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Effect of contact line dynamics on the thermocapillary motion of a droplet on an inclined plate George Karapetsas, Kirti Chandra Sahu, and Omar Kamal Matar Langmuir, Just Accepted Manuscript • DOI: 10.1021/la4014027 • Publication Date (Web): 20 Jun 2013 Downloaded from http://pubs.acs.org on June 23, 2013
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Effect of contact line dynamics on the thermocapillary motion of a droplet on an inclined plate George Karapetsas,∗,† Kirti Chandra Sahu,‡ and Omar K. Matar¶ Department of Mechanical Engineering, University of Thessaly, Volos 38334, Greece, Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Yeddumailaram 502 205, Andhra Pradesh, India, and Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK E-mail:
[email protected] Abstract We study the two-dimensional dynamics of a droplet on an inclined, non-isothermal solid substrate. We use lubrication theory to obtain a single evolution equation for the interface, which accounts for gravity, capillarity, and thermo-capillarity, brought about by the dependence of the surface tension on temperature. The contact line motion is modelled using a relation that couples the contact line speed to the difference between the dynamic and equilibrium contact angles. The latter are allowed to vary dynamically during the droplet motion through the dependence of the liquid-gas, liquid-solid, and solid-gas surface tensions on the local contact line temperature, thereby altering the local substrate wettability at the two edges ∗ To
whom correspondence should be addressed of Mechanical Engineering, University of Thessaly, Volos 38334, Greece ‡ Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Yeddumailaram 502 205, Andhra Pradesh, India ¶ Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK † Department
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of the drop. This is an important feature of our model, which distinguishes it from previous work wherein the contact angle was kept constant. We use finite-elements for the discretisation of all spatial derivatives and the implicit Euler method to advance the solution in time. A full parametric study is carried out in order to investigate the interplay between Marangoni stresses, induced by thermo-capillarity, gravity, and contact line dynamics in the presence of local wettability variations. Our results, which are generated for constant substrate temperature gradients, demonstrate that temperature-induced variations of the equilibrium contact angle give rise to complex dynamics. This includes enhanced spreading rates, non-monotonic dependence of the contact line speed on the applied substrate temperature gradient, as well as ‘stick-slip’ behaviour. The mechanisms underlying this dynamics are elucidated herein.
Introduction The motion of liquid droplets over liquid and solid substrates has attracted the interest of many researchers in the past because of its numerous practical applications and scientific challenges (see, for example, the reviews by de Gennes 1 and Bonn et al. 2 and references therein). It is well known that the application of a body force or external gradients can be used as a mechanism for driving the motion of liquid drops and the ability to control these properties can play a key role in many industrial applications that involve coating processes and microfluidic devices. In the present work, we focus on the migration of droplets on inclined solid substrates due to the presence of thermocapillary effects. A temperature gradient along the substrate causes another along the interface which may in turn induce surface tension gradients driving liquid flow from warmer to colder regions. Early experimental work by Bouasse 3 has shown that this effect can be used to force a drop to climb a tilted wire, against gravity, by heating its lower end. More recent studies on horizontal plates demonstrated that under certain conditions it was possible to get a steady migration of droplets with a fixed shape; for a temperature gradient below a certain threshold, the drop may not move due to the effect of contact angle hysteresis. 4,5 It appears, however, that the effect of contact angle hysteresis is not always important. Pratap 2 ACS Paragon Plus Environment
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et al. 6 performed experiments using decane drops on Polydimethylsiloxane (PDMS)-coated substrates and showed that the effect of contact angle hysteresis was much weaker. An interesting observation in the experiment of Pratap et al. 6 was the fact that there was a significant dependence of the contact angle on temperature and this was clearly demonstrated by the significant distortion of the footprint of the drop from a circular shape. Pratap et al. 6 have also reported that the experimentally measured migration velocity of the drop was not constant but decreases as the drop moves towards colder regions. They have attributed this effect mainly to the increasing viscosity of the drop, and secondarily to the reduction in drop size due to evaporation. However, since in their experiments there was a significant dependence of the contact angle on temperature, it is reasonable to ask whether the variation of the velocity can also be due to the change of wettability as the drop moves to colder regions. One should note that wettability gradients have been proven to be very efficient in driving flow inside liquids. 7,8 Daniel et al. 7 were able to achieve remarkably rapid movement of liquid drops by appropriate manipulation of temperature gradients which altered the wettability of the solid surface. One common factor between the experiments of Pratap et al. 6 and Daniel et al. 7 is that wettability and temperature gradients coexist. In the present study, the wettability gradients are not imposed externally but are a natural consequence of the variation of temperature along the solid surface. The thermocapillary motion of droplets has been the subject of several theoretical studies in the literature. Brochard 9 examined the motion of droplets in the presence of chemical or thermal gradients. She assumed that the shape of the drop is a wedge and employed force balance and energy arguments to deduce the wetting characteristics in terms of the spreading coefficient. Ford and Nadim 10 generalized the work of Brochard 9 to allow for arbitrary shapes of the drop, and also allowed the contact angles to be different at the two ends. Ehrhard and Davis 11 used lubrication theory to describe the spreading of a droplet on a uniformely heated plate, and Anderson and Davis 12 took into account the effect of evaporation. The latter effect was also studied recently by Karapetsas et al. 13 Chen and coworkers took into account the effect of buoyancy convection 14 and studied the phenomenon of thermocapillary nonwetting. 15
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Lubrication theory was also used by Smith 16 in the presence of thermal gradients to derive quasi-steady solutions employing a dynamic boundary condition at the contact line, which relates the velocity of the contact line to the dynamic contact angle, taking into account the effect of contact angle hysteresis. He showed that only two possible steady states exist: either a motionless drop, or a steady migration of the droplet with a fixed shape towards colder regions. Very recently, Gomba and Homsy 17 revisited this problem using lubrication theory in combination with a precursor model to relieve the contact line singularity. The profile of the droplet was allowed to change dynamically with time in the presence of a parametrically-varying constant contact angle. They were able to identify three different regimes depending on the contact angles. For small contact angles, the drop spreads with a capillary ridge whereas for large contact angles the drop translates with a fixed shape. For intermediate contact angles, they found a transition regime with rather complex dynamics involving break-up of the drop into smaller droplets. The results of Gomba and Homsy 17 elucidate the importance of the contact angle in the dynamics of the thermocapillary motion of droplets. The effect of contact line dynamics can be enhanced if we also take into account the effect of the variation of wettability due to the variation of temperature along the solid surface. All of the aforementioned works considered a constant contact angle along the substrate. It is important to investigate what would be the effect of a dynamically-varying contact angle which may have a significant impact in cases where there is a strong dependence of the surface energy of the solid substrate on temperature. To the best of our knowledge, this problem has not been addressed in the literature, and this will be one of the aims of this paper. In addition, we will also consider the case of an inclined surface to investigate the interplay of Marangoni stresses, contact line dynamics, and gravity. The rest of the paper is organized as follows. In section II, we describe the details of the derivation of the evolution equations for the drop profile and temperature, and the numerical method used for their numerical solution. Results are presented and discussed in section III, followed by concluding remarks in section IV.
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Problem formulation We consider the dynamics of a drop of an incompressible, Newtonian fluid with constant density ρ, viscosity µ, specific heat capacity C p and thermal conductivity λ , which has been deposited on a inclined, rigid and impermeable solid substrate subjected to a constant temperature gradient (see Fig. 1). The angle of inclination to the horizontal is denoted by α. The surface tensions of the liquid-gas, liquid-solid and solid-gas interfaces are σlg , σls and σsg , respectively. We assume that initially the drop has a maximal thickness H and a half-width L. In the present work, we consider the drop to be very thin and therefore L is assumed to greatly exceed H so that the drop aspect ratio, ε = H/L, is assumed to be very small. The latter assumption permits the use of lubrication theory, which will be employed below to derive a set of evolution equations that govern the spreading process.
Figure 1: Schematic diagram of the drop on an inclined plate (not to scale). Tw = To + γx, where γ is the constant temperature gradient applied at the solid wall and To is the temperature at (x, z) = (0, 0).
Governing equations We use a Cartesian coordinate system, (x, z), to model the dynamics and the velocity field, u = (u, w) where u and w correspond to the horizontal and vertical components of the velocity field, respectively. The liquid-gas interface is located at z = h(x,t) whereas the liquid-solid and the solidgas interfaces are located at z = 0. The spreading dynamics are governed by the equations of the
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conservation of mass, momentum and energy, given below:
∇ · u = 0,
(1)
ρ(ut + u · ∇u) + ∇p − µ∇2 u − ρg = 0,
(2)
ρC p (Tt + u · ∇T ) − λ ∇2 T = 0,
(3)
where p is the pressure and T is the temperature, while ∇ denotes the gradient operator. Unless stated otherwise, the subscripts x, z and t denote partial differentiation with respect to x, z and t, respectively, where t denotes time. Solutions of the above equations are obtained subject to the following boundary conditions. Along the free surface, the velocity field should satisfy a local force balance between surface tension and viscous stresses in the liquid, setting the pressure in the surrounding gas to zero (datum pressure) without loss of generality. Taking the tangential and normal to the free surface components of this force balance, we obtain
n · τ · t = t · ∇s σlg ,
(4)
n · τ · n = 2κσlg ,
(5)
where n = (−hx , 1)/(1 + h2x )1/2 and t = (1, hx )/(1 + h2x )1/2 denote the outward unit normal and unit tangential vectors on the interface, respectively, ∇s is the surface gradient operator, 2κ is the mean curvature of the free surface, defined as
2κ = −∇s · n,
(6)
∇s = (I − nn) · ∇,
(7)
and
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and τ is the total stress tensor: τ = −pI + µ(∇u + ∇uT ),
(8)
where I is the identity tensor. In addition, along the moving interface we impose the kinematic boundary condition,
ht + uhx = w,
(9)
and the following thermal flux condition
n · ∇T =
hg (T |h − Tg ), λ
(10)
where hg denotes the heat transfer coefficient at the liquid-gas interface and Tg is the temperature of the ambient gas; for simplification we will assume that Tg = To , where To = T (x = 0, z = 0). At the liquid-solid interface, we apply the usual no-penetration condition in the vertical direction: w = 0.
(11)
In the horizontal direction, the usual no-slip condition is replaced by the Navier slip condition 18 to avoid the stress singularity, which would otherwise arise at the moving contact line:
u = β uz ,
(12)
where β is a slip length. We also assume that temperature of the solid surface is fixed and given by the following expression: Tw = T |0 = To + γx,
(13)
where γ = dTw /dx is the value of the constant temperature gradient applied at the solid wall. To complete the description, a constitutive equation that describes the dependence of the inter-
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facial tensions on the temperature is required. To this end, we use a simple linear relation
σi = σi,o +
dσi (Ts − To ), (i = lg, ls, sg), dT
(14)
where Ts is the temperature at the corresponding interface, and σi,o , (i = lg, ls, sg) denotes the surface tension of all interfaces at the reference temperature, To . Scaling The governing equations and boundary conditions are made dimensionless using the following scalings (tildes denote dimensionless variables): ˜ t = L t˜, (u, w) = U(u, ˜ ε w), ˜ (x, z, h) = L(x, ˜ ε z˜, ε h), U µUL p= p, ˜ T = T˜ To , σi = σlg,o σ˜i , (i = lg, ls, sg), H2 where U = −
dσlg To dT µ
(15)
is a characteristic velocity. The dimensionless numbers that arise are the
Bond number Bo = ρgH 2 /µU, the Biot number Bi = hg H/λ , the dimensionless thermal gradient Γ = γL/To , and the dimensionless slip parameter B = β /H. The tildes are henceforth suppressed. Substitution of these scalings into the momentum and mass conservation governing equations and boundary conditions, using the lubrication approximation (ε
