Permeation of Water through Hydrogels with Controlled Network

Nov 10, 2017 - the blob size (ξ) proposed by Tanaka and de Gennes (f ∼ ξ−2. ). However, the experiments on equilibrium-swollen hydrogels are not...
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Permeation of Water through Hydrogels with Controlled Network Structure Takeshi Fujiyabu,† Xiang Li,‡ Mitsuhiro Shibayama,‡ 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 ABSTRACT: Friction between polymer chains and water is considered to govern the water retention and permeation in hydrogels. This concept has been examined by means of water permeation measurements through hydrogels. Previous studies used hydrogels in the equilibrium swollen state and supported the scaling relationship between the friction coefficient (f) and the blob size (ξ) proposed by Tanaka and de Gennes ( f ∼ ξ−2). However, the experiments on equilibrium-swollen hydrogels are not enough, and those on as-prepared hydrogels are needed to fully examine the validity of the scaling relationship. In this study, we established a novel water permeation apparatus, which well prevented the swelling of gels and enabled the water permeation experiments on gels in the as-prepared state. Using this novel apparatus, we measured f of a model polymer network system, i.e., Tetra-PEG gels. By comparing f measured by the water permeation experiment and ξ measured by a small-angle neutron scattering experiment, we confirmed the scaling relationship f ∼ ξ−2 even in the as-prepared state. This result strongly supports the availability of the current model in the equilibrium swollen state and thus strengthens the validity of the model.

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INTRODUCTION Even when a hydrogel is compressed to the point of breaking, the gel hardly allows water to escape from itself. In other words, to squeeze out the solvent from a gel, much higher pressure is required. Despite the difficulty in squeezing out the solvents, most of water molecules are not hydrated to polymer chains1 and diffuse at a similar manner with that in bulk water.2−4 According to the model proposed by T. Tanaka, the friction between a polymer network and water molecules is the source of extremely slow flow of water across the gel.5−7 This concept has been examined by many researchers8−13 and is considered the most promising. The friction coefficient between polymer network and water (f) has been discussed based on Darcy’s law (eq 1),6,7,11−13 which was originally developed to describe the permeability of fluid through porous materials.14 The value of f is estimated by water permeation experiment; a hydrostatic pressure (P) is imposed from the top of a hydrogel membrane with a thickness of d, and the quantity of water permeated through the membrane (v) is measured (Figure 1). f=

P vd

Figure 1. Schematic illustration of the principle for measuring the friction coefficient between a gel and water.

and is allowed to swell. Thus, a hydrogel in the as-prepared state swells during the experiments, which prevents accurate measurement of v. Due to the unavoidable swelling, in the most of the previous works, the hydrogels were used in the equilibrium swollen state.6,7,11−13 The relationship between f and polymer volume fraction in the equilibrium swollen state (ϕe) was found to be f ∼ ϕe1.5.7,13 Theoretically, de Gennes predicted f based on the picture that water flows through the gels by passing through the pores of concentration blobs as15,16

(1)

It should be noted that the f in eq 1, which is always defined as “friction coefficient” with units of Ns/m4, differs from the conventional dimensionless friction coefficient. In a water permeation experiment, a hydrogel always contacts with water © XXXX American Chemical Society

Received: August 21, 2017 Revised: November 10, 2017

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DOI: 10.1021/acs.macromol.7b01807 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules f≈

η ∼ ξ −2 ξ2

Tetra-PEG gel 20K, 68 mM of buffer solution (pH 3.8) was used. 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: 0.034−0.081) and 64 mM (pH 3.4) was used for the higher polymer volume fraction (ϕ0: 0.096). In the case of Tetra-PEG gel 5K, 68 mM of buffer solution (pH 3.8) was used for the lowest polymer volume fraction (ϕ0: 0.034), 64 mM (pH 3.4) was used for the lower polymer volume fraction (ϕ0: 0.050), and 60 mM (pH 3.0) was used for the higher polymer volume fractions (ϕ0: 0.066, 0.081). Equal amounts of two prepolymer solutions were mixed, and the resulting solution was poured into the mold. At least 12 h was allowed for the completion of the reaction before the following experiment was performed. In the case of r-tuned Tetra-PEG gels, the prepolymer solutions were mixed in nonstoichiometric ratio (r = [Tetra-PEG-MA]/([TetraPEG-MA] + [Tetra-PEG-SH])).19 Three kinds of r-tuned Tetra-PEG gels were formed from prepolymers with Mw = 10 kg/mol and ϕ0 = 0.050 (r = 0.5, 0.45, and 0.375). The molar concentration of buffer solution was 68 mM (pH 3.8). These solutions were mixed and waited for the reaction completion. Equilibrium Swelling Experiment. Tetra-PEG gels were formed in the glass tubes with rounded part in the center (WA: the weight of apparatus) (Figure 2). The initial total weight of the gel and the glass

(2)

where η is the solvent viscosity and ξ is the blob size. The blob size is related to the osmotic pressure (πos) in semi-dilute polymer solution as17 πos ≈ ξ −3kT

(3)

with Boltzmann constant (k) and absolute temperature (T). Equation 3 indicates that πos is proportional to the number density of blobs (∼ξ−3). By applying the scaling between πos and polymer volume fraction (ϕ) in good solvent (πos ∼ ϕ2.25) into eqs 2 and 3, we obtain an expression observed in the previous experimental works: f ∼ ϕ1.5

(4)

Although the previous works supported the molecular picture underlying eq 2,7,13 those agreements are not enough to fully validate eq 2. In the equilibrium swollen state, various physical parameters obey the power law of ϕe. For example, the ϕe dependence of elastic modulus (G) is described by the same scaling relationship with πos (G ∼ ϕe2.25).16,17 Even if there were other factors influencing f, it is difficult to identify the true factor influencing f in the equilibrium swollen state. On the other hand, initial polymer volume fraction (ϕ0) dependences of G and πos are described by different scalings as G ∼ ϕ0 and πos ∼ ϕ02.25, respectively.16 Thus, experiments in the asprepared state will provide us a better understanding of f. However, the experimental difficulty prohibited the examination using hydrogels in the as-prepared state. In this study, we investigated water permeation through hydrogels in the “as-prepared state” using a new water permeation apparatus, which consists of a custom-made glass tube with a round part in its center. Hydrogels were formed in the round part of the glass tube, and the swelling of the gels was significantly suppressed due to the geometry of the glass tube. Tetra-PEG gels were used as a model gel system for water permeation experiments. Tetra-PEG gel was formed by mixing two aqueous solutions of tetra-armed prepolymers with mutually reactive end groups (thiol (−SH) and maleimide (−MA)).18 Our previous small-angle neutron scattering (SANS) measurements revealed the extremely low structural heterogeneity in Tetra-PEG gels.19−21 In addition, the relationship between the mechanical properties and network structure is well-known.20 Thus, Tetra-PEG gel system is promising as a model polymer gel system. We tuned the prepolymer molecular weight (Mw) and the initial polymer volume fraction (ϕ0) and investigated the effect on f and ξ. The direct comparison between f and ξ and the validation of eq 2 in the as-prepared state were performed for the first time.

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Figure 2. The gel was prepared in the glass tube rounded in the center (a), and then swollen in water (b).

tube (W0) was measured after the reaction completion. The glass tubes where a gel sample was formed were immersed in H2O at room temperature. After waiting for 2 days, the total weight of gel and glass tube (W) was measured, and the swelling ratio (Q) was calculated by eq 5. The thickness of gel sample (d) was measured at the same time. Because of the concavity at the interface between the gel and water, d was defined as the average length of the longest part and the shortest part of the concavity.

Q=

W − WA W0 − WA

(5)

Water Permeation Experiment. The schematic picture of the water permeation apparatus is shown in Figure 3. The glass tube with round part at the center and polypropylene (PP) tubes (BHJ, BGE, Japan) with different inner diameters (4, 3, and 1 mm) were assembled as shown in Figure 3. The water pressure was applied by a water pillar with the length of h and the diameter of rt4. The pressure (P) generated by the water pillar to a part of gel with the sectional area of πrA2 is given by

MATERIALS AND METHODS

Materials. Tetra-maleimide-terminated poly(ethylene glycol) (Tetra-PEG-MA) and Tetra-thiol-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 Tetra-PEG Gels. Details of Tetra-PEG precursors have been reported elsewhere.18−21 Constant amounts of Tetra-PEGMA 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. The Tetra-PEG gels formed using prepolymers with a Mw of 5, 10, and 20 kg/mol are called Tetra-PEG gel 5K, TetraPEG gel 10K, and Tetra-PEG gel 20K, respectively. In the case of

⎛ r ⎞2 P = ρgh⎜ t4 ⎟ ⎝ rA ⎠

(6)

Here, ρ is the water density, g is the acceleration of gravity, rt4 is the inner radius of 4 mm PP tube (2 mm), and rA is the inner radius of the tube of the sphere-type apparatus (1.25 mm). The velocity of water B

DOI: 10.1021/acs.macromol.7b01807 Macromolecules XXXX, XXX, XXX−XXX

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polymer volume fraction for a bulk gel. Because this error was small enough compared to the difference in the initial set of ϕ0, we decided to treat the difference between ϕ0 and ϕ′ as the experimental error decreasing ϕ0, for avoiding any analytic complication caused by correction of ϕ0 to ϕ′. Notably, the hydrogel specimen has heterogeneous mechanical stress distribution because the swelling of gel is suppressed by an anisotropic apparatus. However, this heterogeneous mechanical stress hardly results in heterogeneous distribution in polymer segment in bulk because the osmotic pressure is much higher than elastic pressure. On the other hand, swelling occurs on the interface between gel and water. To minimize the effect of swollen surface compared to that of the bulk, the value of d should be enlarged. However, enlarged d prolonged the period for an experiment and brought some complications such as evaporation of water. Thus, we decided to set d ≈ 5 mm, considering the balance of these complications. Measurement of Water Permeation through Hydrogel in the As-Prepared State. We checked the invariance of the water pressure (P) and absence of water leakage or gel deformation during the water permeation experiment. The amount of water loss in water pillar caused by the permeation and evaporation was negligible, reflecting almost constant P during the experiments (Figure 5a). Figure 5b shows a typical result of meniscus movement. The meniscus position moved fast at the beginning of the experiment and then reached steady velocities (v′), which is consistent with the previous works.7,8,11−13 We calculated the velocity of water flow through the gel (v) from v′ by eq 7 and plotted v as a function of P/d (Figure 6). The relationship between v and P/d was linear and agreed with the prediction of eq 1 in the small P/d region. In this region, we can assume that water molecules flow through a region on which the water pillar stands (the sectional area of πrA2). There may be flow in the vicinity of this area, which make the net sectional area larger. This net sectional area is most likely constant in the linear region. In contrast, slight upper deviations from the linear relationship were observed in the region above P/d ≈ 3.0 × 106 Pa/m. This nonlinearity may stem from the enlargement of net sectional area or conformation change by high pressure.12 The water permeation experiments were conducted around P/d = 2.5 × 106 Pa/m to minimize the effect of nonlinear effect and the experimental error caused by long experimental period. For the analysis, we ignored the flow outside the area of πrA2 obeying the most of previous works6,7,10−12 because this effect has little influence on the scaling relationships discussed below. On the basis of these results, we confirmed that water permeation experiments using hydrogels in the as-prepared state were well performed. The friction coefficients (f) of Tetra-PEG gel 5K, 10K, and 20K are shown as a function of ϕ0 in Figure 7. The value of f increased with an increase in ϕ0, suggesting that the denser polymer network strongly retarded the water permeation. When we focused on Tetra-PEG gels with the same ϕ0 but different Mws, Tetra-PEG gels with “larger” molecular weight between cross-links retarded the water permeation more. The similar counterintuitive results were observed in Tetra-PEG gel formed with nonstoichiometric prepolymer ratio (r-tuned Tetra-PEG gel, Table 1). The value of f increased with an increase in deviation from the stoichiometry; the heterogeneous polymer network strongly retarded the water permeation. When we focused on the power law relationships between f and ϕ0 (f ∼ ϕ0x), Tetra-PEG gel 5K, 10K, and 20K showed x =

Figure 3. Schematic illustration of the water permeation apparatus and the principle behind the measurement system of f. meniscus (v′) was measured at room temperature. The velocity of water flow through the gels (v) was calculated by eq 7

⎛ r ⎞2 v = ⎜ t1 ⎟ × v′ ⎝ rA ⎠

(7)

Here, rt1 is the inner radius of 1 mm PP tube (0.5 mm). The value of f was estimated from d, P, and v, according to eq 1.

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RESULTS AND DISCUSSION Restriction of Swelling in Water Permeation Apparatus. To confirm the inhibition of swelling in the apparatus, a glass tube containing a gel in their round part (Figure 2) was immersed in the water bath, and the gel was allowed to swell. After confirming the gel reached its maximum change in weight, we measured the swelling ratio in the apparatus (Q, Figure 4).

Figure 4. ϕ0 dependence of Q of Tetra-PEG gel with different Mw (Mw: 5 kg/mol, squares; 10 kg/mol, circles; 20 kg/mol, triangles) in the round part of the glass tubes.

Most of the values of Q were lower than 1.05. Notably, the swelling ratio of these hydrogels without confinement was up to 3.1.19 Thus, this apparatus could prevent gel from swelling and realize the water permeation experiment in a pseudo asprepared condition. However, at the same time, our apparatus had a limitation in suppression of the swelling of gels especially in a higher concentration region (Q ≈ 1.1). Swelling of gels mainly influences two parameters: thickness of gel specimens (d) and the polymer volume fraction (ϕ′). As for d, we measured the values in the equilibrium state in the apparatus and used in the analysis. We also confirmed the invariance of d during the water permeation measurements. As for ϕ′, the maximum deviation from ϕ0 caused by swelling is estimated to be 10%, when we assumed the homogeneous C

DOI: 10.1021/acs.macromol.7b01807 Macromolecules XXXX, XXX, XXX−XXX

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Figure 5. (a) Time course of P during the water permeation experiments for a Tetra-PEG gel (Mw: 10 kg/mol, ϕ0: 0.034) at different pressures (P: 7100 Pa, circles; 9700 Pa, squares; 13 000 Pa, triangles; 15 000 Pa, full circles; 19 000 Pa, full squares; 20 000 Pa, full triangles). (b) Time course of meniscus position during the water permeation experiments for a Tetra-PEG gel (Mw: 10 kg/mol, ϕ0: 0.034) at different P (P: 7100 Pa, circles; 9700 Pa, squares; 13 000 Pa, triangles; 15 000 Pa; full circles; 19 000 Pa; full squares; 20 000 Pa, full triangles).

2.4, 2.5, and 2.1, respectively. These scaling relationships were completely different from that in the equilibrium swollen states ( f ∼ ϕ01.5).7,13 This deviation in the scaling exponents may stem from the condition of hydrogel. Effect of Blob Size on Friction Coefficient. In our previous study using small-angle neutron scattering (SANS), we estimated ξ of Tetra-PEG gel 5K, 10K, and 20K in the asprepared and the equilibrium swollen state.21 Figures 8a and 8b show the ϕe dependence and ϕ0 dependence of ξ, respectively. As for the equilibrium swollen state, Tetra-PEG gel 20K and a part of Tetra-PEG gel 10K obeyed a scaling relationship showing ξ ∼ ϕe−0.75, corresponding well to the scaling prediction for good solvent in semi-dilute regime.7,13 On the other hand, Tetra-PEG gel 5K and most of Tetra-PEG gel 10K did not obey this scaling relationship. In the case of the asprepared state, each Tetra-PEG gel showed different relationship (5K, ξ ∼ ϕ0−1.36; 10K, ξ ∼ ϕ0−1.25; 20K, ξ ∼ ϕ0−0.84), which is completely different from that in the equilibrium swollen state. At a constant ϕ0, Tetra-PEG gels with “larger” Mw tend to have smaller ξ. As for r-tuned Tetra-PEG gels, ξ increased with an increase in the deviation from the stoichiometry (Table 1). Taken together, the “loosely” cross-linked polymer network has a smaller blob size. These counterintuitive results are because the formation of loosely cross-linked locally heterogeneous hydrogel is much more likely to occur under dilute conditions, resulting in increase in the blob size.21,22 Finally, to examine eq 2, we plotted f against ξ in Figure 9. All the data fall onto a guide showing the scaling f ∼ ξ −2 (shown as a dotted line), except for Tetra-PEG gel 5K. This correspondence strongly suggests the validity of eq 2 even in the as-prepared state. This result suggests that water molecules pass through the path with the size ξ both in the as-prepared state and in the equilibrium swollen state. The deviation observed in Tetra-PEG gel 5K may be due to the local heterogeneity.21,22 Because the ϕe dependence of ξ of TetraPEG gel 20K in the equilibrium swollen state well obeyed the scaling prediction for semi-dilute region (ξ ∼ ϕ0.75), we can expect that Tetra-PEG gel 20K will show the classical scaling relationship found in previous studies (f ∼ ϕ1.5). Because the units of f and η/ξ2 are the same, eq 2 becomes an identity formula with a dimensionless constant, a.

Figure 6. P/d dependence of v of Tetra-PEG gel (Mw: 10 kg/mol; ϕ0: 0.034).

Figure 7. ϕ0 dependence of f of Tetra-PEG gel with different Mw (Mw: 5 kg/mol, squares; 10 kg/mol, circles; 20 kg/mol, triangles).

Table 1. r dependence of Q, F, and ξ of r-tuned Tetra-PEG gel (Mw: 10 kg/mol, ϕ0: 0.050)a r 0.375 0.45 0.5

Q

f [Ns/m4]

ξ [Å]

1.05 1.02 1.02

2.6 × 10 2.5 × 1014 2.1 × 1014

19.7 21.1 23.8

14

ξ was estimated by refitting the data in our previous study with the Ornstein−Zernike function. a

f=a D

η ξ2

(8) DOI: 10.1021/acs.macromol.7b01807 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Figure 8. (a) ϕe dependence of ξ of Tetra-PEG gel with different Mw (Mw: 5 kg/mol, squares; 10 kg/mol, circles; 20 kg/mol, triangles). (b) ϕ0 dependence of ξ of Tetra-PEG gel with different Mw (Mw: 5 kg/mol, squares; 10 kg/mol, circles; 20 kg/mol, triangles).

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (T.S.). ORCID

Xiang Li: 0000-0001-6194-3676 Mitsuhiro Shibayama: 0000-0002-8683-5070 Takamasa Sakai: 0000-0001-5052-0512 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. Masayuki Tokita for giving us an advise about the water permeation apparatus. 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, and the Grants-in-Aid for Young Scientists (A) Grant 23700555 to T.S. and Scientific Research (S) Grant 16746899 to U.C.

Figure 9. ξ dependence of f of Tetra-PEG gel with different Mw and rtuned Tetra-PEG gel (Mw: 5 kg/mol, squares; 10 kg/mol, circles; 20 kg/mol, triangles; r-tuned, crosses).

We plotted the value of η/ξ 2 using the viscosity of water (η) at 25 °C (8.9 × 10−4 Ns/m2) as the dashed line in Figure 9. Calculated η/ξ2 was very close to f, indicating a was near unity (a = 0.68). We will further investigate on this point in near future because no direct comparison between η/ξ2 and f has been done in previous studies.

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REFERENCES

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CONCLUSION We established the novel water permeation apparatus, which inhibited the swelling of gel and enabled the water permeation experiment in pseudo as-prepared state. Using this apparatus, we estimated the friction coefficient between polymer chain and water ( f) of Tetra-PEG gels with different molecular weights of prepolymers (Mw), polymer volume fractions (ϕ0), and imbalanced stoichiometries (r). The scaling relationships between f and ϕ0 did not agree with that estimated for the equilibrium swollen states, and elastically effective network or cross-kinks are not directly correlated to friction coefficient. On the other hand, the relationship between f and blob size estimated by SANS (ξ) obeyed the relationship represented by eq 2 even in the as-prepared state. Thus, it is strongly suggested that the permeation of water through hydrogels is governed by the osmotic blob in both the as-prepared and the equilibrium swollen states. Because ξ scales with the osmotic pressure (πos) as πos ∼ ξ−3, the permeation of water molecules through hydrogels is, in other words, governed by πos, which indicates the water-retention ability. These findings will help understand the interaction between solvent molecules and polymer chains in polymer gels. E

DOI: 10.1021/acs.macromol.7b01807 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.7b01807 Macromolecules XXXX, XXX, XXX−XXX