Problem Description

This article examines methods for determining the droplet displacement that results from thermocapillary pumping in a closed microchannel. Thermocapillary pumping (TCP; see Figure 1) occurs when variations of surface tension and differences between contact angles at both ends of a droplet or liquid film contribute to an effective pressure difference across the liquid.

The droplet calculations described here have practical utility for the design and operation of microdevices, such as micro heat engines, actuators, sensors and lab-on-a-chip technology, that rely on accurate control of droplet motion within microchannels.

The droplet is positioned between two air pockets in a closed circular microchannel. There is a uniformly distributed cyclic heat source around the droplet. During heat input, thermocapillary forces induce fluid motion from left to right. Heat transfer to one end of the droplet leads to temperature variations within the liquid. The purpose of this article is to determine the displacement of a droplet, when subjected to a cyclic heating source.

Solution Procedure

The microchannel is modeled in two sections: the droplet/substrate region (on the left side) and the droplet/air region (on the right side). The two regions are assumed to be a quasi one-dimensional semi-infinite region. Each region is assumed to have its own heat source so that the heat transfer on the right and left sides are equal. The heat transfer between the substrate and the environment is neglected because only the axial direction is considered in this analysis. During the heating cycle, the pressure from the air on the right side of the channel opposes both the frictional force from within the droplet and the air force from the air pocket on the left side. The resultant force can be written as

Using the ideal gas law:

where A is the cross sectional area of the channel, m is the mass of air, R is the universal gas constant, V is the volume, and T is the temperature. The frictional force on the droplet is determined on the basis of the following reduced form of the Navier-Stokes equation, using the Poiseuille flow assumption:
The thermocapillary force on the droplet is then determined based on changes in the internal pressure of the droplet, as follows:

where G = 4 for a circular microchannel, and θ is the contact angle. The change of surface tension σ is estimated based on the temperature change across the droplet, σ = A – BT , where A = 0.07583N/m and B = 4.177×10-4 N/mK for water.

The net force on the droplet is then estimated as:

The change in pressure across the channel is the force per unit cross sectional area of channel:The velocity and displacement of the droplet can then be determined by integrating the equation of motion of the droplet, thereby yielding:

where mdroplet is the mass of the droplet, and u° and x° are the velocity and displacement at the previous time step.

Results and Discussion

The droplet motion in the channel is analyzed for a 16 μm microchannel, to observe the effects of heat input on the displacement of a droplet (see Table 1 for problem parameters). Figure 2 shows a comparison between the predicted results (based on the model in previous section) and the measured data. The speed of the droplet after the initial heat input to the channel is rapid, but decreases once the effect of back pressure from the opposite end of the closed channel increases. The initial speed of the droplet over a period of 2.3 seconds is about 13 μm/s. Once the droplet reaches a uniform velocity, the displacement is predicted accurately by the model.

However, during the transition period between the initial acceleration and the subsequent stage of uniform velocity, the model overpredicts the droplet displacement because the droplet motion has not yet reached a uniform velocity.

Conclusions

The difference in thermocapillary pressure across the droplet generates droplet motion within a closed microchannel. The effect of the pressure change can activate a membrane at the end of the microchannel, to generate electricity via piezoelectric materials. The displacement of the droplet shows that the microchannel also can be used as either a micro pump or an actuator.

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