PHYSICAL REVIEW E 89, 032401 (2014)
Effect of charge regulation on the stability of electrolyte films Christiaan Ketelaar* and Vladimir S. Ajaev† Department of Mathematics, Southern Methodist University, Dallas, Texas 75275, USA (Received 24 May 2013; revised manuscript received 13 January 2014; published 10 March 2014) The stability of a thin liquid film of an electrolyte on a solid substrate is investigated. In the framework of the Debye-H¨uckel approximation, we show that the commonly used approximation of fixed potential at the solid-liquid interface does not lead to predictions of film rupture. To reconcile the model with experimental observations, we consider the constant charge density approximation for the solid substrate and then proceed to systematically investigate the effects of charge regulation based on a linear relationship between charge density and potential. Stability criteria are formulated in terms of charge regulation parameters and electrolyte properties, resulting in different types of stability diagrams. Critical thickness below which the film ruptures is shown to decrease as the charge regulation at the solid-liquid interface becomes stronger. DOI: 10.1103/PhysRevE.89.032401
PACS number(s): 68.15.+e, 47.20.Ma, 47.61.Fg
I. INTRODUCTION
Viscous flows in thin liquid layers have been studied both experimentally and theoretically by a number of authors, following the pioneering work of Reynolds on the theory of lubrication [1]. A comprehensive review of the literature on thin-film flows driven by gravity, capillarity, thermocapillarity, solutocapillarity, and disjoining pressure with or without phase change can be found in Oron et al. [2]. In a more recent review, Craster and Matar [3] discuss the latest developments in the studies of thin films. Investigations of stability of thin liquid films on solid substrates are important for a number of applications such as flotation [4], dynamics of the tear film in the eye [5], and microscale heat transfer [6,7]. Film instabilities often occur in regimes where inertia is negligible and other physical effects such as viscosity, surface tension, electrostatic forces, and London–van der Waals dispersion forces are significant. Instabilities can lead to film rupture when the deformable interface between the liquid and the gas phase above it touches the solid substrate, resulting in the formation of dry spots. A driving mechanism of rupture that has received widespread attention in the literature is due to the London–van der Waals dispersion forces which are significant only in very thin films, typically of thickness below 100 nm. Spontaneous rupture of such thin liquid films on a planar solid wall was studied experimentally by Scheludko [8] and theoretically by Ruckenstein and Jain [9]. In the latter study, linear stability criteria are established for a liquid layer under the action of London–van der Waals forces. Further refinements of the linear theory of [9], e.g., by including the electrostatic effects and viscoelastic properties of the film, are discussed in Chap. 8 of [10]. Nonlinear models of instability and rupture of thin liquid films are often based on the assumption that the film thickness is much smaller than the wavelength of the instability [2]. This lubrication-type approach has been used by several authors to derive general evolution equations governing the spatiotemporal behavior of thin liquid films. Williams and
* †
[email protected] [email protected]
1539-3755/2014/89(3)/032401(8)
Davis [11] and Burelbach et al. [12] have examined the nonlinear evolution of a liquid film on a solid substrate under the action of London–van der Waals forces and found that the nonlinearities of the system accelerate rupture. Zhang and Lister [13] have found similarity solutions near the point of van der Waals rupture of a thin film on a solid substrate and carried out simulations of nonlinear evolution of the film thickness. All three studies have shown that as the instability grows, the nonlinear behavior leads to hole opening in the film in a finite time. While theoretical studies usually focus on film rupture driven by the London–van der Waals dispersion forces, an alternative mechanism is often encountered in experiments and is due to electrostatic effects [14]. Many liquids (e.g., aqueous solutions) contain ions and electrical double layers form near charged interfaces. We note that the presence of electric charges not only at solid-liquid but also at liquid-air interfaces has been established in the studies of Graciaa et al. [15] and Li and Somasundaran [16]. Electric charges at interfaces can be controlled by adding supplementary electrodes within the solid structures as proposed by Steffes et al. [17]. In order to develop the stability criteria for an electrolyte film on a charged substrate, it is first important to understand the basic physics of the interaction of two static charged surfaces separated by a film of electrolyte. Early models of such interaction considered both surfaces to be flat and of uniform charge densities [18–22]. The result was based on solving the Debye-H¨uckel equation for the electrostatic potential in the film and usually represented in terms of interaction force or interaction energy between the surfaces. Application of the same ideas to films of aqueous solutions with a deforming interface can be found in several studies [23–26]. Both the Debye-H¨uckel and Poisson-Boltzmann equations have been solved and the results were compared with experiments on film drainage [27,28] and films in constrained vapor bubble experiments [23]. The results are usually represented as an additional term in the stress tensor in the film, referred to as the electrostatic component of disjoining pressure. It is important to note that the electrostatic interaction in all these studies is assumed to be repulsive and the films are stable. Since the electric charges at solid-liquid and liquid-gas interfaces usually appear as a result of chemical reactions at interfaces, they actually depend on the local electric field, a
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©2014 American Physical Society
CHRISTIAAN KETELAAR AND VLADIMIR S. AJAEV
PHYSICAL REVIEW E 89, 032401 (2014)
physical effect known as charge regulation. This effect was investigated in the context of interaction of biological cells [29] and amphoteric solid surfaces [30,31]. Carnie and Chan [32] derived a simplified model of charge regulation based on a linear relation between the surface charge density and electric potential and applied it to interaction of flat solid surfaces and spherical particles. Further studies of charge regulation were carried out by Netz [33], Periset-Camara et al. [34], and Gupta and Sharma [35]. Detailed comparison with experiments by Gupta and Sharma [35] showed that the effects of charge regulation are important for achieving reasonable agreement between experiment and theory for stresses in thin electrolyte films. Several authors applied the formulas for the electrostatic component of disjoining pressure to studies of stability of thin films of aqueous solutions on solid substrates (see [36] and the references therein) and of coating films inside a pipe [37]. However, in these studies only simple boundary conditions of either fixed charge or fixed potential at the interfaces are considered. The objective of the present study is to incorporate more realistic relations between the electric charge density and potential at the interfaces into the model of a thin liquid film on a substrate and to investigate both linear stability and strongly nonlinear evolution of the fluid interface. Of particular interest are the conditions of film rupture due to the effect of electric charges at the solid-liquid and liquid-gas interfaces. We also note that the assumption of fixed interfacial potential at a deforming interface, used, e.g., in Conroy et al. [37], implies no electrically induced tangential stress at the deforming interface. In the present work, we incorporate the effect of such tangential stress. The article is organized as follows. In Sec. II, we describe the model of an electrolyte film and in Sec. III we derive the interface shape evolution equation using a lubrication-type approach. The linearized version of the evolution equation is also discussed. In Sec. IV, we present the results of the stability analysis of the model with a fixed potential at the solid wall and a constant charge density at the liquid-gas interface and show that the film does not rupture under these conditions regardless of its thickness. In Sec. V, we present results of the linear and nonlinear stability analysis of the model with constant charge densities at both interfaces and show that the film can rupture in finite time. In Sec. VI we consider a more general model based on linear charge regulation at the substrate and investigate how the choice of regulation parameters affects rupture conditions. Transition to the limiting cases of constant charge density and constant potential is also discussed.
q* 2
gas
d
liquid electrolyte
q* 1
x*
solid
FIG. 1. Sketch of a thin film of liquid electrolyte of viscosity μ on a flat solid substrate.
where is the electric permittivity of the liquid, kB is the Boltzmann constant, and T is the temperature. The solid surface shown in Fig. 1 is electrically charged as a result of the chemical reactions of dissociation of surface groups. A linearized relation between the dimensional charge density q1∗ and the electrostatic potential ψ1∗ at the solid surface is written as q1∗ = S1 − K1 ψ1∗ ,
(2)
where S1 and K1 depend on the density of surface groups and the dissociation constants, as discussed by Carnie and Chan [32]. Here and below the subscript “1” refers to the solid surface, while the subscript “2” is reserved for quantities describing the liquid-air interface. The applicability of the linearized charge regulation model was discussed in [32] by comparison with nonlinear models for different electrolyte concentrations and parameter values corresponding to a range of solid surfaces such as SiO2 , Fe2 O3 , Al2 O3 , with the general conclusion that the error of the approximation is typically not more than a few percent. In the limit of a very large film thickness (d λD ), the electric potential of the solid surface, ψ¯ 1∗ , can be related to the charge density using the linearized Grahame equation [39]. Combining the latter with Eq. (2) leads to the formula ψ¯ 1∗ =
S1 K1 + λ−1 D
.
(3)
The origin of electric charges and the effects of charge regulation at liquid-air interfaces are not as well understood as for the solid surfaces in contact with liquid. While the assumption of constant electric charge density q2∗ at the liquidair interface is most likely appropriate for the majority of cases, we keep the formulation more general here by incorporating possible effects of linear charge regulation, so that q2∗ = S2 − K2 ψ2∗ .
(4)
Let us define the nondimensional charge densities and potentials by
II. FORMULATION
Consider a thin liquid film of viscosity μ and thickness d on a flat solid surface, as shown in Fig. 1. The surface tension at the air-liquid interface is σ . The liquid is an electrolyte with N different types of ions of valencies zk and bulk concentrations n(0) k (k = 1,2, . . . ,N). Using the standard definition of the Debye length [38,39], we can write kB T (1) λD = N (0) 2 , 2 e k=1 nk zk
y*
qi =
qi∗ (ψ ∗ ,ψi∗ ) , (ψ,ψi ) = , i = 1,2. S1 ψ¯ 1∗
(5)
Then Eq. (2) can be expressed in nondimensional terms as q1 = 1 −
C1 ψ1 , 1 + C1
(6)
where C1 = K1 λD / has the physical meaning of the ratio of the regulation capacitance of the surface to the electrical diffuse layer capacitance. A nondimensional version of (4) is
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EFFECT OF CHARGE REGULATION ON THE STABILITY . . .
written in the form ˆ + C2 ) − C2 ψ2 q(1 , 1 + C1
(7)
S2 (1 + C1 ) K2 λ D , C2 = . S1 (1 + C2 )
(8)
q2 = where qˆ =
To study linear and nonlinear stability of the film, we will consider perturbations of the base state corresponding to a film of uniform thickness d. The dimensional Cartesian coordinates along the solid and normal to it are denoted by x ∗ and y ∗ , respectively. Following the standard lubricationtype approach [40,41], applicable for the horizontal scale of interface deformations being much larger than the film thickness, and neglecting the effects of gravity, we introduce vertical and horizontal length scales defined by d and Ca−1/3 d, respectively, where Ca = μU/σ is the capillary number. The velocity and pressure scales, U =
3/2
(d) μσ 1/2
ψ¯ 1∗3 , λ3D
ψ¯ ∗2 P = 21 , λD
(10)
u∗ p∗ v∗ Ut∗ . (11) , p= , v= , t= 1/3 U P Ca U 3dCa−1/3 If the capillary number is small, the use of our scaling simplifies the Navier-Stokes equations to a system of lubrication-type equations while the electrostatic potential in the liquid satisfies the Debye-H¨uckel equation. Thus, the system of governing equation is of the form u=
px = uyy + ψψx ,
(12)
py = ψψy ,
(13)
ux + vy = 0,
(14)
ψyy = κ ψ,
d κ= . λD
(15)
Liquid flow in this model is coupled to the electric field through the electrostatic body force term in the momentum balance as well as through the interfacial boundary conditions discussed below. The scaled components of this force, ψψx and ψψy , appear on the right-hand sides of Eqs. (12) and (13), respectively. At the solid wall (y = 0) we apply the no-slip and nopenetration boundary conditions for the flow. At the interface, y = h(x,t), the normal stress boundary condition includes the contribution from the Maxwell stress tensor, p0 − p = hxx −
1 ψy2 , 2 κ2
where p0 is the scaled atmospheric pressure. Since we are interested in situations when film dynamics is dominated by electrostatic effects, other possible contributions to interfacial stress balances, e.g., due to London–van der Waals dispersion forces, are neglected. The liquid shear stress at the interface is proportional to the tangential electric field component, leading to the second interfacial stress condition of the form κ 2 uy + ψy (ψx + ψy hx ) = 0.
(16)
(17)
The kinematic boundary condition at the liquid-air interface is 1 h 3 t
+ uhx = v.
(18)
Application of the divergence theorem for the electric field allows one to obtain relationships between the local charge density and the derivative of the potential at each interface. Together with Eqs. (6) and (7), these relationships lead to the following boundary conditions for the electric potential,
(9)
are obtained by considering the balance between the electrostatic, capillary, and viscous contributions to the stresses in the film as it deforms. The dimensional horizontal and vertical velocity components u∗ and v ∗ , the pressure p∗ , and time t ∗ can now be used to define the corresponding nondimensional variables according to
2
PHYSICAL REVIEW E 89, 032401 (2014)
ψy |y=0 = −κ(1 + C1 − C1 ψ1 ),
(19)
ˆ + C2 ) − C2 ψ2 ]. ψy |y=h = κ[q(1
(20)
We note that these equations do not include contributions from the electric field outside the film since we assume that relative dielectric permittivity of the electrolyte r is large, i.e., effectively take the limit of r → ∞. The commonly used boundary conditions of constant potential and constant charge densities are the special cases of Eqs. (19) and (20). The constant potential corresponds to the limit of Ci → ∞ (i = 1,2), while the constant charge density condition corresponds to Ci → 0. Thus, it is the relative magnitude of the regulation capacitance to the diffuse layer capacitance that controls whether an interface behaves as a constant potential, constant charge density, or chargeregulating surface. Following Carnie and Chan [32], we characterize the linear charge regulation in terms of the nondimensional parameters −1 i 1 defined by i =
Ci − 1 , Ci + 1
i = 1,2.
(21)
Note that the constant potential limit then corresponds to i = 1, while the constant charge limit corresponds to i = −1. III. NONDIMENSIONAL EVOLUTION EQUATION
Solving Eq. (15) subject to the boundary conditions (19) and (20) leads to the following expression for the electric potential in the film, ˆ −κy ψ = (eκh − 1 2 e−κh )−1 [(eκh − 1 q)e + (qˆ − 2 e−κh )eκy ].
(22)
Note that for a film of uniform thickness (h = 1) this equation reduces to the nondimensional version of the formula obtained by Carnie and Chan [32] for the case of two flat surfaces with charge regulation. We consider the case of constant charge density at the liquid-air interface, i.e., assume 2 = −1, for the rest of the present article, since the effects of charge regulation at the deforming interface are not expected to be significant. We
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verified that adding some realistic charge regulation correction at that interface does not alter the main conclusions of the present study, while the formulas become significantly more cumbersome. Using Eq. (22) with 2 = −1, the potential ψ2 at the liquid-air interface can be expressed as a function of the interface shape, ψ2 =
ˆ κh − 1 e−κh ) 2 + q(e . eκh + 1 e−κh
(24)
Combining the horizontal velocity profile given by (24) with (14), (16), and the kinematic boundary condition (18), we obtain ˆ 2 (qh ˆ + κ −1 ψ2 )x x = 0. (25) ht + h3 hxx + 12 ψ22 x − 32 qh Note that while this equation only involves the potential at the deforming interface, the film evolution depends on the electrical properties of both interfaces since ψ2 comes out of the solution of the Debye-H¨uckel equation. Note that while Eq. (25) is nonlinear, it has a simple steadystate solution corresponding to a uniform film thickness, h(x,t) = 1. Behavior of a small perturbation ζ (x,t) to this solution is described by the linearized version of (25), ζt + ζxxxx + Gζxx = 0, where G≡
3 3qˆ dψ2 − qˆ 2 ψ2 − 2κ dh 2 h=1
G2 , 4 and is reached at the wavelength √ 2 2π λ= √ . G
(28)
(29)
(30)
Substituting the solution for the electric potential given by (23) into the definition of G given by (27), we can rewrite the instability condition in the form Aqˆ 2 + B qˆ + C > 0,
B = −κ
(31)
where the coefficients A, B, and C are functions of the scaled film thickness κ and the charge regulation
1 2
(eκ − 1 e−κ )2 − 21
(32)
+ 34 (eκ − 1 e−κ )(eκ + 1 e−κ ) = 0, C = −κ(eκ − 1 e−κ ).
(33) (34)
ˆ plane for a fixed 1 are given The stability branches in the q-κ by √ −B ± B 2 − 4AC ˆ , (35) q(κ) = 2A as long as this equation has real solutions; i.e., the discriminant B 2 − 4AC is nonnegative. IV. FIXED SUBSTRATE POTENTIAL
If we consider a constant potential at the solid wall (1 = 1), the formula for the potential at the liquid-air interface, Eq. (23), reduces to ψ2 =
1 + qˆ sinh(κh) . cosh(κh)
(36)
The coefficients A, B, and C defined by Eqs. (32)–(34) simplify to
(27)
Perturbations of all wave numbers decay for G < 0, indicating stability. However, the film is unstable when the condition G > 0 is satisfied. The growth rate of the fastest instability mode then corresponds to the maximum value of the function γ (k), equal to γmax =
− 38 (eκ + 1 e−κ )3 ,
(26)
and ψ2 is now treated as a function of a single variable h, as defined by Eq. (23). Based on Eq. (26), the growth rate γ of a small sinusoidal perturbation of a wave number k is given by γ (k) = k 2 (G − k 2 ).
A = 1 κ(eκ − 1 e−κ ) − 32 1 (eκ + 1 e−κ )
(23)
From Eq. (13), we observe that p − ψ 2 /2 is a function independent of y. The horizontal flow velocity is then found by integrating Eq. (12) with the boundary condition (17) and the no-slip condition at the wall, resulting in ˆ qh ˆ x + κ −1 ψ2x )y. u = 12 (px − ψψx )(y 2 − 2yh) − q(
parameter 1 :
A = 2κ sinh κ − 3 cosh κ − 3 cosh3 κ,
(37)
B = −2κ(sinh2 κ − 1) + 3 sinh κ cosh κ,
(38)
C = −2κ sinh κ.
(39)
Remarkably, when these expressions are substituted into the condition (31), we find that the electrolyte film is stable for all values of the scaled film thickness since the function on the left-hand side of (31) is always negative. Thus, even though the assumptions of fixed substrate potential and fixed charge density at the fluid interface are often mentioned as a natural framework for studies of electrolyte films on solid substrates [36], we find that a model based on these assumptions fails to predict film break-up by growth of interfacial perturbations observed, e.g., in [14]. While film stability at large values of κ can be explained by the lack of sufficient overlap between the electrical double layers of the two interfaces, it is rather puzzling that no rupture threshold is found at small values of ˆ To gain deeper physical insight κ regardless of the value of q. into the behavior of perturbations of the film surface at small κ, let us consider the limiting case of κ = 0. When the fluid interface is flat, the configuration is that of a capacitor with the uniform electric field confined between the two interfaces. The electrostatic energy in the liquid then has constant density, denoted by E. Now let us consider the effect of a small interfacial perturbation ζ . In the framework of the linear approximation in ζ , the electric field remains uniform in the domain 0 y ζ with the same energy density. The total change in the electrostatic part of the system’s energy due to
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PHYSICAL REVIEW E 89, 032401 (2014)
perturbation is
15
∞ −∞
Eζ dx = E
∞
−∞
ζ dx = 0,
(40)
where we used the condition of global mass conservation for the incompressible liquid in the film. Thus, there is no energy reduction due to electrostatic effect needed to overcome the stabilizing effect of surface tension despite the fact that the two interfaces are oppositely charged. One of the limitations of our model is the assumption of fixed charge density at the fluid interface. It is natural to ask whether the conclusions of the present section are still valid when the effect of interfacial charge transport is incorporated into the model by replacing the fixed value of qˆ with the ˜ evolving scaled charge density q(x,t). The standard interfacial charge transport equation (discussed, e.g., in [37]) for the new variable q˜ has to be solved together with the evolution equation for film thickness. Thus, the film dynamics is governed by a system of two coupled partial differential equations. However, when the system is linearized, the leading-order stability criteria turn out to be unaffected by the interfacial charge transport. Thus, the conclusions of the present section remain valid. In order to explain experimentally observed rupture phenomena, in the next two sections we modify the assumption of fixed solid substrate potential. It is first replaced by a condition of constant charge density, followed by a discussion of a more general charge regulation condition. V. FIXED SUBSTRATE CHARGE
Let us consider another commonly used model of interacting charged interfaces in which the effects of charge regulation are completely neglected by assuming the dimensional charge densities to be constant, equal to S1 and S2 at the lower and upper boundaries of the film, respectively. In our nondimensional formulation this implies setting 1 = −1 in Eq. (23). The resulting expression for the potential at the liquid-air interface can then be written as ψ2 =
qˆ cosh(κh) + 1 . sinh(κh)
(41)
Note that qˆ in this case has the simple meaning of the ratio of charge densities at the two interfaces. The general coefficients defined by Eqs. (32)–(34) simplify to A = −2κ cosh κ + 3 sinh κ − 3 sinh3 κ,
(42)
B = −2κ(cosh2 κ + 1) + 3 sinh κ cosh κ,
(43)
C = −2κ cosh κ.
(44)
The stability diagram based on this equation is shown in Fig. 2. We note that accounting for tangential stresses at the interface, as is done in the models of thermocapillary and solutocapillary instabilities, is essential for the correct description of film dynamics. For example, in the limit of κ → 0, the electrostatic contribution to the pressure in the film has a stabilizing effect, so the instability is entirely due to the tangential electric field which arises when a small perturbation of the initially flat interface is introduced.
unstable
10
5
stable qˆ
0
−5
unstable −10
−15
0
0.1
0.2
0.3
0.4
0.5
κ FIG. 2. Linear stability diagram for a model with constant charge densities at both interfaces (1 = 2 = −1).
According to our stability criteria, the film is always stable if its scaled thickness κ > 0.530 12, which is consistent with experimental results showing that instability is generally not expected if there is no significant overlap between the interfacial electric double layers. Under these conditions, the film is not destabilized even for large values of the parameter ˆ in fact, the asymptotic limit of large qˆ corresponds to the q; critical value of the scaled thickness of 0.5051 (the value at which the coefficient A passes through zero). The diagram in Fig. 2 indicates that the film can be linearly unstable at both negative and positive values of the charge ˆ While instability at negative qˆ can density ratio parameter, q. be easily explained in terms of attraction of oppositely charged interfaces, the prediction of linear instability at positive qˆ may seem rather surprising. To better understand this result, we note that it can be interpreted as a manifestation of the wellknown phenomenon of the instability of a charged deformable interface in an electric field, which in our case is generated by the charges at the substrate and in the liquid rather than external electrodes (as is often done in experiments showing this type of instability). This phenomenon has been observed and described under a wide range of conditions, including different electrical conductivities of fluids separated by the charged interface (see, e.g., [42] and references therein). Another way to interpret our result is to point out that the deformable interface interacts not only with the charge at the solid surface but also with the cloud of charge in the solution, so at least some of these interactions always involve charges of opposite signs. The linear stability model breaks down as soon as the amplitude of the perturbation of the base state is no longer much smaller than unity. To investigate perturbation dynamics in this regime and make conclusions about the possibility of film rupture, we solved the full nonlinear Eq. (25) with ψ2 from (41) numerically using the method of lines with time stepping by Gear’s BDF method. As is common in film rupture studies [13], the size of the computational domain L is dictated by the wavelength of the fastest growing perturbation: we
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1.2
1.2
t=0.3 t=0.5 t=0.6
1.15
1
1.1 0.8
1.05
h
a
1
0.6
0.95 0.4
0.9 0.2
0.85 0.8
0
0.2
0.4
0.6
0.8
0
1
x/L FIG. 3. Snapshots of the interface shape for κ = 0.4, qˆ = −2.
choose L = λ/2, where λ is given by Eq. (30). The boundary conditions are hx (0,t) = hx (L,t) = 0,
(45)
hxxx (0,t) = hxxx (L,t) = 0,
(46)
and the initial perturbation is sinusoidal, of the amplitude ζ0 = 10−2 . Note that the nonlinear system considered here depends on two nondimensional parameters, the ratio of the linearized surface charge densities qˆ and the film thickness scaled by the Debye length, κ. The plot in Fig. 3 shows snapshots of the interface at different times for κ = 0.4 and qˆ = −2. Clearly, the interface shapes are no longer sinusoidal due to the nonlinear effects and the film evolution tends to speed up. In order to better understand the time evolution of the liquid-air interface, it is convenient to plot the minimum film thickness (always reached at x = 0 with our choice of the initial condition) as a function of time, in nondimensional terms. The linear stability theory discussed at the end of Sec. III predicts that the scaled minimum thickness, denoted by a, should evolve according to a = 1 − ζ0 e G
2
t/4
.
(47)
This result is represented by a dot-dashed line in Fig. 4. The solid line, corresponding to the numerical solution of the nonlinear evolution equation (25), follows the predictions of the linear stability analysis initially, but then deviates from it significantly. The film ruptures in a finite time in a manner similar to the well-known case of the van der Waals driven rupture [12]. We carried out similar studies for several different values of qˆ and observed the same dynamics. Thus, the model described in the present section is indeed capable of describing not only the linear destabilization of the liquid film but also its rupture in a finite time.
0
0.1
0.2
0.3
t
0.4
0.5
0.6
0.7
FIG. 4. Evolution of the scaled minimum thickness (denoted by a) for κ = 0.4, qˆ = −2. Solid line is obtained from the numerical solution of Eq. (25); dot-dashed line is the prediction of the linear stability theory. VI. EFFECT OF CHARGE REGULATION AT THE SOLID WALL
The stability results in the previous two sections clearly show qualitatively different behavior, but the only difference between the models is the degree of charge regulation at the solid surface, from perfect regulation to no regulation. It is natural to ask how the degree of charge regulation affects the stability of the film. Addressing this issue is the objective of the present section. If the charge regulation effects are neglected, i.e., 1 = −1, the stability diagram is that shown in Fig. 2. As the value of the regulation parameter is increased slightly, both stability branches are moving downward in the stability diagram, but the overall picture remains qualitatively the same. At the values of 1 near −0.5, the lower stability branch starts shifting downward significantly, eventually disappearing to infinity as 1 approaches zero. Typical stages of this process are shown in Fig. 5. The upper stability branch starts deforming so that the region of instability shrinks gradually until in disappears completely in the regime of perfect charge regulation, i.e., constant potential. This corresponds to the limiting case of absolute stability that was discussed in Sec. IV. Comparison between stability diagrams such as the ones ˆ the shown in Fig. 5 shows that for a fixed value of q, values of κ corresponding to the instability threshold are decreasing with increase in 1 . Thus, the critical dimensional thickness for the onset of instability for a given electrolyte properties will decrease as the strength of charge regulation is increased. VII. CONCLUSIONS
We investigated the stability of a thin electrolyte film on a solid substrate. Both normal and tangential stress conditions at the film surface include contributions from the electric field. A linear stability analysis under the assumption of fixed substrate
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8
8
6
6
unstable
4
2
qˆ
qˆ
2 0
stable
−2
0
stable
−2 −4
−4
unstable
−6 −8
unstable
4
0
0.1
−6
0.2
κ
0.3
0.4
0.5
−8
unstable 0
0.05
0.1
0.15
0.2
κ
0.25
0.3
0.35
0.4
FIG. 5. Stability regions for 1 = −0.5 (left side) and 1 = −0.3 (right side).
potential shows that the film is always stable. A simple physical interpretation of this conclusion is provided by considering the change in the electrostatic energy as a result of interfacial perturbation of an initially flat film in the limiting case when film thickness is much smaller than the Debye length of the electrolyte solution. In order to describe the experimentally observed rupture by electrostatic effects, we considered the approximation of fixed charge density at the solid substrate and then more general models incorporating the effects of interfacial charge regulation. The latter is characterized by a nondimensional parameter 1 which depends on the ratio of the solid surface regulation capacitance to the capacitance of the diffuse layer formed in the liquid near the solid. We showed that ˆ variables different types of linear stability diagrams in q-κ are possible depending on the value of 1 (here qˆ is a nondimensional parameter which characterizes the ratio of
electrostatic properties of the two interfaces and κ is the film thickness scaled by the Debye length). The region of instability shrinks as the value of 1 is increased, in accordance with the conclusion of stability under constant substrate potential, i.e., perfect charge regulation. We carried out nonlinear simulations of the film evolution for the case of constant charge densities and observed that linear destabilization of the interface eventually leads to film rupture, in qualitative agreement with experimental data on rupture of electrolyte films.
[1] O. Reynolds, Proc. R. Soc. London 40, 191 (1886). [2] A. Oron, S. H. Davis, and S. G. Bankoff, Rev. Mod. Phys. 69, 931 (1997). [3] R. V. Craster and O. K. Matar, Rev. Mod. Phys. 81, 1131 (2009). [4] J. Israelachvili, Intermolecular and Surface Forces (Academic Press, London, 2011). [5] R. Braun, Annu. Rev. Fluid Mech. 44, 267 (2012). [6] O. Kabov, D. Zaitsev, V. Cheverda, and A. Bar-Cohen, Exp. Therm. Fluid Sci. 35, 825 (2011). [7] D. Zaitsev, M. Lozano Aviles, H. Auracher, and O. Kabov, Microgravity Sci. Technol. 19, 71 (2007). [8] A. Scheludko, Proc. K. Ned. Akad. Wet. B 65, 76 (1962). [9] E. Ruckenstein and R. K. Jain, J. Chem. Soc. Faraday Trans. 70, 132 (1974). [10] I. Ivanov, Thin Liquid Films: Fundametals and Applications (Dekker, New York, 1988).
[11] M. Williams and S. Davis, J. Colloid Interface Sci. 90, 220 (1982). [12] J. Burelbach, S. G. Bankoff, and S. H. Davis, J. Fluid Mech. 195, 463 (1988). [13] W. Zhang and J. Lister, Phys. Fluids 11, 2454 (1999). [14] H. Schulze, K. St¨ockelhuber, and A. Wenger, Colloids and Surfaces A: Physicochem. Eng. Aspects 192, 61 (2001). [15] A. Graciaa, G. Morel, P. Saulner, J. Lachaise, and R. Schechter, J. Colloid Interface Sci. 172, 131 (1995). [16] C. Li and P. Somasundaran, J. Colloid Interface Sci. 146, 215 (1991). [17] C. Steffes, T. Baier, and S. Hardt, Colloids and Surfaces A 376, 85 (2011). [18] B. Derjaguin, N. Churaev, and V. Muller, Surface Forces (Springer, New York, 1987). [19] E. Verwey and J. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948).
ACKNOWLEDGMENTS
This work was supported by the National Science Foundation under Grant No. CBET-0854318. We thank the anonymous referees for a number of useful suggestions.
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CHRISTIAAN KETELAAR AND VLADIMIR S. AJAEV [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
PHYSICAL REVIEW E 89, 032401 (2014)
H. Kruyt, Colloid Science (Elsevier, Amsterdam, 1953). V. A. Parsegian and D. Gingell, Biophys. J. 12, 1192 (1972). J. Gregory, J. Colloid Interface Sci. 51, 44 (1975). R. Mazzoco and P. Wayner, J. Colloid Interface Sci. 214, 156 (1999). D. Chan, R. Dagastine, and L. White, J. Colloid Interface Sci. 236, 141 (2001). R. Pushkarova and R. Horn, Langmuir 24, 8726 (2008). B. Derjaguin and N. Churaev, J. Colloid Interface Sci. 49, 249 (1974). D. Hewitt, D. Fornasiero, J. Ralston, and L. R. Fisher, J. Chem. Soc., Faraday Trans. 89, 817 (1993). R. Manica and D. Chan, Phys. Chem. Chem. Phys. 13, 1434 (2011). B. Ninham and V. A. Parsegian, J. Theor. Biol. 31, 405 (1971). D. Chan, J. Perram, R. White, and T. Healy, J. Chem. Soc., Faraday Trans. 71, 1046 (1975). D. Chan, T. Healy, and L. White, J. Chem. Soc., Faraday Trans. 72, 2844 (1976).
[32] S. Carnie and D. Chan, J. Colloid Interface Sci. 161, 260 (1993). [33] R. Netz, J. Phys. Condens. Matter 15, S239 (2003). [34] R. Pericet-Camara, G. Papastavrou, S. Behrens, and M. Borkovec, J. Phys. Chem. B 108, 19467 (2004). [35] A. Gupta and M. Sharma, J. Colloid Interface Sci. 149, 407 (1992). [36] N. Churaev, Adv. Colloid Interface Sci. 103, 197 (2003). [37] D. T. Conroy, R. V. Craster, O. K. Matar, and D. T. Papageorgiou, J. Fluid Mech. 656, 481 (2010). [38] R. Probstein, Physicochemical Hydrodynamics, 2nd ed. (Wiley, New York, 1994). [39] B. Kirby, Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices (Cambridge University Press, Cambridge, 2010). [40] V. S. Ajaev, Interfacial Fluid Mechanics: A Mathematical Modeling Approach (Springer, New York, 2012). [41] V. S. Ajaev, T. Gambaryan-Roisman, and P. Stephan, J. Colloid Interface Sci. 342, 550 (2010). [42] R. V. Craster and O. K. Matar, Phys. Fluids 17, 032104 (2005).
032401-8
Effect of charge regulation on the stability of electrolyte films Christiaan Ketelaar* and Vladimir S. Ajaev† Department of Mathematics, Southern Methodist University, Dallas, Texas 75275, USA (Received 24 May 2013; revised manuscript received 13 January 2014; published 10 March 2014) The stability of a thin liquid film of an electrolyte on a solid substrate is investigated. In the framework of the Debye-H¨uckel approximation, we show that the commonly used approximation of fixed potential at the solid-liquid interface does not lead to predictions of film rupture. To reconcile the model with experimental observations, we consider the constant charge density approximation for the solid substrate and then proceed to systematically investigate the effects of charge regulation based on a linear relationship between charge density and potential. Stability criteria are formulated in terms of charge regulation parameters and electrolyte properties, resulting in different types of stability diagrams. Critical thickness below which the film ruptures is shown to decrease as the charge regulation at the solid-liquid interface becomes stronger. DOI: 10.1103/PhysRevE.89.032401
PACS number(s): 68.15.+e, 47.20.Ma, 47.61.Fg
I. INTRODUCTION
Viscous flows in thin liquid layers have been studied both experimentally and theoretically by a number of authors, following the pioneering work of Reynolds on the theory of lubrication [1]. A comprehensive review of the literature on thin-film flows driven by gravity, capillarity, thermocapillarity, solutocapillarity, and disjoining pressure with or without phase change can be found in Oron et al. [2]. In a more recent review, Craster and Matar [3] discuss the latest developments in the studies of thin films. Investigations of stability of thin liquid films on solid substrates are important for a number of applications such as flotation [4], dynamics of the tear film in the eye [5], and microscale heat transfer [6,7]. Film instabilities often occur in regimes where inertia is negligible and other physical effects such as viscosity, surface tension, electrostatic forces, and London–van der Waals dispersion forces are significant. Instabilities can lead to film rupture when the deformable interface between the liquid and the gas phase above it touches the solid substrate, resulting in the formation of dry spots. A driving mechanism of rupture that has received widespread attention in the literature is due to the London–van der Waals dispersion forces which are significant only in very thin films, typically of thickness below 100 nm. Spontaneous rupture of such thin liquid films on a planar solid wall was studied experimentally by Scheludko [8] and theoretically by Ruckenstein and Jain [9]. In the latter study, linear stability criteria are established for a liquid layer under the action of London–van der Waals forces. Further refinements of the linear theory of [9], e.g., by including the electrostatic effects and viscoelastic properties of the film, are discussed in Chap. 8 of [10]. Nonlinear models of instability and rupture of thin liquid films are often based on the assumption that the film thickness is much smaller than the wavelength of the instability [2]. This lubrication-type approach has been used by several authors to derive general evolution equations governing the spatiotemporal behavior of thin liquid films. Williams and
* †
[email protected] [email protected]
1539-3755/2014/89(3)/032401(8)
Davis [11] and Burelbach et al. [12] have examined the nonlinear evolution of a liquid film on a solid substrate under the action of London–van der Waals forces and found that the nonlinearities of the system accelerate rupture. Zhang and Lister [13] have found similarity solutions near the point of van der Waals rupture of a thin film on a solid substrate and carried out simulations of nonlinear evolution of the film thickness. All three studies have shown that as the instability grows, the nonlinear behavior leads to hole opening in the film in a finite time. While theoretical studies usually focus on film rupture driven by the London–van der Waals dispersion forces, an alternative mechanism is often encountered in experiments and is due to electrostatic effects [14]. Many liquids (e.g., aqueous solutions) contain ions and electrical double layers form near charged interfaces. We note that the presence of electric charges not only at solid-liquid but also at liquid-air interfaces has been established in the studies of Graciaa et al. [15] and Li and Somasundaran [16]. Electric charges at interfaces can be controlled by adding supplementary electrodes within the solid structures as proposed by Steffes et al. [17]. In order to develop the stability criteria for an electrolyte film on a charged substrate, it is first important to understand the basic physics of the interaction of two static charged surfaces separated by a film of electrolyte. Early models of such interaction considered both surfaces to be flat and of uniform charge densities [18–22]. The result was based on solving the Debye-H¨uckel equation for the electrostatic potential in the film and usually represented in terms of interaction force or interaction energy between the surfaces. Application of the same ideas to films of aqueous solutions with a deforming interface can be found in several studies [23–26]. Both the Debye-H¨uckel and Poisson-Boltzmann equations have been solved and the results were compared with experiments on film drainage [27,28] and films in constrained vapor bubble experiments [23]. The results are usually represented as an additional term in the stress tensor in the film, referred to as the electrostatic component of disjoining pressure. It is important to note that the electrostatic interaction in all these studies is assumed to be repulsive and the films are stable. Since the electric charges at solid-liquid and liquid-gas interfaces usually appear as a result of chemical reactions at interfaces, they actually depend on the local electric field, a
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physical effect known as charge regulation. This effect was investigated in the context of interaction of biological cells [29] and amphoteric solid surfaces [30,31]. Carnie and Chan [32] derived a simplified model of charge regulation based on a linear relation between the surface charge density and electric potential and applied it to interaction of flat solid surfaces and spherical particles. Further studies of charge regulation were carried out by Netz [33], Periset-Camara et al. [34], and Gupta and Sharma [35]. Detailed comparison with experiments by Gupta and Sharma [35] showed that the effects of charge regulation are important for achieving reasonable agreement between experiment and theory for stresses in thin electrolyte films. Several authors applied the formulas for the electrostatic component of disjoining pressure to studies of stability of thin films of aqueous solutions on solid substrates (see [36] and the references therein) and of coating films inside a pipe [37]. However, in these studies only simple boundary conditions of either fixed charge or fixed potential at the interfaces are considered. The objective of the present study is to incorporate more realistic relations between the electric charge density and potential at the interfaces into the model of a thin liquid film on a substrate and to investigate both linear stability and strongly nonlinear evolution of the fluid interface. Of particular interest are the conditions of film rupture due to the effect of electric charges at the solid-liquid and liquid-gas interfaces. We also note that the assumption of fixed interfacial potential at a deforming interface, used, e.g., in Conroy et al. [37], implies no electrically induced tangential stress at the deforming interface. In the present work, we incorporate the effect of such tangential stress. The article is organized as follows. In Sec. II, we describe the model of an electrolyte film and in Sec. III we derive the interface shape evolution equation using a lubrication-type approach. The linearized version of the evolution equation is also discussed. In Sec. IV, we present the results of the stability analysis of the model with a fixed potential at the solid wall and a constant charge density at the liquid-gas interface and show that the film does not rupture under these conditions regardless of its thickness. In Sec. V, we present results of the linear and nonlinear stability analysis of the model with constant charge densities at both interfaces and show that the film can rupture in finite time. In Sec. VI we consider a more general model based on linear charge regulation at the substrate and investigate how the choice of regulation parameters affects rupture conditions. Transition to the limiting cases of constant charge density and constant potential is also discussed.
q* 2
gas
d
liquid electrolyte
q* 1
x*
solid
FIG. 1. Sketch of a thin film of liquid electrolyte of viscosity μ on a flat solid substrate.
where is the electric permittivity of the liquid, kB is the Boltzmann constant, and T is the temperature. The solid surface shown in Fig. 1 is electrically charged as a result of the chemical reactions of dissociation of surface groups. A linearized relation between the dimensional charge density q1∗ and the electrostatic potential ψ1∗ at the solid surface is written as q1∗ = S1 − K1 ψ1∗ ,
(2)
where S1 and K1 depend on the density of surface groups and the dissociation constants, as discussed by Carnie and Chan [32]. Here and below the subscript “1” refers to the solid surface, while the subscript “2” is reserved for quantities describing the liquid-air interface. The applicability of the linearized charge regulation model was discussed in [32] by comparison with nonlinear models for different electrolyte concentrations and parameter values corresponding to a range of solid surfaces such as SiO2 , Fe2 O3 , Al2 O3 , with the general conclusion that the error of the approximation is typically not more than a few percent. In the limit of a very large film thickness (d λD ), the electric potential of the solid surface, ψ¯ 1∗ , can be related to the charge density using the linearized Grahame equation [39]. Combining the latter with Eq. (2) leads to the formula ψ¯ 1∗ =
S1 K1 + λ−1 D
.
(3)
The origin of electric charges and the effects of charge regulation at liquid-air interfaces are not as well understood as for the solid surfaces in contact with liquid. While the assumption of constant electric charge density q2∗ at the liquidair interface is most likely appropriate for the majority of cases, we keep the formulation more general here by incorporating possible effects of linear charge regulation, so that q2∗ = S2 − K2 ψ2∗ .
(4)
Let us define the nondimensional charge densities and potentials by
II. FORMULATION
Consider a thin liquid film of viscosity μ and thickness d on a flat solid surface, as shown in Fig. 1. The surface tension at the air-liquid interface is σ . The liquid is an electrolyte with N different types of ions of valencies zk and bulk concentrations n(0) k (k = 1,2, . . . ,N). Using the standard definition of the Debye length [38,39], we can write kB T (1) λD = N (0) 2 , 2 e k=1 nk zk
y*
qi =
qi∗ (ψ ∗ ,ψi∗ ) , (ψ,ψi ) = , i = 1,2. S1 ψ¯ 1∗
(5)
Then Eq. (2) can be expressed in nondimensional terms as q1 = 1 −
C1 ψ1 , 1 + C1
(6)
where C1 = K1 λD / has the physical meaning of the ratio of the regulation capacitance of the surface to the electrical diffuse layer capacitance. A nondimensional version of (4) is
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EFFECT OF CHARGE REGULATION ON THE STABILITY . . .
written in the form ˆ + C2 ) − C2 ψ2 q(1 , 1 + C1
(7)
S2 (1 + C1 ) K2 λ D , C2 = . S1 (1 + C2 )
(8)
q2 = where qˆ =
To study linear and nonlinear stability of the film, we will consider perturbations of the base state corresponding to a film of uniform thickness d. The dimensional Cartesian coordinates along the solid and normal to it are denoted by x ∗ and y ∗ , respectively. Following the standard lubricationtype approach [40,41], applicable for the horizontal scale of interface deformations being much larger than the film thickness, and neglecting the effects of gravity, we introduce vertical and horizontal length scales defined by d and Ca−1/3 d, respectively, where Ca = μU/σ is the capillary number. The velocity and pressure scales, U =
3/2
(d) μσ 1/2
ψ¯ 1∗3 , λ3D
ψ¯ ∗2 P = 21 , λD
(10)
u∗ p∗ v∗ Ut∗ . (11) , p= , v= , t= 1/3 U P Ca U 3dCa−1/3 If the capillary number is small, the use of our scaling simplifies the Navier-Stokes equations to a system of lubrication-type equations while the electrostatic potential in the liquid satisfies the Debye-H¨uckel equation. Thus, the system of governing equation is of the form u=
px = uyy + ψψx ,
(12)
py = ψψy ,
(13)
ux + vy = 0,
(14)
ψyy = κ ψ,
d κ= . λD
(15)
Liquid flow in this model is coupled to the electric field through the electrostatic body force term in the momentum balance as well as through the interfacial boundary conditions discussed below. The scaled components of this force, ψψx and ψψy , appear on the right-hand sides of Eqs. (12) and (13), respectively. At the solid wall (y = 0) we apply the no-slip and nopenetration boundary conditions for the flow. At the interface, y = h(x,t), the normal stress boundary condition includes the contribution from the Maxwell stress tensor, p0 − p = hxx −
1 ψy2 , 2 κ2
where p0 is the scaled atmospheric pressure. Since we are interested in situations when film dynamics is dominated by electrostatic effects, other possible contributions to interfacial stress balances, e.g., due to London–van der Waals dispersion forces, are neglected. The liquid shear stress at the interface is proportional to the tangential electric field component, leading to the second interfacial stress condition of the form κ 2 uy + ψy (ψx + ψy hx ) = 0.
(16)
(17)
The kinematic boundary condition at the liquid-air interface is 1 h 3 t
+ uhx = v.
(18)
Application of the divergence theorem for the electric field allows one to obtain relationships between the local charge density and the derivative of the potential at each interface. Together with Eqs. (6) and (7), these relationships lead to the following boundary conditions for the electric potential,
(9)
are obtained by considering the balance between the electrostatic, capillary, and viscous contributions to the stresses in the film as it deforms. The dimensional horizontal and vertical velocity components u∗ and v ∗ , the pressure p∗ , and time t ∗ can now be used to define the corresponding nondimensional variables according to
2
PHYSICAL REVIEW E 89, 032401 (2014)
ψy |y=0 = −κ(1 + C1 − C1 ψ1 ),
(19)
ˆ + C2 ) − C2 ψ2 ]. ψy |y=h = κ[q(1
(20)
We note that these equations do not include contributions from the electric field outside the film since we assume that relative dielectric permittivity of the electrolyte r is large, i.e., effectively take the limit of r → ∞. The commonly used boundary conditions of constant potential and constant charge densities are the special cases of Eqs. (19) and (20). The constant potential corresponds to the limit of Ci → ∞ (i = 1,2), while the constant charge density condition corresponds to Ci → 0. Thus, it is the relative magnitude of the regulation capacitance to the diffuse layer capacitance that controls whether an interface behaves as a constant potential, constant charge density, or chargeregulating surface. Following Carnie and Chan [32], we characterize the linear charge regulation in terms of the nondimensional parameters −1 i 1 defined by i =
Ci − 1 , Ci + 1
i = 1,2.
(21)
Note that the constant potential limit then corresponds to i = 1, while the constant charge limit corresponds to i = −1. III. NONDIMENSIONAL EVOLUTION EQUATION
Solving Eq. (15) subject to the boundary conditions (19) and (20) leads to the following expression for the electric potential in the film, ˆ −κy ψ = (eκh − 1 2 e−κh )−1 [(eκh − 1 q)e + (qˆ − 2 e−κh )eκy ].
(22)
Note that for a film of uniform thickness (h = 1) this equation reduces to the nondimensional version of the formula obtained by Carnie and Chan [32] for the case of two flat surfaces with charge regulation. We consider the case of constant charge density at the liquid-air interface, i.e., assume 2 = −1, for the rest of the present article, since the effects of charge regulation at the deforming interface are not expected to be significant. We
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verified that adding some realistic charge regulation correction at that interface does not alter the main conclusions of the present study, while the formulas become significantly more cumbersome. Using Eq. (22) with 2 = −1, the potential ψ2 at the liquid-air interface can be expressed as a function of the interface shape, ψ2 =
ˆ κh − 1 e−κh ) 2 + q(e . eκh + 1 e−κh
(24)
Combining the horizontal velocity profile given by (24) with (14), (16), and the kinematic boundary condition (18), we obtain ˆ 2 (qh ˆ + κ −1 ψ2 )x x = 0. (25) ht + h3 hxx + 12 ψ22 x − 32 qh Note that while this equation only involves the potential at the deforming interface, the film evolution depends on the electrical properties of both interfaces since ψ2 comes out of the solution of the Debye-H¨uckel equation. Note that while Eq. (25) is nonlinear, it has a simple steadystate solution corresponding to a uniform film thickness, h(x,t) = 1. Behavior of a small perturbation ζ (x,t) to this solution is described by the linearized version of (25), ζt + ζxxxx + Gζxx = 0, where G≡
3 3qˆ dψ2 − qˆ 2 ψ2 − 2κ dh 2 h=1
G2 , 4 and is reached at the wavelength √ 2 2π λ= √ . G
(28)
(29)
(30)
Substituting the solution for the electric potential given by (23) into the definition of G given by (27), we can rewrite the instability condition in the form Aqˆ 2 + B qˆ + C > 0,
B = −κ
(31)
where the coefficients A, B, and C are functions of the scaled film thickness κ and the charge regulation
1 2
(eκ − 1 e−κ )2 − 21
(32)
+ 34 (eκ − 1 e−κ )(eκ + 1 e−κ ) = 0, C = −κ(eκ − 1 e−κ ).
(33) (34)
ˆ plane for a fixed 1 are given The stability branches in the q-κ by √ −B ± B 2 − 4AC ˆ , (35) q(κ) = 2A as long as this equation has real solutions; i.e., the discriminant B 2 − 4AC is nonnegative. IV. FIXED SUBSTRATE POTENTIAL
If we consider a constant potential at the solid wall (1 = 1), the formula for the potential at the liquid-air interface, Eq. (23), reduces to ψ2 =
1 + qˆ sinh(κh) . cosh(κh)
(36)
The coefficients A, B, and C defined by Eqs. (32)–(34) simplify to
(27)
Perturbations of all wave numbers decay for G < 0, indicating stability. However, the film is unstable when the condition G > 0 is satisfied. The growth rate of the fastest instability mode then corresponds to the maximum value of the function γ (k), equal to γmax =
− 38 (eκ + 1 e−κ )3 ,
(26)
and ψ2 is now treated as a function of a single variable h, as defined by Eq. (23). Based on Eq. (26), the growth rate γ of a small sinusoidal perturbation of a wave number k is given by γ (k) = k 2 (G − k 2 ).
A = 1 κ(eκ − 1 e−κ ) − 32 1 (eκ + 1 e−κ )
(23)
From Eq. (13), we observe that p − ψ 2 /2 is a function independent of y. The horizontal flow velocity is then found by integrating Eq. (12) with the boundary condition (17) and the no-slip condition at the wall, resulting in ˆ qh ˆ x + κ −1 ψ2x )y. u = 12 (px − ψψx )(y 2 − 2yh) − q(
parameter 1 :
A = 2κ sinh κ − 3 cosh κ − 3 cosh3 κ,
(37)
B = −2κ(sinh2 κ − 1) + 3 sinh κ cosh κ,
(38)
C = −2κ sinh κ.
(39)
Remarkably, when these expressions are substituted into the condition (31), we find that the electrolyte film is stable for all values of the scaled film thickness since the function on the left-hand side of (31) is always negative. Thus, even though the assumptions of fixed substrate potential and fixed charge density at the fluid interface are often mentioned as a natural framework for studies of electrolyte films on solid substrates [36], we find that a model based on these assumptions fails to predict film break-up by growth of interfacial perturbations observed, e.g., in [14]. While film stability at large values of κ can be explained by the lack of sufficient overlap between the electrical double layers of the two interfaces, it is rather puzzling that no rupture threshold is found at small values of ˆ To gain deeper physical insight κ regardless of the value of q. into the behavior of perturbations of the film surface at small κ, let us consider the limiting case of κ = 0. When the fluid interface is flat, the configuration is that of a capacitor with the uniform electric field confined between the two interfaces. The electrostatic energy in the liquid then has constant density, denoted by E. Now let us consider the effect of a small interfacial perturbation ζ . In the framework of the linear approximation in ζ , the electric field remains uniform in the domain 0 y ζ with the same energy density. The total change in the electrostatic part of the system’s energy due to
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perturbation is
15
∞ −∞
Eζ dx = E
∞
−∞
ζ dx = 0,
(40)
where we used the condition of global mass conservation for the incompressible liquid in the film. Thus, there is no energy reduction due to electrostatic effect needed to overcome the stabilizing effect of surface tension despite the fact that the two interfaces are oppositely charged. One of the limitations of our model is the assumption of fixed charge density at the fluid interface. It is natural to ask whether the conclusions of the present section are still valid when the effect of interfacial charge transport is incorporated into the model by replacing the fixed value of qˆ with the ˜ evolving scaled charge density q(x,t). The standard interfacial charge transport equation (discussed, e.g., in [37]) for the new variable q˜ has to be solved together with the evolution equation for film thickness. Thus, the film dynamics is governed by a system of two coupled partial differential equations. However, when the system is linearized, the leading-order stability criteria turn out to be unaffected by the interfacial charge transport. Thus, the conclusions of the present section remain valid. In order to explain experimentally observed rupture phenomena, in the next two sections we modify the assumption of fixed solid substrate potential. It is first replaced by a condition of constant charge density, followed by a discussion of a more general charge regulation condition. V. FIXED SUBSTRATE CHARGE
Let us consider another commonly used model of interacting charged interfaces in which the effects of charge regulation are completely neglected by assuming the dimensional charge densities to be constant, equal to S1 and S2 at the lower and upper boundaries of the film, respectively. In our nondimensional formulation this implies setting 1 = −1 in Eq. (23). The resulting expression for the potential at the liquid-air interface can then be written as ψ2 =
qˆ cosh(κh) + 1 . sinh(κh)
(41)
Note that qˆ in this case has the simple meaning of the ratio of charge densities at the two interfaces. The general coefficients defined by Eqs. (32)–(34) simplify to A = −2κ cosh κ + 3 sinh κ − 3 sinh3 κ,
(42)
B = −2κ(cosh2 κ + 1) + 3 sinh κ cosh κ,
(43)
C = −2κ cosh κ.
(44)
The stability diagram based on this equation is shown in Fig. 2. We note that accounting for tangential stresses at the interface, as is done in the models of thermocapillary and solutocapillary instabilities, is essential for the correct description of film dynamics. For example, in the limit of κ → 0, the electrostatic contribution to the pressure in the film has a stabilizing effect, so the instability is entirely due to the tangential electric field which arises when a small perturbation of the initially flat interface is introduced.
unstable
10
5
stable qˆ
0
−5
unstable −10
−15
0
0.1
0.2
0.3
0.4
0.5
κ FIG. 2. Linear stability diagram for a model with constant charge densities at both interfaces (1 = 2 = −1).
According to our stability criteria, the film is always stable if its scaled thickness κ > 0.530 12, which is consistent with experimental results showing that instability is generally not expected if there is no significant overlap between the interfacial electric double layers. Under these conditions, the film is not destabilized even for large values of the parameter ˆ in fact, the asymptotic limit of large qˆ corresponds to the q; critical value of the scaled thickness of 0.5051 (the value at which the coefficient A passes through zero). The diagram in Fig. 2 indicates that the film can be linearly unstable at both negative and positive values of the charge ˆ While instability at negative qˆ can density ratio parameter, q. be easily explained in terms of attraction of oppositely charged interfaces, the prediction of linear instability at positive qˆ may seem rather surprising. To better understand this result, we note that it can be interpreted as a manifestation of the wellknown phenomenon of the instability of a charged deformable interface in an electric field, which in our case is generated by the charges at the substrate and in the liquid rather than external electrodes (as is often done in experiments showing this type of instability). This phenomenon has been observed and described under a wide range of conditions, including different electrical conductivities of fluids separated by the charged interface (see, e.g., [42] and references therein). Another way to interpret our result is to point out that the deformable interface interacts not only with the charge at the solid surface but also with the cloud of charge in the solution, so at least some of these interactions always involve charges of opposite signs. The linear stability model breaks down as soon as the amplitude of the perturbation of the base state is no longer much smaller than unity. To investigate perturbation dynamics in this regime and make conclusions about the possibility of film rupture, we solved the full nonlinear Eq. (25) with ψ2 from (41) numerically using the method of lines with time stepping by Gear’s BDF method. As is common in film rupture studies [13], the size of the computational domain L is dictated by the wavelength of the fastest growing perturbation: we
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1.2
1.2
t=0.3 t=0.5 t=0.6
1.15
1
1.1 0.8
1.05
h
a
1
0.6
0.95 0.4
0.9 0.2
0.85 0.8
0
0.2
0.4
0.6
0.8
0
1
x/L FIG. 3. Snapshots of the interface shape for κ = 0.4, qˆ = −2.
choose L = λ/2, where λ is given by Eq. (30). The boundary conditions are hx (0,t) = hx (L,t) = 0,
(45)
hxxx (0,t) = hxxx (L,t) = 0,
(46)
and the initial perturbation is sinusoidal, of the amplitude ζ0 = 10−2 . Note that the nonlinear system considered here depends on two nondimensional parameters, the ratio of the linearized surface charge densities qˆ and the film thickness scaled by the Debye length, κ. The plot in Fig. 3 shows snapshots of the interface at different times for κ = 0.4 and qˆ = −2. Clearly, the interface shapes are no longer sinusoidal due to the nonlinear effects and the film evolution tends to speed up. In order to better understand the time evolution of the liquid-air interface, it is convenient to plot the minimum film thickness (always reached at x = 0 with our choice of the initial condition) as a function of time, in nondimensional terms. The linear stability theory discussed at the end of Sec. III predicts that the scaled minimum thickness, denoted by a, should evolve according to a = 1 − ζ0 e G
2
t/4
.
(47)
This result is represented by a dot-dashed line in Fig. 4. The solid line, corresponding to the numerical solution of the nonlinear evolution equation (25), follows the predictions of the linear stability analysis initially, but then deviates from it significantly. The film ruptures in a finite time in a manner similar to the well-known case of the van der Waals driven rupture [12]. We carried out similar studies for several different values of qˆ and observed the same dynamics. Thus, the model described in the present section is indeed capable of describing not only the linear destabilization of the liquid film but also its rupture in a finite time.
0
0.1
0.2
0.3
t
0.4
0.5
0.6
0.7
FIG. 4. Evolution of the scaled minimum thickness (denoted by a) for κ = 0.4, qˆ = −2. Solid line is obtained from the numerical solution of Eq. (25); dot-dashed line is the prediction of the linear stability theory. VI. EFFECT OF CHARGE REGULATION AT THE SOLID WALL
The stability results in the previous two sections clearly show qualitatively different behavior, but the only difference between the models is the degree of charge regulation at the solid surface, from perfect regulation to no regulation. It is natural to ask how the degree of charge regulation affects the stability of the film. Addressing this issue is the objective of the present section. If the charge regulation effects are neglected, i.e., 1 = −1, the stability diagram is that shown in Fig. 2. As the value of the regulation parameter is increased slightly, both stability branches are moving downward in the stability diagram, but the overall picture remains qualitatively the same. At the values of 1 near −0.5, the lower stability branch starts shifting downward significantly, eventually disappearing to infinity as 1 approaches zero. Typical stages of this process are shown in Fig. 5. The upper stability branch starts deforming so that the region of instability shrinks gradually until in disappears completely in the regime of perfect charge regulation, i.e., constant potential. This corresponds to the limiting case of absolute stability that was discussed in Sec. IV. Comparison between stability diagrams such as the ones ˆ the shown in Fig. 5 shows that for a fixed value of q, values of κ corresponding to the instability threshold are decreasing with increase in 1 . Thus, the critical dimensional thickness for the onset of instability for a given electrolyte properties will decrease as the strength of charge regulation is increased. VII. CONCLUSIONS
We investigated the stability of a thin electrolyte film on a solid substrate. Both normal and tangential stress conditions at the film surface include contributions from the electric field. A linear stability analysis under the assumption of fixed substrate
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PHYSICAL REVIEW E 89, 032401 (2014)
8
8
6
6
unstable
4
2
qˆ
qˆ
2 0
stable
−2
0
stable
−2 −4
−4
unstable
−6 −8
unstable
4
0
0.1
−6
0.2
κ
0.3
0.4
0.5
−8
unstable 0
0.05
0.1
0.15
0.2
κ
0.25
0.3
0.35
0.4
FIG. 5. Stability regions for 1 = −0.5 (left side) and 1 = −0.3 (right side).
potential shows that the film is always stable. A simple physical interpretation of this conclusion is provided by considering the change in the electrostatic energy as a result of interfacial perturbation of an initially flat film in the limiting case when film thickness is much smaller than the Debye length of the electrolyte solution. In order to describe the experimentally observed rupture by electrostatic effects, we considered the approximation of fixed charge density at the solid substrate and then more general models incorporating the effects of interfacial charge regulation. The latter is characterized by a nondimensional parameter 1 which depends on the ratio of the solid surface regulation capacitance to the capacitance of the diffuse layer formed in the liquid near the solid. We showed that ˆ variables different types of linear stability diagrams in q-κ are possible depending on the value of 1 (here qˆ is a nondimensional parameter which characterizes the ratio of
electrostatic properties of the two interfaces and κ is the film thickness scaled by the Debye length). The region of instability shrinks as the value of 1 is increased, in accordance with the conclusion of stability under constant substrate potential, i.e., perfect charge regulation. We carried out nonlinear simulations of the film evolution for the case of constant charge densities and observed that linear destabilization of the interface eventually leads to film rupture, in qualitative agreement with experimental data on rupture of electrolyte films.
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ACKNOWLEDGMENTS
This work was supported by the National Science Foundation under Grant No. CBET-0854318. We thank the anonymous referees for a number of useful suggestions.
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PHYSICAL REVIEW E 89, 032401 (2014)
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