Molecular diffusion replaces capillary pumping in phase-change-driven nanopumps

Abstract

Inspired by the capillary-driven heat transfer devices, we present a phase-change-driven nanopump operating almost isothermally. Computational experiments on different-sized nanopumps revealed efficient operation of the pump despite the reduction in system size that extinguishes capillary pumping by annihilating the liquid meniscus structures. Measuring the density distribution of liquid near evaporating and condensing liquid/vapor interfaces, we discovered that phase-change-induced molecular-scale mass diffusion mechanism replaces the capillary pumping in the absence of meniscus structures as long as the liquid wets the walls of the capillary conduit. Therefore, proposed pumps can serve as a part of both nanoelectromechanical and microelectromechanical systems with similar working efficiencies.

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Acknowledgements

Authors thank Prof. BoHung Kim of University of Ulsan for the helpful discussions. Y.A. acknowledges the financial support of ASELSAN Inc. under scholarship program for postgraduate studies. Computations were carried out using high-performance computing facilities of Center for Scientific Computation at Southern Methodist University.

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Correspondence to Ali Beskok.

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Appendices

Appendix 1: Pumping performance prediction in continuum scale

The mass flow rate of a pressure-induced laminar liquid flow in a channel of height, h, and width, w, is given by the following:

$$\begin{aligned} \dot{m} = -\frac{h^3 w}{12 \nu } \, \frac{{\text {d}}p}{{\text {d}}x}, \end{aligned}$$
(1)

where \(\nu\) and p are kinematic viscosity and pressure of the liquid, respectively. If the flow is driven by a capillary pressure gradient resulting from the asymmetric menisci at both ends of the channel, pressure gradient can be approximated using Young–Laplace equation:

$$\begin{aligned} \frac{{\text {d}}p}{{\text {d}}x} \simeq \frac{\sigma }{l} \left( \frac{1}{R_{{\text {cond}}}} -\frac{1}{R_{{\text {evap}}}} \right) \ , \end{aligned}$$
(2)

where \(\sigma\) and l are the surface tension coefficient and length of the channel, respectively. Radius of curvature, R, can be written as the functions of apparent contact angle, \(\theta\), and channel height, h:

$$\begin{aligned} R= \frac{h/2}{\cos \theta } \ . \end{aligned}$$
(3)

Combination of Eqs. (1), (2), and (3) yields the mass flow rate, \(\dot{m}\), as the functions of apparent contact angles formed at the condenser and evaporator regions:

$$\begin{aligned} \dot{m} \simeq \frac{h^2 w}{6 \nu } \frac{\sigma }{l} \left[ \cos {{\theta }_{{\text {evap}}}} - \cos {{\theta }_{{\text {cond}}}} \right] \ . \end{aligned}$$
(4)

Maximum theoretical mass flow rate, \(\dot{m}_{{\text {max}}}^{{\text {therotical}}}\), which would be achieved if the heat conduction along the channel axis, and heat loss from side walls are zero, requires that all of the heat input is utilized to evaporate the liquid from interface:

$$\begin{aligned} \dot{m}_{\text {max}}^{\text {therotical}} = \frac{\dot{q}_{\text {evap}}}{h_{\text {fg}}} \ . \end{aligned}$$
(5)

The ratio of actual mass flow rate to the theoretically maximum one is a good measure of the performance of the pump:

$$\begin{aligned} \eta \equiv \frac{\dot{m}}{\dot{m}_{\text {max}}^{\text {therotical}}} \ . \end{aligned}$$
(6)

When Eqs. (4) and (5) are inserted to Eq. (6), performance of the pump can be demonstrated as the functions of both heat inputs and system geometry. After some algebraic manipulations, the pump performance can be shown as follows:

$$\begin{aligned} \eta \cong \frac{1}{12} \left( \frac{\dot{q}_{\text {evap}}/2ew}{\sigma h_{\text {fg}} / \nu } \right) ^{-1} \left( \frac{e}{h} \right) ^{-1} \left( \frac{l}{h}\right) ^{-1} \left[ \cos {{\theta }_{\text {evap}}} - \cos {{\theta }_{\text {cond}}} \right] , \end{aligned}$$
(7)

where e is the length of the heat addition region. Numerator of the second term at the right-hand side is simply the applied heat flux to the liquid. Denominator of this term is a group of physical properties of the fluid. For a system with small temperature variations, this group can be considered as constant and utilized to non-dimensionalize the heat flux. Thus, the second term at the right-hand side can be considered as non-dimensional heat flux, \(\dot{q}_{\text {evap}}''^{^*}\), applied to the system. The third and fourth terms, on the other hand, are scaled heat addition length, \(e^*\), and aspect ratio of the channel, \(l^*\), respectively. Therefore, Eq. (7) can be written in terms of these non-dimensional parameters as follows:

$$\begin{aligned} \eta \cong (\dot{q}_{\text {evap}}''^{^*})^{-1} \left[ \frac{\cos {{\theta }_{\text {cond}}} - \cos {{\theta }_{\text {evap}}} }{12e^* l^*} \right] . \end{aligned}$$
(8)

Equation (8) simply implies that if the applied heat flux and geometric similarity of the system are preserved, the pump should exhibit identical performance. However, this prediction is restricted with the continuum scale. When nanoscale effects are present, physical properties such as density (Ghorbanian et al. 2016) and viscosity (Vo et al. 2015) or geometrical parameters such as contact angle (Barisik and Beskok 2013) exhibit considerable variations, which can affect the pump performance.

Appendix 2: Uncertainty analysis

Density and velocity are sampled at every \(2\, \text {ns}\), and all the data collected between two measurements are averaged, which yields a certain measurement uncertainty, \(\varepsilon\), for each time averaged data, \(\langle ..\rangle\):

$$\begin{aligned} \rho= & \langle \rho \rangle \pm \varepsilon _{\rho } \end{aligned}$$
(9)
$$\begin{aligned} u= & \langle u \rangle \pm \varepsilon _{u} \ . \end{aligned}$$
(10)

Uncertainties associated with density (\(\varepsilon _{\rho }\)) and velocity (\(\varepsilon _{u}\)) are estimated by calculating the standard error of measurements, which is evaluated by dividing the standard deviation of measurements to the number of samples.

Mass flow is simply the product of mass flux and flow area. Estimation of mass flux has associated uncertainty due to the time averaging of density and velocity:

$$\begin{aligned} \rho u = \langle \rho u \rangle \pm \varepsilon _{\rho u} . \end{aligned}$$
(11)

Uncertainty of mass flux is expressed in terms of the uncertainties of density and velocities as follows:

$$\begin{aligned} \varepsilon _{\rho u} = \sqrt{(\varepsilon _{\rho } u)^2+(\varepsilon _{u} \rho )^2} . \end{aligned}$$
(12)

Multiplication of mass flux with flow area yields the mass flow:

$$\begin{aligned} \dot{m} = \langle \rho u \rangle A \pm \varepsilon _{\rho u}A, \end{aligned}$$
(13)

where flow area (A) is simply the product of channel height (h) and channel depth (w).

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Akkus, Y., Beskok, A. Molecular diffusion replaces capillary pumping in phase-change-driven nanopumps. Microfluid Nanofluid 23, 14 (2019). https://doi.org/10.1007/s10404-018-2185-7

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Keywords

  • Nanopump
  • Phase change
  • Molecular dynamics
  • Nanoscale fluid transport
  • Evaporating meniscus
  • Capillary pumping