Interdisciplinary Applied Mathematics

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The analytical models of Vinogradova have been used with success in fitting the force curves in several SFA measurements, e.g., (Zhu and Granick, 2001; Zhu and Granick, 2002), but the expression for f * does not depend on the shear rate. However, in several SFA experiments it was shown convincingly that there is a strong dependence of the response on the driving speed, and thus these data do not agree with Vinogradova’s theory (Spikes and Granick, 2003). To this end, a new model was proposed by (Spikes and Granick, 2003) based on the observation that the experimental results may represent onset of slip at a fixed shear stress Tco rather than slip at a constant slip length b. Because in the SFA a sphere interacts with a plane, the surface shear stress is zero at the center and also away from it with maximum values in between; see Figure 10.19. This, in turn, implies that there exists an annular region around the contact point where slip occurs. The proposed new model in (Spikes and Granick, 2003), combines both this critical shear stress and the slip at constant b, so the shear stress when boundary slip occus is


_    . M


TcTco V vs.


b


The corresponding pressure gradient for the case of one slippery surface only is


dp    . f 6pWr f 6pWr    6b    ( Tco    3p,Wrl


dr = ~ mm h3 ’ P (h + 46) V X +    h3)) J ’


where min denotes the minimum of the two quantities, and W is the squeeze velocity; h is the gap height at radial distance r. The influence of Tco may not be realized in some applications, including cases in which it is constant, as in microchannel pressure-driven or Couette flow. However, it provides a correction for low shear stress configurations and also for the surface force apparatus and the atomic force microscope as well as in surfaces with roughness.

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