Interdisciplinary Applied Mathematics

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In (Zhu and Granick, 2001), SFA measurements with molecularly smooth surfaces of mica were obtained for water (polar) and tetradecane (non polar), an oil with low viscosity close to water. In particular, three different liquid-solid systems were considered with increasing contact angle: (1) tetradecane against adsorbed surfactant; (2) tetradecane against a methyl-terminated self-assembled monolayer (SAM), and (3) water against a methyl-terminated SAM. In the last two cases a monolayer of octadecyl-triethoxysiloxane (OTE) was used on which the (advancing) contact angle of water was 110° and that of tetradecane was 44°. It was found that for cases (2) and (3) with the OTE layer causing partial wetting of the surface, boundary slip was obtained for film thickness less than about 100 pm. However, case (1)    gave    no slip    at    the    solid interface.    The    results    of    (Zhu

and Granick, 2001) led to similar conclusions as in (Craig et al., 2001), in that the slip length depends strongly on the approach (driving) rate and is largest for water: the largest value is b « 35 nm at shear rates about 10s_1. Below a threshold value of the approach rate (and thus shear rate) the no-slip boundary condition is valid.

The effect of roughness on boundary slip was examined in a follow-up paper by the same researchers (Zhu and Granick, 2002). Roughnesses with rms values up to 6 nm were fabricated using self-assembled OTE monolayers and OTS (octadecyltrichlorosilane) layers. The advancing contact angle was similar    for    all    cases,    but    the receding    contact    angle    was    a decreasing

function of surface roughness. It was hypothesized that large roughness will decrease the slip length, although cases with an increase in slip length have also been reported (Bonaccurso et al., 2003; Ponomarev and Meyerovich, 2003). The results of (Zhu and Granick, 2002) are summarized in Figure 10.21, where atomic force microscopy images of roughness on a 3 /am x3 /am area are also shown. The data of (Zhu and Granick, 2002) show that even the case of largest slip length (b « 35 nm) for water produces no slip at the wall if the roughness rms height exceeds 6-nm. The critical shear rate for onset of slip seems to depend linearly on the roughness rms height and is independent of its wavelength. For 6-nm roughness a value of shear rate

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