Typically, hydrogels are three-dimensional networks of either chemically or physically crosslinked polymers that swell upon the addition of water. The ratio of the swollen volume to the dry volume of the polymer matrix is often referred to as the “degree of swelling,” and is a common parameter for describing many hydrogels. The degree of swelling of poly [acrylamide-stat-(acrylic acid)] hydrogels, for example, may be as high as 20,000, which indicates a volumetric expansion greater than four orders of magnitude. The degree of swelling is dependent on polymer composition and architecture, as well as inherent properties of the aqueous solution such as temperature or ionic strength.
Equilibrium Swelling Theory
The equilibrium swelling theory of neutral, isotropic polymer networks in the presence of small molecules was first described by Flory and Rehner. The Flory–Rehner model was proposed for the structure of a crosslinked polymer network immersed in a good solvent where the free energy of mixing from osmotic pressure induces solvent migration into the network. The model assumes both a Gaussian distribution of polymer chain lengths and average crosslinks to be tetrafunctional. The theory considers forces arising from three sources:
- the entropy change associated with mixing of polymer and solvent
- the entropy change arising from reduced polymer conformations upon swelling
- the enthalpy of mixing the polymer and solvent
The entropy change from polymer-solvent mixing is positive and favors swelling, while the entropy change from chain stretching (reduction of the number of possible conformations) is negative and opposes swelling. The enthalpy of mixing, which is dependent on the gel composition, can be either positive (opposing mixing), negative (favoring mixing), or zero.
Swelling of the gel is a function of elastic retractive forces of the polymer chains and the expansive thermodynamic contribution of mixing of polymer and solvent. From this, the free energy of a neutral hydrogel in the absence of charged species can be expressed as:
where ΔGel is the free energy of elastic retractive forces and ΔGmix is the free energy of polymer and solvent mixing. The term ΔGmix is a measure of the compatibility of the polymer with the surrounding solvent molecules. At equilibrium conditions, the net chemical potential (μ) must equal zero:
Therefore, any changes in the chemical potential due to mixing (μmix) are balanced by elastic retractive forces (μel) of the network. The change in chemical potential due to such forces can be expressed by the theory of rubber elasticity proposed by Edwards.
Polymer Volume Fraction
Among the many parameters used to characterize hydrogels, the polymer volume fraction in the swollen state (ν2,s), the molecular weight of the polymer chain between crosslinks ( ), and the mesh size of the gel (ξ) are among the most informative, especially for drug delivery applications (Fig. 1). These three parameters can be determined using the equilibrium swelling theory of rubber elasticity modified by Flory.
Figure 1. Schematic representation of a polymeric hydrogel where the molecular weight of the polymer chain between crosslinks ( ), and the mesh size of the gel (ξ) are represented.
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The swollen state polymer volume fraction ν2,s (analogous to the degree of swelling) is the ratio of the polymer volume (Vp) to the swollen gel volume (Vg). These terms are related to the volumetric swollen ratio (Q), which is dependent on the densities of the solvent (ρ1) and polymer (ρ2), and the mass swollen ratio (Qm) as in the following equation:
The architecture of a swollen hydrogel can be quantified by the molecular weight between crosslinks in the absence of solvent and is expressed by:
where is the number average molecular weight of the polymer chains prepared in the absence of crosslinkers. and V1 are the specific and molar volume of polymer and solvent, respectively. ν2,s is the volume fraction of polymer in the swollen mass, and χ1 is the Flory–Huggins polymer-solvent dimensionless interaction term.
Equation (4) describes gels that are swollen from the dry state; however, many gels are prepared in the presence of water. A new term was introduced by Peppas and Merrill to account for the presence of water and subsequent changes in chemical potential, which describes the volume fraction density of the chains during crosslinking. The original Flory–Rehner model was revised to incorporate a term describing the polymer in the relaxed, unstretched state (ν2,r):
When ionic groups are present in the network, the swelling equilibrium becomes more complicated. In addition to the entropic contributions described in eq (1), a contribution from ions to the total change in Gibbs free energy is:
At equilibrium, the net chemical potential still equals zero, and expressions for the ionic contribution to this potential are added to eq (5). Terms for the ionic strength, I, and the dissociation constants Ka and Kb have been added to account for the strong dependency on the ionic strength of the surrounding medium and the nature of these ions.[22-24]
For polyelectrolyte gels, eqs (7) and (8) are equivalent expressions for anionic and cationic hydrogels prepared in the presence of a solvent. Artificially engineered protein-based hydrogels present a challenge in polymer physics; however, work by Kim et al. reviews some of the progress on the physics of the dynamic intermolecular interactions of protein hydrogels.
Dynamic Swelling of Hydrogels
The theories described above address the equilibrium swelling state of hydrogels; however, an understanding of the swelling dynamics through volume phase transitions may be useful for predicting the behavior of the hydrogel with time. There are many various models that simulate volume transitions, some of which have been recently reviewed.[26, 27] Recently, work by Li et al. and Lai et al. investigated the swelling dynamics of hydrogels by the multieffect-coupling ionic-strength-stimulus model (MECis), which integrates the Nernst–Planck equation for mobile ion concentrations and the Poisson equation for the electric potential associated with the fixed charges to describe the mechanical displacement of a deformable network and the phase field variable. The MECis model is designed to study hydrogels stimulated by changes in ionic strength in two dimensions.
Both linear and nonlinear theories have been developed to describe the swelling kinetics and the processes of mass transport and mechanical deformation. Tanaka et al. derived a linear diffusion equation for polyacrylamide gels treated as a mixture of solid and liquid.[30, 31] A second linear theory by Scherer treats the gel as a continuum phase with pore pressure as a state variable.[32, 33] Recent work by Yoon et al. examines experimental swelling kinetics of thin layers of poly(N-isopropylacrylamide) with the model of linear poroelasticity. Through fluorescent particle tracking with an optical microscope, these experiments were able to accurately monitor thickness changes on the order of 100 μm in good agreement with a linear model of poroelasticity. The linear theory works well for small deformations; however, Hong et al. formulated a nonlinear theory for coupled mass transport and large deformations on the macroscopic scale. Recent work by Bouklas et al. presents a good comparison between the linear theory and the more recent nonlinear theory of poroelasticity for polymer gels. In this work, the dynamics of swelling of a gel affixed to a substrate as well as free swelling are both addressed.
Calculation of the Mesh Size
Many of the target applications for the stimuli-responsive gels outlined in this review are for drug delivery, where mesh size, sometimes referred to as the “pore” size, becomes important. The mesh size is described by the correlation length (ξ) which is defined as the linear distance between two adjacent crosslinks (Fig. 1) and is calculated by:
where α is the elongation ratio of the polymer chains in any direction and is the root-mean-squared, unperturbed end-to-end distance of the polymer chain between crosslinks.[17, 37] For gels swollen to isotropic equilibrium, the elongation ratio can be related to the swollen polymer volume fraction, ν2,s, by:
Swelling, and therefore mesh size, is affected by numerous physiochemical conditions and structural factors.[38-42]
Many ionic hydrogels exhibit a first-order volume-phase transition where the degree of swelling can change dramatically with only a small change in conditions or application of a stimulus (Fig. 2).
The swelling transition has a first-order response that involves the coexistence of two gel phases, the swollen and unswollen volume components. Polyelectrolyte gels often swell to a significantly greater extent when compared with neutral gels because of the additional osmotic pressure arising from mobile counter-ions, as well as the electrostatic repulsion of anchored ionizable groups. Charged particles inside the gel fail to distribute evenly between the inside of the gel and the outside medium. This difference in mobile ion concentrations between the gel and the surrounding solution causes the gel to swell to a greater extent and is governed by the Donnan equilibrium. For highly ionized polyelectrolyte gels, Donnan theory was modified by Rička and Tanaka to include an osmotic pressure ion contribution term to better approximate volume transitions due to the additional equilibrium swelling pressure. As a result, modifications of these theories have emerged to describe hydrogel swelling behaviors.[22, 44-46] Long-range repulsive electrostatic interactions between the polyelectrolyte segments favor swelling, while attractive electrostatic interactions between counter-ions and charged polymer segments favor shrinking. Counter-ion condensation leads to anisotropic charge distribution and the formation of ion pairs. The non-Gaussian conformation of charged polymer segments, Debye screening contributions, and other gel responses are also described.[20-22]
In the interest of practicality, fast response action is required for most stimuli-responsive thin-film devices. The volume phase transition response is the most common mechanism among hydrogel-based “smart” devices. This transition is driven by a diffusion-limited process; therefore, at least one dimension of the hydrogel must be sufficiently small so that the response behavior occurs on the order of seconds to minutes. Early work by Tanaka and Fillmore shows the volumetric response time of a gel is proportional both to the square of the smallest linear dimension of a gel and to the diffusion coefficient of the gel network, D, which governs response time by:
where E is the longitudinal bulk modulus of the network, and f is the coefficient of friction between the network and the solvent. Stimuli-responsive materials that respond on the order of seconds are typically 10 μm or less in their smallest linear dimension, reducing the physical restrictions to response time. This is not a strict geometric limitation; however, noteworthy examples of gels with a larger dimension, but second to minute response times, are also highlighted. Furthermore, not all of the stimuli-responsive devices outlined in this review are governed by diffusion-limited processes.