Foam-based microfluidics: experiments and modeling with lumped elements

Abstract

A new way of modeling unsteady flows through a deformable polymeric foam is proposed, based on dedicated lumped elements and elementary flow experiments. The electrical analogy commonly used in the modeling of channel microfluidics and biological flows is extended here with the aim to model transient flows in 2-D open-cell foams deformed by a normal stress externally applied. Lumped (foam) elements are defined in order to take account of both the deformability of open-cell foams and the mechanical coupling with internal liquid flows. Supporting experiments are numerically modeled, based on liquid flows generated in foam elements from external means: a syringe pump, a uniaxial compression or a peristaltic actuation.

Appendices

Appendix 1: The case of a pre-deformed foam element

One foam element of a foam-based system can be found to be pre-deformed under the local effect of a lateral containment. In such a case, its ability to deform under a time-dependent flow is modified. As illustrated in Fig. 13, the expansion of a foam element is typically limited by the presence of a plate placed above. Here, whatever the pressure inside the foam element may be, the latter is not able to expand beyond a threshold and its capacity stays constant. The foam element still absorbs or ejects fluid but within a prescribed range of accessible deformation.

Such a deformation-limited condition is not as straight forward to model as a threshold in pressure because the charge of the associated capacitor must remain constant as soon as a voltage (pressure) onset is reached: The model must take into account a range of strains without the need for updating the code. To program the condition of a deformation threshold with SPICE, there is the need to know the dependence of the deformation $\varepsilon$ on the pressure inside the foam, $P_{ \mathrm{foam} }$, and to evaluate what is the pressure $P_{ \mathrm{limit} }$ which corresponds to the prescribed deformation threshold, here referred to as $\lambda$.

The first step is to measure the stress–strain dependence of the foam element from a texture analyzer (see Fig. 1). By assuming the hysteresis effect to be negligible, one single stress/strain curve can be approximated by considering only the compression curve (see Appendix 2, Fig. 1). The dependence of the strain on the pressure applied upon the foam can be described from a curve fitting based on the following test function:

$$\begin{aligned} \varepsilon = g(P_{ \mathrm{foam} }) = \beta _{1} + \left( \beta _{2} + \beta _{1}\right) \cdot \left( \frac{\delta }{1+10^{\left( \psi _{1}-P_{ \mathrm{foam} }\right) h_{1}}} + \frac{1 - \delta }{1+10^{\left( \psi _{2}-P_{ \mathrm{foam} }\right) h_{2}}}\right) , \end{aligned}$$

(20)

with $\beta _{1}$, $\beta _{2}$, $\delta$, $\psi _{1}$, $\psi _{2}$, $h_{1}$ and $h_{2}$, a series of matching parameters to be adjusted. By making use of a least square method, for instance, a correlation coefficient as good as 0.98 can be achieved. Despite the fact it is not really based on a physical background, the previous test function of class $C{^1}$ is continuously differentiable, which prevents issues due to numerical instabilities (the use of piecewise functions must be prohibited).

The second step is related to the lumped model for the foam element (Fig. 4): The capacitor C1 is programmed as follows:

• if $P_{ \mathrm{foam} } < P_{ \mathrm{limit} }$ then $\varepsilon = g(P_{ \mathrm{foam} })$,

• if $P_{ \mathrm{foam} }$ $\ge$ $P_{ \mathrm{limit} }$ then $\varepsilon = \lambda$,

with $P_{ \mathrm{limit} }$, the pressure threshold defined as $P_{ \mathrm{limit} } = g^{-1}(\lambda )$ (see insert in Fig. 1). The stress–strain curve is considered as long as the pressure threshold $P_{ \mathrm{limit} }$ is above the pressure experienced by the foam element, $P_{ \mathrm{foam} }$. If $P_{ \mathrm{foam} }$ reaches the pressure threshold, the deformation $\epsilon$ is set to $\lambda$ even if $P_{ \mathrm{foam} }$ still increases. The capacity of the foam element is therefore capped.

Fig. 13

Deformation of a foam element imposed to a specified value $\lambda$ by an external plate

Appendix 2: Foam hysteresis

Among mechanical characteristics of the foam, a hysteresis feature is made evident when the foam recovers its initial shape after being compressed. Measurements of the stress–strain curve are conducted, in a way similar to the experiment depicted in Fig. 1, but this time, the maximum value of the deformation imposed, $\varepsilon _{\mathrm{max}}$, is made to change (Fig. 14). It is clearly established here that the curve measured during decompression stage strongly depends on $\varepsilon _{\mathrm{max}}$, while the compression curve remains unaffected. Decompression stage is therefore responsible for the hysteresis which is not taken into account by the present model only based on the compression curve.

For further improvement, a constitutive law, $\varepsilon =G({P_{\text {foam}}},\varepsilon _{\mathrm{max}})$, needs to be established in order to reproduce the dependency of the stress–strain characteristic on the parameter $\varepsilon _{\mathrm{max}}$. One key for future dynamic simulations could be the selection of the stress–strain curve depending on whether the foam is being compressed or relaxed. This could be achieved by considering the sign of the slope ${\left. {\frac{{\partial G}}{{\partial {P_{\mathrm{foam}}}}}} \right| _{{\varepsilon _{\max }}}}$ (compression: ${\left. {\frac{{\partial G}}{{\partial {P_{\mathrm{foam}}}}}} \right| _{{\varepsilon _{\max }}}}>0$; decompression: ${\left. {\frac{{\partial G}}{{\partial {P_{\mathrm{foam}}}}}} \right| _{{\varepsilon _{\max }}}}<0$).

Considering now Eq. (17), the magnitude of the capacity can be expected larger during the decompression stage than the one considered in this paper, only based on the compression stage (${\left. {\frac{{\partial g}}{{\partial {P_{\mathrm{foam}}}}}} \right| _{\mathrm{compression}}} < {\left. {\frac{{\partial g}}{{\partial {P_{\mathrm{foam}}}}}} \right| _{\mathrm{decompression}}}$). This marked discrepancy is most likely responsible for the symmetry breakup made evident on the peristaltic flow rate (Fig. 12).

Fig. 14

Strain–stress characteristics as measured for a series of $\varepsilon _{\mathrm{max}}$ values (insert : stress–strain characteristic; color online)

Appendix 3: Foam deformation as given from an auxiliary circuit

According to Eqs. (14–17), the deformation of a foam element can be derived as,

$$\begin{aligned} \varepsilon = \frac{1}{{{C_\mathrm{H}}}}\frac{{\mathrm{d}g}}{{\mathrm{d}{P_{\mathrm{foam}}}}}{q_\mathrm{e}}, \end{aligned}$$

where the electric charge accumulated by the capacitor in the lumped element is defined as: $q_\mathrm{e} = \int {I(C1)\mathrm{d}t}$ with I(C1), the current delivered by the capacitor. SPICE, as many other (free) RLC circuit simulators, disables access to the electric charge. To overcome this difficulty, we consider an auxiliary circuit only devoted to integrate the current I(C1) delivered by the capacitor (see insert in Fig. 15). A simple current generator I2 is connected to a resistor R3 whose value is set so as to depend on the current I(C1). The initial resistance of the auxiliary circuit, R3, is very small ($r_{0} = 1 f \Omega$) in order to prevent errors when starting the simulation. According to Eq. (16), the voltage of the resistor directly provides us with the charge of the capacitor provided that the shift due to $r_{0}$ is negligibly small, while the current generator I2 delivers a steady 1A current.

Fig. 15

Auxiliary circuit devoted to the measurement of the electric charge of the capacitor C1. Insert: The resistance of the resistor R3 is set so as to depend on the current of interest in the capacitor, I(C1). The function “idt(I(C1))” corresponds to a time integration of I(C1). The latter is normalized by a reference voltage V0 in order to get the mechanical deformation of the foam element