Modeling capillary forces for large displacements

Abstract

Originally applied to the accurate, passive positioning of submillimetric devices, recent works proved capillary self-alignment as effective also for larger components and relatively large initial offsets. In this paper, we describe an analytic quasi-static model of 1D capillary restoring forces that generalizes existing geometrical models and extends the validity to large displacements from equilibrium. The piece-wise nature of the model accounts for contact line unpinning singularities ensuing from large perturbations of the liquid meniscus and dewetting of the bounding surfaces. The superior accuracy of the generalized model across the extended displacement range, and particularly beyond the elastic regime as compared to purely elastic models, is supported by finite element simulations and recent experimental evidence. Limits of the model are discussed in relation to the aspect ratio of the meniscus, contact angle hysteresis, tilting and self-alignment dynamics.

Appendix: Model derivation

In this section, we present the full derivation of the analytical model presented in the main text.

With reference to the geometry sketched in Fig. 1, we subsume the partial wetting of the surfaces of the pads in considering $0<{\theta}_{{\rm t}}<{\theta}_{{\rm b}}<\pi /2$, yielding $u_1=h\cot {\theta}_{{\rm b}}$ and $u_2=h\cot {\theta}_{{\rm t}}$ according to Eq. (3). The alternative case of $0<{\theta}_{{\rm b}}<{\theta}_{{\rm t}}<\pi /2$ differs only in the sequence of contact line unpinnings over the pads, its formulation being the same upon mutual replacement of t with b. Figure 8 provides the reference geometries for the estimation of the updated values $h'$ and $h''$ of $h(u)$ upon transitions between regimes under conservation of meniscus volume.

The following holds under model assumptions:

• 1. the surface energy $E(u)$ coincides with the total free energy of the capillary system, and it is invariant under swapping of the surface energies of pads;

• 2. partial wettability of the surfaces of the pads determines the existence of finite relative displacements $u_i$ causing the sequential unpinning of the external contact lines (i.e., those whose vertical projection lies outside the opposite pads’ surface).

Fig. 8

Sketches (not to scale) for geometrical estimation of $h'$ and $h''$ under model assumptions upon transitions between adjacent regimes: a from $R_1$ to $R_2$, b from $R_2$ to $R_3$

The analytical formulation of the model proceeds from the calculation of the energy $E_j(u)$ of the system (Lienemann et al. 2004) for each regime $R_j$ determined by sequential unpinning discontinuities. The lateral capillary force $F_j(u)$ and stiffness $k_j(u)$ of the meniscus are computed by subsequent partial derivatives over $u$ of the energy function.

The energy of the global equilibrium state $R_{0}$ is (up to an additional arbitrary constant):

$$E_{0} = E(u_{0}=0) = \underbrace{L^2(\gamma_{{\rm sl}}^{\rm t} + \gamma_{{\rm sl}}^{\rm b})}_{{\rm pads}} + \underbrace{4\frac{V}{L}\gamma }_{{\rm meniscus}}$$

(6)

For the deformed states $R_1$ and $R_2$:

$$\begin{aligned} E_1&= E(0 \le u \le u_1) \\ &= \underbrace{L^2(\gamma_{{\rm sl}}^{\rm t} + \gamma_{{\rm sl}}^{\rm b})}_{{\rm pads}} + \underbrace{2\frac{V}{L}\gamma }_{{\rm lateral}} + \underbrace{2\gamma L \sqrt{h^2+u^2}}_{{\text {front}}\, \&\,{\text{rear }}} \\ &= E_{0} - 2\frac{V}{L}\gamma + 2\gamma L \sqrt{h^2+u^2} \end{aligned}$$

(7)

$$\begin{aligned} E_2&= E(u_1 \le u \le u_2)\\ &= \underbrace{L^2\gamma_{{\rm sl}}^{\rm t}}_{{\rm top}\,{\rm pad}} + \underbrace{L(u-u_1)\gamma_{{\rm sv}}^{\rm b} + L[L-(u-u_1)]\gamma_{{\rm sl}}^{\rm b}}_{{\rm bottom}\,{\rm pad}}\\ &\quad +\,\underbrace{2\gamma \frac{V}{L}}_{{\rm lateral}} + \underbrace{\gamma L \sqrt{h'^2+u_1^2}}_{{\rm rear}} + \underbrace{\gamma L \sqrt{h'^2+u^2}}_{{\rm front}}\\ &= E_1(u_1) + \gamma L \left(\sqrt{h'^2+u^2} - \sqrt{h'^2+u_1^2}\right)\\ &\quad+ (\gamma_{{\rm sv}}^{\rm b}-\gamma_{{\rm sl}}^{\rm b})L(u-u_1) \\ &\cong {\rm const} +\,\gamma L \sqrt{h^2+u^2} + \gamma L (u-u_1)\cos ({\theta}_{{\rm b}}) \end{aligned}$$

(8)

using Eq. (1) and the following approximation for the constancy of $h$ (see Fig. 8a):

$$\begin{aligned} h'&= h\frac{L}{L-\frac{u-u_1}{2}} \\ &= h\left(1+\frac{u-u_1}{2L}\right)+o^2(u-u_1) \\ &\cong h \,{\text {for}}\,u \ll 2L+u_1 = 2L+h\cot {\theta}_{{\rm b}} \\ \end{aligned}$$

(9)

Similarly for $R_3$:

$$\begin{aligned} E_3&= E(u_2 \le u < u_{{\rm max}}) \\&= \underbrace{2\frac{V}{L}\gamma }_{{\rm lateral}} + \underbrace{\gamma L\sqrt{h''^2+u_1^2}}_{{\rm rear}} + \underbrace{\gamma L\sqrt{h''^2+u_2^2}}_{{\rm front}} \\ &\quad+ \underbrace{L(u-u_1)\gamma_{{\rm sv}}^{\rm b} + L[L-(u-u_1)]\gamma_{{\rm sl}}^{\rm b}}_{{\rm bottom}\,{\rm pad}} \\ &\quad+ \underbrace{L(u-u_2)\gamma_{{\rm sv}}^{\rm t} + L[L-(u-u_2)]\gamma_{{\rm sl}}^{\rm t}}_{{\rm top}\,{\rm pad}} \\&\cong E_2(u_2) + \gamma L (u-u_2)(\cos {\theta}_{{\rm b}} + \cos {\theta}_{{\rm t}}) \end{aligned}$$

(10)

using the approximation (see Fig. 8b):

$$\begin{aligned} h''&= h\frac{L}{L-(u-\frac{u_1+u_2}{2})} \\&\cong h \,{\text {for}}\,\, u \ll L+\frac{u_1+u_2}{2} = L+\frac{h}{2}(\cot {\theta}_{{\rm b}} + \cot {\theta}_{{\rm t}}) \\ \end{aligned}$$

(11)

Equation (11), more stringent than Eq. (9), sets the strict limit of validity of the model over $u$ under the assumptions of constant $V$ and $h$. This condition assumes and is consistent with choices of coupled pairs of $\theta_*$ satisfying the condition set by Eq. (5) for overflow-less transition between adjacent capillary regimes. Equation (11) defines $u_{{\rm max}}(\theta_*,h)$ and relates it to the pad size $L$ rather than to $h$ ($L \gg h$ in general) as in purely elastic models. Given $L=1 \,\hbox {mm}, h$ and $\theta_*$, the relative errors in capillary force estimates between analytical and numerical models for each of the cases reported in Table 1 of the main text were evaluated for the corresponding value:

$$u_{{\rm max}}= \frac{1}{10}\left[ L+\frac{h}{2}(\cot {\theta}_{{\rm b}} + \cot {\theta}_{{\rm t}})\right] .$$

(12)

From Eqs. (7), (8) and (10) it follows, respectively:

$$R_1 \left\{ \begin{array}{ll} \begin{aligned} F_1(u) &= -\frac{\partial E_1(u)}{\partial u} = -2\gamma L \frac{u}{\sqrt{h^2+u^2}} \\ k_1(u) &= -\frac{\partial F_1(u)}{\partial u} = \frac{\partial ^2 E_1(u)}{\partial u^2} \\ &= 2 \gamma L \left(\frac{1}{\sqrt{h^2+u^2}}-\frac{u^2}{(h^2+u^2)^{\frac{3}{2}}}\right) \end{aligned} \end{array}\right.$$

(13)

$$R_2 \left\{ \begin{array}{ll}\begin{aligned} F_2(u) &= -\frac{\partial E_2(u)}{\partial u} = -\gamma L (\frac{u}{\sqrt{h^2+u^2}} + \cos {\theta}_{{\rm b}}) \\ k_2(u) &= -\frac{\partial F_2(u)}{\partial u} = \frac{\partial ^2 E_2(u)}{\partial u^2} \\ &= \gamma L (\frac{1}{\sqrt{h^2+u^2}}-\frac{u^2}{(h^2+u^2)^{\frac{3}{2}}}) \\ &= \frac{k_1(u)}{2}\end{aligned} \end{array}\right.$$

(14)

$$R_3 \left\{ \begin{array}{ll}\begin{aligned} F_3(u) &= -\frac{\partial E_2(u)}{\partial u} = -\gamma L (\cos {\theta}_{{\rm b}} + \cos {\theta}_{{\rm t}}) \\ k_3(u) &= -\frac{\partial F_3(u)}{\partial u} = \frac{\partial ^2 E_3(u)}{\partial u^2} \\ &= 0 \end{aligned}\end{array}\right.$$

(15)

The formulation is consistent with energy and force continuity across adjacent domains, since $E_{i+1}(u_i)=E_i(u_i)$ and $F_{i+1}(u_i)=F_i(u_i)$ hold for all $i = 0,1,2$. Note that for $R_1|_{0<u<h}$ the small displacement values $F_1=-2\gamma L \frac{u}{h}$ and $k_1=2 \gamma \frac{L}{h}$ of linear elastic models are recovered (Tsai et al. 2007; Mastrangeli et al. 2010; Lambert et al. 2010; Berthier et al. 2010). Conversely, the absence of elastic work in $R_3$ is evidenced by the null constant value of $k_3$. Also, ${\theta}_{{\rm t}} \rightarrow {\theta}_{{\rm b}}$ implies a singular domain for $R_2$ as $u_2 \rightarrow u_1$. Particularly, for the limiting case of full wetting of both pads—i.e., for ${\theta}_{{\rm b}} = {\theta}_{{\rm t}} = 0$—$u_1 \rightarrow \infty$, i.e., the domain of $R_1$ extends indefinitely. The purely elastic regime is thus recovered, whereby partial dewetting of the surface of the pads is not possible. In this ideal condition, further model convergence is given by $F_1|_{u \gg h} = F_3|_{{\theta}_{{\rm b}}={\theta}_{{\rm t}}=0}$. Plots of energy, gap and restoring force versus $u$ for $\hbox {AR} = 1/20$ and $1/10$ ($h = 50$ and $100 \,\upmu \hbox {m}$, respectively) are shown in Figs. 9, 10, 11, 12, 13 and 14—complementing those for ${\rm AR} = 1/40$ presented in the main text.

Fig. 9

SE simulations for $h_{0} = 50\,\upmu\hbox{m}$ $(\hbox {AR} = 1/20$) parameterized by ${\theta}_{{\rm b}}$–${\theta}_{{\rm t}}$ pairs. a Total energy $E(u)-E(0)$ versus $u.$ b Normalized gap $(h-h_{0})/h_{0}$ versus $u$

Fig. 10

SE simulation and analytical fit of $F(u)$ versus $u$ for $h_{0} = 50 \,\upmu\hbox{m}$ $(\hbox {AR} = 1/20$) with ${\theta}_{{\rm b}} = 65^{\circ }$ and ${\theta}_{{\rm t}} = 50^{\circ}$. The relative error for $F(u=u_{{\rm max}})$ is $7.3 \,\%$

Fig. 11

Capillary forces predicted for h = 50 μm (AR = 1/20) and various combinations of ${\theta}_{{\rm b}}$ and ${\theta}_{{\rm t}}$ values. a Perfect wetting (no contact line unpinning). b The three sequential regimes, first unpinning on bottom pad. c Absence of second regime for ${\theta}_{{\rm b}} = {\theta}_{{\rm t}}$ ($u_{1}=u_{2}=u_{d},\,F_{1}(u_{d})=F_{2}(u_{d})=F_{3}(u_{d})$). d The three sequential regimes, first unpinning on top pad

Fig. 12

SE simulations for $h_{0} = 50\,\upmu\hbox{m}$ $(\hbox {AR} = 1/20$) parameterized by ${\theta}_{{\rm b}}$–${\theta}_{{\rm t}}$ pairs. a Total energy $E(u)-E(0)$ versus $u.$ b Normalized gap $(h-h_{0})/h_{0}$ versus $u$

Fig. 13

SE simulation and analytical fit of $F(u)$ versus $u$ for $h_{0} = 100\,\upmu\hbox{m}$ $(\hbox {AR} = 1/10$) with ${\theta}_{{\rm b}} = 65^{\circ }$ and ${\theta}_{{\rm t}} = 50^{\circ}$. Relative error for $F(u=u_{{\rm max}})$ is $7.4 \,\%$

Fig. 14

Capillary forces predicted for h = 100 μm (AR = 1/10) and various combinations of ${\theta}_{{\rm b}}$ and ${\theta}_{{\rm t}}$ values. a Perfect wetting (no contact line unpinning). b The three sequential regimes, first unpinning on bottom pad. c Absence of second regime for ${\theta}_{{\rm b}} = {\theta}_{{\rm t}}$ ($u_1=u_2=u_d,\,F_1(u_d)=F_2(u_d)=F_3(u_d)$). d The three sequential regimes, first unpinning on top pad

Fig. 15

Transitions to overflow regimes: a from $R_1$ to $R_2^{{\rm of}}$, b from $R_2$ to $R_3^{{\rm of}}$

Finally, by considering meniscus confinement within the pads by chemical contrast (Fig. 2b) rather than by topographical step, the formal derivation shown above can be adapted to account for the case of liquid bridge overflow (Lienemann et al. 2004). Overflow is here supposed to take place beyond the edge of the bottom pad onto an adjacent and less wettable surface (see Fig. 15). The case of overflow beyond the top pad is energetically equivalent. We assume that the two (pad and adjacent) surfaces are at the same level and that the energetic barrier to overflow is only chemical in nature. For the less wettable surface, the validity of a specific Young–Dupré equation is also assumed, yielding a contact angle ${\theta}_{{\rm b}}^{{\rm of}}>\pi /2>{\theta}_{{\rm b}}$. The overflow happens when the edge angle of the meniscus reaches the advancing value of the contact angle on the adjacent surface (hereby again assumed to coincide with its static value ${\theta}_{{\rm b}}^{{\rm of}}$), prompting the unpinning of the contact line toward the adjacent surface. Unpinning takes place for $u \ge u_{{\rm of}}$ and signals the transition to a regime akin to either the mixed ($R_2^{{\rm of}}$) or the full sliding one ($R_3^{{\rm of}}$). $u_{{\rm of}}$ can be either larger or smaller than $u_1$, yet not larger than $u_2$ because in $R_3$ the inclinations of both perpendicular sides of the meniscus remain constant. For the former case of $0<u_{{\rm of}}<u_1$ (Fig. 15a):

$$R_2^{{\rm of}} \left\{ \begin{array}{ll}\begin{aligned} F_2^{{\rm of}}(u) &= -\frac{\partial E_2^{{\rm of}}(u)}{\partial u} = -\gamma L \left(\frac{u}{\sqrt{h^2+u^2}} - \cos {\theta}_{{\rm b}}^{{\rm of}}\right) \\ k_2^{{\rm of}}(u) &= -\frac{\partial F_2^{{\rm of}}(u)}{\partial u} = \frac{\partial ^2 E_2^{{\rm of}}(u)}{\partial u^2} \\ & = \gamma L \left(\frac{1}{\sqrt{h^2+u^2}}-\frac{u^2}{(h^2+u^2)^{\frac{3}{2}}}\right) \\ &= \frac{k_1(u)}{2}\end{aligned} \end{array}\right.$$

(16)

For the latter case of $u_1<u_{{\rm of}}$ (Fig. 15b):

$$R_3^{{\rm of}} \left\{ \begin{array}{ll}\begin{aligned} F_3^{{\rm of}}(u) &= -\frac{\partial E_2^{{\rm of}}(u)}{\partial u} = -\gamma L (\cos {\theta}_{{\rm b}} - \cos {\theta}_{{\rm b}}^{{\rm of}}) \\ k_3^{{\rm of}}(u) &= -\frac{\partial F_3^{{\rm of}}(u)}{\partial u} = \frac{\partial ^2 E_3^{{\rm of}}(u)}{\partial u^2} \\ &= 0 \end{aligned}\end{array}\right.$$

(17)