Double Lattice Model Research Articles

Physical properties of polymer blends and their theoretical consideration

The physical properties of polymer blends and their theoretical consideration based on the double-lattice model (DLM) and a simple rheological equation of state were investigated. Cloud points of poly(ϵ-caprolactone) (PCL)/polystyrene (PS) blends were determined using the thermooptical analysis (t.o.a.) technique. PCL ( M w = 125 000)/PS( M w = 6200) blends exhibit an upper critical solution temperature (UCST). The adjustable parameter values are r 2 = 844.859, ϵ/ k = −313.092 K, and δϵ/ k = 1381.824 K. For PCL ( M w = 15000)/PS ( M w = 840) blends, r 2 = 177.560, ϵ/ k = −254.163 K, and δϵ/ k = 1195.827 K. The DLM successfully describes and predicts the phase behaviour of polymer blends. Steady-state shear viscosity of PCL/linear low-density polyethylene (LLDPE) blends was also investigated. D.s.c. results showed that there is no miscibility between PCL and LLDPE. A simple rheological equation of state successfully describes and predicts the flow behaviour of the proposed polymer blend.

Estimation of vapor-liquid equilibria for polymer solutions by a group-contribution method

A new group-contribution method is presented for prediction of vapor-liquid equilibria of polymer solutions. It is based on a close-packed lattice model developed previously; this model contains a revised Flory-Huggins entropy and a series expression for excess internal energy, as well as a double-lattice model to account for specific, oriented interactions. The group-contribution method includes three contributions: the combinatorial and free-volume contribution which is primarily responsible for entropy, the van der Waals energy contribution responsible for dispersion and polar forces, and the specific energy contribution from hydrogen bonding. Most of the model parameters can be estimated using pure-component properties, either from experimental data or from group-contribution methods in the literature. The group-contribution method described in this work gives parameters for the cross specific interaction energy from hydrogen bonding between segments of polymers and segments of solvents. Parameters for pairs of groups are obtained by correlating experimental data for vapor pressures and infinite-dilution activity coefficients of solvents in polymer solutions. The method shown here gives improved results for prediction when compared with those of other group-contribution models.

Molecular thermodynamic theory for polymer systems. I. A close-packed lattice model

A molecular thermodynamic theory for multicomponent polymer solutions and polymer blends is presented which is composed of two contributions: an equation of state and a close-packed lattice model served as a high-pressure limit. In this paper, we introduce a close-packed lattice model which is derived by using an effective chain-insertion probability for the entropy and a series expansion for the energy. The former is used to improve the mean-field Flory-Huggins entropy term. While the latter is adopted to account for interactions between more than two segments. For practical application, a double-lattice model is introduced to account for the specific oriented interactions which automatically gives a temperature dependence for the energy parameter. The coefficients in the model are determined by a few computer-simulation data and by referring to the rigorous Freed's theory. The agreement with computer-simulation results for spinodals and binodals is shown to be excellent, much better than the classical Flory-Huggins theory. Comparisons with calculated binodals and critical coordinates by the Freed's theory for binaries and ternaries coverd a wide range of chain length indicate that the two theories give almost the same results while the present work is much simpler and suitable for engineering use. Illustrative examples are presented for liquid-liquid equilibria of polymer solutions and polymer blends.

Molecular thermodynamics of polymer solutions

Abstract The previously revised Freed lattice model is used to replace the Flory-Huggins mean-field theory for binary polymer/solvent mixtures. Liquid-liquid binodals calculated for model systems are in good agreement with those from computer simulation and those from Freed's rigorous lattice cluster theory. A double lattice model is used to account for oriented interactions such as hydrogen bonding. To account for free-volume effects, an essentially empirical, two-step process is adopted. First, pure components are mixed to form a close-packed polymer solution. Then, holes are Introduced to mix with the close-packed solution which is considered to be a pseudo-pure substance. The revised Freed model is applied for both steps. A size parameter C r accounts for the composition dependence of effective chain lengths. Two binary energy parameters are used: ϵ is the conventional interchange energy of a nearest-neighbor i-J segment-segment pair, and ξ corrects the quadratic rule for mixing the two components to form a pseudo-pure substance. Pure-component parameters, Γ i o and ϵ ii , are obtained by fitting experimental pure-component vapor pressure and pVT data. A few examples indicate that the semi -empirical model can describe satisfactorily a variety of liquid-liquid equilibria Including UCST, LCST, XXXiscibility-loop and hour-glass-shaped phase diagrams, as well as the pressure dependence and the molar-mass dependence of liquid-liquid binodals.

Binary liquid-liquid equilibria from a double-lattice model

Hu Y., Liu H., Soane D.S. and Prausnitz J.M., 1991. Binary liquid-liquid equilibria from a double-lattice model. Fluid Phase Equilibria, 67: 65-86. Freed's lattice field theory is used to establish a double-lattice model for the Helmholtz energy of mixing for strongly non-ideal binary liquid mixtures. The coefficients of Freed's expansion terms are adjusted such that the calculated coexistence curve for the simple Ising lattice is in excellent agreement with that calculated from the Padé-approximant coefficients for spontaneous magnetization proposed by Scesney. For a variety of binary systems, two parameters (ϵ/ k and r 2) are obtained from experimental critical consolute points; here (ϵ/ k is the interchange energy and r 2 is the number of lattice sites required by molecule 2, with r 1=1. Calculated coexistence curves are generally in good agreement with experiment for fluid mixtures without specific (oriented) interactions, much better than those calculated using Flory-Huggins theory, especially near critical consolute points. Introducing a secondary lattice to account for highly oriented interactions (hydrogen bonding), requires an additional energy parameter δϵ/ k and an empirical parameter c 10. With these parameters coexistence curves are fitted well for systems having a miscibility loop with both lower and upper critical solution temperatures.

Double-lattice model for binary polymer solutions

A double-lattice model based on Freed's theory is introduced. «Ordinary» polymer solutions are described by the primary lattice, while a secondary lattice is introduced as a perturbation to account for oriented interactions. With this model, coexistence curves can be reproduced for systems having an UCST, a LCST or a missibility loop with both LCST and UCST