Modification of Pascal's triangle for calculating size distributions in polymers

1970 ◽  
Vol 48 (9) ◽  
pp. 1432-1435 ◽  
Author(s):  
S. G. Whiteway
Keyword(s):  
Size Distribution ◽  
Probability Function ◽  
Molecular Size ◽  
Size Distributions ◽  
Binomial Probability ◽  
Alternative Concept ◽  
Pascal’S Triangle ◽  
All Solutions ◽  
Molecular Size Distribution ◽  
Pascal's Triangle

Pascal's triangle, which is well known to yield all solutions to the factorial part of the Bernoulli binomial probability function, is modified to yield the correct factorial term in expressions for the molecular size distribution in polymers. This approach provides an alternative concept of the meaning of these factorial numbers.

Molecular Size Distribution in Random Polyfunctional Condensation with or without Ring Formation: Computer Simulation

1974 ◽  
Vol 52 (18) ◽  
pp. 3285-3295 ◽  
Author(s):  
Michael Falk ◽  
Ruth E. Thomas
Keyword(s):  
Size Distribution ◽  
Functional Group ◽  
Infinite System ◽  
Molecular Size ◽  
Critical Value ◽  
Ring Formation ◽  
Polymerization Process ◽  
Size Distributions ◽  
Polymer Size ◽  
Molecular Size Distribution

We have re-examined the polymer size distribution in a model system initially composed of N0 monomeric units RAƒ, where each unit R carries ƒ identical functional groups A. Self-condensation is postulated to proceed at random by the formation of [Formula: see text] bonds until the fraction α of A groups have reacted. The controversy regarding the applicability of the Flory–Stockmayer (F.S.) and the Whiteway–Smith–Masson (W.S.M.) expressions to this model has been resolved. Two variants of this model are distinguished, depending on whether ring formation is allowed or forbidden. It is shown that in the limit of an infinite system the incidence of rings in the ringsallowed model tends to zero below the critical value of α, αc = 1/(ƒ − 1 ). Consequently, the limiting polymer size distributions for the rings allowed and ringsforbidden models coincide over the range [Formula: see text], and are both described by the F.S. expression. For the ringsforbidden model, the F.S. equation fails above αc, but for the rings allowed model it continues to apply over the entire accessible range [Formula: see text] The W.S.M. equation represents the limiting distribution for the polymerization process in which one functional group per monomer is singled out (A*) and bonding is restricted to [Formula: see text] pairs. In that process, and in the limit of an infinite system, rings are not formed over the entire attainable range of α, from 0 to 2/ƒ.

Determination of the Size Distribution of Dissolved Organics in Effluents

1973 ◽  
Vol 8 (1) ◽  
pp. 1-15 ◽  
Author(s):  
L.A. Addie ◽  
K.L. Murphy ◽  
J.L. Robertson
Keyword(s):  
Size Distribution ◽  
Biological Treatment ◽  
Oxygen Demand ◽  
Molecular Size ◽  
Monitoring Systems ◽  
Clean Surface ◽  
Sephadex Column ◽  
Dissolved Organics ◽  
Molecular Size Distribution

Abstract The importance of removing the small amounts of residual organics is increasing as the sources of clean surface water decrease. Knowledge of the nature of these soluble residual organics will be needed in order to assess the type of treatment required for their removal. Residual organics in three different biological treatment plants were analyzed and compared. An attempt was made to characterize these organics by a molecular size distribution on a Sephadex column monitored by differential ultraviolet and refractive index detectors. The organic carbon and chemical oxygen demand of the fractions collected from the column was also determined. An investigation of some of the problems inherent in the monitoring systems was conducted.

Molecular Size Distribution of Lignin in Wood

Nature ◽  
1967 ◽  
Vol 214 (5086) ◽  
pp. 410-411 ◽  
Author(s):  
W. BROWN ◽  
S. I. FALKEHAG ◽  
E. B. COWLING
Keyword(s):  
Size Distribution ◽  
Molecular Size ◽  
Molecular Size Distribution

A Study of the Malpighian Tubules of the Pill Millipede, Glomeris marginata (Villers)

1974 ◽  
Vol 60 (1) ◽  
pp. 41-51
Author(s):  
PATRICIA ANNE FARQUHARSON
Keyword(s):  
Size Distribution ◽  
Fluid Transport ◽  
Molecular Size ◽  
Malpighian Tubules ◽  
Inverse Relation ◽  
Fluid Production ◽  
Molecular Size Distribution ◽  
Molecular Sieving ◽  
Tubule Fluid ◽  
Tubule Wall

1. Tubule fluid:medium ratios (TF/M) have been measured for inulin, glucose, LMWD and HMWD. These TF/M ratios were surprisingly high. 2. The tubule appears to act as a molecular filter; that is to say, molecules move through the tubule wall in inverse relation to their size. This is best illustrated using polyvinyl pyrrolidone as a tracer. The molecular size distribution of PVP fractions present in tubule fluid differs markedly from the molecular size distribution of PVP in the bathing Ringer. 3. No correlation can be made between the inulin and glucose TF/M and the rate of fluid production. However, the inverse relationship between TF/M and rate of fluid production for dextrans indicates a molecular sieving effect. 4. The significance of these results is discussed with reference to models of fluid transport.

Multidetectors for Analysis of the Composition of Copolymers as a Function of Molecular Size Distribution by Steric Exclusion Chromatography

1973 ◽  
Vol 46 (2) ◽  
pp. 449-463 ◽  
Author(s):  
D. J. Harmon ◽  
V. L. Folt
Keyword(s):  
Liquid Chromatography ◽  
Size Distribution ◽  
Oil Content ◽  
Molecular Size ◽  
The Other ◽  
Additional Information ◽  
Steric Exclusion ◽  
Gel Permeation ◽  
Exclusion Chromatography ◽  
Molecular Size Distribution

Abstract Analysis of molecular size distribution of polymers by steric exclusion liquid chromatography (GPC) is well known. Problems exist, however. These problems involve copolymers and polymer blends. The objectives of the research were to develop methods of analyzing comonomer distribution in copolymers, to study the breakdown of one polymer independent of another in a polymer blend, and to obtain any additional information as might be available. The separations were performed on a Waters Model 200 Gel Permeation Chromatograph. Detectors employed were a Waters R-4 differential refractometer, a Wilks Miran-1 infrared analyzer, and a Beckman Model 144 UV photometer. Examples are given of analysis of average styrene, styrene distribution, and oil content of oil extended SBR. The data is compared with that obtained by other methods. In general the agreement is good. The ability to examine one polymer of a blend independent of the other is also demonstrated. Since elastomers are frequently used as blends, this becomes very important to such studies as milling and extrusion behavior.

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