scholarly journals Molecular size distributions in multichain polymers: application of polymer theory to silicate melts

1970 ◽  
Vol 48 (1) ◽  
pp. 201-202 ◽  
Author(s):  
C. R. Masson ◽  
I. B. Smith ◽  
S. G. Whiteway
Keyword(s):  
Entire Range ◽  
Maximum Degree ◽  
Molecular Size ◽  
Degree Of Polymerization ◽  
Branched Polymers ◽  
Silicate Melts ◽  
Size Distributions ◽  
Polymer Theory ◽  
Good Agreement ◽  
Binary Silicate

Expressions developed previously by Flory and by the authors for molecular size distributions in branched polymers were used to derive theoretical curves of activity vs. composition in binary silicate melts. In contrast with curves based on the Flory distribution those derived from our result were in good agreement with experiment over the entire range of compositions up to the maximum degree of polymerization allowed by the theory. The comparison is illustrated for the systems PbO–SiO2 and SnO–SiO2.

Activities and Ionic Distributions in Cobalt Silicate Melts

1971 ◽  
Vol 49 (5) ◽  
pp. 683-690 ◽  
Author(s):  
I. B. Smith ◽  
C. R. Masson
Keyword(s):  
Silica Content ◽  
Silicate Melts ◽  
Molecular Weights ◽  
Polymer Theory ◽  
Linear Chains ◽  
Experimental Values ◽  
The Relationship ◽  
Gas Chromatographic Separation ◽  
Good Agreement ◽  
Binary Silicate

Activities of CoO in CoO–SiO2 melts were measured at 1450–1500 °C by equilibrating the melts, held in Pt–Rh containers, with atmospheres of known oxygen potential. Activities were calculated by the relationship[Formula: see text]where aCo, the activity of cobalt in the container, was determined in separate experiments.The results were compared with theoretical activity–composition curves based on the application of polymer theory to silicate melts. The results were in good agreement with theoretical curves calculated on the assumption of linear chains. In contrast, for all other binary silicate melts so far investigated the results are best represented in terms of theory in which all chain configurations are allowed. Ionic distributions and number average and weight average molecular weights were calculated as functions of the silica content from the experimental data. The calculated proportions of monomeric ion, SiO44−, dimer Si2O76−, and trimer Si3O108− were in reasonable agreement with experimental values based on trimethylsilylation and gas-chromatographic separation of the ionic constituents in quenched melts.

Activities and ionic distributions in liquid silicates: application of polymer theory

1970 ◽  
Vol 48 (9) ◽  
pp. 1456-1464 ◽  
Author(s):  
C. R. Masson ◽  
I. B. Smith ◽  
S. G. Whiteway
Keyword(s):  
Binary Systems ◽  
Molecular Size ◽  
Thermodynamic Activity ◽  
Silicate Melts ◽  
Size Distributions ◽  
Chain Molecules ◽  
Linear Chains ◽  
High Silica ◽  
The Mean ◽  
Binary Silicate

Theoretical expressions derived previously for molecular size distributions in multichain polymers are applied to binary silicate melts. The treatment is an extension of a previous approach which was limited to the consideration of linear chains. When all configurations of the chain molecules are taken into consideration, the predicted variation of thermodynamic activity with composition agrees with experiment for all binary systems for which data are available. The effect of allowing for all chain configurations is largely to improve the fit between theory and experiment at high silica contents. Calculated ionic distributions for the system 'FeO'–SiO2 do not differ markedly from those previously reported. The mean chain length is unaffected. The results support previous views that principles of polymer chemistry can be applied usefully to silicate melts and glasses.

Theory of molecular size distribution In multichain polymers

1970 ◽  
Vol 48 (1) ◽  
pp. 33-45 ◽  
Author(s):  
S. G. Whiteway ◽  
I. B. Smith ◽  
C. R. Masson
Keyword(s):  
Functional Groups ◽  
Entire Range ◽  
Molecular Size ◽  
Degree Of Polymerization ◽  
Probable Distribution ◽  
Molecular Sizes ◽  
Intramolecular Condensation ◽  
Meaningful Existence ◽  
Molecular Size Distribution ◽  
Most Probable Distribution

Expressions are derived for the most probable distribution of molecular sizes in multi-chain polymers formed by the self-condensation of the monomer A—R—Xf − 1, where A and X are functional groups and X may be either A or B. It is assumed that all functional groups of the same kind are chemically equivalent and that intramolecular condensation may be neglected. For the case A—R—Bf − 1 the results are identical with those of Flory, although it is shown that this is fortuitous and due to a cancellation of two errors in Flory's method. For the case R—Af the results differ significantly from expressions derived by Flory and sources of error in previous work are discussed. In theory, the mole and weight fractions of individual x-mers vary continuously with the extent of reaction α over the entire range up to αmax = 2/f. The ratio of the weight average to the number average degree of polymerization is finite for all values of α below αmax. The critical point for the formation of infinitely large (wall-to-wall) molecules occurs, not at α = 1/(f − 1) as predicted by Flory, but at α = 2/f. The prediction of actual gel points is discussed in terms of the largest molecule which can have a physically meaningful existence at any fixed value of α.

A Compact Model for Spherical Rough Contacts

2004 ◽  
Author(s):  
M. Bahrami ◽  
M. M. Yovanovich ◽  
J. R. Culham
Keyword(s):  
Pressure Distribution ◽  
Entire Range ◽  
Present Model ◽  
Numerical Solutions ◽  
Contact Radius ◽  
Bulk Deformation ◽  
Dimensional Parameters ◽  
Maximum Contact ◽  
Good Agreement ◽  
Key Parameter

The contact of rough spheres is of high interest in many tribological, thermal, and electrical fundamental analyses. Implementing the existing models is complex and requires iterative numerical solutions. In this paper a new model is presented and a general pressure distribution is proposed that encompasses the entire range of spherical rough contacts including the Hertzian limit. It is shown that the non-dimensional maximum contact pressure is the key parameter that controls the solution. Compact expressions are proposed for calculating the pressure distribution, radius of the contact area, elastic bulk deformation, and the compliance as functions of the governing non-dimensional parameters. The present model shows the same trends as those of the Greenwood and Tripp model. Correlations proposed for the contact radius and the compliance are compared with experimental data collected by others and good agreement is observed.

scholarly journals On the Shape–Slope Relation of Drop Size Distributions in Convective Rain

2005 ◽  
Vol 44 (7) ◽  
pp. 1146-1151 ◽  
Author(s):  
Axel Seifert
Keyword(s):  
Size Distribution ◽  
Gamma Distribution ◽  
Drop Size ◽  
Empirical Relation ◽  
Size Distributions ◽  
Shape Parameters ◽  
Raindrop Size Distribution ◽  
Raindrop Size ◽  
Drop Size Distributions ◽  
Good Agreement

Abstract The relation between the slope and shape parameters of the raindrop size distribution parameterized by a gamma distribution is examined. The comparison of results of a simple rain shaft model with an empirical relation based on disdrometer measurements at the surface shows very good agreement, but a more detailed discussion reveals some difficulties—for example, deviations from the gamma shape and the overestimation of collisional breakup.

Transient Nucleation in Co-Zr Metallic Glasses

1985 ◽  
Vol 57 ◽  
Author(s):  
Uwe Köster ◽  
Margret Blank-Bewersdorff
Keyword(s):  
Electron Microscopy ◽  
Metallic Glasses ◽  
Crystal Size ◽  
Time Lag ◽  
Size Distributions ◽  
Quantitative Electron Microscopy ◽  
Crystal Size Distributions ◽  
Different Temperatures ◽  
Transient Nucleation ◽  
Good Agreement

AbstractCrystallization kinetics and crystal size distributions in Co33Zr67-glasses have been analyzed by quantitative electron microscopy. The polymorphic crystallization of spherical CoZr2 crystals is very suitable reaction for such an analysis. Calculated crystal size distributions at different temperatures were compared to those experimentally revealed. Parameters controlling crystallization were varied within reasonable limits until theoretically calculated and experimentally observed crystal size distributions were in good agreement. It has been found that crystal size distribution can be explained by transient nucleation; the time lag and its temperature dependence can be evaluated. These results are discussed in the light of recent theories on transient nucleation.

Random high-dimensional orders

1995 ◽  
Vol 27 (01) ◽  
pp. 161-184 ◽  
Author(s):  
Béla Bollobás ◽  
Graham Brightwell
Keyword(s):  
Partial Order ◽  
Rapid Growth ◽  
Entire Range ◽  
Maximum Degree ◽  
Threshold Function ◽  
High Dimensional ◽  
Sharp Threshold ◽  
Threshold Functions

The random k-dimensional partial order P k (n) on n points is defined by taking n points uniformly at random from [0,1] k . Previous work has concentrated on the case where k is constant: we consider the model where k increases with n. We pay particular attention to the height H k (n) of P k (n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in P k (n): if k – (t/log t!) log n tends to + ∞ then the probability that P k (n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of H k (n) for the entire range of k(n). We also consider the maximum degree of P k (n). We show that, for each fixed d ≧ 2, is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k. We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of P k (n) for large k(n).

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