Living polymers such as wormlike micelles have attracted considerable experimental and theoretical interest over the past three decades, but the differential-integral equations that describe the joint processes of reversible scission and stress relaxation were only recently developed and have not yet been solved. Here, we introduce a numerical method that is simple, stable, accurate, flexible, and fast compared to alternatives. After validating the method and its predictions, we provide a preliminary discussion on previously unquantified sources of uncertainty in a popular stochastic approach to modeling the same problem.
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If one’s goal is to quantitatively compare against experimental data, then the inverse cutoff time
should be larger (e.g., by a factor of 10) than the lessor of (a) the frequency at which
has a local minimum or (b) the maximum frequency for which experimental measurements was taken. When applying these heuristics, be mindful of the fact that
in the present paper is dimensionless, scaled by
.
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Without rescaling, the PBE calculations for
in Fig. 4(b) are shifted down and to the right. This will result in good agreement at high frequencies but worse agreement at low frequencies including near the crossover frequency. The “best fit” criterion from the Pointer algorithm—the algorithm that suggested
as a best fit—can sometimes give a higher weighting to the quality of fit in the vicinity of the crossover frequency, to our understanding.
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