Group expansion

The loosely anchored indigenous particles can be categorized into two major groups expansive authigenic clay minerals and nonexpansive clay minerals. A simple kinetic expression is set up for the generation of these indigenous fines. The variable r represents the fraction of expansive clay minerals with respect to the total releasable fines, ap. ... 

Thus, the system s statistical sum can be written in the form of cluster (group) expansion... 

A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

Takamizawa et al. developed a general ring-expansion reaction of heterocycles that, applied to thiazolium salts, yields 1,4-thiazines (496, 497) thiamine (220) reacts with dialkyl acylphosphonates (221) to give the tricyclic 1,4-thiazine (222) (498), which is easily hydrolyzed to dihydro-1,4-thiazinone (223) (499) (Scheme 106). In the case of thiazolium slats containing no functional groups (224), 1,4-thiazine derivatives (226) were directly obtained in fairly good yields (Scheme 107). 

More recendy, however, the technology of introducing a new functional group to the double bond of allyl alcohol has been developed. Adyl alcohol is accordingly used as an intermediate compound for synthesizing raw materials such as epichlorohydrin and 1,4-butanediol, and this development is bringing about expansion of the range of uses of allyl alcohol. 

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