Energy Dissipation Due to Viscosity During Deformation of a Capillary Surface Subject to Contact Angle Hysteresis

Physica B 435 (2014) 28–30

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Physica B

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Energy dissipation due to viscosity during deformation of a capillary surface subject to contact angle hysteresis

Bhagya Athukorallage n, Ram Iyer

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA article info abstract

Available online 25 October 2013 A capillary surface is the boundary between two immiscible fluids. When the two fluids are in contact Keywords: with a solid surface, there is a contact line. The physical phenomena that cause dissipation of energy Capillary surfaces during a motion of the contact line are hysteresis in the contact angle dynamics, and viscosity of the Contact angle hysteresis fluids involved. Viscous dissipation In this paper, we consider a simplified problem where a liquid and a gas are bounded between two Calculus of variations parallel plane surfaces with a capillary surface between the liquid–gas interface. The liquid–plane Two-point boundary value problem interface is considered to be non-ideal, which implies that the contact angle of the capillary surface at – Navier Stokes equation the interface is set-valued, and change in the contact angle exhibits hysteresis. We analyze a two-point boundary value problem for the fluid flow described by the Navier–Stokes and continuity equations, wherein a capillary surface with one contact angle is deformed to another with a different contact angle. The main contribution of this paper is that we show the existence of non-unique classical solutions to this problem, and numerically compute the dissipation. & 2013 Elsevier B.V. All rights reserved.

1. Introduction remains the same until a critical angle called the receding angle θr is reached. A capillary surface is the boundary between two immiscible Consider a liquid drop, which has a spherical cap shape, on fluids. When the two fluids are in contact with a solid surface, there a solid surface. Let V, R, and d be the volume of the drop, the radius is a contact line. The physical phenomena that cause dissipation of of the sphere, and the diameter of the disc which forms the energy during a motion of the contact line are hysteresis in the contact region, respectively. Let θ be the contact angle, and δp contact angle dynamics, and viscosity of the fluids involved. For a be the difference in pressure between the inside and outside of specific combination of fluids, these two phenomena might have the drop. We assume that the drop is small enough that the widely differing contribution to the total energy loss. In this paper, pressure inside the drop is uniform (that is, the effect of gravity is we start our investigation of contact line motion by studying the negligible). Then, we have the equations: the dissipation of energy due to viscosity when a capillary surface d 2γ 2πR3 is deformed from one shape to another due to the motion of the sin ðθÞ¼ ; δp ¼ ; V ¼ ð1 cos 3ðθÞÞ; 2R R 3 boundary. where γ is the surface tension of the liquid. Fig. 1 shows the variation of the contact angle θ with the diameter d, while Fig. 2 1.1. Contact angle hysteresis δ shows the variationR of ( p) with the volume V is. The integral δpdV computed on any path in Fig. 2 represents Consider a liquid droplet on a solid surface with a contact angle the work that is done either by or on the droplet. Thus, if the of θ. Experiments show that if the liquid is carefully added to the droplet begins with a certain contact diameter and contact angle droplet, the volume and contact angle of the droplet will increase at point A, and the volume is increased until point B is reached, without changing the diameter d of the contact disk until the followed by a decrease in volume to points C and D, and then an θ contact angle reaches a critical value a - called the advancing angle increase in volume back to point A, then the area of the hysteresis [1]. Similarly, if the liquid is removed from a droplet, volume and loop ABCDA is the net work that must be done by the droplet (or contact angle of the droplet decrease, but the contact diameter the external agent) against the surrounding to overcome contact angle hysteresis.

n The important point to note is that if the contact line does not Corresponding author. Tel.: þ1 806 317 4853. E-mail addresses: [email protected] (B. Athukorallage), move then there is no loss due to contact angle hysteresis. However, [email protected] (R. Iyer). there might be losses due to viscous dissipation.

height of the capillary surface f(x) is speciﬁed with respect to the depth where the liquid pressure equals the atmospheric pressure. Consider two plates of unit-width as in Fig. 3(a). The xz plane corresponds to the depth where the pressure in the liquid is equal to the atmospheric pressure. Let the liquid volume between the

xz plane and under the capillary surface be V0. Let ρ denote the γ ﬁ β ; β density of the liquid, denote its surface tension coef cient, 1 2 denote the relative adhesion coefﬁcients for the two walls, and g denotes the magnitude of gravitational acceleration. The necessary condition for the energy minimization [4] lead to the following systems:

f ″ðxÞ ρgf ðxÞγ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þλ ¼ 0in½0; L; ð1Þ θ ′ Fig. 1. Plot of the contact angle versus contact diameter d for a drop on a solid ð1þf 2ðxÞÞ3 surface. The advancing angle is θa, and the receding angle is θr. Z L f ðxÞ dx ¼ V 0 ð2Þ x 104 2 0

1.8 ′ð Þ ′ð Þ β þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 0 ¼ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif L β ¼ : ð Þ 1 0 2 0 3 ′ ′ 1.6 1þf 2ð0Þ 1þf 2ðLÞ 1.4

1.2

0.8

0.6

0.4

0.2 0 0.5 1 1.5 2 2.5 x 10−7

Fig. 2. Hysteresis curves for a liquid drop with R¼0.02 m. δp indicates the pressure difference between the liquid–gas interface of a liquid drop. Points B and C θ ¼ 1 θ ¼ 1 correspond to the advancing ( a 40 ) and receding ( d 10 ) contact angles of the droplet.

2. Model formulation for capillary surface

To make the problem amenable to analysis, we consider a simplified problem where a liquid and a gas are bounded between two parallel plane walls with a capillary surface between the liquid–gas interface. The relative importance of the phenomena causing dissipation of energy may be determined by analyzing this system. 0.99 Using calculus of variations, we obtain the mathematical model for a capillary surface at equilibrium, by minimizing the total 0.98 energy subject to a constant volume constraint [2–4]. In Ref. [3], the author considers a liquid drop between two vertical plates, and neglects the potential energy due gravity. The presence of gravity 0.97 makes the problem significantly different as it causes a pressure change with depth in the fluid – mathematically, Eqs. (2) and (3) 0.96 form a two point boundary value problem which is well-posed due y(cm) to the fact that g a0. 0.95 In our analysis, we consider a liquid meniscus that formed between two vertical plates, and assume the invariance of the 0.94 liquid surface in the z-direction (see Fig. 1(a)), which effectively makes the problem a two-dimensional one, and the walls to be hydrophilic. Furthermore, the potential energy due to the gravita- 0 0.02 0.04 0.06 0.08 0.1 tion is also included in our total energy functional. The capillary x(cm) surface f(x)defined over the interval ½0; L satisfies a second-order ordinary differential equation (1). Specification of the contact Fig. 3. (a) The y-axis is placed along the vertical plate, and the x-axis is perpendicular to the plates. Gravitational acceleration g, acts along the y direction. (b) Initial angles at the two walls (θ and θ ), which are related to f ′ð0Þ 1 2 (dashed) and final (solid-bold) capillary surface profiles. The intermediate curves ′ð Þ and f L , yields a two point boundary value problem that we solve (solid-thin) are computed by solving a two-point boundary value problem for the numerically using the modified simple shooting method [5]. The Navier–Stokes and continuity equations for N¼10 and ε¼5000. 30 B. Athukorallage, R. Iyer / Physica B 435 (2014) 28–30

The Lagrange multiplier λ may be calculated by integrating Eq. (1) Table 1 ð Þ¼ θ ð Þ¼ θ θ θ Viscous energy dissipation and initial kinetic energy variations with different ε and using f x 0 cot 1, f x L cot 2,where 1 and 2 denote the contact angles between the liquid meniscus and the plates 1 values. and 2, respectively. Hence, one can relate the value of β that is given i N ε D 10 4 (erg) KEini (erg) in the constraint equation (3) with the corresponding contact angle

θi, which yields to 10 5000 3.64 0.0210 7000 3.54 0.0201 β ¼ θ ; β ¼ θ : ð Þ 1 cos 1 2 cos 2 4

First, we obtain the initial capillary surface (yi(x)) by solving (1) with λ ¼ 0 together with the boundary conditions: f ′ð0Þ¼ which yields Z cot θ and f ′ðLÞ¼cot θ . We calculate the corresponding liquid 1 ∂ ð ; Þ 1 2 ð Þ¼ ð Þþ ð ; ð ; Þ; Þ y x t ð ; Þ : ð Þ yf x yi x v x y x t t ∂ u x t dt 7 volume V0 using Eq. (2), and it serves as the constraint on the 0 x volume for the subsequent deformations of the initial surface. The By using the series expansions of vðx; y; tÞ and y ðxÞy ðxÞ,werewrite second surface (y (x)) is numerically obtained by solving (1) and f i f Eq. (7) in the following form: (2) with the boundary conditions: f ð0Þ and f(L), where f ðLÞ¼y ðLÞ. i ! Due to the contact angle hysteresis phenomenon [1], the initial N π N π a0 þ ∑ k x ¼ c0 þ ∑ k x and final capillary surfaces have different contact angle values ak cos ck cos 2 k ¼ 1 L 2 k ¼ 1 L (and heights) at the walls 1 and 2. Z 1 α ð Þ N π þ 0 t þ ∑ α ð Þ k x ; k t cos dt 0 2 k ¼ 1 L R 1 where we assume that the term: ðπ=L cos ðπx=LÞyðx; tÞu ðtÞ 3. Analysis of the fluid flow leading to deformation of the 0 0 þ∂yðx; tÞ=∂xuðx; tÞÞ dt has the Fourier series c =2þ∑N c capillary surface 0 k ¼ 1 k cos ðkπx=LÞ. Then, the resulting system of equations is solved together fi α ð Þ Z : In this section, we solve for the initial velocity field of the liquid using a modi ed simple shooting method [5] to obtain k t , k 0 ðÞ fi ðÞ that takes an initial capillary surface yi to a nal one yf while – obeying the Navier Stokes and continuity equations. We consider 4. Numerical results and discussion a Newtonian, incompressible fluid, with dynamic viscosity μ, and ¼ð ð ; ; Þ; ð ; ; ÞÞ velocity u u x y t v x y t . As the liquid has zero velocity in The non-uniqueness in the solutions are due to (a) choice of the ¼ ¼ ð ; ; Þ¼ ð Þ ðπ = Þ the x direction at x 0 and x L, a choice u x y t u0 t sin x L form for uðx; y; tÞ, (b) choice of the form for vðx; y; tÞ, and (c) choice of leads to one class of solutions. ξðtÞ.Thefirst two reasons lead to theoretically non-unique solutions. – From the x-component of the N Sequation[6],wehave Even after the form of u and v are fixed, the last reason leads to non- ð Þ¼ μ=ρðπ2=L2Þt ðρ= Þ 2ð Þ ð π = Þπ= ¼∂ =∂ both u0 t e u0 and 2 u0 t sin 2 x L L p x. uniqueness in numerically computed solutions. This is because no fl ð ; ; Þ¼ðρ= Þ 2ð Þ Therefore, the pressure inside the uid is p x y t 4 u0 t matter what ξ is, Eq. (6) is always satisfied. However, the choice of ξ ð π = Þþ ð ; Þ cos 2 x L p y t . The continuity equation [6] yields can yield numerically different solutions unless one is careful in Z π π y selecting ε and the number of modes N for a solution. For discretiza- ð ; ; Þ¼ ð ; Þ ð Þ x ϕð Þ ; v x y t A x t u0 t cos s ds tion parameters δt and δx fortimeandthespatialcoordinate L L 0 respectively, we select N rL=10δx and 5N2ηrεr lnð0:5Þ=δt: For for some function ϕ. The choice ϕðsÞ¼1 yields one solution. Next, we δt ¼ 10 4 and δx ¼ 10 3, the choice of N and two different ε values apply techniques of Fourier analysis to the y-component of the N–S shown in the table (Table 1) above show similar results. It must be equation. Assume that Aðx; tÞ is given by noted that for a choice of ε ¼ 2000 and N¼10, which does not satisfy 1 the inequalities presented above, we found the kinetic energy to be α0ðtÞ kπx Aðx; tÞ¼ þ ∑ αkðtÞ cos ð5Þ KE¼0.0075 ergs, and the energy dissipated due to viscosity to be 2 L k ¼ 1 D ¼ 2:21 10 4 ergs. This shows that there exist numerically non- ξ over the interval ½L; L; where for each t,itisalsoassumedthat unique solutions depending on the choice of . Aðx; tÞ¼Aðx; tÞ.WesubstituteintheseriesforAðx; tÞ into the y The numerical results are shown in Fig. 3(b). Other numerical v component of the N–Sequationandequatetermsoneachsidefor solutions corresponding to different choices of u and needs to be each k.Fork¼0, we have done in the future.

α′ ðtÞ π π2 ∂ 0 ð Þα ð Þþ 2ð Þ ¼ 1 p : ð Þ u0 t 1 t u0 t y g 6 5. Conclusions 2 L L2 ρ ∂y

If, for some function ξ, the functions α0, α1, u0, and p satisfy In this paper, we investigated the dynamic motion of a capillary α′ ð Þ= ðπ= Þ ð Þα ð Þ¼ξð Þ ðπ2 2Þ 2ð Þ þ þ =ρ ∂ =∂ ¼ 0 t 2 L u0 t 1 t t and L u0 t y g 1 p y surface that forms between two vertical plates. We considered the ξðtÞ then they automatically satisfy (6). Thus, different choices Navier–Stokes and continuity equations to set up a two-point bound- of ξ lead to different solutions. We choose ξðtÞ¼εα0ðtÞ for ε40, ary value problem for fluid motion that deforms the capillary surface. as otherwise the y-component of the N–S equations lead to an We showed that there exist non-unique solutions to the problem. unstable system. Consider the boundary value problem defined by the initial and References fi ð Þ ð Þ the nal capillary surfaces yi(x)andyf(x), and suppose yf x yi x ¼ = þ∑N ð π = Þ a0 2 k ¼ 1ak cos k x L . By using the kinematic free surface [1] P.G. de Gennes, F. Brochard-Wyart, D. Quere, Capillarity and Wetting Phenomena: boundary condition, the meniscus profile at time t,thatisyðx; tÞ,may Drops, Bubbles, Pearls, Waves, Springer, 2003. be expressed using [2] R. Finn, Equilibrium Capillary Surfaces, Springer-Verlag, 1986. Z [3] T.I. Vogel, SIAM J. Appl. Math. 47 (3) (1987) 516. t ∂yðx; sÞ [4] I. Gelfand, S. Fomin, Calculus of Variations, Dover Publications, 2000. ð ; Þ¼ ð Þþ ð ð ; ð ; Þ; Þ ð ; ÞÞ [5] R. Holsapple, R. Venkataraman, D. Doman, J. Guid. Control Dyn. 27 (2004) 301. y x t yi x v x y x s s ∂ u x s ds 0 x [6] W. Deen, Analysis of Transport Phenomena, Oxford University Press, 1998.