Multiscale Dynamics of Inhomogeneity-Free Polymer Gels

Jan 16, 2014 - Department of Bioengineering, School of Engineering, The University of Tokyo, ..... was performed by SNS-NSE (BL15) at SNS, Oak Ridge, ...
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Multiscale Dynamics of Inhomogeneity-Free Polymer Gels Takashi Hiroi,† Michael Ohl,‡ Takamasa Sakai,§ and Mitsuhiro Shibayama*,† †

Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan Jülich Center for Neutron Science at the Spallation Neutron Source, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States § Department of Bioengineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ‡

S Supporting Information *

ABSTRACT: For precise understanding of the dynamics of gels, it is necessary to distinguish the effect of inherent cross-linking from accompanying inhomogeneity. This separation is realized by the use of inhomogeneity-free gel such as Tetra-PEG gel. We investigated the dynamics of Tetra-PEG gel by quasi-elastic scattering. Mesoscopic (length scale: ∼100 nm) motion was measured by dynamic light scattering (DLS). In addition to this scale, we used neutron spin echo (NSE) to measure microscopic (length scale: ∼1 nm) motion. From these measurements, it is revealed that the gels with no connectivity/topological inhomogeneities show the transition from Zimm mode to collective diffusion mode in larger length scale, even beyond the q-range of NSE. In addition to this, the absence of spatial inhomogeneities is reflected as disappearance of nondecay component in the intermediate dynamic structure factor. Through the combination analysis of DLS and NSE, the multiscale dynamics of gels is elucidated.

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INTRODUCTION Polymer gels attract great attention for their various functions and applications. For example, hydrogels are expected to be applied as biomaterials because of their soft and wet nature. As a result, a lot of attractive gels are invented all over the world. To understand their various features, it is necessary to characterize their properties from various aspects. In addition to macroscopic physical properties such as Young’s modulus and stress−strain curves, molecular level structure should also be elucidated, especially for the synthesis of gels with bottomup approach. Here, the structure contains two aspects.1 One is the static structure. To investigate the static structure of gels, small-angle neutron scattering (SANS) is known as one of the most powerful tools.2 SANS is compatible with the research of gels, since one can measure the scattering only from networks by using deuterated solvent. Thanks to its potential to elucidate the network structure, investigations of static structure of gels are quite developed. The other aspect is the dynamic structure. Dynamic properties such as a collective diffusion constant are strongly correlated to the macroscopic physical properties.3 Dynamics of gels is usually measured by dynamic light scattering (DLS). This method is a kind of quasi-elastic scattering, and this probes the fluctuations of the network.4 However, dynamic properties obtained from DLS are limited to mesoscopic (∼100 nm) length scale because visible light is used as a probe. To observe microscopic (∼1 nm) motions, we have to use the source whose wavelength is also of the order of ∼1 nm. One of the promising sources is a neutron. Since neutron source has very low energy compared to light with the same wavelength (X-ray), it has been considered as the best probe to © 2014 American Chemical Society

observe the microscopic motion of gels. However, such a kind of experiment was not developed until the 1980s because of the lack of appropriate experimental methods. Neutron spin echo (NSE) is a method to realize quasi-elastic neutron scattering with high energy resolution (2.1 neV in the case of the instrument we used) in about 1 nm length scale, proposed by Mezei in 1972.5 By using NSE, microscopic dynamics of polymer network system has been investigated. First research about the cross-link dynamics is reported by Richter et al. by using polydimethylsiloxane (PDMS) network.6 After this report, many gels are investigated by NSE.7−13 Those researches aim to elucidate the effect of inhomogeneity on the microscopic motion of gels. Here, spatial inhomogeneities observed by SANS measurements are usually taken into consideration. In addition to this, other kinds of inhomogeneities such as connectivity inhomogeneities and topological inhomogeneities14 can affect the dynamics of gels although there is no report related to this point of view. However, we have to treat the inherent cross-linking effect and accompanying inhomogeneity effect separately to give the clear-cut discussion about this issue. In this meaning, bottleneck of the research for microscopic motion of gels was the lack of inhomogeneity-free gel samples. A breakthrough in the solution was proposed in 2008 by Sakai et al. as the development of a hydrogel called Tetra-PEG gel.15 This gel is made by cross-end coupling of two types of tetra-arm polyethylene glycol (PEG) Received: November 28, 2013 Revised: January 2, 2014 Published: January 16, 2014 763

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Figure 1. Normalized intensity time correlation functions of Tetra-PEG: (a) 20 mg/mL solution, (b) 40 mg/mL solution, (c) 20 mg/mL gel, and (d) 40 mg/mL gel. In each graph, the data obtained at different scattering angles (60, 90, 120, and 150°) are shown.

Figure 2. Probability distribution functions of ξ for Tetra-PEG solutions: (a) 20 mg/mL, (b) 40 mg/mL and for gels, (c) 20 mg/mL, and (d) 40 mg/mL. In each graph, the data obtained at different scattering angles (60, 90, 120, and 150°) are shown.

effect of not only spatial but also connectivity/topological inhomogeneities on the microscopic motions of gels. At the end, we integrate the DLS and NSE results and consider the multiscale motion of gels.

chains. It became evident that this gel is inhomogeneity-free through mechanical character and SANS observations.15,16 Owing to its homogeneity, Tetra-PEG gel has high transparency (∼100%), high deformability (∼900%), and high breaking strength (∼30 MPa).17 In this paper, we investigate the mesoscopic and microscopic motion of Tetra-PEG gels by using DLS and NSE, respectively. Tetra-PEG gels make us to see the inherent cross-linking effect on the mesoscopic/microscopic motions of gels without caring about inhomogeneity. This also means that we can infer the effect of inhomogeneity on the mesoscopic/microscopic motions of gels appear in conventional gels. We found the

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EXPERIMENTAL SECTION

Sample Preparation. Tetraamine-terminated PEG (TAPEG) and tetra-NHS-glutarate-terminated PEG (TNPEG) were prepared from tetrahydroxyl-terminated PEG (THPEG) having equal arm lengths. Here, NHS represents N-hydroxysuccinimide. In this paper, we call one TAPEG or TNPEG unit as “macromer” and one PEG unit as “monomer”. The molecular weight of TAPEG and TNPEG is 20 kg/ mol. The chain overlap concentration of this macromer in water, c*, is

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40 mg/mL.18 Solution samples (20 and 40 mg/mL) were made by dissolving TAPEG in phosphate buffer (pH = 7.4). Gel samples were synthesized by mixing equal amounts of TAPEG and TNPEG (20 and 40 mg/mL) solutions. TAPEG was dissolved in phosphate buffer and TNPEG was dissolved in phosphate-citric acid buffer (pH = 5.9). Resulting solution was poured into an appropriate mold. At least 12 h was spent for the completion of the reaction before the following experiments. We can ignore the remaining sol component since the reaction efficiency is sufficiently high (close to 0.9).19 The details of TAPEG and TNPEG preparation and characterization are reported elsewhere.15 Dynamic Light Scattering (DLS). DLS measurements were carried out at 25 °C by using a DLS/SLS-5000 compact goniometer (ALV, Langen), coupled with an ALV photon correlator. A 22 mW He−Ne laser (wavelength, λ = 632.8 nm) was used as incident beam. We measured each sample with scattering angle, θ, at 60°, 90°, 120° and 150° to obtain q-dependence of relaxation. All of the samples were prepared in a test tube (diameter = 8 mm). Each measurement time was 30 s. The viscosity of the solvents was measured by using a rheometer (MCR501, Anton Paar, Austria) at 25 °C with a cone− plate geometry. The shear rate was 100 s−1. Neutron Spin Echo (NSE). The NSE experiments were performed at NSE-SNS neutron spin−echo spectrometers at the Spallation Neutron Source (SNS) at the Oak Ridge National Laboratory.20,21 The measurements were performed at room temperature in a momentum transfer range 0.054 < q < 0.189 Å −1 and covered a time range from 0.01 to 34 ns. Incident wavelength bands are from 5 to 8 Å. All of the samples were loaded in a sealed aluminum container of size 40 × 30 × 2 mm3. The sample solutions and gels were made by using D2O instead of H2O as a solvent to suppress incoherent scattering from solvent. Measurement time was about 50 h for 40 mg/ mL samples and 60 h for 20 mg/mL samples.

components correspond to translational diffusions of macromers and clusters, respectively.15,23,24 Since the scattering intensity from clusters shows strong q-dependence compared to that from macromers, the ratio of the heights of the fast and the slow components also depends on the scattering angles; the smaller the scattering angle becomes, the stronger the scattering intensity from clusters becomes and the more the proportion of the slow component becomes. In contrast to this, there is only one relaxation mode in the case of gels (Figure 2c and d). To fit the data of gels, we have to notice that the initial amplitude (g(2)(τ = 0) − 1) is not equal to one even if σI2 is assumed to be unity. This depression comes from the nonergodicity of gels.4 For these nonergodic samples, the data is usually analyzed by a partial heterodyne method.4 The data is fitted by the following equation: g(2)(τ ) − 1 = A exp( −2ΓAτ )

where ΓA is the apparent relaxation rate. ΓA can be converted into the genuine relaxation rate, Γ, by using the initial amplitude, A: ΓA =

Γ = Dq2

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⟨Is(τ )⟩T 2

1−A Γ A

(4)

(5)

where q is a momentum transfer, q := 4πn sin θ/λ where n is a retractive index of the solvent. Although the data fit of clusters (Γs) is a little bit rough because of their polydispersity, the decay rates of solutions and gels are almost proportional to q2 as shown in Figure 3, supporting the validity of eq 5. From the slope of Figure 3, the diffusion constants are calculated. The results are shown in Table 1. In the case of the solutions, D is connected to Brownian motion of the solutes. From D, we can calculate the correlation length, ξ, by using the following equation:25

: = g(2)(τ ) (1)

where Is(τ) is the scattering light intensity at time t = τ and ⟨···⟩T means time-averged quantity. Normalized intensity time correlation functions of Tetra-PEG solutions and gels are shown in Figure 1. Each graph contains the data measured at four different scattering angles. From the intensity correlation functions in Figure 1, we calculate the characteristic decay time distribution functions with an inverse Laplace transform program (a constrained regularization program, CONTIN, provided by ALV). Then we convert these distribution functions into the probability distribution functions of the correlation length, G(ξ), by using eq 6 as shown later. The results are shown in Figure 2. As shown in Figure 2a and b, there are fast and slow relaxation modes in the case of solutions. To decompose these modes, we fit the data by using a following equation: g(2)(τ ) − 1 = σI 2[A f exp( −Γf τ ) + A s exp( −Γsτ )]2

1−

These relaxation rates, Γf, Γs, and Γ are related to the diffusion constant, D:

RESULTS AND DISCUSSION 1. DLS: Mesoscopic Motion. To see the mesoscopic motion, we measured DLS of 20 mg/mL (c*/2) and 40 mg/ mL (c*) solutions and gels. From DLS, we obtain a normalized intensity time correlation function:22 ⟨Is(0)Is(τ )⟩T

(3)

D=

kBT 6πηξ

(6)

where kB, T, and η are the Boltzmann constant, absolute temperature, and the viscosity of the solvents (0.911 cP for sol buffer and 0.896 cP for gel buffer), respectively. Calculated ξ values are also shown in Table 1. In the case of solutions, ξ implies approximately the size of solutes. Notice that ξ becomes the hydrodynamic radius in the dilute limit. With respect to the macromer, these values are in good agreement with the previously reported SANS results; ξ becomes smaller as the concentration increases since each macromer is compressed by the neighboring macromers.16 In contrast to this, the existence of the clusters is proved for the first time by DLS since the size of the cluster is too large to detect by the neutron scattering techniques. In the case of gels, D is connected to collective motion and eq 6 works out. Here, ξ is regarded as the size of network mesh when the gel is not near the critical demixing point. In this meaning, it is reasonable that the calculated ξ (shown in Table 1) becomes smaller as the concentration increases since the network becomes more dense. Details of concentration dependence of ξ are shown in the Supporting Information with molecular weight dependence. From the macroscopic point of view, D is connected to the osmotic

(2)

σI is a parameter representing the depression of initial amplitude of the intensity correlation function (g(2)(τ = 0) − 1) derived from finite size of beam (smearing effect). From now, we ignore this effect since σI2 is almost unity in our instrument. Γf implies the decay rate of the fast component, and Γs implies that of the slow component. Af and As represent the proportion of these components (Af + As = 1). These 2

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Here, the osmotic modulus is represented by the differential form as follows:27 Kos = ϕ

∂Π ∂ϕ

(9)

where ϕ is the volume fraction of polymers and Π is the osmotic pressure. Π of solutions has roots in the mixing free energy. In the case of as-prepared gels, Π also has roots only in the mixing free energy. From this point of view, the crosslinking restricts the volume available to the polymer chains and Π of gels is reduced.28 This also means the decrease of Kos of sol gels compared to that of solutions (Kgel os < Kos ) assuming that the ϕ dependence is almost the same between solutions and gels. Since the shear modulus is small because of the low sol concentration of our samples, Kgel os + (4/3)μ < Kos also holds true. This is why the diffusion constant of gels is smaller than that of solutions. This idea is supported by the fact that the diffusion constant of the solution and gel shows approximately the same value in the case of 2c* sample (80 mg/mL); (9.8 ± 0.3) × 10−11 m2 s−1 for the solution and (9.2 ± 0.1) × 10−11 m2 s−1 for the gel (data are not shown). 2. NSE: Microscopic Motion. Figure 4 shows the neutron scattering intensity curves of Tetra-PEG solutions and gels.16

Figure 3. q2-Dependence of the decay rates of Tetra-PEG solutions (upper, Γf; lower, Γs) and gels (upper, Γ). The dashed lines show the best fit by eq 5 for solution samples, and the solid lines show that for gel samples.

Table 1. Diffusion Constants and Correlation Lengths Measured by DLS state

concn/ mg mL−1

solution, fast solution, fast solution, slow solution, slow gel gel

20 40 20 40 20 40

D/m2 s−1 (6.9 (7.9 (1.4 (8.4 (3.7 (5.8

± ± ± ± ± ±

0.1) 0.1) 0.1) 0.6) 0.1) 0.1)

× × × × × ×

ξ/nm 10−11 10−11 10−12 10−13 10−11 10−11

3.5 3.1 (1.8 (2.9 6.7 4.2

± ± ± ± ± ±

0.1 0.1 0.1) × 102 0.2) × 102 0.1 0.1

Figure 4. SANS curves of Tetra-PEG solutions and gels.16 The dashed lines show the curve fits of solutions with the Debye function for the four-arm polymer chain modified by the interpolymer interaction. The solid lines show the curve fits of gels with Ornstein−Zernike functions. The blue region shows the covered q-range of NSE in this research.

modulus, Kgel os , the shear modulus, μ, and the friction constant, f, of the network:3 D=

Kosgel + (4/3)μ f

Notice that there is no upturn for gels even at low-q region. This means there is no inhomogeneity such as entanglements and loops. The q-range covered by NSE is hatched in the figure. SANS results indicate that there is no noticeable structural difference between solutions and gels in this q-range. Compared to this static view, dynamic motion is different at least in the mesoscopic scale measured by DLS. Then, our interest is whether there is the effect of cross-link on dynamic motion even in this q-range. NSE is the only way to access the microscopic motion of gels since required energy resolution is about sub-μeV, which cannot be accessed by other quasi-elastic neutron scattering techniques. By using Tetra-PEG solutions and gels, we are able to explore this issue without caring about the effect of inhomogeneity. In NSE, what we obtain is the intermediate dynamic structure factor, I(q⃗,t):29

(7)

Then, we can interpret that the diffusion constants of gels become larger with the increase of concentration since the elasticity of gels becomes strong. Although most of the gels show larger diffusion constants in the case of gels compared to that of solutions,7 the diffusion constants of Tetra-PEG gels are smaller than those of solutions. This result is consistent with the estimation of osmotic moduli of Tetra-PEG gels by the previous SANS measurement.26 Let us see the reason for this. Compared to the representation of gels (eq 7), the diffusion constant of solutions is represented as follows: D=

Kossol f

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Macromolecules I (q ⃗ , t ) =

1 N

N

Article N

exp[iq ⃗ · rj⃗(t )]⟩ ⃗ ∑ ∑ ⟨exp[−iq ⃗ · ri (0)] i

(10)

j

where N is the number of nuclei in the irradiated volume and ri⃗ (t) is the position of PEG monomer i at time t. In this experiment, we use the usual protonated PEG. In this case, the measured intermediate dynamic structure factor mainly comes from the pair-correlation part (i ≠ j), called the coherent intermediate dynamic structure factor. In general, this factor does not show a single exponential decay. However, it is known that the long-time behavior of the coherent factor can be approximated by the following stretched-exponential form: I(q , t ) ∝ exp[−(Γt )β ]

(11)

The value of β and q-dependence of Γ reflect the types of microscopic motion. These are summarized in Table 2. Notice Table 2. Characterization of Microscopic Motion types of motion

β

Γ

collective diffusion Zimm mode Rouse mode

1 2/3 1/2

∝ q2 ∝ q3 ∝ q4

that we cannot use the linear combination of stretchedexponential forms since these motions do not occur simultaneously. Although eq 11 does not hold true at short times, we apply this equation to all regions for simplicity. This approximation is phenomenologically valid.9,12 Following this approach, we fit the measured normalized intermediate dynamic structure factors by the following form: I (q , t ) = A 0 + A1exp[−(Γt )β ] I(q , 0)

Figure 5. Normalized intermediate dynamic structure factors of TetraPEG solutions and gels. The dashed lines show the best fit by eq 12 for solution samples, and the solid lines show that for gel samples with β = 2/3.

(12)

A0 is added to express the nondecay component, as explained later. Figure 5 shows the normalized intermediate dynamic structure factors, I(q,t)/I(q,0), of Tetra-PEG solutions and gels measured by NSE. Now, we evaluate the effect of crosslinks on microscopic motion from two aspects. I. Types of Motion. First, we fit I(q,t)/I(q,0) by eq 12. As a result, the best fit we received for β is 2/3 for all of the data. Then, we fit the data again with the restriction that β = 2/3. The best fit is shown in Figure 5. Obtained relaxation rates, Γ, are scaled by q3, as shown in Figure 6. The values of Γ show no difference between solutions and gels within the measured qregions. In addition to this, concentration-dependence is not observed. These results suggest that the microscopic motion is Zimm mode in all measured q-regions. Notice that the decay rate of Zimm mode is determined only by the viscosity of solvent. This is unusual behavior compared to the conventional gels. Most of the gels show the transition from Zimm mode (q3-dependence in high-q region) to collective diffusion (q2dependence in low-q region) in this q-range.7,11,12 This transition occurs at the point that the monomers notice they are connected to an infinite network. There are two cues for monomers to realize their connectivity (Figure 7). One is the cross-link point. In conventional inhomogeneous gels, the length between neighboring cross-links is widely distributed. This corresponds to connectivity inhomogeneities. Since the realization of connectivity is achieved by finding of the nearest cross-link,

Figure 6. q-Dependence of decay rate of NSE. The solid line implies q3 .

the transition value of q is higher than the value expected from the average mesh size or correlation length (Figure 7a). In contrast to the conventional gels which have connectivity inhomogeneities, Tetra-PEG gel has highly controlled network. The length between two cross-links is the same over the whole network. This is one of the reasons that only Zimm mode is observed instead of the collective diffusion in the measured qrange despite their correlation length is approximately the same compared to that of conventional gels (Figure 7c). To be more specific, let us compare Tetra-PEG gel to PNIPA (poly(Nisopropylacrylamide)) gel. SANS measurement of the correlation length of NIPA gels which is made from 689 mM NIPA and 22.4 mM BIS (N,N′-methylenebisacrylamide) was reported 767

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disappearance of nondecay component also shows the inhomogeneity-free nature of Tetra-PEG gel. Richter et al. proposed quantitative formulation about this nondecay component.6 They devote this nondecay component to the limited range of cross-link movement. Following their formulation, this nondecay component, called elastic incoherent structure factor (EISF), can be written as a t-independent part in I(q,t)/I(q,0): ⎡ q2⟨l 2⟩ ⎤⎞ ⎡ q2⟨l 2⟩ ⎤ ⎛ I (q , t ) ⎥⎟ ⎥ + ⎜⎜1 − exp⎢ − = exp⎢ − I(q , 0) 4 ⎦ ⎝ 4 ⎦⎟⎠ ⎣ ⎣ exp[− (Γt )β ]

(13)

where ⟨l2⟩ is the square of the width of the Gaussian distribution of the monomer displacement from its equilibrium position. The disappearance of nondecay component means that the cross-links of Tetra-PEG gel travel farther compared to the conventional gels. This fact can be understood from the viewpoint of the inhomogeneity. There are dense and sparse regions in conventional gels. The monomers in dense regions cannot travel so far because of the many cross-links or entanglements around them. On the other hand, the monomers in sparse regions also cannot travel so far because they are stretched compared to the monomers in dense regions from the beginning. An experimentally determined value of l is 1−2 nm.6,12 In contrast to this, Tetra-PEG gel is not affected by such inhomogeneities. As mentioned before, the number of monomers between cross-links of Tetra-PEG gel is about 230. Compared to this large number, the size of network mesh, ξ, is not so large (see Table 2). This means there is room for monomers to travel several-fold length of ξ. 3. Integration of Different Scale Motions. Decay rates of Tetra-PEG solutions and gels measured by light and neutrons are summarized in Figure 8. To see the q-dependence clearly,

Figure 7. Comparison of conventional gels and Tetra-PEG gel from the viewpoint of neurons. In high-q region, monomers do not see other cross-links. As q becomes lower, monomers begin to see the nearest cross-link and notice their connectivity (a). Entanglements are also regarded as cross-links in the time region of NSE (b). In the case of Tetra-PEG gel, it requires lower q (i.e., large distance) to see other cross-links since the length between cross-links is strictly controlled and there are no entanglements (c).

by Shibayama et al.30 After setting the volume fraction of the solute ϕ = 0.0579, ξ of NIPA gel is calculated as 4.5 nm at T = 23 °C by using the Ornstein−Zernike function. ξ of Tetra-PEG gel with ϕ = 0.0531 is calculated as 2.0 nm from the SANS experiment.16 Although the correlation lengths of these gels is the same order of magnitude, the average number of monomers between cross-links is totally different: about 30 for the NIPA gel and about 230 for the Tetra-PEG gel. These difference suggest that there is a lot of sparse region for the NIPA gel which makes the average size of network mesh larger. The other cue is the entanglement. This corresponds to topological inhomogeneities. An entanglement is regarded as one cross-link in the time region observed by NSE. This means that the monomers notice their connectivity (topological connectivity) before they see the genuine cross-link (chemical connectivity) in the conventional gels (Figure 7b). However, this does not matter for Tetra-PEG gels since there are no entanglements. As a result, inhomogeneity-free Tetra-PEG gel shows Zimm mode in a wide q-range. Through this analysis, we found the effect of connectivity/topological inhomogeneities on the dynamics of gels for the first time. II. Disappearance of Nondecay Component. The best fit of Figure 5 by eq 12 shows A0 = 0 for all q and all samples. In contrast to this, all of the reported gels in the literature show nondecay component in low-q region except for physical crosslink gel such as PVA with borax.12 Qualitatively, the nondecay component comes from frozen component observed in SANS profiles. In other words, spatial inhomogeneities make the nondecay component. From this point of view, the

Figure 8. Decay rates of Tetra-PEG solutions and gels. The lines are the fitting results. The slopes of the lines are zero and unity.

the ordinate is shown as a form of Γ/q2. Although the length scale of q is different between DLS and NSE about 2 orders of magnitude, the decay rates of gels seem to be connected smoothly (shown by solid lines in Figure 8). Here, we fit the Γ/ q2 of NSE data by a linear function with taking all of the NSE data (both concentrations of solution and gel samples) into consideration. Such smooth connections are not obtained from solution samples; the relaxation rate obtained from DLS are larger than that estimated from NSE. This can be regarded as 768

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the additional translational motion, which is absent for the gels. The crossover points for gels mean the transition from collective diffusion (q2) to Zimm mode (q3). The crossover of 20 mg/mL gel appears at lower q (large scale in real space) compared to the 40 mg/mL gel: 4.5 × 108 m−1 (0.45 nm−1) and 7.0 × 108 m−1 (0.70 nm−1), respectively. These crossover points, q*, can be converted into the characteristic length scale of transition, l*: 2π l* : = q*

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +81 (0)4 71363418. Fax: +81 (0)4 71346069. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been financially supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology (No. 22245018 to M.S.). The NSE experiment was performed by SNS-NSE (BL15) at SNS, Oak Ridge, TN. The authors gratefully acknowledge the financial support provided by JCNS and the use of the JCNSNSE instrument at the Spallation Neutron Source (SNS), Oak Ridge, USA. Part of the research conducted at SNS was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy. T.H. acknowledges the support from Advanced Leading Graduate Course for Photon Science, Program for Leading Graduate Schools, Japan Society for the Promotion of Science. This work was carried out under the Joint-Use Research Program for Neutron Scattering, Institute for Solid State Physics (ISSP), the University of Tokyo, at the Research Reactor JRR-3, JAEA (Proposal No. 13612).

(14)

The physical meaning of l* is the size of window at the transition from Zimm mode to collective diffusion, as shown in Figure 7c. l* of Tetra-PEG gel is 14 nm for 20 mg/mL and 9.0 nm for 40 mg/mL. This value is approximately twice as large as the correlation length of corresponding samples (6.7 nm for 20 mg/mL and 4.2 nm for 40 mg/mL), which is consistent with the picture of inhomogeneity-free gels (see Figure 7c). From another point of view, q*ξ values for each sample show the order of unity (3.0 for 20 mg/mL and 2.9 for 40 mg/mL). This means that the standard understanding of the transition point from collective diffusion to Zimm mode can also apply to this inhomogeneity-free gels. Notice that we can regard the correlation length approximately as the mesh size obtained by DLS since there is no spatial inhomogeneities in Tetra-PEG gels. The difference of q* between two gel samples is exaggerated since the 20 mg/mL (c*/2) gel sample is deficient in the cross-linking points. To the best of our knowledge, this is the first report of such a combination analysis. This type of analysis can be applied to the general network systems and become a powerful tool for the analysis of inhomogeneity from the viewpoint of dynamics.

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REFERENCES

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CONCLUSION Mesoscopic and microscopic motions of gels are investigated by DLS and NSE. By using Tetra-PEG gel as a sample, we are able to investigate the effect of cross-links on the motion of gels without caring about inhomogeneity. The diffusion constants of polymer solutions measured by DLS are consistent with the result obtained by SANS. Compared to the solutions, the diffusion constants of gels are suppressed because of the reduction of the osmotic modulus. Although there is some difference between the mesoscopic motions of solutions and gels, microscopic motions measured by NSE do not show significant difference. Especially, we found two remarkable points peculiar to this inhomogeneity-free gel. One is the observation of Zimm mode upon the broad q-range. The other is the complete decay of I(q,t)/I(q,0). These results contrast markedly with that of conventional gels. We clarify that the reason for the first one is the absence of connectivity/ topological inhomogeneities and that for the second one is the absence of spatial inhomogeneities. Then, the mesoscopic and microscopic decay rates of gels are compared simultaneously. These decay rates are connected smoothly and the transition between Zimm mode and collective diffusion can be estimated beyond the q-range of NSE.

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ASSOCIATED CONTENT

S Supporting Information *

Concentration/molecular weight dependence of ξ for TetraPEG gels. This material is available free of charge via the Internet at http://pubs.acs.org/. 769

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