Step-weighted lattice walk

SF theory is a statistical thermodynamic model in which chain conformations are formulated as step-weighted random walks in an interfacial lattice (Figure 2). A simple case involves the adsorption of a flexible, linear, homo-disperse, uncharged molecule at a uniform planar surface. Interactions among... 

The quasi-lattice model was developed by Roe (13) and Scheutjens and Fleer (14) (SF theory) The basic analysis considered all chain conformations as step-weighted random walks on a quasi-crystalline lattice that extends in parallel layers from the surface. This is illustrated in Figure 16.2 which shows a possible conformation of a polymer molecule at a flat surface. The partition function was written in terms of the number of chain configurations that were treated as connected sequences of segments. In each layer parallel to the surface, random mixing between the segments and solvent molecules was assumed, i.e. by using... 

Consider one r-th order triangular subgraph of the infinite order graph. It is connected to the rest of the lattice by only three bonds. Our aim to sum over different configurations of the SAW on this subgraph, with a weight x for each step of the walk. These can be divided into four classes, as shown in Fig. 6. Here AW is the sum over all configurations... 

An improvement of this method—the so-called biased sampling [55] (or inversely restrieted sampling)—suggests to look ahead at least one step in order to overcome the attrition. Consider a SAW of i steps on a -coordination number lattice. To add the / + 1st step one first checks which of the = q — neighboring sites are empty. If k qQ > k>0) sites are empty one takes one of these with equal probability 1 /A if A = 0 the walk is terminated and one starts from the beginning. This reduces the attrition dramatically. Now each A-step walk has a probability PAr( i ) = Ylf=i so that dense configurations are clearly more probable. To compensate for this bias, each chain does not count as 1 in the sample but with a weight... 

One of the simplest idealizations of a flexible polymer chain consists in replacing it by a random walk on a periodic lattice, as shown in Fig. I.l. The walk is a sucession of N steps, starting from one end (a) and reaching an arbitrary end point ( >). At each step, the next jump may proceed toward any of the nearest-neighbor sites, and the statistical weight for all these possibilities is the same. The length of one step will be called a. 

For definiteness, we inscribe our chain on a Floiy-Huggins lattice of parameter a. It is then described by a walk ofN steps linking the lattice points r I... Fjv. If a potential U(r) acts on each monomer, the statistical weight associated with this particular realization is... 

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