Lateral Capillary Forces

Abstract

Lateral capillary forces ensuing from perturbed fluid menisci are pivotal to many important technologies, including capillary self-alignment and self-assembly of heterogeneous microsystems. This chapter presents a comprehensive study of the quasi-statics of lateral capillary forces arising from a constrained cylindrical fluid meniscus subjected to small lateral perturbations. After a contextual literature review, we describe a novel experimental apparatus designed to accurately characterize such a fundamental system. We then reproduce our experimental data on lateral meniscus forces and stiffnesses by means of both a novel analytical model and a finite element model. The agreement between our measurements and our models validate earlier reports and provides a solid foundation for the applications of lateral capillary forces to microsystems handling and assembly.

Content of this chapter, including all pictures and tables, originally published in [32], $\copyright$ (2010) IOP Publishing Ltd (available at: http://iopscience.iop.org/0960-1317/20/7/075041/). Presented hereby in revised form with permission from IOP Publishing Ltd.

Appendix

1.1 Bending Stiffness of the Spring

We estimated the bending stiffness $K$ of our double-cantilever sliding spring in 3 different ways, obtaining 5 estimates. All geometrical parameters of the spring were known: cantilever length $L=282\,\mathrm m\mathrm{m}$, width $b=12.7\,\mathrm m\mathrm{m}$, thickness $t=0.102\,\mathrm m\mathrm{m}$; total spring mass (including both beams, shuttle and top pad) $M=18.448\,g$. Multiple alternative estimates were motivated by the uncertainty on the actual Young Modulus $E$ (assumed: $210\, \mathrm G\mathrm {Pa}$) and density $\rho$ (assumed: $7800\, \mathrm K\mathrm g\mathrm{m}^{-3}$) of our steel cantilevers, directly affecting our analytic estimates. A good agreement between all estimates was obtained—as summarized in Table 3.3 and detailed below. Nonetheless, we attributed the highest confidence to the 2 fully-experimental estimates of $K$ (defined below as $K_{3}$ and $K_{4}$), both of which avoided the use of $E$ and $\rho$. Therefore, we assumed for the spring a bending stiffness equal to the average of $K_{3}$ and $K_{4}$, i.e. $K=0.9969\, \mathrm N\mathrm{m}^{-1}$, with a relative uncertainty of $5.96\,\%$.

Table 3.3 Summary of the estimates of the spring’s bending stiffness $K$

We remark that such high sensitivity enabled the spring’s desired force resolution (see requirement (iii) of Sect. 3.2.1) but also its high susceptibility to environmental perturbations (see Sect. 3.4).

1.2 Auxiliary Cantilever Method

First estimates of $K$ involved an auxiliary steel cantilever of known dimensions (Precision Brand, length $l=86.4\, \mathrm m\mathrm{m}$, thickness $t=0.102\, \mathrm m\mathrm{m}$, width $b=12.7\, \mathrm m\mathrm{m}$). The measurement exploited the lateral balance between the elastic forces of the perturbed cantilever and spring (Fig. 3.12). Starting from the rest position, common to both cantilever and spring, a laser-tracked lateral displacement imposed on the cantilever induced a laser-tracked lateral displacement on the spring. After determining the bending stiffness of the cantilever, the stiffness of the spring was obtained from its force-versus-displacement curve by first-order polynomial fitting.

Fig. 3.12

The auxiliary cantilever used to estimate the spring stiffness

We estimated the bending stiffness $k$ of the auxiliary cantilever in 3 ways. Assuming the standard stainless steel’s Young’s Modulus $E$ and density $\rho$, the bending stiffness of a cantilever $k$ for small deformations is given analytically by (see [49]):

$$\begin{aligned} k=\frac{3EI}{L^{3}} \end{aligned}$$

(3.13)

where $I$ is the cantilever’s second moment of the area. We estimated $I$:

• 1. Analytically, as $I=\frac{{bt}^{3}}{12}$. Inserting this in ( 3.13) leads to $k_{1}=1.097\, \mathrm N\mathrm{m}^{-1}$.

• 2. From the knowledge of the first resonance $f_{1}$ of the cantilever, as obtained by solving Euler’s beam equation ( [49], p. 273):

  $$\begin{aligned} I=\frac{2\pi f_{1}\rho }{E\beta _{1}^{4}} \end{aligned}$$

  (3.14)

  where $\beta _{1}=1.875$. We measured the vibration period $t=97\, \mathrm m\mathrm s$ of the cantilever analyzing its laser-tracked oscillations on a digital oscilloscope. Hence, from ( 3.14) and ( 3.13) we got $k_{2}=0.8889\, \mathrm N\mathrm{m}^{-1}$.

We also estimated the bending stiffness of the cantilever by measuring and fitting numerically its tip load-versus-tip displacement curve. This gave us a value of $k_{3}=1.0071\, \mathrm N\mathrm{m}^{-1}$. We considered this the most reliable of our estimates of $k$.

We consequently obtained one value of the bending stiffness $K_{\#}$ from each $k_{\#}$ value: $K_{1}=1.1506\, \mathrm{N/m}$, $K_{2}=0.9323\, \mathrm N\mathrm{m}^{-1}$ and $K_{3}=1.0563\,\mathrm N\mathrm{m}^{-1}$.

1.3 Dynamic Method

From the natural oscillation frequency $f_{1}$ of the double-cantilever spring ($f_{1}=1.266\, \mathrm Hz$ in our case, as measured by oscilloscope), its stiffness $K_{4}$ can be directly estimated according to:

$$\begin{aligned} K_{4}=4\pi ^{2}f_{1}^{2}M_{\mathrm{eff }} \end{aligned}$$

(3.15)

where $M_{\mathrm{eff }}$ is the effective spring mass, including the mass of the shuttle and the kinetic energy-averaged mass of the cantilevers (according to Rayleigh method; see [49], p. 23, and 3.5 for details). We obtained $K_{4}=0.9375\, \mathrm N\mathrm{m}^{-1}$.

1.4 Analytic Method

We also calculated $K$ fully-analytically. We assumed that $K_{5}$ had 2 components: (1) the mechanical stiffness of 2 parallel, coupled cantilevers—with their unclamped extremities constrained by the shuttle to slide along a direction perpendicular to the cantilevers—given by material strength theory [49]; and (2) a component due to the gravitational potential energy, converted into a gravitational stiffness.

The mechanical component was obtained from:

$$\begin{aligned} K_{\mathrm{mech }}=2\cdot \frac{12EI}{L^{3}} \end{aligned}$$

(3.16)

We estimated the gravitational stiffness as (see 3.5 for details):

$$\begin{aligned} K_{\mathrm{grav }}=\frac{6g}{5L}\left( m_{\mathrm{sh }}+\frac{m_{\mathrm{b }}}{2}\right) \end{aligned}$$

(3.17)

where $g$ is the acceleration of gravity. All parameters being known, we got a value of $K_{5}=0.9036\, \mathrm N\mathrm{m}^{-1}$.

1.5 Effective Spring Mass

The mass of the spring’s 2 cantilever ($m_{\mathrm{b }}=5.7\,g$) was not negligible compared to that of the shuttle and top pad ($m_{\mathrm{s }}=12.778\,g$). Therefore, in the dynamic estimation of the spring’s stiffness we introduced an equivalent mass for both beams $m_{\mathrm{eq }}$, which would have the same kinetic energy as the actual cantilevers for the same shuttle velocity $v$ according to:

$$\begin{aligned} \frac{1}{2}m_{\mathrm{eq }}v^{2}=2\cdot \frac{1}{2}\int _{0}^{L}v^{2}(z)dm^{\prime }=2\cdot \frac{1}{2}\lambda \int _{0}^{L}v^{2}(\xi )d\xi \end{aligned}$$

(3.18)

where $dm^{\prime }=\lambda d\xi$, and $\lambda$ has the dimension of mass per unit length. The velocity $v(z)$ of each cantilever element located at a distance $z$ from the clamped extremity was assumed to be proportional to its displacement computed by material strength theory:

$$\begin{aligned} v(z)=\frac{q(z)}{u}v \end{aligned}$$

(3.19)

where the element $q(z)$ is given by:

$$\begin{aligned} q(z)=\frac{F}{EI}\left( \frac{Lz^{2}}{4}-\frac{z^{3}}{6}\right) \end{aligned}$$

(3.20)

and $u=q(L)$. Using ( 3.20), ( 3.19), and ( 3.16) we get:

$$\begin{aligned} v^{2}(z)=v^{2}\left( \frac{9z^{4}}{L^{4}}-\frac{12z^{5}}{L^{5}}+\frac{4z^{6}}{L^{6}}\right) \end{aligned}$$

(3.21)

which, inserted in ( 3.18), leads to:

$$\begin{aligned} m_{\mathrm{eq }}=\frac{13}{35}m_{\mathrm{b }} \end{aligned}$$

(3.22)

Finally, the effective spring mass $M_{\mathrm{eff }}$ is given by:

$$\begin{aligned} M_{\mathrm{eff }}=m_{\mathrm{sh }}+m_{\mathrm{eq }}=14.8171g \end{aligned}$$

(3.23)

1.6 Gravitational Stiffness

The gravitational component of the spring’s stiffness arises from the fact that an horizontal displacement $u$ of the shuttle is concurrent to a vertical parasitic motion $p$ given by Henein ([18], formula 5.13) as³:

$$\begin{aligned} p\approx \frac{3u^{2}}{5L} \end{aligned}$$

(3.24)

Considering that the shuttle undergoes a $p$ upward displacement while each beam’s mass center undergoes a $p/2$ vertical displacement, the gravitational stiffness $K_{{ grav}}$ is defined as follows ($m_{b}$ is the mass of the 2 cantilevers):

$$\begin{aligned} \frac{1}{2}K_{\mathrm{grav }}u^{2}=m_{\mathrm{sh }}gp+m_{\mathrm{b }}g\frac{p}{2} \end{aligned}$$

(3.25)

which together with (3.24) leads to (3.17).