Tank-treading

Chen WD, Chen JZ, An LJ (2013) Tumbling and tank-treading dynamics of individual ring polymers in shear flow. Soft Matter 9 4312-4318... 

In a patent issued to the US Army in the late 1980s, the zinc salts of acrylic acid and methacrylic acid were used to upgrade the properties of hydrogenated nitrile rubber (HNBR) [3]. The goal of this work was to develop abrasion-resistant materials that could be used to extend the service life of tank treads. Zinc dimethacrylate (ZDMA) proved to be the best material in the study for improving the tear strength, abrasion resistance, and high temperature performance of HNBR. 

Fig. 20 Successive snapshots of a star polymer of functionality / = 25 and arm length Lf = 20, which illustrate the tank-treading motion...

For B > 1, there is a stationary solution, with cos(20) = 1/B. This corresponds to a tank-treading motion, in which the orientation of the vesicle axis is time independent, but the membrane itself rotates around the vorticity axis. 

The shear rate y only determines the time scale, but does not affect the tank-treading or tumbling behavior. Therefore, a transition between these two tyjtes of motion can only be induced by a variation of the vesicle shape or the viscosities. 

Effects of Membrane Viscosity Tank-Treading and Tumbling... 

The theory of Keller and Skalak [194] predicts for fluid vesicles a transition from tank-treading to tumbling with increasing viscosity contrast rjm/Tjo- This has been confirmed in recent simulations based on a phase-field model [203],... 

Fig. 28 Dynamical phase diagram as a function of viscosity contrast = Tjin/rjo, for r = 0 and various reduced volumes V, calculated from (105), (106) without thermal noise. The tank-treading phase is located on the left-hand side of the dashed lines. The solid lines represent the tumbling-to-swinging transitions. From [205]...

Fig. 29 Dynamical phase diagram of a vesicle in shear flow for reduced volume V = 0.59. Symbols correspond to simulated parameter values, and indicate tank-treading discocyte and tank-treading prolate (circles), tank-treading prolate and unstable discocyte (triangles), tank-treading discocyte and tumbling (transient) prolate (open squares), tumbling with shape oscillation (diamonds), unstable stomatocyte (pluses), stable stomatocyte (crosses), and near transition (filled squares). The dashed lines are guides to the eye. From [180]...

At higher shear rates, an intermittent behavior has been observed, in which the vesicle motion changes in irregular intervals between tumbling and tank-treading [180]. 

At finite temperature, stochastic fluctuations of the membrane due to thermal motion affect the dynamics of vesicles. Since the calculation of thermal fluctuations under flow conditions requires long times and large membrane sizes (in order to have a sufficient range of undulation wave vectors), simulations have been performed for a two-dimensional system in the stationary tank-treading state [213]. For comparison, in the limit of small deviations from a circle, Langevin-type equations of motion have been derived, which are highly nonlinear due to the constraint of constant perimeter length [213]. 

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