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The structure and material properties of polymer networks can depend sensitively on changes in the environment. There is a great deal of progress in the development of stimuli-responsive hydrogels for applications like sensors, self-repairing materials or actuators. Biocompatible, smart hydrogels can be used for applications, such as controlled drug delivery and release, or for artificial muscles. Numerical studies have been performed on different length scales and levels of details. Macroscopic theories that describe the network systems with the help of continuous fields are suited to study effects like the stimuli-induced deformation of hydrogels on large scales. In this article, we discuss various macroscopic approaches and describe, in more detail, our phase field model, which allows the calculation of the hydrogel dynamics with the help of a free energy that considers physical and chemical impacts. On a mesoscopic level, polymer systems can be modeled with the help of the self-consistent field theory, which includes the interactions, connectivity, and the entropy of the polymer chains, and does not depend on constitutive equations. We present our recent extension of the method that allows the study of the formation of nano domains in reversibly crosslinked block copolymer networks. Molecular simulations of polymer networks allow the investigation of the behavior of specific systems on a microscopic scale. As an example for microscopic modeling of stimuli sensitive polymer networks, we present our Monte Carlo simulations of a filament network system with crosslinkers.

If you put a gummy bear into a glass of water, the next day you will find the same bear grown in volume by more than a factor of five. (The effect depends on the choice of Gummy bears.) This is a very simple example of a polymer network that is strongly responsive to the stimuli of the environment (in this case humidity). Different kinds of

There are two important types of smart polymer networks: (a) Covalently crosslinked networks that swell or shrink if exposed to a stimulus, and (b) networks with a variable topology, in which crosslinks can be broken or created by external stimuli.

Both types of smart polymer networks can change their density, their mechanical properties, and their permeability in response to changes of the environment. The majority of applications use chemically crosslinked polymer networks (see the application references in the first paragraph), however, networks with stimuli-sensitive crosslinks are also used for the development of sensors [

This article is dedicated to numerical studies of smart polymer networks that are sensitive to external stimuli. Modeling methods are discussed on different levels of details, from macroscopic theories to molecular modeling. The introduction to the different techniques is combined with more detailed descriptions of three numerical methods that have been developed by the authors. We think that they are particularly interesting examples of numerical studies of intelligent polymer networks that show novel routes for polymer network modeling [

We start with macroscopic models, in which all material properties are described by continuous fields. Here, we discuss various modeling principles, before we explain, in more detail, a phase field model for chemo-sensitive hydrogels that was recently presented by one of the authors [

Stimulated by external conditions, such as a change of the pH or the chemical composition of the solvent, the volume of hydrogels can change strongly. For the investigation of hydrogel swelling on a macroscopic scale, various macroscopic models have been developed. Overviews on macroscopic hydrogel models have been presented in [

The first model, developed by Flory and Rehner [

The equilibrium ion distribution in the hydrogel and in the bath can be obtained from the Donnan equilibrium and the postulation of electric neutrality, as discussed below.

An important model for hydrogels is the multiphase mixture model, in which the system consists of two or more phases. In the biphasic theory, the system is divided into an incompressible polymer phase and an incompressible fluid phase that is in contact with a fluid reservoir [

Macroscopic hydrogel theories are frequently based on conservation laws and constitutive equations [_{p}_{s}_{1},...,_{N}_{k}_{k}

Frequently, electric neutrality is postulated for the hydrogel system [_{f}

Here, _{f}_{f}_{k}^{(k)}. The mass densities _{k}_{k}_{k}_{k}_{k}_{m,k}_{k}σ_{kλ}_{kλ}_{k}c_{k}_{k}

If the ion concentrations in the solvent are small, so that _{s}_{p}_{k}_{p}_{s}

The flow of ions can be treated more generally by using the Poisson equation instead of the electric neutrality condition. Models that include the Poisson equation have been used, for example, by Wallmersperger

Following the method used in [_{k}^{4}^{−1}, the gas constant ^{−1} ^{−1}, and the temperature _{k}_{μ}_{k}_{μ}_{0} and the relative dielectric constant _{k}_{k}

The osmotic pressure causes a deformation _{μ}P_{μv}_{μ}_{h}E_{λλ}δ_{μv}_{h}E_{μv}^{(osm)}_{μv}_{μv}_{μv}_{h}_{h}_{μv}

For small deformations, the third term in the brackets can be neglected. Solving Equations (6)–(9), numerically, requires appropriate boundary conditions. If the hydrogel is firmly attached to a substrate, the corresponding surface region is fixed. For the remaining surface region, Г_{W}_{μv}n_{v}_{μ}_{W}

Various types of stimuli-induced swelling and deformation of hydrogels have been investigated with transport theories. A numerical study of temperature-dependent swelling is presented in [

A system of particular interest, which is frequently investigated experimentally [

Equations (6)–(9) are typically solved with finite element methods. Alternatively, meshless methods have been used [

The phase field method (PF method in the following) is generally used to study multiphase systems with interfacial dynamics. It allows following the dynamics of the system interfaces inherently coupled to all relevant physical and chemical driving fields. In recent years, there has been a growing number of articles that apply the PF method on polymer systems,

The PF method, which is used to describe the non-equilibrium dynamics of phase boundaries, is based on a free energy functional _{inh}_{1} and _{2}. An overview over phase field methods and their applications are given in [

Here, the flux density is given by:

Considering a flow velocity

The fluid dynamics can be described by a continuity equation and a (advanced) Navier-Stokes equation, which is given by:
_{surf}_{ext}

The connection of the Cahn-Hilliard and the Navier-Stokes equation is called “Model H” in the nomenclature of Hohenberg and Halperin [_{elast}

The dynamics of non-conserved phase field variables _{ξ}

Recently, we have developed a PF model for studying the swelling behavior of a hydrogel as the ion concentration in the surrounding is changed [

The second term is a double well potential as used in Landau theory. The second and third term provide an interface region of finite width. _{p}_{m}_{p}_{m}_{ξ}^{2}(3 − 2_{p}^{(osm)} − ^{(eq)} is the difference between the osmotic pressure and ^{(eq)}, where ^{(eq)} is the osmotic pressure at the initial equilibrium configuration from which the displacement ^{(osm)} is defined as [

The elastic energy density:
_{h}_{s}

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

The bath solution is assumed to be a shear-free liquid. Analogously to Equation (8), we ensure mechanical equilibrium for the strain tensor _{μv}

With ^{(eq)} are the concentration of fixed ions and the phase field variable at the initial equilibrium state. Note that _{f}_{k}_{f}_{μv}

Profiles of the phase field _{bnd}_{bnd}

In the simulation, a hydrogel with negative fixed ions (_{f}^{+} (_{1} = +1) and Cl^{−} (_{2} = −1) ions. At the beginning, the concentration of fixed ions in the hydrogel is _{f}^{−3}. In the bath, the Na^{+} and Cl^{−} ions have a concentration of _{b}^{−3}. The initial values of _{1}, _{2}, and

At the boundary of the system, the displacement and the electric potential is kept zero, while the concentration _{bnd}

In _{bnd}_{bnd}

Evolution of the gel fraction with time. In the shown time period, the boundary ion concentration is linearly decreased from _{bnd}_{bnd}_{bnd}_{bnd}

_{bnd}_{bnd}

Our new phase field model is an alternative approach for studying the dynamics of hydrogel swelling induced by the ion concentration of the bath. With the chosen decrease or increase rate |_{bnd}^{−3} ^{−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.

The macroscopic theories of polymer networks depend on phenomenological, constitutive equations. Microscopic aspects of the polymer system can be studied with molecular simulations but they are restricted to small time and length scales. An alternative method for polymer modeling is the self-consistent field theory. Here, polymers are represented by continuous lines. They interact by steric and attractive interactions, where the latter is weighted by the Flory parameter. The method takes into account the connectivity of polymer chains, the fact that two spatially separated monomers are much more correlated if they belong to the same polymer. For the given model system, the theory starts from an exact partition function. Then, the equilibrium properties are typically determined with the help of a saddle-point approximation. The self-consistent field theory, which has been introduced by Edwards and Helfand [

The model system resembles an incompressible polymer blend with a constant overall density _{0} and three types of polymers _{a}_{a}_{b}_{b}_{c}_{c} ≡ N

We use paths _{j,k}_{j}

Without crosslinks, the partition function of the system is given by:

The Flory-Huggins interaction in the exponent includes a Flory parameter

We apply our extended self-consistent field theory on an _{A}w_{A}_{A}w_{B}_{B}w_{B}_{A}_{B}

Setting _{A}_{B}_{A} − w_{B}_{rel}_{0}/

With the help of the Hubbard-Stratonovich (HS) transformation a free energy can be derived, which can be solved with a saddle-point approximation. We present the resulting free energy expression with dimensionless quantities, indicated by a tilde. Dimensionless lengths are defined as _{g}_{red}_{0}/_{k}

The method has been used to calculate a phase diagram as a function of the relative adhesion strength _{rel}_{a}_{b}_{c}_{χ}N_{c}_{y}_{x}_{x}_{x}_{rel}_{rel}

The self-consistent field theory study shows that the reversibly crosslinked polymer network in an

Nanostructures formed by a polymer blend of A and B homopolymers and reversibly crosslinked AB block copolymers. Shown are sketches of (_{A}

Phase diagram of the _{A}_{A}_{A}_{A}_{A}_{A}

Properties of polymer networks can be studied on a microscopic level with the help of Monte Carlo simulations and molecular dynamic simulations, in which molecular details of the system can be investigated explicitly. Detailed descriptions of molecular simulation techniques can be found in [

Atomistic simulations of polymers are very time consuming. Consequently, only a few atomistic simulations of polymer networks have been performed [_{bl}_{bend}_{tors}_{i, i−1} is the distance vector between the centers of mass of neighbor atoms _{i}_{i+1, i} and _{i, i−1}, while _{i, i+1} is the angle between the vectors _{i, i−1} × _{i+1, i} and _{i+1, i} × _{i+2, i+1}. This type of potentials has been used in Gibbs ensemble MD simulations of swelling polymer networks [

We have used this method by studying a system of rigid filaments that can form filament networks with the help of reversibly binding crosslinkers [

We have studied the system with the help of Monte Carlo simulations. The filaments are represented by long hard spherocylinders, while the crosslinkers are modeled by short spherocylinders with specifically binding ends. A snapshot of the system is given in _{F}_{L}_{j}_{j}_{steric}_{i}_{i}_{j}_{j}_{a}_{C}_{a}

The adhesion is mimicked by a square-well potential:

The filament hydrogel is modeled with the help of a Metropolis algorithm. At each step, position and/or orientation of a filament or a crosslinker is changed randomly by a small amount, and the new configuration is accepted with a probability _{acc}_{B}T

In the simulations, we have studied the occurrence of filament bundles and the percolation behavior of the system. A cluster is defined as a set of filaments that are interconnected by crosslinker bonds. A filament bundle is a cluster in which the filaments are suitably well aligned. We have analyzed the average structure of the filament hydrogel in phase diagrams as a function of the filament volume fraction _{lf}_{s}_{max}_{s}_{s} ∝ s^{τ}_{lr}

The network and bundle formation can also be induced by increasing the crosslink-filament ratio. Another interesting aspect is the dependence of the percolation threshold on the filament length [_{lr}_{t}_{F}_{F}_{lr}_{F}

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

Various macroscopic models and molecular simulation techniques have been developed, which allow the study of various kinds of stimuli-sensitive polymer networks. With these methods, the response of polymer networks on external stimuli can be analyzed and optimized on various time and length scales. The behavior of a hydrogel can depend, in a complex way, on many physical and chemical properties of the environment, such as the pH value, ion concentrations, the concentration of specific molecules, external electric fields, temperature, or illumination. The development and optimization of tailored, stimuli-sensitive polymer networks requires a good understanding of the interplay of all these aspects. Here, numerical modeling and simulations are crucial. Molecular simulations require a comparably low amount of parameters. For atomistic simulations only reasonable interatomic potentials are needed, which, in many cases, are provided by simulation software. However, time scales and system sizes that can be studied with atomistic simulations are small so that only a small number of network meshes can be studied. More coarse-grained simulations require model interaction parameters, which are often not available for the polymer network of interest. The parameters can be extracted from atomistic simulations on a small length scale or by comparing material properties of the model system with experimental measurements. If the parameters are found, very detailed information can be found about system properties like the swelling behavior of the network, the breaking and healing of bonds, changes of the polymer configurations and the network structure or, for networks with suitably small mesh sizes, the elastic and inelastic responses of the network on applied strains. Results of molecular simulations help to interpret the experimental measurements but they can also support macroscopic models, which depend on material properties such as the elastic constants of the polymer network for a given composition of the solvent or the permeability of a hydrogel for a specific molecule. For systems with chain lengths that are too large for molecular simulations, approaches like the self-consistent field theory can be a good alternative, especially, if the system consists of copolymers and forms nanostructures. Our extended self-consistent field theory allows the modeling of polymer systems on an intermediate length scale and helps to fill the gap between macroscopic theories and molecular simulations. Thus far, only a very small number of multi-scale methods have been used for the modeling of stimuli-sensitive networks on various length scales. This is one of the major tasks for the future.

One particularly important aspect of smart polymer networks is the response time. Quantitative predictions of absolute time scales are a big challenge for numerical methods. Many macroscopic models are restricted to stationary or steady-state systems. Macroscopic numerical studies of network dynamics are based on parameters like diffusion constants that have to be taken from molecular simulations or experiments. This shows, once more, the necessity of numerical studies on various levels of detail and of multiscale studies that combine these methods.

Another challenge is the numerical study of polymer networks with spatial inhomogeneities on the microscale rather than on the nanometer scale. Experimentally, such inhomogeneities can be inferred, for example, by differential photo-crosslinking of polymers [

As documented in this article, numerical methods for small and large time scales are available such that in the future multi-scale simulations should play an important role for a comprehensive understanding of these fascinating and multifunctional materials.

The authors like to thank Daming Li for his contributions to the self-consistent field theory of reversibly crosslinked polymer networks and the phase field model of hydrogels. We also like to thank Hong Liu Yang for supporting the phase field studies of hydrogels. The simulations of the systems with stiff filament and crosslinkers have been carried out together with Raghunath Chelakkot and Reinhard Lipowsky, whom we would also like to thank. The work on this article was supported by DFG SPP 1259: “Intelligente Hydrogele, modeling and simulation of hydrogel swelling under strong non-equilibrium conditions using the phase-field and phase-field crystal methods” and the DFG SFB 840 “Von partikulären Nanosystemen zur Mesotechnologie”.

The authors declare no conflict of interest.

_{2}responsive crosslinking/decrosslinking system