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In this chapter we consider a model introduced in Kantor and Kardar [203], where each monomer carries a random charge, and each self-intersection of the polymer is rewarded when the two charges of the associated monomers have opposite sign and is penalized when they have the same sign. This model is a variation on the weakly self-avoiding walk described in Chapters 3 and 4, with a random self-interaction...
A copolymer is a polymer consisting of different types of monomer. In this chapter we consider a two-dimensional directed copolymer, consisting of a random concatenation of hydrophobic and hydrophilic monomers, near a linear interface separating two immiscible solvents, oil and water (see Fig. 9.1). We will be interested in the quenched path measure (of the type defined in (1.3)). We will show that,...
In this chapter we consider a different version of the directed copolymer model analyzed in Chapter 9, namely, one where the linear interface is replaced by a random interface. In particular, rather than putting the oil and the water in two halfplanes, we place them in large square blocks in a random percolation-type fashion. This is a crude model of a copolymer in an emulsion, consisting of oil droplets...
In this chapter we look at pinning and wetting of a polymer by a random substrate. To that end, we think of the substrate as being composed of different types of atoms or molecules, occurring in a random order, and each time the polymer visits the substrate it picks up a reward or a penalty according to the type it encounters (see Fig. 11.1). Alternatively, we may think of a random copolymer consisting...
In Chapters 9-10 we studied a copolymer in the vicinity of a linear, respectively, a random selective interface between solvents. The application we had in mind was a copolymer consisting of a random concatenation of hydrophobic and hydrophilic monomers, living in a medium consisting of oil and water located in two halfspaces, respectively, in mesoscopic droplets arranged in a percolation-type fashion...
In Chapters 3 and 4 we consider a variation of the SAW in which selfintersections are not forbidden but are penalized. We refer to this as a soft polymer. In Chapter 3 will show that the soft polymer has ballistic behavior in d = 1. The proof uses a Markovian representation of the local times of onedimensional SRW (a powerful technique that is useful also for other models), in combination with large...
In this chapter we consider the same model as in Chapter 3, but for d ? 5 instead of d = 1. Our goal is to prove diffusive behavior. The tool to achieve this is the so-called lace expansion, a combinatorial technique that hinges on the idea that in high dimensions the soft polymer can be viewed as a “perturbation” of SRW. For the exposition below, we borrow from van der Hofstad [156], Section 2, and...
In this chapter we take a brief look at a version of the soft polymer where the penalty of the self-intersections decays with the loop length, i.e., the difference between the times at which the self-intersection occurs. This model is called the elastic polymer. Interestingly, it will turn out that this model has diffusive behavior in any d ? 1 as soon as the decay is sufficiently fast, namely, the...
In this chapter we look at the soft polymer (studied in Chapters 3-4) and add to the penalty for self-intersections a reward for self-touchings, i.e., a negative energy is associated with contacts between any two monomers that are not connected to each other within the polymer chain. This is a model of a polymer in a poor solvent: when the polymer does not like to make contact with a solvent it is...
In this chapter we study a polymer in the vicinity of a linear substrate. Each monomer on the substrate feels a binding energy, resulting in an attractive interaction between the polymer and the substrate. We will consider the two situations where the substrate is: (1) penetrable (‘pinning at an interface”), (2) impenetrable (“wetting of a surface”). Our focus will be on the free energy and on the...
A polymer is a large molecule consisting of monomers that are tied together by chemical bonds. The monomers can be either small units (such as CH2 in polyethylene; see Fig. 1.1) or larger units with an internal structure (such as the adenine-thymine and cytosine-guanine base pairs in the DNA double helix; see Fig. 1.2). Polymers abound in nature because of the multivalency of atoms like carbon, silicon,...
In this chapter we consider two basic models for a polymer chain: simple random walk (SRW), describing a polymer with no self-interaction, and selfavoiding walk (SAW), describing a polymer with a self-interaction given by the “excluded-volume effect”, i.e., no site can be occupied by more than one monomer. Our goal is to give a quick summary of what is known for these models in order to set the stage...
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