Behavior of Ideal Polymer Gels
The behavior of neat elastomers is a useful starting point for a discussion of polymer gels since they can be thought of as elastomers swollen with solvent. For a Neo-Hookean material, the strain energy density, U, is given by24
where λx, λy, and λz are the principal extension ratios (for incompressible materials λ λ λ = 1), J1 is I1 − 3 where I1 is the first strain invariant, and E is Young's modulus corresponding to the undeformed state, for example, where J1 = 0. For uniaxial compression, λx = λz = λ where λy is the compression ratio and the nominal stress, σ, (force divided by undeformed cross-sectional area) is equal to . Thus, using eq 1 σ for Neo-Hookean materials can be written as
The Young's modulus for an unentangled, unsolvated (dry) polymer network, denoted by subscript d, can be calculated using simple rubber elasticity theory
where vd is the number density of chains, Rd is the average distance between crosslinks, and R is the mean-squared end-to-end distance for equivalent Gaussian chains between crosslink points.25
By assuming that a polymer gel can be thought of as an elastomer that is affinely swollen with solvent, an equivalent expression for a gel can be written by replacing Ed, vd, and Rd with values in the swollen state (denoted by the subscript s). Both vs and Rs can be related to the values in the dry state through the gel's polymer concentration, ϕp, using vs = vdϕp and R = Rϕ. The modulus of a polymer gel can be written as
As vd = ρ/MNav and only those chains that are elastically active contribute to the modulus, eq 4 can be rewritten as
where Nav is Avogadro's number, f is the fraction of elastically active molecules, M is the polymer molecular weight, and ρ is the polymer density.
The total osmotic pressure for a polymer gel, Π, can be expressed as Π = Πel + Πs where Πel is the elastic contribution that resists swelling and Πs is the contribution from polymer/solvent contacts that drives swelling. Πel is defined by
where V is the swollen volume and V0 is the initial gel volume. For isotropic swelling,
Combining eqs 1, 6, and 8 yields
Thus, for an elastomer swollen to equilibrium
where Esw is the modulus of the gel in its equilibrium swollen state, Eref is the gel modulus in the reference state and λswell is the linear extension ratio required to go from the reference state to the equilibrium swollen state. As λswell is related to the change in polymer concentration, we can write
where ϕ and ϕ are the polymer volume fractions in the initial reference state and equilibrium swollen state, respectively.
An estimate of Πs for a polymer gel can be obtained by assuming that the osmotic pressure of the gel is equivalent to that of a solution of high molecular weight polymer at the same polymer volume fraction as the gel. In the semidilute case, scaling theory26 gives
where ζ is the screening length and a0 is a characteristic monomer size.
For a gel swollen to equilibrium in pure solvent Π = 0. Thus Πel = −Πs, which for a Neo-Hookean materials yields
Solving for the equilibrium polymer volume fraction yields
This coupling of the network elasticity to the osmotic pressure means that network chains stretched due to the formation of the gel structure will have a reduced ability to expand to accommodate solvent, limiting the gel's ability to swell. In extreme cases where network chains are highly extended during network formation, syneresis is observed and the gel expels solvent to gain entropy by allowing less-extended chain conformations.
Although eq 1 is mathematically simple and can be used to described elastomers and gels at low strains, these materials often exhibit significantly more strain hardening than is predicted by the Neo-Hookean model.27–29 This behavior can be described by modifying the Neo-Hookean model with a term that accounts for finite chain extensibility. The response of acrylic triblock alcogels is well described by a strain energy density function28
which is equivalent to the model developed by Fung30 that is often used to describe deformation of blood vessels.31 The only free fitting parameter, J*, accounts for the finite extensibility of the chains in a way that is similar to a model proposed by Gent.32 Comparison of this strain energy density function with the strain energy density for a Neo-Hookean material ( ) reveals that they are equivalent at low strain but the exponential term leads to strain hardening as J1 approaches J*.
Using the expression for the strain energy density given in eq 15, the nominal stress during uniaxial compression can be written as
where λy is the compression ratio. This is equivalent to the Neo-Hookean response given in eq 2 multiplied by the term which accounts for strain hardening.
Πel for strain hardening materials described by eq 15 can be calculated in an analogous manner to the Neo-Hookean case and yields
Similarly to the uniaxial compression response, Πel for the exponentially strain hardening case is equivalent to the Neo-Hookean solution multiplied by the term , which becomes significant as J1 approaches J*. Therefore, the significance of the strain hardening term for a given system will depend both on the extent of swelling, λ, as well as J*.
It is also true for a strain-hardening gel that Πel + Πs = 0 after swelling to equilibrium in pure solvent and ϕ can be calculated in an analogous manner to the Neo-Hookean case. Using eqs 8, 12, and 17 yields,
As , it is not possible to easily arrive at an algebraic expression for ϕ as it was in the Neo-Hookean case. However, it is clear that as J1 approaches J* the predictions of both models for ϕ will diverge with the Neo-Hookean model predicting more swelling (smaller ϕ) than the exponentially strain hardening model. Finally, we note here that the two models presented above assumed ideal polymer networks, for example, that the crosslink points are infinitely small.
Micellar Triblock Copolymer Gels
Micellar triblock gels differ from idealized gels with point like crosslinks in a few important ways. Depending on the relative block fractions and polymer lengths, the volume of the micelle cores, and thus of the crosslinks, can be significant. However, as long as the endblock aggregates remain discrete objects, it is expected that the material response of the gel will be dominated by the deformation of the bridging midblocks. Therefore, rubber elasticity has been used to rationalize the modulus of micellar gels. Micellar triblock gels are not formed via swelling of a dry network and thus, unlike swollen elastomers, Rd is not an experimentally relevant variable. The average distance between micelle centers (d), equivalent to the average distance between crosslinking points, is easily measured from scattering and has been shown to vary as a function of gel concentration.21 Therefore, eq 4 can be recast as
and has been used to understand the modulus of micellar triblock gels with unentangled midblocks.33 In the case of triblock gels, f is the fraction of molecules which form bridges rather than loops. It has been shown that the combined variable and the micelle aggregation number increase with increasing gel concentration for acrylic triblock alcogels.21
In the previous discussion of ideal polymer gels, it was assumed that all the polymer in the gel was osmotically active. For micellar triblock gels which have glassy endblock aggregates, the amount of solvent in the aggregates is insensitive to total gel concentration, which suggest that the endblock should be considered as osmotically inactive when considering gel swelling.23 If we assume as a first approximation that only the midblock is osmotically active, we can rewrite eq 12 for a micellar gel as
where α is the volume fraction of midblock in the triblock.
Rigid Crosslink Volume Effects
It is important to also consider implications of the nonpoint-like crosslinks in these systems. Several approaches that have been advanced for accounting for the contribution of the rigid aggregates in phase separated ABA triblock melts.34, 35 The most common treats the aggregates as rigid fillers that reinforce the elastomeric matrix in a way analogous to classic filled rubbers. It is well known that filling a viscous liquid or an elastomer with rigid particles causes, respectively, the viscosity or the modulus to increase with increasing filler volume fraction.36–40 At relatively low volume fraction of filler, E can be modeled using the Guth–Gold relationship
where E0 is the modulus of the unfilled elastomer and ϕfiller is the volume fraction of rigid filler. The major drawback to this approach is that eq 21 is derived using the assumption that continuum mechanics holds (e.g., that the size scale of the filler is larger than molecular dimensions); as noted by Bard and Chung, this is not the case for phase separated triblock melts.35
Application of this model to triblock melts is complicated by the fact that the low strain modulus in these systems is generally dominated by midblock entanglements.34, 35 However, for the triblock gels studied in this article, the midblocks are unentangled21 removing this difficulty. It should also be noted that while the ratio of gel modulus at a given filler content to the unfilled modulus is relatively straight forward for an elastomer to which rigid particles are added; for the triblock systems, a completely unfilled system is one that contains no aggregated endblocks and, therefore, has no crosslinks and no modulus. Thus, all the gels investigated in this study are “filled” to various extents. The volume fraction occupied by the glassy, yet still solvated, PMMA aggregates in triblock gels can be calculated using
where ϕPMMA and ϕS are the volume fraction PMMA in the gel and the volume fraction of solvent in the core of the micelle, respectively. The last term accounts for the fact that the PMMA will be slightly solvated even at low temperatures and eq 26 can be used to calculate ϕS (at 25 °C, ϕS = 0.24). Lastly, as eq 21 is written, the matrix modulus, E0, is treated as independent of filler content. However, if the topology of the elastomer network is affected by amount of rigid phase, E0 should not be treated as a constant and should be replaced with eq 4. For these reasons, we feel that a treatment based on eq 19 provides a more appropriate description of the gel mechanical response than the simple rigid filler prediction based on eqs 21 and 22.