Polymer extension under flow: Some statistical properties of the work distribution function

Abstract

In an extension of earlier studies from this group on the application of the Jarzynski equality to the determination of the elastic properties of a finitely extensible Rouse model of polymers under flow [A. Ghosal and B. J. Cherayil, J. Chem. Phys. 144, 214902 (2016)], we derive several new theoretical results in this paper on the nature of the distribution function P(w) that governs the long-time limit t>>1 of the fluctuations in the work w performed by the polymer during flow-induced stretching. In particular, we show that an expression for the average of the nth power of the work, ⟨w(t)⟩, can be obtained in closed form in this limit, making it possible to exactly calculate three important statistical measures of P(w): the mean μ, the skewness γ, and the kurtosis γ (apart from the variance σ). We find, for instance, that to leading order in t, the mean grows linearly with t at a constant value of the dimensionless flow rate Wi and that the slope of the μ-t curve increases with increasing Wi. These observations are in complete qualitative agreement with data from Brownian dynamics simulations of flow-driven double-stranded DNA by Latinwo and Schroeder [Macromolecules 46, 8345 (2013)]. We also find that the skewness γ exhibits an interesting inversion of sign as a function of Wi, starting off at positive values at low Wi and changing to negative values at larger Wi. The inversion takes place in the vicinity of what we interpret as a coil-stretch transition. Again, the finding exactly reproduces behavior seen in other numerical and experimental work by the above group Latinwo et al. [J. Chem. Phys. 141, 174903 (2014)]. Additionally, at essentially the same value of Wi at which this sign inversion takes place, we observe that the kurtosis reaches a minimum, close to 1, providing further evidence of the existence of a coil-stretch transition at this location. Our calculations reproduce another numerical finding: a power law dependence on Wi of the rate of work production that is characterized by two distinct regimes, one lying below the putative coil-stretch transition, where the exponent assumes one value, and the other above, where it assumes a second.

Fluctuation relations for flow-driven trapped colloids and implications for related polymeric systems

“…The results of the previous sections have immediate implications for our earlier study of polymer stretching under flow [21][22][23][24], which, as mentioned in the Introduction, was motivated by Schroeder et al's experimental and numerical investigations [17][18][19][20] of the work done when a flexible polymer in an initially unperturbed state is stretched under the action of an extensional flow of strength γ.…”

“…Our interest in this system actually goes beyond its purely heuristic value. Schroeder et al recently showed how one particular fluctuation theorem (the Jarzynski relation) could be used to determine the elasticity of single-stranded DNA from an examination of the molecule's work fluctuations in the presence of extensional flow [17][18][19][20], and we later showed how many of the experimental and simulation results from that study could be satisfactorily described by a path integral treatment of a finitely extensible Rouse chain [21][22][23][24]. But we have now come to realize that some of our earlier calculations were in error.…”

“…These differences have important consequences. In the case of the mean, the old [22] and new functions are qualitatively the same, but they differ quantitatively in the way their work rates, β W R / t, depend on Wi, the Weissenberg number. For both, this dependence can be described by two distinctower laws, one for the region Wi < 1 and one for the region Wi > 1, but the values of the respective power law exponents in these regions are different for each.…”

“…Two other parameters that provide useful information about the nature of the distribution of a random variable z are the skewness γ 1 and the kurtosis γ 2 ; they are conventionally defined as γ 1 = (z − z) 3 /σ 3/2 z and γ 2 = (z − z) 4 /σ 2 z , where z is the mean of z and σ z is its variance. In reference [22], γ 1 and γ 2 were found to be independent of t, and in the case of γ 1 to change from positive to negative at the coil-stretch transition (the presumptive location of which corresponds to Wi = 1), and in the case of γ 2 to exhibit a minimum at the same location. In the work of Schroeder et al, there were indications that the skewness did change sign near the CST, but the results were for systems that transitioned from one nonequilibrium steady-state to another, so it is unclear if they are relevant to our present findings.…”

“…In a follow-up to our original study of chain elasticity, we had also used our finitely extensible (FE) model to calculate various moments of the work done by a polymer during flow-driven stretching [22]. In light of our new findings, the expressions for these moments must also now be reconsidered.…”