Statistical thermodynamics in the framework of the lattice fluid model, 1. Polydisperse polymers of special distribution (original) (raw)
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Thermodynamic properties of lattice polymers: Monte Carlo simulations and mean-field theories
The Journal of Chemical Physics, 2000
Monte Carlo simulations of a lattice polymer melt are used to determine the thermodynamic properties of the system over a range of monomer volume fractions 0 ≤ φ ≤ 0.8 and effective temperatures 3.3 ≤ T * ≤ ∞. The simulations consider chains of length M = 40 and M = 100. The thermodynamic quantities analyzed are the chemical potential, the entropy, the specific heat, the isothermal compressibility, the internal energy, and the pressure. Canonical and grand canonical ensemble methods are employed as independent checks of the simulations for the chemical potential and the pressure. The predictions of Flory-Huggins (FH) theory, lattice cluster theory (LCT), and Guggenheim's random mixing and quasichemical approximations are compared with the simulations. The comparisons greatly extend prior demonstrations of very large errors in the simple FH approximation and show the major improvements provided by Guggenheim's approximations and the LCT.
Statistical Thermodynamics of Polymer Solutions
Macromolecules, 1978
The lattice fluid theory of solutions is used to calculate heats and volumes of mixing, lower critical solution temperatures, and the enthalpic and entropic components of the chemical potential. Results of these calculations are compared with literature data on several polyisobutylene solutions. In most instances the agreement with experiment is favorable and comparable to that obtained with the Flory equation of state theory. Several insights into polymer solution behavior are obtained and include: (1) differences in equation of state properties of the pure components make an unfavorable entropic contribution to the chemical potential that becomes large and dominant as the gas-liquid critical temperature of the solvent is approached; (2) limited miscibility of nonpolar polymer solutions at low and high temperatures is a manifestation of a polymer solution's small combinatorial entropy; and (3) negative heats of mixing in nonpolar polymer solutions are caused by the solvent's tendency to contract when polymer is added. Suggestions on how the theory can be improved are made Freeman and Rowlinson' in 1960 observed that several hydrocarbon polymers dissolved in hydrocarbon solvents phase separated at high temperatures. These nonpolar polymer solutions exhibited what are known as lower critical solution temperatures (LCST), a critical point phenomenon that is relatively rare among low molecular weight solutions. Soon after the discovery of the universality of LCST behavior in polymer solutions, Flory and c o -~o r k e r s~-~ developed a new theory of solutions which incorporated the "equation of state" properties of the pure components. This new theory of solutions, hereafter referred to as the Flory theory, demonstrated that mixture thermodynamic properties depend on the thermodynamic properties of the pure components and that LCST behavior is related to the dissimilarity of the equation of state of properties of polymer and solvent. P a t t e r~o d -~ has also shown that LCST behavior is associated with differences in polymer/solvent properties by using the general corresponding states theory of Prigogine and collaborators.1° Classical polymer solution theory, i.e., Flory-Huggins theory," which ignores the equation of state properties of the pure components, completely fails to describe the LCST behavior.
Statistical thermodynamics of associated polymer solutions
Macromolecules, 1991
The lattice-fluid theory for polymer solutions has been modified to account for strong interactions (hydrogen bonding) between the polymer and solvent. The resultant model is an equation-of-state model able to describe thermodynamic properties of these systems over an extended range of external conditions from the ordinary liquid state up to the critical or supercritical state. The general case is treated where the solvent self-associates and cross-associates with the polymer. The model has been tested against consistent sets of experimental data on vapor pressures, heats of mixing, and volumes of mixing of chloroform + polyether solutions. It is shown that the model is able to predict the sigmoidal-shaped isotherms characteristic of these systems.
Lattice model of living polymerization. I. Basic thermodynamic properties
The Journal of Chemical Physics, 1999
A Flory-Huggins type lattice model of living polymerization is formulated, incorporating chain stiffness, variable initiator concentration r, and a polymer-solvent interaction. Basic equilibrium properties ͓average chain length L, average fraction of associated monomers ⌽, specific heat C P , entropy S, polymerization temperature T p , and the chain length distribution p(N)͔ are calculated within mean-field theory. Our illustrative calculations are restricted to systems that polymerize upon cooling ͓e.g., poly͑␣-methylstyrene͔͒, but the formalism also applies to polymerization upon heating ͑e.g., sulfur, actin͒. Emphasis is given to living polymer solutions having a finite r in order to compare theory with recent experiments by Greer and co-workers, whereas previous studies primarily focused on the r→0 ϩ limit where the polymerization transition has been described as a second order phase transition. We find qualitative changes in the properties of living polymer solutions for nonzero r: ͑1͒ L becomes independent of initial monomer composition m 0 and temperature T at low temperatures ͓L(TӶT p)ϳ2/r͔, instead of growing without bound; ͑2͒ the exponent describing the dependence of L on m 0 changes by a factor of 2 from the r→0 ϩ value at higher temperatures (TуT p); ͑3͒ the order parametertype variable ⌽ develops a long tail with an inflection point at T p ; ͑4͒ the specific heat maximum C P * at T p becomes significantly diminished and the temperature range of the polymer transition becomes broad even for small r ͓rϳO(10 Ϫ3)͔. Moreover, there are three characteristic temperatures for rϾ0 rather than one for r→0: a ''crossover temperature'' T x demarking the onset of polymerization, an r-dependent polymerization temperature T p defined by the maximum in C P ͑or equivalently, the inflection point of ⌽͒, and a ''saturation temperature'' T s at which the entropy S of the living polymer solution saturates to a low temperature value as in glass-forming liquids. A measure of the ''strength'' of the polymerization transition is introduced to quantify the ''rounding'' of the phase transition due to nonzero r. Many properties of living polymer solutions should be generally representative of associating polymer systems ͑thermally reversible gels, colloidal gels, micelles͒, and we compare our results to other systems that self-assemble at equilibrium.
Helmholtz and Gibbs ensembles, thermodynamic limit and bistability in polymer lattice models
Continuum Mechanics and Thermodynamics, 2017
Representing polymers by random walks on a lattice is a fruitful approach largely exploited to study configurational statistics of polymer chains and to develop efficient Monte Carlo algorithms. Nevertheless, the stretching and the folding/unfolding of polymer chains within the Gibbs (isotensional) and the Helmholtz (isometric) ensembles of the statistical mechanics have not been yet thoroughly analysed by means of the lattice methodology. This topic, motivated by the recent introduction of several single-molecule force spectroscopy techniques, is investigated in the present paper. In particular, we analyse the force-extension curves under the Gibbs and Helmholtz conditions and we give a proof of the ensembles equivalence in the thermodynamic limit for polymers represented by a standard random walk on a lattice. Then, we generalize these concepts for lattice polymers that can undergo conformational transitions or, equivalently, for chains composed of bistable or two-state elements (that can be either folded or unfolded). In this case, the isotensional condition leads to a plateau-like force-extension response, whereas the isometric condition causes a sawtooth-like force-extension curve, as predicted by numerous experiments. The equivalence of the ensembles is finally proved also for lattice polymer systems exhibiting conformational transitions.
Thermodynamic behavior of a polymer with interacting bonds on a square lattice
Physical Review E, 2001
Using the transfer matrix technique, finite-size scaling, phenomenological renormalization group, and conformal invariance ideas, the thermodynamic behavior of a polymer with interacting bonds on a square lattice has been studied. In this model, one monomer that belongs to the polymer has an activity xϭe  , while the interactions between bonds of the polymer that are located on opposite edges of elementary squares of the lattice have a statistical weight yϭe Ϫ⑀ , where ⑀ is the interaction energy. Next, the phase diagram of the model in the (x,y) plane was found, which shows three phases, two of them being polymerized. Furthermore, the densities of occupied sites and of bond interactions in each phase were calculated, in order to determine the nature of the transitions between the phases. The results obtained are consistent with a second-order transition line between the nonpolymerized and the regular polymerized phase and a first-order transition between the nonpolymerized and the dense polymerized phase. The boundary between both polymerized phases may be of first or second order, and thus evidence for a tricritical point is found.
REPORT ON POLYMERS AND STATISTICAL THERMODYNAMICS
This report sums up some results about polymers and statistical thermodynamics and used as a background material for "An air-polymer analogy for modeling air flow through rubber-metal interface" research thesis by this author. The subjects reviewed are the basics of statistic thermodynamics, the thermodynamics of chain molecule and the mechanical properties of rubberlike network of chains.
Physica A: Statistical Mechanics and its Applications, 1991
A new statistical mechanics treatment of polymer systems is presented, built on the lattice model as used by Flory and Huggins, but using the pair approximation of the cluster variation method (CVM). The present paper is the first of a series, and shows how the new method compares with the most common theories in the polymer ficld. Phase separation diagrams arc qualitatively similar to those of Flory. A noticcabie differcncc is that our result for x, deviates from the value i proposed by Flory for any coordination number z. WC also calculate the osmotic pressure for the polymer solution using our approximation and some comparison with Flory (F), Flory-Huggins (FH), Bawcndi-Freed (BF) theories and the Monte Carlo (MC) simulations are done for the a thermal case. The qualitative agreement between our results and the FH results and the MC results supports the reliability of the present treatment.