Abstract
Polysaccharides, such as cellulose, are often processed by dissolution in solvent mixtures, e.g. an ionic liquid (IL) combined with a dipolar aprotic co-solvent (CS) that the polymer does not dissolve in. A multi-walker, discrete-time, discrete-space 1-dimensional random walk can be applied to model solvation of a polymer in a multi-component solvent mixture. The number of IL pairs in a solvent mixture and the number of solvent shells formable, x, is associated with n, the model time-step, and N, the number of random walkers. The mean number of distinct sites visited is proportional to the amount of polymer soluble in a solution. By also fitting a polynomial regression model to the data, we can associate the random walk terms with chemical interactions between components and probe where the system deviates from a 1-D random walk. The 'frustration' between solvents shells is given as ln x in the random walk model and as a negative IL:IL interaction term in the regression model. This frustration appears in regime II of the random walk model (high volume fractions of IL) where walkers interfere with each other, and the system tends to its limiting behaviour. In the low concentration regime, (regime I) the solvent shells do not interact, and the system depends only on IL and CS terms. In both models (and both regimes), the system is almost entirely controlled by the volume available to solvation shells, and thus is a counting/space-filling problem, where the molar volume of the CS is important. Small deviations are observed when there is an IL-CS interaction. The use of two models, built on separate approaches, confirm these findings, demonstrating that this is a real effect and offering a route to identifying such systems. Specifically, the majority of CSs - such as dimethylformide - follow the random walk model, whilst 1-methylimidazole, dimethyl sulfoxide, 1,3-dimethyl-2-imidazolidinone and tetramethylurea offer a CS-mediated improvement and propylene carbonate results in a CS-mediated hindrance. It is shown here that systems, which are very complex at a molecular level, may, nonetheless, be effectively modelled as a simple random walk in phase-space. The 1-D random walk model allows prediction of the ability of solvent mixtures to dissolve cellulose based on only two dissolution measurements (one in neat IL) and molar volume.
Highlights
Random walks are models of a walker, be it a drunkard, a molecule, or a stock price, exploring space randomly
We demonstrate that the 1-D random walk model can be used to vastly reduce the number of dissolution experiments required to characterise the cellulose dissolution profile in new organic electrolyte solutions’’ (OESs), such that the model is of great utility to experimentalists
C B x, where x is the number of solvent shells; all ions of ionic liquid (IL) are associated with a solvation shells (SSs), and c scales linearly with wIL
Summary
Random walks are models of a walker, be it a drunkard, a molecule, or a stock price, exploring space (real space for the drunkard and molecule, phase space for the stock price). Studying the motion of a single walker selected from a set can, as the walkers are identical and the walk is selected randomly, be a study of the average properties.[23] order statistics, which is the study of the first walkers to arrive at a point, can model single molecule experiments.[24,25] Another useful property is the size of a random walk, which relates to the radius of gyration for a polymer, for example.[26,27] In 1951 Dvoretzky and Erdos suggested that the number of distinct sites visited by any random walker, when there are N interacting random walkers, was an interesting problem.[28] The first solution was published in 1992 by Weiss et al.,[29,30,31] and was solved by describing the system with generating functions for probability distributions, and approximating the behaviour of a coordinate (Laplace) transform approximation of the generating function at a singular point: this gave solutions that were valid only for the large number of walkers limit and extended time.[30]. We build a 1-D random walk model for the quantity of cellulose that dissolves in an OES with variable CSs, and compare this to a multi-polynomial regression model of the same system. We demonstrate that the 1-D random walk model can be used to vastly reduce (to two) the number of dissolution experiments required to characterise the cellulose dissolution profile in new OESs, such that the model is of great utility to experimentalists
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