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In preparation for a more thorough study based on our own experimental work of the debonding of a thin film gel by stress concentration on the interface with a rigid substrate, in this article we revisit, from the viewpoint of the synergy between mathematics, exper- iments, and finite element simulations, the problem of the swelling of a thin rectangular polyacrylamide gel covalently bonded on the bottom surface to a glass slide. With meth- ods of the calculus of variations and perturbation theory we show that the solution to the corresponding zero-displacement boundary value problem converges, in the thin film limit, to a uniquely defined uniform uniaxial extension on the direction normal to the substrate. Both the experiments and the finite element simulations that we perform confirm that the amount of lateral swelling is very small, with a very good quantitative agreement between the two approaches. The proposed model of minimizing an energy functional comprising both a term for the elastic distortion and the Flory-Huggins expression for the entropy of mixing is thus experimentally and numerically validated, with parameters coming from ex- perimental measurements, including the initial polymer volume fraction of the hydrogel synthesized in the laboratory (which is taken as the reference configuration instead of the dry polymer).
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Gels: Energetics, Singularities, and Cavitation
This article studies equilibrium singular configurations of gels and addresses open questions concerning gel energetics. We model a gel as an incompressible, immiscible and saturated mixture of a solid polymer and a solvent that sustain chemical interactions at the molecu- lar level. We assume that the energy of the gel consists of the elastic energy of its polymer network plus the Flory-Huggins energy of mixing. The latter involves the entropic energies of the individual components plus that of interaction between polymer and solvent, with the temperature dependent Flory parameter, χ , encoding properties of the solvent. In particular, a good solvent promoting the mixing regime, is found below the threshold value χ = 0.5, whereas the phase separating regime develops above that critical value. We show that cavi- ties and singularities develop in the latter regime. We find two main classes of singularities: (i) drying out of the solvent, with water possibly exiting the gel domain through the bound- ary, leaving behind a core of exposed polymer at the centre of the gel; (ii) cavitation, in response to traction on the boundary or some form of negative pressure, with a cavity that can be either void or flooded by the solvent. The straightforward and unified mathematical approach to treat all such singularities is based on the construction of appropriate test func- tions, inspired by the particular states of uniform swelling or compression. The last topic of the article addresses a statistical mechanics rooted controversy in the research commu- nity, providing an experimental and analytic study in support of the phantom elastic energy versus the affine one.
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- Award ID(s):
- 1816740
- PAR ID:
- 10410943
- Date Published:
- Journal Name:
- Journal of Elasticity
- ISSN:
- 0374-3535
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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The excess free energy of mixing Δ G ex governs the phase behavior of mixtures and controls material properties. It is challenging, however, to measure Δ G ex in simulations. Previously, we developed a method that combines molecular dynamics (MD) simulations with thermodynamic integration along the path of transformation of chains to predict the Flory Huggins interaction parameter χ for polymer mixtures and block copolymers. However, this method is best applied when the constituent molecules of the blends are structurally related. To overcome this limitation, we have developed a new method to predict Δ G ex for mixtures. We perform simulations to induce phase separation within a mixture by gradually weakening the interaction between different species. To compute Δ G ex we measure the thermodynamic work required to modify the interactions and the interfacial energy between the separated phases. We validate our method by applying it first to equimolar mixtures of labeled and unlabeled Lennard-Jones (LJ) beads, and labeled and unlabeled benzene, which results in good agreement with ideal solution theory. Then we compute the excess free energy of mixing for equimolar mixtures of benzene and pyridine, using both united-atom (UA) and all-atom (AA) potentials. Our results using UA potentials predict a value for Δ G ex about four times the experimental value, whereas using AA potentials gives results consistent with experiment, highlighting the need for good potentials to faithfully represent mixture behavior.more » « less
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Hillmyer, Marc A (Ed.)A high-salt phase-separation re-entry is observed in mixtures of poly (diallyldimethyl ammonium chloride) (PDADMAC), a strong polycation, and poly (acrylic acid) (PAA), a partially charged polyanion, within the pH range 4.7 to 5.3. This intriguing phenomenon exclusively occurs at salt concentrations exceeding the critical salt concentration required for dissolving the coacervate formed at low salt concentrations, here named the “Upper Critical Salt Concentration” (UCSaC), and at monomer concentrations exceeding 0.1M for each polymer. The transition from associative phase separation at low salt concentration, to a single solution, and ultimately to segregative separation at high salt concentration called the Lower Critical Salt Concentration (LCSaC), arises from the interplay between electrostatic interactions and the hydrophobicity of neutral PAA monomers in a high-salt solvent. To explain this transition, we use a theory combining short-range ion pairing and counterion condensation with long-range electrostatics using the random phase approximation (RPA), and with hydrophobic interactions between PAA neutral monomers and water. The latter is modeled through a Flory-Huggins χ parameter of around 0.6. Literature observations of a continuous transition from associative to segregative phase transition with increasing salt concentration, without a homogeneous single-phase solution at intermediate salt concentration, are also predicted and discussed.more » « less
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We introduce a lattice framework that incorporates elements of Flory–Huggins solution theory and the q-state Potts model to study the phase behavior of polymer solutions and single-chain conformational characteristics. Without empirically introducing temperature-dependent interaction parameters, standard Flory–Huggins theory describes systems that are either homogeneous across temperatures or exhibit upper critical solution temperatures. The proposed Flory–Huggins–Potts framework extends these capabilities by predicting lower critical solution temperatures, miscibility loops, and hourglass-shaped spinodal curves. We particularly show that including orientation-dependent interactions, specifically between monomer segments and solvent particles, is alone sufficient to observe such phase behavior. Signatures of emergent phase behavior are found in single-chain Monte Carlo simulations, which display heating- and cooling-induced coil–globule transitions linked to energy fluctuations. The framework also capably describes a range of experimental systems. Importantly, and by contrast to many prior theoretical approaches, the framework does not employ any temperature- or composition-dependent parameters. This work provides new insights regarding the microscopic physics that underpin complex thermoresponsive behavior in polymers.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10659-023-10000-5
