Spin-contrast-variation small-angle neutron scattering study of fully and partially swollen silica-filled rubber

Noda, Y.; Koizumi, S.; Masui, T.; Kishimoto, H.; Yamaguchi, D.; Kumada, T.; Takata, S.; Ohishi, K.; Suzuki, J.

doi:10.1107/S1600576726000361

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JOURNAL OF
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CRYSTALLOGRAPHY

ISSN: 1600-5767

Volume 59| Part 2| April 2026| Pages 492-512

https://doi.org/10.1107/S1600576726000361

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Spin-contrast-variation small-angle neutron scattering study of fully and partially swollen silica-filled rubber

Yohei Noda,^a,^b ^* Satoshi Koizumi,^a,^b Tomomi Masui,^c Hiroyuki Kishimoto,^c Daisuke Yamaguchi,^d Takayuki Kumada,^d,^e Shin-ichi Takata,^e Kazuki Ohishi ^f and Jun-ichi Suzuki ^b,^f

^aInstitute of Quantum Beam Science, Ibaraki University, Ibaraki, 316-8511, Japan, ^bNeutron Industrial Application Promotion Center, Comprehensive Research Organization for Science and Society (CROSS), Ibaraki, 319-1106, Japan, ^cResearch Center for Advanced Technology & Innovation, Research & Development HQ, Sumitomo Rubber Industries Ltd, Kobe, 651-0072, Japan, ^dMaterials Sciences Research Center, Japan Atomic Energy Agency, Ibaraki, 319-1195, Japan, ^eJ-PARC Center, Japan Atomic Energy Agency, Ibaraki, 319-1195, Japan, and ^fNeutron Science and Technology Center, Comprehensive Research Organization for Science and Society (CROSS), Ibaraki, 319-1106, Japan
^*Correspondence e-mail: [email protected]

Edited by E. P. Gilbert, Australian Centre for Neutron Scattering, ANSTO, Australia (Received 17 October 2025; accepted 14 January 2026; online 20 March 2026)

We have elucidated the polymer adsorption layer structure in filler–rubber systems by conducting spin-contrast-variation small-angle neutron scattering (SANS) on partially and fully swollen filler–rubber samples with and without a silane coupling agent. In spin-contrast-variation SANS, dynamic nuclear polarization (DNP) was used to polarize protons and change their scattering length with respect to polarized neutron beams significantly. SANS measurements were performed in dynamically polarized states using a DNP cryostat (1.2 K and 3.35 T). From SANS profiles obtained at various proton spin polarizations, partial scattering functions (PSFs) for each component were separated by regarding each sample as a three-component system composed of silica, polymer and deuterated toluene. To analyze the obtained PSFs in detail, we built a structure model for the silica aggregates and the surrounding polymer adsorption layer. Numerical calculation based on this model successfully reproduced the experimentally obtained PSFs, providing the structural parameters of the silica aggregates and polymer adsorption layer. The results showed a considerable difference in structural parameters between the partially and fully swollen states. For the sample with the silane coupling agent, the thickness of the polymer adsorption layer decreased as the solvent fraction increased. The difference in polymer volume fraction between the polymer adsorption layer and the outside matrix was very small in less swollen states but significant in the fully swollen state. Furthermore, the scattering contribution of the polymer chains in the solvent was accurately separated via contrast variation. In the swollen silica-filled rubber without the silane coupling agent, the size of the polymer-dense regions was almost constant, regardless of the swelling ratio. By contrast, in the swollen silica-filled rubber with the silane coupling agent, the size of the polymer-dense regions significantly increased by a factor of 2 with an increase in the swelling ratio.

Keywords: small-angle neutron scattering; SANS; contrast variation; dynamic nuclear polarization; silica-filled rubber; bound rubber; swelling; polymer adsorption layer.

1. Introduction

Filler–rubber systems are widely used in industrial products such as pneumatic tires, conveyer belts, shock absorbers and rubber seals. Filler particles, such as carbon and silica particles, are added to rubber to improve tear and wear resistance. A polymer adsorption layer forms around these filler particles and plays an important role in boosting mechanical performance. Due to its industrial value, the polymer adsorption layer has been investigated using various approaches, such as small-angle X-ray scattering and small-angle neutron scattering (SANS) (Baeza et al., 2016 ; Botti et al., 2006 ; Genix & Oberdisse, 2015 ; Genix et al., 2019 ; Iwasaki et al., 2025 ; Jimenez et al., 2019 ; Jouault et al., 2010 ; Liu et al., 2017 ; Mashita et al., 2013 ; Mashita et al., 2016 ; Miyazaki et al., 2007 ; Morfin et al., 2006 ; Nakanishi et al., 2021 ; Nakanishi et al., 2024 ; Presto et al., 2020 ; Shui et al., 2021a ; Shui et al., 2021b ; Staropoli et al., 2019 ; Staropoli et al., 2020 ; Takenaka et al., 2009 ; Takenaka et al., 2012 ; Watanabe et al., 2023 ; Yamaguchi et al., 2017 ), neutron reflectivity (Hori et al., 2017 ; Kumada et al., 2024 ; Shimokita et al., 2024 ; Shimomura et al., 2016 ), neutron spin echo (Jiang et al., 2015 ; Koga et al., 2018 ; Salatto et al., 2021 ), mechanical analysis (Baeza et al., 2014 ; Mujtaba et al., 2014 ), nuclear magnetic resonance (NMR) (Chassé et al., 2013 ; Kishimoto et al., 2023 ; Valentín et al., 2010 ), and atomic force microscopy (Ito et al., 2022 ; Morozov et al., 2012 ; Ueda et al., 2019 ). Many researchers have combined several approaches.

Contrast-variation SANS has been successfully performed on filler–rubber systems swollen with deuterated solvents (d-solvents) (Takenaka et al., 2009; Takenaka et al., 2012; Liu et al., 2017; Nakanishi et al., 2024). Polymer confined to the filler surface absorbs less solvent than the surrounding free polymer. This solvent distribution difference can cause a contrast between the polymer adsorption layer and the surrounding free polymer. Researchers have decomposed SANS profiles at various contrasts into partial scattering functions (PSFs) to evaluate the adsorption layer thickness and the local polymer fraction in the polymer adsorption layer and the surrounding matrix. However, such studies have only been performed on systems in the fully swollen state, not those in the partially swollen state. Filler–rubber systems with different swelling ratios should be investigated for a more detailed understanding of the polymer adsorption layer. The existence of a more complex internal structure of the polymer adsorption layer is being debated. For example, Shui et al. (2021a, 2021b) and Huang et al. (2021 ) proposed a double-layer model.

Spin-contrast-variation SANS combined with d-solvent addition is expected to be an ideal solution for studying partially swollen filler–rubber systems. Deuterium substitution, which utilizes the neutron scattering length difference between a proton and a deuteron, is widely used. As protons and neutrons have spin, the scattering length of a proton (b_H) with respect to a fully polarized neutron significantly depends on the proton spin polarization (P_H) (Sears, 1992 ):

$[b_{\rm H} = (-0.374 + 1.456 P_{\rm H})\times 10^{-12}\ {\rm cm},\eqno(1)]$

where

$[P_{\rm H} = (N_{\rm up} - N_{\rm down}) / (N_{\rm up} + N_{\rm down}).\eqno(2)]$

N_up and N_down mean the number of up and down proton spins, respectively. Scattering lengths can vary significantly (Fig. 1), which can be utilized for contrast variation in SANS. After the pioneering work of Stuhrmann et al. (1986 ), we successfully performed spin-contrast-variation SANS for filler–rubber systems (Noda et al., 2013 , Noda et al., 2016 ).

Figure 1
Neutron scattering lengths of protons and deuterons as a function of proton spin polarization (P_H) in the case of fully polarized protons (P_N = 1).

The use of the conventional approach, namely, control of the solvent H/D ratio, to partially swollen filler–rubber systems is hindered by the following difficulties. (i) The limited amount of solvent in less swollen samples narrows the contrast-variation range. (ii) Samples with solvents with different H/D ratios must be prepared carefully, as fluctuations in the degree of swelling across samples impair the accuracy of decomposed PSFs. (iii) The incoherent scattering background tends to increase, as protonated solvents are needed to expand the contrast-variation range.

Spin-contrast-variation SANS is expected to solve these difficulties. (i) The scattering lengths of polymer chain protons can be controlled in spin-contrast-variation SANS, so contrast can be changed significantly, even for less swollen samples. Spin polarization can diffuse through flip-flops between neighboring proton spins. This process differs from the addition of d-solvents, which cannot modify the scattering length of polymer chain protons. (ii) Only one sample is necessary because a set of SANS profiles with different contrasts can be obtained by controlling P_H. Fluctuations in swelling degree across samples is not a problem. (iii) Contrast can be controlled through P_H, eliminating the need for a protonated solvent. This lowers the incoherent scattering background. In summary, contrast creation via d-solvent introduction and contrast variation via proton spin polarization can be combined to study partially swollen filler–rubber systems effectively.

D-solvent swelling can be used to visualize not only the polymer adsorption layer but also the polymer chain distribution. It has been used to evaluate polymer inhomogeneity and polymer network structure sizes in gel systems. Using this approach, Ikeda et al. (2009 ) studied sulfur-cross-linked rubber with different zinc oxide concentrations. Karino et al. (2007 ) and Suzuki et al. (2010 ) studied peroxide-cross-linked natural rubber containing protein aggregates, finding significant low-q scattering. They performed a contrast-variation study while controlling the solvent H/D ratio and isolated the scattering contribution of the solvent polymer chains from the contribution of protein aggregates. For swollen silica-filled rubber samples, the target material of the current study, the high-q scattering of the polymer network structure (mesh size) can be evaluated easily because the scattering contribution of silica particles is small at high q. However, the low-q scattering contribution of polymer distribution inhomogeneity is difficult to evaluate because of its overlap with the silica particle contribution. In the present study, we attempt to use contrast variation to separate the scattering contribution of polymer chains in swollen silica-filled rubber samples. Specifically, we applied spin-contrast-variation SANS to partially and fully swollen silica-filled rubber samples, focusing on the effect of a silane coupling agent.

2. Experimental

2.1. Sample preparation

We studied the polymer adsorption layer around silica particles by preparing two samples without and with a silane coupling agent [Si266; Fig. 2(c)], labeled R_N and R_CA, respectively. Table 1 shows the sample composition. Both R_N and R_CA contain 5 vol.% silica particles in a styrene–butadiene random copolymer [SBR; Fig. 2(b)]. In addition, sulfur, N-tert-butyl-2-benzothiazole sulfenamide [TBBS; Fig. 2(d)] and 1,3-diphenylguanidine [DPG; Fig. 2(e)] were added for vulcanization. All ingredients were mixed in a milling machine, and the resulting mixture was pressed into molds and heated at 170°C for 12 min. The thickness of the rubber sheet before swelling was approximately 0.6 mm for R_N and about 0.9 mm for R_CA.

Table 1
Composition (vol.%) of rubber samples before swelling

Sample	S-SBR (VSL4720)	Silica (VN3)	Silane coupling agent (Si266)	Sulfur	Accelerator (TBBS)	Accelerator (DPG)
R_N	92.950	5.014	0.0	0.636	0.681	0.720
R_CA	92.230	4.975	0.774	0.631	0.675	0.715

Figure 2
Molecular structure formulas of TEMPO (a), SBR (b), silane coupling agent Si266 (c), TBBS (d) and DPG (e). In (c), the average sulfur chain length (x) is 2.15.

Samples in dynamic nuclear polarization (DNP) experiments should include unpaired electrons. The stable free radical 2,2,6,6-tetramethylpiperidin-1-oxyl [TEMPO; Fig. 2(a)] can be used conveniently as an electron spin source (Bunyatova, 2004 ). In our previous study on filler–rubber systems without solvents, we used vapor sorption, spontaneously diffusing TEMPO vapor into rubber (Noda et al., 2016). TEMPO doping was easily completed by adding TEMPO to the solvent for swelling. The optimal TEMPO concentration is 30 mM for DNP experiments at 3.35 T (Noda et al., 2016), which is much lower than the solvent amount needed for swelling. In addition, the O₂ concentration in a sample should be minimized to obtain a high |P_H| in DNP experiments. O₂ accelerates proton spin relaxation. Considering these issues, we devised the current experimental procedure as follows.

For the swelling of rubber samples, we used deuterated toluene (d-toluene, 99 at.% D; Sigma–Aldrich). To prepare the partially swollen samples, we placed a cut rubber sheet (10.5 × 10.5 mm) of R_N or R_CA in an aluminium metallized film package. Then, a TEMPO/d-toluene solution (73–109 mM) was added into the package using a microsyringe. By adjusting the solution volume (0.03–0.10 ml), we obtained swollen samples with Q_swell ≃ 1.5 for R_N1 and R_CA1 and Q_swell ≃ 2 for R_N2 and R_CA2, where Q_swell is the ratio of the swollen rubber volume to the initial one. Then, we placed an oxygen absorber (A500-HS, AS ONE) in the package and sealed it using a heat sealer. The package containing the swollen sample was kept at room temperature for 12 h before the SANS experiment. The Q_swell values in the saturated state were 5.51 and 4.76 for R_N and R_CA, respectively.

To prepare the fully swollen samples (R_N3 and R_CA3), we placed a cut rubber sheet (7 × 7 mm) of R_N or R_CA in a glass bottle with a sealing cap. Then, an excess amount (approximately 0.25 ml) of a TEMPO/d-toluene solution (48 mM), compared with the cut rubber sheet, was added to the bottle. We placed an oxygen absorber in the package while avoiding its contact with the solution. The glass bottle containing the fully swollen sample was kept at room temperature for 12 h before the SANS experiment.

According to electron spin resonance measurements, the TEMPO concentrations of R_N1, R_N2, R_CA1 and R_CA2 were 34, 28, 31 and 37 mM, respectively. They were close enough to the optimal TEMPO concentration (30 mM) for DNP experiments at 3.35 T. Loss of TEMPO during swelling was insignificant (approximately 15% at most). Given the confirmed reliability of this procedure, the TEMPO concentration evaluation of R_N3 and R_CA3 was omitted. For SANS, we prepared samples in different packages in a similar way.

A sample obtained from each package was immediately placed in a sample-holding unit at the end of the sample stick of a DNP cryostat (Section 2.2). Then, the unit was inserted into the sample chamber of the DNP cryostat, which was filled with liquid He.

2.2. Dynamic nuclear polarization

At thermal equilibrium, proton spins are polarized slightly (P_H = 0.30%), even at 1.2 K and 3.35 T. However, under the same conditions, electron spins are almost fully polarized (95%). In DNP, microwave irradiation stimulates polarization transfer from electron to proton spins to achieve a high P_H (Abragam & Goldman, 1978 ). Here, we used a DNP cryostat (Kumada et al., 2009 ) designed for SANS, which had split-type superconducting magnet coils. Between the coils is a sample chamber, which is filled with liquid ⁴He, and its temperature can be reduced to 1.2 K by evacuating liquid ⁴He. The magnetic field is parallel to the direction of the neutron beam, which passes along the central axis of the magnet coils. Thin aluminium plates form windows through which the neutron beam passes, causing a slight SANS background.

Each sample was placed in the sample-holding unit at the end of the sample stick of the DNP cryostat. The sample-holding unit had a three-turn NMR coil for P_H evaluation. The sample sheet (14 × 14 × ∼1 mm) was inserted in this NMR coil. The NMR signal was proportional to P_H, so we could simultaneously measure NMR during the SANS experiment. The NMR coil was made of a 0.1 mm-thick aluminium sheet and caused a slight SANS background. When protons are polarized via DNP, their NMR signal becomes significant. Then, P_H can be evaluated accurately in relative terms. To obtain absolute P_H values, we needed to calibrate P_H using thermal equilibrium NMR signals (P_H = 0.082% at 4.2 K and 3.35 T, for example). However, measurement of this NMR signal was not completed because of the low sensitivity of our NMR equipment and limited SANS beam time. Nonetheless, neutron transmission depends on P_H. Therefore, we calibrated P_H through neutron transmission (Section 2.3).

Microwave radiation for polarization transfer from electron to proton spins was generated using a Gunn oscillator (94 GHz) placed on the top plate of the DNP cryostat, which irradiated the sample through a stainless-steel pipe with a length of 1 m and an inside diameter of 6 mm. P_H was controlled through microwave frequency tuning, as P_H quickly responds to changes in microwave conditions. The time constant for this response was approximately 3 min.

2.3. Small-angle neutron scattering

SANS experiments were performed using TAIKAN (BL15) (Shinohara et al., 2009 ; Shinohara et al., 2009 ; Takata et al., 2015 ) at the Material and Life Science Experimental Facility (MLF) in the Japan Proton Accelerator Research Complex (J-PARC). The DNP cryostat was placed on the TAIKAN sample stage. The device allocation is described in our previous article (Noda et al., 2016). Because of the DNP cryostat window structure, the available scattering angle 2θ was limited (2θ < 15°) and the detectors could not be fully utilized. A polarized neutron beam was provided by a magnetic supermirror polarizer composed of an Fe/Si multilayer. Since the neutron polarization decreases for short-wavelength neutrons (λ < 4 Å), we employed SANS data with a limited wavelength range (4 < λ < 7.6 Å), where the magnitude of neutron polarization was close enough to 1, resulting in a q range of 0.005 < q < 0.3 Å⁻¹ as in our previous study (Noda et al., 2016). SANS at P_H = 0% does not depend on neutron polarization. We will employ the SANS data at P_H = 0% for the full wavelength range of neutrons (1 < λ < 7.6 Å) to discuss the high-q profile (q < 1 Å⁻¹) later in Section 3.5.

As TAIKAN simultaneously measures SANS and neutron transmission, we used the observed neutron transmission to calibrate P_H. The microscopic total cross sections for H, D, C, N, O, S and Si are σ_tot,H = (81.99 − 66.97P_H) × 10⁻²⁴ cm², σ_tot,D = (7.63 + 3.76P_D) × 10⁻²⁴ cm², σ_tot,C = 5.55 × 10⁻²⁴ cm², σ_tot,N = 13.41 × 10⁻²⁴ cm², σ_tot,O = 4.23 × 10⁻²⁴ cm², σ_tot,S = 1.56 × 10⁻²⁴ cm² and σ_tot,Si = 2.34 × 10⁻²⁴ cm², respectively (Sears, 1992). Here, P_D is deuteron spin polarization. The macroscopic total cross section (Σ_tot) of each ingredient and each sample was calculated using the known chemical compositions (Tables 2 and 3).

Table 2
Calculated scattering length density and total and incoherent scattering cross sections for each ingredient

Ingredient	SLD (×10¹⁰ cm⁻²)	Σ_tot (cm⁻¹)	Σ_inc (cm⁻¹)
SBR (VSL4720)	0.59 + 8.44P_H	5.00 − 3.89P_H	1.52(3 − 2P_H − $[{P}_{\rm H}^2]$ )
Silica (VN3)	3.08	0.208	0.000
Silane coupling agent (Si266)	0.15 + 7.91P_H	4.63 − 3.64P_H	1.45(3 − 2P_H − $[{P}_{\rm H}^2]$ )
Sulfur	1.08	0.059	0.000
Accelerator (TBBS)	1.45 + 6.54P_H	3.98 − 3.01P_H	1.20(3 − 2P_H − $[{P}_{\rm H}^2]$ )
Accelerator (DPG)	2.26 + 6.48P_H	4.03 − 2.98P_H	1.19(3 − 2P_H − $[{P}_{\rm H}^2]$ )
Deuterated toluene (99 at.% D)	5.62 + 0.06P_H + 1.22P_D	0.60 − 0.03P_H + 0.17P_DP_N	0.012(3 − 2P_H − $[{P}_{\rm H}^2]$ ) + 0.046(2 − P_D − $[{P}_{\rm D}^2]$ )
TEMPO	0.27 + 9.19P_H	5.41 − 4.23P_H	1.68(3 − 2P_H − $[{P}_{\rm H}^2]$ )

Table 3
Calculated total and incoherent scattering cross sections for each sample

Sample	Q_swell	Thickness (cm)	Σ_tot (cm⁻¹)	Σ_inc (cm⁻¹)
R_N1	1.69	0.079	3.07 − 2.21P_H + 0.07P_D	0.864(3 − 2P_H − $[{P}_{\rm H}^2]$ ) + 0.018(2 − P_D − $[{P}_{\rm D}^2]$ )
R_N2	2.14	0.098	2.55 − 1.75P_H + 0.09P_D	0.686(3 − 2P_H − $[{P}_{\rm H}^2]$ ) + 0.024(2 − P_D − $[{P}_{\rm D}^2]$ )
R_N3	5.51	0.250	1.38 − 0.72P_H + 0.14P_D	0.281(3 − 2P_H − $[{P}_{\rm H}^2]$ ) + 0.037(2 − P_D − $[{P}_{\rm D}^2]$ )
R_CA1	1.66	0.100	3.11 − 2.25P_H + 0.07P_D	0.879(3 − 2P_H − $[{P}_{\rm H}^2]$ ) + 0.018(2 − P_D − $[{P}_{\rm D}^2]$ )
R_CA2	1.80	0.127	2.91 − 2.07P_H + 0.07P_D	0.809(3 − 2P_H − $[{P}_{\rm H}^2]$ ) + 0.020(2 − P_D − $[{P}_{\rm D}^2]$ )
R_CA3	4.76	0.125	1.49 − 0.82P_H + 0.13P_D	0.321(3 − 2P_H − $[{P}_{\rm H}^2]$ ) + 0.036(2 − P_D − $[{P}_{\rm D}^2]$ )

Protons and deuterons have common spin temperatures, even in dynamically polarized states (de Boer et al., 1974 ). In this case, P_D is related to P_H by

$[\eqalignno{P_{\rm D}& = 4\tanh[(E_{{\rm Z},{\rm D}}/E_{{\rm Z},{\rm H}}) \tanh^{-1}(P_{\rm H})]&\cr &\quad/ \{3+\tanh^2[(E_{{\rm Z},{\rm D}}/E_{{\rm Z},{\rm H}})\tanh^{-1}(P_{\rm H})]\},&(3)}]$

where E_Z,D and E_Z,H are the Zeeman splitting energies of protons and deuterons, respectively, and E_Z,D/E_Z,H is 0.1535. Under our experimental conditions (|P_H| < 30%), tanh⁻¹(P_H) could be approximately regarded as P_H. Hence, we used the approximate equation

$[P_{\rm D} \simeq 0.2047 P_{\rm H}.\eqno(4)]$

Neutron transmission T_N was calculated as

$[T_{\rm N} = \exp(-\Sigma_{\rm tot}t_{\rm s}),\eqno(5)]$

where t_s is sample thickness. We calibrated P_H by comparing the calculated and experimental T_N. In Fig. 3, the filled black circles indicate the experimentally obtained T_N as a function of P_H after calibration, where T_N increased with P_H. The less swollen samples had steeper slopes than the fully swollen samples, as the latter had a higher proton concentration.

Figure 3
Neutron transmission as a function of P_H. The top and bottom rows present the results for the rubber samples without and with the silane coupling agent, respectively. In each panel, the filled black circles indicate the experimental results, and the dased lines indicate the numerical calculation results.

3. Results and discussion

3.1. Small-angle neutron scattering results

Fig. 4 shows SANS profiles observed at various P_H for the silica-filled swollen rubber samples. The horizontal axis is the magnitude of the scattering vector q [= (4π/λ) sin θ]. In each panel, the filled gray symbols indicate SANS profiles before DNP. After DNP, the proton spins are polarized by up to several tens in percentage. The red and blue symbols indicate SANS profiles in positively and negatively polarized states, respectively. A significant profile change is thus successfully demonstrated.

Figure 4
SANS profiles observed at various P_H. The horizontal axis shows the magnitude of the scattering vector q [= (4π/λ) sin θ, where λ is the neutron wavelength and 2θ is the scattering angle]. The top and bottom rows present the results for the rubber samples without and with the silane coupling agent, respectively. In each profile, the incoherent scattering intensity contribution is subtracted.

The profiles in Fig. 4 exclude incoherent scattering contributions. These incoherent scattering contributions are evaluated as follows. The microscopic incoherent scattering cross sections for H and D are σ_inc,H = 26.64(3 − 2 $[P_{\rm H}]$ − $[P_{\rm H}^2]$ ) × 10⁻²⁴ cm² and σ_inc,D = 1.02(2 − $[P_{\rm D}]$ − $[P_{\rm D}^2]$ ) × 10⁻²⁴ cm², respectively (Sears, 1992). Contributions from other atoms are negligible. As P_H increases, incoherent scattering decreases monotonically. The macroscopic incoherent scattering cross section (Σ_inc) of each ingredient and each sample is calculated using the known chemical compositions (Tables 2 and 3). From the observed SANS profiles, we subtract the incoherent scattering intensity I_inc, which is calculated considering multiple scattering (Shibayama et al., 2005 ) as

$[I_{\rm inc} = [\exp(\Sigma_{\rm inc}t_{\rm s})- 1] / (4\pi t_{\rm s}). \eqno(6)]$

For the less swollen samples [Figs. 4(a), 4(b), 4(d) and 4(e)], a prominent scattering contribution is found at low q (q < 0.03 Å⁻¹), which is probably due to silica aggregates. The low-q contribution exhibits quadratic P_H dependence, with its minimum being at around P_H = 0%. A less steep contribution is found at high q (q > 0.03 Å⁻¹), which is due to the polymer chains in the d-solvent. The high-q contribution decreases monotonically as P_H increases. For the fully swollen samples [Figs. 4(c) and 4(f)], the low-q contribution does not change significantly, whereas the high-q contribution decreases monotonically as P_H increases.

The scattering intensity is proportional to the contrast factor, which is the squared difference in scattering length density (SLD) between relevant domains. The coherent scattering lengths for H, D, C, N, O, S and Si are b_H = (−0.374 + 1.456P_H) × 10⁻¹² cm, b_D = (0.667 + 0.27P_D) × 10⁻¹² cm, b_C = 0.665 × 10⁻¹² cm, b_N = 0.936 × 10⁻¹² cm, b_O = 0.580 × 10⁻¹² cm, b_S = 0.285 × 10⁻¹² cm and b_Si = 0.415 × 10⁻¹² cm, respectively (Sears, 1992). The SLD of each ingredient in the swollen filler–rubber samples is calculated using their known chemical compositions (Table 2). The swollen silica-filled rubber is a three-component system composed of silica, polymer and d-toluene. The SLD of each component (ρ_S for silica, ρ_P for polymer, ρ_T for d-toluene) is

$[\rho_{\rm S} = 3.08\times 10^{10}\ {\rm cm}^{-2}, \eqno(7)]$

$[\rho_{\rm P} = (0.59 + 8.38 P_{\rm H})\times 10^{10}\ {\rm cm}^{-2}, \eqno(8)]$

$[\eqalignno{\rho_{\rm T} &= (5.61 + 0.07 P_{\rm H} + 1.21 P_D)\times 10^{10}\ {\rm cm}^{-2}&\cr &\simeq (5.61 + 0.31 P_{\rm H})\times 10^{10}\ {\rm cm}^{-2}. &(9)}]$

To calculate ρ_P, we considered only the contributions of the SBR and sulfur. We ignored the contributions of the minor components (the silane coupling agent, accelerator and TEMPO), whose P_H dependence is close to that of the SBR. Fig. 5 shows the calculated SLDs as a function of P_H. Before swelling, ρ_S and ρ_P match at P_H = 30%.

Figure 5
Neutron SLDs of three components as a function of P_H. The blue, green and orange solid lines represent neutron SLDs of silica (ρ_S), polymer (ρ_P) and d-toluene (ρ_T), respectively. The gray and black dashed lines represent neutron SLDs calculated for a homogeneous mixture of polymer and d-toluene (ρ_P+T) at Q_swell = 2 and Q_swell = 5, respectively.

In Fig. 5, the gray and black dashed lines indicate SLDs for a homogeneous mixture of polymer and d-toluene (ρ_P+T) at Q_swell of 2 and 5, respectively. At Q_swell = 2, the calculated ρ_P+T and ρ_S match at around P_H = 0%. This explains the observed results for the less swollen samples. The minimum intensity of the low-q contribution is at around P_H = 0%. At Q_swell = 5, the calculated ρ_P+T and ρ_S match at around P_H = −75%, which exceeds our experimental P_H range. This explains the observed results for the fully swollen samples, in which the low-q contribution does not change significantly with P_H.

The scattering contribution of the polymer chains in d-toluene is proportional to the contrast factor (ρ_P − ρ_T)². ρ_P and ρ_T match at approximately P_H = 62%, which is out of our experimental P_H range. This explains the observed high-q scattering contribution, which monotonically decreases as P_H increases.

3.2. Partial scattering function

The swollen silica-filled rubber is a three-component system composed of silica, polymer and d-toluene. The spatial distribution function of the SLD is

$[\rho({\bf r}\semi P_{\rm H}) = \rho_{\rm S}\varphi_{\rm S}({\bf r}) + \rho_{\rm P}(P_{\rm H})\varphi_{\rm P}({\bf r}) + \rho_{\rm T}(P_{\rm H})\varphi_{\rm T}({\bf r}), \eqno(10)]$

where φ_S(r), φ_P(r) and φ_T(r) are the spatial distribution functions of silica, polymer and d-toluene, respectively. Then, the scattering intensity, I(q), is calculated as

$[\eqalignno{I\left({q\semi {P_{\rm H}}} \right)& = \int \int \rho \left({\bf{r}}_1\semi {P_{\rm H}} \right)\rho \left({\bf{r}}_{2}\semi{P_{\rm H}} \right)\exp[i{\bf{q}} \cdot ({\bf{r}}_{1} - {\bf{r}}_{2})]\,{\rm d}{\bf{r}}_{1}\,{\rm d}{\bf{r}}_{2},&\cr & = \rho _{\rm S}^2{S_{\rm SS}}\left(q \right) + \rho _{\rm P}^2\left({P_{\rm H}} \right){S_{\rm PP}}\left(q \right) + \rho _{\rm T}^2\left({P_{\rm H}} \right){S_{\rm TT}}\left(q \right)&\cr &\quad + 2{\rho _{\rm S}}{\rho _{\rm P}}\left({P_{\rm H}} \right){S_{\rm SP}}\left(q \right)+ 2{\rho _{\rm S}}{\rho _{\rm T}}\left({P_{\rm H}} \right){S_{\rm ST}}\left(q \right)&\cr &\quad+ 2{\rho _{\rm P}}\left({P_{\rm H}} \right){\rho _{\rm T}}\left({P_{\rm H}} \right){S_{\rm PT}}\left(q \right), &(11)}]$

where S_ij(q) is the PSF between components i and j,

$[S_{ij}(q) = (1/V_{\rm s})\int\int\varphi_i({\bf r}_1)\varphi_j({\bf r}_2) \exp[i{\bf q}\cdot ({\bf r}_1 - {\bf r}_2)]\,{\rm d}{\bf r}_1\,{\rm d}{\bf r}_2. \eqno(12)]$

V_s is sample volume. From the above definition, S_ij(q) = S_ji(q). The observed SANS profiles are decomposed into PSFs following Endo (2006 ) and Endo et al. (2008 ) as follows. Through the incompressibility theorem, the following equation should be satisfied:

$[\varphi_{\rm S}({\bf r}) + \varphi_{\rm P}({\bf r}) + \varphi_{\rm T}({\bf r}) = 1. \eqno(13)]$

Given the definition of PSFs, the following equations are obtained:

$[S_{\rm SS}(q) + S_{\rm SP}(q) + S_{\rm ST}(q) = 0, \eqno(14)]$

$[S_{\rm SP}(q) + S_{\rm PP}(q) + S_{\rm PT}(q) = 0, \eqno(15)]$

$[S_{\rm ST}(q) + S_{\rm PT}(q) + S_{\rm TT}(q) = 0. \eqno(16)]$

Eliminating the PSFs related to the d-toluene component results in the following equation:

$[\eqalignno{I\left(q\semi {P_{\rm H}} \right)& = [{\rho _{\rm S}} - {\rho _{\rm T}}\left({P_{\rm H}} \right)]^2{S_{\rm SS}}\left(q \right)&\cr&\quad + 2 [{\rho _{\rm S}} - {\rho _{\rm T}}\left({P_{\rm H}} \right)][{\rho _{\rm P}}\left({P_{\rm H}} \right) - {\rho _{\rm T}}\left({P_{\rm H}} \right)] {S_{\rm SP}}\left(q \right)&\cr &\quad+ [{\rho _{\rm P}}\left({P_{\rm H}} \right) - {\rho _{\rm T}}\left({P_{\rm H}} \right)]^2{S_{\rm PP}}\left(q \right). &(17)}]$

SANS profiles with various P_H are obtained via contrast-variation experiments. The PSFs are automatically obtained through the procedure described in Appendix A. Consequently, S_SS(q), S_PP(q) and S_TT(q) [= S_SS(q) + S_PP(q) + 2S_SP(q)] are obtained (Fig. 6). S_SS(q) exhibits a moderate slope (∼q⁻²) at low q (q < 0.03 Å⁻¹), whereas a steep slope (∼q⁻⁴) is identified at high q (q > 0.03 Å⁻¹). The low-q slope (∼q⁻²) indicates aggregate formation by silica primary particles. The obtained S_SS(q) profiles are similar for all samples.

Figure 6
The obtained partial scattering functions. The top and bottom rows present the results for the rubber samples without and with the silane coupling agent, respectively. In each panel, the blue squares, green circles and orange diamonds indicate the experimentally obtained S_SS(q), S_PP(q) and S_TT(q), respectively. The S_SS(q), S_PP(q) − S_G(q) and S_TT(q) − S_G(q) profiles numerically calculated from the sphere collection model are shown by the dotted-line curves.

For the less swollen samples (R_N1, R_N2, R_CA1 and R_CA2), S_PP(q) and S_TT(q) exhibit similar q dependence with S_SS(q) at low q. This is attributed to the Babinet principle as follows. We assume that the polymer and d-toluene volume fractions are φ_homo and (1 − φ_homo), respectively, and that they mix homogeneously. Then, we obtain φ_P(r) = φ_homo[1 − φ_S(r)] and φ_T(r) = (1 − φ_homo)[1−φ_S(r)]. Hence, S_PP(q) = $[\varphi_{\rm homo}^2]$ S_SS(q) and S_TT(q) = (1 − φ_homo)²S_SS(q).

According to the profiles of R_N2, R_CA1 and R_CA2, the low-q slope of S_TT(q) is slightly steeper than that of S_SS(q), whereas that of S_PP(q) is slightly less steep than that of S_SS(q). This indicates deviation from the assumed homogeneous distribution of the polymer and d-toluene in the matrix and is attributed to the formation of a polymer adsorption layer (Section 3.3).

Unlike S_SS(q), S_PP(q) and S_TT(q) indicate similar significant scattering contributions at high q. At the length scale of this high-q region, the spatial distribution of the polymer and d-toluene components is in an inverse relationship [φ_P(r) = 1 − φ_T(r)]. Hence, the Babinet principle explains the similarity between S_PP(q) and S_TT(q) at high q.

For the fully swollen samples (R_N3 and R_CA3), the low-q profile of S_TT(q) increases, approaching that of S_SS(q). By contrast, S_PP(q) decreases. This is reasonably understood through the increase in the d-toluene volume fraction and the decrease in the polymer volume fraction outside the silica. The S_SS(q) and S_TT(q) profiles of R_N3 are similar at low q, whereas S_SS(q) and S_TT(q) of R_CA3 differ considerably at low q. This is attributed to the formation of a polymer adsorption layer (Section 3.3).

Compared with the less swollen samples, the fully swollen ones show greater PSF fluctuations. This can be understood by considering the P_H dependence of the SLD. In the less swollen state (Q_swell = 2), ρ_S and ρ_P+T match at around P_H = 0 (Fig. 5). Hence, the scattering intensity significantly changes through its minimum. In the fully swollen state (Q_swell = 5), ρ_S and ρ_P+T match at approximately P_H = −70%, which is beyond our achievable range (|P_H| < 30%). Therefore, the scattering intensity changes insignificantly in relative terms. As for the fully swollen state, additional SANS measurements near the matching point should provide PSFs with fewer fluctuations. For this purpose, SANS measurement at high |P_H| values is beneficial. This can be achieved using a recently developed high-P_H-performance DNP instrument (1.2 K, 6.7 T) designed for BL20 iMATERIA (Noda et al., 2020 ).

3.3. Partial scattering function ratio

The previous section discusses the decomposed PSFs, focusing on low-q behavior. For R_N1, S_PP(q), S_TT(q) and S_SS(q) exhibit common q dependence at low q. Therefore, the polymer and d-toluene are homogeneously mixed in the matrix. The different q dependence of S_SS(q), S_PP(q) and S_TT(q) of R_N2, R_CA1, R_CA2 and R_CA3 at low q suggests the formation of a polymer adsorption layer. This section focuses on the ratio between PSFs as a useful indicator.

The microscopic view of swollen silica-filled rubber in Fig. 7 can be divided into three regions: the silica aggregate (α), polymer adsorption layer (β) and matrix (γ). For these three regions, we define spatial distribution functions φ_α(r), φ_β(r) and φ_γ(r), whose value is 1 within the corresponding region and 0 otherwise. If the polymer volume fraction is φ_L in region β and φ_M in region γ, the spatial distribution functions φ_S(r), φ_P(r) and φ_T(r) are

$[\varphi_{\rm S}({\bf r}) = \varphi_\alpha({\bf r}), \eqno(18)]$

$[\varphi_{\rm P}({\bf r}) = \varphi_{\rm L} \varphi_\beta({\bf r}) + \varphi_{\rm M} \varphi_\gamma({\bf r}), \eqno(19)]$

$[\varphi_{\rm T}({\bf r}) = (1-\varphi_{\rm L}) \varphi_\beta({\bf r}) + (1-\varphi_{\rm M}) \varphi_\gamma({\bf r}). \eqno(20)]$

Hence, for the present three-component system, the PSFs are calculated as

$[{S_{\rm SS}}\left(q \right) = n \langle F_\alpha ^2\left(q \right) \rangle \langle K\left(q \right) \rangle, \eqno(21)]$

$[\eqalignno{S_{\rm PP}(q) &= n[\varphi_{\rm L}^2 \langle F_\alpha ^2\left(q \right) \rangle -2\varphi_{\rm L}\Delta\varphi\langle F_{\alpha}(q)F_{\alpha+\beta}(q)\rangle &\cr &\quad+\Delta\varphi^2\langle F_{\alpha+\beta}^2(q)\rangle]\langle K(q)\rangle +S_{\rm G}(q),&(22)}]$

$[\eqalignno{S_{\rm TT}(q) &= n[(1-\varphi_{\rm L})^2\langle F_\alpha ^2(q)\rangle&\cr &\quad+ 2(1-\varphi_{\rm L})\Delta\varphi \langle F_\alpha(q)F_{\alpha+\beta}(q)\rangle &\cr &\quad + \Delta\varphi ^2\langle F_{\alpha+\beta}^ 2(q)\rangle]\langle K(q)\rangle + S_{\rm G}(q), &(23)}]$

$[S_{\rm SP}(q) = n[-\varphi_{\rm L}\langle F_\alpha ^2(q)\rangle + \Delta \varphi \langle F_\alpha (q)F_{\alpha+\beta}(q)\rangle]\langle K(q)\rangle, \eqno(24)]$

$[S_{\rm ST}(q) = n[-(1-\varphi_{\rm L})\langle F_\alpha ^2(q)\rangle - \Delta \varphi \langle F_\alpha (q)F_{\alpha +\beta}(q)\rangle]\langle K(q)\rangle, \eqno(25)]$

$[\eqalignno{S_{\rm PT}(q)& = n[\varphi_{\rm L}(1-\varphi_{\rm L})\langle F_\alpha ^2(q)\rangle+ (2\varphi_{\rm L}-1)\Delta\varphi \langle F_\alpha (q)F_{\alpha +\beta}(q)\rangle &\cr &\quad - \Delta\varphi^ 2\langle F_{\alpha+\beta}^2(q)\rangle] \langle K(q)\rangle-S_{\rm G}(q). &(26)}]$

Here Δφ is the difference in polymer volume fraction (= φ_L − φ_M). n is the number density of the aggregate. $[\langle F_\alpha^2(q)\rangle]$ is the form factor of region α. $[\langle F_{\alpha+\beta}^2(q)\rangle]$ is the form factor of region α+β. 〈F_α(q)F_α+β(q)〉 is a cross term between regions α and α+β. The angle brackets indicate an ensemble average, accounting for polydispersity in structural parameters. 〈K(q)〉 is a structure factor accounting for the spatial distribution of the aggregate (Appendix B). S_G(q) is the scattering intensity due to the polymer chains in the solvent. The form factors and the cross term are defined as

$[F_\alpha^ 2(q) = (1/V_{\rm s})\int\int \varphi_ \alpha({\bf r}_1)\varphi_\alpha({\bf r}_2) \exp[i{\bf q}\cdot({\bf r}_1 - {\bf r}_2)]\,{\rm d}{\bf r}_1\,{\rm d}{\bf r}_2, \eqno(27)]$

$[\eqalignno{F_{\alpha+\beta}^ 2(q)& = (1/V_{\rm s})\int\int \varphi_{\alpha+\beta}({\bf r}_1)\varphi_{\alpha+\beta}({\bf r}_2)&\cr &\quad\times\exp[i{\bf q}\cdot({\bf r}_1 - {\bf r}_2)]\,{\rm d}{\bf r}_1\,{\rm d}{\bf r}_2, &(28)}]$

$[\eqalignno{F_\alpha(q)F_{\alpha+\beta}(q)& = (1/V_{\rm s})\int\int\varphi_\alpha({\bf r}_1)\varphi_{\alpha+\beta}({\bf r}_2)&\cr &\quad\times\exp[i{\bf q}\cdot ({\bf r}_1 - {\bf r}_2)]\,{\rm d}{\bf r}_1\,{\rm d}{\bf r}_2. &(29)}]$

Here, V_s is the sample volume. φ_α+β(r) is the spatial distribution function of region α+β [= φ_α(r) + φ_β(r)]. S_G(q) is the sum of the Debye–Bueche (DB) and Ornstein–Zernike (OZ) functions (Debye & Bueche, 1949 ; Ornstein & Zernike, 1914 ),

$[S_{\rm G}\left(q \right) = {A_{\rm DB}}/{\left({1 + {q^2}L_{\rm DB}^2} \right)^2} + A_{\rm OZ}/\left({1 + {q^2}L_{\rm OZ}^2} \right), \eqno(30)]$

where L_DB is the size of cross-linking density heterogeneity and L_OZ is the mesh size of the polymer network. For a swollen rubber sample without filler particles, the corresponding PSFs $[S_{\rm PP}'(q)]$ , $[S_{\rm TT}'(q)]$ and $[S_{\rm PT}'(q)]$ are

$[S_{\rm PP}'(q) = S_{\rm G}(q), \eqno(31)]$

$[S_{\rm TT}'(q) = S_{\rm G}(q), \eqno(32)]$

$[S_{\rm PT}'(q) = -S_{\rm G}(q). \eqno(33)]$

These contributions are simply incorporated into S_PP(q), S_TT(q) and S_PT(q) as in previous studies (e.g. Nakanishi et al., 2024).

Figure 7
Schematic diagram of a silica aggregate surrounded by a polymer adsorption layer. The dark gray region (α) is the silica aggregate, the light gray region (β) is the polymer-dense layer, and the white external region (γ) is the matrix.

We propose the ratio −S_SP(q)/S_SS(q) as a useful indicator of the formation of the polymer adsorption layer:

$[-S_{\rm SP}(q) / S_{\rm SS}(q) = \varphi_{\rm L} - \Delta\varphi\langle F_\alpha(q)F_{\alpha+\beta}(q)\rangle/\langle F_\alpha^2(q)\rangle. \eqno(34)]$

Our use of this ratio avoids the effect of 〈K(q)〉 and S_G(q), enabling least-squares fitting in a wider q range with fewer adjustable parameters. In equation (34), the second term on the right side, $[\langle F_\alpha^2(q)\rangle]$ , is an ordinary form factor that decreases according to Porod's law (q⁻⁴) at high q. 〈F_α(q)F_α+β(q)〉 is cross term that is a product of oscillation functions with different frequencies. As q increases, the cross term quickly reduces to a negative value. Hence, 〈F_α(q)F_α+β(q)〉 drops faster than $[\langle F_\alpha^2(q)\rangle]$ at high q. The contribution of the second term diminishes to provide a constant φ_L.

The meaning of equation (34) is clarified using the following approximate equations based on the Guinier approximation, which are satisfied only at low q:

$[\langle F_\alpha ^2\left(q \right) \rangle \approx \langle {V_\alpha } \rangle ^2 \exp(- R_{{\rm g},\alpha} ^2{q^2}/3), \eqno(35)]$

$[\langle F_{\alpha + \beta }^2\left(q \right) \rangle \approx \langle V_{\alpha + \beta } \rangle ^2 \exp(- R_{{\rm g},\alpha + \beta }^2{q^2}/3), \eqno(36)]$

$[\eqalignno{\langle {F_\alpha }\left(q \right){F_{\alpha + \beta }}\left(q \right) \rangle &\approx \langle {V_\alpha } \rangle \langle {V_{\alpha + \beta }} \rangle \exp(- R_{{\rm g},\alpha} ^2{q^2}/6)&\cr &\quad\times\exp(- R_{{\rm g},\alpha + \beta }^2{q^2}/6), &(37)}]$

where 〈V_α〉 and 〈V_α+β〉 are the expected volumes of regions α and α+β, respectively. R_g,α and R_g,α+β are the gyration radii of regions α and α+β, respectively. Applying these to equation (34) yields the approximate equation

$[\eqalignno{-S_{\rm SP}(q) / S_{\rm SS}(q) &\simeq \varphi_{\rm L} - \Delta\varphi(\langle V_\alpha\rangle / \langle V_{\alpha+\beta}\rangle)&\cr &\quad\times\exp[-(R_{{\rm g},\alpha+\beta}^2-R_{{\rm g},\alpha}^2)q^2/6]. &(38)}]$

−S_SP(q)/S_SS(q) is expected to show a flat region (= φ_L) at high q and a depression at low q. This approximate equation provides an opportunity to evaluate Δφ(〈V_α〉/〈V_α+β〉) as the depression depth and ( $[R_{\rm g,\alpha+\beta}^2 - R_{\rm g,\alpha}^2]$ ) by the q dependence of the depression curve. −S_ST(q)/S_SS(q) can be used for the same purpose. However, the sum of −S_SP(q)/S_SS(q) and −S_ST(q)/S_SS(q) is 1. Hence, only one of them should be assessed.

Fig. 8 shows the calculated −S_SP(q)/S_SS(q), with R_N1 exhibiting a flat profile [Fig. 8(a)]. This indicates that the polymer and d-toluene are mixed homogeneously outside the silica. R_N2 exhibits a slight depression at low q, indicating the formation of the polymer adsorption layer [Fig. 8(b)]. As the d-toluene volume fraction increases, the high-q constant value falls. As for R_N3, as the amount of d-toluene increases to full swelling, the polymer fraction falls further [Fig. 8(c)]. No clear depression is identified, partially due to the larger fluctuations compared with those in the case of R_N1 and R_N2.

Figure 8
−S_SP(q)/S_SS(q) profiles. The top and bottom rows present the results for the rubber samples without and with the silane coupling agent, respectively. In each panel, the unfilled gray circles represent the experimentally obtained results, and the gray solid-line curves represent the profiles numerically calculated from the structure model. Here, the smear effect is considered. The black dashed-line curves indicate the numerical calculation results obtained using the approximate equation (38)

As shown in Figs. 8(d) and 8(e), R_CA1 and R_CA2 exhibit slight depressions at low q. As the d-toluene volume fraction increases, the high-q constant value falls and the low-q depression becomes clearer. As for R_CA3, when the d-toluene volume fraction increases to full swelling, a significant low-q depression emerges [Fig. 8(f)]. The validity of the approximate equation (38) is discussed in Section 3.6.

3.4. Numerical calculation based on sphere collection model

In previous contrast-variation studies on swollen filler–rubber systems (Takenaka et al., 2009; Takenaka et al., 2012; Liu et al., 2017), researchers calculated $[\langle F_\alpha^2(q)\rangle]$ and $[\langle F_{\alpha+\beta}^2(q)\rangle]$ from decomposed PSFs and then applied the Beaucage unified equation (Beaucage & Schaefer, 1994 ; Beaucage, 1995 ; Beaucage, 1996 ; Beaucage, 2004 ) to obtain the structural parameters of the filler aggregate and adsorption layer. The Beaucage unified equation is advantageous for investigating filler aggregate systems with multilevel hierarchies. However, this approach does not provide an explicit formula for the cross term 〈F_α(q)F_α+β(q)〉. Therefore, many trials are required to determine the optimal parameter set.

Another traditional approach numerically calculates scattering profiles based on structure models. Nakanishi et al. (2024) developed a structure model that considers the formation of dimers, in which two primary particles contact each other. This approach was effective for their samples, which consisted of large silica particles with a radius of 523 Å. However, our samples, which were composed of silica particles with a radius of 105 Å, were likely to form aggregates composed of more than two primary particles. Hence, we built a structure model considering aggregates formed by numerous primary particles as follows.

In this structure model, silica aggregates are modeled as a collection of spherical particles, and the outer envelope of the polymer adsorption layer is modeled as a large sphere. Although this structure model is designed to reflect actual silica aggregates as much as possible, simplifications are made to reduce calculation difficulty. The structure model is defined by the following rules:

(i) In one aggregate, all primary particles have a common radius (R_p).

(ii) Each particle center is placed at a lattice point of a face-centered cubic lattice. The lattice constant is set to 2 $[\sqrt 2]$ R_p so that particles at neighboring lattice points contact each other.

(iii) Among the possible aggregate configurations formed by N_p particles, we use only the `most compact' one, as determined by the following procedure. The most compact configuration has the largest number of particle pairs with the shortest distance (2R_p). If several configurations satisfy this criterion, then the numbers of particle pairs with the second-shortest distance are compared. If several configurations fulfill this requirement, then the numbers of particle pairs with the third-shortest distance are compared, and so on. Through this procedure, we manually determine the most compact configuration up to N_p = 19. As expected from these criteria, symmetric configuration is advantageous. We adopt a dumbbell for N_p = 2, an equilateral triangle for N_p = 3, a regular tetrahedron for N_p = 4, a regular octahedron for N_p = 6 and a cuboctahedron (including one particle at the center) for N_p = 13. Fig. 9 shows schematic diagrams of configurations up to N_p = 4. For N_p values not mentioned above, less symmetrical configurations are determined manually on the basis of the defined criteria. In the search process, we start from highly symmetrical configurations and then manually add or remove particles one by one.

Figure 9
Schematic diagram of the sphere collection model. The particle configuration up to N_p = 4 is indicated.

(iv) The sum region of the silica aggregate and polymer adsorption layer (α+β) is defined as a single large sphere whose center coincides with the aggregate mass center. The radius of this sum region (L) is

$[L = {R_{\rm agg}} + t_{\rm L}, \eqno(39)]$

where R_agg is the circumsphere radius of the aggregate and t_L is the thickness of the polymer adsorption layer. In Fig. 9, the aggregate circumsphere and the outer envelope of the polymer adsorption layer are schematically indicated by the gray solid-line curves and black dashed-line curves, respectively.

(v) To account for aggregate polydispersity, we assume the distribution functions W_Np(N_p) and W_Rp(R_p). W_Np(N_p) is assumed to be a log-normal distribution,

$[W_{\rm Np}(N_{\rm p}) = {1 \over {\sqrt {2\pi } \sigma _{{\rm{Np}}}N_{\rm{p}}}}\exp\left\{ - {{\left[\ln(N_{\rm{p}}) - \ln\left(N_{{\rm{p}},{\rm{med}}} \right) \right]^2} \over {2\sigma _{\rm{Np}}^2}} \right\}, \eqno(40)]$

where N_p,med is the median value of N_p and σ_Np is the standard deviation of ln(N_p). W_Rp(R_p) is assumed to be a Gaussian distribution:

$[W_{\rm Rp}(R_{\rm p}) = {1 \over {\sqrt {2\pi } \sigma _{\rm{Rp}}}} \exp\left[ - {{\left(R_{\rm{p}} - R_{{\rm{p}},{\rm{mean}}} \right)^2} \over {2\sigma _{\rm{Rp}}^2}} \right], \eqno(41)]$

where R_p,mean is the mean value of R_p and σ_Rp is the standard deviation of R_p.

Thus, the structure model for the silica aggregate and polymer adsorption layer is built. The aggregate configuration is defined in full detail for scattering intensity calculation. However, the adopted configuration is merely a typical one. A practical sample has various configurations that differ from the analyzed one. These slightly different configurations cannot be distinguished, as their subtle difference causes only a slight fluctuation in scattering profile, which is concealed by the effect of polydispersity in practical samples.

For the determined set of N_p particle center positions (c_i,Np, where i = 1, 2,…, N_p), the spatial distribution functions for regions α and α+β are, respectively,

$[\varphi_{\alpha,{\rm Np}}({\bf r}) = \sum\limits_{i = 1}^{{{N_{\rm p}}}} \varphi _{{\rm{sph}}} \left({\bf{r}} - {\bf{c}}_{i,{\rm{Np}}}\semi R_{\rm{p}} \right) , \eqno(42)]$

$[\varphi_{\alpha +\beta ,{\rm Np}}({\bf r}) = \varphi _{\rm{sph}}\left({\bf{r}} - {\bf{c}}_{{\rm{M}},{\rm{Np}}}\semi R_{{\rm{agg}},{\rm{Np}}} + t_{\rm{L}} \right) .\eqno(43)]$

Here c_M,Np is the mass center position of an aggregate formed by N_p particles, which is calculated as

$[{\bf c}_{\rm M,Np} = {1 \over {N_{\rm{p}}}}\sum\limits_{i = 1}^{{N_{\rm p}}} {\bf{c}}_{i,{\rm{Np}}} . \eqno(44)]$

φ_sph(r; R) is the spatial distribution of a sphere and defined as

$[\eqalignno{\varphi_{\rm sph}({\bf r}\semi R)& = 1\ {\rm for}\ |r|\le R,&\cr & = 0\ {\rm for}\ |r|\ \gt\, R.&(45)}]$

The form factors for φ_α,Np(r) and φ_α+β,Np(r) are

$[\eqalignno{F_{\alpha,{\rm Np}}^2(q) &= nF_{\rm{sph}}^2\left(q\semi R_{\rm{p}} \right)\left[N_{\rm{p}} + \sum\limits_{i = 1}^{N_{\rm p}}\sum\limits_{j \ne i}^{N_{\rm p}} \sin \left(qr_{ij,{\rm{Np}}} \right)/\left(qr_{ij,{\rm{Np}}} \right)\right] &\cr &= nF_{{\rm{sph}}}^2\left(q\semi R_{\rm{p}} \right)G_{\rm Np}\left(R_{\rm{p}} \right), &(46)}]$

$[\eqalignno{&F_{\alpha,{\rm Np}}(q)F_{\alpha+\beta,{\rm Np}}(q) &\cr &= nF_{{\rm{sph}}}\left(q\semi R_{\rm{p}} \right)F_{{\rm{sph}}}\left(q\semi R_{{\rm agg},{\rm{Np}}} + t_{\rm{L}} \right)\sum\limits_{i = 1}^{N_{\rm p}} \sin \left(qr_{i{\rm{M}},{\rm{Np}}} \right)/\left(qr_{i{\rm{M}},{\rm{Np}}} \right) &\cr &= nF_{{\rm{sph}}}\left(q\semi R_{\rm{p}} \right)F_{{\rm{sph}}}\left(q\semi R_{{\rm{agg}},{\rm{Np}}} + t_{\rm{L}} \right)H_{{\rm{Np}}}\left(R_{\rm{p}} \right) ,&(47)}]$

$[F_{\alpha+\beta,{\rm Np}}^2(q) = nF_{{\rm{sph}}}^2\left(q\semi R_{{\rm agg},{\rm{Np}}} + t_{\rm{L}} \right). \eqno(48)]$

Here n is the number density of the aggregate [= (f_silica/Q_swell)/〈V_α〉, where f_silica is the silica volume fraction]. r_ij,Np is the distance between particles i and j for the aggregate formed by N_p particles (= |c_i,Np − c_j,Np|). r_iM,Np is the distance between particle i and the aggregate mass center for the aggregate formed by N_p particles (= |c_i,Np − c_M,Np|). F_sph(q; R) is the form amplitude of a sphere of radius R,

$[{F_{\rm sph}}\left(q\semi R \right) = (4\pi {R^3}/3)3[\sin\left({qR} \right) - \left({qR} \right)\cos\left({qR} \right)]/\left({qR} \right)^3. \eqno(49)]$

G_Np(R_p) in equation (46) is the interference term for the particle pairs in the aggregate. Possible r_ij,Np values are 2R_p, $[2\sqrt 2]$ R_p, $[2\sqrt 3]$ R_p, 4R_p and so on, since the structure model assumes that the particle centers are located in a face-centered cubic lattice with the lattice constant of $[2\sqrt 2]$ R_p. For reducing the calculation effort, we rearranged G_Np(R_p) as

$[G_{\rm Np}(R_{\rm p}) = N_{\rm{p}} + \sum\limits_{k = 1}^{k_{\max,{\rm G},{\rm Np}}} A_{k,{\rm{Np}}} \sin\left(qB_{k,{\rm{Np}}} \right)/\left(qB_{k,{\rm{Np}}} \right), \eqno(50)]$

where B_k,Np is defined for sorting the r_ij,Np values and A_k,Np is the occurrence number of B_k,Np during the double summation in equation (46). A_k,Np and B_k,Np parameters calculated for the adopted N_p particle aggregate configuration are listed in Table 4.

Table 4
Parameters A_k,Np (upper values in each cell) and B_k,Np (lower values in each cell) calculated for the adopted N_p particle aggregate configuration

N_p \ k	1	2	3	4	5	6
2	2
	2R_p
3	6
	2R_p
4	12
	2R_p
5	16	4
	2R_p	$[2\sqrt 2]$ R_p
6	24	6
	2R_p	$[2\sqrt 2]$ R_p
7	30	6	6
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p
8	36	8	12
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p
9	42	10	18	2
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p
10	48	14	24	4
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p
11	54	16	32	8
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p
12	62	20	40	10
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p
13	72	24	48	12
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p
14	80	26	56	12	8
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p	$[2\sqrt 5]$ R_p
15	72	24	72	18	24
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p	$[2\sqrt 5]$ R_p
16	96	30	72	18	24
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p	$[2\sqrt 5]$ R_p
17	104	32	80	22	32	2
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p	$[2\sqrt 5]$ R_p	$[4\sqrt 2]$ R_p
18	112	34	88	28	40	4
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p	$[2\sqrt 5]$ R_p	$[4\sqrt 2]$ R_p
19	120	36	96	36	48	6
	2R_p	$[2\sqrt 2]$ R_p	$[2\sqrt 3]$ R_p	4R_p	$[2\sqrt 5]$ R_p	$[4\sqrt 2]$ R_p

H_Np(R_p) in equation (47) is the interference term between the particles and mass center of the aggregate. For reducing the calculation effort, we rearranged H_Np(R_p) as

$[H_{\rm Np}(R_{\rm p}) = \sum\limits_{k = 1}^{k_{\max,{\rm H},{\rm Np}}} C_{k,{\rm{Np}}} \sin\left(qD_{k,{\rm{Np}}} \right)/\left(qD_{k,{\rm{Np}}} \right), \eqno(51)]$

where D_k,Np is defined for sorting the r_iM,Np values and C_k,Np is the occurrence number of D_k,Np during the summation in equation (47). C_k,Np and D_k,Np parameters calculated for the adopted N_p particle aggregate configuration are listed in Table 5.

Table 5
Parameters C_k,Np (upper values in each cell) and D_k,Np (lower values in each cell) calculated for the adopted N_p particle aggregate configuration

N_p \ k	1	2	3	4	5	6	7	8
1	1
	0.000
2	2
	1.000R_p
3	3
	1.155R_p
4	4
	1.225R_p
5	1	4
	1.131R_p	1.442R_p
6	6
	1.414R_p
7	3	3	1
	1.245R_p	1.641R_p	2.100R_p
8	2	2	2	2
	1.118R_p	1.500R_p	1.803R_p	2.062R_p
9	1	2	2	1	1	2
	0.969R_p	1.352R_p	1.648R_p	1.899R_p	2.012R_p	2.222R_p
10	1	4	1	4
	0.849R_p	1.523R_p	1.980R_p	2.173R_p
11	1	2	1	2	2	2	1
	0.315R_p	1.734R_p	1.836R_p	1.933R_p	2.025R_p	2.112R_p	2.197R_p
12	1	1	4	2	4
	0.167R_p	1.833R_p	1.922R_p	2.007R_p	2.088R_p
13	1	12
	0.000	2.000R_p
14	1	4	4	4	1
	0.202R_p	1.863R_p	2.010R_p	2.148R_p	2.626R_p
15	1	1	4	2	4	1	2
	0.267R_p	1.733R_p	1.881R_p	2.018R_p	2.146R_p	2.267R_p	2.647R_p
16	1	3	6	3	3
	0.306R_p	1.759R_p	2.023R_p	2.257R_p	2.663R_p
17	1	1	4	2	4	1	2	2
	0.235R_p	1.764R_p	1.893R_p	2.013R_p	2.127R_p	2.235R_p	2.667R_p	2.838R_p
18	1	4	4	4	1	4
	0.157R_p	1.892R_p	2.006R_p	2.114R_p	2.671R_p	2.833R_p
19	1	12	6
	0.000	2.000R_p	2.828R_p

In addition, we calculate the circumsphere radius R_agg,Np of the adopted N_p particle aggregate configuration, and the obtained R_agg,Np values are listed in Table 6. In addition, the squared gyration radius $[R_{\rm g,agg,Np}^2]$ is calculated for the adopted N_p particle aggregate configuration:

$[R_{\rm g,agg,Np}^2 = \left [{3 \over 5} + {1 \over {N_{\rm{p}}}}\sum\limits_{i = 1}^{N_{\rm p}} \left({{r_{i{\rm M},{\rm{Np}}}} \over {R_{\rm{p}}}} \right)^2 \right]R_{\rm{p}}^2 . \eqno(52)]$

The obtained $[R_{\rm g,agg,Np}^2]$ values are listed in Table 6.

Table 6
Parameters R_agg,Np and $[R_{\rm g,agg,Np}^2]$ calculated for the adopted N_p particle aggregate configuration

N_p	R_agg,Np	$[R_{\rm g,agg,Np}^2]$
1	1.000R_p	0.600 $[R_{\rm p}^2]$
2	2.000R_p	1.600 $[R_{\rm p}^2]$
3	2.155R_p	1.933 $[R_{\rm p}^2]$
4	2.225R_p	2.100 $[R_{\rm p}^2]$
5	2.442R_p	2.520 $[R_{\rm p}^2]$
6	2.414R_p	2.600 $[R_{\rm p}^2]$
7	3.100R_p	3.049 $[R_{\rm p}^2]$
8	3.062R_p	3.350 $[R_{\rm p}^2]$
9	3.222R_p	3.662 $[R_{\rm p}^2]$
10	3.173R_p	3.880 $[R_{\rm p}^2]$
11	3.197R_p	4.137 $[R_{\rm p}^2]$
12	3.088R_p	4.239 $[R_{\rm p}^2]$
13	3.000R_p	4.292 $[R_{\rm p}^2]$
14	3.626R_p	4.559 $[R_{\rm p}^2]$
15	3.653R_p	4.796 $[R_{\rm p}^2]$
16	3.663R_p	5.006 $[R_{\rm p}^2]$
17	3.838R_p	5.251 $[R_{\rm p}^2]$
18	3.833R_p	5.464 $[R_{\rm p}^2]$
19	3.828R_p	5.653 $[R_{\rm p}^2]$

We account for N_p and R_p polydispersity using the equations

$[\langle F_\alpha^2(q) \rangle = {{\sum_{N_{\rm p} = 1}^{N_{\rm p,\max}} W_{{\rm{Np}}} \left(N_{\rm{p}} \right)\int W_{\rm{R}}\left(R_{\rm{p}} \right)F_{\alpha, {\rm{Np}}}^2\left(q\semi R_{\rm{p}} \right)\,{\rm{d}}R_{\rm{p}}} \over {\sum_{N_{\rm p} = 1}^{N_{\rm p,\max}} W_{{\rm{Np}}} \left(N_{\rm{p}} \right)\int W_{{\rm{Rp}}}\left(R_{\rm{p}} \right)\,{\rm{d}} R_{\rm{p}}}}, \eqno(53)]$

$[\eqalignno{&\langle F_\alpha(q)F_{\alpha+\beta}(q) \rangle =&\cr &{{\sum_{N_{\rm p} = 1}^{N_{\rm p,{\max}}} W_{{\rm{Np}}}\left(N_{\rm{p}} \right)\int W_{{\rm{Rp}}}\left(R_{\rm{p}} \right)\left| F_{\alpha, {\rm{Np}}}\left(q\semi R_{\rm{p}} \right)F_{\alpha + \beta, {\rm{Np}}}\left(q\semi R_{\rm{p}} \right) \right|\,{\rm{d}}R_{\rm{p}}} \over {\sum_{N_{\rm p} = 1}^{N_{\rm p,{\max}}} W_{{\rm{Np}}} \left(N_{\rm{p}} \right)\int W_{{\rm{Rp}}}\left(R_{\rm{p}} \right)\,{\rm{d}}R_{\rm{p}}}} , &\cr &&(54)}]$

$[\langle F_{\alpha+\beta}^2(q) \rangle = {{\sum_{N_{\rm p} = 1}^{N_{\rm p,{\max}}} W_{{\rm{Np}}} \left(N_{\rm{p}} \right)\int W_{{\rm{Rp}}}\left(R_{\rm{p}} \right)F_{\alpha + \beta, {\rm{Np}}}^2\left(q\semi R_{\rm{p}} \right)\,{\rm{d}} R_{\rm{p}}} \over {\sum_{N_{\rm p} = 1}^{N_{\rm p,{\max}}} W_{\rm Np} \left(N_{\rm{p}} \right)\int W_{{\rm{Rp}}}\left(R_{\rm{p}} \right)\,{\rm{d}}R_{\rm{p}}}} . \eqno(55)]$

Fig. 10 shows the $[F_{\alpha,{\rm Np}}^2(q)]$ profiles numerically calculated using R_p = 105 ± 25 Å and A_k,Np and B_k,Np (Table 4). A profile with N_p = 1 indicates the form factor of a single sphere. The profile is flat at low q but follows q⁻⁴ at high q. The low-q scattering intensity increases with N_p, whereas the high-q region of q⁻⁴ does not change. The shape of the shoulder at approximately q = 0.02 Å⁻¹ gradually changes from convex to concave.

Figure 10
Numerically calculated profiles of silica aggregates formed by N_p particles [ $[F_{\alpha,{\rm Np}}^2(q)]$ ] with a particle radius of R_p = 105 ± 25 Å (up to N_p = 10). The particle configuration of the silica aggregates was determined according to the criteria described in this article.

By considering N_p polydispersity, we calculate $[\langle F_\alpha^2(q)\rangle]$ , as indicated by the blue dotted-line curves in Fig. 6. The numerical calculations account for the smear effect of BL15 beam collimation (Takata et al., 2015). This effect is only slight at low q. Consequently, the numerically calculated profiles excellently reproduce the experimentally obtained S_SS(q) [= n $[\langle F_\alpha^2(q)\rangle]$ 〈K(q)〉] despite the slight deviation at low q (q < 0.01 Å⁻¹), which is probably due to 〈K(q)〉. The distribution functions W_Rp(R_p) and W_Np(N_p), which are used across all samples, are shown in Figs. 11(a) and 11(b), respectively. In a previous study (Nakanishi et al., 2024), a silane coupling agent improved the dispersion of silica particles. However, such dispersion improvement was not observed in our samples.

Figure 11
Distribution functions used for numerical calculation: W_Rp(R_p) (a) and W_Np(N_p) (b).

We evaluate W_Rp(R_p) and W_Np(N_p) by analyzing S_SS(q). Next, we set t_L to calculate $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ and 〈F_α(q)F_α+β(q)〉 (Fig. 12). The adopted t_L values [50 and 120 Å in Figs. 12(a) and 12(b), respectively] are optimized parameters for fitting the experimental results, as shown later in this section. In Fig. 12, $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ is always higher than $[\langle F_\alpha^2(q)\rangle]$ . The q⁻⁴ slope of $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ continues to lower q (q ∼ 0.01 Å⁻¹) compared with that of $[\langle F_\alpha^2(q)\rangle]$ . Fig. 12 also shows 〈F_α(q)F_α+β(q)〉. At low q, 〈F_α(q)F_α+β(q)〉 is between $[\langle F_\alpha^2(q)\rangle]$ and $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ . As q increases, 〈F_α(q)F_α+β(q)〉 decreases to a negative value more rapidly than $[\langle F_\alpha^2(q)\rangle]$ and then oscillates while decreasing in magnitude.

Figure 12
Numerically calculated $[ \langle F_{\alpha}^2(q)\rangle]$ , $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ and 〈F_α(q)F_α+β(q)〉 for t_L = 50 Å (a) and t_L = 120 Å (b). For both panels, R_p = 105 ± 25 Å and the W_Rp(R_p) and W_Np(N_p) shown in Fig. 11

are used.

We determine φ_L and φ_M, which are necessary for calculating S_PP(q) and S_TT(q) by equations (22) and (23), as follows. φ_L and φ_M are

$[\varphi_{\rm L} = \varphi_{\rm homo} + [f_\gamma / (f_\beta + f_\gamma)] \Delta\varphi , \eqno(56)]$

$[\varphi_{\rm M} = \varphi_{\rm homo} - [f_\beta / (f_\beta + f_\gamma)] \Delta\varphi, \eqno(57)]$

where φ_homo is given by (1 − f_silica)/(Q_swell − f_silica). f_silica is the silica volume fraction before swelling (= 0.05 in this study). f_α, f_β and f_γ, which are the volume fractions of regions α, β and γ, respectively, are

$[f_\alpha = f_{\rm silica} / Q_{\rm swell}, \eqno(58)]$

$[f_\beta = (f_{\rm silica} / Q_{\rm swell})(\langle V_\beta\rangle / \langle V_\alpha\rangle), \eqno(59)]$

$[f_\gamma = 1 - (f_{\rm silica} / Q_{\rm swell})[(\langle V_\alpha\rangle + \langle V_\beta\rangle) / \langle V_\alpha\rangle], \eqno(60)]$

where 〈V_α〉 and 〈V_β〉 are the expected volumes of regions α and β, respectively, and can be calculated by the determined structure model. Therefore, only two parameters (t_L and Δφ) are adjustable in calculating S_PP(q) and S_TT(q). The S_PP(q) and S_TT(q) profiles with optimized parameters are drawn using green and orange dotted-line curves, respectively, in Fig. 6. At low q, the numerically calculated profiles deviate downwards similarly to S_SS(q). This is because the numerical calculation does not consider the structure factor 〈K(q)〉 due to the higher-order structure. In Fig. 8, the numerically calculated ratio −S_SP(q)/S_SS(q) is indicated by the gray curves. With our use of this ratio, the effect of K(q) and S_G(q) can be eliminated, enabling least-squares fitting in a wider q range to determine t_L and Δφ.

The parameters determined by this fitting process are listed in Table 7. A schematic diagram of the microstructure around the silica particle surface of R_CA is shown in Fig. 13. As Q_swell increases, Δφ increases while t_L decreases. The volume of the silane coupling agent in one silica aggregate is 0.26 × 10⁷ Å³, as determined by the composition of the rubber ingredient, assuming that all silane coupling agents react with silica surfaces. This value is used across all R_CA samples. Moreover, the volume of polymer chains in the polymer adsorption layer is calculated as 〈V_β〉/φ_L. A comparison of these values suggests that the polymer adsorption layer contains not only the silane coupling agent but also polymer chains confined by the silane coupling agent. For R_CA1, R_CA2 and R_CA3, the computed volume of polymer chains in the polymer adsorption layer is 43, 23 and 6.8 times larger than that of the silane coupling agent, respectively. The confinement degree is expected to decrease with distance from the silica aggregate surface. It was suggested that in the fully swollen state only tightly confined polymer chains are included in the polymer adsorption layer. By contrast, in a less swollen state, the polymer adsorption layer additionally contains moderately confined polymer chains. NMR relaxation times have shown that an intermediate region forms outside the confined polymer around filler particles (O'Brien et al., 1976 ). The correlation between the results of these two approaches will further elucidate the structure of the polymer adsorption layer.

Table 7
Structural parameters used for the sphere collection model

	R_N1	R_N2	R_N3	R_CA1	R_CA2	R_CA3
Q_swell	1.69	2.14	5.51	1.66	1.80	4.76
φ_homo	0.579	0.4545	0.174	0.590	0.543	0.202
φ_L	0.579	0.4553	0.174	0.591	0.547	0.259
φ_M	0.579	0.4543	0.174	0.589	0.542	0.199
Δφ = (φ_L− φ_M)	0.000	0.001	0.000	0.002	0.005	0.060
t_L (Å)	–	120	–	120	70	50
R_p,mean (Å)	105	105	105	105	105	105
σ_Rp (Å)	25	25	25	25	25	25
N_p,med	2.5	2.5	2.5	2.5	2.5	2.5
σ_Np	0.6	0.6	0.6	0.6	0.6	0.6
$[R_{\rm g,\alpha}^2]$ (10⁴ Å²)	2.02	2.02	2.02	2.02	2.02	2.02
〈V_α〉 (10⁷ Å³)	1.68	1.68	1.68	1.68	1.68	1.68
$[R_{\rm g,\alpha+\beta}^2]$ (10⁴ Å²)	–	6.75	–	6.75	4.94	4.30
〈V_α+β〉 (10⁷ Å³)	–	17.06	–	17.06	10.96	9.03
〈V_β〉 (10⁷ Å³)	–	15.38	–	15.38	9.28	7.35
φ_L〈V_β〉 ( 10⁷ Å³)	–	7.00	–	9.09	5.08	1.90
A_Porod (10³ cm⁻⁵)	3.0	3.0	3.7	0.98	1.1	3.4
A_DB (10⁻²⁰ cm⁻¹)	0.86	1.15	1.9	0.82	1.8	51
L_DB (Å)	130	140	150	170	200	350
A_OZ (10⁻²³ cm⁻¹)	6.5	9.0	30/5.0	6.0	6.5	46/9.0
L_OZ (Å)	1.3	2.0	14/2.0	1.3	2.0	17/4.0
n (10⁻¹⁰ Å⁻³)	17.6	13.9	5.40	17.9	16.5	6.25
n^−1/3 (Å)	828	896	1230	823	846	1170

Figure 13
Schematic diagram of the polymer adsorption layer on the silica aggregate surface for R_CA1 (a), R_CA2 (b) and R_CA3 (c). In each panel, the black dashed-line curve indicates the boundary between the polymer-dense layer and the surrounding matrix. The polymer chains are omitted for simplicity. Outside the silica, the background color indicates the polymer volume fraction.

For R_N, a slight indication of the formation of a polymer adsorption layer emerges in the less swollen state (R_N2). In the fully swollen state (R_N3), little indication of formation is found. Even for the sample without a silane coupling agent, a polymer adsorption layer, reflecting loose confinement around the silica aggregate, is observed. However, polymer adsorption layers reflecting tight confinement around silica aggregates are not observed naturally in samples without silane coupling agents.

3.5. Separation of scattering contribution due to polymer in d-toluene

S_G(q), observed in swollen rubber samples, provides insights into polymer networks and cross-links (Karino et al., 2007; Ikeda et al., 2009; Suzuki et al., 2010). For silica-filled swollen rubber, the OZ contribution at high q can be analyzed easily, as the scattering contributions of filler particles are insignificant in the high-q region. By contrast, the DB contribution at low q is difficult to analyze because the scattering contribution of filler particles is prominent in the low-q region. The contrast-variation approach provides a way to separate S_G(q).

By substituting equations (21), (22) and (24) into equation (17), we obtain

$[\eqalignno{I(q) &= [\rho_{\rm S} - \rho_{\rm P+T}(P_{\rm H})]^2 n\langle F_\alpha^2(q)\rangle\langle K(q)\rangle &\cr &\quad+ 2[\rho_{\rm S} - \rho_{\rm P+T}(P_{\rm H})] [\rho_{\rm P}(P_{\rm H}) - \rho_{\rm T}(P_{\rm H})]&\cr &\quad\times n\Delta \varphi \langle F_\alpha(q)F_{\alpha+\beta}(q)\rangle\langle K(q)\rangle &\cr &\quad+ [\rho_{\rm P}(P_{\rm H}) - \rho_{\rm T}(P_{\rm H})]^2&\cr &\quad\times[n\Delta\varphi^2\langle F_{\alpha+\beta}^2(q)\rangle\langle K(q)\rangle + S_{\rm G}(q)], &(61)}]$

where ρ_P+T is the SLD of the matrix domain, assuming the matrix is homogeneous: ρ_P+T = φ_homoρ_P + (1 − φ_homo)ρ_T. I(q) consists of three terms with different P_H dependence. Both nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 and S_G(q) are included in the final term. A set of three independent PSFs can be determined through contrast variation. From these PSFs, another set of three independent functions [n $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉, nΔφ〈F_α+β(q) × F_α+β(q)〉〈K(q)〉 and nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 + S_G(q)] can be obtained. However, contrast variation itself cannot determine nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 and S_G(q) separately. In cases with an insignificant Δφ (e.g. less swollen or homogeneously swollen states), nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 can be neglected to determine S_G(q) easily. However, in cases with a significant Δφ (e.g. R_CA3), S_G(q) is difficult to determine. In this case, S_G(q) should be evaluated using the following equation, which is derived from equations (21), (22) and (24):

$[\eqalignno{S_{\rm G}(q)& = \varphi_{\rm L}^2 S_{\rm SS}(q) + 2\varphi_{\rm L} S_{\rm SP}(q) + S_{\rm PP}(q)&\cr &\quad- n\Delta\varphi^2\langle F_{\alpha +\beta}^ 2(q)\rangle \langle K(q)\rangle.&(62)}]$

We determine nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 through the structure model analysis in Section 3.4. The evaluated S_G(q) profiles are shown by the unfilled gray circles in Fig. 14. Each panel shows a clear DB contribution at low q. However, the shape of the OZ contribution at high q is not completely covered, because of the limited q range of our setup. This is compensated for by overlaying the SANS profile obtained in the unpolarized state (P_H = 0.1%), as shown by the blue squares in Fig. 14. The low-q side of these unpolarized-state profiles is omitted, and the intensity shifts. These unpolarized-state profiles cover higher q values, as they involve full-range neutron wavelengths (1.0 < λ < 7.6 Å).

Figure 14
S_G(q) profiles due to polymer chains in solvent. The top and bottom rows present the results for the rubber samples without and with the silane coupling agent, respectively. In each panel, the experimentally obtained S_G(q) is shown by the unfilled gray circles. The S_G(q) profiles numerically calculated using equations (30) and (63) are shown by the black solid-line and purple dotted-line curves, respectively. In panels (b), (d), (e) and (f), the experimentally obtained nΔφ² $[\langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 contribution is shown by the gray dashed-line curve. The experimentally obtained profiles in the unpolarized state for compensation in the high-q region are shown by the blue squares. The S_G,approx(q) numerically calculated using equation (65)

is shown by the red crosses. In these numerical calculations, the smear effect is considered.

The profiles are fitted using equation (30), as indicated by the black solid-line curves. In addition, we numerically calculate profiles using the following equation, where the DB function is replaced with the power-law function (q⁻⁴) derived from Porod's law:

$[{S_{{\rm G},{\rm Porod}}}\left(q \right) = {A_{\rm Porod}}{q^{ - 4}} + {A_{\rm OZ}}/\left(1+ {q^2}{L_{\rm OZ}}^2 \right). \eqno(63)]$

Here A_Porod = A_DB/ $[L_{\rm DB}^4]$ . S_G,Porod(q) is shown in Fig. 14 by the purple dotted-line curves. The experimentally obtained S_G(q) is lower than S_G,Porod(q) at low q. This indicates the validity of profile fitting using the DB function, although the observed q range does not cover the full shape of the DB function. The evaluated A_DB, L_DB and A_Porod parameters are listed in Table 7.

For cases with polymer adsorption layer formation, we subtract the nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 term to assess S_G(q) using equation (62). Here, we use the nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 value determined via structure model analysis. Fig. 14 shows nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 as the gray dotted-line curves. For R_N2, R_CA1 and R_CA2, the nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 contribution is much smaller than that of S_G(q). For R_CA3, which has a significant polymer adsorption layer, the calculated S_G(q) is comparable to the nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 contribution at low q. As mentioned above, S_G(q) cannot be determined directly in this case. S_G(q) can only be accurately evaluated when nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 is obtained through structure model analysis.

Fig. 15 shows the determined parameters (L_DB, A_DB, A_Porod, L_OZ and A_OZ) as functions of the local Q_swell in the matrix (= 1/φ_M). Concerning the low-q DB function, which is due to the inhomogeneous distribution of polymer chains, R_N exhibits almost constant L_DB and A_DB values, regardless of the swelling ratio. By contrast, R_CA shows significant increases in L_DB and A_DB with an increase in the swelling ratio. This result is ascribed to the spatial distribution shown in Fig. 16. For R_N, the polymer-dense domains are distributed sparsely. For R_CA, the polymer-dense domains are distributed mostly near silica aggregates. With an increase in solvent fraction, polymer-dense domains enlarge. By increasing Q_swell from 1.66 to 4.76, we find that L_DB increases by a factor of 2.1 (= 350/170) and A_DB increases by a factor of 62 (= 51/0.82). This finding is explained quantitatively as follows. A_DB is proportional to n_PDΔ $[\varphi_{\rm PD}^2 V_{\rm PD}^2]$ , where n_PD is the number of polymer-dense regions per volume, Δφ_PD is the difference in polymer volume fraction between a polymer-dense region and the surrounding matrix, and V_PD is the volume of a polymer-dense domain. Because of the 2.1-times increase in L_DB, V_PD should increase by a factor of 8.7 [= (350/170)³]. Approximately nine polymer-dense domains in less swollen states merge to create larger domains in the fully swollen state. Furthermore, n_PD should decrease by a factor of 1/25.5 {= 1/[(350/170)³ × (1.69/4.96)]}, considering the swelling-induced volume increase. Δ $[\varphi_{\rm PD}^2 ]$ should increase by a factor of 21 (= 62 × 25.5/8.7²) to reproduce the obtained A_DB. The polymer volume fraction in the matrix can be roughly estimated via (Q_swell − f_silica)/(1 − f_silica) to be 0.59 and 0.20 at Q_swell = 1.66 and Q_swell = 4.76, respectively. Assuming that the volume fraction of the polymer-dense layer is 0.7 and is not significantly affected by the swelling ratio, Δ $[\varphi_{\rm PD}^2 ]$ should increase by a factor of 21 to reproduce the observed results. Although the volume fraction of the polymer-dense layer may be affected by the swelling ratio, this microscopic picture roughly reproduces the silica-filled rubber sample containing the silane coupling agent.

Figure 15
Parameters L_DB (a), A_DB (b), A_Porod (c), L_OZ (d) and A_OZ (e) as functions of local Q_swell in the matrix (= 1/φ_M). In each panel, the black circles and gray squares indicate the results for R_CA and R_N, respectively. The dotted lines are guides for the reader.

Figure 16
Schematic diagram of polymer-dense domain distribution for R_N (a) and R_CA (b). In each panel, the dark gray circles at the bottom-left corner indicate the silica aggregates, and the light gray circles indicate the polymer-dense domains. The polymer chains are omitted for simplicity.

The obtained microscopic picture [Fig. 16(b)] may be a result of the presumed chemical process during the vulcanization of R_CA as follows. Silane coupling agents have silanol groups for binding silica particle surfaces, and polysulfide bonds for creating cross-links with surrounding polymers. Therefore, for R_CA, polysulfide bonds generate sulfur radicals concentrated around the silica aggregates, creating nearby cross-links. This can explain the formation of polymer-dense regions [Fig. 16(b)].

Two OZ functions are used for both R_CA3 and R_N3 at high q. Two sets of A_OZ and L_OZ are determined and they both increase with Q_swell. It is natural that the polymer mesh size increases with increasing amount of solvent in the polymer. In the fully swollen state, R_CA exhibits larger A_OZ and L_OZ than R_N. Therefore, R_CA has a sparser cross-link distribution in the matrix than R_N. For R_CA, the merging of polymer-dense domains at the fully swollen state, as discussed in the DB function results, could lead to higher dilution in the matrix.

Nakanishi et al. (2024) significantly increased A_DB by introducing an excessive amount of a silane coupling agent during sample preparation. They attributed the large observed A_DB to the cross-link inhomogeneity formed by the excess silane coupling agent separate from the silica aggregates. By contrast, we used a much smaller amount of silane coupling agent.

3.6. Further discussion on the model-free approach based on the Guinier approximation

As stated in Section 3.3, we use the Guinier approximation [equations (35)–(38)]. The −S_SP(q)/S_SS(q) value numerically calculated using equation (38) is shown by the black dashed-line curve in Fig. 8. It is close to the results calculated through structure model analysis, as shown by the gray solid-line curve in Fig. 8, although a slight deviation is found in the fully swollen case (R_CA3). The approximate equation is useful for rough estimation. Here, we extend this approach further to S_G(q) estimation.

From equations (35)–(37), we obtain the approximate equation

$[\langle F_\alpha ^2\left(q \right) \rangle \langle F_{\alpha + \beta }^2\left(q \right) \rangle \approx \langle {F_\alpha }\left(q \right){F_{\alpha + \beta }}\left(q \right)\rangle ^2. \eqno(64)]$

By applying this to equations (21), (22), (24) and (62), we develop another approach to approximating S_G(q):

$[{S_{{\rm G},{\rm approx}}}\left(q \right) \approx {S_{\rm PP}}\left(q \right) - S_{\rm SP}^2\left(q \right) / {S_{\rm SS}}\left(q \right). \eqno(65)]$

However, this equation is only satisfied at low q. Nonetheless, a benefit of this approach is that it eliminates the need to evaluate the nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 contribution separately. As shown by the red crosses in Fig. 14, the calculated S_G,approx(q) values for all samples coincide well with the S_G(q) values obtained in the previous section. The S_G,approx(q) value computed using equation (65) is useful for DB-term parameter evaluation. This approach can avoid the need for numerical calculation based on the sphere collection model. As mentioned earlier, the DB term is difficult to assess accurately because significant scattering due to filler particles overlaps at low q.

The difference between S_G,approx(q) and S_G(q) is

$[\eqalignno{S_{{\rm G},{\rm approx}}(q) - S_{\rm G}(q) &\simeq n\Delta\varphi ^2[\langle F_\alpha(q) F_{\alpha+\beta}(q)\rangle ^2/\langle F_{\alpha+\beta}^2(q)\rangle&\cr &\quad- \langle F_{\alpha+\beta}^2(q)\rangle] \langle K(q)\rangle. &(66)}]$

Hence, if equation (64) is satisfied, then S_G,approx(q) becomes equal to S_G(q). The difference is proportional to Δφ². Hence, S_G,approx(q) and S_G(q) should coincide well for less swollen or homogeneously swollen states (Δφ << 1).

On the basis of this approximation, an equation that yields nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 directly from PSFs can be derived as

$[n\Delta\varphi^ 2\langle F_{\alpha+\beta}^2(q)\rangle \langle K(q)\rangle \simeq [S_{\rm SP}(q) + \varphi_{\rm L}S_{\rm SS}(q)]^2 / S_{\rm SS}(q). \eqno(67)]$

The nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 contribution obtained from this approximate equation is indicated by the unfilled black circles in Fig. 17. The nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 contribution numerically calculated from the sphere collection model is indicated by gray dashed-line curves. The curves coincide well at low q. However, the profile obtained by the approximate equation deviates downwards as q increases, as the Guinier approximation is only valid at low q.

Figure 17
nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 profiles. In each panel, the gray dashed-line curves indicate the nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 profiles obtained through numerical calculation based on the sphere collection model. The unfilled black circles indicate the nΔφ² $[ \langle F_{\alpha+\beta}^2(q)\rangle]$ 〈K(q)〉 contribution obtained using the approximate equation (67)

. The profiles obtained by the two approaches coincide well at low q but deviate at high q.

4. Conclusion

Spin-contrast-variation SANS was applied to silica-filled rubber with and without a silane coupling agent in partially and fully swollen states. SANS profiles were obtained at various P_H using a DNP cryostat (1.2 K and 3.35 T). Each sample was regarded as a system composed of silica, polymer and d-toluene components, and the PSF of each component was evaluated. Analytically, −S_SP(q)/S_SS(q) is proposed as a useful indicator, providing the polymer volume fraction (φ_L) as a flat region at high q and as a depression at low q in the presence of a polymer adsorption layer. Furthermore, we built a sphere collection model for silica aggregates and the surrounding polymer adsorption layer. Although this model includes simplifications for reduced calculation effort, the obtained profiles excellently reproduced the experimental results. Consequently, the radius distribution of the primary silica particles and the particle number distribution in a silica aggregate were determined. Furthermore, on the basis of this structure model, t_L and Δφ at various Q_swell were calculated. For R_CA, t_L decreased and Δφ increased as Q_swell increased. Based on the structural parameters, the computed ratios of the polymer volume in the polymer adsorption layer to the volume of the silane coupling agent were 43, 23 and 6.8 for R_CA at Q_swell values of 1.7, 1.8 and 4.8, respectively. In the fully swollen state, the polymer adsorption layer contained only tightly confined polymer chains. In a less swollen state, it additionally contained moderately confined polymer chains. Furthermore, the S_G(q) contribution was accurately evaluated. The obtained S_G(q) profiles indicated a clear difference between R_N and R_CA. R_CA formed polymer-dense domains distributed mostly around the silica aggregates. As Q_swell increased, the polymer-dense domains merged to create larger domains. By contrast, R_N formed sparsely distributed polymer-dense domains. In addition, on the basis of the Guinier approximation (satisfied only at low q), we derived several approximate equations. With these approximate equations, rough estimates could be obtained without requiring structure-model-based analysis. The validity of the approximate equations was investigated using the present study results. Finally, we found reasonable alignment between these approaches. The developed analytical approaches can be used not only in spin-contrast-variation SANS studies but also in conventional contrast-variation ones.

APPENDIX A

Decomposition of the partial scattering function

Contrast-variation experiments were conducted to obtain SANS profiles with different P_H,i (i = 1,…, N_prof). From equation (17), the obtained profiles at various P_H constitute a system of linear equations, as described by the matrix

$[\eqalignno{&\left[\matrix{ I\left(q\semi P_{\rm{H,1}} \right) \cr I\left(q\semi P_{\rm{H,2}} \right) \cr I\left(q\semi P_{\rm{H,3}} \right) \cr\vdots \cr I\left(q\semi P_{\rm{H,N_{prof}}} \right) \cr } \right] =&\cr &\left[\matrix{ {\Delta}_{\rm{SS}}\left(P_{\rm{H,1}} \right) & {\Delta}_{\rm{SP}}\left(P_{\rm{H,1}} \right) & {\Delta}_{\rm{PP}}\left(P_{\rm{H,1}} \right) \cr {\Delta}_{\rm{SS}}\left(P_{\rm{H,2}} \right) & {\Delta}_{\rm{SP}}\left(P_{\rm{H,2}} \right) & {\Delta}_{\rm{PP}}\left(P_{\rm{H,2}} \right) \cr {\Delta}_{\rm{SS}}\left(P_{\rm{H,3}} \right) & {\Delta}_{\rm{SP}}\left(P_{\rm{H,3}} \right) & {\Delta}_{\rm{PP}}\left(P_{\rm{H,3}} \right) \cr\vdots &\vdots &\vdots \cr {\Delta}_{\rm{SS}}\left(P_{{\rm H},N_{\rm prof}} \right) & {\Delta}_{\rm{SP}}\left(P_{{\rm H},N_{\rm prof}} \right) & {\Delta}_{\rm{PP}}\left(P_{{\rm H},N_{\rm prof}} \right) \cr } \right]\left[\matrix{ S_{\rm{SS}}\left(q \right) \cr S_{\rm{SP}}\left(q \right) \cr S_{\rm{PP}}\left(q \right) \cr } \right]&\cr &\equiv {\bf{M}}\left[\matrix{ S_{\rm{SS}}\left(q \right) \cr S_{\rm{SP}}\left(q \right) \cr S_{\rm{PP}}\left(q \right) \cr } \right],&(68)}]$

where the N_prof × 3 matrix in the second line is denoted by M. The matrix elements are contrast factors:

$[\Delta_{\rm SS}(P_{{\rm H},i}) = [\rho_{\rm S} - \rho_{\rm T}(P_{{\rm H},i})]^2, \eqno(69)]$

$[\Delta_{\rm SP}(P_{{\rm H},i}) = 2[\rho_{\rm S} - \rho_{\rm T}(P_{{\rm H},i})] [\rho_{\rm P}(P_{{\rm H},i}) - \rho_{\rm T}(P_{{\rm H},i})], \eqno(70)]$

$[\Delta_{\rm PP}(P_{{\rm H},i}) = [\rho_{\rm P}(P_{{\rm H},i}) - \rho_{\rm T}(P_{{\rm H},i})]^2. \eqno(71)]$

In our experiments, where the number of obtained profiles (N_prof) is larger than 3, M is obtained as a nonsquare matrix (N_prof × 3). Therefore, instead of a simple inverse matrix, a Moore–Penrose pseudo-inverse matrix M⁺ is needed,

$[{\bf M}^+ = ({\bf M}^{\rm T}{\bf M})^{-1}{\bf M}^{\rm T}, \eqno(72)]$

where M^T is the transposed matrix of M. M⁺ is known to give the shortest-length least-squares solution for equation (68):

$[{\bf{M}}^ + \left[\matrix{ I\left(q\semi P_{\rm{H,1}} \right) \cr I\left(q\semi P_{\rm{H,2}} \right) \cr I\left(q\semi P_{\rm{H,3}} \right) \cr\vdots \cr I\left(q\semi P_{{\rm H},N_{\rm prof}} \right) \cr } \right] = \left[\matrix{ S_{\rm{SS}}\left(q \right) \cr S_{\rm{SP}}\left(q \right) \cr S_{\rm{PP}}\left(q \right) \cr } \right]. \eqno(73)]$

APPENDIX B

Effect of the higher-order structure formed by aggregates

Considering the effect of the higher-order structure formed by aggregates, the spatial distribution functions $[\varphi_{\alpha}^\prime({\bf r})]$ and $[\varphi_{\alpha+\beta}^\prime({\bf r})]$ for regions α and α+β are calculated as

$[\varphi_\alpha^\prime ({\bf r}) = (1/N_{\rm agg}) \sum \limits_{i = 1}^{N_{\rm{agg}}} {\varphi}_{\alpha}\left({\bf{r}} - {\bf{c}}_{i,{\rm{agg}}} \right), \eqno(74)]$

$[\varphi_{\alpha+\beta}^\prime ({\bf r}) = (1/N_{\rm agg}) \sum \limits_{i = 1}^{N_{\rm{agg}}} {\varphi}_{\alpha + \beta}\left({\bf{r}} - {\bf{c}}_{i,{\rm{agg}}} \right),\eqno(75)]$

where c_i,agg is the aggregate center position, i is the labeling number of aggregates, and N_agg is the total number of aggregates in a sample. φ_α(r) and φ_α+β(r) are the spatial distributions of regions α and α+β considering the aggregates and the polymer adsorption layer and without considering a further higher-order structure. Hence, the form factors [ $[F_\alpha^{\prime\ 2}(q)]$ and $[F_{\alpha+\beta}^{\prime\ 2}(q)]$ ] and the cross term [ $[F_\alpha^{\prime}(q) F_{\alpha+\beta}^{\prime}(q)]$ ] considering this higher-order structure are calculated as

$[F_\alpha^{\prime\ 2} (q) = F_\alpha^2 (q)(1/N_{\rm agg}^2) \sum \limits_{i = 1}^{N_{\rm{agg}}} \sum \limits_{j = 1}^{N_{\rm{agg}}} \exp\left[i{\bf{q}} \cdot \left({\bf{c}}_{i,{\rm{agg}}} - {\bf{c}}_{j,{\rm{agg}}} \right) \right], \eqno(76)]$

$[F_{\alpha+\beta}^{\prime\ 2} (q) = F_{\alpha+\beta}^2 (q)(1/N_{\rm agg}^2) \sum \limits_{i = 1}^{N_{\rm{agg}}} \sum \limits_{j = 1}^{N_{\rm{agg}}} \exp\left[i{\bf{q}} \cdot \left({\bf{c}}_{i,{\rm{agg}}} - {\bf{c}}_{j,{\rm{agg}}} \right) \right],\eqno(77)]$

$[\eqalignno{&F_\alpha^{\prime}(q) [qF_{\alpha+\beta}^\prime(q)] =&\cr &\quad F_\alpha (q)F_{\alpha+\beta}(q)(1/N_{\rm agg}^2) \sum \limits_{i = 1}^{N_{\rm{agg}}} \sum \limits_{j = 1}^{N_{\rm{agg}}}\exp\left[i{\bf{q}} \cdot \left({\bf{c}}_{i,{\rm{agg}}} - {\bf{c}}_{j,{\rm{agg}}} \right) \right].&(78)}]$

Here, we define the structure factor 〈K(q)〉 as

$[\langle K(q)\rangle = (1/N_{\rm agg} ^2)\left\langle \sum \limits_{i = 1}^{N_{\rm{agg}}} \sum \limits_{j = 1}^{N_{\rm{agg}}}\exp\left[i{\bf{q}} \cdot \left({\bf{c}}_{i,{\rm{agg}}} - {\bf{c}}_{j,{\rm{agg}}} \right) \right]\right\rangle.\eqno(79)]$

The form factors considering this higher-order structure are calculated using 〈K(q)〉 as

$[\langle F_\alpha^{\prime\ 2}(q)\rangle = \langle F_\alpha^2(q)\rangle \langle K(q)\rangle, \eqno(80)]$

$[\langle F_{\alpha+\beta}^{\prime\ 2}(q)\rangle = \langle F_{\alpha+\beta}^2(q)\rangle \langle K(q)\rangle, \eqno(81)]$

$[\langle F_{\alpha}^\prime(q)F_{\alpha+\beta}^ \prime(q)\rangle = \langle F_\alpha(q)F_{\alpha+\beta}(q)\rangle \langle K(q)\rangle. \eqno(82)]$

Here, we assume that the aggregate spatial distribution is not affected by polydispersity in the aggregate structure. In this way, 〈K(q)〉 in equations (21)–(26) is derived.

Supporting information

Table S1 List of the symbols used in this study. DOI: https://doi.org/10.1107/S1600576726000361/ge5181sup1.pdf

Acknowledgements

The neutron scattering experiment at MLF in J-PARC was performed under the user program (Proposal Nos. 2014B0166, 2015A0269). We appreciate the support of the MLF instrument safety team, sample environment team and low-temperature center.

Funding information

This study was financially supported by a Grant-in-Aid for Young Scientists (A) (grant No. 25706033) of the Japan Society for the Promotion of Science.

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JOURNAL OF
APPLIED
CRYSTALLOGRAPHY

ISSN: 1600-5767

Volume 59| Part 2| April 2026| Pages 492-512

https://doi.org/10.1107/S1600576726000361

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research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Spin-contrast-variation small-angle neutron scattering study of fully and partially swollen silica-filled rubber

1. Introduction

2. Experimental

2.1. Sample preparation

2.2. Dynamic nuclear polarization

2.3. Small-angle neutron scattering

3. Results and discussion

3.1. Small-angle neutron scattering results

3.2. Partial scattering function

3.3. Partial scattering function ratio

3.4. Numerical calculation based on sphere collection model

3.5. Separation of scattering contribution due to polymer in d-toluene

3.6. Further discussion on the model-free approach based on the Guinier approximation

4. Conclusion

APPENDIX A

Decomposition of the partial scattering function

APPENDIX B

Effect of the higher-order structure formed by aggregates

Supporting information

Acknowledgements

Funding information

References

research papers