Schulz distribution function and the polydispersity of the binary suspension of charged macroions

Pandey, Rakesh Kumar; Tripathi, Dharmendra

doi:10.1016/s0927-7757(01)00707-5

Cited by 10 publications

(13 citation statements)

References 27 publications

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“…If the average string length is large enough, the coordination dipoles of a string end may all belong to the inner parts of other strings, which lead to the termination of string formation. This kind of topologic termination reminds the termination effect during the polymerization processes, where the chain length distribution is described by the Schulz distribution [43][44]. Based upon the above discussion, the Schulz distribution of the string length seems to be appropriate.…”

Section: Appendix IImentioning

confidence: 86%

“…As shown in Appendix II, the probability, n g , that a dipole is located in an n-string is not only determined by the intra-string interaction 1 H but also by both the coordination number z and the directional angle of the string θ. For 2 = z and 2 π θ = , the statistic dynamics methods described in Appendix II give the probability n g for a dipole of the system to belong to an n-string as , an expression similar to Flory's well-known molecular weight distribution function [43,44]. Notice that 2…”

Section: Iii-2 String-length Distributionmentioning

confidence: 99%

“…For arbitrary values of z and θ, and also under the condition that the average string length is 10 long enough, n g becomes the Schulz distribution [43][44] Shown in Fig.1a are the calculated results for n g , at different temperatures, plotted as a function of n for 0 .…”

Section: Iii-2 String-length Distributionmentioning

confidence: 99%

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Superdipole liquid scenario for the dielectric primary relaxation in supercooled polar liquids

Huang

Wang

Riande

2005

The Journal of Chemical Physics

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We propose a dynamic structure of coupled dynamic molecular strings for supercooled small polar molecule liquids and accordingly we obtain the Hamiltonian of the rotational degrees of freedom of the system. From the Hamiltonian, the strongly correlated supercooled polar liquid state is renormalized to a normal superdipole liquid state. This scenario describes the following main features of the primary or alpha-relaxation dynamics in supercooled polar liquids: (1) the average relaxation time evolves from a high temperature Arrhenius to a low temperature non-Arrhenius or super-Arrhenius behavior; (2) the relaxation function crosses over from the high temperature exponential to low temperature nonexponential form; and (3) the temperature dependence of the relaxation strength shows non-Curie features. According to the present model, the crossover phenomena of the first two characteristics arise from the transition between the superdipole gas and the superdipole liquid. The model predictions are quantitatively compared with the experimental results of glycerol, a typical glass former.

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Section: Appendix IImentioning

confidence: 86%

Section: Iii-2 String-length Distributionmentioning

confidence: 99%

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Superdipole liquid scenario for the dielectric primary relaxation in supercooled polar liquids

Huang

Wang

Riande

2005

The Journal of Chemical Physics

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“…with I c (0) / hL 2 i/hLi and J 1 (x) being the first-order Bessel function of the argument x. In the present work, it was assumed that the distribution of cylinder lengths Z(L) could be described by the Schulz function (Pandey & Tripathi, 2001;Riello et al, 2003), often assumed in aggregation of linear polymers, which can be written as…”

Section: Scattering From Cylindrical Particles With a Length Distribumentioning

confidence: 99%

Rod-like particles growing in sol–gel processing of 1:1 molar mixtures of 3-glycidoxypropyltrimethoxysilane and tetraethoxysilane

et al. 2017

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The growth kinetics and the structure of organic/silica hybrids prepared from acid hydrolysis of 1:1 molar mixtures of 3‐glycidoxypropyltrimethoxysilane and tetraethoxysilane were studied by small‐angle X‐ray scattering (SAXS) at 315, 325 and 335 K. The evolution of the SAXS intensity is compatible with the growth of silica‐rich domains by aggregation from a fixed number of primary particles. Two distinct growth regimes could be identified by analyzing the relation I(0) ∝ RgD between the intensity extrapolated to zero I(0), the radius of gyration Rg of the aggregates and the exponent D, which gives information on the geometry and the mechanism of growth of the aggregates. An initial period was attributed to the growth of rod‐like particles with approximately the same radius and variable length. At more advanced degrees of aggregation the process was controlled by the growth of larger aggregates with higher‐order dimensionality. A narrow distribution of cylinder lengths given by the Schulz function fitted the experimental SAXS data well during the most part of the initial regime of cylindrical particle growth. These particles were later found as rod‐like subunits of the larger aggregates grown at more advanced degrees of aggregation. Some condensation possibilities yielding the formation of structures compatible with those inferred from the present study are discussed.

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“…Tata et al [7] have calculated the coordination number for different concentrations of impurity ions to study the effect of impurity ions on ordered structures prevailing in aqueous macroionic suspension using the method of integration of region r 2 g(r). Pandey and Tripathi [8] have calculated the coordination numbers using this method to study the ordered behavior of binary suspension of macroions by changing controlling parameters of the suspension. Recently, Pandey has used the integration method of region r 2 g(r) in the limit 0 -r min to calculate the coordination number in order to ascertain the ordering nature in the plasmas [9] and biomacroions [10].…”

Section: Introductionmentioning

confidence: 99%

Parametric Dependence of Coordination Number Obtained Using Derjaguin, Landau, Verwey and Overbeek DLVO Pair Interaction Potential for Ordered Macroions Suspended into Water

Pandey¹

2014

OALib

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Coordination numbers have been calculated for the ordered structures prevailing in aqueous suspension of macroions for its different governing parameters viz. sizes, charges, impurity ion concentrations and temperatures by integration of region r 2 g(r) in the limit 0-rmin around the main peak in g(r)-r curve in order to examine the effects of controlling parameters on the coordination number. The failure of RMSA method has been reported in case of variation of temperature and effective charge over macroions in terms of volume fraction. Orderings of FCC/HCP type with coordination number nearly 12 amongst the macroions suspended into water have been reported to exist.

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Schulz distribution function and the polydispersity of the binary suspension of charged macroions

Cited by 10 publications

References 27 publications

Superdipole liquid scenario for the dielectric primary relaxation in supercooled polar liquids

Superdipole liquid scenario for the dielectric primary relaxation in supercooled polar liquids

Rod-like particles growing in sol–gel processing of 1:1 molar mixtures of 3-glycidoxypropyltrimethoxysilane and tetraethoxysilane

Parametric Dependence of Coordination Number Obtained Using Derjaguin, Landau, Verwey and Overbeek DLVO Pair Interaction Potential for Ordered Macroions Suspended into Water

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