Particle-Based Simulations

We use the grand-reaction method as presented in chapter 5.

Especially, we employ the particle exchange moves (5.18), (5.19), (5.20), (5.21) and the identity exchange moves, (5.25) and (5.26) in order to be able to simulate low salt concentrations and minimize finite size-effects. For ensuring chemical equilibrium we use reactions (5.28), (5.29), (5.30), (5.31) and the particle swapping move (5.35). The simulation protocol is similar to the one presented in Section 5.5. We perform extensive equilibration of the particle exchanges and reactions in the system prior to producing data. We simulate at different prescribed volumes and calculate the pressure in the gel to obtainPVcurves. The difference to the previous simulations of strong gels is that we now consider weak acidic monomers.

Periodic Gel SimulationsThe peridodic gel simulations are set up and used as described in Section 2.6.2.

Single-Chain Cell-Gel ModelThe single-chain cell-gel model (CGM) is set up and used as described in chapter 3.

6.1.2 Poisson-Boltzmann Cell-Gel Model

The Poisson Boltzmann cell-gel model (CGM) is set up and used as described in chapter 3. In contrast to the previous simulations,

we now consider weak acidic monomers and couple the cell to a more complicated reservoir containing H⁺,OH⁻,Na⁺,Cl⁻ions.

As outlined in Section2.4, coupling a reservoir to a system which is modeled via Poisson-Boltzmann theory is straightforward, given the reservoir concentrations. In PB theory, we set the reservoir electric potential to zero which means that we assume zero excess chemical potentials in the reservoir. Therefore, we can use the results acquired in note 12. Neglecting OH⁻ions can only be done safely at low pH but not at high pH, where OH⁻ ions are ubiquitous. Neglecting OH⁻ions at high pH results in false predictions of the swelling equilibria.

We treat pH-dependent reactions using the charge regulation approach by Ninham et al. [5,23]. The charge regulation de-termines the degree of dissociation and, therefore, the space charge density of the penetrable rod. For a weak polyelectrolyte, monomers may be either neutral or charged (HA−)−−−−*−A^–+H⁺).

The dissociation constant in the Poisson-Boltzmann framework is given by:

K= a_H⁺aA⁻

aHA = cH⁺(~r)cA⁻(~r)

cHA(~r)c , (6.2) where we used Equation (2.34) to introduce the position depen-dence.

The concentration of titratable monomers, A^– or HA, isc0(~r)= cA⁻(~r)+cHA(~r) and is distributed with probability densityp(~r), i.e.c0(~r) =Np(~r). This givescHA(~r) = Np(~r)−cA⁻(~r) which we substitute into Equation (6.2) and solve forcA⁻(~r):

cA⁻(~r)= c0(~r)Kc

c^res_H+exp(−e0ψ(~r)/(kBT))+Kc . (6.3)

The fixed charge density used in the PB Equation (3.14) is then given byρf(~r)=−e0cA⁻(~r).

Equation (6.3) is equivalent to a space dependent degree of dissociationα(~r), given by:

α(~r)= cA⁻(~r)

c0(~r) = Kc

c^res_H+exp(−e0ψ(~r)/(kBT))+Kc . (6.4) Using the charge regulation approach above, we can perform PB CGM simulations of weak polyelectrolyte gels.

6.2 pH-dependent Swelling of Gels

To compute the equilibrium swelling, we compute, as outlined before in Section 2.6.2, pressure-volume curves. The pressure as a function of the end-to-end distance, for two different pH values and a fixed salt concentration, is shown in Figure 6.2for the i) periodic gel MD model, ii) the single-chain CGM and iii) the PB CGM. The resultsP(Re) of all three models agree very well. When the gel is compressed, the pressure is greater than the pressure of the reservoir (P^gel−P^res >0) and when the gel is stretched the pressure of the gel is smaller than the pressure of the reservoir (P^gel−P^res<0). Changing the pH value of the supernatant solution increases the pressure in the gel. This can be seen in Figure 6.2: The P(Re) curves are shifted upwards for higher pH. The reason for this increased gel pressure is an increased dissociation of the gel charges: at high pH the weak acid is more dissociated. This results in a) an increased osmotic pressure of the ions and b) an increased repulsion of monomers

along the polymer backbone. As for the strong gels before, the zero crossing of the pressure curve is the equilibrium end-to-end distanceReq. As we can see in Figure6.2, the three models agree in tendency. For pH = 11 there are differences between the single-chain CGM and the periodic gel model visible left to the zero crossing of theP−Re curve. In this pH range the weak polyelectrolyte gel is highly charged and compressing the gel results in strong electrostatic inter-chain repulsions increasing the pressure of the system. Since this chain-chain repulsion is not present in the single chain model, the correspondingP−Re

curve of the single-chain model is below the one for the periodic gel model. We expect that the chain-chain repulsion decreases, the more the gel is stretched and indeed Figure6.2shows that both particle-based models have roughly the same zero crossing.

For pH=3 the gel is alsmost uncharged and theP−Recurve of the single-chain CGM and the periodic gel model agree within errorbars. Around pH=3 theP−Recurve of the PB CGM model however differs from the ones of the two particle-based models.

In this pH range the gel is almost uncharged (see Figure 6.3b)).

In this case the stretching pressure is the dominating pressure contribution and a deviation from the particle-based models reflects that the expression used for the stretching pressure is based on very simplistic arguments (see Section 3.5.1).

In Figure 6.3a) we show the equilibrium extension of the chains Reqof a weak polyelectrolyte gel as a function of pH. All models predict that the gel is collapsed at low pH. With increasing pH the dissociation of the monomers increases (see Figure 6.3 b). This increased charge of the gel results in a stretching of the gel which can be seen in Figure 6.3 a). Around pH = 9 the maximum degree of dissociationαis reached and the gel

0.0 0.2 0.4 0.6 0.8 1.0

equilibrium extension R

/R

8 6 4 2 0 2 4 6 8 10

p re ss ur e P

P

[b ar ]

pH=3

pH=11 iii) PB CGM i) periodic gel model ii) single-chain CGM

Figure 6.2: Pressure curves of the different gel models (pK = 4, c^res_salt = 0.01mol/L) at pH = 3 and pH = 11 as a function ofRe/Rmax,whereRmax =(N−1)bis the contour length of the chain. From these curves the equilibrium end-to-end distances in swelling equilibriumReqare found via linear interpolation. Errors inRe/Rmaxare typically smaller than symbol size.

swelling plateaus. A further increase of pH in the reservoir adds more Na⁺and OH⁻ ions. On the one hand, this increases the electrostatic screening in the gel and reduces the repulsion of like charged monomers. On the other hand, further addition of Na⁺and OH⁻ions increases the (osmotic) pressure excreted by the reservoir. Both effects deswell the gel again around pH=11.

These findings in Figure 6.3a) are in good qualitative agreement with experiments [135,150] and in analogy to the findings for the end-to-end distance of polyelectrolytes in solution (see Section 5.6).

In Figures 6.3, we observe that the swelling and dissociation behavior of the two particle-based models are very similar. How-ever, for small pH value the PB CGM shows slightly lower

1 3 5 7 9 11 13 pH

0.0 0.2 0.4 0.6 0.8 1.0

equilibrium extension Req/Rmax

i) periodic gel model ii) single-chain CGM iii) PB CGM

(a)

1 3 5 7 9 11 13 pH

0.0 0.2 0.4 0.6 0.8 1.0

degree of dissociation

i) periodic gel model ii) single-chain CGM iii) PB CGM ideal

(b)

Figure 6.3: a) Swelling of the gel as a function of pH (pK=4, c^res_salt = 0.01mol/L). b) degree of dissociation with pH

equilibrium extensionsReq. From figures 6.3b), we see that the gel is only slightly charged in this pH range. Again, the stretching pressure is the dominant contribution here, and we attribute the deviation of the PB CGM to the other models to the simplistic arguments used in deriving the stretching pressure (see Section 3.5.1).

The dissociation behavior in Figure 6.3b) shows an even more pronounced difference of the PB CGM to the two particle-based models: the degree of dissociation in the PB CGM is constantly in-creased compared to the two particle-based models. We attribute this to the fact that the dissociation equilibria are determined differently in the PB CGM compared to the particle-based mod-els. In the PB framework only the local concentrations enter the calculation which determinesα(compare Equation (6.4)). In the particle-based models also an energetic contribution enters in the reaction-ensemble acceptance probability (compare (2.67)).

This energetic contribution penalizes states with high electro-static repulsion, resulting in a decreased degree of dissociation

compared to the PB CGM.