Viscoelastic Properties of Poly(vinyl alcohol) Hydrogels Having

May 16, 2013 - microrheology based on diffusing-wave spectroscopy (DWS), classical rheology and single dynamic light scattering (DLS), and compared wi...
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Viscoelastic Properties of Poly(vinyl alcohol) Hydrogels Having Permanent and Transient Cross-Links Studied by Microrheology, Classical Rheometry, and Dynamic Light Scattering Tetsuharu Narita,†,* Koichi Mayumi,† Guylaine Ducouret,† and Pascal Hébraud‡ †

Laboratoire PPMD-SIMM, UPMC-ESPCI ParisTech-CNRS UMR7615, 10 rue Vauquelin, 75005 Paris, France IPCMS/CNRS UMR 7504, 23 rue du Loess PB43 67034 Strasbourg Cedex, France

‡

ABSTRACT: Dynamics of poly(vinyl alcohol) (PVA) hydrogels having chemical and physical transient cross-links simultaneously (dual cross-link PVA gels) were studied by microrheology based on diffusing-wave spectroscopy (DWS), classical rheology and single dynamic light scattering (DLS), and compared with those of corresponding chemical and physical PVA gels. Three different relaxation modes (fast, intermediate and slow modes) are observed for physical gels, while one mode (fast mode) is found for chemical gels, and two (fast and intermediate) for dual cross-link gels. The three modes are attributed respectively to Brownian diffusion of PVA polymer or collective diffusion of the network or gel mode (fast mode), macroscopic stress relaxation (intermediate mode whose characteristic time shows q0 dependence) and Brownian diffusion of aggregates (slow mode). Microrheological measurements are in good agreement with macrorheological showing segmental Rouse mode dynamics in the high frequency range. For physical gels, we found Maxwell type viscoelasticity characterized by a crossover frequency (maximum of G″) and G′ ∼ ω2 and G″ ∼ ω1 in the lower frequency range. The chemical gels displayed an elastic plateau with low G″ at low frequency. For the dual cross-link gel a maximum of G″ was observed, and its characteristic time agrees with that of the intermediate mode measured by DLS. We show that this relaxation mode corresponds to the associative Rouse mode characterized by G′ = G″ ∼ ω0.5, depending on the dissociation rate of the reversible transient cross-links. We propose a stress relaxation mechanism of the PVA chains in the presence of elastically inactive but associative transient cross-links which induces incomplete stress relaxation.

1. INTRODUCTION A hydrogel, a polymeric network swollen in water, can be a good candidate for various biomedical and bioengineering applications. The major problem of most of the synthetic hydrogels is their low fracture toughness. Many efforts to improve the fracture toughness have been made in the last 10 years, and various hydrogels having different chemical architectures have been successfully synthesized.1−5 One promising strategy for the mechanical reinforcement is the introduction of breakable physical cross-links. The physical cross-links based on various noncovalent interactions can be introduced to the chemical network and their ability to break and reform allows the network to dissipate a lot of fracture energy and recover its original structure and properties. Such gels which can be called “self-healing gels”, using hydrogen bonding with fillers,2,6 ionic interactions between chains,7 or hydrophobic interactions,8,9 have been studied experimentally and theoretically. A small amount of chemical cross-links by covalent bonding can be added in order to prevent the network from plastic flow, or the network has a finite shape permanently at rest, facilitating mechanical tests. Although the dynamics of physical and chemical gels are well understood, new questions arise when the two cross-links © XXXX American Chemical Society

coexist, in particular, how the relaxation mechanisms, such as bond breaking and formation, and Rouse modes, compete in the stress relaxation processes. Hui and Long proposed a constitutive model considering a network having primary unbreakable bonds and secondary transient bonds which reform rate-dependently.10 They showed that their model described well the large deformation suppression of strain softening at sufficiently low loading rates in a uniaxial tensile test as well as hysteresis and plasticity in a loading−unloading cycle. The description of the linear properties of these gels and in particular of their dissipation mechanisms is a prerequisite to the understanding of their large deformation properties. The key parameters in these systems are the rates of bond breaking and reforming, which give to the system rate and history dependent mechanical behaviors. It is important to use a good model system having well-defined measurable bond breaking/reforming rates. Here we report on viscoelastic properties of physically and chemically cross-linked poly(vinyl alcohol) hydrogels. The Received: March 22, 2013 Revised: April 29, 2013

A

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chemically cross-linked PVA gels by glutaraldehyde complexed with borate ions which create transient cross-links were used. These systems are good model gels thanks to single characteristic time and their simple preparation. The corresponding chemical and physical PVA gels have been intensively studied and their rheological properties are wellknown.11−15 For the “dual crosslink” gels, cross-linked chemically and physically simultaneously, swelling behaviors have been studied previously and compared with the viscosity of the physical gels,16,17 while rheological properties of the dual cross-link gels have not been studied. Effects of the transient nature of the physical cross-link need to be correlated to the rheological properties, and in order to characterize the dynamics of the transient cross-link, rheological characterizations in a large frequency range in a linear regime are required. We study the dynamics of dual cross-link PVA gels studied by microrheology based on diffusing-wave spectroscopy, DWS. One of the earliest works on the microrheological characterizations of chemically cross-linked gels has been done with PVA chemical gels.18 Though a good agreement between microrheology and macrorheology for the elastic modulus at the plateau was demonstrated, no dynamic characterization was reported. Here we show that DWS microrheology was successfully applied to the dual cross-link PVA gels as well as to the corresponding chemical and physical gels, and we demonstrate the presence of the associative Rouse mode due to the transient physical cross-link in the dual cross-link gel. Both physical and dual cross-link gels are analyzed in analogy with associative polymer solutions which present the associative Rouse mode previously studied theoretically.19

Figure 1. Dissociation of sodium tetraborate in water and complexation equilibria of borate ion with PVA chains. Dual cross-link PVA gels, having chemical and physical cross-links simultaneously, were prepared from corresponding chemical PVA gels by diffusing borax molecules from solution in contact with the chemical gel. The volume of borax solution is the same as that of the chemical gel, and the system was equilibrated for 2 weeks. Dynamic Light Scattering. Dynamic light scattering measurements were performed with an ALV CGS-3 goniometry system (ALV, Langen, Germany), equipped with a cuvette rotation/translation unit (CRTU) and a He−Ne laser (22 mW at λ = 632.8 nm). The system measures the time-averaged autocorrelation function of the scattered light intensity at a scattering vector q defined as q = ((4πn)/λ) sin(θ/ 2), where n is the refractive index, θ the scattering angle. For ergodic samples, it is possible to explore the whole phase space in the course of the experimental time at a fixed sampling position when the duration of acquisition is sufficiently long compared with the slowest characteristic time of the system. Thus, the time averaged autocorrelation function is equivalent to the ensemble averaged one: experimentally, we obtain a constant value of the amplitude of the autocorrelation function at t = 0, and it goes to zero for at long times. In the case of nonergodic samples such as chemically cross-linked gel, due to the static concentration gradient associated with the presence of frozen disorder in the system, the time averaged autocorrelation function does not probe all the configuration of the system, and is different from the ensemble averaged one.22 In order to circumvent the nonergodicity, the sample cell was vertically translated steadily while time-averaging. Thus, g(2) coincides with the ensemble-averaged autocorrelation function, up to a cutoff delay time fixed by the sample translation speed. The averaging duration was more than 3600 s depending on the slowest relaxation modes found in the autocorrelation functions. The obtained field correlation functions were fitted with the stretched exponential functions having one to three relaxation modes. These functions fit well empirically the autocorrelation functions.23 Gels for DLS measurements were prepared in test tubes (10 mm in diameter), prefiltered solutions of PVA, borax, or GA, and HCl, as well as water, were mixed in a test tube prior to chemical/physical gelation. Syringe filters of 0.22 μm (and 0.8 μm for PVA solution) were used. Microrheology by Diffusing-Wave Spectroscopy. Microrheological measurements based on diffusing-wave spectroscopy (DWS) were performed using a laboratory-made setup. The coherent source was a Spectra-Physics Cyan CDRH laser, operated at λ = 488

2. EXPERIMENTAL SECTION Materials. Poly(vinyl alcohol) (PVA; molecular weight: 89000 g/ mol, degree of deacetylation: 99%) was purchased from Janssen Chimica (Beerse, Belgium). Glutaraldehyde (GA; 25% aqueous solution), sodium tetraborate decahydrate (borax), and hydrochloric acid were purchased from Sigma-Aldrich. Polystyrene microspheres (diameter: 500 nm) were purchased from PolySciences, Inc. All chemicals were used as received. Gels Preparation. PVA stock solution (8.8%, or 2 M for repeating unit) was prepared by dissolving PVA powder in water at 95 °C under vigorous stirring. Chemically cross-linked PVA gels were prepared by cross-linking PVA with GA at acidic condition. The PVA concentration was adjusted to be 4.4%. Unless otherwise noted, the GA concentration was 0.2 mM, giving cross-linking ratio (Rc, defined as the molar ratio of cross-linker to repeating units of PVA) of 0.002. Physically cross-linked PVA gels were prepared by complexing PVA with borate ions. One mol of sodium tetraborate dissociates into 2 mol of borate ion and boric acid. (Figure 1). As shown in Figure 1, the PVA-borate ion complexation mechanism has two equilibria, monodiol complexation, and didiol complexation. The latter is considered as physical transient cross-link in this paper. Lin et al. reported the binding isotherm of borate ion onto PVA.20 The monodiol complexation was found to be Langmuir type binding, starting to saturate about 30 mM of borate ion. They measured the equilibrium constant of the monodiol complexation, K1 = 9 M−1. Thus, at the borax concentration studied in this work (0.5−5 mM), about 82 mol % of the borate ion was considered to be bound by monodiol complexation, proportionally to the borax concentration, Cb. Boric acid shows no didiol complexation thus does not contribute to the physical cross-link.21 The cross-linking ratio Rc is defined in the same manner as chemical cross-link, as the molar ratio of borate ion to repeating units of PVA. The efficiency of the physical cross-link taking account of the equilibria will be discussed later in this paper. B

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nm with an output power of 50 mW. The laser beam was expanded to approximately 1 cm at the sample. The diffused light was collected by an optical fiber placed in the transmission geometry. Gels samples for DWS measurements were prepared in cuvettes for fluorescence spectroscopy (L = 4 mm in path length). The 1% polystyrene microspheres used as probes were dispersed in gels. The PVA-microsphere mixtures were stable, no aggregation or precipitation was observed for all gels tested in this work after 6 months. In order to obtain ensemble-averaged autocorrelation function, a diffuser glass plate was placed in front of the optical fiber and rotated at a constant rate. The obtained intensity autocorrelation function g(2)(t) was converted into the field autocorrelation function g(1)(t) by the Siegert relation, g(2)(t) = β[1 + g(1)(t)]2, then the mean square displacement was calculated by solving numerically the following equation:24,25

stress-controlled Haake RS600 rheometer using rough cone−plate geometry (diameter: 35 mm, angle 2°, gap 103 μm). Frequency sweep (between 0.01 and 100 rad/s) was performed with fixed amplitude in the linear domain (2 or 3 Pa). All the measurements were performed at 25 °C, using a laboratory-made antievaporation system. For the chemical gel, the gelation took place in the geometry. The gelation time was estimated by the crossover of G′ and G″ at 1 Hz, found to be about 4 × 104 s.

3. RESULTS AND DISCUSSION 3.1. Dynamic Light Scattering. Figure 2 shows typical examples of intensity autocorrelation function g(2) of PVA

⎡L ⎡z ⎤ 4⎤ ⎡z 2⎤ ⎧ g(1)(t ) = ⎢ + ⎥/⎢ 0 + ⎥ × ⎨sinh⎢ 0 k 0 2⟨Δr 2(t )⟩ ⎥ ⎣ l* ⎦ ⎣ l* 3 ⎦ ⎣ l* 3⎦ ⎩ ⎡ ⎤ z 2 + k 0 2⟨Δr 2(t )⟩ cosh⎢ 0 k 0 2⟨Δr 2(t )⟩ ⎥ ⎣ l* ⎦ 3 ⎛ ⎞ ⎡L ⎤ 4 k 2⟨Δr 2(t )⟩ ⎥ / ⎜1 + k 0 2⟨Δr 2(t )⟩⎟ sinh⎢ ⎣ l* 0 ⎦ ⎝ ⎠ 9 ⎡ ⎤ L 2 + k 0 2⟨Δr 2(t )⟩ cosh⎢ k 2⟨Δr 2(t )⟩ ⎥ ⎣ l* 0 ⎦ 3

}

{

}

Here L is the sample thickness, l* is the sample transport mean free path of the scattered light, z0 is the distance the light must travel through the sample before becoming randomized, here it is set z0 = l*,26 k0 is defined as k0 = 2π/λ. The only unknown parameter, l*, is obtained for each sample by comparing with the known value of l* and IT in water used as reference. For this work l* was found about 220 μm. By using the mean square displacement of a probe particle of radius R, the frequency dependence of the complex shear modulus can be obtained from the generalized Stokes−Einstein equation: G̃(s) =

Figure 2. Autocorrelation function g(2) of the PVA solution (Cp = 4.4%, dashed line) and physical PVA gel (Cp = 4.4%, Cb = 2.5 mM, solid line) measured by DLS. Diffusion angle: 30°.

solution and physical gel (PVA concentration: 4.4%) measured at 30°. For both the signal of the scattered light was found to be ergodic. For the solution we observed two modes of decorrelation. The fast one has a characteristic time of 10−4 s (fast mode), while the slow one decays at 0.1 s. The amplitude of the slow mode is high, more than 0.8. This bimodal function has been reported previously.13,18,27 For the physical gel containing 2.5 mM of borax, three modes (fast, intermediate and slow modes) were found. The amplitude of the intermediate mode is very low. The fast mode is similar to that of the polymer solution, and the slow mode characteristic time is almost 3 orders of magnitude higher than that of the uncross-linked solution. In order to characterize the nature of the decorrelation of these three modes, the diffusion vector q dependence of the characteristic time τ of the each mode for the filed autocorrelation function g(1)(t) was measured (Figure 3). The characteristic time of the fast mode, τf, varies as τf ∼ qα, where α = −1.8 for the solution and −1.9 for the physical gel. For the slow mode, the exponent α was between −1.9 and −2.4, depending on the borax concentration. For various polymer solutions, similar bimodal autocorrelation functions have been observed. Generally, for semidilute solutions of polymers, two relaxation modes can be observed. They can be attributed to diffusion of the polymer chains and of large polymer clusters. We expect α = −2, thus we calculated the diffusion coefficient of the polymer and its hydrodynamic radius Rh from τs−1 = Dq2 and D = kBT/6πηRh, where D is the diffusion coefficient, kBT the thermal energy, η the viscosity. For large clusters having Rh comparable to 1/q, α can vary between −2 and −3 depending on the polymer concentration. For entangled and/or crosslinked polymer solutions the fast mode can be attributed to gel

kBT πRs⟨Δr 2̃ (s)⟩

Here G̃ (s) is the viscoelastic spectrum as a function of Laplace frequency s, kB is the Boltzmann constant, T is the temperature, and ⟨Δr̃2(s)⟩ is the Laplace transform of the mean square displacement ⟨Δr2(t)⟩. This equation can be expressed in the Fourier frequency domain G*(ω) =

kBT πRiω -⟨r 2(t )⟩

where -⟨Δr 2(t )⟩ is the Fourier transform of the mean square displacement. Assuming a local power law form for the mean square displacement, we can express G*(ω) as a function of the experimentally measured mean square displacement ⟨Δr2(t)⟩: G*(ω) =

kBT 2

πR⟨Δr (1/ω)⟩Γ[1 + α(ω)]

G′(ω) = G*(ω) cos[πα(ω)/2]

and

G″(ω) = G*(ω) sin[πα(ω)/2] where t = 1/ ω, α(t) = ∂[ln⟨Δr2(t)⟩]/∂[ln t] is the slope of the graph of ⟨Δr2(t)⟩ plotted against time to logarithmic scales evaluated at t. Γ is the gamma function which serves as the conversion factor of the transform. With our setup at the experimental conditions written above, G′ and G″ can be measured in the range between 10 Pa and several kilopascals for 10−1 to 105 rad/s with good precision compared with the classical macrorheology. Classical Macrorheology. Macroscopic rheological measurements of chemical and physical PVA gels were performed with a C

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Figure 4. Autocorrelation function g(2) of the PVA chemical gel (Cp = 4.4%, Rc = 0.002, dashed line) and PVA dual cross-link gel (Cp = 4.4%, Rc = 0.002, Cb = 2.5 mM, solid line) measured by DLS. Diffusion angle: 90°. Inset: q dependence of the characteristic times (chemical gel, open circles; dual cross-link gel, filled symbols).

Figure 3. Three characteristic times of the PVA solution (Cp = 4.4%) and physical PVA gel (Cp = 4.4%, Cb = 2.5 mM) as a function of diffusion vector q. Open circles: fast mode of the solution. Open squares: slow mode of the solution. Closed circles: fast mode of the physical gel. Closed triangles: intermediate mode of the physical gel. Closed squares: slow mode of the physical gel.

mode, having τf ∼ q−2; here we found −1.7. This slightly low value of the exponent is presumably due to the high heterogeneity of the gel cross-linked at a low polymer concentration close to the concentration of entanglement, Ce. It also explains partly the very low amplitude of the fast mode, indicating that the frozen-in, nonfluctuating intensity is high. The presence of the clusters in the gel also contributes a lot to the increase in the frozen-in intensity. It should be noted that there is a very clear elastic plateau even at 1 s. For the dual cross-link gel, having chemical and physical cross-links simultaneously, we observed two modes of decorrelation, fast and intermediate modes. The fast mode is similar to that of the chemical gel, τf is about 10−4 s at this angle and the exponent α was found to be −2.2. The intermediate mode is found at around 0.1 s similarly to that of the corresponding physical gel. Since we do not see this mode in the chemical gel, we attribute this mode to the dynamics due to the PVA-borate ion interaction. Although the amplitude of the intermediate mode is always as low as that in the physical gel, we can characterize this mode with sufficient precision even at borax concentration as low as 1 mM, because there is no other mode near this mode, confirmed by the presence of the very clear elastic plateau up to 1 s in the corresponding chemical gel. We found that for τi independent of q (inset of Figure 4), the exponent α is −0.0. We found also that the amplitude of this intermediate mode was proportional to the borax concentration (data not shown). Shibayama et al. reported DLS experiments on the same dual cross-link PVA gels.16 Although they characterized the gel mode in detail, they did not report the intermediate mode. In Figure 5a, we summarized borax concentration dependence of the characteristic time at 90° of the three modes found in the physical and dual cross-link gel. For the slow mode found only in the physical gels, its characteristic time τs increases exponentially with the borax concentration, similarly to the viscosity (data not shown). From about Cb = 1 mM, increase in τs is stronger, suggesting that the percolation of the transient network occurs from this concentration. For the intermediate and fast modes, borax concentration dependence is less significant compared with the slow mode. The value of τi is in the order of 0.1 s similarly to the literature (Figure 5b).13 It increases with borax concentration, and the difference between the physical and dual cross-link gel is small.

−2

mode having τ ∼ q , explained as collective fluctuations of the entangled polymer chains characterized by the diffusion coefficient Dc = (τfq2)−1. For the case of PVA, having the strong interchain hydrogen bonding, formation of large cluster, or aggregate of PVA chain due to presumably partial crystallization, is induced. Often this aggregate formation depends on the preparation, age and the thermal history of the solution. Even at ambient temperature, this interchain interaction can induce formation of cluster, the turbidity of the solution increases slowly with time. Thus, the corresponding mode in the DLS autocorrelation function is not reproducible.18 By using the value of the complex viscosity (30 Pa s) obtained by dynamic viscoelasticity measurements, RH of the aggregate is estimated to be 75 nm. In diluted solution without borax, RH was measured as 100 nm, thus these results suggest that the slow mode corresponds to Brownian diffusion of small amounts of large clusters. It should be noted that the light scattering is very sensitive to even very small amount of big cluster, it will not influence the rheological measurements in the following section. As for the intermediate mode, which is found only for the physical gels, its characteristic time τi does not depend strongly on q, it varies as τf ∼ q−0.4. Here we do not discuss the detail since the amplitude of the intermediate mode is very low and it cannot be detected for the physical gels having borax less than 2 mM, and τs can be very close to τi, thus the separation is difficult. Nemoto et al. reported intensive dynamic light scattering studies of PVA−borax systems.13 Although they did not show trimodal autocorrelation functions, their slow mode showed both q−2 and q0 dependences depending on the conditions. We consider that they are equivalent to the fast and intermediate modes observed in this work. Figure 4 shows intensity autocorrelation functions of PVA chemical and dual cross-link gels. Contrary to the solution and the physical gel, these chemically cross-linked gels are nonergodic, and the ensemble averaged autocorrelation functions do not go to zero at infinite time but reach a nonzero asymptotic value which represent the static inhomogeneity of the permanent chemical network. For the chemical gel we observed one mode of decorrelation (fast mode) at around 10−4 s for this angle. This fast mode is typical of the gel D

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Figure 6. G′ and G″ of the PVA physical gel as a function of angular frequency. Borax concentration: 2.5 mM. Solid line: G′ by DWS microrheology. Dashed line: G″ by DWS microrheology. Closed circles: G′ by classical rheometry. Open circles: G″ by classical rheometry. Figure 5. (a) Three characteristic times of the PVA physical and dual cross-link gels as a function of borax concentration. Diffusion angle: 90°. (b) Intermediate mode. (c) Fast mode. Closed symbols: physical gel. Open symbols: dual cross-link gel.

The borax concentration dependence will be discussed later in this article. For the fast mode, the values of τf show little dependence on borax concentration (Figure 5c). For the solution and the physical gels at low Cb up to 1 mM, this fast mode gives Rh of the PVA chain. We found for the solution Rh = 12 nm, a comparable value was found in the literature.28 We observed a slight increase in τf from Cb = 1 mM to 1.5 mM. From 1.5 mM the fast mode corresponds to the gel mode. For the dual crosslink gel it is also gel mode and τ f decreases with Cb monotonously. The correlation length of the gel was calculated to be 10−14 nm. For dual cross-link gels, τf decreases monotonously. This can be explained by the ionic nature of the physical cross-linking agent, borate ion. When complexed with borate ions, PVA becomes a polyelectrolyte. The gel mode of polyelectrolyte is influenced by the effect of the charge as well as physical cross-link. Both results in decrease of τf. Indeed, cross-linking decreases the chain length between the two crosslinking point and the shorter chains show faster dynamics. The dynamics of a polyelectrolyte is coupled with its counterion whose dynamics is faster than that of the polyelectrolyte, or with the increase in the degree of ionization of the polymer, τf decreases. Shibayama et al. observed similar monotonous variation of collective diffusion coefficient.16 3.2. Macro- and Microrheology. Figure 6 shows storage and loss moduli, G′ and G″, of the physically cross-linked PVA gel as a function of the angular frequency ω measured by classical macrorheology and DWS-based microrheology. The superposition of macro- and microrheology was satisfactory at the overlapped frequency range. The observed viscoelastic behavior seems Maxwellian type in the low frequency region: at lower frequency than the crossover frequency about 10 rad/s, it is found that G′ < G″ with G′ ∼ ω2, and G″ ∼ ω1. At higher frequency, showing Rouse like behavior, we observed G′ > G″, and both moduli continue to increase and another crossover was found around 105 rad/s. The comparison of macro- and microrheology for chemically cross-linked PVA gel is shown in Figure 7. Little difference of

Figure 7. G′ and G″ of PVA chemical gel as a function of angular frequency. Rc = 0.002. Solid line: G′ by DWS. Dashed line: G″ by DWS microrheology. Closed circles: G′ by classical rheometry. Open circles: G″ by classical rheometry.

the absolute value of G′ was found for this gel, while the frequency dependence is identical for the two rheological techniques. The good agreement for G″ might be accidental. Dasgupta et al. applied DWS microrheology for a chemically cross-linked polyacrylamide gel and observed similar difference in the absolute value of moduli from macrorheology.29 For nonergodic systems such as chemically cross-linked gels correct measurements of ensemble average are essential and small error can lead to shifting of the moduli (we find empirically factor of 2 at a maximum). The moduli show the frequency dependence typical for chemical gel: at low frequency the storage modulus has an elastic plateau or G′ is almost independent of ω. The gel studied here has a very small amount of chemical cross-linkers close to gel point, and was very soft, the value of G″ is not negligible. The loss modulus showed a power-law behavior G″ ∼ ω0.5 at most of the frequency range measured. At higher frequency we found crossover around 105 rad/s similarly to the physical gel. Generally, the storage and loss moduli at the gel point exhibit a power-law type frequency dependence G′ ∼ G″ ∼ ωn, and the physical origin of n is the fractal structure of the network at the percolation. Experiments on the curing of epoxy resins show that the loss modulus exhibits the same power-law G″ ∼ ωn in the entire postgel regime up to the fully cured network, while the storage modulus becomes frequency E

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independent.30 Our macro-rheological measurements during gelation process showed n = 0.6 at the gel point, then it decreases slightly to 0.5 (data not shown). The value of n = 0.6 at the gel point agrees well with the similar experiments of PVA chemical gels in the literature.15 Figure 8a shows the frequency dependence of G′ and G″ of dual cross-link gel measured by microrheology. We found that

difficult.31 Still, for swollen dual cross-link gels, the main features of the moduli mentioned above were confirmed (data not shown here). In order to examine separately the contributions of the physical and chemical cross-links in the dual cross-link gel to the viscoelasticity of the system, we assume that the two types of cross-link contribute independently. In the literature it is shown experimentally and theoretically that two types of transient cross-links can be considered as independent and described by two-mode models, expressed as the sum of the contribution of the two modes.32,33 In our system, although one of the two cross-links is chemical by nature, we apply the twocomponent model, since the chemical permanent cross-link can be considered as a transient cross-link having an infinite characteristic time. We suppose that the moduli of the dual cross-link gels (measured) have two components, one is that of the chemical cross-link which is the same as that of the chemical gel (measured separately), and the other is that of the physical cross-link in the chemical network (which is not supposed to be identical to the physical gel modulus and is thus unknown). In order to evaluate the contribution of the physical cross-link, the values of moduli of the chemical gel, GC, were subtracted from those of the dual cross-link gels, GD. The static heterogeneity of the chemical network structure can be ignored as far as we compare the dual cross-link gels with the chemical gel from which the dual cross-link gels had been prepared. As shown in Figure 9, the differences, G′D − G′C and G″D − G″C, which

Figure 8. (a) G′ and G″ of PVA dual cross-link gel as a function of angular frequency measured by DWS microrheology. Borax concentration: 2.5 mM. Chemical cross-linking ratio Rc = 0.002. Solid line: G′. Dashed line: G″. (b) G′ and (c) G″ of PVA dual cross-link gel of different borax concentrations measured by DWS microrheology. Black solid line: Cb = 2.5 mM. Black dashed line: Cb = 1.5 mM. Gray solid line: Cb = 0 mM (chemical gel).

G′ > G″ at the whole frequency range studied except for at higher frequency than 105 rad/s. G″ is characterized by a peak around 10 rad/s similarly to the physical gel. G′ increases with ω, showing a plateau around 100 rad/s. Borax concentration dependence of moduli is shown in Figure 8, parts b and c. For the storage modulus (Figure 8b), it is shown that the addition of physical cross-links by borax increases the modulus at whole frequency range measured, especially, the sharp increase around 10 rad/s becomes pronounced. The value at low frequency does not seem influenced by the borax concentration, suggesting that it is determined only by the chemical crosslinks at very low frequency range since the transient physical cross-links dissociate at low frequency. For the loss modulus (Figure 8c), although very little effect of borax is observed at high frequency, we clearly see a large increase at low frequency. Apparition of clear peak at 10 rad/s indicates that the PVA− borate ion interaction results in large dissipation at this frequency. More importantly, we observed difference between the frequency dependence of G″ for the dual cross-link gel and physical gel: at frequency lower than the peak of G″, for dual cross-link gel, we observed G″ ∼ ω0.5 while G″ ∼ ω1 for the physical gel. The latter indicates Maxwellian behavior, while the former shows a different dynamics. The coexistence of the two different cross-links does change the dynamics. Note that we did not realize macrorheological measurements at the same condition for superposing with the microrheological results: complexation with borate ion swells the chemical gel, thus preparation of dual cross-link gel for macrorheology by keeping the same PVA concentration as the chemical gel as prepared is

Figure 9. G′D − G′C and G″D − G″C as a function of angular frequency for the dual cross-link gel measured by DWS microrheology. Borax concentrations: 2.5 mM (black curves), 1.5 mM (gray curves). Chemical cross-linking ratio: 0.2 mol %. Solid line: G′. Dashed line: G″.

signify the contribution of the physical cross-links, superpose at frequencies lower than 10 rad/s, varying as G′ = G″ ∼ ω0.5. At frequencies higher than 10 rad/s, the value of G″ decreases as G″ ∼ ω−1 toward the minimum at around 103 rad/s. Increasing the borax concentration results in an increase in the moduli. At high frequency range, if the segmental Rouse mode is not influenced by the cross-links, we expect a plateau for G′ and zero G″. We observed an increase in the both moduli especially for Cb = 2.5 mM, suggesting that the complexion with borate ion modifies the segmental dynamics of PVA chains, presumably due to the introduction of charges from borate ions. Further detailed experiments are required to understand the effect of the complexation on the high frequency behavior. Our physically cross-linked systems can be considered as solutions of associative polymers, which have “stickers” to F

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associate with other polymer chains by interactions having association and dissociation time constants. The physical crosslinker borate ion serves as mobile and reversible sticker. It should be noted that unlike the usual stickers such as hydrophobic segments which associate each other, a borate ion connects two chains alone, or, borate ions themselves do not associate with. According to the theoretical studies by Indei and Takimoto,19 the presence of reversible association can slow down the segmental Rouse dynamics of the chains between two entanglement points. This “associative” Rouse mode scales with frequency in the same manner as the segmental Rouse mode, G′ = G″ ∼ ω0.5. The associative Rouse mode is found between two characteristic times: the shorter time scale is the lifetime of the association, τx, and the longer time is the characteristic time of reptation, or the time where the associating chains start to flow, τA. The key to observe associative Rouse mode is that these two characteristic times are well separated. In our physical gels we did not observe the signature of the associative Rouse mode, G′ = G″ ∼ ω0.5 at studied conditions. On the contrary in our dual cross-link gel system, we observed the associative Rouse mode at all the conditions studied. In the dual cross-link gels, no reptation occurs due to the chemical cross-link; thus, τA can be considered as infinite. Thus, the associative Rouse mode is found between the fast plateau (physical and chemical cross-links) and the second plateau (only chemical cross-links). To our knowledge, it is the first identification of the associative Rouse mode in chemically cross-linked gels. Polyacrylamide hydrogels formed by hydrophobic interactions in the presence of surfactants showed similar linear rheology.9,34 The authors suggested that presence of the power-law behavior (G′ ∼ ω0.46, G″ ∼ ω0.44) in their gels was due to the self-similar fractal structure close to gel point.34 We believe that their gels also show the associative Rouse mode, since the presence of the elastic plateau at higher frequency than the power −law regime and its high value (G′ ∼ 1000 Pa) do not suggest the fractal network close to gel point. In order to better discuss the Cb dependence of the rheological properties of the associative Rouse mode in the dual cross-link gels, those of the physical gels were also investigated. The scaling of τA and τx for the physical gels was discussed. Indei and Takimoto assumed that τA is a time necessary for the chain center to move a distance equal to its gyration radius, Rg, thus τA ∼ Rg2/D. They showed the scaling τA ∼ τxN2, where D is the diffusion coefficient of the chain, N is the average number of bound stickers per chain.19 For the scaling discussion we took D = kBT/(6πηRg), and we estimated the values of τA by using the values of complex viscosity at plateau, η, and gyration radius of noncross-linked chain (Rg = 16 nm).28 The values of τx were determined from the crossover frequency of G′ and G″. In Figure 10 we plotted τA and τx as a function of the borax concentration Cb. We remark that both τA and τx increase with Cb, scaling as τA ∼ Cb4.3 and τx ∼ Cb2.3; thus, τA scales as τA ∼ τxCb2, as predicted by Indei and Takimoto. This result indicates that our physical gel behaves as associative polymer solution, though the power law behavior (G′ = G″ ∼ ω0.5) was not observed between the two characteristic times. We found also that with decrease in Cb, the two times approach and seem to coincide with each other at about Cb = 1 mM. This concentration is supposed to be the gel point of the transient network. The gel point of the chemical gel also coincides with this cross-linking ratio (Rc = 0.002) for this PVA concentration. A detailed study on gelation will be reported elsewhere.

Figure 10. Terminal relaxation time τA calculated from the complex viscosity and gyration radius, and the network relaxation time τx for physical gels measured by classical rheometry as a function of the borax concentration.

In order to discuss in more detail the effect of Cb on τx, τx measured by DWS microrheology and classical macrorheology (reciprocal of the frequency at the peak position of G″) was plotted as a function of Cb in Figure 11 for the physical gels

Figure 11. Intermediate mode characteristic time τi measured by DLS and the mechanical relaxation time τx measured by DWS microrheology and classical rheometry, (a) of the physical gel, and (b) of the dual cross-link gel, plotted as a function of the borax concentration. Open circles: τi by DLS. Closed circles: τM by DWS microrheology. Open squares: τM by classical rheometry. Dashed line is the guide for the eye (dual cross-link gel measured by DWS).

(Figure 11a) and the dual cross-link gels (Figure 11b). For the comparison, the characteristic time of the intermediate mode observed by DLS was also plotted. For the physical gels, the values of these characteristic times were in good agreement, indicating that the DLS intermediate mode corresponds to the macroscopic stress relaxation due to the dissociation of PVA− borate ion interaction, thus no q-dependence. Nemoto et al. observed the same agreement for DLS and macrorheology, for PVA physical gels at fixed physical cross-linking ratio.13 We found the same agreement for the dual cross-link gels. We found also that the Cb dependence of τx for the dual cross-link gels is different from that of the physical gels: τx is almost constant or increases slightly with Cb. At lower Cb, τ is higher for the dual cross-link gel than that for the physical gel, at higher Cb, the values are practically the same. In order to study the efficiency of the chemical and physical cross-links, the plateau elastic modulus was evaluated as a G

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function of chemical and physical cross-linking ratio Rc (which is calculated with the total amounts of both cross-linkers, without taking account of the possible inefficient cross-linking or the chemical equilibrium of the borate ions). For the dual cross-link gels only the contribution of the physical cross-links, or the value of G′D − G′C at the plateau is plotted in Figure 12

For the efficiency of the physical cross-links, we made a rough estimation taking account of the thermodynamics of the borate ion binding. As shown in the Experimental Section, at given concentration, 82 mol % of the borate ion is estimated to be bound to a PVA chain from the binding constant (at equilibrium with 18 mol % free borate ion in the gel without binding). From the elastic modulus, the effective physical crosslink by didiol complexation was estimated to be about 10 mol %. This rough estimation shows that most of borate ions in the physical gel are bound (by monodiol complexation) but elastically inactive. For the dual cross-link gel of Rc = 0.002 at Cb = 5 mM, on a chain between two chemical cross-links consisting of 3000 vinyl alcohol monomers, in average 2.5 molecules of borate ion are bound, and among them 0.3 are elastically active. We underline that for physical gels, these inefficient crosslinkers are potentially associative, or, these elastically inactive borate ions are able to associate with any neighboring chain. As a consequence the rate of association can be high. When the rate of association is high, the chain dissociated from a physical cross-link can associate with another physical cross-linker before the constraint is relaxed. This imperfect relaxation can increase τx. From these results we propose a stress relaxation mechanism in the presence of reversible transient cross-link schematically illustrated in Figure 13. The stress relaxation due to dissociation of an elastically active transient cross-link having the characteristic time τx corresponds to the dynamics for a chain formerly associated with another chain via a transient cross-link to dissociate then reassociate to another chain. The dissociated chain can be associated with any other elastically inactive borate

Figure 12. Plateau modulus G0 of the PVA gels measured by DWS microrheology as a function of the cross-linking ratio Rc (chemical or physical). Closed squares: physical component of the dual cross-link gel. Open squares: physical gel. Open circles: chemical gel.

and compared with that of the physical gels and chemical gels. We found the physical cross-links in the dual cross-link gels give practically the same value of modulus with those in the physical gels, indicating that the small amount of chemical cross-link does not influence the complexation of borate ion to the PVA chains. We also found that for the chemical and physical cross-links the curves superpose. Because of the difference of the nature of bonding, the two cross-links are not necessarily expected to superpose, still, the physical cross-links are elastically active and give the same elasticity with the chemical cross-links in the linear regime at a shorter time scale than the lifetime of the complexation, τx. It is expected that the modulus increases with the cross-linking ratio as G0 ∼ Rc1. At higher Rc, the modulus is proportional to Rc, while at lower Rc, modulus showed lower value than this tendency. This can be explained as imperfect network formation close to the gel point, which is about Rc = 0.002 for both chemical and physical gels. The length of elastically active chain, ξ, is estimated from the plateau value of the elastic modulus of the gel: G′ = kBT/ξ3. For the chemical gel of Rc = 0.002, we found ξ = 47 nm. By using the values of the Kuhn length (b = 0.62 nm)35 and the exponent in the Marks−Houwink relation (0.63)36 for PVA in water found in the literature, the number of PVA monomers between two elastically active chemical cross-links, N, was estimated to be about 3000 (ξ = bN(0.63+1)/3). For the tetrafunctional chemical cross-links of Rc = 0.002, this value of N indicates that only 8 mol % of the cross-linker work efficiently. At this low PVA concentration close to Ce, the crosslinking reaction is very inefficient, and a loosely cross-linked network is formed having inefficient cross-links or elastically inactive cross-links such as failure to formation of tetrafunctional cross-links (cross-linker not reacted or only with one chain), as well as loop and dangling chain formation. The fact that the modulus of the physically cross-linked gel is of the same order of that of the chemically cross-linked gel suggests that the physical cross-linking is as inefficient as the chemical cross-linking. We estimated about 8−12 mol % of physical cross-links are efficient for the dual cross-link gels.

Figure 13. Schematic illustration of chain relaxation of the physical gel and dual cross-link gel. The imperfect relaxation is pronounced when more elastically inactive physical cross-links and/or physical cross-links are available in the distance ξ (indicated as circle of radius ξ). Upper row: at a low Cb, ξ is longer than the distance between the chemical cross-link, and imperfect relxation for the dual cross-link gels is mainly governed by the chemical cross-links, leading to higher τX than for the physical gel. Lower row: at a high Cb, the elastically inactive physical cross-links are dominant, the chemical cross-links do not change the breakage/reforming mechanism, and τX is practically same for the both gels. H

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ion found within the distance ξ, determined by the density of elastically active cross-link (chemical and physical). Increase in the physical cross-link concentration, Cb, has two effects. One is increase in the elastically active physical cross-link which decreases ξ thus decreases also τx. The other is the increase in the elastically inactive borate ions which increases the rate of the association thus increases τx too. Our experiments indicate that the competition of the two effects results in increase in τx, increase in Cb increases the number of the elastically inactive transient cross-linker and induces the incomplete relaxation, or the chain reassociates before complete stress relaxation. For the dual cross-link gel, this characteristic length is mainly controlled by the chemical cross-link especially at lower Cb. The chemical cross-link is permanent and presumably has stronger effect on imperfect relaxation, thus the characteristic time τx is higher than that in the physical gels. At higher Cb, we expect less difference between the dual cross-link gel and the physical gels, as observed in Figure 11. We emphasize here the difference between our physical (and dual cross-link) gels based on binding of borate ions to PVA chains and the hydrophobically modified associative polymer solutions which have been well studied. In the case of borate ions, only one borate ion is involved in an elastically active cross-link, while at least two hydrophobic stickers are required for the hydrophobic associative polymers. We expect relatively higher amount of non associating (thus elastically inactive) borate ions which can associate with any available chain, than non associating hydrophobic stickers (which are very insoluble in water) which need to associate with available stickers. Detailed experiments and theoretical modeling to clarify the difference and role of physical cross-linker mobility would be required.

cross-linkers, which can associate with any PVA chain in a certain characteristic distance determined by the elastically active cross-links (chemical and physical). It is necessary to develop a detailed theory to describe the dynamics of these PVA−borate ion hydrogels.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: (T.N.) [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is financed by Agence Nationale de Recherche (ANR) of France (ANR 2010 JCJC 0808 “Dynagel”). The authors thank François Lequeux, Christian Frétigny, and Tsutomu Indei for helpful discussion and Marine Desbois for preliminary experiments.

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REFERENCES

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4. CONCLUSIONS Three different types of poly(vinyl alcohol) (PVA) hydrogels (physical gel, chemical gel and dual cross-link gel having chemical and physical cross-links simultaneously) were successfully prepared with glutaraldehyde as chemical crosslinker and borate ion as physical cross-linker, and the dynamics of these gels were studied by microrheology, classical rheology and single dynamic light scattering (DLS). By DLS we observed a q independent mode for physically cross-linked systems (physical and dual cross-link gels) which corresponds to the macroscopic dissipation measured by macro- and/or microrheology. This relaxation derives from dissociation of transient physical cross-link (PVA-borate ion complexation). Microrheological measurements are in good agreement with macrorheological measurements giving additional information on high frequency range showing segmental Rouse mode dynamics. Compared with the physical gels, which showed Maxwell type viscoelasticity characterized by a crossover frequency (maximum of G″) and G′ ∼ ω2 and G″ ∼ ω1, the dual cross-link gel showed a particular dynamics characterized by G″ ∼ ω0.5. This relaxation mode corresponds to the associative Rouse mode, identified for the first time in a chemically cross-linked hydrogel. Dynamic behaviors of the physical gel are well described by a model proposed by Indei and Takimoto.19 We found relatively strong transient crosslinker concentration dependence of the dissociation time, τx, suggesting strong incomplete relaxation. We proposed here a stress relaxation mechanism of the physically cross-linked PVA chain. We estimate a high rate of association of the physical cross-link due to the bound but elastically inactive physical I

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