# Simulation of Stimuli-Responsive Polymer Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Macroscopic Models, Finite Element and Phase Field Methods

_{p}and ϕ

_{s}are the volume fractions of the polymer network and the solvent phase, while ϕ

_{1},...,ϕ

_{N}are the volume fractions of the N ion phases. The volume fractions ϕ

_{k}are proportional to corresponding concentrations c

_{k}. In the absence of chemical reactions, the concentrations obey the conservation law:

_{f}, electric neutrality requires:

_{f}and c

_{f}are the valence number and the concentration of fixed ions, attached to the polymer network and z

_{k}is the valence number of the mobile ion species k. In several models, the dynamics of the system is studied with the help of momentum equations [37,38,39,40,41,43,46]. For the solvent and the ion phases, the forces induced by the chemical potential of solvent molecules, and the electrochemical potentials of the ions, are balanced by the friction forces between the different phases that flow with different velocities. The equations are of the form:

^{(k)}. The mass densities ρ

_{k}are given by ρ

_{k}= m

_{k}c

_{k}, where m

_{k}is the mass of a solvent molecule (k = s) or an ion (k = 1,…, N). The constants α

_{m,k}are friction coefficients between phases k and m. For the polymer phase, one has:

_{k}σ

_{kλ}= 0

_{kλ}are the components of the stress tensor. Furthermore, several constitutive equations are used: Usually, the chemical potential is expressed by a linear combination of the ion concentrations and the hydrostatic pressure, while the electrochemical potentials of the ions can be expressed as a function of the chemical activity term, which is proportional to ln(γ

_{k}c

_{k}) with an activity coefficient γ

_{k}, and a term that is proportional to the electric potential [39]. Furthermore, a linear or nonlinear relation between the stress tensor of the polymer network and its elastic strain tensor is considered [28].

_{s}and ϕ

_{p}are distinctly larger than the volume fractions of the mobile ions ϕ

_{k}(k = 1,..., N), Equation (1) can be approximated by ϕ

_{p}+ ϕ

_{s}= 1. Then, the volume fraction of water molecules is determined by the polymer volume fraction and does not have to be considered explicitly.

_{k}of mobile ions of type k can be described by the Nernst-Planck equation:

^{4}Cmol

^{−1}, the gas constant R ≃ 8.314Jmol

^{−1}K

^{−1}, and the temperature T. Ions of species k have diffusion constants D

_{k}. The Nernst-Planck equation corresponds to the continuity equation Equation (2) under the assumption that the ion flow is determined by the concentration gradient and the electric potential . For systems that show chemical reactions or a non-ideal mixing behavior of the solvent, extended versions of Equation (5) have been used, which include a term for the chemical activity or other source terms for the ions [49]. In [39], a convection term −∇

_{μ}(c

_{k}V

_{μ}), with a flow velocity V, is added to the right side of Equation (5). The electric potential obeys the Poisson equation:

_{0}and the relative dielectric constant ∊. The osmotic pressure depends on the differences c

_{k}− between the ion concentrations c

_{k}in the hydrogel and the respective concentrations at the external boundary of the hydrogel:

_{μ}P

_{μv}= ∇

_{μ}(λ

_{h}E

_{λλ}δ

_{μv}+ 2μ

_{h}E

_{μv}− p

^{(osm)}δ

_{μv}) = 0

_{μv}and δ

_{μv}are components of the full pressure tensor in the hydrogel region and the unit tensor, respectively. The coefficients λ

_{h}and μ

_{h}are the first and second Lamé coefficients of the hydrogel. The Green-Lagrangian strain tensor E

_{μv}is given by:

_{W}, one has:

_{μv}n

_{v}= W

_{μ}at Г

_{W}

#### 2.1. Phase Field Theory

_{inh}(∇Ψ))

_{1}and c

_{2}. An overview over phase field methods and their applications are given in [62]. Usually, one has an extended region in space in which the phase field variable is Ψ ≃ 1 and another region in which the phase is absent and Ψ ≃ 0. Both regions are separated by an interface region of finite width, in which the phase field decreases smoothly. If Ψ is conserved, the dynamics of the phase field is described by a Cahn-Hilliard equation [63]:

_{surf}and considers the surface tension from the interfaces [59]. The term k

_{ext}represents external forces, such as gravity, while denotes the elastic response of polymers to a flow field, leading to viscoelastic behavior [67]. An alternative description of viscoelasticity is the Oldroyd-B model [68], which has been used, together with the PF method, to study the elongation and burst of viscoelastic droplets [69] and the coalescence of polymer drops and interfaces [61,70].

_{elast}part is neglected so that the polymer phase is described as a Newtonian fluid. In principle, all terms on the right hand side (beside P) can be coupled to the phase field variable Ψ, as the involved material parameters may depend on the composition. The model H has been used to study spinodal decomposition with and without external velocity fields and Rayleigh-Taylor instabilities [57,59,72]. The described method with slightly different Cahn-Hilliard free energy has been used to study the shape of fluid films on a dewetting substrate [73].

_{ξ}is the mobility coefficient and F consists again of a bulk free energy density and a gradient term. In various studies, the Allen-Cahn equation has been used to investigate the growth of polymer crystals [60,75,76]. In the following example, it is used to characterize the extension of the hydrogel [20].

#### 2.2. Phase Field Model of a Hydrogel

_{p}and V

_{m}are the potential height and the molar volume of the mixture, respectively. In alloy systems, the coefficients W

_{p}/V

_{m}and K

_{ξ}can be extracted from the surface energy and the interface width. For the function h(ξ) ≡ ξ

^{2}(3 − 2ξ) one has h(0) = 0, h(1) = 1, and h′(0) = h′(1) = 0, so that the derivative of the first part of F is only finite in the interface region. The term ∆

_{p}≡ p

^{(osm)}− p

^{(eq)}is the difference between the osmotic pressure and p

^{(eq)}, where p

^{(eq)}is the osmotic pressure at the initial equilibrium configuration from which the displacement u is measured. The osmotic pressure p

^{(osm)}is defined as [20]:

_{h}and in the surrounding bath region λ

_{s}.

**Figure 1.**Sketch of a hydrogel system with a polymer network, a surrounding bath, ions attached to the network, and mobile cations and anions. In the phase field method, all system components are represented by continuous field variables. The extension of the hydrogel is characterized by a phase field variable, which is ξ = 1 inside the hydrogel region (light green area). At the interface to the bath, the phase field decreases smoothly to ξ = 0 [20].

_{μv}by an equation:

^{(eq)}are the concentration of fixed ions and the phase field variable at the initial equilibrium state. Note that ∫ dV c

_{f}(r, t) = ∫ dV at all times. In the bath region, the concentration of fixed charges is zero inside the bath region while it is a finite constant inside the hydrogel region. The fields ξ, c

_{k}, c

_{f}, , and E

_{μv}are coupled via Equations (19)–(23), which are solved simultaneously.

**Figure 2.**Profiles of the phase field ξ(x) that represents the hydrogel region for boundary ion concentrations (

**a**) c

_{bnd}≃ 0,108c* and (

**b**) c

_{bnd}≃ 1.892c* [Springer, Colloid Polymer Science, 289, 2011, 513-521, Phase field model simulations of hydrogel dynamics under chemical stimulation, Li, D., Yang, H.L., Emmerich, H., Figure 4, © Springer-Verlag 2011] with kind permission from Springer Science+Business Media B.V.

_{f}= −1) is studied in a bath solution with Na

^{+}(z

_{1}= +1) and Cl

^{−}(z

_{2}= −1) ions. At the beginning, the concentration of fixed ions in the hydrogel is c

_{f}= 4 mol m

^{−3}. In the bath, the Na

^{+}and Cl

^{−}ions have a concentration of c

_{b}= 2 mol m

^{−3}. The initial values of c

_{1}, c

_{2}, and in the hydrogel region are determined by the Donnan equilibrium and electric neutrality conditions

_{bnd}of mobile ions is continuously changed with time. The time step is chosen to be 0.05s.

_{bnd}≃ 0.108c* (Figure 2a) the gel is strongly swollen and shows a rather quadratic shape, due to the quadratic boundary of the simulation cell. At large ion concentration c

_{bnd}≃ 1.892c* (Figure 2b) the hydrogel region becomes more circular in order to minimize the interface energy.

**Figure 3.**Evolution of the gel fraction with time. In the shown time period, the boundary ion concentration is linearly decreased from c

_{bnd}= 1c* to c

_{bnd}≃ 0.108c*, then increased to c

_{bnd}≃ 1.892c*, and decreased again to c

_{bnd}= 1c* [Springer, Colloid Polymer Science, 289, 2011, 513-521, Phase field model simulations of hydrogel dynamics under chemical stimulation, Li, D., Yang, H.L., Emmerich, H., Figure 7, © Springer-Verlag 2011] with kind permission from Springer Science+Business Media B.V.

_{bnd}≃ 0.108c* (time step 54,000), increased to c

_{bnd}≃ 1.892c* (time step 163,500), and decreased again to the initial value. The gel fraction follows the changes of the boundary ion concentration with a short delay, which is too small to be visible in Figure 2 but has been investigated in the numerical study [20]. In general, the equilibrium gel volume is monotonously shrinking with increasing ion concentration in the bath. The result is in qualitative agreement with experiments, in which the pH is kept constant and the ion concentration is varied [35,77].

_{bnd}| ≃ 71 mol m

^{−3}s

^{−1}for the boundary ion concentration, the hydrogel swells and shrinks without significant hysteresis. The method goes without explicit boundary conditions at the hydrogel surface and can be easily adapted to other hydrogel systems. For example, one can study a hydrogel shell that includes an active agent. Then, a concentration field of drug molecules can be added to the given model in order to study the drug release kinetics.

## 3. Self-Consistent Field Theory

_{0}and three types of polymers j = a, b, c. The system consists of (i) n

_{a}homopolymers of type a with A monomers and a length N

_{a}, (ii) n

_{b}homopolymers of type b with B monomers and a length N

_{b}, (iii) n

_{c}polymers of type c, which are symmetric diblock copolymers AB that have a length of N

_{c}≡ N. The system stoichiometry is determined by the volume fraction of copolymers:

_{j,k}(s) with s ∈ [0, N

_{j}] to parametrize the configuration of the kth polymer of type j. All chains have a Gaussian shape distribution . A and B monomers have a local density that can be divided into the density from type a homopolymers and the density from the A part of the copolymers. Analogously, one has for the density of B monomers.

#### Self-Consistent Field Theory for Reversibly Crosslinked Polymer Networks

_{A}w

_{A}, zw

_{A}w

_{B}, or zw

_{B}w

_{B}, respectively. Here, z is the crosslink strength and w

_{A}and w

_{B}are weighting factors, considering the types of the crosslinked monomers. The partition function can then be written as [21,22,23]:

_{A}+ w

_{B}= 1, we use the crosslink strength z and the crosslink asymmetry ∆w ≡ w

_{A}− w

_{B}to specify the crosslink properties. For convenience, the crosslink strength is represented by the parameter z

_{rel}= zρ

_{0}/χ.

_{g}of a free Gaussian polymer of length N. Dimensionless densities are given by , while we use dimensionless free energies of the form for a system with d dimensions. We use z

_{red}= zρ

_{0}/χ for the crosslink strength. The dimensionless mean field free energy can be divided into three terms [21]:

_{k}is the partition function of a single polymer chain of type k fluctuating in the auxiliary fields (see [20]). It can be calculated numerically by solving modified diffusion equations. In the saddle point approximation, the auxiliary fields and and, finally, the desired monomer densities can be determined by considering that the variations of the free energy with respect to , , and must vanish. An algorithm is described in [21].

_{rel}, the crosslink asymmetry ∆w, and the relative A fraction of homopolymers The investigated system has homopolymers of lengths N

_{a}= N

_{b}= N/5 with N ≡ N

_{c}, a Flory parameter of

_{χ}N = 11.3 and a volume fraction ϕ

_{c}= 0.6 of copolymers. In the studied parameter range, the system can form a lamellar phase or a hexagonal phase, as shown in the sketches and the calculated A monomer densitities in Figure 4. The phase diagram has been determined by studying unit cells with a box length ratio of L

_{y}/L

_{x}= . The geometry supports the formation of a hexagonal as well as a lamellar phase. We have varied L

_{x}in the range of 3.0 ≤ L

_{x}≤ 4.8 and determined the phase with the lowest free energy density. We find that for and z

_{rel}> 0, only lamellar structures are stable. Otherwise, the hexagonal phase is stable if z

_{rel}and |∆w| are suitably small. In Figure 5, the phase boundaries between the hexagonal and the lamellar phase are shown for various A fractions of the homopolymers in the range of .

**Figure 4.**Nanostructures formed by a polymer blend of A and B homopolymers and reversibly crosslinked AB block copolymers. Shown are sketches of (

**a**) the hexagonal phase and (

**b**) the lamellar phase. Blue and yellow lines represent the A and B parts of the copolymers, yellow circles denote crosslinkers. A and B homopolymer densities are indicated by the light blue and yellow background colors. Results of self-consistent field theory calculations for the monomer density ρ

_{A}(x) in (

**c**) a hexagonal, and (

**d**) a lamellar structure [22]. Figure (a) and (b) reproduced with permission from Thomas Gruhn, Heike Emmerich: Phase behavior of polymer blends with reversible crosslinks–A self-consistent field theory study. Journal of Material Research; Materials Research Society, in preparation, doi:10.1557/jmr.2013.315.

**Figure 5.**Phase diagram of the AB + A + B polymer blend: The colored curves denote the phase boundaries between the hexagonal phase on the inside (small |∆w|) and the lamellar phase on the outside. Phase boundaries are shown for ϕ

_{A}= 0.75 (dark blue), ϕ

_{A}= 0.7 (red), ϕ

_{A}= 0.65 (green), ϕ

_{A}= 0.6 (brown), ϕ

_{A}= 0.59 (magenta), ϕ

_{A}= 0.58 (cyan) [22]. Reproduced with permission from Thomas Gruhn, Heike Emmerich: Phase behavior of polymer blends with reversible crosslinks–A self-consistent field theory study, Journal of Material Research; Materials Research Society, in preparation, doi:10.1557/jmr.2013.315.

## 4. Monte Carlo and Molecular Dynamic Simulations of Polymer Networks

_{bl}, which depends on the distance of neighboring beads, a bending potential u

_{bend}, which considers the angle between neighboring bond vectors, and a potential u

_{tors}for the local torsion of the polymer chain. Then, the potential energy of a linear polymer with N beads is given by:

_{i, i−1}is the distance vector between the centers of mass of neighbor atoms i−1 and i, θ

_{i}is the angle between r

_{i+1, i}and r

_{i, i−1}, while ϕ

_{i, i+1}is the angle between the vectors r

_{i, i−1}× r

_{i+1, i}and r

_{i+1, i}× r

_{i+2, i+1}. This type of potentials has been used in Gibbs ensemble MD simulations of swelling polymer networks [104]. In a more coarse-grained approach, each bead can represent a chain sequence that is distinctly longer than the persistence length of the polymer. In this case, the angular dependent parts of the polymer potential can be neglected. The remaining bond length potential is typically modeled by a harmonic spring or a finitely extensible nonlinear elastic (FENE) potential [105,106,107]. The resulting chains are rather flexible and can be used to represent suitably long polymers on the respective length scale. If the persistence length is comparable with the polymer length, which is often the case for short biopolymers, a wormlike chain model is frequently used, which includes the bond-length and the bending potential but neglects the torsion [108,109]. In the case of very stiff polymers, one can also neglect the bending and bond-length fluctuations and represent the whole polymer by a rigid rod. While this approach neglects many internal degrees of freedom, it enables very efficient sampling of the configuration space. We elucidate the method and facilities of rigid rod simulations, using the example of our filament network simulations [24,25].

#### Monte Carlo Simulation of Filament Networks with Reversibly Binding Crosslinkers

_{F}and L = l

_{L}for filaments and crosslinkers, respectively. The configuration of a filament j is determined by the center of mass r

_{j}and a unit vector u

_{j}that is parallel to the cylinder axis. Steric interactions between spherocylinders i and j are considered by a potential U

_{steric}(r

_{i}, u

_{i}, r

_{j}, u

_{j}) that gets infinite if the two spherocylinders overlap. Each crosslinker, i, has adhesive sites at the both ends, with which it can physically bind to a neighboring filament, j. The definition of the adhesion potential is based on the shortest distances between the axis of filament, j, and the adhesive sites of the crosslinker, i:

_{a}away from the crosslinker’s center of mass. We use l

_{C}= 2D and l

_{a}= 1.35D, so that the adhesion sites lie inside the hemispheres of the crosslinkers.

_{acc}= min(1, exp(−∆U/(k

_{B}T)), where ∆U is the change of the configurational energy.

_{lf}, and the adhesion strength ∊. The phase diagram region, in which more than 20% of the filaments are part of a bundle, is called the bundle region. The percolation threshold is determined by studying the cluster size distribution n

_{s}and the fraction of filaments in the largest cluster a

_{max}: At the percolation threshold, the fraction n

_{s}of clusters containing s filaments is of the form n

_{s}∝ s

^{τ}with the Fisher exponent τ. For a sufficiently large system, one has at the percolation threshold. In Figure 6a, phase diagram of the system is shown as a function of the adhesion strength ∊ /T and the filament volume fraction ϕ for fixed crosslinker-filament ratio n

_{lr}= 2. A percolated network forms if the filament volume fraction and the adhesion strength are large enough. This means, that a percolated network can be created and destroyed, reversibly, by changing the hydrogel volume or the temperature. Furthermore, the system forms bundles for suitably low filament volume fractions. This can be explained as follows: The system favors crosslinkers that are bound to filaments on both ends. Parallel filaments can be interconnected by a large number of crosslinks forming a ladder-like structure. At low filament concentrations, only aligned groups of filaments make it possible that large numbers of crosslinkers can bind on both ends. At high filament concentrations, crosslinkers can bind on both ends, even if the filament network is disordered. The phase diagram shows that percolated network may form with and without pronounced bundling and bundling can form with and without the formation of a percolated network.

_{lr}= 2 and a fixed filament volume fraction, the binding strength ∊

_{t}at which percolation sets in, is rather independent of the filament length that has been varied in the range between l

_{F}= 10D and l

_{F}= 25D. The results could be reproduced qualitatively with a simple analytic model. Note, that the independence of the length is given at a fixed ratio n

_{lr}of filament to crosslink number density. In practice, the ratio r of crosslinkers to filament monomers can be more important. For suitably large filament lengths one has so that the percolation threshold is independent of the filament length if rl

_{F}is kept fixed.

**Figure 6.**Model network with stiff filaments (blue) and short crosslinkers (yellow) with adhesive ends (red), studied with Monte Carlo simulations. (

**a**) Snapshot of a network structure. (

**b**) Phase diagram as a function of the adhesion strength over temperature ∊/T and the filament volume fraction ϕ. Figures from (R. Chelakkot, R. Lipowsky and T. Gruhn, Soft Matter, 2009, 5, 1504)—Reproduced with permission of The Royal Society of Chemistry.

## 5. Summary and Conclusions

## Acknowledgments

## Conflict of Interest

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Gruhn, T.; Emmerich, H.
Simulation of Stimuli-Responsive Polymer Networks. *Chemosensors* **2013**, *1*, 43-67.
https://doi.org/10.3390/chemosensors1030043

**AMA Style**

Gruhn T, Emmerich H.
Simulation of Stimuli-Responsive Polymer Networks. *Chemosensors*. 2013; 1(3):43-67.
https://doi.org/10.3390/chemosensors1030043

**Chicago/Turabian Style**

Gruhn, Thomas, and Heike Emmerich.
2013. "Simulation of Stimuli-Responsive Polymer Networks" *Chemosensors* 1, no. 3: 43-67.
https://doi.org/10.3390/chemosensors1030043