Lattice model of equilibrium polymerization. VI. Measures of fluid “complexity” and search for generalized corresponding states

Predicting polymer solubility from phase diagrams to compatibility: a perspective on challenges and opportunities

Polymer processing, purification, and self-assembly have significant roles in the design of polymeric materials. Understanding how polymers behave in solution (e.g., their solubility, chemical properties, etc.) can improve our control over material properties via their processing-structure-property relationships. For many decades the polymer science community has relied on thermodynamic and physics-based models to aid in this endeavor, but all rely on disparate data sets and use-case scenarios. Hence, there are still significant challenges to predict a priori the solubility of a polymer, whether it is for selecting sustainable solvents, obtaining thermodynamic parameters for phase separation, or navigating the coexistence curve. This perspective aims to discuss the different approaches of applying computational tools to predict polymer solubility, with a significant focus on machine learning techniques to capture the rapid progress in that space. We examine challenges and opportunities that remain for creating a comprehensive solubility toolset that can accelerate the design of a broad range of applications including films, membranes, and pharmaceuticals.

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.

Evidence of Many-Body Interactions in the Virial Coefficients of Polyelectrolyte Gels

Simulation studies of aqueous polymer solutions, and heuristic arguments by De Gennes for aqueous polyethylene oxide polymer solutions, have suggested that many-body interactions can give rise to the ‘anomalous’ situation in which the second osmotic virial coefficient is positive, while the third virial coefficient is negative. This phenomenon was later confirmed in analytic calculations of the phase behavior and the osmotic pressure of complex fluids exhibiting cooperative self-assembly into extended dynamic polymeric structures by Dudowicz et al. In the present study, we experimentally confirm the occurrence of this osmotic virial sign inversion phenomenon for several highly charged model polyelectrolyte gels (poly(acrylic acid), poly(styrene sulfonate), DNA, hyaluronic acid), where the virial coefficients are deduced from osmotic pressure measurements. Our observations qualitatively accord with experimental and simulation studies indicating that polyelectrolyte materials exhibit supramolecular assembly in solution, another symptomatic property of fluids exhibiting many-body interactions. We also find that the inversion in the variation of the second (A₂) and third (A₂) virial coefficients upon approach to phase separation does not occur in uncharged poly(vinyl acetate) gels. Finally, we briefly discuss the estimation of the osmotic compressibility of swollen polyelectrolyte gels from neutron scattering measurements as an alternative to direct, time-consuming and meticulous osmotic pressure measurements. We conclude by summarizing some general trends and suggesting future research directions of natural and synthetic polyelectrolyte hydrogels.

Volume Phase Transition in Gels: Its Discovery and Development

 The history of volume phase transition of responsive gels from its theoretical prediction to experimental discovery was described and the major role of mixing Gibbs energy function in theoretical models was stressed. For detailed analysis and fine tuning of the volume phase transition, the generalized Flory-Huggins model with concentration and temperature dependent interaction function coupled with Maxwell construction as a tool is very suitable. Application of expansive stresses can uncover the potential of various swelling gels for volume phase transition. Experimentally, the abrupt, equilibrium-controlled phase transition is often hard to achieve due to passage of gel through states of mechanical instability and slow relaxation processes in macroscopic objects.

Lattice theory of competitive binding: Influence of van der Waals interactions on molecular binding and adsorption to a solid substrate from binary liquid mixtures

The reversible binding of molecules to surfaces is one of the most fundamental processes in condensed fluids, with obvious applications in the molecular separation of materials, chromatographic characterization, and material processing. Motivated in particular by the ubiquitous occurrence of binding processes in molecular biology and self-assembly, we have developed a lattice type theory of competitive molecular binding to solid substrates from binary mixtures of two small molecule liquids that interact between themselves by van der Waals forces in addition to exhibiting binding interactions with the solid surface. The derived theory, in contrast to previously existing theoretical frameworks, enables us to investigate the influence of van der Waals interactions on interfacial binding and selective molecular adsorption. For reference, the classic Langmuir theory of adsorption is recovered when all van der Waals interaction energies between the molecules in the bulk liquid phase and those on the surface are formally set to zero. Illustrative calculations are performed for the binding of molecules to a solid surface from pure liquids and from their binary mixtures. The properties analyzed include the surface coverage θ, the binding transition temperature T_bind, the individual surface coverages, θ_A and θ_C, and the relative surface coverages, σ_AC≡θ_A/θ_C or σ_CA≡θ_C/θ_A. The latter two quantities coincide with the degrees of adsorption directly determined from experimental adsorption measurements. The Langmuir theory is shown to apply formally under a wide range of conditions where the original enthalpies (Δh or Δh_A and Δh_C) and entropies (Δs or Δs_A and Δs_C) of the binding reactions are simply replaced by their respective "effective" counterparts (Δh_eff or Δh_A^eff and Δh_C^eff and Δs_eff or Δs_A^eff and Δs_C^eff), whose values depend on the strength of der Waals interactions and of the "bare" free energy parameters (Δh or Δh_A and Δh_C, and Δs or Δs_A and Δs_C). Numerous instances of entropy-enthalpy compensation between these effective free energy parameters follow from our calculations, confirming previous reports on this phenomenon obtained from experimental studies of molecular binding processes in solution.

Mixtures of two self- and mutually-associating liquids: Phase behavior, second virial coefficients, and entropy-enthalpy compensation in the free energy of mixing

The theoretical description of the phase behavior of polymers dissolved in binary mixtures of water and other miscible solvents is greatly complicated by the self- and mutual-association of the solvent molecules. As a first step in treating these complex and widely encountered solutions, we have developed an extension of Flory-Huggins theory to describe mixtures of two self- and mutually-associating fluids comprised of small molecules. Analytic expressions are derived here for basic thermodynamic properties of these fluid mixtures, including the spinodal phase boundaries, the second osmotic virial coefficients, and the enthalpy and entropy of mixing these associating solvents. Mixtures of this kind are found to exhibit characteristic closed loop phase boundaries and entropy-enthalpy compensation for the free energy of mixing in the low temperature regime where the liquid components are miscible. As discussed by Widom et al. [Phys. Chem. Chem. Phys. 5, 3085 (2003)], these basic miscibility trends, quite distinct from those observed in non-associating solvents, are defining phenomenological characteristics of the "hydrophobic effect." We find that our theory of mixtures of associating fluids captures at least some of the thermodynamic features of real aqueous mixtures.

Simple liquids' quasiuniversality and the hard-sphere paradigm

This topical review discusses the quasiuniversality of simple liquids' structure and dynamics and two possible justifications of it. The traditional one is based on the van der Waals picture of liquids in which the hard-sphere system reflects the basic physics. An alternative explanation argues that all quasiuniversal liquids to a good approximation conform to the same equation of motion, referring to the exponentially repulsive pair-potential system as the basic reference system. The paper, which is aimed at non-experts, ends by listing a number of open problems in the field.

The osmotic virial formulation of the free energy of polymer mixing

We derive an alternative formulation of the free energy of polymer mixing in terms of an osmotic virial expansion. Starting from a generalized free energy of mixing, and the assumption that the internal energy of mixing is analytic in the polymer composition variable, we demonstrate that the free energy of mixing can be represented as an infinite series in the osmotic virial coefficients. This osmotic virial formulation is consistent with, but more general than, a relationship derived for polymer blends with structured monomers by Dudowicz, Freed, and Douglas [J. Chem. Phys. 116, 9983 (2002)] and Douglas, Dudowicz, and Freed [J. Chem. Phys. 127, 224901 (2007)].

Phase field method for nonequilibrium dynamics of reversible self-assembly systems

Phase field methods are extended to describe the nonequilibrium dynamics of reversible self-assembly systems, an extension that is complicated by the mutual coupling of many non-conserved order parameters into a set of highly nonlinear partial differential equations. Further complications arise because the sum of all non-conserved order parameters equals a conserved order parameter. The theory is developed for the simplest model of reversible self-assembly in which no additional constraints are imposed on the self-assembly process since the extension to treat more complex self-assembly models is straightforward. Specific calculations focus on the time evolution of the cluster size distribution for a free association system that is rapidly dropped from one ordered state to a more ordered state within the one-phase region. The dynamics proceed as expected, thereby providing validation of the theory which is also capable of treating systems with spatial inhomogeneities.

Isomorphs, hidden scale invariance, and quasiuniversality

This paper first establishes an approximate scaling property of the potential-energy function of a classical liquid with good isomorphs (a Roskilde-simple liquid). This "pseudohomogeneous" property makes explicit that-and in which sense-such a system has a hidden scale invariance. The second part of the paper gives a potential-energy formulation of the quasiuniversality of monatomic Roskilde-simple liquids, which was recently rationalized in terms of the existence of a quasiuniversal single-parameter family of reduced-coordinate constant-potential-energy hypersurfaces [J. C. Dyre, Phys. Rev. E 87, 022106 (2013)]. The new formulation involves a quasiuniversal reduced-coordinate potential-energy function. A few consequences of this are discussed.

NVU perspective on simple liquids' quasiuniversality

The last half-century of research into the structure, dynamics, and thermodynamics of simple liquids has revealed a number of approximate universalities. This paper argues that simple liquids' reduced-coordinate constant-potential-energy hypersurfaces constitute a quasiuniversal family of compact Riemannian manifolds parametrized by a single number; from this follows the quasiuniversalities.

Lattice cluster theory of associating telechelic polymers. III. Order parameter and average degree of self-assembly, transition temperature, and specific heat

The lattice cluster theory of strongly interacting, structured polymer fluids is applied to determine the thermodynamic properties of solutions of telechelic polymers that may associate through bifunctional end groups. Hence, this model represents a significant albeit natural extension of a diverse array of prior popular equilibrium polymerization models in which structureless "bead" monomers associate into chain-like clusters under equilibrium conditions. In particular, the thermodynamic description of the self-assembly of linear telechelic chains in small molecule solvents (initiated in Paper II) is systematically extended through calculations of the order parameter Φ and average degree <N> of self-assembly, the self-assembly transition temperature T(p), and the specific heat C(V) of solutions of telechelic molecules. Special focus is placed on examining how molecular and thermodynamic parameters, such as the solution composition φ, temperature T, microscopic interaction energies (ε(s) and ε), and length M of individual telechelic chains, influence the computed thermodynamic quantities that are commonly used to characterize self-assembling systems.

Equilibrium polymerization models of re-entrant self-assembly

As is well known, liquid-liquid phase separation can occur either upon heating or cooling, corresponding to lower and upper critical solution phase boundaries, respectively. Likewise, self-assembly transitions from a monomeric state to an organized polymeric state can proceed either upon increasing or decreasing temperature, and the concentration dependent ordering temperature is correspondingly called the "floor" or "ceiling" temperature. Motivated by the fact that some phase separating systems exhibit closed loop phase boundaries with two critical points, the present paper analyzes self-assembly analogs of re-entrant phase separation, i.e., re-entrant self-assembly. In particular, re-entrant self-assembly transitions are demonstrated to arise in thermally activated equilibrium self-assembling systems, when thermal activation is more favorable than chain propagation, and in equilibrium self-assembly near an adsorbing boundary where strong competition exists between adsorption and self-assembly. Apparently, the competition between interactions or equilibria generally underlies re-entrant behavior in both liquid-liquid phase separation and self-assembly transitions.

Lattice model of equilibrium polymerization. VII. Understanding the role of "cooperativity" in self-assembly

Cooperativity is an emergent many-body phenomenon related to the degree to which elementary entities (particles, molecules, organisms) collectively interact to form larger scale structures. From the standpoint of a formal mean field description of chemical reactions, the cooperativity index m, describing the number of elements involved in this structural self-organization, is the order of the reaction. Thus, m for molecular self-assembly is the number of molecules in the final organized structure, e.g., spherical micelles. Although cooperativity is crucial for regulating the thermodynamics and dynamics of self-assembly, there is a limited understanding of this aspect of self-assembly. We analyze the cooperativity by calculating essential thermodynamic properties of the classical mth order reaction model of self-assembly (FAm model), including universal scaling functions describing the temperature and concentration dependence of the order parameter and average cluster size. The competition between self-assembly and phase separation is also described. We demonstrate that a sequential model of thermally activated equilibrium polymerization can quantitatively be related to the FAm model. Our analysis indicates that the essential requirement for "cooperative" self-assembly is the introduction of constraints (often nonlocal) acting on the individual assembly events to regulate the thermodynamic free energy landscape and, thus, the thermodynamic sharpness of the assembly transition. An effective value of m is defined for general self-assembly transitions, and we find a general tendency for self-assembly to become a true phase transition as m-->infinity. Finally, various quantitative measures of self-assembly cooperativity are discussed in order to identify experimental signatures of cooperativity in self-assembling systems and to provide a reliable metric for the degree of transition cooperativity.

Gel-like behavior in aggrecan assemblies

Aggrecan, a large biological polyelectrolyte molecule with a bottlebrush shape, forms complexes with hyaluronic acid (HA) that provide compressive resistance in cartilage. In solutions of aggrecan alone, the concentration dependence of the osmotic pressure Pi is marked by self-assembly of the molecules into aggregates. When HA is added to the solution at low aggrecan concentration c, the osmotic pressure is reduced, but in the physiological concentration range this trend is reversed. The osmotic modulus c partial differentialPi partial differentialc, which determines load bearing resistance, is enhanced in the HA-containing solutions. Dynamic light scattering (DLS) measurements show that the aggregates behave like microgels and that they become denser as the aggrecan concentration increases. The degree of densification is greatest at large distance scales in the microgels, but decreases at short distance scales. Measurements at higher resolution, involving small angle neutron scattering and small angle x-ray scattering (SAXS), confirm that at length scales shorter than 1000 angstroms, the density is independent of the concentration and that the individual bottlebrushes in the microgels retain their identity. The absence of collective diffusion modes in the relaxation spectrum, measured by DLS and neutron spin echo, corroborates the lack of interpenetration among the aggrecan subunits in the microgel. Complexation with HA modifies the long-range spatial organization of the microgels. Comparison of the scattering pattern of the individual aggrecan molecules obtained from SAXS measurements with that of the complexes measured by DLS shows that the aggrecan-HA structure is denser and is more uniform than the random microgels. This enhanced space-filling property allows higher packing densities to be attained, thus, optimizing resistance to osmotic compression.