Minimum multiplicity

Polydispersity. If a liquid is injected into polydisperse cellular foam (a process which decreases the foam multiplicity) until the foam becomes spherical with the same size distribution of bubbles, then the obtained spherical foam is said to be equivalent. The equivalent foam is characterized by the multiplicity called the minimum multiplicity Kmjn. Obviously, the minimum multiplicity of polydisperse foam is larger than that of monodisperse foam, since in gaps between densely packed spheres of the same largest size, spheres of smaller sizes can be located. Thus the value K a can be used as a quantitative measure of polydispersity [214], While K a = 3.86 for monodisperse foam, for actual polydisperse foam we have 10 to 15 in practice Kma never exceeds 20 [480],... 

From these considerations, we expect that there is no surfactant-mediated process if the minimum multiplicity is ko = 2, i.e., if there is no gap in the size distribution. Under... 

To demonstrate these ideas, we calculate from (10.6) the sol-gel transition concentration as a function of the concendation of the added surfactant. Figure 10.2(b) shows the result for telecheUc (/=2) polymers. To clarify the effect of the minimum multiplicity, ko is varied from curve to curve, while the maximum multiplicity is fixed at = 8. It is clear that the sol-gel concentration c y monotonically increases with the surfactant concentration for ko = 2 (no lower bound). Gelation is blocked by the surfactant. But if there is a gap between k = 1 (unassociated) and k = ko, a minimum in c j- appears where gelation is easiest, as can be seen for > 3 in Figure 10.2(b). The surfactant concentration at which becomes minimum, referred to as the surfactant-mediated gel point (SMG), increases as the gap becomes larger. 

Rg. 10.3 Isotherm of surfactant molecules bound by bifunctional (/ = 2) polymers. The fraction 0 of the adsorbed surfactant molecules is plotted against their total concentration. The minimum multiplicity of the junctions is changed from curve to curve with fixed /km = 8. The curve starts from a finite value and shows a peak at the SMG concentration. (Reprinted with... 

Abstract This paper reviews possible phase diagrams of associating polymer solutions in which phase separation and molecular association interfere. Paying special attention on the structure and reorganisation of the network junctions, we study competition between phase separation and gelation. The molecular structure of associating micelles, or multiple cross-link junctions, in the networks is analyzed from the sol/gel transition lines. The effect of added surfactants on the formation of reversible gels in hydrophobically modified polymer solutions is also studied under the assumption of the existenee of a minimum multiplicity required for stable cross links. To describe... 

Minimum multiplicity model. This model allows junctions to take multiphcity greater than Sq. We have... 

When So = 2, this model reduces to the variable multiplicity model in which junctions of arbitrary multiplicity can coexist at the probability determined by the thermodynamic balance. In the case of micro-crystalline junctions, for instance, it is natural to assume that a minimum number Sq greater than 2 of the crystalline chains is required for a junction formation. This is because, the surface energy terms will prevent small-k units from being stable, leading to the existence of the critical multiplicity for the nucleation of the crystallites. Similarly, a minimum aggregation number is required for the stability of micelles formed by hydrophobes on water-soluble polymers. As we will see later, surfactants added to the solution cause complex interaction with hydrophobically modified polymers due to the existence of this minimum multiplicity. 

Minimum risk estimates are sometimes used to quantify either maximum exposure in monetary terms or, in the case of an annual work plan containing multiple projects, to help determine the proportion of firm projects. Firm projects are those which have budget cover even if costs overrun. A minimum risk estimate is one with little or no probability of overrun, and can be used to reflect the risk associated with very complex or novel projects. 

Ripoll D R and H A Scheraga 1989. On the Multiple-Minimum Problem in the Conformational Anal of Polypeptides. IB. An Electrostatically Driven Monte Carlo Method Tests on met-Enkepha journal of Protein Chemistry 8 263-287. 

Fold. Strength of concentrated flavoring materials. The concentration is expressed as a multiple of a standard, eg, citms oil is compared to cold pressed oil. In the case of vanilla, folded flavors are compared to a standard extract with minimum bean content. 

A low temperature of approach for the network reduces utihties but raises heat-transfer area requirements. Research has shown that for most of the pubhshed problems, utility costs are normally more important than annualized capital costs. For this reason, AI is chosen eady in the network design as part of the first tier of the solution. The temperature of approach, AI, for the network is not necessarily the same as the minimum temperature of approach, AT that should be used for individual exchangers. This difference is significant for industrial problems in which multiple shells may be necessary to exchange the heat requited for a given match (5). The economic choice for AT depends on whether the process environment is heater- or refrigeration-dependent and on the shape of the composite curves, ie, whether approximately parallel or severely pinched. In cmde-oil units, the range of AI is usually 10—20°C. By definition, AT A AT. The best relative value of these temperature differences depends on the particular problem under study. 

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