Lattice model polymer adsorption

In our previous discussion of the lattice model of adsorption, we have already made use of the components of the radius of gyration as excellent measures for the globular compactness of polymer conformations parallel [see Figs. 13.9(a,b)] and perpendicular [Figs. 13.9(c,d)] to the surface, respectively. These components are particularly helpful for the identification of structural changes induced by the presence of an attractive substrate. 

Of particular interest has been the study of the polymer configurations at the solid-liquid interface. Beginning with lattice theories, early models of polymer adsorption captured most of the features of adsorption such as the loop, train, and tail structures and the influence of the surface interaction parameter (see Refs. 57, 58, 62 for reviews of older theories). These lattice models have been expanded on in recent years using modem computational methods [63,64] and have allowed the calculation of equilibrium partitioning between a poly-... 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

Silberberg47) used a quasi-crystalline lattice model for the adsorption of flexible macromolecules. If it is assumed that an adsorbed polymer chain with P segments consists of ma trains of length i and mBi loops of length i, the total number of configurations of the chains is given by... 

In this review, we introduce another approach to study the multiscale structures of polymer materials based on a lattice model. We first show the development of a Helmholtz energy model of mixing for polymers based on close-packed lattice model by combining molecular simulation with statistical mechanics. Then, holes are introduced to account for the effect of pressure. Combined with WDA, this model of Helmholtz energy is further applied to develop a new lattice DFT to calculate the adsorption of polymers at solid-liquid interface. Finally, we develop a framework based on the strong segregation limit (SSL) theory to predict the morphologies of micro-phase separation of diblock copolymers confined in curved surfaces. 

Lattice models play a central role in the description of polymer solutions as well as adsorbed polymer layers. All of the adsorption models reviewed so far assume a one-to-one correspondence between lattice random-walks and polymer configurations. In particular, the general scheme was to postulate the train-loop or train-loop—tail architecture, formulate the partition function, and then calculate the equilibrium statistics, e.g., bound fraction, average loop... 

Note that x does not explicitly enter [3.4.56]. The effect of x indirectly accounted for through its influence on the profile 0 z). As a matter of fact, [3.4.56] is not an exclusive result of the SF-theoiy earlier polymer adsorption models such as those of Roe ) and Helfemd ) give exactly the same form for x as a functional of the profile, although these theories predict a different profile as a function of x cmd X (and, hence, lead to different numerical results for tc). Equation [3.4.56] may be considered as the lattice version of a density functioned. This equation does not only apply to adsorbed homopolymer layers, but it is also valid for brushes (where... 

It is, however, the self-consistent field method that has been most extensively applied to polymer adsorption and in particular the lattice-based discretisation of the Edwards modified diffusion equation associated with Scheutjens and Fleer (1979, 1980). In this model the solution up to the impenetrable adsorbing surface is modelled as a lattice of equal volume cells. All of the lattice layers... 

Several theories exist that describe the process of polymer adsorption, which have been developed either using a statistical mechanical approach or quasi-lattice models. In the statistical mechanical approach, the polymer is considered to consist of three... 

The lattice models have been used frequently in theoretical studies of adsorption. The wide popularity of this approach results from its flexibility and simplicity. It may be applied for systems of various dimensionahties, for many lattice geometries, for different models of adsorbate-adsorbate molecular interactions, and many spatially varying external fields. Most studies have focused on a two-dimensional or three-dimensional cubic lattice, with only isotropic nearest-neighbor couplings. The isomorphism between the Ising model and the classical lattice gas or a coarse model of binary mixture is well known and very helpfijl for theoretical analysis. The lattice models can be also appHed to describe the systems involving polymers. 

Eventually, let us compare the adsorption behavior with what we had found in Chapter 13 for simplified hybrid lattice models of polymers and peptides near attractive substrates. The adhesion of the jjeptides at the Si(lOO) substrate exhibits very similar features. Exemplified for peptide S3, Fig. 14.13 shows the plot of the canonical probability distributionpcan E, q) 8 E — E(X))S(q — q(X))) at room temperature (T = 300 K). The peak at E, q) (80.5 kcal/mole, 0.0) corresponds to conformations that are not in contact with the substrate. It is separated from another peak near (E, q) (74.5 kcal/mole, 0.2) and belongs to conformations with about 17% of the heavy atoms with distances < 5 A from the substrate surface (compare with Fig. 14.12). That means adsorbed and desorbed conformations coexist and the gap in between the peaks separates the two pseudophases in g -space, which causes a kinetic free-energy barrier. Thus, the adsorption transition is a first-order-like pseudophase transition in q, but since both structural phases (adsorbed and desorbed) coexist almost at the same energy, the transition in E space is weakly of first order. ... 

The simplest model of polymers comprises random and self-avoiding walks on lattices [11,45,46]. These models are used in analytical studies [2,4], in particular in the numerical implementation of the self-consistent field theory [4] and in studies of adsorption of polymers [35,47-50] and melts confined between walls [24,51,52]. 

To date, very little experimental and modeling work has been done on the adsorption of oxygen in the presence of water and polymer electrolyte although it is strongly anticipated that water, polymer, and specifically adsorbed species have a great impact on adsorption of oxygen. Most experimental and simulation studies were carried out in so-called vapor phase. In these simulations, only a few gas molecules and a small-size lattice of the catalyst were considered. 

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