Abstract
The paper compares the statistical description of physical-metallurgical processes and ceramic-polycrystalline evolutions, termed the normal grain growth (NGG), as adopted to soft- and chemically-reactive grains, with a Smoluchowski's population-constant kernel cluster-cluster aggregation (CCA) model, concerning irreversible chemical reaction kinetics. The former aiming at comprehending, in a semi-quantitative way, the volume-conservative (pressure-drifted) grain-growth process which we propose to adopt for hydrogel systems at quite low temperature (near a gel point). It has been noticed, that by identifying the mean cluster size $<k>$ from the Smoluchowski CCA description with the mean cluster radius' size $R_D$, from the NGG approach of proximate grains, one is able to embark on equivalence of both frameworks, but only under certain conditions. For great enough, close-packed clusters, the equivalence can be obtained by rearranging the time domain with rescaled time variable, where the scaling function originates from the dispersive (long-tail, or fractal) kinetics, with a single exponent equal to $d+1$ (in $d$-dimensional (Euclidean) space). This can be of interest for experimenters, working in the field of thermoresponsive gels formation, where crystalline structural predispositions overwhelm.
Highlights
In 1916, Marian Smoluchowski proposed a case of constant-kernel cluster–cluster aggregation (CCA), for which it is manageable to find analytically, by employing scaling arguments, a solution in terms of the cluster size (k) distribution function, n(k) [1, 2]
The clustering arguments, first introduced by Smoluchowski [1], are applicable to a statistical description of physical-metallurgical processes and ceramic-polycrystalline evolutions, termed the normal grain growth (NGG), in which bigger clusters grow at the expense of their smaller neighboring counterparts due to preferentially capillary conditions [3]
It is proposed that upon identifying k from the Smoluchowski CCA description with the mean cluster radius’ size RD, from the NGG approach of proximate soft-and-reactive grains, one is able to embark on their equivalence
Summary
In 1916, Marian Smoluchowski proposed a case of (populationally fixed) constant-kernel cluster–cluster aggregation (CCA), for which it is manageable to find analytically, by employing scaling arguments, a solution in terms of the cluster size (k) distribution function, n(k) [1, 2]. Met mostly in either inappropriate (harsh) or typically low temperature conditions, act as badly as possible while interacting with a polymer chain, making it shrunken or obstructed in its capability of gaining more space around, an opposite physicochemical scenario prevails. The former is termed a coil effect whereas the latter is known as a globule counter-effect. It applies in particular to the ones of bioreactive gels and/or living-matter involving contexts where microgels with swollen microcrystalline domains exist
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