5.3.1 Gel model
Weak polyelectrolyte gels were modelled as a diamond-like network composed of 16 chains, each of which withN=50 monomer units. The network was assumed to be in equilibrium with a reservoir of an aqueous solution at a specific concentration csof positive and negative monovalent ions, denoted as Na+and Cl−.
Each monomer unit of the network carries an acidic pendant, which can be either protonated or deprotonated according to the following reaction:
HA −−↽K−−⇀A−+H+ (5.1)
whereKis the ionization equilibrium constant.
The fraction of charged monomer units (ionization degree)αis defined as:
α= [A−]
[A−] + [HA]. (5.2)
Simultaneously, each monomer unit carries a hydrophobic pendant, which is characterized by a hydrophobicity parameter. Two approaches were used to model the system, namely the coarse-grained model and the mean-field model. Depending on the model, we chose eitherε(the parameter of interparticle interaction potential) orχ(the Flory – Huggins parameter) as the hydrophobicity parameter.
5.3.2 Coarse-grained model
In this model, the mechanical movement of all particles present in system was ac- counted for by Langevin dynamics, and the ionization reaction equilibrium was as- sessed using Monte Carlo procedure. Short runs of Langevin dynamics were inter- spersed with short runs of Monte Carlo simulations.
Electrostatics Long-range electrostatic interactions were considered using the Par- ticle-Particle-Particle-Mesh (P3M) electrostatic solver. This Fourier-based Ewald sum- mation method calculates Coulomb potentials as follows:
VEL(r) =lBkBTq1q2
r , (5.3)
wherelB is the Bjerrum length, q1 and q2 are the charges of the ionized segments, andr is the distance between them. lB = 2σ = 0.7 nm, which corresponds to the value for water at 300 K.
The Langevin thermostat [100] was used to ensure that the system was in thermal equilibrium with the heat bath at a temperature of 300 K.
Monte Carlo Ionization reactions of gel monomer units were accounted for by fol- lowing the procedure described in our previous study [99]. Short runs of Monte Carlo simulations models the following chemical reactions: (i) ionization of monomer units (Eq.5.1), and (ii) exchange of Na+and Cl−ion pairs with the reservoir:
∅ −−−↽K−−−NaCl⇀Na++Cl−. (5.4) At neutral pH, the availability of H+ions for the reaction Eq.5.1is very low. Thus, this reaction was assumed to occur only by sodium ion exchange for a hydrogen ion.
H+ K
−−⇀′
↽−−Na+,K′ =eβ(µH+−µNa+) (5.5) where the reaction constantK′ accounts for the ion exchange, 1/β = kBT, andµH+ andµNa+ are the chemical potentials of the corresponding ions, which were com- puted in separate simulations (for details see [99]). In our simulations,µNa+ =µCl−. The input data of this simulation consist of the volume of the simulation box Vgel, the hydrophobicity parameter, ε, the chemical potentials of the ions, and the ionization constantK. The output data are the average pressure and the number of ionsNNa+ andNCl− in the simulation box; and the ionization degree,αEq.5.2.
Once the pressure is determined, the osmotic pressure of ions is subtracted, thereby assessing the partial pressure of the gel,p,i. e., the pressure that must be ap- plied to the hydrogel using a piston permeable only to solvent and ions, to achieve the desired gel densityφ. The osmotic pressure of ions was calculated by running a separate simulation in which a pure ionic gas exchanged ions with the same bath as that used in the simulation with the gel.
5.3.3 Mean-field model
The mean-field approximation was used based on the classical lattice Flory theory of polymers. In such an approximation, a single chain of the gel was considered in a mean-fieldproduced by the other components of the system: water, salt ions, and the rest of the gel [14]. The free energy of a chain consists of three independent terms:
the conformational free energy of a uniformly extended chain, excluded volume in- teractions, which account for the hydrophobicity of the gel network, and the ionic contribution.
The conformational free energy accounts for the finite extensibility of a chain [101]:
Fconf kBT = 3
2
[︄ R2/(Nb2)−1 (1−R2/(N2b2))d −ln
(︃ R2 Nb2
)︃]︄
, (5.6)
where the first term includes Gaussian elasticity, and the logarithmic term accounts for the effect of chain compression,d, a non-negative parameter characterizing the
divergence behavior of the stretching energy,b, the chain Kuhn length,R, an end-to- end distance of a chain defined as
R=
(︃ANb3 φ
)︃1/3
, (5.7)
where A is a topological parameter, whose value, 3√
3/4, corresponds to the net- work with a diamond-like geometry,φis the chain density.
The excluded volume interactions are represented by the entropy of solvent molecules with density φ0 and by the energy of polymer-solvent interactions defined by the Flory-Huggins parameterχ:
Fint kBT = N
φ [φ0lnφ0+χφφ0], (5.8) where the volume of mobile ions (with densitiesφ+and φ−) was accounted for by assumingφ0 =1−φ−φ+−φ−.
The ionic term was defined by the entropy and osmotic pressure of mobile ions as following:
Fion
kBT= 2csN φ
⎡
⎢
⎣1−
⌜
⃓
⃓
⎷1+ (︄
αφ 2cs
)︄2⎤
⎥
⎦+Nln(1−α) (5.9) FintandFiondepend on the degree of gel ionizationα, which in turn is calculated from the electroneutrality condition and Donnan equilibrium via
α
1−α10−(pH−pK) =
⌜
⃓
⃓
⎷1+ (︄
αφ 2cs
)︄2
− αφ
2cs (5.10)
The entire formula for the free energy is a function of the gel densityφand con- tainscs,χand pH−pKas parameters.
By taking a derivative of the hydrogel free energy with respect to the chain molar volume, 1/φ, the hydrogel partial pressure is assessed using the following equation:
p(φ) =− ∂F(φ)
∂(1/φ). (5.11)
5.3.4 Maxwell construction
Under some conditions, the solution of Eq.5.11and results of simulations lead to an unphysical outcome, that is, the increase of gel compression with the decrease in ap- plied pressure, as shown by the loop between white triangles of the black pressure- extension curve in Fig.5.1. This behavior resembles that of real gases model upon their compression, where competition between attractive and repulsive interactions results in phase separation. This similarity allows us to draw an analogy between the hydrogel phase transition theory and the van der Waals theory of the liquid- vapor phase transition [102,103] when the analytical description of liquid/vapor (in our case hydrogels) behavior is represented by unrealistic, non-monotonic curves of applied pressure, temperature, or other state functions on the volume. At this point, the free energy of the system is concave, with two local minima and one local maximum. The minima describe the two equilibrium densities of the gel (swollen
and collapsed), whereas the slope of the line that connects the two minima is the applied pressure with a negative sign. One of the treatments of the free energy con- vexity is the Maxwell construction [104,105], which makes it possible to project the experimental behavior of the system.
Maxwell construction is helpful when only derivatives of free energy are avail- able (in most cases). In this method, the van der Waals loops are replaced by hori- zontal lines, which are drawn so that the loops in the pressure-volume curve define equal areas above and below these lines. The result of the Maxwell construction is the red curve shown in Fig.5.1. Thexaxis is in logarithmic scale and corresponds to the gel density (reciprocal value of the gel volume).
The code (python script) is available onGitHub.