Diffusion Behavior of Water Molecules in Hydrogels with Controlled

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Diffusion Behavior of Water Molecules in Hydrogels with Controlled Network Structure Takeshi Fujiyabu,† Xiang Li,‡ Ung-il Chung,† and Takamasa Sakai*,† †

Department of Bioengineering, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ‡ Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan

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S Supporting Information *

ABSTRACT: Diffusion behavior of particles in hydrogels is important both for the fundamental understanding of mass transport and for practical applications and has been investigated for a long time. There are three major theories describing the diffusion behavior of small particles in a polymer network: obstruction, hydrodynamic, and free volume theories. Although many researchers have examined these three theories, their applicability is still unclear due to ambiguity stemming from the heterogeneity of conventional hydrogels. Recently, we have developed a near-ideal hydrogel called Tetra-PEG gel that provides a unique possibility to correlate gel structure with properties. In this study, we measured the diffusion coefficient of water molecules (D) in Tetra-PEG gels by pulsed field gradient spin-echo 1H NMR. By comparing D and the correlation length of a polymer network (ξ) measured by small-angle neutron scattering, we observed an identity formula similar to the hydrodynamic theory (D/D0 = exp(−d/ξ), where D0 is the diffusion coefficient of particles in the absence of polymers and d is the diameter of a particle). This result suggests that the diffusion behavior of small particles in hydrogels is determined by the characteristic sizes of a particle (d) and a polymer network (ξ).

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model based on this theory was proposed by Cukier.8 By considering the screening constant (κ) that is related to the friction experienced by a polymer network, D/D0 is given as

INTRODUCTION Hydrogels are open systems where substances can diffuse into and out. This unique property has attracted significant academic and industrial interest.1 Because of the importance of the diffusion in gels, many researchers have investigated this phenomenon and established various models.2−12 Current models describing the diffusion behavior of hard spherical particles in a polymer network are mainly based on three major theories.2−4 First is the obstruction theory, in which a polymer network is considered to behave as an obstacle preventing the diffusion of particles (Figure 1a). There are many models based on this theory: Maxwell−Fricke, Mackie−Meares, and Ogston models.2,3,5−7 In the Ogston model, the polymers are assumed to be randomly oriented straight fibers, and the diffusion coefficient of a particle (D) is given by the possibility of a particle penetrating through the suspension as D = exp( −Kϕ) D0

D = exp( −κR ) D0

where R is the radius of a particle. By substitution of the scaling relationship between κ, the correlation length of a polymer network (ξ), and ϕ (κ ∼ ξ−1 ∼ ϕ0.75),13 eq 2 is transformed as i Ry D ≈ expjjjj− zzzz D0 k ξ{

(3)

D ≈ exp( −Rϕ0.75) D0

(4)

The data of previous studies investigating the solute diffusion behavior in miscible gels was consistent with eq 4.2 The model was modified by Phillies and employed for examining varioussized particles in wide concentration region.9 As a result, it was confirmed that the model described the diffusion behavior of particles in a polymer network.3,9 However, the model still contains undefined scaling parameters.2,3

(1)

where D0 is the diffusion coefficient of a particle in the absence of polymers, K is the retardation coefficient depending on the shape of a particle, and ϕ is the polymer volume fraction. Some experimental results of immiscible gels well agreed with this model, especially for small particles in the dilute region.2,3 Second is the hydrodynamic theory, in which the existence of polymer causes the hydrodynamic friction and decreases the diffusion rate of particles (Figure 1b). The representative © XXXX American Chemical Society

(2)

Received: November 21, 2018 Revised: February 6, 2019

A

DOI: 10.1021/acs.macromol.8b02488 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Schematic illustrations of three major theories ((a) obstruction, (b) hydrodynamic, and (c) free volume theories) describing the diffusion behavior of hard sphere particles in polymer networks.

In this study, we investigated the diffusion behavior of water molecules in Tetra-PEG gels by pulsed field gradient spin-echo 1 H nuclear magnetic resonance (PGSE-NMR). PGSE-NMR is a representative method to measure the diffusion coefficients of particles including water molecules in hydrogels.2−4,23−25 We tuned the molecular weight between cross-links (Mw), polymer volume fractions (ϕ), and imbalanced stoichiometries (r) and investigated the effect on D. The validity of the three major theories and some new models was tested, and the conceptual understanding of the diffusion behavior of particles in hydrogels was discussed.

Third is the free volume theory, in which particles diffuse passing through free volumes of solvent and polymer (Figure 1c). The concept was formulated by Cohen and Turnbull14 and was first applied to the particle diffusion in gels by Yasuda et al.2,10 In Yasuda’s model, D/D0 is given by the possibility of a particle finding a proper free volume as ij xR i ϕ yzyz D zzz = expjjj− jjjj j V 1 − ϕ zzzz D0 {{ k fs k

(5)

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where x is a constant and Vfs is the free volume of a solvent. Although models based on this theory contain more parameters than those on other theories, they describe the diffusion of small particles in hydrogels with high water content.2,3 Many researchers have investigated the applicability of the three major theories.2,3 A critical review suggested a better applicability of the hydrodynamic theory than other theories for describing the diffusion behavior of small particles in miscible hydrogels.2 The models based on the hydrodynamic theory described not only the diffusion behavior of small particles but also that of large molecules.15,16 Another critical review introduced somewhat new models based on different concepts from three major theories.3 A representative of new models is Petit model which approximates a polymer network as a square lattice and considers the frequency that a particle overcomes the energy barrier.11 Another is the obstructionscaling model proposed by Amsden, which combines the obstruction and the hydrodynamic theories.12 Although the applicability of these new models has been tested, experimental results are limited compared to three major theories.17,18 Therefore, the most appropriate model or the proper condition of each model is still unclear. One of the reasons causing this situation is the unclear structure−property relationship of the conventional hydrogels due to their inherent heterogeneity. To conquer this situation and to understand the diffusion behavior of particles in hydrogels, a model gel system with a clear structure−property relationship is necessary. Recently, we have developed a near-ideal hydrogel, TetraPEG gel, which is formed by two tetra-armed prepolymers with mutually reactive end groups (thiol (−SH) and maleimide (−MA)).19 Our previous small-angle neutron scattering (SANS) measurements revealed the extremely low structural heterogeneity in Tetra-PEG gels.20−22 Also, the relationship between the physical properties and network structure is wellknown.22 Thus, Tetra-PEG gel system is highly suitable as a model polymer gel system.

MATERIALS AND METHODS

Materials. Tetramaleimide-terminated poly(ethylene glycol) (Tetra-PEG-MA) and Tetrathiol-terminated poly(ethylene glycol) (Tetra-PEG-SH) were purchased from NOF Co. (Tokyo, Japan), and all the other reagents were purchased from WAKO. All materials were used without further purification. Fabrication of the As-Prepared Tetra-PEG Gels. Details of Tetra-PEG precursors have been reported elsewhere.19−22 Constant amounts of Tetra-PEG-MA and Tetra-PEG-SH were dissolved in phosphate−citric acid buffer. To control the reaction rate, the optimal ionic strength and pH of buffers were chosen. Tetra-PEG gels formed using prepolymers with a molecular weight (Mw) of 10 and 20 kg/mol are called Tetra-PEG gel 10K and Tetra-PEG gel 20K, respectively. In the case of Tetra-PEG gel 10K, 68 mM of buffer solution (pH 3.8) was used for the lower polymer volume fractions (ϕ0: 0.017−0.081), 64 mM (pH 3.4) for the middle polymer volume fraction (ϕ0: 0.096), and 60 mM (pH 3.0) for the higher polymer volume fractions (ϕ0: 0.124, 0.150). In the case of Tetra-PEG gel 20K, 68 mM of buffer solution (pH 3.8) was used for the lower polymer volume fractions (ϕ0: 0.017−0.096) and 64 mM (pH 3.4) for the middle polymer volume fraction (ϕ0: 0.124, 0.150). Equal amounts of two prepolymer solutions were mixed, and the resulting solution was poured into a mold. At least 12 h was allowed for the completion of the reaction before the following experiments were performed. In the case of the r-tuned Tetra-PEG gels, the prepolymer solutions were mixed in a nonstoichiometric ratio (r = [Tetra-PEG-MA]/ ([Tetra-PEG-MA] + [Tetra-PEG-SH])).20 The r-tuned Tetra-PEG gels were formed in three different conditions. First samples were formed from prepolymers with Mw = 10 kg/mol and ϕ0 = 0.050 (r = 0.5, 0.45, 0.375, 0.25, 0). Second samples were formed from prepolymers with Mw = 20 kg/mol and ϕ0 = 0.034 (r = 0.5, 0.45, 0.375, 0.25, 0). Third samples were formed from prepolymers with Mw = 20 kg/mol and ϕ0 = 0.124 (r = 0.5, 0). The molar concentration of buffer solution was the same as stoichiometric Tetra-PEG gels for all samples. These solutions were mixed and waited for the reaction completion. Fabrication of the Equilibrium-Swollen Tetra-PEG Gels. As for Tetra-PEG gel 10K and 20K with ϕ0 = 0.050, 0.096, and 0.124, the swelling experiment was conducted to form the equilibriumswollen Tetra-PEG gels. Tetra-PEG gels were prepared as discs B

DOI: 10.1021/acs.macromol.8b02488 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules (height: 1.0 mm; diameter: 25 mm). The initial weights of Tetra-PEG gels (W0) were measured. The prepared gel samples were immersed in H2O at room temperature. After 2 days, the gel reached its maximum change in weight, and the weights of Tetra-PEG gels (W) were measured. The swelling ratio (Q) and the equilibrium swollen polymer volume fraction (ϕe) were calculated by eqs 6 and 7, respectively. Q=

ϕe =

W W0

(6)

ϕ0 Q

(7) 1

Pulsed Field Gradient Spin-Echo H Nuclear Magnetic Resonance (PGSE-NMR). PGSE-NMR measurements were performed on a TD-NMR Minispec mq20 (Bruker). The NMR signal from the particles is related to their diffusion coefficient by Stejskal− Tanner equation as26 ln

i S δy = − G2γ 2δ 2jjjΔ − zzzD 3{ S0 k

Figure 2. Relationship between ln(S/S0) and G2γ2δ2(Δ − δ/3) of Tetra-PEG gel (circles) and Tetra-PEG sol (triangles) with Mw = 20 kg/mol and ϕ0 = 0.124.

(8)

On the other hand, we observed two slopes in the sol state; both diffusions of water molecules and PEG were detected. On the basis of eq 9, the slopes in the small and large G2γ2δ2(Δ − δ/3) region correspond to D (≈ 109 m2/s) and DPEG (≈ 1011 m2/s).28 Because D was much larger than DPEG and the slopes were constant in the region G2γ2δ2(Δ − δ/3) < 1.0 × 109 s/m2 for all samples (Figure S1), it can be considered that the influence of DPEG on the estimation of D was negligible in the region G2γ2δ2(Δ − δ/3) < 1.0 × 109 s/m2. Therefore, we estimated D from the slope in the region G2γ2δ2(Δ − δ/3) < 1.0 × 109 s/m2. Effect of Structural Parameters on Diffusion of Water Molecules. We measured the diffusion coefficient of water molecules (D) by tuning three network structure parameters: the molecular weight between cross-links (Mw), the initial and equilibrium swollen polymer volume fractions (ϕ0 and ϕe), and the imbalanced stoichiometries (r). The value of r is related to the connectivity of the polymer network (p), which is the probability that one arm of the tetrafunctional polymers is connected to another arm, as p ≈ 2r.29 In this study, we tuned r = 0, 0.25, 0.375, 0.45, and 0.5, corresponding to p = 0, 0.5, 0.75, 0.9, and 1.0, respectively. Only the sample with r = 0 was in the sol state, while all the other samples were in the gel state. In Figure 3a, we plotted D against the polymer volume fraction (ϕ). In both the as-prepared and the equilibrium swollen Tetra-PEG gels, D decreased with an increase in ϕ roughly obeying an exponential function. On the other hand, the effect of Mw on D was small. These results suggest that ϕ is the essential factor to control the diffusion behavior of water molecules in hydrogels. In other words, the existence of the polymer prevents the diffusion of water molecules. Figure 3b shows the relationship between D and r. The value of D decreased with an increase in the deviation from the stoichiometry (with a decrease in p) and did not change drastically before or after the gelation threshold. This result suggests that p is the second factor controlling the diffusion of water molecules in hydrogels. Notably, the result is counterintuitive, because the formation of the network leads to increase in the mobility of water molecules, suggesting that the pore size is not an essential factor governing the diffusion of water molecules. Applicability of Theoretical Models to Experimental Results. We first examined the applicability of three major

where S is the NMR echo signal intensity with the field gradient pulses, S0 is the NMR echo signal intensity without the field gradient pulses, G is the strength of the magnetic field gradient, γ is the magnetogyric ratio of the probe molecule, δ is the gradient pulse length, Δ is the gradient pulse interval, and D is the diffusion coefficient of a particle. In this study, δ, Δ, and γ were 5.00 × 10−4 s, 3.00 × 10−2 s, and 2.68 × 108 (T s)−1, respectively. The values of S and S0 were measured by changing G (≈ 0.3−4.0 T/m) at 25°C. By plotting ln(S/S0) against G2γ2δ2(Δ − δ/3), we can assess D as a slope of a line. In the system containing two particles, D’s are described by the sum of eq 8.27 i i S δy y = fA expjjjj− G2γ 2δ 2jjjΔ − zzzDA zzzz + (1 − fA ) S0 3{ { k k ij 2 2 2ij y y δ × expjjj− G γ δ jjΔ − zzzDBzzzz 3{ { k k

(9)

where fA is the fraction of the NMR echo signal from particle A, DA is the diffusion coefficient of particle A, and DB is the diffusion coefficient of particle B. When DA is much larger than DB (DA ≫ DB), DA and DB are represented as slopes of lines in smaller and larger G2γ2δ2(Δ − δ/3) regions, respectively.

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RESULTS AND DISCUSSION Estimation of Diffusion Coefficient of Water Molecule. We conducted PGSE-NMR experiment to estimate the diffusion coefficient of water molecules in Tetra-PEG gels (D). PGSE-NMR detects the NMR signal from the proton of diffusing particles.23,24 On the basis of eqs 8 and 9, a diffusion coefficient is determined from the slope in the plot of ln(S/S0) against G2γ2δ2(Δ − δ/3). Tetra-PEG gels contain mainly two types of protons: the protons in water molecules and poly(ethylene glycol) (PEG). To estimate D in Tetra-PEG gels, we need to decompose D and the diffusion coefficient of PEG (DPEG). In Figure 2, ln(S/S0) of Tetra-PEG gel and Tetra-PEG sol is plotted against G2γ2δ2(Δ − δ/3). In the gel state, only one slope was observed in most cases. This signal was assigned to that of water because the diffusion coefficient was close to that of pure water. On the contrary, it was often difficult to observe any line in the large G2γ2δ2(Δ − δ/3) region, suggesting that the diffusion of PEG was too small to be detected under our experimental condition. All the lines passed through the origin, suggesting the applicability of eq 8 for the estimation of D. C

DOI: 10.1021/acs.macromol.8b02488 Macromolecules XXXX, XXX, XXX−XXX

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Figure 3. (a) Relationship between D and ϕ of Tetra-PEG gels with different Mw (Mw: 10 kg/mol; circles; 20 kg/mol, triangles) in the as-prepared (open symbols) and in the equilibrium swollen state (full symbols). (b) Relationship between D and r of Tetra-PEG gels (Mw = 10 kg/mol and ϕ0 = 0.050, squares; Mw = 20 kg/mol and ϕ0 = 0.034, rhombuses).

classical theories; the ϕ and Mw dependences were successfully reproduced, while the r dependence was not (Figure S2). Relationship between Diffusion Coefficient of Water Molecules and Correlation Length of Polymer Network. To explain the deviation observed in the r dependence, we focused on the correlation length of a polymer network (ξ) estimated by our previous small-angle neutron scattering measurements (SANS).20,21 The scattering curves of TetraPEG gels were well described by Ornstein−Zernike (OZ) function, and the values of ξ were estimated by the fit. The reanalyzed ϕ dependence of ξ is shown in Figure 5a.21 In both the as-prepared and the equilibrium swollen state, ξ decreased with an increase in ϕ, and the relationship between ξ and ϕ roughly agreed with the well-known scaling relationship for a semidilute good solvent (ξ ∼ ϕ−0.75).13 The reanalyzed relationship between ξ and r is shown in Figure 5b.20 The value of ξ decreased with an increase in the deviation from the stoichiometry, suggesting that ξ has not only the ϕ dependence but also the r dependence. This result is not predicted by the scaling theory for semidilute solution. Therefore, the discrepancy between the experimental results and the model predictions is most likely caused by the breakage of the scaling for semidilute solution in the r-tuned system. Thus, for further discussion, we directly compare D and ξ. We examined some representative plots and found the linear relationship between D/D0 and 1/ξ (Figure 6). Notably, all the data including the r dependence fall onto the master relationship in this plot. The function of the master relationship is the following:

theories to the results. In Figure 4, the normalized diffusion coefficient of water molecules (D/D0) is plotted against polymer volume fraction (ϕ), where D0 is the diffusion coefficient of pure water in the absence of polymers. The lines in Figure 4 show the fits of the representative models based on each theory (eqs 1, 4, and 5). All three models reproduced the ϕ and Mw dependences of D/D0, and the difference among the models was negligible in this region. In this study, we investigated D in the small ϕ region due to the experimental limitation. In the small ϕ region, the difference between eqs 1 and 5 is negligible because (1 − ϕ) is close to 1, and the effect of the difference in the exponents of ϕ in eq 1 (1) and eq 4 (0.75) is small. Therefore, it is impossible to test the applicability of three major theories based on these experimental results. On the other hand, the r dependence of D/D0 was not reproduced by any models (Figure 4). The reason for this deviation is clear; eqs 1, 4, and 5 only have ϕ as a variable and do not represent the effect of p. This deviation suggests that the model should include the effect of p. Notably, the deviation from three representative models was mainly observed in the low p region (p < 0.5). This deviation has not been observed in previous studies, most likely because the connectivity of the polymer network has not been changed intentionally. We also checked the applicability of the models proposed by Petit et al. and Amsden.11,12 The fit results were the same with

i 2.9 yz D zz = 0.98 expjjjj− z D0 k ξ {

(10)

Interestingly, the formula is almost same as that of hydrodynamic theory eq 3. In addition, the front factor is almost unity, and 2.9 Å is close to the diameter of a water molecule (2.74−3.30 Å).30 On the basis of the results, we propose a semiempirical equation as i dy D = expjjjj− zzzz D0 k ξ{

Figure 4. Relationship between D/D0 and ϕ of Tetra-PEG gels with different Mw (Mw: 10 kg/mol, circles; 20 kg/mol, triangles) in the asprepared (open symbols) and in the equilibrium swollen state (full symbols) and r-tuned Tetra-PEG gels (Mw = 10 kg/mol and ϕ0 = 0.050, squares; Mw = 20 kg/mol and ϕ0 = 0.034, rhombuses).

(11)

where d is the diameter of a particle. The importance of the term of d/ξ, which is the ratio of the characteristic sizes of a particle and a polymer network, has been suggested by D

DOI: 10.1021/acs.macromol.8b02488 Macromolecules XXXX, XXX, XXX−XXX

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Figure 5. (a) Relationship between ξ and ϕ of Tetra-PEG gels with different Mw (Mw: 10 kg/mol, circles; 20 kg/mol, triangles) in the as-prepared (open symbols) and in the equilibrium swollen state (full symbols). (b) Relationship between ξ and r of Tetra-PEG gels (Mw = 10 kg/mol and ϕ0 = 0.050, squares; Mw = 20 kg/mol and ϕ0 = 0.034, rhombuses).

Figure 6. Relationship between D/D0 and ξ of Tetra-PEG gels with different Mw (Mw: 10 kg/mol, circles; 20 kg/mol, triangles) in the asprepared (open symbols) and in the equilibrium swollen state (full symbols) and r-tuned Tetra-PEG gels (Mw = 10 kg/mol and ϕ0 = 0.050, squares; Mw = 20 kg/mol and ϕ0 = 0.034, rhombuses).

Figure 7. Relationship between D/D0 and ξ of Tetra-PEG gels with different Mw (Mw: 10 kg/mol, circles; 20 kg/mol, triangles) in the asprepared (open symbols) and in the equilibrium swollen state (full symbols) and r-tuned Tetra-PEG gels (Mw = 10 kg/mol and ϕ0 = 0.050, squares; Mw = 20 kg/mol and ϕ0 = 0.034, rhombuses). The solid, dotted, and dashed lines are the fits of our semiempirical equation (eq 11), Petit model (eq 12), and Amsden model (eq 13), respectively.

Tokita.25 According to the idea, the diffusion behavior of small particles in hydrogels is considered to be determined by the ratio of the characteristic sizes. We also checked the applicability of Petit (eq 12) and Amsden (eq 13) models, which are expressed as follows:11,12 D 1 = D D0 1 + 02 ÄÅ É 2Ñ ÅÅ i Ñ ÅÅ jj R + rf yzz ÑÑÑ D ÑÑ = expÅÅÅ−π jj z ÅÅ jk ξ + 2rf zz{ ÑÑÑ D0 ÅÇ ÑÖ kξ

are the important factors and that eq 11 is the appropriate equation to reproduce our results. Our results strongly support the validity of the concept of hydrodynamic theory, of which scaling form was previously confirmed for various kinds of particles.2,3,9,15,16,25 Therefore, it is expected that eq 11 can be applied not only to water but also to other particles. Similar good correlation between the dynamics of water molecules and ξ corresponds to our previous research investigating the permeation of water molecules through hydrogels.31 For the further understanding, it needs other methodologies such as low-field NMR (LF-NMR), fluorescence recovery after photobleaching (FRAP), and quasi-elastic neutron scattering (QENS).18,32,33 Furthermore, we should test the applicability of eq 11 for other small molecules (other solvents and small polymers) and larger molecules (large polymers, dye probes, and proteins). We will further investigate these points in the near future.

(12)

(13)

where k is the jump frequency and rf is the polymer chain radius. Although eqs 12 and 13 have more parameters than eq 11, they may consist with the experimental results because they include ξ. In Figure 7, we plotted D/D0 against ξ. Our semiempirical equation (solid line) showed the best reproducibility of the results, and d estimated by the fit (≈ 3.2 Å) was close to the diameter of a water molecule (2.74−3.30 Å).30 Amsden model also reproduced the results; while R estimated by the best fit (≈ −1.1 Å) was negative and rf estimated by the best fit (≈ 11 Å) was larger than the values calculated in previous research. 12 When we fix R = 1.5 Å, the correspondence between the fit (dashed line) and the results is still reasonable, and rf (≈ 5.3 Å) was similar to the previous research.12 Besides, Petit model (dotted line, k = 4.6 × 109 s−1) showed the poorer reproducibility compared to the other two models. These results suggest that the exponential form and ξ

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CONCLUSION We estimated the diffusion coefficient of water molecules (D) in Tetra-PEG gels with different molecular weights of crosslinks (Mw), polymer volume fractions (ϕ), and imbalanced stoichiometries (r) by PGSE-NMR. The ϕ and Mw dependences of D well agreed with the representative models based on three major theories describing the diffusion behavior of small particles in a polymer network: obstruction, hydrodynamic, E

DOI: 10.1021/acs.macromol.8b02488 Macromolecules XXXX, XXX, XXX−XXX

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(6) Mackie, J. S.; Meares, P. The Diffusion of Electrolytes in a Cation-Exchange Resin Membrane. II. Experimental. Proc. R. Soc. A Math. Phys. Eng. Sci. 1955, 232 (1191), 510−518. (7) Ogston, A. G. The Spaces in a Uniform Random Suspension of Fibres. Trans. Faraday Soc. 1958, 54, 1754. (8) Cukier, R. I. Diffusion of Brownian Spheres in Semidilute Polymer Solutions. Macromolecules 1984, 17 (2), 252−255. (9) Phillies, G. D. Universal Scaling Equation for Self-Diffusion by Macromolecules in Solution. Macromolecules 1986, 19 (9), 2367− 2376. (10) Yasuda, H.; Ikenberry, L. D.; Lamaze, C. E. Permeability of Solutes through Hydrated Polymer Membranes. Part II. Permeability of Water Soluble Organic Solutes. Makromol. Chem. 1969, 125 (1), 108−118. (11) Petit, J. M.; Roux, B.; Zhu, X. X.; Macdonald, P. M. A New Physical Model for the Diffusion of Solvents and Solute Probes in Polymer Solutions. Macromolecules 1996, 29 (18), 6031−6036. (12) Amsden, B. Modeling Solute Diffusion in Aqueous Polymer Solutions. Polymer 2002, 43 (5), 1623−1630. (13) De Gennes, P. G. Dynamics of Entangled Polymer Solutions. II. Inclusion of Hydrodynamic Interactions. Macromolecules 1976, 9 (4), 594−598. (14) Cohen, M. H.; Turnbull, D. Molecular Transport in Liquids and Glasses. J. Chem. Phys. 1959, 31 (5), 1164−1169. (15) Cherdhirankorn, T.; Best, A.; Koynov, K.; Peneva, K.; Muellen, K.; Fytas, G. Diffusion in Polymer Solutions Studied by Fluorescence Correlation Spectroscopy. J. Phys. Chem. B 2009, 113 (11), 3355− 3359. (16) Thévenot, J.; Cauty, C.; Legland, D.; Dupont, D.; Floury, J. Pepsin Diffusion in Dairy Gels Depends on Casein Concentration and Microstructure. Food Chem. 2017, 223, 54−61. (17) Wang, Y. J.; Therien-Aubin, H.; Baille, W. E.; Luo, J. T.; Zhu, X. X. Effect of Molecular Architecture on the Self-Diffusion of Polymers in Aqueous Systems: A Comparison of Linear, Star, and Dendritic Poly(Ethylene Glycol)S. Polymer 2010, 51 (11), 2345− 2350. (18) Hadjiev, N. A.; Amsden, B. G. An Assessment of the Ability of the Obstruction-Scaling Model to Estimate Solute Diffusion Coefficients in Hydrogels. J. Controlled Release 2015, 199, 10−16. (19) Sakai, T.; Matsunaga, T.; Yamamoto, Y.; Ito, C.; Yoshida, R.; Suzuki, S.; Sasaki, N.; Shibayama, M.; Chung, U. I. Design and Fabrication of a High-Strength Hydrogel with Ideally Homogeneous Network Structure from Tetrahedron-like Macromonomers. Macromolecules 2008, 41 (14), 5379−5384. (20) Matsunaga, T.; Sakai, T.; Akagi, Y.; Chung, U. I.; Shibayama, M. Structure Characterization of Tetra-PEG Gel by Small-Angle Neutron Scattering. Macromolecules 2009, 42 (4), 1344−1351. (21) Matsunaga, T.; Sakai, T.; Akagi, Y.; Chung, U. I.; Shibayama, M. SANS and SLS Studies on Tetra-Arm PEG Gels in as-Prepared and Swollen States. Macromolecules 2009, 42 (16), 6245−6252. (22) Sakai, T. Experimental Verification of Homogeneity in Polymer Gels. Polym. J. 2014, 46 (9), 517−523. (23) Matsukawa, S.; Ando, I. A Study of Self-Diffusion of Molecules in Polymer Gel by Pulsed-Gradient Spin−Echo 1 H NMR. Macromolecules 1996, 29 (22), 7136−7140. (24) Matsukawa, S.; Yasunaga, H.; Zhao, C.; Kuroki, S.; Kurosu, H.; Ando, I. Diffusion Processes in Polymer Gels as Studied by Pulsed Field-Gradient Spin-Echo NMR Spectroscopy. Prog. Polym. Sci. 1999, 24 (7), 995−1044. (25) Tokita, M.; Miyoshi, T.; Takegoshi, K.; Hikichi, K. Probe Diffusion in Gels. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1996, 53 (2), 1823−1827. (26) Stejskal, E. O.; Tanner, J. E. Spin Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient. J. Chem. Phys. 1965, 42 (1), 288−292. (27) Tanner, J. E.; Stejskal, E. O. Restricted Self-Diffusion of Protons in Colloidal Systems by the Pulsed-Gradient, Spin-Echo Method. J. Chem. Phys. 1968, 49 (4), 1768−1777.

and free volume theories. However, the r dependence of D could not be reproduced by any models. On the other hand, the relationship between D and the correlation length of a polymer network (ξ) estimated by SANS obeyed the identity formula based on the hydrodynamic theory (D/D0 = exp(−d/ ξ), where D0 is the diffusion coefficient of particles in the absence of polymers and d is the diameter of a particle) including all ϕ, Mw, and r dependences. Among the models tested, our semiempirical equation showed the best fit with reasonable parameters. These results suggest that the diffusion behavior of small particles in hydrogels is determined by the characteristic sizes of a particle (d) and a polymer network (ξ); hydrodynamic friction caused by blobs retards the diffusion of particles. These findings will help to understand the diffusion behavior of particles in polymer gels.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b02488.

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Figures S1 and S2 (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. ORCID

Takeshi Fujiyabu: 0000-0002-4686-1047 Xiang Li: 0000-0001-6194-3676 Takamasa Sakai: 0000-0001-5052-0512 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. Koichi Mayumi for assisting and advising us with the PGSE-NMR analysis. This work was supported by the Japan Society for the Promotion of Science (JSPS) through the Grants-in-Aid for the Graduate Program for Leaders in Life Innovation (GPLLI), the International Core Research Center for Nanobio, Core-to-Core Program A. Advanced Research Networks, the Grants-in-Aid for Scientific Research (B) Grant 18H02027 to T.S., and Scientific Research (S) Grant 16746899 to U.C., and the Japan Science and Technology Agency (JST) through Center of Innovation (COI).

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