Modeling of Polyelectrolyte Gels in Equilibrium with Salt Solutions

Article pubs.acs.org/Macromolecules

Modeling of Polyelectrolyte Gels in Equilibrium with Salt Solutions Peter Košovan,*,† Tobias Richter,‡ and Christian Holm‡ †

Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University in Prague, Hlavova 8 128 00 Praha 2, Czech Republic ‡ Institute for Computational Physics, University Stuttgart, Allmandring 3 70569 Stuttgart, Germany S Supporting Information *

ABSTRACT: We use hybrid molecular dynamics/Monte Carlo simulations and coarse-grained polymer models to study the swelling of polyelectrolyte gels in salt solutions. Besides existing industrial applications, such gels have been recently proposed as a promising agent for water desalination. We employ the semi-grand canonical ensemble to investigate partitioning of the salt between the bulk solution and the gel and the salt-induced deswelling of the gels under free swelling equilibrium and under compression. We compare our findings to the analytic model of Katchalsky and Michaeli which explicitly accounts for electrostatic effects. The partitioning of small ions predicted by the model well captures the deviations from the simple Donnan approximation observed in the simulation data. In contrast, the original model highly overestimates the gel swelling, predicting even chain stretching beyond contour length. With a modified model, where we replace the Gaussian elasticity with the Langevin function for finite extensibility, we obtain nearly quantitative agreement between theory and simulations both for the swelling ratio and for the partitioning of salt, across the whole range of studied gel parameters and salt concentrations. The modified model thus provides a very good description of swelling of polyelectrolyte gels in salt solutions and can be used for theoretical predictions of water desalination using hydrogels. These predictions are much less computationally demanding than the simulations which we used to validate the model.

■

INTRODUCTION Polyelectrolytes are polymers composed of monomers that dissociate upon dissolution in water, releasing small counterions (e.g., Na+) into the solution. Polyelectrolyte hydrogels are a network of cross-linked polyelectrolytes. Their prominent feature is a huge swelling capacity in aqueous solution up to 103 times their dry volume. Because of their ability to absorb water, they are widely used as superabsorbers for hygiene products1 as well as biomedical2−9 and agricultural10−12 applications. In contrast to bulk materials, polyelectrolyte micro- and nanogels are being investigated as nanoreactors13−15 or as carriers for controlled drug release.16−20 Recently, polyelectrolyte hydrogels were proposed to desalinate saline water.21,22 The swelling of polyelectrolyte hydrogels is determined by the balance of several factors. On one hand, there is the electrostatic repulsion among the fixed charges of the polymer and the osmotic pressure of its counterions. Both the above contributions promote gel swelling. This is counteracted by the elasticity of the polymer chains, which balances out the other two contributions at a particular volume of the swollen gel. Electrostatic interactions may be screened by the addition of salt, which results in deswelling of the gels as compared to the salt-free case. The salt content in the gel, however, is different from that in the bulk, which is the key observation for the proposed use of hydrogels for desalination.22 Deswelling of the gels can be induced by the presence of specific short-range © XXXX American Chemical Society

hydrophobic interactions, which prefer polymer−polymer over polymer−solvent contacts. These are termed hydrophobic interactions in the context of water-soluble compounds23 or poor solvent conditions in the context of the Flory−Huggins theory.24 Experimentally, it is very difficult to prepare a gel with welldefined properties such as the length of polymer chains between the cross-links and the degree of ionization. Therefore, a comparison of experimental data with theoretical predictions typically concerns the main trends, while a quantitative comparison is difficult because of the lack of detailed information about the system morphology. In contrast to experiments, systems studied in simulations are well-defined by construction and provide the possibility of quantitative tests of theoretical predictions. Existing theoretical predictions for gel swelling usually rely on the Donnan partitioning of salt between the gel and solution and the Flory−Rehner approach24,25 or on the scaling description of chain extension.26−28 Electrostatic interactions are often neglected, apart from their implications for the electroneutrality and for the Donnan partitioning. However, chains connecting the network nodes in polyelectrolyte gels are Received: July 4, 2015 Revised: September 18, 2015

A

DOI: 10.1021/acs.macromol.5b01428 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

network, where N is the number of monomer units connecting two junctions (nodes). The end-to-end distance of a free chain in solution, R0, and its maximum extension, Rmax, are then

typically short (10−100 monomer units). This has two important consequences: the applicability of scaling arguments to such short chains is questionable, and their finite extensibility often plays an important role. This has been observed e.g. in the recent work of Rumyantsev et al.,29 where the authors have shown the necessity of introducing finite-extensible chains in order to avoid unphysical stretching of their gels. With the finite-extensible model they predicted the formation of a hollow region inside charged microgels, which they confirmed by simulations. Interestingly, while numerous experiments and theories have examined the swelling of polyelectrolyte gels, molecular simulations of gel swelling are not so abundant. Much of the literature stems from four groups: the Lund group (Linse and co-workers),30,31 the Madison group (de Pablo, Olvera de la Cruz, and co-workers),32−34 the Mainz group (Kremer, Holm, and co-workers),28,35−37 and more recently the group from ́ Linares/Granada (Quesada-Pérez, Martin-Molina, and coworkers).38−40 A more detailed overview can be found elsewhere.41 Gel swelling in salt solutions has been studied by Edgecombe et al.31 for a series of systems at free swelling. They have shown that the salt concentration inside the gel as a function of salt concentration in the bulk is higher than predicted by the ideal Donnan equilibrium. They also observed significant deviations of the simulation results from predictions of the Flory−Rehner theory24,25 coupled to the Donnan equilibrium. Yin et al.33 studied swelling of polyelectrolyte (PE) gels by mono- and divalent ions. They observed a nonmonotonic dependence of the salt concentration in the gel on the reservoir salt concentration: first it increases, but at high reservoir concentrations it decreases again. They also observed a more pronounced deswelling of the gel and the formation of nanoheterogeneities when multivalent salt ions were used. ́ Recently, Martin-Molina and co-workers42−44 published several papers, investigating polyelectrolyte microgels in salt solutions at physiological salt concentrations, showing that in collapsed gels the salt concentration inside the gel can be lower than predicted by the Donnan approach. In the present work, we study the swelling of polyelectrolyte gels in equilibrium with a salt reservoir, in some aspects similar to earlier studies, especially.31,45 We focus on the deviations of the salt partitioning from the Donnan prediction mentioned above in various gels at free swelling equilibrium as well as under compression or extension. With a broad data set systematically covering several chain lengths, degree of ionization, and salt concentrations, we compare the simulation results to an analytic model.46

R 0 = abN ν ,

R max = b(N − 1)

(1)

where b is monomer size (average bond length), a is a dimensionless prefactor, and ν is the scaling exponent. For an ideal polymer without excluded volume ν = 1/2 and a = (lp/ b)1/2, where lp is the persistence length. For a polymer in athermal solvent, ν ≈ 0.588 and a can be approximately determined from the second virial coefficient and the persistence length.47 For a particular polymer model, a straightforward approach to obtain a is to fit the dependence of Re(N) obtained from simulations. In our simulations the number of monomers is rather small, so that it becomes important to calculate the contour length with the actual number of bonds N − 1. Formally we can introduce the volume per chain as R e 3 = AV

(2)

where Re is the end-to-end distance. The prefactor A is derived from the network topology. For diamond lattice used in the simulations described below, we obtain for fully stretched chains A = √27/4 ≈ 1.30. For a neutral gel the classical theory of Flory and Rehner24,25 predicts that the chain dimensions in the swollen gel are equal to its dimensions in free solution. Assuming that A is independent of chain extension (affine deformation), we may express the volume per chain, V0, in the neutral gel as

R 0 3 = AV0

(3)

In the following, the volume of the swollen neutral gel, V0, will be used as a reference state. The volume per chain at arbitrary swelling can then be expressed in terms of chain extension: ⎛ R e ⎞3 V ⎜ ⎟ = V0 ⎝ R0 ⎠

(4)

The swelling equilibrium is attained at the minimum of the free energy as a function of the gel volume, i.e., when its derivative is equal to zero. Noting that the latter has the dimension of pressure, for situations where it is nonzero, we identify it as an external pressure applied on the gel, Pext: Pext = −

■

∂F ∂V

(5)

Consider now a charged gel, where α is the fraction of charged monomer units of the polymer, in equilibrium with a reservoir of salt solution of 1:1 electrolyte, with a given concentration, cbs . To obtain Pext for such a gel, we need an expression for the free energy as a function of gel swelling or, equivalently, of chain extension. As is common in polyelectrolyte theory,48−51 we split the total free energy into individual contributions, arising from chain stretching, Fstr, electrostatic interactions, Fel, and short-ranged interactions (excluded volume), Fint, and define the corresponding pressure terms:

THEORETICAL MODEL The theoretical model that we apply in this work is based on the model originally derived by Katchalsky and Michaeli in 1955.46 As will be seen from comparison with simulation data, the original model fails to correctly predict the gel swelling, even on a qualitative level. This discrepancy can be resolved by introducing a finite extensibility of the chains. This, together with other modifications, ensures consistent behavior of the model in various limits as described in the next subsection. Original Model of Katchalsky and Michaeli. To introduce the notation and to enable discussion of our modifications, we briefly outline the model of Katchalsky and Michaeli for the swelling of polyelectrolyte gels in salt solutions.46 Consider a chemically cross-linked polymer

Pstr = −

∂Fstr , ∂V

Pel = −

∂Fel , ∂V

Pint = −

∂Fint ∂V

(6)

In addition to the above, we identify the osmotic pressure contribution, Posm, which arises from a concentration difference B


Article

Macromolecules

and water molecules, Nw, assuming additivity of partial molar volumes

of free ions in the gel and in the bulk. We now formally state the condition of mechanical equilibrium of the gel Pext = Pel + Pstr + Posm + Pint

(7)

VNA = NanVan̅ + NcatVcat ̅ + NVm̅ + NwVw̅ ≈ NwVw̅

where expressions for individual contributions will be discussed below. As we are dealing here with highly swollen gels, we neglect the short-range interaction term, setting Pint = 0. We note, however, that this term might be important in collapsed gels under poor solvent conditions or at very high salt concentrations.27,43,44 For the stretching free energy per chain, Katchalsky and Michaeli used the Gaussian approximation for an ideal polymer: 2 ⎡ ⎤ Fstr 3 ⎢⎛ R e ⎞ = ⎜ ⎟ − 1⎥ ⎥⎦ kBT 2 ⎢⎣⎝ R 0 ⎠

where NA is the Avogadro number, V denotes the partial molar volume, and the subscripts “cat”, “an”, “m”, and “w” refer to anions, cations, monomer units, and water, respectively. With the above approximation the following expression is obtained for the mechanical equilibrium (eq 12 of ref 46 written in our current notation): g xan

(8)

(9)

i

=−

−1/3 2 1 (αN ) λB ξ ⎛ V ⎞ ⎜ ⎟ ∑i Ni R 0 1 + ξ ⎝ V0 ⎠

6(αN )2 λB 1 ∑i Ni κR 0 2(1 + ξ)

(17)

From this point, the derivation of Katchalsky and Michaeli continues, approximating the volume per chain of the dry gel as V0 = NVm/NA. Unfortunately, this approximation is inconsistent with how V0 was introduced in eq 3. The neutral gel at free swelling equilibrium, with its chain dimensions comparable to the extension of free chains in solution, contains a significant amount of solvent, which is actually the dominant component. Consequently, its volume is much larger than the volume of a dry gel without any solvent. For this reason, we do not further follow the original derivation. Before introducing the modifications, we cast the equations in a more convenient form. Assuming that the swollen gel is dilute enough, we approximate the total volume from eq 14 as

(11)

VNA ≈ NwVw̅

(18)

This allows us to replace mole fractions by concentrations and write eq 15 as (12)

where cp denotes the polymer concentration inside the gel. Knowing the salt concentration in the gel, we can express the osmotic pressure between the gel and the bulk solution Posm = kBT (2csg + αc p − 2csb)

(16)

b g g xcat = − − ln xan 2 ln xan

where Nan, Ncat, and Nsalt are the numbers of anions, cations, and penetrating salt ions, respectively. If the counterions of the polymer are of the same type as the salt (e.g., Na+), then the distinction between the counterions and salt ions is only a mathematical construction. The number of penetrating salt ions depends on the salt concentration in the bulk as well as on the gel parameters and its swelling state. In the ideal gas approximation, the salt concentration in the gel, cgs = Nsalt/V, is given by the Donnan approximation, as discussed in detail elsewhere:22 ⎛⎛ αc p ⎞ 2 ⎞1/2 αc p b 2 ⎟ + (c ) ⎟ csg = ⎜⎜ − s 2 ⎝⎝ 2 ⎠ ⎠

(15)

For the salt partitioning, the following expression is obtained (eq 14 of ref 46 written in our current notation):

(10)

It is important to note that for eq 9 we need the electrostatic screening length inside the gel. Therefore, the sum in eq 10 runs over all charged polymer segments, their counterions, and the additional ions from the salt solution that penetrate the gel. If we assume that the polymer-bound charges are fully compensated by their counterions, we obtain for monovalent ions (z = 1)

∑ zi 2Ni = Nan + Ncat = 2(Nsalt + αN )

2/3 −1/3 ⎧ Vw̅ ⎪⎛ V ⎞ (αN )2 λB ⎛ V ⎞ ⎨ = ⎜ ⎟ + ⎜ ⎟ NAV ⎪ 3R 0 ⎝ V0 ⎠ ⎩⎝ V0 ⎠

1/3 6 ⎛V ⎞ ξ= ⎜ ⎟ κR 0 ⎝ V0 ⎠

where λB is the Bjerrum length and κ is the inverse Debye screening length: i

−

2xsb

where x denotes the mole fraction and the subscripts and superscripts have the same meaning as in eqs 12 and 14. The variable ξ has been introduced to simplify the notation:

2

κ = (4πλBV −1 ∑ zi 2Ni)1/2

+

g xcat

⎤⎫ ⎡ 2.5ξ ⎪ ×⎢ − ln(1 + ξ)⎥⎬ ⎦⎪ ⎣1 + ξ ⎭

For the electrostatic free energy, they used the free energy of a stretched macromolecule derived in a preceding work52,53 Fel 6R e ⎞ (αN ) λB ⎛ ⎟ ln⎜1 + = kBT Re κR 0 2 ⎠ ⎝

(14)

2csg + αc p − 2csb =

2 cp ⎧ ⎪⎛ R ⎞ (αN )2 λB ⎨⎜ e ⎟ + N⎪ 3R e ⎩⎝ R 0 ⎠

⎡ 2.5ξ ⎤⎫ ⎪ ⎬ ×⎢ − ln(1 + ξ)⎥⎪ ⎣1 + ξ ⎦⎭

(13)

where cgs denotes the salt concentration in the gel, to be distinguished from the salt concentration in the bulk, cbs . In their derivation of the theoretical model, Katchalsky and Michaeli46 proceed by differentiating the different free energy terms with respect to the number of anions, Nan, cations, Ncat,

(19)

where we immediately identify the left-hand side as the osmotic pressure difference, and noting that the first term on the righthand side comes from differentiating Fstr and the second one comes from differentiating Fel, we identify them as the C


Article

Macromolecules

where 3(x) is the Langevin function. With this, we obtain a modified expression for the stretching contribution to the pressure c polR e −1⎛ R e ⎞ Pstr = −kBT 3 ⎜ ⎟ ⎝ bN ⎠ (26) 3bN

(negative) stretching and electrostatic pressure contributions, respectively: Pel = −kBT

⎤ c p (αN )2 λB ⎡ 2.5ξ − ln(1 + ξ)⎥ ⎢ ⎦ N 3R e ⎣ 1 + ξ

cp ⎛ R ⎞ Pstr = −kBT ⎜ e ⎟ N ⎝ R0 ⎠

(20)

where the value of the inverse Langevin function, 3−1(R e/bN ), has to be calculated numerically. The introduction of finite extensibility makes the restoring force, and consequently also Pstr, diverge as the chain extension approaches Rmax. This is important to ensure consistent results at high extensions. In the other extreme at low extensions, both eq 21 and eq 26 predict nonzero Pstr at Re = R0. This is of little practical significance, since in such situations Pstr is negligible when compared to Posm. We fix this by simply subtracting the pressure at R0 from the predicted value of Pstr, using either eq 26 or eq 21. It might seem more natural to use Pstr(Re − R0), but in this way we would also shift the divergence of Pstr from Rmax to Rmax + R0. In the remaining text, we will use the term original model of Katchalsky and Michaeli for the equation of state eq 7 with individual terms represented by eqs 20, 21, and 13. We will use the term modif ied model of Katchalsky and Michaeli, for eq 7, where for we use eq 26 for Pstr instead of eq 21. In both cases, we use eq 22 with the constant C given by eq 23 to determine the cgs as a function of cbs .

2

(21)

Similarly, we can rewrite eq 17 in the form ⎛⎛ α c p ⎞ 2 ⎞1/2 αc p b 2 ⎜ ⎟ =⎜ + (cs ) C ⎟ − ⎝ ⎠ 2 ⎝ 2 ⎠

(22)

⎛ cp 6α 2NλB ⎞ ⎟⎟ C = exp⎜⎜ − g 2 ⎝ 2cs + αc p κR 0 (1 + ξ) ⎠

(23)

csg

which for C = 1 is identical to the Donnan formula (eq 12), while deviations from the ideal behavior are comprised in the term C. We use the following numerical procedure to solve the equations. For a series of chain extensions uniformly distributed between R0 and Rmax, we first obtain cgs from eq 22 using the same iterative procedure as Katchalsky and Michaeli.46 We then use cgs to obtain the value of Pext from eq 7. In this way, we construct the whole P(V) dependence and identify the free swelling equilibrium as Pext = 0. Modification of the Original Model. We already mentioned that in the model of Katchalsky and Michaeli, V0 has been used inconsistently as the volume occupied by a free neutral chain in solution and at the same time as the dry volume, i.e., the sum of monomer volumes. In addition, the true dry volume is very difficult to measure experimentally, since even after extensive drying the gels often contain a significant amount of water. Similarly, in simulations with implicit solvent and soft interactions, such as we describe in the following section, it is impossible to define a dry state, and some arbitrary typical spacing between the particles has to be chosen. To avoid this problem, we use the volume of the neutral swollen gel, V0, as the reference state. This state is accessible and well-defined for simulation, theory, and also experiments. Consequently, we define the swelling ratio as

Q=

V V0

■

SIMULATION MODEL AND METHOD Simulation Model. To simulate hydrogel swelling in saline solutions, we use the same bead−spring polymer model as in our preceding publication.22 It consists of linear chains of N

Figure 1. Simulation snapshot of a highly swollen PE network to illustrate the polymer model. The cube defines the actual simulation box. Several periodic images are included (semitransparent rendering) to emphasize the quasi-infinite connectivity. Charged monomers are shown in red and uncharged in blue. Small ions have been removed for clarity.

(24)

Strictly speaking, the change of the reference state is not really a modification of the original model, as we could still use its intermediate result, eq 15, to predict the gel swelling. If the original model is used as is it allows for chain extension beyond Rmax, provided that α is sufficiently high and cbs low. As follows from the Results and Discussion section, this proves to be crucial, as the chain stretching in swollen gels is significant and beyond the applicable range of the commonly employed entropic spring stretching. To remedy this, we introduce the Langevin function for the relation between chain extension and the elastic restoring force:45,47 ⎡ Re 1 ⎤ = ⎢coth(−bPstr) + ⎥ ≡ 3(bPstr) bN bPstr ⎦ ⎣

monomer units (beads), connected by tetrafunctional nodes to form a diamond-like topology, similar to models constructed in earlier simulations by Mann and co-workers35,37,45 and Linse and co-workers.30,31 A fraction, α, of the polymer beads are charged, with valency z = −1, and an equal number of oppositely charged small ions (counterions) are present in the simulation box. Additional salt ions (cation + anion ion pairs) are present as well. Their number is fluctuating, as described in more detail below. The distinction between the counterions and cations from the salt is artificial, as their interaction

(25) D


Article

Macromolecules

separate series of simulations, employing the Widom particle insertion method.54 We use the ESPResSo software package55 to carry out the simulations. We treat the electrostatic interactions using the P3M algorithm, as implemented in ESPResSo, tuned to an accuracy of 10−4. For each box size we use seven independent simulation runs which differ only in the initial conformation. The fact that they yield identical results (within estimated statistical error) ensures that the simulations have converged. The statistical errors are estimated using the binning analysis.56 Additionally, we checked that the simulation time is ∼102 longer than the estimated autocorrelation time of the slowest evolving variable, which we identify as Re. Chemical Potential of the Salt Solution. To study the gel in equilibrium with a salt solution, it has to be equilibrated with a salt solution of a given chemical potential, which depends on the salt concentration. This can be obtained e.g. from the extended Debye−Hückel formula, using parameters of NaCl. Prior to the simulations of PE hydrogels in equilibrium with salt solution, we carried out a series of simulations of pure salt solutions with our model in order to determine how its chemical potential depends on cbs . Our goal here is to tune the parameters such that the model reasonably represents a simple salt, such as NaCl. It turns out that the concentration dependence of the chemical potential is sensitive to the balance of two important length scales in the system: the monomer size, σ, and the Bjerrum length, λB. The former determines the range of the repulsive excluded volume interactions, while the latter determines the strength of the cohesive electrostatic interactions. On the other hand, there is no clear consensus in the literature regarding the effective ion size.23 In Figure 2, we

parameters are identical. We study polymers with the following parameters (all available combinations): n = 16 chains per simulation box (one diamond unit cell), chain length N = {39, 59, 79}, degree of ionization α = {1/2, 1/4, 1/8}, where for α = 1/x we use a repeating sequence of x − 1 uncharged beads followed by one charged. The chain lengths are chosen such that for all studied combinations of N and α the charge distribution is symmetric with respect to the central bead. We study all these gels in a range of salt concentrations cbs = {0.01, 0.02, 0.05, 0.1, 0.2 M}, which covers the range of salt concentrations from potable water to seawater, motivated by the potential application of hydrogels in water desalination.21 All charged beads (polymer, counterions, salt) are univalent. For the electrostatic interactions we set the Bjerrum length λB = 0.71 nm, which corresponds to distilled water at 300 K. The excluded volume interactions are modeled using the Weeks−Chandler−Andersen (WCA) potential between all pairs of particles: ⎧ ⎛⎛ ⎞12 ⎛ ⎞6 σ σ 1⎞ 1/6 ⎪ ⎪ 4ϵ⎜⎜ ⎟ − ⎜ ⎟ + ⎟ for r ≤ 2 σ ⎝r⎠ 4⎠ U (r ) = ⎨ ⎝⎝ r ⎠ ⎪ ⎪ ⎩0 for r ≥ 21/6σ

(27)

where we choose ϵ = 1 kBT and σ = 0.35 nm = λB/2, which defines the effective particle size. To model the bonds between the polymer beads, we employ the finite-extensible nonlinear elastic (FENE) potential: ⎡ ⎛ r ⎞2 ⎤ 1 2 ⎢ U (r ) = − kFrF ln 1 − ⎜ ⎟ ⎥ ⎢⎣ 2 ⎝ rF ⎠ ⎥⎦

for r < rF (28)

with kF = 80 kBT/nm ≈ 10ϵ/σ and rF = 0. 53 nm ≈ 1.5σ, which in combination with the WCA potential provides the mean bond length of b = 1.03σ and the numerical prefactor a = 1.18 (see Figure 2 of the Supporting Information). Simulation Method. To ensure that the chemical potential of the salt inside the hydrogel is equal to that of the external salt solution, we combine Langevin dynamics in an [NVT] ensemble with a Monte Carlo (MC) scheme for insertion and deletion of salt ion pairs in the simulation box. Overall, we thus work in an ensemble with a fixed number of polymer segments, chemical potential of salt, volume, and temperature. The MC move is executed every 100 integration steps. First, a particle deletion or insertion is chosen with equal probability, and then a randomly selected ion pair is deleted or an ion pair is inserted at random positions. The acceptance probability of the new configuration, p, is given by 2

2

⎡ (c b)2 ⎤ ins pacc = min⎢1, s exp[−β( −μex + ΔE)]⎥ ⎢⎣ canccat ⎥⎦

(29)

⎡ c c ⎤ del pacc = min⎢1, anbcat2 exp[−β(μex + ΔE)]⎥ ⎢⎣ (cs ) ⎥⎦

(30)

Figure 2. Chemical potential of an ion pair of in our simulation as a function of the salt concentration compared to experimental data for NaCl.57,58 Results for several different sizes of the model ions illustrate the range, within which the excluded volume and electrostatic contributions can be appropriately balanced in the coarse-grained model. Results for the variation of osmotic pressure with cbs are provided in Figure 1 of the Supporting Information.

where can = Nan/V and ccat = Ncat/V are the (instantaneous) concentrations of the free anions and cations in the gel, respectively, β = 1/kBT, ΔE = Enew − Eold is the interaction energy change upon the attempted insertion or deletion, and μex is the excess chemical potential of the salt at the desired bulk concentration, cbs . The dependence of μex and of the salt osmotic pressure on cbs for our model was determined from a

compare the dependence of the excess chemical potential of our model salt solution, μex = μs − μideal on cbs with reference data on NaCl57,58 and with the extended Debye−Hückel formula (eq 22 of ref 59). The sensitivity of μex to the size of the ions is obvious when comparing the results for three different ion sizes: σ = 0.5λB, σ = λB, and σ = 2λB. In the E


Article

Macromolecules relevant range of concentrations, the simulation results for σ = 0.5λB agree reasonably well with both the Debye−Hückel prediction and the experimental data on NaCl. In contrast, in the results for the two bigger ion sizes we observe an increasing trend of μex at rather low cbs as a consequence of stronger excluded volume effects. On the basis of the above, we conclude that with σ = λB/2 = 0.35 nm we get satisfactory agreement of the chemical potential of our salt model with experimental results on NaCl, and the model is suitable for simulations of gel swelling equilibria. Further optimization of the model might be necessary for cbs > 0.1 M. Finding the Swelling Equilibrium. To find the swelling equilibrium, we carry out a series of simulations at various fixed box sizes and measure the mean pressure in each of them. After subtracting the osmotic pressure of the salt solution for the given cbs , we obtain the dependence of excess pressure, Pext, on the simulation box size, L, as illustrated in Figure 3. To find the

Figure 4. Salt concentration in the gel at free swelling equilibrium, cgs , for selected values of the bulk salt concentration, cbs , as a function of the concentration of fixed charges in the gel, αcpol. Full points: simulation data; empty points: corresponding solutions to eqs 22 and 23. Different degrees of ionization are distinguished by color and chain lengths by symbol shape. The lines represent the Donnan theory which depends on cbs but not on the polymer parameters. The upper set of points and curves corresponds to cbs = 0.1 M and the lower set to cbs = 0.02 M. Plots for other values of cbs are provided in the Supporting Information.

salt concentration in the gel, cgs , at free swelling equilibrium (Pext = 0) as a function of the polymer concentration in the gel, scaled by its degree of ionization, αcpol. The theory results (empty points in Figure 4) are solutions to to eqs 22 and 23. They are located at slightly different values of αcpol than the simulation results, corresponding to polymer concentrations at free swelling equilibrium determined from numerically solving eq 7 for the modif ied model with finite chain extensibility. For a given value of cbs , the simple Donnan model (eq 12) predicts cgs to be a universal function of αcpol (black line), irrespective of the structural details of the gel. While eq 12 captures the general qualitative trend, it fails on the quantitative level. Clearly, both simulation data and theoretical predictions significantly deviate from the Donnan prediction and agree rather well with each other. The deviation increases with α, and the Donnan model presumably represents a lower bound to cgs . The last observation is in agreement with experimental data of ref 46. The increasing nonideality with increasing α could be expected, as the assumption of a homogeneous distribution of ions in the system breaks down. It is consistent with our own earlier observation on a smaller data set,22 where the deviations have been attributed to the inhomogeneity of charge distribution. The inhomogeneity becomes particularly strong at α ≥ 1/2, where the limit for Manning condensation is reached. The correction to the Donnan picture of salt partitioning has important implications for the possible applications in desalination: the nonideality of the gel + salt system has roughly 2 times lower efficiency in desalination than predicted by the ideal Donnan model. In Figure 5 we show the same quantities as in Figure 4, but now including all gel volumes (simulation box sizes) that were simulated to find the free swelling equilibrium. In other words, Figure 5 additionally includes gels under compression or extension. Here we observe that the data for nine different gels collapse on three master curves, each of which corresponds to a different value of α. A change in the chain length, N, at constant α keeps the data points on the same universal curve. The theoretical predictions of eq 15 (colored lines) quantitatively

Figure 3. Dependence of pressure in the simulation box on the box volume (points) for a given salt concentration and several gels with different parameters indicated in the legend. The swelling equilibrium corresponds to the intersection of the P(V) dependence with the osmotic pressure of the bulk salt solution (horizontal line). The lines are quadratic fits using the average fit parameters as determined by the bootstrap method (see text for details).

box volume that corresponds to free swelling equilibrium, we fit the simulation data with a quadratic function, weighting each of the data points with its respective statistical error. The quadratic function is used as the simplest available option without any physical significance. The fit parameters are constrained to always yield a convex curve. Any intersections of the fit function with the Pext = 0 line that occur outside the range of interpolated data are discarded as artifacts of the quadratic fit (e.g., α = 1/8 in Figure 3). There is no reason to expect an increase in Pext(L) at such high chain extensions (see also discussion of various contributions to the pressure in the Results and Discussion section, Figure 7). To ensure that the procedure is robust and to obtain the error estimate of the equilibrium swelling volume, we further apply the bootstrap method, removing randomly 10% of the data points and fitting the remaining 90%. The bootstrap procedure is repeated 104 times, and the statistical uncertainty of the equilibrium value of L is obtained as the standard deviation of the individual fit results.

■

RESULTS AND DISCUSSION Partitioning of Salt between the Gel and Solution. In Figure 4, we show the simulation results (full points) for the F


Article

Macromolecules

Michaeli46 show excellent agreement between theory and experiment, similar curves in the plots of swelling ratio as a function of α represent just guides to the eye. The predictions of gel swelling from both the original and the modif ied theory are discussed in detail in the following section. Swelling of PE Gels in Salt Solutions. As is shown in Figure 6, the numerical solution of eq 15 strongly overshoots the gel swelling compared to simulation data. The disagreement becomes particularly strong at high values of α and low values of cbs . At sufficiently high α (illustrated in Figure 3(top right) in the Supporting Information, for N = 39 and α = 0.625), the

Figure 5. Salt concentration in the gel as a function of the concentration of fixed charges in the gel under compression and extension. The solid lines represent the Donnan theory (eq 12), which predicts a universal dependence irrespective of the polymer parameters. The colored lines represent the prediction of Katchalsky and Michaeli46 for the given value of α (corresponding color). The predictions for different N (different line styles) are nearly indistinguishable for low α. Plots for other values of cbs are available in the Supporting Information.

capture the salt partitioning, predicting a strong influence of α and much smaller influence of N. Comparison of theoretical predictions and simulation data for all studied salt concentrations (see Figure 4 of the Supporting Information) reveals that the agreement between theory and simulation deteriorates with increasing cbs , α, or N. At high cbs , eq 17 predicts higher values of cgs than the simulation data. This is expected since the Debye−Hückel picture, which was used in the derivation of the theory, breaks down at high salt concentrations. Similarly, it breaks down at high α, when the charges are concentrated on the chains and their distribution in space becomes inhomogeneous. Nevertheless, for all studied parameters the agreement remains fair even at cbs = 0.2 M, and eq 15 provides much better predictions than the ideal Donnan picture. Ultimately, the success of the rather old theory of Katchalsky and Michaeli46 in predicting the partitioning of salt between the gel and solution could have been expected from the agreement of the theory and experiment presented in the original work. One could argue that the gels were not well characterized, and their structural parameters were obtained indirectly from the swelling of neutral gels in a different solvent (chloroform). In this context, the simulation data present a quantitative confirmation of the theory on a system with well-defined parameters. While the curves in their plots of salt concentration as a function of α in the original work of Katchalsky and

Figure 6. Swelling ratio of various PE gels at free swelling equilibrium as a function of salt concentration compared to the predictions of our modification of the model of Katchalsky and Michaeli46 (solid lines) and to the original model (dashed lines). The empty data points are the results of Mann45 for cbs = 0 (plotted arbitrarily as cbs = 2 × 10−4 M to allow for logarithmic scale). G


Article

Macromolecules

stretching pressure, P*str, and the solution to the original * . Similar plots for all studied model with Gaussian elasticity, Pext systems are provided in Figure 6 of the Supporting Information. Among the individual contributions, Posm is strictly positive, and with increasing Re it monotonically decreases toward zero. This directly follows from the Donnan equation (eq 12) and on the qualitative level is not changed by the correction in eq 15. On the quantitative level, as follows from Figure 4, the correction in eq 15 leads to a higher salt content in the gel and significantly higher Posm than predicted by the Donnan model. Therefore, the correction to the Donnan partitioning is an essential ingredient of the pressure balance. The electrostatic contribution, Pel, is negative through the whole range of chain extensions, which at first sight contradicts the intuitive notion that it describes the screened electrostatic repulsion among the charges on the polymer. To understand the apparent discrepancy, we need to realize that the first term in eq 20 originates from differentiating Fel with respect to Re and is indeed strictly positive (see Supporting Information for full details on the derivation). The second term originates from differentiation with respect to κ, which implicitly also depends on Re. The second term is negative in the relevant range and dominates over the first term, making Pel negative as well. Last but not least, we address one more apparent contradiction: if Pel would be disregarded, the gel would swell more, in contrast with the intuition that with electrostatic interactions disregarded, the gel has to swell less. However, disregarding Pel is not equivalent to disregarding electrostatics. There is also an electrostatic contribution in the correction to Posm (eq 23). If we disregard both Pel and the electrostatic correction to Posm (using C = 1), the model, in agreement with intuition, predicts lower swelling than with these terms fully included. This demonstrates that the inclusion of electrostatic interactions is vital for quantitative predictions, but at the same time if they are included only partially, completely erroneous results may be obtained. The stretching contribution to pressure, Pstr, is zero at Re = R0, initially features a rather steep drop, followed by a very weak decrease over a broad range of Re/Rmax, and finally diverges as Re approaches Rmax. Comparing this to the Gaussian stretching, we observe a qualitative difference: instead of diverging at Re = Rmax, in the Gaussian approximation Pstr passes through a minimum and eventually starts increasing. In the limit of high extension, the stretching force on the chain increases proportional to Re, while the surface increases as Re2, which results in the Gaussian stretching pressure approaching zero as 1/Re. Apparently, this is the origin of the failure of the original model in predicting the gel swelling and the ultimate reason why it allows for chain extension beyond Rmax. We should also note here that modern scaling theories60 predict a much more complex response of polyelectrolyte chains to an applied stretching force. Several different scaling regimes can be identified, depending on the balance of chain extension, Debye screening length, λD, and the electrostatic coupling, λB. For short chains in our simulations, different scaling regimes can hardly be distinguished, as the available range of extensions from R0 to Rmax only spans about 1 decade. Even for much longer chains, the transition between different scaling regimes is surprisingly smooth.61 In the case of short chains, before the different scaling regimes come into play, the divergence of Pstr due to finite extensibility overwhelms other contributions.

original theory predicts chain stretching beyond Rmax at low salt concentration. Obviously, introducing finite extensibility is desired here, and indeed Figure 6 shows a significant improvement and nearly quantitative predictions of the free swelling are achieved when the Langevin function is used to account for finite chain extensibility in Pstr. Similar to the case of salt partitioning, the agreement is very good at low α and worse at high α. Unlike the salt partitioning, the agreement seems to deteriorate with increasing N. In addition, the modified theory reasonably matches the simulation results of Mann45 for the limit cbs = 0, shown as empty points in Figure 6. The contrast between the original theory, the modified theory, and the simulation data in the no salt limit can be better seen in the double-logarithmic representation in Figure 3 of the Supporting Information. There are several reasons for the occurrence of deviations. First of all, as Q ∼ Re3, even a small disagreement in the estimation of equilibrium Re gets greatly amplified when converted to Q. Second, we know from Figures 4 and 5 that the amount of salt in the gel, and hence the osmotic pressure, is overestimated at high α. Last, but not least, the electrostatic free energy assumes only weakly perturbed homogeneous charge distribution, which gets increasingly violated with increasing α and increasing Re. Eventually, all the assumptions of the theory break down when the system crosses the Manning condensation threshold at α ≈ 1/2. Various Contributions to the Pressure. In this section, we discuss how the excess pressure in the gel depends on its volume (or equivalently on chain extension)the P(V) behavior. The equilibrium swelling ratio of a gel as well as its behavior under compression or extension is a nontrivial result of a balance of different contributions. This can be seen from Figure 7, where we plot the dependence of the total excess pressure in the gel (eq 7) on chain extension scaled by the contour length, Re/Rmax, and individual contributions to pressure: Pstr (eq 26), Posm (eq 13), and Pel (eq 20). For further discussion, we also show the osmotic pressure obtained * (eq 12), the Gaussian from the Donnan equation, Posm

Figure 7. Various contributions to the pressure as a function of the chain end to end distance, scaled by its contour length, Rmax. The data points are numerical solutions to the modif ied model of Katchalsky and Michaeli.46 For comparison, the dashed lines show the osmotic * ), the pressure obtained from the Donnan approximation (Posm * ), and the Gaussian approximation of the stretching pressure (Pstr external pressure from the original model of Katchalsky and Michaeli * ). The plot shows intermediate values of N = 59, α = 1/4, and cbs = (Pext 0.05 M. H


Article

Macromolecules Nevertheless, it is interesting to see where our system is in this parameter space. In Figure 8 we plot the ratio λD/Re as a

Figure 9. Comparison of simulation data and theory for the pressure− extension curves. The solid lines are predictions of the extended swelling model, and dashed lines are predictions of the original model of Katchalsky and Michaeli.46 Data points are simulation result showing the pressure in the simulation box minus the osmotic pressure of the bulk salt solution at cbs (equivalent to Pext) as a function of chain extension scaled by its contour length.

Figure 8. Variation of the Debye screening length in the gel with the chain extension scaled by its maximum extension. The plot represents one particular value of cbs and N. Points represent results of eq 22 for the given parameters.

function of chain extension. We observe that λD ≈ 0.1Re and depends only weakly on Re. This trend appears quite robust irrespective of the system parameters, as can be seen from Figure 5 in the Supporting Information. To understand this apparently strange behavior, we need to recall that the salt concentration in the gel increases, while the total concentration of ions in the gel decreases with increasing Re. Hence the Debye screeninig length, λD, increases with increasing Re. Apparently, these two effects cancel to a large extent, keeping the ratio λD/Re nearly constant across the whole range of swelling. With respect to the scaling theories, we note that in all investigated systems λD < Re, but λD and Re have values that assume the same order of magnitude. Unlike a free chain in solution, for which λD is independent of extension, our chains in the gel never extend too far beyond λD. Finally, the total pressure in Figure 7 looks qualitatively similar to Pstr, but at low extensions it is dominated by the other contributions. Consequently, Pext is positive at small extensions, crosses the Pext = 0 value, then features a weak decrease with an inflection point, and eventually decreases steeply, following the behavior of Pstr. At high extensions Pext is entirely dominated by Pstr, while both Pel and Posm converge to zero. In contrast, at low extensions the dominant term is Posm, opposed by Pel, while the value of Pstr is smaller in absolute value than the other two. At high salt (see Figure 6 of the Supporting Information) the Pel term becomes comparable to Posm, which may even result in nonmonotonic behavior of Pext(Re), resembling the van der Waals loops corresponding to two-phase coexistence in real gases. In this range, however, the Debye−Hückel approximation breaks down and other assumptions of the theory as well. Therefore, we consider this behavior an artifact of the model rather than a physical effect. This is also supported by our simulation data, which show no sign of such behavior. A direct comparison of the predicted dependence of Pext on Re from the modified model of Katchalsky and Michaeli with the simulation data shown in Figure 9 reveals fair agreement across a rather broad range of chain extensions (Figure 8 of the Supporting Information provides such plots for all studied systems). The agreement becomes worse at high α and high cbs . For comparison, we also show the Pext obtained from the

original model, eq 17, as dashed lines. At small extensions (under compression) it yields agreement comparable with the modified model, but around Pext = 0 and especially for gels under extension (Pext < 0) it fails both on a quantitative and qualitative level. This failure is a direct consequence of the use of eq 21 for the stretching contribution, as discussed in the paragraph above. As can be seen from Figure 7 of the Supporting Information, the equilibrium swelling is typically in the range of the quasi-plateau in Pstr, where the qualitative discrepancy between different approaches to Pstr has a huge influence on the obtained value of Q. Again, this illustrates that the finite extensibility is a necessary ingredient of a quantitative description of the swelling of PE gels. Assumption of Affine Deformation. Apparently, the agreement between theory and simulation results in terms of pressure−extension relations is well satisfied. However, there is no a priori guarantee that the assumption of affine deformation, which allows us to write eq 4, is fulfilled. The value of A which we used for eq 3 has been obtained in the limit of full stretching, i.e., taking A = Rmax3/Vmax. To see the applicability of this assumption, we plot in Figure 10 the geometry prefactor A as a function of the simulation box volume scaled by the box volume at equilibrium. While there is a notable spread of various data sets, we can observe a general correlation: the value of A is significantly larger than the expected A ≈ 1.3 especially in gels under compression, while at free swelling equilibrium and in gels under extension it is reasonably close. The values of A at free swelling equilibrium are compiled in Figure 11, showing deviations from the ideal value between +20% and +50%. At the moment we do not have a reasonable approximation to predict the variation of A with Re and the structural parameters of the network. As the predicted value of Q is directly proportional to A, on the basis of Figure 11 we argue that a similar difference in Q values between the theory and our simulations should be expected and may account for a large fraction of the deviations in Figure 6.

■

CONCLUSION In this work we studied swelling of polyelectrolyte gels in aqueous salt solutions. We used computer simulations to obtain I


Article

Macromolecules

explicit consideration of electrostatic interactions in the description of salt partitioning that goes beyond the Donnan approximation is essential for a quantitative prediction of the osmotic pressure. However, once electrostatic interactions are considered in salt partitioning, it is vital to include them also in other terms in order to obtain quantitative description of the gel swelling. Last but not least, we observed that with increasing chain extension the concentration in the gel increases, the total concentration of ions in the gel increases, and consequently the Debye length in the gel increases such that the ratio λD/Re stays almost constant. This leads to a different chain stretching scenario than in the case of polyelectrolyte chain in free solution. Besides the free swelling equilibria, with the modified model we found reasonable agreement with simulations for the whole pressure−extension curve. This will allow us to use the model to study the mechanical properties of gels under compression and partitioning of salt under such conditions, which is desirable for the proposed application in desalination. Studying such systems via MD simulations using bead−spring models and explicit ions would be computationally very expensive. Such studies will therefore benefit from the availability of an improved analytic model as presented in this article.

Figure 10. Geometric prefactor as a function of the gel expansion (relative to free swelling equilibrium) for a selected value of cbs = 5 × 10−2 mol dm−3. The horizontal line represents the ideal value for the diamond lattice (1.30). Plots for all other values of cbs are provided in the Supporting Information.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01428. Excess osmotic pressure of pure salt solution as a function of concentration; end-to-end distance of a free chain as a function of chain length; swelling ratios of all studied gels as a function of salt concentration; salt concentration in the gel; Debye length inside the gel and pressure−extension curves for all studied gels; step-bystep derivation of the original equations of Katchalsky and Michaeli and of our modifications introduced into the equations (PDF)

Figure 11. Geometric prefactor A (ratio of the chain end-to-end distance cubed to the volume per chain) as a function of the bulk salt concentration for all studied gels at free swelling equilibrium. The horizontal line represents the ideal value for the diamond lattice (1.30).

■

AUTHOR INFORMATION

Corresponding Author

the pressure−extension relations for a series of gels with various structural parameters (chain length, degree of ionization), each of them in equilibrium with a series of salt solutions spanning 2 decades in salt concentrations. We compared the simulation results with the theoretical predictions of Katchalsky and Michaeli46 for the partitioning of salt between the gel and bulk solution. This theory provides a correction to the simple Donnan relation, and we found good agreement between the theory and simulations across the studied range of parameters. We also compared simulation results for the free swelling equilibria and the full pressure−extension curves to the abovementioned theory46 and to its modification, where we introduced the correction for finite extensibility. We found good agreement of the swelling ratio obtained from simulations with the predictions of the modified model. In contrast, the original model predicts much higher swelling, in some situations even beyond the physical bounds. We discussed in detail the origin of this discrepancy, ascribing it to the finite extensibility. Since the chain lengths in a cross-linked polyelectrolyte gel never approach the limit of an infinite chain and their stretching is significant, the finite extensibility is essential for a quantitative description of gel swelling. Similarly,

*E-mail: [email protected] (P.K.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS P.K. acknowledges E. B. Zhulina for fruitful discussions regarding the finite extensibility and for pointing out the context of the Debye length. The authors gratefully acknowledge financial support from the DFG under Grant HO 1108/ 26-1. P.K. acknowledges support from the Ministry of Education, Youths and Sports of the Czech Republic (grant LK21302), and the Czech Science Foundation (Grant P208/ 14-23288J).

■

REFERENCES

(1) Zohuriaan-Mehr, M. J.; Kabiri, K. Iran. Polym. J. 2008, 17, 451− 477. (2) Peppas, N. A.; Bures, P.; Leobandung, W.; Ichikawa, H. Eur. J. Pharm. Biopharm. 2000, 50, 27. (3) Samsonov, G. V.; Kuznetsova, N. P. Adv. Polym. Sci. 1992, 104, 1−50. J


Article

Macromolecules (4) Jia, X.; Kiick, K. L. Macromol. Biosci. 2009, 9, 140−156. (5) Jagur-Grodzinski, J. Polym. Adv. Technol. 2009, 21, 27−47. (6) Meenach, S.; Anderson, K.; Hilt, J. In Safety of Nanoparticles; Webster, T. J., Ed.; Nanostructure Science and Technology; Springer: New York, 2009; pp 131−157. (7) Tamura, H.; Furuike, T.; Nair, S.; Jayakumar, R. Carbohydr. Polym. 2011, 84, 820−824. (8) Sannino, A.; Esposito, A.; Rosa, A. D.; Cozzolino, A.; Ambrosio, L.; Nicolais, L. J. Biomed. Mater. Res. 2003, 67A, 1016−1024. (9) Zavan, B.; Cortivo, R.; Abatangelo, G. Hydrogels; Springer: Milan, 2009; pp 1−8. (10) Zohuriaan-Mehr, M. J.; Omidian, H.; Doroudiani, S.; Kabiri, K. J. Mater. Sci. 2010, 45, 5711−5735. (11) Kazanskii, K.; Dubrovskii, S. Polyelectrolytes Hydrogels Chromatographic Materials; Springer: Berlin, 1992; Vol. 104. (12) Harland, R.; Prud’homme, R. Polyelectolyte Gels; American Chemical Society: Washington, DC, 1992. (13) Kozlovskaya, V.; Kharlampieva, E.; Chang, S.; Muhlbauer, R.; Tsukruk, V. V. Chem. Mater. 2009, 21, 2158−2167. (14) Mohan, Y. M.; Lee, K.; Premkumar, T.; Geckeler, K. E. Polymer 2007, 48, 158−164. (15) Diaz, D. D.; Kuehbeck, D.; Koopmans, R. J. Chem. Soc. Rev. 2011, 40, 427−448. (16) Oh, J. K.; Drumright, R.; Siegwart, D. J.; Matyjaszewski, K. Prog. Polym. Sci. 2008, 33, 448−477. (17) Hamidi, M.; Azadi, A.; Rafiei, P. Adv. Drug Delivery Rev. 2008, 60, 1638−1649. (18) Makino, K.; Idenuma, R.; Murakami, T.; Ohshima, H. Colloids Surf., B 2001, 20, 355−359. (19) Tada, D.; Tanabe, T.; Tachibana, A.; Yamauchi, K. J. Biosci. Bioeng. 2005, 100, 551−555. (20) Wang, Q.; Li, S.; Wang, Z.; Liu, H.; Li, C. J. Appl. Polym. Sci. 2009, 111, 1417−1425. (21) Höpfner, J.; Klein, C.; Wilhelm, M. Macromol. Rapid Commun. 2010, 31, 1337. (22) Höpfner, J.; Richter, T.; Košovan, P.; Holm, C.; Wilhelm, M. In Intelligent Hydrogels; Sadowski, G., Richtering, W., Eds.; Progress in Colloid and Polymer Science; Springer International Publishing: Berlin, 2013; Vol. 140, pp 247−263. (23) Israelachvili, J. Intermolecular and Surface Forces; Academic: London, 1992. (24) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (25) Flory, P. J.; Rehner, J. J. Chem. Phys. 1943, 11, 512. (26) Quesada-Pérez, M.; Maroto-Centeno, J. A.; Forcada, J.; HidalgoAlvarez, R. Soft Matter 2011, 7, 10536−10547. (27) Polotsky, A. A.; Plamper, F. A.; Borisov, O. V. Macromolecules 2013, 46, 8702−8709. (28) Mann, B. A.; Everaers, R.; Holm, C.; Kremer, K. Europhys. Lett. 2004, 67, 786−792. (29) Rumyantsev, A. M.; Rudov, A. A.; Potemkin, I. I. J. Chem. Phys. 2015, 142, 171105. (30) Schneider, S.; Linse, P. Eur. Phys. J. E: Soft Matter 2002, 8, 457− 460. (31) Edgecombe, S.; Schneider, S.; Linse, P. Macromolecules 2004, 37, 10089−10100. (32) Yin, D.-W.; Horkay, F.; Douglas, J. F.; de Pablo, J. J. J. Chem. Phys. 2008, 129, 154902. (33) Yin, D.-W.; Olvera de la Cruz, M.; de Pablo, J. J. J. Chem. Phys. 2009, 131, 194907. (34) Longo, G. S.; Olvera de la Cruz, M.; Szleifer, I. Macromolecules 2011, 44, 147−158. (35) Mann, B. A.; Holm, C.; Kremer, K. J. Chem. Phys. 2005, 122, 154903. (36) Mann, B. A.; Holm, C.; Kremer, K. Macromol. Symp. 2006, 237, 90−107. (37) Mann, B. A.; Lenz, O.; Kremer, K.; Holm, C. Macromol. Theory Simul. 2011, 20, 721−734.

(38) Quesada-Pérez, M.; Ibarra-Armenta, J. G.; Martín-Molina, A. J. Chem. Phys. 2011, 135, 094109. (39) Quesada-Pérez, M.; Ramos, J.; Forcada, J.; Martín-Molina, A. J. Chem. Phys. 2012, 136, 244903. (40) Quesada-Pérez, M.; Maroto-Centeno, J. A.; Martin-Molina, A. Macromolecules 2012, 45, 8872−8879. (41) Košovan, P.; Richter, T.; Holm, C. In Intelligent Hydrogels; Sadowski, G., Richtering, W., Eds.; Progress in Colloid and Polymer Science; Springer International Publishing: Berlin, 2013; Vol. 140, pp 205−221. (42) Quesada-Pérez, M.; Martín-Molina, A. Soft Matter 2013, 9, 7086−7094. (43) Quesada-Pérez, M.; Ahualli, S.; Martín-Molina, A. J. Polym. Sci., Part B: Polym. Phys. 2014, 52, 1403−1411. (44) Quesada-Pérez, M.; Ahualli, S.; Martín-Molina, A. J. Chem. Phys. 2014, 141, 124903. (45) Mann, B. A. The Swelling Behaviour of Polyelectrolyte Networks. Ph.D. Thesis, Johannes Gutenberg-University, Mainz, Germany, 2005. (46) Katchalsky, A.; Michaeli, I. J. Polym. Sci. 1955, 15, 69. (47) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: Oxford, UK, 2003. (48) Holm, C.; Hofmann, T.; Joanny, J. F.; Kremer, K.; Netz, R. R.; Reineker, P.; Seidel, C.; Vilgis, T. A.; Winkler, R. G. Adv. Polym. Sci. 2004, 166, 67−112. (49) Dobrynin, A. V.; Rubinstein, M. Prog. Polym. Sci. 2005, 30, 1049−1118. (50) Dobrynin, A. V. Curr. Opin. Colloid Interface Sci. 2008, 13, 376− 388. (51) Borisov, O. V.; Zhulina, E. B.; Leermakers, F. A.; Ballau, M.; Müller, A. H. E. In Self Organized Nanostructures of Amphiphilic Block Copolymers I; Müller, A. H. E., Borisov, O., Eds.; Adv. Polym. Sci.; Springer: Heidelberg, 2011; Vol. 241, pp 1−55. (52) Katchalsky, A.; Lifson, S.; Mazur, J. J. Polym. Sci. 1953, 11, 409− 423. (53) Lifson, S.; Katchalsky, A. J. Polym. Sci. 1954, 13, 43−55. (54) Widom, B. J. Chem. Phys. 1963, 39, 2802−2812. (55) Arnold, A.; Lenz, O.; Kesselheim, S.; Weeber, R.; Fahrenberger, F.; Röhm, D.; Košovan, P.; Holm, C. ESPResSo 3.1 − Molecular Dynamics Software for Coarse-Grained Models. Meshfree Methods for Partial Differential Equations VI. 2013; pp 1−23. (56) Janke, W. Quantum Simulations of Complex Many-Body Systems: from Theory to Algorithms 2002, 423−445. (57) Truesdell, A. H. Science 1968, 161, 884−886. (58) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid Phase Equilibria; Prentice Hall: Upper Saddle River, NJ, 1999. (59) Olivares, W.; McQuarrie, D. Biophys. J. 1975, 15, 143−162. (60) Zhulina, E. B.; Rubinstein, M. Soft Matter 2012, 8, 9376−9383. (61) Stevens, M. J.; McIntosh, D. B.; Saleh, O. A. Macromolecules 2013, 46, 6369−6373.

K


Modeling of Polyelectrolyte Gels in Equilibrium with Salt Solutions

Recommend Documents