Diffusion of Polyelectrolytes in Polyelectrolyte Gels - ACS Publications

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Diffusion of Polyelectrolytes in Polyelectrolyte Gels Anand Rahalkar‡,§ and Murugappan Muthukumar*,† †

Polymer Science and Engineering, University of Massachusetts Amherst, 120 Governors Drive, Amherst, Massachusetts 01003, United States ‡ Material Measurement Laboratory, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, Maryland 20899, United States § Chemical and Biomolecular Engineering, Rice University, 6100 Main MS-362, Houston, Texas 77005, United States ABSTRACT: We have studied diffusion of polyelectrolyte probe molecules in a similarly charged gel matrix. A variety of factors that affect the polymer diffusion inside the gel were explored such as the charge density and the cross-link density of the gel as well as the solvent quality and the salt concentration of the medium. In our study, the probe is sodium polystyrenesulfonate and the gel matrix is poly(acrylamide-co-sodium acrylate) gel. Using swelling studies and rheology, we characterize these synthesized gels under various salt concentrations. The diffusion of the probe was deduced based on dynamic light scattering studies of the polymer alone and the gel alone and then on the polymer diffusing inside the gel matrix. We show that in our system the reduced diffusion coefficient for the polymer inside the gel D ∼ exp(−ϕ), where ϕ is the volume fraction of the equilibrated gel. Effects of the charge density and the cross-link density of the gel and the salt concentration and the quality of the solvent on the probe diffusion appear only through ϕ.

1. INTRODUCTION Hydrogels form an important class of gels as they resemble biological systems and have many applications in biological environments. Hydrogels are used in biological applications because of their high water content, biocompatibility, and ability to tune the properties specific to particular applications.1 Understanding transport properties of biomacromolecules in these biological materials is of interest in order to gain insights into their functioning.2 Diffusion of polymers in gels is a vital aspect of controlled drug delivery systems3 as well as other applications such as separation processes. In this paper, we study diffusion of polyelectrolytes in polyelectrolyte gels using model systems of sodium polystyrenesulfonate and polyacrylamide-co-poly(acrylic acid) gels (PAM−PAA gel) to understand the effect of confinement and electrostatics on the diffusion of the polyelectrolyte inside the gel matrix. The gel is a viscoelastic material, and usually for chemically cross-linked networks, its elastic component is dominant. Many experiments and simulations have been performed to understand the properties of polymer gels.4−21 In polyelectrolyte gels, along with the factors that contribute to gel swelling of an uncharged gel, the osmotic pressure of counterion ions contributes significantly to gel swelling. Depending on the nature of the gel, the network can be swollen orders of magnitude from its dry state. Polyelectrolyte gels swell more than uncharged gels and also undergo a drastic volume transition compared to an uncharged gel. Hydrogels can undergo volume phase transitions, changing the volume of the gel by orders of magnitude, in response to external stimuli such as solvent quality, temperature, pH, electric field, and salt for ionic gels.4−8 This volume transition can either be continuous © XXXX American Chemical Society

over the range of stimuli applied or discontinuous where the gel collapses at a critical stimulus level. The dynamics of polymers inside the gel is controlled by the local environment of the gel network. The properties of the gel network are dictated by the mesh size, which in turn depends on the extent of the gel swelling. Many factors contribute to gel swelling and deswelling, such as the solvent quality, cross-link density and charge density of the gel, salt concentration for ionic gels, etc.5,6,14,15 The swelling and volume phase transition of a polymer gel leading to collapse of the gel is essentially described by balancing the entropic, enthalpic, and elastic forces of an uncharged gel and additional component of osmotic pressure of ions for a charged gel. Tanaka pioneered the work in establishing volume transition and swelling ratios for ionic and nonionic gels.4−10 They observed that the gels undergo volume phase transition in the presence of an unfavorable solvent and when salt is added to ionic gels. For PAM−PAA gels, the ratio of water−acetone at which the gels collapsed was related to the charge density of the gel. In the absence of charges, the gel would undergo a continuous volume transition. In experiments performed by McCoy et al.,15 ionic gels collapsed in acetone−water mixtures when salt was added to the solution, increasing the effective χ parameter of the gel− solvent interaction. There are many models that describe dynamics of polymers in solutions and in confined environments.22−26 These can be broadly classified into three categories, namely, reptation, Received: June 20, 2017 Revised: September 20, 2017

A

DOI: 10.1021/acs.macromol.7b01310 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules πmix + πel + πion = 0

sieving, and entropic barrier models. In the reptation model, the diffusion of the chain is assumed to occur by essentially one-dimensional random walk along a tube generated by the constraints of the gel matrix. In this model, the diffusion coefficient D scales with the number of segments N of the polymer chain as D ∼ N−2. The theories based on the sieving concept address the obstruction effect of the gel network. The theory for effect on diffusion of the probe molecule due to the network strands was proposed by Ogston et al., where they used the size of the solute and the size of the network mesh to estimate the reduced diffusion coefficient.23 Deen and coworkers and Basak et al. used the Ogston model to predict the normalized diffusivity of the probe inside the gel.30,31 Phillies also proposed a similar model to that of Ogston, where he used the concentration of the network to describe the reduced diffusion coefficient.25 He proposed the empirical form Dt = D0 exp(−αcβ) to fit the experimental data.32,33 Cukier also proposed a similar model for decrease in diffusion of the solute in the network, using similar parameters of dry gel content and solute radius.24 Another theory put forward by Muthukumar and Baumgartner describes the process of polymer diffusion in confined environments as an entropic barrier process.26 In this theory, the polymer chain is partitioned into various compartments, and to diffuse from one location to another, it must overcome an entropic barrier which is related to the sizes of the polymer and the compartment. In this model, D = (A/N) exp(−BN), where A and B depend on the size of the compartments and the bottlenecks connecting the compartments. Predictions of the entropic barrier model were also supported by the experiments for probe diffusion in uncharged gels.30,34 Depending on the size of the polymer and the level of confinement, the polymer diffusion process in the gel matrix can be described by the following models. The Ogston model describes the weak confinement effected by the gel matrix on the polymer chains. The entropic barrier theory describes diffusion processes in which the polymer has stronger confinement than the Ogston model but weaker than the reptation model.34−37 For polyelectrolyte gels, three terms contribute to the total osmotic pressure of the gel:6,21,38

From Flory’s theory, πel is related to the shear modulus (for ϕ1/3 ≫ ϕ) as

πel = −G G = πmix + πion

⎛1 Gv1 α2 ⎞ 2 =⎜ −χ+ ⎟ϕ kBT 4v1csa ⎠ ⎝2

2 2

αϕ +

4v12cs 2

− 2v1cs

(7)

In this paper, we study diffusion of polyelectrolytes in polyelectrolyte gels. First, to study the gel properties independent of the polymer, we have synthesized the polyelectrolyte gels with controlled charge density and crosslink density of the gel. We determined the effect of solvent quality, salt concentration, and cross-link density on the gel swelling, shear modulus of the gel, and the gel diffusion coefficient. Finally, to examine the effect of gel composition on the diffusion of polymer inside the gel, we synthesized the gel and the polymer together and equilibrated the gel in the polymer solution. Then using dynamic light scattering, we probed the effect of solvent quality, salt concentration, crosslink density of the gel, and charge density of the gel on the polymer diffusion inside the gel.

2. MATERIALS AND METHODS Materials. Acrylamide (40% w/v) and bis(acrylamide) (2% w/v) solutions were bought from Amresco and used without further purification. Tetramethylethylenediamine (TEMED), ammonium persulfate, and sodium acrylate were purchased from Sigma-Aldrich and used as received. Sodium polystyrenesulfonate (NaPSS) of different molecular weights was purchased from Scientific Polymers (catalog numbers 621, 622, 624, and 574), and solutions were prepared using deionized water, acetone, and sodium chloride (purchased from Sigma-Aldrich). Four different molecular weights of NaPSS were used in this study (Mw = 16K, 33K, 70K, and 127K g/ mol) ranging more than order of magnitude in molecular weight. In the rest of the paper, NaPSS 127K g/mol molecular weight will be abbreviated as NaPSS127k. Gel Synthesis. Polyacrylamide-co-sodium acrylate gels were synthesized using free radical polymerization as described elsewhere.15 The total weight of monomers (acrylamide (WAc), sodium acrylate (WNaAc), and bis(acrylamide) (WBis)) in these gels per 100 mL of the solution (%T) was fixed to 4%. The cross-link density of the gel (%C) and the molar ratio of the cross-linker bis(acrylamide) (MBis) to total moles of the monomers (acrylamide (MAc), sodium acrylate (MNaAc), and bis(acrylamide) (MBis)) were varied as required. The charge density of the gel, defined as the molar ratio of the sodium acrylate to the acrylamide, was varied as required.

(1)

(2)

osmotic pressure due to mobile ions (πion) πionv1 = kBT

(6)

When 2v1cs ≫ αϕ and ϕ ≪ 1, we get from eqs 1−6

osmotic pressure due to elasticity of the network (πel) πelv1 ϕ⎞ 1⎛ = − ⎜α02ϕ1/3 − ⎟ kBT N⎝ 2⎠

(5)

.Hence

osmotic pressure due to mixing (πmix) πmixv1 = −ln(1 − ϕ) − ϕ − χϕ2 kBT

(4)

%T = (3)

WAm + WNaAc + WBis 100

%C =

MAm

MBis + MNaAc + MBis

(8) Millipore water (18.2 MΩ) is stirred with 24.6 mL of 40% (w/v) of acrylamide solution, 6.7 mL of 20% (w/w) of sodium acrylate solution, and 33.0 mL of 2% (w/v) of bis(acrylamide) solution, and 300 mg of NaPSS was added. The total volume of the solution is 300 mL. Millipore water (235.7 mL) is added to the monomer and cross-linker solution to bring the volume to 300 mL. For light scattering experiments, it is important to filter and remove dust from all pregel samples. NaPSS solutions in water were filtered using 200 nm cellulose acetate filters, and acetone was filtered using PTFE filters. Filtered solutions were then mixed to make required solutions. This pregel

The expression for the net osmotic pressure (π) is given by addition of eqs 1−3, where ϕ = volume fraction of the gel, χ = interaction parameter between the polymer and solvent, N = average number of Kuhn segments between cross-links, α03 = volume fraction of the gel where the network strands obey the Gaussian statistics, α = degree of ionization, v1 = molecular volume of the solvent, and cs = salt concentration of the system (mol/L). In equilibrium, π = 0. Hence B

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Macromolecules For a molecule undergoing Brownian diffusion

solution is bubbled with nitrogen gas for 15 min to remove any dissolved oxygen which can inhibit the reaction. To polymerize the reaction, TEMED (450 μL) and ammonium persulfate (1.5 mL) are added to the pregel solution. The solution is stirred for 1 min after addition of the initiator and poured into a tray (1 mm thickness) and casted for 2 h at room temperature. The tray is covered with a top plate to prevent wrinkling of the gel. The gel is removed from the tray and cut into pieces. The gel pieces are placed in reservoirs of NaPSS (same concentration as that in gel) and NaCl concentration to equilibrate the gel. A light scattering tube was thoroughly rinsed with soap water first and then with acetone to remove any dust in the tube. The equilibrated gel was added to the tube. Filtered solution of NaPSS (same concentration as in gel) and NaCl was added to the tube in excess amount on top of the gel to maintain equilibrium between the gel and the solution. In this work three gels are used. The nomenclature and its corresponding charge density and cross-link density are described in Table 1.

2

g1(τ ) = e−Dq τ

where D is the diffusion coefficient. The diffusion coefficient is related to the hydrodynamic radius from Stokes−Einstein relationship k T D = 6πηB R . On the other hand, the gel network does not undergo h

Brownian diffusion as in the case of a polymer solution. However, the electric field correlations are instead related to the correlations of the displacement vector of the network (Uq). According to the theory of Tanaka et al.17 2

g1(τ ) ∼ ⟨Uq(0)Uq(t )⟩ = ⟨Uq(0)⟩2 e−Mq t / f

charge density (mol %)

cross-link density (mol %)

10C2.7Xl 10C3.5Xl 5C2.7Xl

10 10 5

2.7 3.5 2.7

g2(τ ) = X2(g1(τ ))2 + 2X(1 − X )g1(τ )

Swelling Ratio. The gels were characterized by using the swelling ratio measurements to calculate the volume fraction of the polymer in the gels and estimate the average molecular weight between cross-links using the Flory−Rehner (F−R) theory with an additional component of Donnan potential.21 The volume fraction of the gel (ϕ) is given by ϕ=

Vdry Vswollen

−1 ⎡ ρp ⎛ M ⎞ ρp ⎤ = ⎢1 + ⎜ s ⎟ − ⎥ ⎢⎣ ρs ⎝ Md ⎠ ρs ⎥⎦

where X =

N=

− ln(1 − ϕ) − ϕ − χϕ2 +

4π n λ

sin

⟨IF⟩ is the intensity of the fluctuating component,

g2(τ ) = 1 + β exp(− 2DA q2τ )

(14)

where DA is the apparent elastic diffusion coefficient related to the true elastic diffusion coefficient (Dgel) through the factor X

(9)

DA = Dgel /(2 − X )

(15)

For the gels investigated here, X is about 0.56. In general, the correlation functions are fitted with a multiexponential function as shown in eq 16, where τ is the delay time in measuring the correlation function.

ϕ 2 2 2

g1(τ ) = A1 exp(−Γ1τ ) + A 2 exp(−Γ2τ )

α ϕ + 4v12cs 2 − 2v1cs

(16)

where Γ1 and Γ2 are the characteristic inverse decay times for the two modes where A1 and A2 are amplitudes. The corresponding diffusion coefficients can be determined by using eq 17 for Γ1 and Γ2 vs q2.

(10) For 10C2.7Xl gel swollen in water, α = 0.1, χ = 0.48, α0 = 0.74,v1 = 18 mL/mol, and cs is the salt concentration in mol/m3. Rheology. Shear modulus experiments were performed on the gels to measure their shear storage and shear loss moduli. A straincontrolled rheometer (ARES TA Instruments) with 40 mm diameter parallel plate was used to measure the modulus. The measurements were performed in the linear elastic region by applying a frequency sweep of 0.1−1 Hz with a constant shear strain of 1.0%. The frequency sweep was performed in this limited range due to instability of the gel at higher frequency and water loss at lower frequencies as has been reported.25 Dynamic Light Scattering. For light scattering experiments, an ALV-5000/E correlator was used. The scattering wave vector (q) is defined by q =

⟨IF⟩ , ⟨IF⟩ + ⟨Is⟩

(13)

and ⟨Is⟩ is the intensity of the static component. Thus

where vdry = volume of the dry gel, vswollen = volume of swollen gel, ρp = density of the polymer, ρs = density of the solvent, ms = weight of the swollen gel, and md = weight of the dry gel. And we can calculate molecular weight between cross-links (N) from eqs 1−4.

α0 2ϕ1/3 −

(12)

where M is the longitudinal modulus and f is the friction coefficient between the gel strands and the solvent. Because of similarity between eqs 11 and 12, we can identify a “diffusion coefficient” for the gel, where Dgel = M/f. The longitudinal modulus M is related to the bulk modulus K and the shear modulus G as M = K + (4/3)G. For polymer gels, due to the presence of the permanent cross-links in the network and the unavoidable structural inhomogeneity created during the gel synthesis, the gel can be nonergodic. The gel is constrained due to these inhomogeneities, and hence the chains between cross-links can only fluctuate about their mean position. Because of this pseudofixed position of the scattering elements in the system, the scatterers in the constrained regions cannot fully explore phase space. These frozen-in domains contribute to a static scattered intensity ⟨Is⟩, which results in a modified expression for relation of intensity correlation function (g2(τ)) to the electric field correlation function (g1(τ)):17,40−48

Table 1. Nomenclature of the Gels Used in This Work and Their Corresponding Charge Density and Cross-Link Density nomenclature

(11)

Γ1 = D1q2

and

Γ2 = D2q2

(17)

The data fitting is done using minimization of error between the fit prediction and the data. The obtained correlation function (g2(τ) − 1) was fitted in MATLAB using multiterm exponential function as shown in eqs 16 and 17.49 We obtain residuals from the difference between the original data and the fitting curve (iterations were performed until the fitting curve obtained was within the tolerance limit). It is important to analyze the data by using multiple exponential fits and ensure the residuals are not biased; that is, they are randomly distributed about the mean of zero and not have systematic fluctuations about its mean.

( θ2 ), where λ is 514.5 nm, the wavelength of

argon laser used, n is the refractive index of the solution, and θ is the angle between the incident and scattered light. The experimentally measured intensity correlation function (g2(τ)) is related to the electric field correlation function (g1(τ)) by the Siegert relationship:39 g2(τ ) = 1 + β(g1(τ ))2 where β is the coherence factor.

3. RESULTS AND DISCUSSION We find that the gel volume fraction (ϕ) controls the diffusion of the probe polymer inside the gel. The mesh size of the gel is inversely related to the gel volume fraction and can be tuned C

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Macromolecules with charge density and the cross-link density of the gel as well as by changing the solvent quality (acetone−water system) and the salt (NaCl) concentration of the solution. Thus, we synthesize gels with controlled charge density and cross-link density and performed swelling studies to determine the effect solvent quality, salt concentration, and cross-link density of the gel on the swelling of the gel. We have used rheological measurements to establish the elastic nature of these gels and observe the gel characteristics for frequency sweep measurements. Finally, we report dynamic light scattering studies on the gel alone and the polymer alone in different solvent qualities and salt concentrations and later with polymer diffusing inside the gel for the same solvent qualities and salt concentrations. We observe that the effect of gel charge density, gel cross-link density, solvent quality, and the salt concentration of the system converges to the gel volume fraction, and that controls the polymer diffusion inside the gel. 3.1. Swelling Ratio. The primary purpose of the swelling measurements is to establish the dry polymer concentration of the equilibrated gel matrix when the probe polymer undergoes diffusion. As has been discussed earlier, for polyelectrolyte gels, multiple factors contribute to gel swelling. Primary factors are the charge density of the gel, the cross-link density of the gel, χ parameter between the network and the solvent, and the salt concentration of the system as seen in eqs 1−4. At zero salt concentration, due to the Donnan equilibrium, the mobile counterions exert pressure on the gel which leads to gel swelling. When the salt is added to the system, the ion pressure decreases, resulting in deswelling of the gel. Also, when the cross-link density of the gel increases, the molecular weight between cross-links decreases. Thus, due to elastic constraints on the network strands, higher cross-link density gels do not swell as much. The gel swelling is inversely related to volume fraction of the gel, which can be experimentally measured using eq 9. In Figure 1, we show the effect of solvent quality on the gel volume fraction, and in Figure 2, we show the effect of cross-link density of the gel on the gel volume fraction.

Figure 2. Effect on swelling of gel as a function of NaCl concentration in 55% acetone solution for gels of same charge density (10%) but different cross-link density (2.7% and 3.5%). As more salt is added to the system, the Donnan contribution decreases, resulting in higher volume fraction (ϕ) of the gel. Higher cross-link density gel has a higher volume fraction at all NaCl concentrations. The black and red arrows indicate the threshold values of NaCl concentration at which the gel collapses for the 2.7% and 3.5% cross-link density, respectively.

synthesized (ϕ0 is constant). As the gel swells, ϕ decreases; hence, ϕ/ϕ0 is less than 1. As salt concentration increases, the gel deswells and ϕ increases; hence, the ratio ϕ/ϕ0 increases. However, upon gel collapse, the gel excludes almost all the solvent previously present inside the network, and its volume shrinks; consequently, ϕ/ϕ0 is greater than 1. The 10C2.7Xl PAM−PAA gel collapses at 11 mM NaCl concentration in 55% acetone solution but does not collapse at the same NaCl concentration in water as shown in Figure 1. The gel volume fraction at 10 mM NaCl in water is 3.6 × 10−3, and it increases to 9.6 × 10−3 in 10 mM NaCl concentration in 55% acetone solution. Since acetone is a poor solvent for the acrylamide gel backbone, the gel deswells in the presence of acetone. Thus, for same NaCl concentration, the gel in acetone−water mixture is less swollen than in water and hence has a higher volume fraction. The ratio of the gel volume fraction at 10 mM NaCl compared to its reference state (ϕ/ϕ0) is 0.27, and it drastically increases to 3.74 at 11 mM NaCl concentration in in 55% acetone solution, as the gel collapses. As the gel collapses, it expels the solvent from the gel network. Consequently, its volume fraction increases drastically compared to a gel yet to undergo collapse transition. Hence, the ratio of ϕ/ϕ0, is greater than 1 when the gel collapses. In Figure 2, we show the dependence of gel volume fraction on the added salt concentration as well as the effect of crosslink density of the gel. When the salt concentration of the solution increases, the contribution of the osmotic pressure from the ions in the gel decreases. Thus, the gel deswells, resulting in higher volume fraction of the gel. Increasing the cross-link density of the gel imposes restriction on gel swelling; consequently, a higher cross-link density gel does not swell as much compared to lower cross-link density gel. Hence, the volume fraction of 10C3.5Xl gel (3.5% cross-link density) is higher than the 10C2.7Xl gel (2.7% cross-link density) as seen in Figure 2. 3.2. Rheology. The goal of these rheological measurements is to ensure that the synthesized gels are robust and mechanically stable and exhibit all the expected characteristics of a gel. Typical characteristics of gel observed in rheological experiments are G′ > G″, and the shear modulus is independent of the frequency at which it is measured. To determine the shear modulus of the polyelectrolyte gels, we performed frequency sweep measurements. As seen in Figure 3, we

Figure 1. Volume transition of 10C2.7Xl PAM−PAA gel (10% charge density and 2.7% cross-link density) in the presence of 55% acetone solution at 11 mM NaCl concentration. The gel does not collapse at 11 mM NaCl concentration in the presence of water as solution. Inset shows the expanded scale of the volume fraction of the gel when the gel is not yet collapsed (ϕ0 = 0.4).

It was shown by Ohmine et al.5 that these gels undergo volume phase transitions in the presence of unfavorable solvent and at a salt concentration and later also by McCoy et al.15 In Figure 1, we plot the ratio of ϕ/ϕ0 vs NaCl concentration, where ϕ represents the volume fraction of the gel at the given salt concentration and ϕ0 (= α03) represents the volume fraction of the gel in the reference state in which the gel was D

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observe two diffusive modes in light scattering of a gel + polymer system in high salt concentrations: a polymer mode, which is affected due to the presence of the matrix surrounding it, and a gel mode, which is not affected by the polymer chains inside the gel due to very small concentration of the polymer. Next, we present light scattering data for the water−acetone system, where inevitably we deal with lower salt concentration solutions (since the gel collapses at higher salt concentration in the water−acetone mixture). In these low salt concentration solutions, we observe the multiple modes for the gel due to its heterogeneities and for the NaPSS solutions due to the presence of aggregates. Since the presence of the slow mode in the gel or in the NaPSS solution is dependent on the salt concentration, we present examples of light scattering in high salt conditions and low salt conditions to address and identify these multiple modes in each case. 3.3.1. Gels. As discussed earlier, light scattering of gels captures the fluctuations of the network, and as a result, the diffusion coefficient of the gel is proportional to the longitudinal modulus of the gel and inversely proportional to the friction coefficient between the gel strands and the solvent. The correlation functions were analyzed using multiexponential fits. Correlation functions of gels often do not show a singleexponential decay because of the heterogeneities present in the gel. Hence, we first discuss these gel modes and their diffusive and nondiffusive behavior. The diffusive mode persists at all angles and has q2 dependence, while the nondiffusive mode is not necessarily present at all angle and does not have a q2 dependence. In the light scattering analysis of gels, the modes corresponding to heterogeneities in the system also contribute to the scattering intensity and the corresponding intensity correlation function. These modes appear at longer time scales than the characteristic relaxation mode of the gel and does not have a q2 dependence. All correlation functions presented in this work are shown without any normalization. For gels, the amplitude of the intensity correlation function g2(τ) is often less than that observed for a polymer solution because of builtin heterogeneities. At higher salt concentrations, as the gel deswells the mesh size decreases, leading to a more homogeneous gel structure (this does not mean the gel becomes homogeneous; it is more homogeneous compared to its homogeneity without salt). Thus, the non-q2-dependent slow mode which occurs due to heterogeneity disappears at high salt concentrations. As seen in the Figure 4, at higher NaCl concentration, the correlation function can be fitted using single-exponential decay for the correlation function and the residuals confirm this result.

Figure 3. Effect on modulus of 10C2.7Xl gel as a function of NaCl concentration. Increase in the concentration of NaCl increases the modulus of the gel; G′ (filled circles) and G′′ (filled squares).

observe that the storage modulus is an order of magnitude greater than the loss modulus, indicating the elastic nature of these gels. The storage modulus is also independent of frequency as is expected for a chemically cross-linked network.50 As more salt is added to the system, the gel deswells and its volume fraction increases, leading to an increase in the modulus of the gel. We have observed similar characteristics for the elastic behavior of the gel for the 10C3.5Xl gel. As an example, the G′ = 678 Pa for the 10C3.5Xl gel and G′ = 466 Pa for the 10C2.7Xl gel for salt-free conditions. As the cross-link density increases, the gel becomes stiffer and hence higher cross-link density gel has higher modulus. 3.3. Dynamic Light Scattering. In this section, we report the slowing down of the probe diffusion in the gel in comparison with its diffusion in the corresponding solution without the gel network. We find that the key factor in controlling the probe polymer diffusion inside the gel is the gel volume fraction (ϕ). We tune ϕ with charge density and the cross-link density of the gel as well as by changing the solvent quality (acetone−water system) and the salt (NaCl) concentration of the solution. In general, multiple modes are observed in dynamic light scattering of these systems. In order to correctly identify and quantify the diffusive behavior of the probe inside the gel, we have systematically conducted DLS measurements on the polymer alone and the gel alone under different conditions of the solvent quality and the salt concentration and the polymer + gel system under the same solvent quality and salt concentration condition. We first present light scattering data on gels and probe individually, and when mixed in high NaCl concentration solution in water, to clearly interpret the modes. We clearly

Figure 4. (a) Correlation function for the 10C2.7Xl gel + 100 mM NaCl solution for the scattering angle of 30°. The correlation function is fitted with a single exponential decay. (b) The accuracy of the fit can be seen from the residuals. (c) The plot of Γ vs q2 is shown. The slope of the fit is the diffusion coefficient of the gel. E

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the gel takes place as seen in Figure 6. This was also observed by McCoy et al. for the same gel system, where the phenomena are attributed to the structural changes in the gel prior to the imminent volume phase transition.15 As the cross-link density of the gel increases, the collapse transition occurs at higher salt concentration as shown in Figure 1. Thus, for the 10C3.5Xl gel, the gel diffusion coefficient decreases by 2 orders of magnitude at higher NaCl concentration compared to the 10C2.7Xl gel as seen in Figure 6. For the 10C3.5Xl gel, the volume phase transition is observed 16 mM NaCl as indicated by the red arrow in Figure 2. 3.3.2. NaPSS Solutions. We characterize the probe alone, gel alone, and the probe diffusing inside the gel at the same NaCl concentration individually to understand the impact on decreased diffusion coefficient of the probe inside the gel. An advantage of using high salt concentration for light scattering of NaPSS solution is the avoidance of the polyelectrolyte slow mode.51,52 3.3.2.1. High NaCl Concentration. In high salt concentrations, the slow polyelectrolyte mode disappears, and only one mode is observed. Thus, using high NaCl concentration, where the gel alone has only one mode (which is q2 dependent) and the probe alone also has only one mode (which is q2 dependent), helps in understanding the gel + probe system better as discussed below. One mode is present when light scattering measurement is performed on the polymer alone in 100 mM NaCl solution as seen in Figure 7. Similarly, the gel alone in 100 mM NaCl solution also has only one mode as has been discussed earlier. Now we discuss light scattering data of NaPSS in low salt concentrations. 3.3.2.2. Low NaCl Concentration. In case of low salt concentrations, an aggregate mode is observed in addition to the fast mode observed in Figure 7. The correlation function and the q2 dependence of the two modes for 6 mM NaCl concentration in 55% acetone are shown in Figure 8. At low NaCl concentrations, the gel alone sample has two modes in the presence of 55% acetone solution. One mode is q2 dependent while the other mode does not have a q 2 dependence. In the NaPSS only sample, two modes which are both q2 dependent are observed. Hence, when we perform light scattering experiments on diffusion of polymer inside the gel at low NaCl concentrations, the presence of multiple modes in the gel and polymer by themselves convolutes the analysis. To understand this complex scenario, we first discuss experiments on polymer diffusion inside the gel at high NaCl concentration in the next section. 3.3.3. Polymer Inside the Gel. 3.3.3.1. High NaCl Concentration. Two diffusive modes are present when the probe polymer is added to the gel as seen in Figure 9. Both the probe polymer and the 10C2.7Xl gel in 100 mM NaCl solution show only one mode (shown in Figures 7 and 4, respectively). In Figure 9, the correlation functions of the 10C2.7Xl gel + NaPSS in 100 mM NaCl solution, the residuals from the fit, and the Γ vs q2 for the gel mode and probe polymer mode are shown. We observe that an additional mode is present when the probe polymer is added to the gel. Other researchers have observed an additional mode when a probe molecule was introduced to the gel. Shibayama et al. observed only one mode for PNIPAM gels, and on addition of probe to the gel, another diffusive mode at longer time scales was observed, which they attributed to probe diffusion inside the gel.37 Pajevic and coworkers studied the diffusion of PMMA in PMMA gel; in

At low salt concentrations, multiple decay times can be observed at each angle. However, as shown in Figure 5, only

Figure 5. Correlation function for the 10C2.7Xl gel + 55% acetone + 6 mM NaCl solution. (a) The correlation function for the scattering angle of 30° is fitted with two exponential decays. (b) The accuracy of the fit can be seen from the residuals. (c) The fast mode Γ1 has a q2 dependence, and (d) the slow mode Γ2 does not have a q2 dependence.

one diffusive mode is observed, and this fast mode corresponds to the gel diffusion coefficient. The slower, non-q2-dependent mode is present at lower salt concentration. This extra mode may arise due to the inhomogeneity present in the gel or because of an unfavorable solvent for the gel such as acetone. This has been reported previously in the literature.51,52 At lower salt concentrations in water, such as 1−10 mM, we observe multiple modes for the gel. The presence of multiple modes for a gel is strongly dependent on the salt concentration and hence is observed in low salt concentrations for both water and acetone−water mixtures. Of importance to us is the q2-dependent fast mode. In Figure 6, we discuss this mode for the effect of salt concentration and the cross-link density of the gel. The 10C2.7Xl gel undergoes volume phase transition at 11 mM NaCl concentration as shown in Figure 1 (and indicated by the black arrow in Figure 2). The diffusion coefficient of the gel decreases by about 2 orders of magnitude just before the volume phase transition of

Figure 6. Elastic diffusion coefficient for the 10C2.7Xl gel + 55% acetone + NaCl solution and the 10C3.5Xl gel + 55% acetone + NaCl solution. The elastic diffusion coefficient decreases as NaCl is added to the solution. Just before the volume phase transition, the diffusion coefficient of the gel decreases by 2 orders of magnitude. Higher crosslink density gel collapses at higher salt concentration. F

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Figure 7. Light scattering data for the NaPSS127k + 100 mM NaCl. (a) Correlation function for the scattering angle of 30° exhibits one diffusive mode. Black line corresponds to the correlation function, and the red line is the data fitting for single mode. (b) Residuals of the single mode data fitting. (c) The diffusion coefficient is calculated from the slope of the linear fit of Γ vs q2 plot.

Figure 8. NaPSS127k + 55% acetone + 6 mM NaCl. (a) Correlation function for the scattering angle of 30° shows two diffusive modes. Black line corresponds to the correlation function, and the red line is the data fitting for two modes. (c, d) The fast mode Γ1 and the slow mode Γ2 both have a q2 dependence. The diffusion coefficient is calculated from the slope of the linear fit of Γ vs q2 plot for each mode.

Table 2. Comparison of Diffusion Coefficient from Light Scattering Experiments for the 10C2.7Xl Gel in 100 mM NaCl Solution, NaPSS in 100 mM NaCl Solution, and 10C2.7Xl Gel + NaPSS in 100 mM NaCl Solution sample

D1 × 10−7 (cm2/s)

D2 × 10−7 (cm2/s)

gel +100 mM NaCl NaPSS + 100 mM NaCl gel + NaPSS + 100 mM NaCl

4.5 2.7 4.1

0.1

not affected by the probe polymer chains inside the gel due to very small concentration of the probe polymer. 3.3.3.2. Low NaCl Concentration. At high salt concentrations, the gel and the NaPSS polymer do not exhibit a slow mode. However, at lower salt concentrations, the gel as well as the NaPSS polymer exhibits a fast mode and a slow mode. Now we discuss the diffusion of the polymer in the gel at lower salt concentrations. From Figure 10, we unsurprisingly observe that the gel + NaPSS sample has two modes that are q2 dependent and hence diffusive in nature. The fast mode corresponds to the gel diffusion coefficient in the sample as we had seen in the high salt case (Figure 9). The slower mode of the gel + NaPSS sample thus corresponds to the probe polymer chains diffusing in the gel. The diffusion coefficient of the gel and the probe polymer is ascertained from the slope of the Γ vs q2 plot. Also, the slow mode seen in NaPSS solutions, argued (ref 53 and references therein) to be due to aggregates of about 100 nm size, is absent when NaPSS is present inside a gel due to the mesh size being smaller than the aggregate size. Thus, the probe polymer diffusion inside the gel is affected by the constraints from the gel matrix. The probe polymer diffusion coefficient in the solution (fast mode) is D0 = 4.4 × 10−7 cm2/s. The probe polymer diffusion decreases by an order of magnitude inside the gel matrix (slow mode) to D2 = 1.5 × 10−8 cm2/s. Now we show the dependence of the probe polymer diffusion inside the gel on the charge density and the

Figure 9. Light scattering data for the 10C2.7Xl gel + NaPSS127k + 100 mM NaCl. (a) Correlation function for the scattering angle of 30° shows two diffusive modes. Black line corresponds to the correlation function, and the red line is the data fitting for two modes. (b) Residuals of the two mode data fitting. (c, d) The gel mode Γ1 is faster and has a q2 dependence. The probe polymer mode Γ2 is slower inside the gel and has a q2 dependence. The diffusion coefficient is calculated from the slope of the linear fit of Γ vs q2 plot for both modes.

addition to the modes observed from the gel, they observed an additional mode which they attributed to the probe diffusion.27,28 It is also important to note the two diffusion coefficients from the gel + NaPSS light scattering data are an order of magnitude apart. The diffusion coefficient of the second mode is an order of magnitude different from the diffusion coefficient of the gel alone or the probe alone, which has also been observed by others.27−29,37 Diffusion coefficients from all three samples are summarized in Table 2. We expect two diffusive modes in light scattering of a gel and polymer system. A polymer mode, which is affected due to the presence of the matrix surrounding it, and a gel mode, which is G

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Figure 10. Comparison of correlation function and diffusive modes for the gel and the probe individually and when combined in 55% acetone + 6 mM NaCl concentration. (a) The correlation functions for samples 10C2.7Xl + 55% acetone + 6 mM NaCl, (d) NaPSS127k + 55% acetone + 6 mM NaCl, and (g) 10C2.7Xl + NaPSS127k + 55% acetone + 6 mM NaCl. The second and third columns are Γ vs q2 plots for the same samples. First row: 10C2.7Xl + 55% acetone + 6 mM NaCl. (a) Correlation function with two modes. Black line corresponds to the correlation function, and the red line is the data fitting for two modes. (b, c) The diffusion coefficient is calculated from the slope of the linear fit of Γ vs q2 plot. The second mode does not have a q2 dependence. Second row: NaPSS127k + 55% acetone + 6 mM NaCl. (d) Correlation function with two diffusive modes. Black line corresponds to the correlation function, and the red line is the data fitting for two modes. (e, f) The diffusion coefficient is calculated from the slope of the linear fit of Γ vs q2 plot for each mode. Third row: 10C2.7Xl + NaPSS127k (5 mg/mL) + 55% acetone + 6 mM NaCl. (g) Correlation function with two diffusive modes. Black line corresponds to the correlation function, and the red line is the data fitting for three modes. (h, i) The diffusion coefficient was calculated from the slope of the linear fit of Γ vs q2 plot for each mode (third mode does not have a q2 dependence and is not shown). The scattering angle is 30° in (a), (d), and (g).

weight of the probe. Thus, we infer that the 10C2.7Xl gel does not impose strong confinement on the probe polymer diffusion. To measure the effect of cross-link density, the probe diffusion coefficient in the polyelectrolyte gel is measured inside gels of two different cross-link densities (2.7% and 3.5%) while maintaining the charge density constant (10%), ensuring that the electrostatic interaction between the negatively charged probe polymer chain and the gel matrix backbone which is also negatively charged remains constant. As the cross-link density of the gel increases, the mesh size decreases for a constant charge density. Thus, 10C3.5Xl has smaller mesh size compared to 10C2.7Xl. The probe diffuses slower in higher cross-link density 10C3.5Xl gel (smaller mesh gel) compared to the 10C2.7Xl gel. The size of the probe compared to mesh size of the gel is important in deciding whether the probe faces any barriers moving through the matrix. As seen in the lower cross-link density gel, in Figure 11a, the probe diffusion in the gel is not sensitive to the molecular weight of the probe. But when the cross-link density of the gel is increased, the effect of molecular weight on the probe is seen in Figure 11b. The mesh of the 10C2.7Xl gel is not small enough to impose sufficient constraints on probe polymer diffusion on the gel. But on increasing the cross-link density of the gel, the 10C3.5Xl gel, which has smaller mesh size, imposes more constraints on probe polymer diffusion that the effect of molecular weight of the probe is observed. The percent

cross-link density of the gel as well as the solvent quality and the salt concentration of the solution. 3.4. Diffusion Coefficient of the Polymer in the Gel. In this section, we discuss the effect of cross-link density, charge density, and volume fraction of the gel as well as the molecular weight of the probe polymer on the diffusion coefficient of the probe polymer inside the gel. The diffusion coefficient of the second mode (D2) obtained from light scattering of gel + NaPSS is discussed here. Figure 11a shows diffusion coefficient of the probe inside the gel as a function of molecular weight of the probe. The probe diffusion inside the gel decreases as NaCl is added. As NaCl concentration is increased, the gel deswells due to Donnan equilibrium. Thus, addition of NaCl effectively decreases the mesh size of the gel, resulting in greater hindrance for the diffusion of the probe and in slower diffusion in the gel matrix. The effect of molecular weight of the probe on the diffusion coefficient of the probe inside the gel is also shown in Figure 11. As the molecular weight of the probe increases, the probe size increases. For a fixed NaCl concentration, the mesh size of a gel remains constant. Thus, changing the molecular weight of the probe for fixed NaCl concentration results in increased levels of confinement for the higher molecular weight probes. One can expect the probe diffusion to decrease because of more confinement on the higher molecular weight probe. But from Figure 11, as the size of the probe increases, its diffusion inside the gel has a weak or negligible dependence on the molecular H

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The diffusion of NaPSS in 5C2.7Xl is slower than in 10C2.7Xl at all molecular weights of the probe and all NaCl concentrations. As the 5C2.7Xl gel has lower charge density, the gel does not swell as much as 10C2.7Xl. Hence, it has higher dry gel content, resulting in smaller mesh size. 3.5. Effect of Volume Fraction. Various models have been proposed to explain diffusion of probe polymers in concentrated solutions or in confined systems. These models are dependent upon the concentration of the polymer−solvent system or that of confinement. There are three main theories that describe diffusion of polymer in constrained environments. The Ogston model applies to lowest confinement level. For higher level of confinement than the Ogston regime, the entropic barrier model describes the diffusion process. And for even higher confinements imposed on polymer diffusion, the reptation model is used to describe the diffusion process. From Figure 11, we observe that the probe polymer diffusion inside the gel has a weak dependence on the polymer molecular weight. Furthermore, the hydrodynamic radius Rh of NaPSS in NaCl solutions is in the range of 3−5 nm for the four NaPSS molecular weights, as discerned from DLS data and using the Stokes−Einstein law. On the other hand, the estimated value (ref 20) of the mesh size is about 17 nm for the 10C2.7X1 gel, as an example. As a result, we are inclined to describe the diffusion process for the given system using the Ogston model. Ogston et al. used solute size as well as size of the gel mesh to estimate slowing of solute diffusion inside the gel,23 where the diffusion coefficient of the probe in the gel (D2) is given by D2 = D0 exp(−αϕ), where ϕ represents volume fraction of the dry gel content. Many experiments using FCS, to study diffusion of small molecules in swollen gel networks, used similar model to explain their data.33 From Figure 12, we observe that the normalized diffusion coefficient of the NaPSS in the gel can be explained by an

Figure 11. Diffusion coefficient of NaPSS diffusion in the gel in 55% acetone + NaCl solution. Effect of molecular weight of NaPSS and NaCl concentration of probe diffusion is shown for different gel compositions. (a) Diffusion coefficient of NaPSS in the 10C2.7Xl gel in 55% acetone + NaCl solution. (b) Effect of cross-link density: diffusion coefficient of NaPSS diffusion in the 10C3.5Xl gel in 55% acetone + NaCl solution. (c) Effect of charge density: diffusion coefficient of NaPSS diffusion in the 5C2.7Xl gel in 55% acetone + NaCl solution.

decrease in the higher cross-link density gel (10C3.5Xl) is greater than in the lower cross-link density gel (10C2.7Xl). To measure the effect of the charge density, the probe diffusion coefficient in the polyelectrolyte gel is measured inside gels of two different charge densities (5% and 10%) while maintaining the cross-link density of the gel constant (2.7%), so that the molecular weight between cross-links remains constant. The effect of the electrostatic repulsion between the negatively charged polymer chain and the gel matrix backbone which is also negatively charged is observed in Figure 11c. Chain entropy and electrostatics both contribute to the decrease in the diffusion coefficient of the probe polymer chain inside the gel mesh. Changing charge density of the gel affects the probe diffusion through two processes: change in mesh size of the gel due to Donnan equilibrium and change in electrostatic repulsion between the probe and gel matrix (as both are negatively charged). Since the Debye length is extremely small at the given salt concentration and the solvent composition, the contribution due to electrostatics is negligible. Thus, the mesh size of the gel plays an important role in dictating the diffusion of the probe polymer chain inside the polymer gel.

Figure 12. Diffusion coefficient of NaPSS in the gel (D2) in NaCl solution normalized by diffusion coefficient of NaPSS in the NaCl solution (D0). The normalized diffusion coefficient is linear to volume fraction ϕ in the semilog plot. The normalized probe diffusion coefficient represents different gel systems: (●) the 10C2.7Xl gel in the presence of acetone−water mixture and NaCl; (■) the 10C2.7Xl gel in the presence of NaCl; (▲) the 10C3.5Xl gel in the presence of acetone−water mixture and NaCl; and (▼) the 5C2.7Xl gel in the presence of acetone−water mixture and NaCl. Plotted line serves as a guide to the eye.

equation proposed by Ogston, D2 = D0 exp(−αϕ). The power dependence of D2/D0 vs ϕ is not commented on because of the small range of ϕ that is experimentally accessible. Thus, the diffusion process in the weak confinement regime is explained by the Ogston model. I

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(5) Ohmine, I.; Tanaka, T. Salt effects on the phase transition of ionic gels. J. Chem. Phys. 1982, 77 (11), 5725. (6) Tanaka, T. Collapse of Gels and the Critical Endpoint. Phys. Rev. Lett. 1978, 40, 820. (7) Shibayama, M.; Tanaka, T. Small angle neutron scattering study on weakly charged poly(Nisopropyl acrylamidecoacrylic acid) copolymer solutions. J. Chem. Phys. 1995, 102 (23), 9392. (8) English, A. E.; Tanaka, T.; Edelman, E. R. Equilibrium and nonequilibrium phase transitions in copolymer polyelectrolyte hydrogels. J. Chem. Phys. 1997, 107 (5), 1645. (9) Ricka, J.; Tanaka, T. Swelling of ionic gels: quantitative performance of the Donnan theory. Macromolecules 1984, 17 (12), 2916. (10) Tanaka, T. Gels. Sci. Am. 1981, 244, 124−138. Tanaka, T.; Fillmore, D.; Sun, S.-T.; Nishio, I.; Swislow, G.; Shah, A. Phase Transitions in Ionic Gels. Phys. Rev. Lett. 1980, 45, 1636 DOI: 10.1103/PhysRevLett.45.1636. (11) Hua, J.; Mitra, M. K.; Muthukumar, M. Theory of volume transition in polyelectrolyte gels with charge regularization. J. Chem. Phys. 2012, 136, 134901. (12) Horkay, F.; Hecht, A.-M.; Geissler, E. Small Angle Neutron Scattering in Poly(vinyl alcohol) Hydrogels. Macromolecules 1994, 27 (7), 1795. (13) Hecht, A. M.; Geissler, E. Dynamic light scattering from polyacrylamide-water gels. J. Phys. 1978, 39, 631 DOI: 10.1051/ jphys:01978003906063100. Geissler, E.; Hecht, A. M. Dynamic light scattering from gels in a poor solvent. J. Phys. 1978, 39, 955 DOI: 10.1051/jphys:01978003909095500. (14) Skouri, R.; Schosseler, F.; Munch, J. P.; Candau, S. J. Swelling and Elastic Properties of Polyelectrolyte Gels. Macromolecules 1995, 28 (1), 197. (15) Mccoy, J. L.; Muthukumar, M. Dynamic light scattering studies of ionic and nonionic polymer gels with continuous and discontinuous volume transitions. J. Polym. Sci., Part B: Polym. Phys. 2010, 48 (21), 2193. (16) Shibayama, M.; Norisuye, T.; Nomura, S. Cross-link Density Dependence of Spatial Inhomogeneities and Dynamic Fluctuations of Poly(N − isopropylacrylamide) Gels. Macromolecules 1996, 29, 8746. (17) Tanaka, T.; Hocker, L. O.; Benedek, G. B. Spectrum of light scattered from a viscoelastic gel. J. Chem. Phys. 1973, 59 (9), 5151. (18) Shibayama, M.; Ikkai, F.; Inamoto, S.; Nomura, S.; Han, C. C. pH and salt concentration dependence of the microstructure of poly(Nisopropylacrylamidecoacrylic acid) gels. J. Chem. Phys. 1996, 105 (10), 4358. (19) Takata, S.; Norisuye, T.; Shibayama, M. Preparation Temperature Dependence and Effects of Hydrolysis on Static Inhomogeneities of Poly(acrylamide) Gels. Macromolecules 1999, 32, 3989. (20) Shibayama, M.; Toyoichi, T. Adv. Polym. Sci. 1993, 109, 1−62. (21) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (22) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (23) Ogston, A. G.; Preston, B. N.; Wells, J. D. On the Transport of Compact Particles Through Solutions of Chain-Polymers. Proc. R. Soc. London, Ser. A 1973, 333 (1594), 297. (24) Cukier, R. I. Diffusion of Brownian spheres in semidilute polymer solutions. Macromolecules 1984, 17 (2), 252. (25) Phillies, G. D. J. Universal scaling equation for self-diffusion by macromolecules in solution. Macromolecules 1986, 19 (9), 2367. (26) Muthukumar, M.; Baumgaertner, A. Effects of entropic barriers on polymer dynamics. Macromolecules 1989, 22 (4), 1937. (27) Pajevic, S.; Bansil, R.; Konak, C. Diffusion of linear polymer chains in gels. J. Non-Cryst. Solids 1991, 131-133, 630. (28) Pajevic, S.; Bansil, R.; Konak, C. Diffusion of linear polymer chains in methyl methacrylate gels. Macromolecules 1993, 26 (2), 305. (29) Lodge, T. P.; Rotstein, N. A. Tracer diffusion of linear and star polymers in entangled solutions and gels. J. Non-Cryst. Solids 1991, 131-133, 671.

4. CONCLUSION Diffusion of NaPSS in a polyelectrolyte gel was studied using light scattering to understand the effect of the chain confinement and the electrostatic interactions on the diffusion of the probe through the matrix. Two diffusive modes were observed in the light scattering experiments. The first mode corresponds to the gel dynamics while the second mode corresponds to the diffusion of the probe polymer inside the gel. The dynamics of the gel are unaffected by the presence of polyelectrolyte chains inside the gel as the probe polymer concentration inside the gel is less than 2.5 wt %. The dynamics of the probe polymer are significantly affected by the gel network. The probe diffusion constant decreased as the crosslink density of the gel increases due to stronger confinement imposed by the mesh size of the gel. Also, we observed that the diffusion of the probe is slower in a lower charge density gel than a higher charge density gel of the same cross-link density because of the reduced mesh size. The mesh size of the gel decreases as the salt concentration is increased. As a result, the probe diffusion coefficient decreases when the salt concentration is increased. It was observed that mesh size relative to the probe size dictates diffusion of the probe inside the gel. The reduced diffusion coefficient of the trapped probe polymer obeys an exponential dependence on the volume fraction of the gel, in accordance with the sieving mechanism.23 Thus, in our system, the mesh size of the gel governed by the volume fraction of the gel is an important factor in determining the diffusion of the probe inside the gel compared to the electrostatic interaction between the probe and the matrix. We have shown the utility of dynamic light scattering in obtaining diffusion coefficient of the probe despite appearance of multiple modes in dynamic light scattering for these systems. More direct measurements of the tracer diffusion coefficient using FCS, FRS, or PFG NMR would be desirable for charged polymers undergoing diffusion in similarly charged gel matrices. It is of further interest to investigate longer probe chains to explore mechanisms of macromolecular diffusion under stronger confinements.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected], Tel 413-577-1212 (M.M.). ORCID

Murugappan Muthukumar: 0000-0001-7872-4883 Notes

The authors declare no competing financial interest. A.R.: Guest Researcher at NIST.

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ACKNOWLEDGMENTS Acknowledgment is made to the National Science Foundation (DMR-1504265) and AFOSR Grant FA9550-14-1-0164. REFERENCES

(1) Lee, K. Y.; Mooney, D. J. Hydrogels for Tissue Engineering. Chem. Rev. 2001, 101 (7), 1869. (2) Ellis, R. J. Macromolecular crowding: an important but neglected aspect of the intracellular environment. Curr. Opin. Struct. Biol. 2001, 11 (1), 114. (3) Peppas, N. A.; Bures, P.; Leobandung, W.; Ichikawa, H. Hydrogels in pharmaceutical formulations. Eur. J. Pharm. Biopharm. 2000, 50 (1), 27. (4) Hirotsu, S.; Hirokawa, Y.; Tanaka, T. Volume phase transitions of ionized Nisopropylacrylamide gels. J. Chem. Phys. 1987, 87 (2), 1392. J

DOI: 10.1021/acs.macromol.7b01310 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules (30) Johnson, E. M.; Berk, D. A.; Jain, R. K.; Deen, W. M. Hindered diffusion in agarose gels: test of effective medium model. Biophys. J. 1996, 70 (2), 1017. (31) Basak, S.; Chattopadhyay, K. Fluorescence Correlation Spectroscopy Study on the Effects of the Shape and Size of a Protein on Its Diffusion Inside a Crowded Environment. Langmuir 2013, 29 (47), 14709. (32) Reina, J. C.; Bansil, R.; Konak, C. Dynamics of probe particles in polymer solutions and gels. Polymer 1990, 31 (6), 1038. (33) Fatin-Rouge, N.; Wilkinson, K. J.; Buffle, J. Combining Small Angle Neutron Scattering (SANS) and Fluorescence Correlation Spectroscopy (FCS) Measurements To Relate Diffusion in Agarose Gels to Structure. J. Phys. Chem. B 2006, 110 (41), 20133. (34) Slater, G. W.; Wu, S. Y. Reptation, Entropic Trapping, Percolation, and Rouse Dynamics of Polymers in “Random” Environments. Phys. Rev. Lett. 1995, 75, 164. (35) Easwar, N. Diffusion of linear polystyrene in controlled pore glasses. Comparison of experimental data with a theoretical model of entropic barriers. Macromolecules 1989, 22 (8), 3492. (36) Guo, Y.; Langley, K. H.; Karasz, F. E. Hindered diffusion of polystyrene in controlled pore glasses. Macromolecules 1990, 23 (7), 2022−2027. (37) Shibayama, M.; Isaka, Y.; Shiwa, Y. Dynamics of Probe Particles in Polymer Solutions and Gels. Macromolecules 1999, 32 (21), 7086. (38) Dusek, K.; Patterson, D. Transition in swollen polymer networks induced by intramolecular condensation. J. Polym. Sci: Part A-2 1968, 6, 1209. (39) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Courier Dover Publications: 2000. (40) Xue, J. Z.; Pine, D. J.; Milner, S. T.; Wu, X. L.; Chaikin, P. M. Nonergodicity and light scattering from polymer gels. Phys. Rev. A: At., Mol., Opt. Phys. 1992, 46, 6550. (41) Joosten, J. G. H.; McCarthy, J.; Pusey, P. N. Dynamic and static light scattering by aqueous polyacrylamide gels. Macromolecules 1991, 24, 6690. (42) Fang, L.; Brown, W. Dynamic light scattering by permanent gels: heterodyne and nonergodic medium methods of data evaluation. Macromolecules 1992, 25, 6897. (43) Norisuye, T.; Tran-Cong-Miyata, Q.; Shibayama, M. Dynamic Inhomogeneities in Polymer Gels Investigated by Dynamic Light Scattering. Macromolecules 2004, 37, 2944. (44) Pusey, P. N.; Van Megen, W. Dynamic Light scattering by nonergodic media. Phys. A 1989, 157, 705. (45) Fang, L.; Brown, W.; Konk, C. Dynamic properties of polyacrylamide gels and solutions. Polymer 1990, 31 (10), 1960. (46) Pusey, P. N. Dynamic Light-Scattering by Non-Ergodic Media. Macromol. Symp. 1994, 79, 17−30. (47) Norisuye, T.; Tran-Cong-Miyata, Q.; Shibayama, M. Dynamic Inhomogeneities in Polymer Gels Investigated by Dynamic Light Scattering. Macromolecules 2004, 37 (8), 2944. (48) Xue, J. Z.; Pine, D. J.; Milner, S. T.; Wu, X. I.; Chaikin, P. M. Nonergodicity and light scattering from polymer gels. Phys. Rev. A: At., Mol., Opt. Phys. 1992, 46, 6550. (49) Rausch, K.; Reuter, A.; Fischer, K.; Schmidt, M. Evaluation of Nanoparticle Aggregation in Human Blood Serum. Biomacromolecules 2010, 11 (11), 2836. (50) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; John Wiley & Sons: 1980. (51) Sedlak, M. The ionic strength dependence of the structure and dynamics of polyelectrolyte solutions as seen by light scattering: The slow mode dilemma. J. Chem. Phys. 1996, 105, 10123. (52) Forster, S.; Schmidt, M. Polyelectrolytes in Solution. Adv. Polym. Sci. 1995, 120, 51−133. (53) Muthukumar, M. Ordinary-extraordinary transition in dynamics of solutions of charged macromolecules. Proc. Natl. Acad. Sci. U. S. A. 2016, 113, 12627−12632.

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