Single-contact approximation

The single-chain structure factors calculated in the previous sections correspond to the infinite dilution limit. This limit also corresponds to zero scattering intensity and is not useful so that concentration effects have to be included in the modeling of polymer solutions. First, Zimm s single-contact approximation [5] is reviewed for dilute polymer solutions then, a slight extension of that formula which applies to semidilute solutions, is discussed. 

Since 1986, theories have been developed which employ the concept of direct correlation functions. The total correlation is factorized into a contribution from intermolecular correlations, Ceff(q), and intramolecular correlations, P(q), the form factor of the polyion. Benmouna etal. [48,49] presented a generalization of the single contact approximation due to the Zimm [50] to express the direct correlation function as... 

In addition to an estimate for P(q,cX interpretation data on S(q,c) requires an evaluation of H(q, c). Experience su ests that H(q, c) 1 for dflute solutions (e.g., solutions with [t]]c <0.1), in which case the single-contact approximation applks, and data on S(q, c) vs u = (qRc) at different c form a series of paraltel curves [1-6, 12, 19]. However, data are now available showing that in moderately concentrated solutions, H(q, c) deviates tignificantly from unity under both Flory Theta condhions [18] ai in good solvents [46,48, 49]. The experimental evaluation of H(q,c) is ampler under Flory Theta conditions as Rg may be considered to be independent coiK tration in that case. The behavior for H(q, c) under Flory Theta ctmditions calculated from... 

In concluding this paper, we remark that if we limit our calculations to a single contact approximation and pretend as though the result to be useable to all N, then we have the following "self-consistent values for a , the expansion parameter. 

For a mixture of solutes all having the same specific refractive increment, e.g., a polymer heterogeneous in molecular weight, the reciprocal scattering function in Eq. (A5), for the single-contact approximation, is replaced... 

Figure 3.22. The schematic representation of (a) the exponential model (EM) and (b) contact approximation. In EM the reaction sphere of radius a is transparent for particles, which leave it by a single jump with the rate ksep. In contact calculations, the same sphere surrounds an excluded volume and recombination takes place only at its surface, or more precisely in a narrow spherical layer around it.

Zimm s approximation [5] assumes that inter-chain interactions occur only through single contacts. Given two monomers i and j that belong to two different chains (say, called 1 and 2), the two-chain distribution function is ... 

Hence in a single contact experiment the gain in signal intensity is approximately... 

The continuum theory of interfacial rubbing temperature described in the preceding section requires perfect contact of the two bodies over the entire nominal rubbing area but we know that true contact of real surfaces is at the asperities. If the continuum treatment is to have any relevance for real rubbing temperatures, it must be an acceptable approximation to what actually occurs physically. One such possibility, advocated by Archard [3] as applicable for closely spaced asperities and slow sliding speeds, is to treat the aggregate true area of the asperity contacts as the equivalent area of a single contact. 

These problems have been improved in recent years by the microfabrication of sharp tips with radii less than 10 nm, the observation in an SEM or STEM of the exact radius before and after the experiment, the use of robust carbon-nanotube probes, and general improvements in control electronics. However, another method used initially was the attachment of a small colloid particle in place of the AFM tip. These particles were considered a reasonably good approximation to a single-asperity contact their radii were accurately known and remained the same for the duration of the experiment. Such probes have also been used to investigate colloids where surface roughness is an important aspect of the colloid interaction. 

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