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Interfaces: Adsorption, Reactions, Films, Forces, Measurement Techniques, Charge Transfer, Electrochemistry, Electrocatalysis, Energy Production and Storage
Ringlike Migration of a Droplet Propelled by an Omnidirectional Thermal Gradient Qingwen Dai, Wei Huang, Xiaolei Wang, and Michael M. Khonsari Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b04259 • Publication Date (Web): 12 Mar 2018 Downloaded from http://pubs.acs.org on March 12, 2018
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Langmuir
Ringlike Migration of a Droplet Propelled by an Omnidirectional Thermal Gradient
Qingwen Dai1, Wei Huang1, Xiaolei Wang1*, M.M. Khonsari2 1
College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing, 210016, China
2
Department of Mechanical and Industrial Engineering, Louisiana State University, 3283 Patrick Taylor Hall, Baton Rouge, LA 70803, USA
Abstract The interfacial phenomenon associated with the ringlike motion of a liquid droplet subjected to an omnidirectional thermal gradient is investigated. An experimentally verified model is proposed for estimating the droplet’s migration velocity. It is shown that the unbalanced interfacial tension acting on the liquid in the radial direction provides the necessary propulsion for the migration while the internal force acting on the adjoining liquid contributes to the equilibrium condition in the circumferential direction. This study puts forward the understanding of the interfacial spreading phenomenon, the knowledge of which is important in applications where the liquid lubricants are encountered with directionally unstable thermal gradients.
Keywords: Ringlike migration; Omnidirectional thermal gradient; Interfacial tension; Lubrication approximation
*
Author for correspondence: Xiaolei Wang, Tel/Fax: +86-25-84893630, e-mail:
[email protected] 1
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1. Introduction Thermocapillary migration of a droplet over a solid substrate is a part of the larger topic known as the Marangoni flows in which the surface tension due to both temperature and chemical composition at the interface provides the driving force. These flows have relevance to a large number of technological applications such as microfluidics, ink-jet printing, miniature rolling bearings and so on.1-11 Naturally, when dealing with thermocapillary action alone, it is the thermal gradient in the surface tension of a liquid that provides the driving force to induce propelling motion from the high temperature to low-temperature regions.12-14 This thermally-driven flow involves the interaction of three phases (solid-liquid-gas) through coupling by heat conduction with the substrate and convection within the droplet.15-17 The existing literature contains rich volumes of theoretical research devoted to the interfacial migration phenomenon.18-27 Among noteworthy and relevant contribution to the subject of this paper is the work of Brochard28 who developed theoretical models to describe the unidirectional migration of droplets with various cross-section shapes caused by either chemical or thermal gradients. Inspired by that work, Ford and Nadim29 presented a derivation for droplets with an arbitrary height profile, which takes the slip condition in the vicinity of the contact lines into consideration. Given the fact that the liquid viscosity is significantly affected by temperature, Dai et al.30 incorporated an appropriate viscosity-temperature relationship into the derivation. These studies provide valuable insights into the unidirectional migration phenomenon induced by the thermal gradient. Nevertheless, they are also suitable for evaluating the unidirectional motion of droplets induced by surfaces wettability gradient31-32 or vibration.33-34 In treating the unidirectional migration, it is typically assumed that the droplet has a 2-D arclike shape and that the thin-film lubrication approximation applies, or that it takes on the shape of
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a spherical-cap in a 3-D model.35 In applications where the liquid flow is driven by its proximity to a heat source, the temperature distribution on the surface can become heterogeneous and the thermal gradient directionally unstable, so that the migration direction becomes uncertain. Research reveals that under a unidirectional thermal gradient the droplet of paraffin oil tends to migrate from the high temperature to low-temperature regions, i.e. from left to right as shown in Fig. 1a.30 In the case of the omnidirectional thermal gradient, flow originates from a single heat source and temperature decreases uniformity in all directions. Accordingly, the droplet tends to migrate radially outward from the center to the surrounding regions, forming a ringlike motion as shown in Fig. 1b.
Fig. 1. Thermocapillary migration of paraffin oil droplets on the smooth surface induced by (a) unidirectional thermal gradient30 and (b) omnidirectional thermal gradient. Examination of the available literature indicates that in contrast to unidirectional migration, there are currently no suitable models available to describe the motion of omnidirectional migration. Hence, we report the results of an experimentally-verified model for predicting the ringlike migration of liquid droplets on a solid surface. The theoretical model is based on the thin-
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film lubrication flows and enables one to develop an effective approach for estimating the theoretical migration velocity.
2. Theoretical It has been proven that the motion of the droplet under a thermal gradient can be recognized as a balance of two forces acting on the droplet:28 the viscous resistance force (Fv) and the driving force (Fd), which is directly related to the gradients of the solid/liquid interfacial tension. Noting that the existing models for describing forces in unidirectional cases do not apply, we now proceed to derive the governing equations of the forces under an omnidirectional thermal gradient. Fig. 2 shows the geometrical configuration of a migrated droplet with the internal forces within the droplet. A cylindrical coordinate system (r, φ, z) is used to describe the flow in which (r,
φ) and z represent the horizontal and vertical coordinates, respectively. The following theoretical derivation does not hold for the early stages of spreading, i.e., the stages before the ringlike motion is formed since the very small amount of liquid remaining in the middle region is ignored. Our previous research30 has revealed that at high temperatures, the contact angles of paraffin oil droplets on the stainless steel surface are very small, so the difference of contact angles is ignored in this study. We assume that the droplet footprint is circular in shape and the height profile of the droplet is denoted by h(r). Since the width (L) and height (h) of the cross-section decrease with increasing the radius of ring-shape, the cross-section of the liquid in one z-r section will always change with time. Nevertheless, for a specified h, one can predict the migration velocity by assuming that the motion at each time interval is quasi-steady. The internal force (FInternal) acting on the adjoining liquid contributes to the inner balance, while the viscous resistance force (Fv) and the driving force
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(Fd) are deduced as follows.
Fig. 2. Side and plan views of the footprint of a droplet migrating on a solid surface.
2.1 Viscous resistance force
= Vφ 0,= Vz 0 , the Applying the thin-film lubrication approximation theory and assuming Navier-Stokes equation reduces to:
∂ 2Vr 1 ∂P = ∂z 2 µ ∂r
(1)
where Vr is the flow velocity in r direction, P is the pressure field, and µ is the dynamic viscosity,
µ=ρυ (ρ is the density and υ is the kinematic viscosity). The reference coordinate system is considered to be traveling with the droplet so that the substrate appears to move toward the center with a velocity U. The boundary conditions at z=h(r) and z=0 are:
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∂Vr 1 ∂γ , z h(r ) = = ∂z µ ∂r V = 0 r −U , z =
(2)
Integrating eq. 1, applying the boundary conditions, eq. 2, and solving for the velocity field, we have:
Vr (z) =
1 1 ∂P 2 ( z − 2 zh) − U C z + γ T T 2 ∂r µ
where γT is the surface tension coefficient, γ T =
(3)
∂γ , and the thermal gradient is assumed to be ∂T
∂T ∂x
constant, i.e.,= const = CT . Since no volumetric flux passes through the vertical cross-section of the droplet in the zdirection, we can write: h
∫ V ( x, y,z)dz = 0 0
r
(4)
Then, the pressure gradient satisfying this constraint is given by: ∂P 3µ 3 = − 2U+ CT γ T ∂r h 2h
(5)
The viscous stress σ zr at the liquid/solid interface (z= 0) is:
1 ∂u 3µ σ = µ = U − γ T CT 2 ∂z h
r = zr (z 0)=z 0
(6)
The hydrodynamic force (Fv) exerted by the solid surface is obtained by integrating the viscous stress: = Fv
2π
RB
0
RA
{σ } dA ∫ ∫ ∫∫= rz (z = 0)
A
σ rz rdrdφ
6πµ ( RB ) π = U − γ T CT ( RB 2 − RA 2 ) h 2
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(7)
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2.2 Driving force The Young’s equation defines the interfacial tension forces balance and contact angle in the vicinity of three-phase contact line:36
γ= γ SL + γ cos θ SG
(8)
Where θ is the equilibrium contact angle. The unbalanced interfacial tension forces subjected to an omnidirectional thermal gradient constitutes the driving force (fd) for per unit length is:
f d = ( γ SG − γ SL ) B − ( γ SG − γ SL ) A
(9)
Combining eqs. 8-9, we arrive at the following expression for the driving force per unit length:
= f d γ T CT ( RB − RA ) cos θ
(10)
Then, the driving force (Fd) exerted on the droplet is:
= Fd
∫
2π
0
f d ( RB − RA )dφ
= 2π cos θ ( RB − RA ) γ T CT 2
(11)
Note that the liquid viscosity (µ) is temperature-dependent, and drops rapidly with increasing T.37 The following viscosity-temperature equation is employed to describe this relationship:
µ (T ) = µ0 e−b(T −T0 )
(12)
where the µ(T) is the viscosity at the temperature T, µ0 is the viscosity at the reference temperature T0, and b is viscosity-temperature coefficient. Because the droplet film is extremely thin, the change of viscosity in the height direction, z, can be ignored. While in the radial direction, the viscosity is changed with increasing of RA and RB, and it can be considered as a function of the position r: µ ( x ) = µ0 e
− b (TS −CT r ) −T0
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(13)
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where TS is the temperature at the starting position. As mentioned above, the theoretical derivation is based on the assumption that the migration at each time interval is quasi-steady. To make the prediction more accurate, instead of substituting the viscosity-temperature equation into the integration formula (eq. 7) directly, the viscosity at the middle position of outside and inside diameters of the ring, i.e., at the position of r=(RA+RB)/2, is employed in the calculation. Using the forces balance Fv=Fd, and combining eqs. 7, 11 and 13, the migration velocity U can be derived as: U=
where µT = µ0 e
γ T CT 4 cos θ ( RB − RA ) + ( RB + RA ) Q A 12 µT ( RB + RA )
R + RA − b TS −CT A −T0 2
(14)
, Q (constant value, 5 µL in this study) is the volume of the droplet,
and A is the apparent area of the annulus liquid. As described in the experimental section, the area A and the position of RA and RB during a migration process are measured. Since the contact angle of the paraffin oils on the solid surface is very small (
