Huggins Molecular Models

Huggins Molecular Models.—In a series of, so far, 14 papers, Huggins has developed an approach to solution thermodynamics that essentially builds on lattice-graph models, removing phenomenological parameters by re-expressing them in terms of molecular quantities obtained, as far as is possible, by independent experimental means. Thus far the theory has been applied mainly to non-polymer systems. The most recent part considers benzene solutions of n-alkanes as model oligomers for polyethylene. 

In its simplest form the model assumes perfectly random mixing of various kinds of molecular contacts. In this form it has been applied to polystyrene-cyclohexane with partial justification from data on cyclohexane-toluene mixtures. A refinement to the random mixing assumption corrects for the influence of local surroundings on the average randomness of orientation of solvent molecules and polymer segments. With these corrections the free energy of mixing can be written  

Kleintjens et al. have obtained spinodal and critical point expressions from equation (12) and applied these to data for polystyrene in cyclohexane. They obtain values for ko and kp which can be related to empirical correction terms introduced by Flory et al. Few conclusions could be drawn, however, as to either the effectiveness or the underlying molecular nature of the orientation correction parameters.  

Koningsveld and Stepto used an extension of Silberberg s derivation (see p. 301) of the ideal entropy of mixing in order to attach some molecular interpretation to randomness of orientation corrections. They conclude that these terms 

Huggins has developed a more genoal model which considers non-randomness of mixing but as yet this appears to have not been applied to polymers. It could no doubt be useful in analysing the wealth of data collected by Starkweather on non-random mixing (clustering) in polymer solutions. 

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Polymer simulations can be mapped onto the Flory-Huggins lattice model. For this purpose, DPD can be considered an off-lattice version of the Flory-Huggins simulation. It uses a Flory-Huggins x (chi) parameter. The best way to obtain % is from vapor pressure data. Molecular modeling can be used to determine x, but it is less reliable. In order to run a simulation, a bead size for each bead type and a x parameter for each pair of beads must be known. 

Linking this molecular model to observed bulk fluid PVT-composition behavior requires a calculation of the number of possible configurations (microstmctures) of a mixture. There is no exact method available to solve this combinatorial problem (28). ASOG assumes the athermal (no heat of mixing) FIory-Huggins equation for this purpose (118,170,171). UNIQUAC claims to have a formula that avoids this assumption, although some aspects of athermal mixing are still present in the model. 

For the solubility of TPA in prepolymer, no data are available and the polymer-solvent interaction parameter X of the Flory-Huggins relationship is not accurately known. No experimental data are available for the vapour pressures of dimer or trimer. The published values for the diffusion coefficient of EG in solid and molten PET vary by orders of magnitude. For the diffusion of water, acetaldehyde and DEG in polymer, no reliable data are available. It is not even agreed upon if the mutual diffusion coefficients depend on the polymer molecular weight or on the melt viscosity, and if they are linear or exponential functions of temperature. Molecular modelling, accompanied by the rapid growth of computer performance, will hopefully help to solve this problem in the near future. The mass-transfer mechanisms for by-products in solid PET are not established, and the dependency of the solid-state polycondensation rate on crystallinity is still a matter of assumptions. 

Current thermodynamic theories for polymer systems are combinations of the Flory -Huggins, Guggenheim, and Equations-of-State approaches. All of these theories make use of empirical parameters and are based on assumptions about the underlying molecular model. 

This coefficient can be predicted by molecular modeling using an off-lattice Flory-Huggins simulation approach [GIL 09a, GIL 10, VIT 10],... 

Considering a binary blend composed of homopolymer A and B with volume fraction /a and /b = 1 - /a and assuming monomers of equal molecular size, the Flory-Huggins lattice model delivers an expression for the Gibbs free energy of... 

Flory-Huggins interaction parameter versus weight fraction of PVA in PVA/chitosan blends. (From Jawalkar, S. S., Raju, K. V. S. N., Halligudi, S. B., Sairam, M., and Aminabhavi, T. M. 2007. Molecular modeling simulations to predict compatibility of poly(vinyl alcohol) and chitosan blends A comparison with experiments. Journal of Physical Chemistry B 111 2431-2439.)... 

Most theoretical approaches to polymer blends are extensions of models for polymer solutions and thus focus on the liquid state. They aim to describe experimental phase behaviour and most authors claim to be able to predict trends in phase behaviour outside the range of measurements. Current theories are combinations of Flory-Huggins, Guggenheim, and Equations of State approaches. All of these theories make use of empirical parameters and are based on assumptions about the underlying molecular model. 

Compatibility of polymer blends is often achieved through favorable specific interaction such as hydrogen bonding. Although a fundamental understanding of the pertinent thermodynamics plays a crucial role in the preparation of blends, there are few useful molecular thermodynamic models for polymer blends with specific interactions, a major exception is the classical incompressible model developed by Flory and Huggins [7-8]. The objective of this work is to develop an approximate but theoretically based molecular model for predicting compatibility of polymer blends within the framework of a lattice model. 

Both Flory [143] and Huggins [144] in 1941 addressed themselves to this problem with the initial aim of describing solutions of linear polymers in low molecular weight solvents. Both used lattice models, and their initial derivations considered only polymer length (rather than shape, i.e. branching, etc.) The derivation given here will also limit itself to differences in molecular size, but will be based on an available volume approach. 

Recent developments in the theory of polymer solutions have been reviewed by Berry and Casassa (32), and by Casassa (71). Casassa, who has contributed very largely to these developments, has adopted a statistical mechanical approach using molecular distribution functions, as first outlined by Zimm (72), rather than using a lattice model like that used by Flory, Huggins, and many later workers. 

If we turn from phenomenological thermodynamics to statistical thermodynamics, then we can interpret the second virial coefficient in terms of molecular parameters via a model. We pursue this approach for two different models, namely, the excluded-volume model for solute molecules with rigid structures and the Flory-Huggins model for polymer chains, in Section 3.4. 

Taking into account the modes in which the water can be sorbed in the resin, different models should be considered to describe the overall process. First, the ordinary dissolution of a substance in the polymer may be described by the Flory-Huggins theory which treats the random mixing of an unoriented polymer and a solvent by using the liquid lattice approach. If as is the penetrant external activity, vp the polymer volume fraction and the solvent-polymer interaction parameter, the relationship relating these variables in the case of polymer of infinite molecular weight is as follows ... 

Micelle formation in solutions of an AB diblock in low-molecular-weight A homopolymer has been considered by Leibler et al. (1983), using Flory-Huggins theory to determine the free energy of mixing of micelles. This model is discussed in detail in Section 3.4.2. 

Rigby et al. (1985) showed using a Flory-Huggins model that for symmetric blends, the spinodal and critical temperatures decrease linearly with increasing content of a symmetric diblock for blends with equal volume fractions of homopolymers (with the same molecular weight). The condition for a linear decrease of the binodal was less restrictive, not requiring equal concentrations of homopolymer in the blend. 

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