A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading

Abstract

The motion of a contact line is examined, and comparisons drawn, for a variety of models proposed in the literature. Pressure and stress behaviours at the contact line are examined in the prototype system of quasistatic spreading of a thin two-dimensional droplet on a planar substrate. The models analysed include three disjoining pressure models based on van der Waals interactions, a model introduced for polar fluids, and a liquid–gas diffuse-interface model; Navier-slip and two non-linear slip models are investigated, with three microscopic contact angle boundary conditions imposed (two of these contact angle conditions having a contact line velocity dependence); and the interface formation model is also considered. In certain parameter regimes it is shown that all of the models predict the same quasistatic droplet spreading behaviour.

Appendices

Appendix A. Solutions for slip models in Moffatt wedge geometry

The stream function, velocity, stress, and pressure behaviours for the Moffatt wedge geometry, for Navier-slip in (15), are found to be

$$\begin{aligned}&\psi \sim -\frac{r^2 U}{4\lambda }\left( 1-\frac{\theta }{\theta _\mathrm{c}} + \frac{\sin (2\theta )\cos (2\theta _\mathrm{c})}{\sin (2\theta _\mathrm{c})}-\cos (2\theta ) \right) ,\nonumber \\&u \sim -\frac{rU}{4\lambda }\left( -\frac{1}{\theta _\mathrm{c}} + \frac{2\cos (2\theta )\cos (2\theta _\mathrm{c})}{\sin (2\theta _\mathrm{c})}+2\sin (2\theta ) \right) ,\qquad v \sim -\frac{2\psi }{r}, \qquad p \sim \frac{U}{\lambda \theta _\mathrm{c}} \log r, \\&T_{rr} \sim \frac{U}{2\lambda }\frac{\sin (2\theta _\mathrm{c})-2\theta _\mathrm{c}\cos (2(\theta _\mathrm{c}-\theta ))}{\theta _\mathrm{c}\sin (2\theta _\mathrm{c})} ,\qquad T_{r\theta } \sim \frac{U}{\lambda }\frac{\sin (2(\theta -\theta _\mathrm{c}))}{\sin (2\theta _\mathrm{c})} ,\qquad T_{\theta \theta } \sim -T_{rr}. \nonumber \end{aligned}$$

(64)

These results show clearly how the Navier-slip model removes the singularity in the stress as the contact line is approached but a logarithmic singularity in the pressure remains.

Similarly for the general slip case in (15), we obtain

$$\begin{aligned}&\psi \sim r^{2+n}\left( C_{1\mathrm{s}}\sin (n\theta )+C_{2\mathrm{s}}(\cos (n\theta )-\cos ((2+n)\theta )) + C_{3\mathrm{s}}\sin ((2+n)\theta ) \right) , \nonumber \\&u \sim r^{1+n}\left( n[C_{1\mathrm{s}}\cos (n\theta )-C_{2\mathrm{s}}\sin (n\theta )] +(2+n)[C_{3\mathrm{s}}\cos ((2+n)\theta ) + C_{2\mathrm{s}}\sin ((2+n)\theta )]\right) , \nonumber \\&v \sim -\frac{(2+n)\psi }{r} ,\qquad p \sim 4(1+n) r^n C_{1\mathrm{s}} \frac{\cos (n(\theta _\mathrm{c}-\theta ))}{\cos (n\theta _\mathrm{c})}, \qquad T_{\theta \theta } \sim -T_{rr}, \nonumber \\&T_{rr} \sim 2(1+n) r^n \left( (2+n)[C_{3\mathrm{s}}\cos ((2+n)\theta ) + C_{2\mathrm{s}}\sin ((2+n)\theta )] + \frac{C_{1\mathrm{s}}n\cos (n(\theta _\mathrm{c}-\theta ))}{\cos (n\theta _\mathrm{c})} \right) ,\nonumber \\&T_{r\theta } \sim 2(1+n) r^n \left( {C_{2\mathrm{s}}\frac{(2+n)\sin ((2+n)(\theta _\mathrm{c}-\theta ))}{\sin ((2+n)\theta _\mathrm{c})} -C_{1\mathrm{s}} \frac{n\sin (n(\theta -\theta _\mathrm{c}))}{\cos (n\theta _\mathrm{c})} }\right) , \end{aligned}$$

(65)

where

$$\begin{aligned} C_{1\mathrm{s}} = \frac{U\lambda ^{-(1+n)}\sin ^n\theta _\mathrm{c}}{4(1+n)\tan (n\theta _\mathrm{c})},\qquad C_{2\mathrm{s}} = - C_{4\mathrm{s}} = -C_{1\mathrm{s}}\tan (n\theta _\mathrm{c}),\qquad C_{3\mathrm{s}} = \frac{C_{2\mathrm{s}}}{\tan ((2+n)\theta _\mathrm{c})}. \end{aligned}$$

(66)

We see from this solution that $n=0$ does not recover the Navier-slip solution because this is a degenerate solution of the biharmonic equation. However, the two cases considered by Haley and Miksis [36], corresponding to $n=1$ and $n=2$ here, will resolve both the pressure and stress singularities. In contrast to the Navier-slip model, they both give zero pressure and stresses as the contact line is approached—the pressure and stresses being respectively $O(r)$ and $O(r^2)$ in the two cases.

There is, however, a possibility of removing the logarithmic pressure singularity but retaining an $O(1)$ stress behaviour. Previously in (5) we suggested that for the stresses to be $O(1)$, the stream function must be prescribed by $\psi \sim r^2 F(\theta )$, with $F(\theta )$ to be determined. We have also seen that this form, which corresponds to the Navier-slip condition, yields $O(\log r)$ pressure as $r\rightarrow 0$ due to the $1/r$ terms in the divergence of the stress. However, it is also possible to cancel the logarithmic pressure terms by relaxing the assumption of constant surface tension, thereby obtaining a finite pressure solution. More specifically, from (11), where $\varvec{\nabla }_{\mathrm{s}}\sigma$ is the surface gradient of $\sigma$ and which, given in terms of the usual gradient operator, is $\varvec{\nabla }_{\mathrm{s}}\sigma = (\mathbf{I}-\mathbf n \otimes \mathbf n )\varvec{\nabla }\sigma$, we can determine the dynamic boundary condition as

$$\begin{aligned} T_{r\theta } = -\frac{{\partial {\sigma }}}{{\partial {r}}} \qquad \text{ on }\quad \theta =\theta _c. \end{aligned}$$

(67)

Now using the pressure equations from (6), with $\psi \sim r^2 F(\theta )$, gives

$$\begin{aligned} \frac{{\partial {p}}}{{\partial {r}}} = \frac{1}{r}(F'''+4F'), \qquad \frac{{\partial {p}}}{{\partial {\theta }}} = 0, \end{aligned}$$

(68)

and thus (at leading order in $r$) $p=p(r)$. To have a pressure without a singularity requires $F'''+4F'=0$, and this third-order ODE has the general solution

$$\begin{aligned} F = B_1+B_2\sin (2\theta )+B_3\cos (2\theta ), \end{aligned}$$

(69)

where the $B_i$ are constant coefficients. From (69) and (9), we thus impose the stream function form

$$\begin{aligned} \psi \sim r^2(B_1+B_2\sin (2\theta )+B_3\cos (2\theta ))&+r^3(C_1\sin \theta +C_2\cos \theta +C_3\sin (3\theta )+C_4\cos (3\theta )), \end{aligned}$$

(70)

keeping two orders in $r$ so as to obtain the leading-order term in the pressure behaviour. We find after applying the boundary conditions [including (67)] with Navier-slip

$$\begin{aligned}&\psi \sim -\frac{1}{4}\frac{Ur^2}{\lambda }\left( 1-\tan \theta _\mathrm{c}\sin (2\theta )-\cos (2\theta ) \right) \nonumber \\&\qquad \,\, \,\,-\frac{1}{16}\frac{Ur^3}{\lambda ^2}\left( \sin \theta - \tan \theta _\mathrm{c}\cos \theta - \frac{(2\cos (2\theta _\mathrm{c})-1)\sin (3\theta )}{2\cos (2\theta _\mathrm{c})+1} + \tan \theta _\mathrm{c}\cos (3\theta )\right) , \nonumber \\&u \sim \frac{1}{2}\frac{Ur}{\lambda } \left( \tan \theta _\mathrm{c}\cos (2\theta ) - \sin (2\theta ) \right) , \qquad v \sim \frac{1}{2}\frac{Ur}{\lambda }\left( 1-\tan \theta _\mathrm{c}\sin (2\theta )-\cos (2\theta ) \right) ,\nonumber \\&T_{rr} \sim \frac{U}{\lambda }\left( \tan \theta _\mathrm{c}\cos (2\theta ) - \sin (2\theta ) \right) ,\qquad T_{r\theta } \sim -\frac{U}{\lambda }\left( \tan \theta _\mathrm{c}\sin (2\theta )+\cos (2\theta ) \right) ,\nonumber \\&T_{\theta \theta } \sim -T_{rr} ,\qquad p \sim -\frac{1}{2}\frac{Ur}{\lambda ^2}\frac{\cos (\theta _\mathrm{c}-\theta )}{\cos \theta _\mathrm{c}}, \end{aligned}$$

(71)

displaying the mathematical possibility of such a flow. We note here as part of the result the form of the surface tension was forced to satisfy

$$\begin{aligned} \frac{{\partial {\sigma }}}{{\partial {r}}} \sim \frac{U}{\lambda } \quad \Rightarrow \quad \sigma \sim \sigma _\mathrm{c} + \frac{Ur}{\lambda } \qquad \text{ as } r\rightarrow 0, \end{aligned}$$

(72)

which gives a reasonable result for the surface tension $\sigma$, that it is a constant to leading order when $r\ll 1$, the regime in which we are interested. This also gives a motivation as to why a model which allows for variable surface tension, i.e. surface tension relaxation, may be able to predict a flow which has no logarithmic pressure singularity but finite and non-zero stresses at the contact line. Such surface tension relaxation is a feature of the interface formation model [59], where the surface tension is given as $\sigma =\sigma (\rho ^s)$ and the surface density is a function of position and time, i.e. $\rho ^s=\rho ^s(\mathbf x ,t)$.

Appendix B. Intermediate region for quasistatic spreading

Motivated by the analysis of [27] and the matching conditions (52), we introduce the intermediate scalings

$$\begin{aligned} \Phi = C_a\xi \phi , \qquad \hat{\zeta }= \hat{\epsilon }\ln \left( \mathrm{e}^{C_a^2C_b}C_a\xi \right) , \end{aligned}$$

(73)

where $\hat{\epsilon }= 1/|\ln \delta |$. Neglecting exponentially small terms, (44) transforms to

$$\begin{aligned} \frac{{\partial {}}}{{\partial {\hat{\zeta }}}}\left\{ \mathrm{e}^{\hat{\zeta }/\hat{\epsilon }-C_a^2C_b}\left[ C_a\dot{a}\phi + C_a^4\phi ^3\left( \hat{\epsilon }^3\frac{{\partial ^3{\phi }}}{{\partial {\hat{\zeta }}^3}}-\hat{\epsilon }\frac{{\partial {\phi }}}{{\partial {\hat{\zeta }}}} \right) \right] \right\} =0, \end{aligned}$$

(74)

so that to leading order in $\hat{\epsilon }$ we have

$$\begin{aligned} \phi = \left( \mathcal {K} + \frac{3\dot{a}\hat{\zeta }}{C_a^3\hat{\epsilon }} \right) ^{1/3}, \end{aligned}$$

(75)

where $\mathcal {K}$ is a constant of integration. Expanding these for $\mathcal {K} \gg {3\dot{a}\hat{\zeta }}/{(C_a^3\hat{\epsilon })}$ and rewriting in inner variables, we have

$$\begin{aligned} \frac{{\partial {\Psi }}}{{\partial {\xi }}} \sim C_a\mathcal {K}^{1/3} + \dot{a}\mathcal {K}^{-2/3}\left[ {C_b + \ln (\text{ e }\, C_a \xi )/C_a^2} \right] , \end{aligned}$$

(76)

which gives $\mathcal {K}=1$ to match with (52). To match this intermediate solution to the outer solution, (75) clearly suggests that matching should be performed with the cube of the free-surface slope, and indeed we find from (75) using scalings (73) (with our result that $\mathcal {K}=1$) that

$$\begin{aligned} -\left( {\frac{{\partial {h}}}{{\partial {x}}}} \right) ^3 \sim C_a^3 + 3\dot{a}\left( {C_a^2 C_b + \ln \left[ {\frac{\text{ e }\, C_a(a-x)}{\delta }} \right] } \right) , \end{aligned}$$

(77)

which is the same result as would be found directly from the inner-region behaviour (52). From the outer-region behaviour (42) we have

$$\begin{aligned} -\left( {\frac{{\partial {h}}}{{\partial {x}}}} \right) ^3 \sim \frac{27}{a^6} + 3\dot{a}\ln \left( \frac{\text{ e }^3(a-x)}{2a}\right) , \end{aligned}$$

(78)

and thus the matching of these provides the spreading rate result given in (54).