Model Of Equilibrium Polymerization Research Articles

Entropy of polydisperse chains: Solution on the Husimi lattice

We consider the entropy of polydisperse chains placed on a lattice. In particular, we study a model for equilibrium polymerization, where the polydispersity is determined by two activities, for internal and endpoint monomers of a chain. We solve the problem exactly on a Husimi lattice built with squares and with arbitrary coordination number, obtaining an expression for the entropy as a function of the density of monomers and mean molecular weight of the chains. We compare this entropy with the one for the monodisperse case, and find that the excess of entropy due to polydispersity is identical to the one obtained for the one-dimensional case. Finally, we obtain a distribution of molecular weights with a rather complex behavior, but which becomes exponential for very large mean molecular weight of the chains, as required by scaling properties, which should apply in this limit.

Competition between self-assembly and surface adsorption

We investigate a minimal equilibrium polymerization model for the competition between self-assembly on a boundary and in solution that arises when an assembling system is in the presence of an adsorbing interface. Adsorption generally occurs upon cooling, but assembly (equilibrium polymerization) may arise either upon cooling or heating. Both cases are shown to exhibit a coupling between adsorption and self-assembly. When both assembly and adsorption proceed upon cooling, a change in the ratio of the enthalpy of adsorption to the enthalpy of assembly in solution can switch the system between a predominance of self-assembly in solution to assembly on the substrate. If assembly is promoted by heating and adsorption by cooling, as in many self-assembling proteins in aqueous solution, then a self-assembly analog of a closed loop phase boundary is found. In particular, the order parameter for assembly on the surface exhibits a peak as a function of temperature. As demonstrated by illustrative examples, the coupling between surface adsorption and self-assembly provides a powerful means of switching self-assembly processes on and off. Understanding and controlling this switching phenomenon will be useful in designing and directing self-assembly processes on surfaces for applications to nanomanufacturing and in developing treatments for diseases arising from pathological adsorption-induced assembly.

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.

Entropy of polydisperse chains: Solution on the Bethe lattice

We consider the entropy of polydisperse chains placed on a lattice. In particular, we study a model for equilibrium polymerization, where the polydispersivity is determined by two activities, for internal and endpoint monomers of a chain. We solve the problem exactly on a Bethe lattice with arbitrary coordination number, obtaining an expression for the entropy as a function of the density of monomers and mean molecular weight of the chains. We compare this entropy with the one for the monodisperse case and find that the excess of entropy due to polydispersivity is identical to the one obtained for the one-dimensional case. Finally, we obtain an exponential distribution of molecular weights.

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

Particle association in "complex" fluids containing charged, polar, or polymeric molecular species often leads to deviations from the corresponding state description of "simple" fluids in which the molecules are assumed to have relatively symmetric interactions and shapes. This fundamental problem is addressed by developing a minimal thermodynamic model of activated equilibrium polymerization solutions that incorporates effects associated with the competition between van der Waals and associative interactions, as well as features related to molecular anisotropy and many-body interactions. As a dual purpose, we focus on thermodynamic signatures that can be used to identify the nature of dynamic clustering transitions and the interaction parameters associated with these rounded thermodynamic transitions. The analysis begins by examining "singular" features in the concentration dependence of the osmotic pressure Pi that generically characterize the onset of particle association. Because molecular self-assembly can strongly couple with fluid phase separation, evidence is also sought for associative interactions in the behavior of the second A(2) and third A(3) osmotic virial coefficients. In particular, the temperatures T(Theta2) and T(Theta3) where A(2) and A(3), respectively, vanish are found to contain valuable information about the relative strength of the associative and van der Waals interactions. The critical temperature T(c) for phase separation, the critical composition phi(c), and the rectilinear diameter A(d), describing the asymmetry of the coexistence curve for phase separation, along with the average cluster mass L(c) and extent of polymerization Phi(c) at the critical point, further specify the relevant interaction parameters of our model. Collectively, these characteristic properties provide a thermodynamic metric for defining fluid complexity and in developing a theoretically based corresponding state relation for complex fluids.

Self-assembly of patchy particles into polymer chains: A parameter-free comparison between Wertheim theory and Monte Carlo simulation

The authors numerically study a simple fluid composed of particles having a hard-core repulsion, complemented by two short-ranged attractive (sticky) spots at the particle poles, which provides a simple model for equilibrium polymerization of linear chains. The simplicity of the model allows for a close comparison, with no fitting parameters, between simulations and theoretical predictions based on the Wertheim perturbation theory. This comparison offers a unique framework for the analytic prediction of the properties of self-assembling particle systems in terms of molecular parameters and liquid state correlation functions. The Wertheim theory has not been previously subjected to stringent tests against simulation data for ordering across the polymerization transition. The authors numerically determine many of the thermodynamic properties governing this basic form of self-assembly (energy per particle, order parameter or average fraction of particles in the associated state, average chain length, chain length distribution, average end-to-end distance of the chains, and the static structure factor) and find that predictions of the Wertheim theory accord remarkably well with the simulation results.

Lattice model of equilibrium polymerization. V. Scattering properties and the width of the critical regime for phase separation

Dynamic clustering associated with self-assembly in many complex fluids can qualitatively alter the shape of phase boundaries and produce large changes in the scale of critical fluctuations that are difficult to comprehend within the existing framework of theories of critical phenomena for nonassociating fluids. In order to elucidate the scattering and critical properties of associating fluids, we consider several models of equilibrium polymerization that describe widely occurring types of associating fluids at equilibrium and that exhibit the well defined cluster geometry of linear polymer chains. Specifically, a Flory-Huggins-type lattice theory is used, in conjunction with the random phase approximation, to compute the correlation length amplitude xi(o) and the Ginzburg number Gi corresponding, respectively, to the scale of composition fluctuations and to a parameter characterizing the temperature range over which Ising critical behavior is exhibited. Our calculations indicate that upon increasing the interparticle association energy, the polymer chains become increasingly long in the vicinity of the critical point, leading naturally to a more asymmetric phase boundary. This increase in the average degree of polymerization implies, in turn, a larger xi(o) and a drastically reduced width of the critical region (as measured by Gi). We thus obtain insight into the common appearance of asymmetric phase boundaries in a wide range of "complex" fluids and into the observation of apparent mean field critical behavior even rather close to the critical point.

Equilibrium polymerization in the Stockmayer fluid as a model of supermolecular self-organization

A diverse range of molecular self-organization processes arises from a competition between directional and isotropic van der Waals intermolecular interactions. We conduct Monte Carlo simulations of the Stockmayer fluid (SF) with a large dipolar interaction as a minimal self-organization model and focus on basic thermodynamic properties that are needed to characterize the polymerization transition that occurs in this fluid. In particular, we determine the polymerization transition lines from the maximum in the specific heat, C(v), and the inflection point in the extent of polymerization, Phi. We also characterize the geometry (radius of gyration R(g), chain length L, chain topology) of the clusters that form in this associating fluid as a function of temperature, T, and concentration, rho . The pressure, P, and the second virial coefficient, B2, were determined, since these properties contain essential information about the strength of the isotropic (van der Waals) interactions. Our simulations indicate that the locations of the polymerization lines are quantitatively consistent with a model of equilibrium polymerization with the enthalpy of polymerization ("sticking energy") fixed by the minimum in the intermolecular potential. The polymerization transition in the SF is accompanied by a topological transition from predominantly linear to ring polymers upon cooling that is driven by the minimization of the dipolar energy of the clusters. We also find that the basic interaction parameters describing polymerization and phase separation in the SF can be estimated based on the existing theory of equilibrium polymerization, but the theory must be refined to account for ring formation in order to accurately describe the configurational properties of this model self-organizing fluid.

Flory-Huggins model of equilibrium polymerization and phase separation in the Stockmayer fluid.

The competition between chain formation and phase separation in the Stockmayer fluid (SF) of dipolar particles is analyzed using a renormalized Flory-Huggins model of equilibrium polymerization. Calculated critical properties (T(c), phi(c), Z(c)) for the SF compare favorably with simulations over a wide range of the dimensionless dipolar (or "sticking") energy mu*. We find that the polymerization transition preempts phase separation for a large mu*, i.e., (mu*)2 >22.

Lattice model of equilibrium polymerization. IV. Influence of activation, chemical initiation, chain scission and fusion, and chain stiffness on polymerization and phase separation

The influence of thermal activation, chemical initiation, chain fragmentation, and chain stiffness on basic thermodynamic properties of equilibrium polymerization solutions is systematically investigated using a Flory–Huggins type lattice model. The properties treated include the average chain length L, extent of polymerization Φ, Helmholtz free energy F, configurational entropy S, specific heat CV, polymerization transition temperature Tp, osmotic pressure Π, and the second and third virial coefficients, A2 and A3. The dependence of the critical temperature Tc and critical composition φc (volume fraction of associating species) on the enthalpy Δhp and entropy Δsp of polymerization and on the strength εFH of the FH effective monomer–solvent van der Waals interaction (χ=εFH/T) is also analyzed as an illustration of the strong coupling between phase separation and polymerization. For a given polymerization model, both Tc and φc, normalized by their values in the absence of polymerization, are functions of the dimensionless “sticking energy” hε≡(|Δhp|/R)/(2εFH) (where R is the gas constant) and Δsp.

Phase transitions within the isolated polymer molecule: Coupling of the polymer threading a membrane transition to the helix-random coil, the collapse, the adsorption, and the equilibrium polymerization transitions

The polymer threading a membrane transition (PTM), which is a first-order thermodynamic phase transition for an isolated linear polymer in the limit of infinite molecular weight, is coupled to the other four phase transitions of the isolated polymer molecule. They are (1) the helix–random coil (HR) phase transition which can be diffuse (polypeptides), second-order (DNA) or first-order (collagen) depending on the number of strands, (2) the collapse (C) transition which is usually second-order but can be first-order for polymeric solvents, (3) adsorption onto a surface (SA) which is second-order, (4) a model of equilibrium polymerization (P) which is first-order. In each case an exact expression for the partition function of the coupled pair is given as a one-dimensional summation over products of the individual partition functions corresponding to sides 1 and 2. Using a procedure analogous to evaluation of the grand canonical ensemble the summation can be performed and the character of the transition elucidated in the limit of infinite molecular weight. Given that the solutions on either side are sufficiently diverse there are 15 possible translocation pair couplings. They are PTM–PTM, HR–HR, C–C, SA–SA, P–P, PTM–HR, PTM–C, PTM–SA, PTM–P, HR–C, HR–SA, HR–P, C–SA, C–P, SA–P. The PTM–P coupling is most interesting because one can create polymer in the PTM side even though the P side is in the depolymerization regime. For HR–HR there are eight possible translocation modes. For example, as we raise the temperature we can have H1→H2→R1→R2 in obvious notation. These exact model solutions provide a thermodynamic base for the study of the kinetics of significant technological problems such as the translocation of DNA through pores imbedded in membranes. They also throw light on the nature of polymer–membrane–pore interactions in living cells and viruses.

Self-assembly and structure transformations in living polymers forming fibrils

The classical isodesmic one-dimensional model for equilibrium polymerization is generalized in order to describe self-assembly in systems forming fibrils. The model was applied to peptide solutions forming -sheet tapes which can further aggregate into stacks of various thickness: double tapes and fibrils (several double tapes stacked together). We found that in some cases the model yields several step-like transitions as the concentration increases: first from monomers to single or double tapes, and then to fibrils. The abruptness of the first transition is controlled by the free energy penalty for transformation of a peptide from random coil to a straight -strand conformation (the latter is characteristic for tapes). If both single and double tapes are allowed, the length of the aggregates after the first transition can be very large with high scission energies. For very low energies of attraction between double tapes, the transition from double tapes to fibrils happens separately (above the first transition), and it is even more abrupt and produces extremely long fibrils. The theoretical findings are used to extract the characteristic molecular parameters for the self-assembly of the de novo peptide DN1 forming polymeric -sheets in water.

Equilibrium polymerization: towards a numerical description of the dynamics of wormlike micelles

The self-diffusion of surfactant in a system of cylindrical, polymer-like micelles is studied for high concentrations in the framework of the equilibrium polymerization model. The different types of coarse grained dynamic equilibrium-polymers algorithms are first briefly reviewed and commented. The results suggest that the self-diffusion is dominated by the smallest species, in marked contrast with rheological experiments showing the behaviour of the slowest (i.e. longest) chains. The self-diffusion coefficient D is described using Phillies' formalism. An analysis of the experimental and simulational results suggests that the dependence D ∼ Φ -1.5 is found without the ansatz of the presence of entanglements.

New statistical mechanical treatment of systems near surfaces. IV. Surface and surface-induced capillary transitions in polymer solutions

We apply a recently developed analytic but approximate method to study surface and surface-induced capillary transitions in a solution of polydisperse linear and/or branched polymers confined between two infinite but identical surfaces. We use an equilibrium polymerization model where various densities in the system are controlled by the corresponding activities. The bulk region, i.e., the central region between the surfaces may or may not be in the bulk equilibrium state (see the text). We find a line of first-order transitions, commonly known as a prewetting transition line, passing through the phase separation point in the bulk equilibrium state, at which the surface undergoes a first-order transition. For a certain range around the bulk equilibrium transition point, the bulk region undergoes a stable-metastable transition. This presents us with the possibility of being able to prepare the system in a bulk metastable state, no matter how far apart the two surfaces are. This range is found to be identical to the range of metastability for the bulk equilibrium state. Outside of this range, the transition line behaves like a prewetting transition line and, as usual, terminates at surface critical points, one on each side of the above range.

Tetracritical and novel tricritical points in a model magnet

A generalized three-dimensional spherical model of magnetism is shown to exhibit a rich variety of phase equilibria and critical phenomena, including critical points not found in any classical Landau theory, but observed recently in a model of equilibrium polymerization. The model exhibits a tetracritical point at which four phases, two nonmagnetic and two magnetically ordered, become identical, as well as novel tricritical points at which the magnetic and wing critical lines meet at a cusp. The model also exhibits additional transitions, not found in the polymerization model.

Polymer model with annealed dilution on the square lattice: A transfer-matrix study.

We study a lattice model for equilibrium polymerization with annealed dilution. The model considered is an Ising lattice-gas monomer-solvent system where the polymers are represented by mutually and self-avoiding walks constrained to pass through sites occupied by monomers. Numerical results on the square lattice using transfer-matrix techniques and finite-size scaling are reported. The values obtained for the tricitical exponent ${\ensuremath{\nu}}_{\mathit{t}}$ are in agreement with the tricritical Ising exponent (8/11) for high values of the monomer fugacity, but the accuracy we obtained does not rule out the possibility of ${\ensuremath{\nu}}_{\mathit{t}}$ being equal to the critical self-avoiding-walk (SAW) value 3/4. Our results indicate that a crossover occurs in the tricritical behavior of the model. This crossover corroborates the equivalence between a particular limit of the diluted model and the self-attracting polymer system proposed recently.

Thermal and percolative transitions and the need for independent symmetry breakings in branched polymers on a Bethe lattice

We consider a very general model of equilibrium polymerization of branched polymers. Our model contains, as a special case, the ‘‘a priori equal probability’’ model considered by Flory and Stockmayer. In this limit, the model exhibits only percolation transition. We solve our general model in the interior of a Bethe lattice. There are thermal as well as percolation transitions in the model. Each of the two transitions requires an independent spontaneous symmetry breaking; neither implies the other. Without spontaneous symmetry breaking, the transitions do not manifest themselves. Thermal transitions correspond to singularities in the equation of state. Percolation transitions, on the other hand, do not correspond to any singularity in the equation of state. We also discuss the failure of a topological identity, valid for any finite Cayley tree, in the interior of the Bethe lattice. We consider various different cases to show the usefulness of our model. In particular, we argue that one must distinguish between the ‘‘tree approximation’’ of Flory on a general lattice and our exact solution on the Bethe lattice. The former, in general, allows for loop formation, whereas there are no loops allowed in the latter solution.

Equilibrium polymerization with annealed crosslinks

We show that the magnetic n-vector model with two- and four-spin interactions in the limit when n→0 is equivalent to an equilibrium polymerization model with annealed crosslinks between chains. In the mean field solution, the polymerization transition is of first order for sufficiently high values of the four-spin interaction, a tricritical point being present in the phase diagram. This feature is also present in the solutions of the model on the Bethe and Husimi lattices. In the particular case of the Husimi lattice with coordination number equal to four, a qualitatively different phase diagram is obtained, with two ordered phases, a critical endpoint and a tricritical point. Finally, the model is solved on the square lattice in the particular case of vanishing two-spin interactions, and, as expected, a first order transition is observed.

Chain-length distribution in a model of equilibrium polymerization.

Using renormalization-group techniques, I derive a general scaling form of the chain-length distribution for the equilibrium-polymerization model of des Cloizeaux [J. Phys. (Paris) 36, 281 (1975)]. The result allows more freedom than was assumed in some previous work. This disproves arguments suggesting that in semidilute polymer solutions there exists a phase related to some anomaly of the zero-component field theory. The scaling function of the chain-length distribution is calculated to first order in \ensuremath{\epsilon}. It varies with the overlap of the chains and, in general, differs somewhat from a Schultz distribution. No anomaly related to the semidilute limit is found. Some rather nontrivial aspects of the result are well understood in terms of de Gennes ``blob'' concept.

Quantitative structural analysis of dispersed vanadia species in TiO 2(anatase)-supported V 2O 5

V 2O 5-TiO 2(anatase) catalysts have been studied under oxidizing and reducing conditions using in situ laser Raman spectroscopy (LRS) and temperature-programmed reduction (TPR) and oxidation (TPO). Quantitative Raman and TPO analysis of the oxidized samples show that these materials are comprised of a distribution of monomeric vanadyls, polymeric vanadates, and crystallites of V 2O 5. At low loadings, the predominant species are monomeric vanadyls, with the remaining vanadia being present in the form of polymeric vanadates. As the surface concentration of V 2O 5 increases, a maximum in the concentration of the polymeric vanadates is detected. Crystallites of V 2O 5 form at the expense of the polymeric vanadates as the loading is raised above the dispersive capacity of the TiO 2(anatase) support. An equilibrium polymerization model is proposed to account for the observed concentration of vanadia species, which leads to an initial polymer size of ∼2 at low loadings, consistent with the formation of dimeric species. Raman and TPR/TPO studies of the reduction process indicate that the terminal V = 0 groups of the monomeric and polymeric vanadia species are removed preferentially to the bridging oxygen atoms of the polymeric species. The maximum loss of oxygen upon reduction is one oxygen atom per vanadium atom.