Cell cluster theory

Theories of the Liqmd State 140. 8 Cell Cluster Theory 141. 

Formula (7.8.13) expresses the configurational partition function (7.8.6) as the sum of contributions from all possible subdivisions of the iV-ceUs into cell clusters. A similar expression is well known in the theory of imperfect gases where Q is expressed as a sum of contributions from all possible molecular clusters (cf. for example Mayer and Mayer [1940], p. 277). 

So far, no calculation has been publidied for the condensed states. It is dear that if sufiSdently large cell clusters are retained a correct result must be obtained. The situation is the same as in multiple cell occupation theories if sufifidently large cells are considered. The difficulty in both cases is that the calculation of the exact partition... 

An interesting and practically valuable result was obtained in [21] for PE + N2 melts, and in [43] for PS + N2 melts. The authors classified upper critical volumetric flow rate and pressure with reference to channel dimensions x Pfrerim y Qf"im-Depending on volume gas content

channel entrance (pressure of 1 stm., experimental temperature), x and y fall, in accordance with Eq. (24), to tp 0.85. At cp 0.80, in a very narrow interval of gas concentrations, x and y fall by several orders. The area of bubble flow is removed entirely. It appears that at this concentration of free gas, a phase reversal takes place as the polymer melt ceases to be a continuous phase (fails to form a continuous cluster , in flow theory terminology). The theoretical value of the critical concentration at which the continuous cluster is formed equals 16 vol. % (cf., for instance, Table 9.1 in [79] and [80]). An important practical conclusion ensues it is impossible to obtain extrudate with over 80 % of cells without special techniques. In other words, technology should be based on a volume con-... 

This compares well with an estimation of the minimum size of the dipolar cluster formed around nearest neighbors Nbs+ ion given by [M>5+]-1 = [0.38] 1 2.63. At elevated temperatures (T > 354 K), the Nb5+ ions hop at random between equivalent minima of their potential wells within their site in the unit cell. As the phase transition is approached, these ions form a dipolar cluster around the Cu+ ions that are randomly distributed far apart from each other. These clusters are at first of a minimum size containing only the Nbs+ ions that are closest to the Cu+ ion. As the phase transition is approached, the cluster size around the Cu+ ion grows accordingly. These represent the rearranging clusters of the A G theory. 

Hess ST, Gould TJ, Gudheti MV, Maas SA, Mills KD, Zimmer-berg J. Dynamic clustered distribution of hemagglutinin resolved at 40nm in living cell membranes discriminates between raft theories. Proc. Natl. Acad. Sci. 2007 104 17370-17375. 

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