Droplet generation in a microchannel with a controllable deformable wall

Abstract

We report the droplet generation behavior of a microfluidic droplet generator with a controllable deformable membrane wall using experiments and analytical model. The confinement at the droplet generation junction is controlled by using external pressure, which acts on the membrane, to generate droplets smaller than junction size (with other parameters fixed) and stable and monodispersed droplets even at higher capillary numbers. A non-dimensional parameter, i.e., controlling parameter K _p, is used to represent the membrane deformation characteristics due to the external pressure. We investigate the effect of the controlled membrane deformation (in terms of K _p), viscosity ratio λ and flow rate ratio r on the droplet size and mobility. A correlation is developed to predict droplet size in the controllable deformable microchannel in terms of the controlling parameter K _p, viscosity ratio λ and flow rate ratio r. Due to the deflection of the membrane wall, we demonstrate that the transition from the stable dripping regime to the unstable jetting regime is delayed to a higher capillary number Ca (as compared to rigid droplet generators), thus pushing the high throughput limit. The droplet generator also enables generation of droplets of sizes smaller than the junction size by adjusting the controlling parameter.

Appendix

In order to find an expression for Δh _max in terms of the local pressure gradient p _d(z) − p _c(z) across the membrane, we perform force balance on an infinitesimally small strip of the membrane wall of length dz, as shown in Fig. 12b. Depending upon the pressure gradient across the membrane, it will deflect in the direction of lesser pressure. In this case, here we are taking a case where pressure in the droplet channel is higher than in controlling channel, which results in the outward deflection of the membrane increasing the height of the channel. The force acting on the membrane due to the pressure difference across the membrane F _p is balanced by the restoring force F _r that holds the membrane on the bulk PDMS. The lateral component of this restoring force F _r cos α cancels out due to symmetry (as there is no lateral movement of the membrane). The vertical component of the force due to pressure difference F _p and the restoring force F _r sin α balance each other as follows,

$$\left[ {p_{\text{d}} \left( z \right) - p_{\text{c}} \left( z \right)} \right]\,\left( {w\,dz} \right) \approx - \sigma_{m} \,\left( {2\,dz\,t_{\text{m}} } \right)\sin \alpha = - \sigma_{m} \,\left( {2\,dz\,t_{\text{m}} } \right)\left. {\frac{\partial (\Delta h)}{\partial y}} \right|_{{y = {w \mathord{\left/ {\vphantom {w 2}} \right. \kern-0pt} 2}}}$$

(14)

Fig. 12

a Parabolic shape of deformed thin wall at a positive discharge condition. b Forces and c stresses acting on an infinitesimal strip of membrane wall

Here, the pressure force F _p is calculated as pressure p(z) times the elemental membrane area wdz and the restoring force F _r is calculated as the membrane stress σ _m time the cross section of the membrane around both the rims 2dzt _m (which is twice the membrane thickness t _m time the elemental length dz). For small angles α, the sine is approximated as the tangent which is the slope of the membrane at its edges and can be found by calculating the derivative of the deflection curve at the rim of the membrane as ∂ (Δh)/ ∂y| _y=w/2 = −4(Δh _max)/w. Next, by solving for the pressure p(z), we get

$$p_{\text{d}} (z) - p_{\text{c}} (z) = \frac{{8\sigma_{m} t_{\text{m}} \Delta h_{\hbox{max} } }}{{w^{2} }}$$

(15)

The stress in the membrane wall (Fig. 12c) is a combination of residual stress σ ₀ which may be already present when there is no deflection of the membrane and the stress σ _d due to Hooke’s law generated by the deflection of the membrane (Schomburg 2011). Assuming zero residual stress in the membrane, we get

$$\sigma_{m} = \sigma_{d}$$

(16)

The stresses due to membrane deflection σ _d can be calculated from the strains ɛ _y and ɛ _R in the transverse and radial directions, respectively (since, L ≫ w, the strain in the longitudinal direction is neglected). Now, according to Hooke’s law, the strains ɛ _y and ɛ _R can be expressed as

$$\varepsilon_{y} = \frac{1}{{E_{\text{m}} }}\left( {\sigma_{y} + \nu_{m} \sigma_{R} } \right)$$

(17)

$$\varepsilon_{R} = \frac{1}{{E_{\text{m}} }}\left( {\sigma_{R} + \nu_{m} \sigma_{y} } \right)$$

(18)

where ν _m and E _m denote Poisson’s ratio and Young’s modulus of the membrane, respectively. For thin membranes, transverse strain is assumed to be constant over the entire membrane (Schomburg 2011) which can be estimated by the extension of the membrane along the neutral fiber of the membrane. The length of the resulting parabola in the deflected state of the membrane is given as (Bronstein and Semendjajew 1976)

$$L_{pa} \approx w\,\left[ {1 + \frac{8}{3}\left( {\frac{{\Delta h_{\hbox{max} } }}{w}} \right)^{2} - \frac{32}{5}\left( {\frac{{\Delta h_{\hbox{max} } }}{w}} \right)^{4} } \right]$$

(19)

From the above expression of the extended length of the parabola and undeformed width of the membrane w and by using the assumption Δh _max ≪ y ₀, the transverse strain ɛ _y can be obtained as

$$\varepsilon {}_{y}\, \approx \frac{{L_{pa} - w}}{w} = \frac{8}{3}\left( {\frac{{\Delta h_{\hbox{max} } }}{w}} \right)^{2}$$

(20)

The fact that the radial strain is still unknown is resolved using one of the two assumptions (Schomburg 2011) (1) transverse and radial strains are equal throughout the membrane (2) transverse strain is zero throughout the membrane. The second assumption is invalid in the present case as the membrane has a deflection profile along the y-direction thus is subjected to nonzero transverse strain. Assuming that the tangential and radial strain are equal in magnitude throughout the membrane (transverse strain is tensile in nature but radial stress is compressive in nature, which results in ɛ _R  = −ɛ _y ), from Eqs. 17 and 18, we get,

$$\sigma_{y} = \varepsilon_{y} \left( {\frac{{E_{\text{m}} }}{{1 - \nu_{m} }}} \right) = \sigma_{m}$$

(21)

Finally, using Eqs. 15, 20 and 21, we get

$$p_{\text{d}} (z) - p_{\text{c}} (z) \approx \frac{64}{3}\,\frac{{t_{\text{m}} \,E_{\text{m}} }}{{w\,\left( {1 - \nu_{m} } \right)}}\,\left( {\frac{{\Delta h_{\hbox{max} } }}{w}} \right)^{3}$$

(22)

By rearranging the terms in the above equation, we get the expression relating ∆h _max with the pressure gradient across the membrane wall at the location, which is as follows.

$$\Delta h_{\hbox{max} } = \left( {\frac{{3\,w^{4} \,\left( {1 - \nu_{m} } \right)\,\left[ {p_{\text{d}} \left( z \right) - p_{\text{c}} \left( z \right)} \right]}}{{64\,t_{\text{m}} \,E_{\text{m}} }}} \right)^{{\frac{1}{3}}}$$