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Chemosensors 2013, 1(3), 43-67; doi:10.3390/chemosensors1030043
Abstract: 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 smart polymer networks are sensitive to different stimuli, such as temperature changes [1,2], illumination [3,4], or properties of the solvent, such as the pH value [1,5,6,7], the ion concentration [7,8], or the chemical composition [5,8,9,10]. Some materials are responsive to more than one kind of stimuli [11,12]. Reversible networks can be switched back and forth between different states by turning the stimulus on and off. Smart polymer networks can be used for various applications, such as chemical sensors [8,13], biosensors [13,14], actuators [7,15], artificial muscles , or for drug transport and release [1,6,16]. For the development and optimization of these devices, it is crucial to investigate the network behavior, both experimentally and theoretically. In this article, we focus on numerical methods for studying intelligent polymer networks and describe modeling techniques on different scales, ranging from macroscopic methods to molecular modeling.
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  and for controlled drug delivery and release [5,17,18]. One important aspect of reversibly crosslinked polymer networks is their ability to rebuild broken bonds and to rearrange in a completely different topology and shape. This way, the polymer network can heal if it has been plastically deformed or broken . On the other hand, a fixed network topology has its own advantages: Even strongly swollen, the gummy bear still keeps its shape.
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 [20,21,22,23,24,25]. We use these examples to describe important aspects of modeling methods on different levels of detail. For a comprehensive description of our numerical studies, we refer to the original articles [20,21,22,23,24,25].
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 . Afterwards, we give a short introduction to self-consistent field theory. The starting point of the self-consistent field theory is the exact statistically physical description of a polymer model system so that constitutive equations are not required. It has been used, for many decades, for the studying of polymer melts. We elucidate an extension of the method that allows modeling reversibly crosslinked polymer networks with switchable microphases [21,22,23]. Finally, we discuss molecular modeling techniques and present, as an example, a Monte Carlo study of a network formed by stiff polymers and physical crosslinkers that, depending on external parameters, can change its structural and topological properties [24,25].
2. Macroscopic Models, Finite Element and Phase Field Methods
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 [26,27,28].
The first model, developed by Flory and Rehner [29,30], is based on the entropy changes in a swelling polymer network. There are various extensions of this statistical approach [31,32,33,34]. The equilibrium state of a hydrogel in a given environment can be calculated by minimizing the free energy of the total system. Relevant contributions to the free energy are the interaction and the entropy of polymer chains, the entropy of ions in the hydrogel and in the bath, and the Coulomb interactions. Thermodynamic hydrogel models have been used, for example, by Caykara et al. , and Kramarenko et al. .
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 . The method has been extended to three-phase (or multiphase) mixture models that include a polymer phase, a solvent phase, and one or several ionic phases, formed by the mobile ions [38,39,40,41]. A multiphase approach, based on the theory of porous media, has been developed by Freiboth et al. .
Macroscopic hydrogel theories are frequently based on conservation laws and constitutive equations [39,40,41,43]. If the ion, solvent, and polymer phases are incompressible, mass conservation leads to:
Frequently, electric neutrality is postulated for the hydrogel system [41,43,44,45]. If negatively charged ions are attached to the polymer network with a concentration cf, electric neutrality requires:
Here, zf and cf are the valence number and the concentration of fixed ions, attached to the polymer network and zk 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:
If the ion concentrations in the solvent are small, so that ϕs and ϕp are distinctly larger than the volume fractions of the mobile ions ϕk (k = 1,..., N), Equation (1) can be approximated by ϕp + ϕs = 1. Then, the volume fraction of water molecules is determined by the polymer volume fraction and does not have to be considered explicitly.
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 et al. , Li et al. , and Chen et al. , to study the swelling and shrinking of hydrogels induced by chemical and electric stimuli.
Following the method used in , here we discuss a typical set of equations used in combination with the Poisson equation. The time dependence of the concentration ck of mobile ions of type k can be described by the Nernst-Planck equation:
The osmotic pressure causes a deformation u(x) of the initial geometry of the hydrogel. Strong, non-linear deformations can be considered by finite strain theory [45,46,48]. If the deformations are small, they fulfill the linear equation:
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, ГW, one has:
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 . For hydrogel spheres in water, the radial polymer density distribution has been calculated as a function of swelling time . Cheng et al. have studied the curling effect of articular cartilage with non-uniform fixed charge density .
A system of particular interest, which is frequently investigated experimentally , consists of a hydrogel strip in water between two electrodes. The electric field from the electrodes is perpendicular to the strip area. The strip is fixated at one end, or in the middle. The diffusion of ions, induced by the electric field, leads to a bending of the non-uniformly swelling strip. The deformation of the strip, which can be used for sensors and actuators in biological and other aqueous environments, could be reproduced and analyzed in detail in various numerical studies [39,43,52].
Equations (6)–(9) are typically solved with finite element methods. Alternatively, meshless methods have been used [39,40,50,53,54,55,56], which are conceptually more complex but avoid the definition and adaptation of a mesh grid during the simulation. In finite element and meshless methods, boundary conditions have to be defined on the hydrogel surface, which typically moves at each time step so that the hydrogel surface has to be traced during the swelling process. Recently, we have developed a phase field model, which avoids the precise localization of the phase boundaries and can be easily coupled with other phase fields that may describe additional phases in the system.
2.1. Phase Field Theory
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, i.e., systems in which at least one phase is a polymer melt or a polymer solution (see, for example, [57,58,59,60,61]). A phase field model for a hydrogel that has been developed in our group  is elucidated in detail below. To that end, we first introduce the concept of the PF method and describe how it has been applied on polymer systems. Our motivation for the model’s development was to introduce a new modeling concept that can be easily adapted to different network properties and external conditions.
The PF method, which is used to describe the non-equilibrium dynamics of phase boundaries, is based on a free energy functional F. If there is only one phase field variable Ψ(r, t), the free energy is of the form:
Here, the flux density is given by:
Considering a flow velocity v, Equation (15) can be extended to the convective Cahn-Hilliard equation with a noise term :
The fluid dynamics can be described by a continuity equation and a (advanced) Navier-Stokes equation, which is given by:
The connection of the Cahn-Hilliard and the Navier-Stokes equation is called “Model H” in the nomenclature of Hohenberg and Halperin . In many cases, the kelast 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 .
The dynamics of non-conserved phase field variables ξ(r, t) is described by the Allen-Cahn equation :
2.2. Phase Field Model of a Hydrogel
Recently, we have developed a PF model for studying the swelling behavior of a hydrogel as the ion concentration in the surrounding is changed . In the following, some details are described to demonstrate the concept. The system is sketched in Figure 1. The extension of the hydrogel is characterized by a phase field variable ξ(r, t). The dynamics of ξ is controlled by Equation (18), with a free energy 
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. Wp and Vm are the potential height and the molar volume of the mixture, respectively. In alloy systems, the coefficients Wp/Vm and Kξ can be extracted from the surface energy and the interface width. For the function h(ξ) ≡ ξ2(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 :
The elastic energy density:
The bath solution is assumed to be a shear-free liquid. Analogously to Equation (8), we ensure mechanical equilibrium for the strain tensor Eμv by an equation:
With and . The concentration of fixed charges is defined by:
In the simulation, a hydrogel with negative fixed ions (zf = −1) is studied in a bath solution with Na+ (z1 = +1) and Cl− (z2 = −1) ions. At the beginning, the concentration of fixed ions in the hydrogel is cf = 4 mol m−3. In the bath, the Na+ and Cl− ions have a concentration of cb = 2 mol m−3. The initial values of c1, c2, and in the hydrogel region are determined by the Donnan equilibrium and electric neutrality conditions
At the boundary of the system, the displacement and the electric potential is kept zero, while the concentration cbnd of mobile ions is continuously changed with time. The time step is chosen to be 0.05s.
In Figure 2, profiles of the phase field ξ(x) are shown. At low ion concentration cbnd ≃ 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 cbnd ≃ 1.892c* (Figure 2b) the hydrogel region becomes more circular in order to minimize the interface energy.
Figure 3 demonstrates the swelling, shrinking and swelling dynamics of the hydrogel as the boundary ion concentration is decreased to cbnd ≃ 0.108c* (time step 54,000), increased to cbnd ≃ 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 . 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].
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 |ċbnd| ≃ 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
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 [78,79], is a well-established method for polymer chains without crosslinks [80,81,82,83,84]. Polymer networks with permanent random crosslinks have been studied with a replica field theory [85,86,87], but the application of the method in numerical studies is difficult, since the fields are defined in the replica space. Recently, we have developed an extended self-consistent field theory, which allows studying smart polymer networks with reversible crosslinks [21,22,23]. In Section 2.1, an application of the method for a blend of reversibly crosslinked ABdiblock copolymers, mixed with A and B homopolymers, is described. The monomers A and B are assumed to have a low compatibility so that the correspnding homopolymers would demix macroscopically in the absence of the AB copolymers. In the AB + A + B mixture, various nanostructures may form. Without reversible crosslinks, the system has been studied in detail theoretically [81,88,89] and experimentally [90,91]. Many aspects of the model can be transferred to other systems with amphiphilic AB copolymers dissolved in a mixture of incompatible solvents. We assume that the AB copolymers can bind with non-covalent crosslinks, as in .
The model system resembles an incompressible polymer blend with a constant overall density ρ0 and three types of polymers j = a, b, c. The system consists of (i) na homopolymers of type a with A monomers and a length Na, (ii) nb homopolymers of type b with B monomers and a length Nb, (iii) nc polymers of type c, which are symmetric diblock copolymers AB that have a length of Nc ≡ N. The system stoichiometry is determined by the volume fraction of copolymers:
We use paths Rj,k(s) with s ∈ [0, Nj] 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.
Without crosslinks, the partition function of the system is given by:
The Flory-Huggins interaction in the exponent includes a Flory parameter χ. The delta function considers the incompressibility of the polymer blend. So far, the ansatz corresponds to that of Edwards and Helfand [78,79]. We have developed an extended model, which considers the formation of reversible crosslinks. The result is the first self-consistent field theory for reversibly crosslinked polymer networks. The method is described in detail in . In the following, we explain the basic concept.
4. Monte Carlo and Molecular Dynamic Simulations of Polymer Networks
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  and . Special simulation methods for polymer systems are described in . In canonical Monte Carlo (MC) simulations, small changes of the molecule configurations are performed randomly at each step and the new configuration is accepted with a probability depending on the energy change. The acceptance criterion produces a sequence of configurations with a Boltzmann distribution. Other ensembles, such as a grand-canonical ensemble or a constant-pressure ensemble can be sampled by allowing changes of system parameters like the number of molecules or the volume of the system. In molecular dynamics (MD) simulations, the positions and velocities of the atoms are obtained from equations of motion that depend on external and interatomic forces. The method allows direct investigation of the dynamical properties of the system. Similar simulation techniques are Brownian dynamic simulations that include a random noise term and diffusive particle dynamic (DPD) simulations, in which the dynamics of interpenetrating spheres resemble the flow behavior of different species in the system (see [93,94]). DPD simulations have been used to investigate the release of colloids from stimuli-responsive microgel capsules . Polyelectrolytes in electric fields have been studied with Brownian dynamics . Compared to field theories, molecular simulations provide more details but are restricted to much smaller time and length scales. In multiscale simulations, material parameters obtained in molecular simulations are used in field-based or other coarse-grained methods on larger scales.
Atomistic simulations of polymers are very time consuming. Consequently, only a few atomistic simulations of polymer networks have been performed [98,99,100,101]. In a multiscale simulation of curing reactions in epoxy networks, atomistic simulations have been combined with DPD simulations . The simulation effort can be reduced by using a coarse-grained model, such as the united atom model . In this case, a polymer is approximated by a chain of beads, where each bead represents a monomer or a sequence of neighboring monomers. A model potential for a polymer chain is often composed of a bond length potential ubl, which depends on the distance of neighboring beads, a bending potential ubend, which considers the angle between neighboring bond vectors, and a potential utors for the local torsion of the polymer chain. Then, the potential energy of a linear polymer with N beads is given by:
5. Summary and Conclusions
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  or by applying an irradiation pattern on a thermosensitive gel .
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”.
Conflict of Interest
The authors declare no conflict of interest.
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