Force balance clusters

In order to evaluate the descending velocity of the cluster layer, an alternative approach proposed originally by Glicksman (1988) is adopted. As shown in Fig. 22, clusters are assumed to sweep along the surface and then accelerate downwards under the action of gravity, until they reach a maximum velocity, Fcma,. According to an analysis of force balance, the cluster layer descending velocity can be expressed as... 

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. 

Both synthetic and natural polymers have superstructures that influence or dictate the properties of the material. Many of these primary, secondary, tertiary, and quaternary structures are influenced in a similar manner. Thus, the primary structure is a driving force for the secondary structure. Allowed and preferred primary and secondary bondings influence structure. For most natural and synthetic polymers, hydrophobic and hydrophilic domains tend to cluster. Thus, most helical structures will have either a hydrophobic or hydrophilic inner core with the opposite outer core resulting from a balance between secondary and primary bonding factors and steric and bond angle constraints. Nature has used these differences in domain character to create the world around us. 

Clearly, U is the biggest number in the cycle and is the main driving force for the formation of ionic compounds. Nevertheless, the other factors can tip the balance one way or another. For example, AHSub is particularly large for the transition metals niobium, tantalum, molybdenum, tungsten, and rhenium, with the result that, in their lower oxidation states, they do not form simple ionic compounds such as ReCl3 but rather form compounds that contain clusters of bonded metal atoms (in this example, Re3 clusters are involved, so the formula is better written ResClg). 

In summary, the incorporation of Ceo into artificial bilayer membranes, despite being successful in principle, gives rise to a number of unexpected complications. In consideration of the strong aggregation forces among fullerene cores, it is imperative to separate the individual fullerene moieties. Only the adequate hydrophilic-hydrophobic balance of the host matrix is an appropriate means to hinder the spontaneous cluster formation [88]. 

The cluster is not infinitely stable. At some temperature the cluster decomposes, and at that temperature the elastic forces and the electrostatic forces just balance each other. Since the elastic force was calculated per chain and the electrostatic force per ion pair, Wch can be set equal to W at thus... 

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