Exact solution of the thermodynamics and size parameters of a polymer confined to a lattice of finite size: Large chain limit

Abstract

We extend the exact solutions of the Di Marzio-Rubin matrix method for the thermodynamic properties, including chain density, of a linear polymer molecule confined to walk on a lattice of finite size. Our extensions enable (a) the use of higher dimensions (explicit 2D and 3D lattices), (b) lattice boundaries of arbitrary shape, and (c) the flexibility to allow each monomer to have its own energy of attraction for each lattice site. In the case of the large chain limit, we demonstrate how periodic boundary conditions can also be employed to reduce computation time. Advantages to this method include easy definition of chemical and physical structure (or surface roughness) of the lattice and site-specific monomer-specific energetics, and straightforward relatively fast computations. We show the usefulness and ease of implementation of this extension by examining the effect of energy variation along the lattice walls of an infinite rectangular cylinder with the idea of studying the changes in properties caused by chemical inhomogeneities on the surface of the box. Herein, we look particularly at the polymer density profile as a function of temperature in the confined region for very long polymers. One particularly striking result is the shift in the critical condition for adsorption due to surface energy inhomogeneities and the length scale of the inhomogeneities; an observation that could have important implications for polymer chromatography. Our method should have applications to both copolymers and biopolymers of arbitrary molar mass.

Lattice cluster theory for dense, thin polymer films

“…Previous self-consistent field theories of polymers at interfaces [19][20][21][22][23] are formulated in terms of the configurations of an ensemble of single chains 24 in the self-consistent environment of their surroundings that are composed of the same set of chains. The present extension of the LCT to describe thin polymer films, on the other hand, follows the treatment in the LCT of the thermodynamics of polymers in the bulk by being predicated on a wholly different approach.…”

Surface Interaction Parameter Measurement of Solvated Polymers via Model End-Tethered Chains

“…8 This model has recently been extended to operate at a nearly atomistic level of detail. 9,10 The ability to extend and experimentally apply these models to new polymer structures would represent an immediate catalyst to method development in interaction based separations for synthetic polymers, and broaden adoption of this promising method of macromolecular characterization. In Guttmann’s model, the partition function for isolated chains in a pore is calculated via a Di Marzio-Rubin 11 lattice model, and is used to predict elution curves via the Casassa 12 method.…”

“…Guttmann et al have presented a model that can reproduce, across all these regimes, the elution curves of a polymer given the polymer’s structure . This model has recently been extended to operate at a nearly atomistic level of detail. , The ability to extend and experimentally apply these models to new polymer structures would represent an immediate catalyst to method development in interaction-based separations for synthetic polymers and broaden adoption of this promising method of macromolecular characterization.…”

A Simple Method for Complex Monomer Creation in the Matrix Method for the Statistics and Thermodynamics of a Confined Polymer Chain

“…For an analytical partition function or averaging method, we use the traditional integration technique to sum over all states. 10 In the matrix method discussed in this paper, 11,12 the chain is confined to a lattice. In the matrix method for a polymer chain, the matrix creates all possible states to step between two sites on the lattice for a given monomer.…”

“…15−19 matrix theory, we obtained exact equations for the thermodynamic properties including chain density of a linear polymer molecule made up of one step monomers confined within an infinite rectangular cylinder. 12 The chain was allowed to interact with the wall with variable spatial interactions. In this theory, the boundary could be of arbitrary shape and the attraction of the monomers for the sites could be an arbitrary function of each site.…”