Spontaneous Capillary Rise in Hydrophobic Capillaries

A further confirmation of the adsorption of surfactant molecules in front of the moving meniscus on the bare hydrophobic substrate is a phenomenon of the spontaneous capillary rise of surfactant solutions in hydrophobic capillaries. 

Let us consider a spontaneous capillary rise of surfactant solutions in hydro-phobic capillaries. 

The maximum height of the capillary rise, L , is reached after the concentration on the meniscus, C, becomes equal to the concentration at the capillary entrance, Cq. After that the capillary rise stops. Using Eqnation 5.37, is determined as 

In the following text the problem of the spontaneous capillary rise of surfactant solntions in hydrophobic capillaries is considered in the case when concentration at the capillary inlet is below CMC. In this case, the transport of surfactant molecules is described by Equation 5.15 and Eqnation 5.16, and boundary conditions (5.17) throngh (5.19). The snbstantial difference from the spontaneons capillary imbibition is that now the relation between l(t) and the concentration on the moving meniscus, C, is given by relation (5.37), which shows that is an unknown function of time. Using these equations and boundary conditions, we show in the following text that l(t) dependency on time can be calculated, and it is proportional to the sqnaie root of time at the initial stage of the capillary rise (see Appendix 1 for details). 

The solution in Appendix 1 shows that, at the initial stage of capillary rise, 1(0 develops as 

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Equation 5.38 shows that the case under consideration is governed by a completely different mechanism as compared with the case of the horizontal imbibition (where remains constant over time). In the case of the spontaneous capillary rise in hydrophobic capillaries, CJt) does not remain constant but must increase as the capillary rise progresses. The comparison of Figure 5.11 and Figure 5.10 shows that the time scale of the spontaneous capillary rise is around 100 times bigger than the corresponding time scale in the case of the capillary imbibition into horizontal capillaries. 

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