The swelling equilibrium of Olympic gels, which are composed of entangled cyclic polymers, is studied by Monte Carlo Simulations. In contrast to chemically crosslinked polymer networks, we observe that Olympic gels made of chains with a \emph{larger} degree of polymerization, $N$, exhibit a \emph{smaller} equilibrium swelling degree, $Q\propto N^{-0.28}ϕ_{0}^{-0.72}$, at the same polymer volume fraction $ϕ_{0}$ at network preparation. This observation is explained by a desinterspersion process of overlapping non-concatenated rings upon swelling.
In this note we present a scaling theory for the elasticity of olympic gels, i.e., gels where the elasticity is a consequence of topology only. It is shown that two deformation regimes exist. The first is the non affine deformation regime where the free energy scales linear with the deformation. In the large (affine) deformation regime the free energy is shown to scale as $F \propto λ^{5/2}$ where $λ$ is the deformation ratio. Thus a highly non Hookian stress - strain relation is predicted.
Concatenation and equilibrium swelling of Olympic gels, which are composed of entangled cyclic polymers, is studied by Monte Carlo Simulations. The average number of concatenated molecules per cyclic polymers, $f_n$, is found to depend on the degree of polymerization, $N$, and polymer volume fraction at network preparation, $ϕ_0$, as $f_n ~ ϕ_0^{ν/(3ν-1)}N$ with scaling exponent $ν = 0.588$. In contrast to chemically cross-linked polymer networks, we observe that Olympic gels made of longer cyclic chains exhibit a smaller equilibrium swelling degree, $Q ~ N^{-0.28} ϕ_0^{-0.72}$, at the same po…
The patterns arising from the differential swelling of gels are investigated experimentally and theoretically as a model for the differential growth of living tissues. Two geometries are considered: a thin strip of soft gel clamped to a stiff gel, and a thin corona of soft gel clamped to a disk of stiff gel. When the structure is immersed in water, the soft gel swells and bends out of plane leading to a wavy periodic pattern which wavelength is measured. The linear stability of the flat state is studied in the framework of linear elasticity using the equations for thin plates. The flat state i…
We derive a multiphysics model that accounts for network elasticity with spontaneous strains, swelling and nematic interactions in liquid crystal gels (LCGs). We discuss the coupling among the various physical mechanisms, with particular reference to the effects of nematic interactions on chemical equilibrium and that of swelling on the nematic-isotropic transition. Building upon this discussion and using numerical simulations, we explore the transient phenomena involving concurrent swelling and phase transition in LCGs subject to a temperature change. Specifically, we demonstrate separation i…
We derive a large-strain plate model that allows to describe transient, coupled processes involving elasticity and solvent migration, by performing a dimensional reduction of a three-dimensional poroelastic theory. We apply the model to polymer gel plates, for which a specific kinematic constraint and constitutive relations hold. Finally, we assess the accuracy of the plate model with respect to the parent three-dimensional model through two numerical benchmarks, solved by means of the finite element method. Our results show that the theory offers an efficient computational framework for the s…
Different methods for creating Olympic gels are analyzed using computer simulations. First ideal reference samples are obtained from freely interpenetrating semi-dilute solutions and melts of cyclic polymers. The distribution of pairwise concatenations per cyclic molecule is given by a Poisson-distribution and can be used to describe the elastic structure of the gels. Several batches of linear chains decorated with different selectively binding groups at their ends are mixed in the "DNA Origami" technique and network formation is realized. While the formation of cyclic molecules follows mean f…
We present a pedagogical review of the swelling thermodynamics and phase transitions of polymer gels. In particular, we discuss how features of the volume phase transition of the gel's osmotic equilibrium is analogous to other transitions described by mean-field models of binary mixtures, and the failure of this analogy at the critical point due to shear rigidity. We then consider the phase transition at fixed volume, a relatively unexplored paradigm for polymer gels that results in a phase-separated equilibrium consisting of coexisting solvent-rich and solvent-poor regions of gel. Again, the …
We present two successive mean-field approximations for describing the mechanical properties and the swelling equilibrium of polyelectrolyte gels in contact with a salt solution. The first mean-field approximation reduces the many-chain problem of a gel to a corresponding single chain problem. The second mean-field step integrates out the degrees of freedom of the flexible chain and the ions. It replaces the particle-based description of the polyelectrolyte with suitable charge distributions and an effective elasticity term. These simplifications result in a computationally very efficient Pois…
Polymer gels such as jellies and soft contact lenses are soft solids consisting of three-dimensional polymer networks swollen with a large amount of solvent. For approximately 80 years, the swelling of polymer gels has been described using the Flory--Huggins mean-field theory. However, this theory is problematic when applied to polymer gels with large solvent contents owing to the significant fluctuations in polymer concentration. In this study, we experimentally demonstrate the superiority of the semidilute scaling law over the mean-field theory for predicting the swelling of polymer gels. Us…
Polymeric Solvents Control Swelling-Induced Surface Creasing
Beihang University（北京航空航天大学）；Hangzhou International Innovation Institute, Beihang University（北京航空航天大学杭州国际创新学院）；Institute of Mechanics, Chinese Academy of Sciences（中国科学院力学研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Univ. Bordeaux（波尔多大学）；CNRS（法国国家科学研究中心）；LOMA, UMR 5798（LOMA实验室，UMR 5798）；Gulliver UMR 7083 CNRS, PSL Research University, ESPCI Paris（PSL研究大学、巴黎高等物理化工国立理工学院、CNRS Gulliver联合研究实验室，UMR 7083）；IPGG；Peng Huanwu Collaborative Center for Research and Education, Beihang University（北京航空航天大学彭恒武理论物理协同创新中心）
Soft membranes are commonly employed in shape-morphing applications, where the material is programmed to achieve a target shape upon activation by an external trigger, and as coating layers that alter the surface characteristics of bulk materials, such as the properties of spreading and absorption of liquids. In particular, polymer gel membranes experience swelling or shrinking when their solvent content change, and the non-homogeneous swelling field may be exploited to control their shape. Here, we develop a theory of swelling material surfaces to model polymer gel membranes and demonstrate i…
Non-uniform swelling of polyelectrolyte hydrogels: effects of charge regulation
Wenzhou University（温州大学）；Zhejiang Key Laboratory of Soft Matter Biomedical Materials, Wenzhou Institute, University of Chinese Academy of Sciences（中国科学院大学温州研究院软物质生物医用材料浙江省重点实验室）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；School of Physics and Astronomy, & Center for Physics and Chemistry of Living Systems, Tel Aviv University（特拉维夫大学物理与天文学学院及生命系统物理化学中心）；School of Sino-German Engineering, Shanghai Technical Institute of Electronics and Information（上海电子与信息工程技术学院中德工程学院）
We develop a double mean-field theory for charged macrogels immersed in electrolyte solutions in the spirit of the cell model approach. We first demonstrate that the equilibrium sampling of a single explicit coarse-grained charged polymer in a cell yields accurate predictions of the swelling equilibrium if the geometry is suitably chosen and all pressure contributions have been incorporated accurately. We then replace the explicit flexible chain by a suitably modeled penetrable charged rod that allows to compute all pressure terms within the Poisson-Boltzmann approximation. This model, albeit …
We use molecular dynamics simulations to study the swelling of randomly end-cross-linked polymer networks in good solvent conditions. We find that the equilibrium degree of swelling saturates at Q_eq = N_e**(3/5) for mean strand lengths N_s exceeding the melt entanglement length N_e. The internal structure of the network strands in the swollen state is characterized by a new exponent nu=0.72. Our findings are in contradiction to de Gennes' c*-theorem, which predicts Q_eq proportional N_s**(4/5) and nu=0.588. We present a simple Flory argument for a self-similar structure of mutually interpenet…
Colloidal spheres synthesized from polymer gels swell by absorbing molecules from solution. The resulting change in size can be monitored with nanometer precision using holographic video microscopy. When the absorbate is chemically similar to the polymer matrix, swelling is driven primarily by the entropy of mixing, and is limited by the surface tension of the swelling sphere and by the elastic energy of the polymer matrix. We demonstrate though a combination of optical micromanipulation and holographic particle characterization that the degree of swelling of a single polymer bead can be used …
We study theoretically using scaling arguments the behavior of polyelectrolyte gels in poor solvents. Following the classical picture of Katchalsky, our approach is based on single chain elasticity but it accounts for the recently proposed pearl-necklace structure of polyelectrolytes in poor solvents. The elasticity both of gels at swelling equilibrium and of partially swollen, non equilibrium, gels is studied when parameters such as the ionic strength or the fraction of charged monomers are varied. Our theory could be useful to interpret recent experiments performed in Strasbourg that show th…
We present an experimentally guided, multi-phase, multi-species polyelectrolyte gel model to make quantitative predictions on the electro-chemical properties of articular cartilage. The mixture theory consists of two different types of polymers: Poly(ethylene gylcol) (PEG), Chondrotin sulfate (ChS), water (acting as solvent) and several different ions: H$^+$, Na$^+$, Cl$^-$. The polymer chains have covalent cross-links modeled using Doi rubber elasticity theory. Numerical studies on polymer volume fraction and net osmolarity (difference in the solute concentration across the gel) show the inte…
We propose a theory based on non-equilibrium thermodynamics to describe the mechanical behavior of an active polymer gel created by the inclusion of molecular motors in its solvent. When activated, these motors attach to the chains of the polymer network and shorten them creating a global contraction of the gel, which mimics the active behavior of a cytoskeleton. The power generated by these motors is obtained by ATP hydrolysis reaction, which transduces chemical energy into mechanical work. The latter is described by an increment of strain energy in the gel due to an increased stiffness. This…
We present an experimentally guided, multi-phasic, multi-species ionic gel model to compare and make qualitative predictions on the rheology of mucus of healthy individuals (Wild Type) versus those infected with Cystic Fibrosis. The mixture theory consists of the mucus (polymer phase) and water (solvent phase) as well as several different ions: H+, Na+ and Ca++. The model is linearized to study the hydration of spherically symmetric mucus gels and calibrated against the experimental data of mucus diffusivities. Near equilibrium, the linearized form of the equation describing the radial size of…
In order to allow the simultaneous transport of hydrophilic and hydrophobic substances, polymeric networks with finely distributed hydrophilic and hydrophobic components are very suitable. When designing new soft materials like coatings, in addition to the structure in the volume phase, the structure at the interface plays a critical role. In this study, two alternating tetra-arm star polymers poly(ε-caprolactone) (tetra-PCL-Ox) and amino-terminated poly(ethylene glycol) (tetra-PEG-NH2) form an amphiphilic co-polymer network. They serve as a model system for controlling and understanding the s…
The swelling dynamics of polymer gels are characterized by the (collective) diffusion coefficient $D$ of the polymer network. Here, we measure the temperature dependence of $D$ of polymer gels with controlled homogeneous network structures using dynamic light scattering. An evaluation of the diffusion coefficient at the gelation point $D_{\mathrm{gel}}$ and the increase therein as the gelation proceeds $ΔD\equiv D-D_{\mathrm{gel}}$ indicates that $ΔD$ is a linear function of the absolute temperature with a significantly large negative constant term. This feature is formally identical to the re…
