Inhomogeneous deformation of elastomer gels in equilibrium under saturated and unsaturated conditions

doi:10.1016/j.jmps.2010.12.013

Journal of the Mechanics and Physics of Solids

Volume 59, Issue 3, March 2011, Pages 561-582

https://doi.org/10.1016/j.jmps.2010.12.013 Get rights and content

Abstract

A variational method is employed to obtain governing equations and boundary conditions describing finite strain equilibrium configurations of elastomeric gels. Three situations are considered: a liquid saturated gel, an unsaturated gel, and a gel in equilibrium with a vapor of its own liquid. Surface tractions can lead to equilibrium transitions between these cases. The liquid saturated gel is regarded as immersed in a liquid bath. If this bath becomes depleted, then the gel is unsaturated. The degree of unsaturation – a measure of the amount of liquid that would restore a state of saturation – affects the subsequent mechanical behavior. If the unsaturated system is further allowed to condense or evaporate its liquid component at the gel surface, then a new state of equilibrium is achieved. The transition between the unsaturated case and the case of being in equilibrium with the vapor phase corresponds to the chemical potential variable of the gel changing its value from one that is determined by a volume constraint to the value of the chemical potential in the vapor phase. A finite element method is created on the basis of the variational method and demonstrated in the context of eversion, a deformation that imposes very large finite strains. Liquid migration within the gel is not modeled as our focus is on equilibrium states that occur after all such non-equilibrium processes come to rest.

Introduction

A gel is a mixture of crosslinked polymer chains and an interpenetrating fluid. Because of their solid-like properties and the ability to contain a large amount of a liquid, gels have numerous applications including areas such as tissue engineering (Lee and Mooney, 2001, Nguyen and West, 2002) and drug delivery (Qiu and Park, 2001, Gupta et al., 2002). When an external force is applied to such a gel, the fluid diffuses in or out from the swollen solid and the volume of the body changes. Eventually, when the diffusion process is completed, the body maintains equilibrium with the external force by virtue of the elastic stresses arising from the deformation. Many of these stresses are entropic in nature as they arise both from the mixing of the solid and liquid constituents, and from the configurational entropy changes of the crosslinked polymer chains. From these observations, Treloar (1975) pointed out that the equilibrium mechanical properties of the gel correspond to those of a compressible material made up of a single constituent, even when the individual constituents – the polymer and the fluid – are each incompressible.

To describe the mechanical behavior of an elastomeric gel in a quantitative fashion, theoretical models have been developed using variational methods based on the thermodynamics of mixtures (see for example: Huggins, 1942; Flory and Rehner, 1943; Treloar, 1950; Bowen, 1980, Bowen, 1982; Gandhi et al., 1987, Gandhi et al., 1989; Rajagopal and Tao, 1996; Baek and Srinivasa, 2004a; Hong et al., 2008; Duda et al., 2010). In particular, such a gel is in equilibrium only if its free energy is a minimum with respect to both changes in the mass fraction of the individual constituents and to the deformation of the solid constituent. For large deformation the latter type of variation naturally lends itself to a hyperelastic treatment, whereas the former type of variation involves liquid flux across the gel boundary and liquid redistribution within the gel.

If the amount of liquid is limited it can then be the case that certain loadings cause all of the liquid component to enter into the gel. If this is the case then the gel is no longer saturated and such a gel is referred to here as unsaturated. Specifically, if an additional small amount of liquid is brought into contact with such an unsaturated gel then that liquid, upon being absorbed by the gel, allows the system to establish a lower free energy for the same state of mechanical loading. Under such a process the gel remains unsaturated under repeated introduction of a small amount of liquid so long as the liquid continues to be absorbed. Once equilibrium is attainable with no additional liquid absorption then the gel is once again saturated. Such considerations apply to both homogeneous and inhomogeneous deformation. For homogeneous deformation the liquid is uniformly distributed within the gel, so that any liquid introduced to the unsaturated gel also becomes uniformly distributed after liquid diffusion is complete and equilibrium subsequently attained. In contrast, for inhomogeneous deformation the liquid is generally not uniformly distributed.

This concept of saturation is discussed in the book by Rajagopal and Tao (1996), and the notion of saturation boundary conditions is developed by Rajagopal, Wineman and collaborators in a variety of papers in the 1980s and 90s.¹ They do not, however, systematically consider the notion of a transition between equilibrium states, one of which is saturated and the other of which is not saturated. Nor do they examine the effect of what one may regard as the amount of undersaturation due to differing overall quantities of liquid in an unsaturated gel. Such issues have recently been studied by Deng and Pence, 2010a, Deng and Pence, 2010b, where it is shown how, generally, a loss-of-saturation transition renders the system mechanically stiffer than that of a corresponding saturated system. Such stiffness increases with the amount of undersaturation.

One of the difficulties in developing a theoretical framework for modeling gels under such circumstances is related to the standard and reasonable approximation that all of the volume change in the gel is due exclusively to fluid mass transfer. In the usual variational treatment this constraint results in a Lagrange multiplier which then appears in the expressions for the stress and the chemical potential. As discussed by Baek and Srinivasa (2004a), the use of the Lagrange multiplier technique can obscure the physical interpretation of the resulting balance expressions and hinders the identification of appropriate boundary conditions. To clarify these issues, the treatment in Baek and Srinivasa (2004a) reformulates the variational problem so that it is initially without such a constraint. Then a limiting process is employed such that departure from the constraint is increasingly penalized. The resulting variational formulation – which applies to the notion of a saturated gel as discussed above – takes the system to be the gel with its surrounding fluid bath. By including the fluid bath, the limiting procedure provides a means for obtaining the equilibrium boundary conditions at the interface between the gel and the bath.

In this work, we employ this variational approach to treat both saturated and unsaturated gels in equilibrium and subject to loading at the gel surface. Body forces are neglected and so in particular the effect of gravity is not addressed. We also restrict attention to isothermal processes so that temperature either need not enter the treatment or, more generally, is regarded as a parameter. The constitutive theory requires knowledge of the free energy of the gel $ψ_{g}$ and of the surrounding fluid $ψ_{f}$ . Since we confine attention to equilibrium states, there is no need to specify constitutive entities associated with the dynamical processes that take place prior to attaining equilibrium (e.g., diffusion) and for this reason it is not necessary to specify a constitutive form for the rate of dissipation as is done in Baek and Srinivasa (2004a). Here a variational formulation is obtained for three separate and related situations. The first is that of a saturated gel, meaning a gel that is in equilibrium with applied surface tractions while also within a liquid bath (the saturated gel referred to above). The second is that of an unsaturated gel in the sense already described, meaning that there is no longer a source for additional liquid to enter into the gel even though maintaining saturation under surface tractions would require such a liquid source. The third is that of gel subject to surface tractions that is in equilibrium with a vapor of the same substance as that which constitutes the liquid component of the gel. Both the liquid bath in the first situation, and the surrounding vapor in the third situation, will generally have an associated pressure. Different boundary conditions at the gel surface are obtained from the variational procedure for these different situations. In particular, the procedure clarifies the role of the gel chemical potential in each of these differing situations. In addition, we show how the theoretical framework employed in Deng and Pence, 2010a, Deng and Pence, 2010b emerges naturally from the variational procedure for the first two situations enumerated above. This is especially useful since the framework as employed in Deng and Pence, 2010a, Deng and Pence, 2010b proceeded on the basis of hyperelasticity and specifically involved no explicit mention of the chemical potential concept. Thus, even though the framework of Deng and Pence, 2010a, Deng and Pence, 2010b does not specifically identify a chemical potential, it remains consistent with mixture theory treatments that specifically identify such an entity. All of these considerations are developed in 3 Interactions with a vapor phase and loss of saturation, 4 Application: Modeling of an everted swollen tube.

The variational formulation is then used to construct a finite element formulation for two situations of a saturated gel and an unsaturated gel. The accuracy of the finite element procedure is demonstrated for the special case of an everted gel cylinder (turning a short tube inside-out). The special case of eversion with specialized end tractions that make the everted shape a perfect cylinder was reduced to the consideration of ODEs in Deng and Pence (2010a) and solved numerically. The finite element procedure presented here faithfully replicates these cylindrically symmetric solutions for both the saturated and the unsaturated case. Then the finite element procedure is demonstrated on eversion problems that do not maintain perfect cylinders upon eversion (and which are thus not treatable by the ODE method in Deng and Pence, 2010a). In particular, everted cylinders that are completely free of surface tractions will generally flare out a bit at the top and bottom. This is confirmed by the finite element procedure, and the effect of differing amounts of gel undersaturation is quantified in this regard. Finally, the finite element method (FEM) is used to simulate the full eversion sequence itself by prescribing a set of cap displacements that have the effect of turning the cylinder inside-out.

Section snippets

The liquid saturated gel: A gel in contact with a fluid bath

As stated in the Introduction, a gel is mixture of solid (polymer) and liquid constituents. Fig. 1 shows a liquid saturated gel occupying domain $Ω_{g}$ . It is surrounded by a bath of the same fluid which occupies domain $Ω_{f}$ . The surface of the gel is $Γ_{g}$ which is therefore also the interface between the gel and the fluid bath. This surface is defined by locations of the solid polymer component, thus fluid may transfer between the gel and the bath at $Γ_{g}$ . The boundary of the fluid bath consists of $Γ_{g}$

Interactions with a vapor phase and loss of saturation

The volume additivity constraint as considered in the previous section was developed in the context of a fluid saturated gel that is surrounded by a bath of the pure fluid. The pure fluid itself was taken to be incompressible. One could alternatively consider the case where the saturated gel is removed from this fluid bath and is instead placed in an environment where it makes direct contact with a gaseous phase consisting of the gel fluid in vapor form. Here we consider modifications to the

Application: Modeling of an everted swollen tube

As demonstration we consider equilibrium deformations corresponding to the axisymmetric eversion of a cylindrical gel tube, meaning that the tube is turned inside-out. This is easily accomplished for example in short pieces of rubber hosing. The eversion deformation has a long history of study in conventional hyperelasticity beginning with Rivlin in the context of the incompressible theory (Rivlin, 1949) and then leading through more recent work (Haughton and Orr, 1997, Haughton and Orr, 1996,

Finite element formulation

A weak form for a saturated gel can be obtained by finding x which satisfies the following equation for all admissible $δ x$ , $δ Π = \int_{Ω_{R}} \frac{\partial \hat{Ψ}}{\partial C} : δ C + \hat{H}' (J) \frac{\partial J}{\partial C} : δ C dV - \int_{Γ_{t}} t \cdot δ x da = 0,$ where the last term represents the work done by the traction t on the boundary $Γ_{t}$ in the current configuration. Using nda=JF^−TN dA and defining t^o=JTF^−TN on $Γ_{R}$ (the inverse image of $Γ_{t}$ on the boundary of $Ω_{R}$ ) this surface integral can be rewritten with respect to the computational domain as $\int_{Γ_{R}} t^{o} \cdot δ x dA$ . To solve various boundary value

Eversion simulation by means of the FEM

For numerical simulation, consider a hollow cylinder such that R^s_o=2R^s_i and L^s=R^s_o. These ratios are preserved by the free swelling (65). In addition, the deformation (68) which establishes locations in the computational domain then gives R_o=2R_i. The lateral surfaces are the particle points that are at R=R_i and R=R_o in $Ω_{R}$ . The upper cap are the particles at $Z = ζ L^{s}$ and the lower cap are the particles at Z=0 in $Ω_{R}$ . As discussed in Wineman and Rajagopal (1992) and Deng and Pence (2010a) the

Summary

In this work we examine swollen deformations for elastomeric gels subject to surface tractions. The gel is described as a mixture in which both the porous solid constituent and the interstitial liquid co-occupy each point of the material. Swelling is due to an increase in the amount of liquid constituent. These aspects are all quite standard in continuum mixture theory. Attention is focussed upon equilibrium states, so that there is no motion of the liquid constituent with respect to the solid

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Inhomogeneous deformation of elastomer gels in equilibrium under saturated and unsaturated conditions

Abstract

Introduction

Section snippets

The liquid saturated gel: A gel in contact with a fluid bath

Interactions with a vapor phase and loss of saturation

Application: Modeling of an everted swollen tube

Finite element formulation

Eversion simulation by means of the FEM

Summary

Int. J. Non-Linear Mech.

Int. J. Non-Linear Mech.

Int. J. Eng. Sci.

Int. J. Eng. Sci.

Int. J. Eng. Sci.

J. Mech. Phys. Solids

Int. J. Eng. Sci.

Drug Discovery Today

Int. J. Solids Struct.

J. Mech. Phys. Solids

Int. J. Solids Struct.

Biomaterials

Adv. Drug Deliver. Rev.

Int. J. Eng. Sci.

The insight of mixtures theory for growth and remodeling

Z. Angew. Math. Phys.