Theory of thermoreversible gelation and anomalous concentration fluctuations in polyzwitterion solutions

We present a theoretical framework to investigate thermoreversible phase transitions within polyzwitterion systems, encompassing macrophase separations (MPS) and gelation. In addition, we explore concentration fluctuations near critical points associated with MPS, as well as tricritical and bicritical points at the intersection of MPS and gelation. By utilizing mean-field percolation theory and field theory formalism, we derive the Landau free energy in terms of polyzwitterion concentration with fixed dipole strengths and other experimental variables, such as temperatures and salt concentrations. As the temperature decreases, the dipoles can form cross-links, resulting in polyzwitterion associations. The associations can grow to a gel network and enhance the propensity for MPS, including liquid-liquid, liquid-gel, and gel-gel phase separations. Remarkably, the associations also impact critical behaviors. Using the renormalization group technique, we find that the critical exponents of the polyzwitterion concentration correlation functions significantly deviate from those in the Ising universality class due to the presence of polyzwitterion associations, leading to crossover critical behaviors.

Skeletal Network Enabling New-Generation Thermoplastic Vulcanizates

Upcycling of cross-linked rubbers is pressing. The introduction of dynamic covalent bonds into the networks is a popular tactic for recycling thermosetting polymers, but it is very challenging to integrate engineering performance and continuous yet stable reprocessability. Based on traditional rubber formulations, herein, a straightforward strategy is presented for constructing a skeletal network (SN) through interfacial crosslinking and percolation of rubbery granules in a rubber matrix. Rapid exchange reactions involving dynamic interfacial sulfides realize repeated "fragmentation and healing" in the solid-state and consequent reconfiguration of the SN topology of the elastomer, thus endowing the resultant SN elastomer with continuous yet stable re-extrudability. These SN elastomers with hierarchical structures exhibit high gel contents, high resilience, low creep, and reinforcibility competitive to traditional vulcanizates. Specifically, SN elastomers exhibit better overall performance than commercial thermoplastic vulcanizates (TPVs) materials. Overall, a new concept of thermoplastic vulcanizates is proposed, which will promote the sustainable development of rubbers.

Distribution Cutoff for Clusters near the Gel Point

The mechanical and dynamic properties of developing networks near the gel point are susceptible to the distribution of clusters coexisting with percolating networks. The distribution of cluster numbers follows a broad power law, wrapped by a cutoff function that rapidly decays at a characteristic size. The form of the cutoff function has been speculated based on known results from lattice percolation and, in certain cases, solved. We obtained this cutoff function from simulated dynamic clusters of polymeric precursor chains using a hybrid Monte Carlo algorithm. The results obtained from three different precursor chain lengths are consistent with each other and are consistent with the expectation from lattice percolation.

Dynamical heterogeneity in the gelation process of a polymer solution with a lower critical solution temperature

The gelation of a hydrophobically modified hyaluronic acid aqueous solution which shows a lower critical solution temperature of about 25 °C was investigated by multi-particle tracking microrheology. The linear viscoelasticity of the gelling system is converted from the microrheological data. The critical gelling temperature T_gel = 36.3 °C was determined from the loss tangent by the Winter-Chambon criterion. The critical exponent n = 0.62 was determined from the shift factors of the time-cure superposition. The length scales of the dynamic heterogeneity of the gelling system were analyzed using a proposed framework where single-particle and multi-particle non-Gaussian parameters were compared. The length scale of the dynamic heterogeneous regions monotonically decreases during the gelation process, consistent with the nucleation-and-growth mechanism of phase separation. Distributions of local viscosity in the gelling system were extracted from the observed distributions of particle displacement as a time-dependent fingerprint of the dynamic heterogeneity of the gelling system. The results and analyzing methods proposed in the present work can be applied to other microrheological studies.

Hierarchical molecular dynamics of bovine serum albumin in concentrated aqueous solution below and above thermal denaturation

The dynamics of proteins in solution is a complex and hierarchical process, affected by the aqueous environment as well as temperature. We present a comprehensive study on nanosecond time and nanometer length scales below, at, and above the denaturation temperature Td. Our experimental data evidence dynamical processes in protein solutions on three distinct time scales. We suggest a consistent physical picture of hierarchical protein dynamics: (i) self-diffusion of the entire protein molecule is confirmed to agree with colloid theory for all temperatures where the protein is in its native conformational state. At higher temperatures T > Td, the self-diffusion is strongly obstructed by cross-linking or entanglement. (ii) The amplitude of backbone fluctuations grows with increasing T, and a transition in its dynamics is observed above Td. (iii) The number of mobile side-chains increases sharply at Td while their average dynamics exhibits only little variations. The combination of quasi-elastic neutron scattering and the presented analytical framework provides a detailed microscopic picture of the protein molecular dynamics in solution, thereby reflecting the changes of macroscopic properties such as cluster formation and gelation.

Percolation threshold and scattering power law of gelatin gels

Gelation of gelatin was broadly studied by experimental and theoretical methods. Power laws observed on the gel point-mainly obtained by dynamic light scattering (DLS)-are considered to be the signature of some special dynamic phenomena ascribed to the appearance of a percolation cluster. We present here experimental (DLS and rheometric measurements) and Monte Carlo simulation studies showing that the percolation threshold and DLS power-law decay occur on different times. We ascribe the percolation point to the time where the scattering medium mode diverges. This mode is sensitive to the clusters' growth and diverges when the system attains the percolation threshold. The power-law behavior is obtained only in the postpercolation regime.

Interfaces and fluctuations in confined polymeric liquid mixtures: from immiscible to near critical systems

In this paper the structure of the interface between polymer films is discussed to elucidate fluctuations and confinement effects in fluid polymer mixtures. The neutron reflectivity technique has been employed to investigate the dependence of the structure of the interface on the degree of immiscibility of the polymers over a wide range, as criticality is approached, and to characterize it in terms of intrinsic width, as calculated by mean field theories, and capillary fluctuations. For more immiscible systems, as the degree of incompatibility between the polymers is decreased, the width of the interface increases slowly, and it is independent of the molecular weight of the polymers. Closer to the critical point the dependence on the degree of miscibility becomes stronger and the way in which the interfacial width diverges, as criticality is approached, is related to both chain length and Flory-Huggins interaction parameter (χ). The results have been compared to the predictions of mean field theories. Self-consistent field numerical calculations, with the additional contribution due to capillary waves, provide a good description of the width of the interface between two polymer bulk phases, in particular at higher and intermediate degrees of immiscibility-the product of the Flory-Huggins interaction parameter χ and the number N of monomers of the chain, χN. For more miscible systems a crossover is observed to a region where the square gradient theory in the weak segregation limit better approximates the experimental results. Moreover, the mechanisms by which confinement affects the interface have been investigated. To understand the relative importance of the long ranged van der Waals forces and short ranged 'truncation forces' in modifying thermally excited fluctuations at the polymer/polymer interface, the thickness dependence of the interfacial width has been studied for different degrees of miscibility, approaching criticality. The results show a gradual transition from a region where long ranged dispersion forces are dominant in influencing the capillary wave spectrum, for higher degrees of immiscibility, to a region where short ranged forces-connected to the presence of the walls-become more important and the dependence of the interfacial width on film thickness is stronger. The thickness dependence of the interfacial width was also studied for systems with different molecular weights in different conditions of miscibility, to investigate the effects of a confined geometry on polymers with different chain lengths as criticality is approached.

Critical viscoelastic behavior of polydimethylsiloxane networks around the sol-gel threshold

Seven model polydimethylsiloxane (PDMS) networks were obtained by hydrosilation of a difunctional vinyl-terminated PDMS prepolymer with a SiH-containing cross-linker. Viscoelastic experiments, completed by size exclusion chromatography and static light scattering experiments, were performed in order to study the influence of molecular parameters on the dynamic properties around the sol-gel threshold. The dynamic critical parameter u was determined from experiments close to and above the sol-gel threshold. Our results show that the growth mechanism of PDMS clusters and the viscoelastic behavior are a function of the ratio NN_{e} , where N and N_{e} are, respectively, the numbers of Kuhn monomers between branch points and between entanglements. For NN_{e}<1 , the growth mechanism of clusters is the critical percolation and u=0.69 , and for NN_{e}>1 , the growth mechanim of clusters is the diffusion-limited cluster aggregation and u=0.76 .

Dynamic scaling in entangled mean-field gelation polymers

We present a simple reaction kinetics model to describe the polymer synthesis used by Lusignan et al. [Phys. Rev. E 60, 5657 (1999)] to produce randomly branched polymers in the vulcanization class. Numerical solution of the rate equations gives probabilities for different connections in the final product, which we use to generate a numerical ensemble of representative molecules. All structural quantities probed in the experiments are in quantitative agreement with our results for the entire range of molecular weights considered. However, with detailed topological information available in our calculations, our estimate of the "rheologically relevant" linear segment length is smaller than that estimated from the experimental results. We use a numerical method based on a tube model of polymer melts to calculate the rheological properties of such molecules. Results are in good agreement with experiment, except that in the case of the largest molecular weight samples our estimate of the zero-shear viscosity is significantly lower than the experimental findings. Using acid concentration as an indicator for closeness to the gelation transition, we show that the high-molecular-weight polymers considered are at the limit of mean-field behavior--which possibly is the reason for this disagreement. For a truly mean-field gelation class of model polymers, we numerically calculate the rheological properties for a range of segment lengths. Our calculations show that the tube theory with dynamical dilation predicts that, very close to the gelation limit, the contribution to viscosity for this class of polymers is dominated by the contribution from constraint-release Rouse motion and the final viscosity exponent approaches a Rouse-like value.

Power law polydispersity and fractal structure of hyperbranched polymers

Using the complementary approaches of Flory theory and the overlap function, we study the molecular weight distribution and conformation of hyperbranched polymers formed by the melt polycondensation of A-R(N)(0)-B(f - 1) monomers in their reaction bath close to the mean field gel point p(A) = 1, where p(A) is the fraction of reacted A groups. Here f > or = 3, N(0) is the degree of polymerisation of the linear spacer linking the A group and the f-1 B groups and condensation occurs exclusively between the A and B groups. For Epsilon tripe bond (1-pA) <<1, we assume that the number density of hyperbranched polymers with degree of polymerisation N generally obeys the scaling form n(N) =N(- tau)f(N/Nl) and we explicitly show that this scaling assumption is correct in the mean field regime (here Nl is the largest characteristic degree of polymerisation and the function f (N/Nl) cuts off the power law sharply for N>Nl). We find the upper critical dimension for this system is d(c) = 4, so that for d> or = dc the mean field values for the polydispersity exponent and fractal dimension apply: tau=3/2, d(f) = 4. For d = 3, mean field theory is still correct for Epsilon > Epsilon G where Epsilon G approximately equal N -1 0 is the Ginzburg point; for Epsilon < Epsilon G, mean field theory applies on small mass scales N< N c but breaks down on larger mass scales N> N c where N c approximately equal N 3 0 is a cross-over mass. Within the Ginzburg zone (i.e., d< d(c) < Epsilon G), we show that the hyperbranched chains on mass scales N> N(c) are non-Gaussian with fractal dimension given by d(f) = d (for d = 2,3,4). Our results are qualitatively different from those of the percolation model and indicate that the polycondensation of AB(f-1), unlike polymer gelation, is not described by percolation theory. Instead many of our results are similar to those for a monodisperse melt of randomly branched polymers, a consequence of the fact that tau < 2 so that polydispersity is irrelevant for excluded volume screening in hyperbranched polymer melts.

Viscoelasticity near the gel point: a molecular dynamics study

We report on extensive molecular dynamics simulations on systems of soft spheres of functionality f, i.e., particles that are capable of bonding irreversibly with a maximum of f other particles. These bonds are randomly distributed throughout the system and imposed with probability p. At a critical concentration of bonds, p(c) approximately 0.2488 for f=6, a gel is formed and the shear viscosity eta diverges according to eta approximately (p(c)-p)(-s). We find s approximately 0.7 in agreement with some experiments and with a recent theoretical prediction based on Rouse dynamics of phantom chains. The diffusion constant decreases as the gel point is approached but does not display a well-defined power law.

Renormalization-group approach to the vulcanization transition

The vulcanization transition-the cross-link-density-controlled equilibrium phase transition from the liquid to the amorphous solid state-is explored analytically from a renormalization-group perspective. The analysis centers on a minimal model which has previously been shown to yield a rich and informative picture of vulcanized matter at the mean-field level, including a connection with mean-field percolation theory (i.e., random graph theory). This minimal model accounts for both the thermal motion of the constituents and the quenched random constraints imposed on their motion by the cross-links, as well as particle-particle repulsion which suppresses density fluctuations and plays a pivotal role in determining the symmetry structure (and hence properties) of the model. A correlation function involving fluctuations of the amorphous solid order parameter, the behavior of which signals the vulcanization transition, is examined, its physical meaning is elucidated, and the associated susceptibility is constructed and analyzed. A Ginzburg criterion for the width (in cross-link density) of the critical region is derived and is found to be consistent with a prediction due to de Gennes. Inter alia, this criterion indicates that the upper critical dimension for the vulcanization transition is 6. Certain universal critical exponents characterizing the vulcanization transition are computed, to lowest nontrivial order, within the framework of an expansion around the upper critical dimension. This expansion shows that the connection between vulcanization and percolation extends beyond mean-field theory, surviving the incorporation of fluctuations in the sense that pairs of physically analogous quantities (one percolation related and one vulcanization related) are found to be governed by identical critical exponents, at least to first order in the departure from the upper critical dimension (and presumably beyond). The relationship between the present approach to vulcanized matter and other approaches, such as those based on gelation-percolation ideas, is explored in the light of this connection. To conclude, some expectations for how the vulcanization transition is realized in two dimensions, developed with H. E. Castillo, are discussed.

Viscoelasticity of randomly branched polymers in the vulcanization class

We report viscosity, recoverable compliance, and molar mass distribution for a series of randomly branched polyester samples with long linear chain sections between branch points. Molecular structure characterization determines tau=2.47+/-0.05 for the exponent controlling the molar mass distribution, so this system belongs to the vulcanization (mean-field) universality class. Consequently, branched polymers of similar size strongly overlap and form interchain entanglements. The viscosity diverges at the gel point with an exponent s=6.1+/-0.3, that is significantly larger than the value of 1.33 predicted by the branched polymer Rouse model (bead-spring model without entanglements). The recoverable compliance diverges at the percolation threshold with an exponent t=3.2+/-0.2. This effect is consistent with the idea that each branched polymer of size equal to the correlation length stores k(B)T of elastic energy. Near the gel point, the complex shear modulus is a power law in frequency with an exponent u=0.33+/-0.05. The measured rheological exponents confirm that the dynamic scaling law u=t/(s+t) holds for the vulcanization class. Since s is larger and u is smaller than the Rouse values observed in systems that belong to the critical percolation universality class, we conclude that entanglements profoundly increase the longest relaxation time. Examination of the literature data reveals clear trends for the exponents s and u as functions of the chain length between branch points. These dependencies, qualitatively explained by hierarchical relaxation models, imply that the dynamic scaling observed in systems that belong to the vulcanization class is nonuniversal.