm is shown in Figure 4(a) immaterial at all for the background subtraction method. (28) The differential scattering cross section ∂Σ/∂Ω is defined as the energy flux of the scattered beam per the unit solid angle subtended by the detector set at the scattering angle θ per unit energy flux of the incident beam. (29) Teixeira, J. Small-angle scattering by fractal systems. J. Appl. Crystallogr. 1988, 21, 781−785. (30) Wong, P.-Z.; Bray, A. J. Small-angle scattering by rough and fractal surfaces. J. Appl. Crystallogr. 1988, 21, 786−794. (31) The second term in the right-hand side of eq 9 describes the scattering cross section per unit area of the smooth surface S2 (in eq 1.5 of ref 27) measured with the scale larger than l2. (32) Debye, P.; Bueche, A. M. Scattering by an Inhomogeneous Solid. J. Appl. Phys. 1949, 20, 518−525. (33) p(r) is given by reρe(r) sin γ for X-ray and ks,02α(r) sin γ for light, where sin2 γ = (1 + cos2 θ)/2, re is the classical radius of electron, ρe(r) is the electron density, ks,0 is the wave number of light in air, and α(r) is the polarizability density. (34) Lin, Y.-C.; Chen, C.-Y.; Chen, H.-L.; Hashimoto, T.; Chen, S.A.; Li, Y.-C. Hierarchical self-assembly of nanoparticles in polymer matrix and the nature of the interparticle interaction. J. Chem. Phys. 2015, 142, 214905-1−214905-14. (35) Lin, Y.-C.; Chen, C.-Y.; Chen, H.-L.; Hashimoto, T.; Chen, S.A.; Li, Y.-C. Erratum: Hierarchical self-assembly of nanoparticles in polymer matrix and the nature of the interparticle interaction. [J. Chem. Phys. 142, 214905 (2015)]. J. Chem. Phys. 2015, 143, 249901− 1. (36) In ref 34, K was given by π3 because of the integration of T(q) in eq A11 with respect to qx, qv, and qz being conducted only over the volume specified by 0 ≤ qx, qv, qz ≤ ∞, i.e., only 1/8 of the whole qspace. More correctly, the integration should be done over the whole q-space, which gives K = 8π3. (37) Hashimoto, T.; Tanaka, H.; Koizumi, S.; Naka, K.; Chujo, Y. A combined small-angle scattering study of a chemical reaction at specific sites and reaction-induced self-assembly as a problem in open nonequilibrium phenomena. J. Appl. Crystallogr. 2007, 40, s73−s77. (38) Debye, P.; Anderson, H. R.; Brumberger, H. Scattering by an Inhomogeneous Solid. II. The Correlation Function and Its Application. J. Appl. Phys. 1957, 28, 679−683. (39) See for example: Guinier, A. In X-ray Diffraction; W.H. Freeman and Co.: San Francisco, 1963. (40) Witten, T. A.; Sander, L. M. Diffusion-limited aggregation. Phys. Rev. B: Condens. Matter Mater. Phys. 1983, 27, 5686−5697. (41) Meakin, P. Diffusion-Limited Aggregation in Three Dimensions: Results from a New Cluster-Cluster Aggregation Model. J. Colloid Interface Sci. 1984, 102, 491−504. (42) From eqs A2 and A3, pcluster(r) is given by 2

∼ pcluster (r) =

∫V p∞(u)σ(u)p∞ (r + u)σ(r + u) du ir

= Vir⟨σ(u)σ(r + u)p∞ (u)p∞ (r + u)⟩r where ∫ Vir...du is the volume integral over Vir under a given fixed r, and ⟨...⟩r is the average of the quantity inside the brackets overall volume T

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Macromolecules Vir under a given r. Since it is reasonable to assume that the shape σ(r) and the internal SLD distribution p(r) are independent, we obtain 2

∼ pcluster (r) = Vir⟨σ(u)σ(r + u)r ⟩⟨p∞ (u)p∞ (r + u)⟩r =

∫V σ(u)σ(r + u) d u ∫ p∞ (u)p∞ (r + u) d u/Vir ir

2

2

2

pcluster (r) = σ∼(r)∼ p∞ (r)/Vir . Thus, we obtain ∼

U

DOI: 10.1021/acs.macromol.7b01052 Macromolecules XXXX, XXX, XXX−XXX