Cluster integrals

The quantity b has the dimension of a volume and is known as the excluded volume or the binary cluster integral. The mean force potential is a function of temperature (principally as a result of the soft interactions). For a given solvent or mixture of solvents, there exists a temperature (called the 0-temperature or Te) where the solvent is just poor enough so that the polymer feels an effective repulsion toward the solvent molecules and yet, good enough to balance the expansion of the coil caused by the excluded volume of the polymer chain. Under this condition of perfect balance, all the binary cluster integrals are equal to zero and the chain behaves like an ideal chain. 

In the equations developed by Reilly and Wood (15) from the cluster Integral model (1 6), y+ is calculated in complex solutions from excess properties of single salt solutions. Note that the cluster Integral approach 1s based upon terms which represent the contributions of pair-wise ion interactions 1n various types of clusters to the potential interaction energy. Then, the partition function and the excess properties of the solution can be evaluated. The procedure is akin to the vlrial expansion 1n terms of clusters. 

Fig. 10.15 Pharmacogenomic response to a multifactorial therapy in Alzheimer s disease (AD) according to a trigenic cluster integrated by the apolipoprotein E (APOE), presenUin 1 (PSl), and presenilin 2 (PS2) genes. (Adapted from refs. 13 0, and 291.)...

Based on the cluster integral method [27], Pitzer [19] divided the difference terms 0mn into two parts... 

Here ft is the two-dimensional binary cluster integral defined by... 

According to the definition, Eq. (38), the cluster integral /5 may be roughly evaluated as... 

A and B are short- and long-range interaction parameters (A is also termed the unperturbed chain dimension), < R2)o/2 is the mean-square end-to-end distance of the chain in the unperturbed state, P the binary cluster integral, and nio is the molecular weight of a segment. According to the Kurata-Fukatsu-Sotobayashi-Yamakawa theory 65), ho(z) is related to z by... 

One of the newer theoretical treatments, based on the pioneering statistical thermodynamic work of McMillan and Mayer (6) y as mathematically formulated by Friedman W, does appear to hold significant promise as a theory of sufficient generality that it may eventually embody other working theories as demonstrated special cases. This theory, known as the cluster integral expansion theory (j ) or simply as cluster theory (9)y has been developed to the point where applications have been made to calculating... 

This was accomplished by expanding an existing model based on the cluster integral expansion theory of electrolyte solution structure into a comprehensive thermodynamic model describing the major and minor components of brines, including metals present in trace amounts. 

CLSM = confocal laser scanning microscopy see CSLM cluster integral 1.3.65... 

The essential feature of the ideal, or unperturbed, state resides in that two chain atoms do not interact if their separation along the chain sequence is sufficiently large. This will be expressed by saying that the sum of the binary cluster integral / and of a repulsive three-body contribution is zero at the ideal temperature 7=0 [6], We have [3], in k T units. 

The Zimm Cluster Integral. Zimm was the first to apply the KB theory to the solvent clustering in binary solvent (1) + polymer (protein) (2) mixtures. On the basis of the KB theory, he was the first to derive the following expression for the KB integral (KBI),... 

Further, Zimm and Zimm and Lundberg introduced the notion of cluster integral G lv to characterize the solvent clustering in the systems solvent (1)—polymer (protein) (2). They called G v a cluster integral , because the quantity q> G lv is the mean number of type 1 molecules in excess of the mean concentration of type 1 molecules in the neighborhood of a given type 1 molecule thus, it measures the clustering tendency of the type 1 molecules . 

Using eqs 3 and 4, one can write the following expression for the cluster integral G lv ... 

The excess (deficit) of a solvent around a central solvent molecule (p Gulv provided by the Zimm cluster integral is related to the true excess (or deficit) A nvia the expression... 

Figure 1. Cluster integral, G lv, versus solvent volume fraction (the solid line was calculated by us) (A) the toluene (1) + polystyrene (2) system (, ref 1), (B) the water (1) + collagen (2) system (, ref 12), (C) the water (1) + serum albumin (2) system, (D) the water (1) + hydroxypropyl cellulose (2) system, and (E) the water (1) + Pluronic PI05 (2) system.

Toluene (1) + Polystyrene (2). The cluster integral Gn/ v for this system was calculated by Zimm and Lundberg and... 

In this case, the cluster integral of Zimm and Lundberg (10) for water-water interactions is high enough to approach the vapor pressure of pure water as a limit and fugacity equal to 1. 

The cluster itself was also recovered in a racemic form (140). This result means that the photo-induced racemization of the chiral catalyst is faster than the hydrosilylation, and it may be used as an argument for or against cluster integrity throughout the catalytic process. 

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