Adsorption, polymer self-consistent field theory

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]. 

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. 

Although Xs is referred to as an energy parameter and Ua are adsorption energies it is clear that they are actually viewed as enthalpies by the original authors of the self-consistent field theory of polymer adsorption. This needs to be appreciated in reading the literature since great confusion can arise otherwise. 

Cyclic polymers are unable to form tails and hence the conformational energy change on adsorption is less for the cyclic polymer at low relative molecular masses when the surface concentration is small. As the relative molecular mass increases, cyclics form larger loops that reduce the entropy change on adsorption whereas in the linear polymer the contribution of the tails to the entropy change becomes diluted at higher relative molecular masses. Figure 5.17 shows the volume fraction profiles calculated from self-consistent field theory, with the separate contributions from loops and tails. 

Figure 1. Theoretically calculated adsorption isotherm for a polymer of 100 segments adsorbing from a theta solvent (x=0.5). The isotherm was obtained from the self-consistent field theory of Scheutjens and Fleer. Two arrows indicate the values of the concentration difference Cb-Cs between bulk and surface zone one for adsorption (high cb ) and one for desorption (cb =0). The length of the arrows is a measure of the rate of the corresponding processes.

The development of adsorption theory provides the explanation of the macromolecule behavior in the adsorption layer and provides the basis of arguments on the experimental results. A few theoretical models, describing the adsorbed macromolecule, are widely used now. Self-consistent field theory or mean field approach is used to calculate the respective distribution of trains, loops, and tails of flexible macromolecule in the adsorption layer [22-26]. It allows one to find the segment density distribution in the adsorption layer and to calculate the adsorption isotherms and average thickness of the adsorption layer. Scaling theory [27-29] is used to explain the influence of the macromolecule concentration in the adsorption layer on the segment density profile and its thickness. Renormalization group theory [30-33] is used to describe the excluded volume effects in polymer chains terminally attached to the surface. The Monte Carlo method has been used for the calculation of the density profile in the adsorption layer [33-35]. 

Molecular-based theories are useful for developing rational stabilizer design criteria and investigating the correlation with bulk phase behavior for stabilizers in supercritical fluids. Molecular theories of polymer adsorption, such as the lattice self-consistent field (SCF) theory of Scheutjens and Fleer[69], allow chain structure, adsorption energy, solubility, length, and concentration to be varied independently. Simulation, while more computationally intensive, offers the additional advantages of... 

Adsorption of polymers on surfaces plays a key role in numerous technological applications and is also relevant to many biological processes. During the last three decades it has been constantly a focus of research interest. The theoretical studies of the behavior of polymers interacting with solid substrate have been based predominantly on both scaling analysis [49] as well as on the self-consistent field (SCF) approach [50]. The close relationship between theory and computer experiments in this field [27, 51] has proved especially fruitful. Most investigations focus on the determination of the critical adsorption point (CAP) location and on the scaling behavior of a variety of quantities below, above, and at the CAP. 

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