Langmuir 2000, 16, 6719-6723
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Charge-Induced Modification of Contact Angle: The Secondary Electrocapillary Effect R. Digilov* Department of Education in Technology and Science, Technion-Israel Institute of Technology, 32000, Haifa, Israel Received October 5, 1999. In Final Form: May 3, 2000 The Gibbs-Johnson variational approach was applied to thermodynamic treatment of a charged liquid drop placed on a charged substrate. The line tension effect and the presence of charges on the three-phase contact line were taken into consideration. As a result a novel form of the generalized Young equation, containing an electric driving force at the three-phase contact line, was obtained. Application of this relationship to analysis of the electrowetting phenomenon enables us to conclude that the potentialinduced change in the contact angle is the secondary electrocapillary effect, caused by redistribution of the charges on the three-phase contact line.
Introduction The phenomenon of change in the contact angle caused by applied potential to the interface of a three-phase system, the so-called electrowetting effect, has long been of interest1-3 and received renewed attention in the past few years4-8 with its potential applicability in industrial situations. This phenomenon was interpreted quantitatively in terms of the Young and Lippman equations.3-8 Young’s equation relates the cosine of the equilibrium contact angle to the three interfacial tensions by
γsv - γsl cos θ ) γlv
(1)
where subscripts (sv), (sl), and (lv) refer to solid-vapor, solid-liquid, and liquid-vapor interfaces respectively (Figure 1). When the potential is applied to the (sl) interface, γsv and γlv are assumed to be potential independent and γsl varies with the potential according to the Lippmann equation:
γqsl
) γsl -
∫φ
φsl 0 sl
σsldφsl
(2)
Here γsl is the solid-vapor interface tension at the potential of zero charge φsl0; φsl is the potential on the φ solid-liquid interface; σsl ) ∫φsl0slCsldφsl is the surface charge density and Csl is the differential capacitance of the interface. Substitution of eq 2 into eq 1 gives5
cosθq ) cosθ +
1 γlv
∫φφ ∫Csl[dφsl]2 0 sl
sl
(3)
where cosθ is the cosine of the contact angle in the absence * Email:
[email protected]. (1) Nakamura, Y.; Kamada, K.; Katoh, Y.; Watanabe, A. J. Colloid Interface Sci. 1973, 44, 517. (2) Nakamura, Y.; Matsumoto, M.; Nishizawa, K.; Kamada, K.; Watanabe, A. J. Colloid Interface Sci. 1977, 59, 201. (3) Sparnaay, M. J. Surf. Sci. 1964, 1, 213. (4) Vallet, M.; Berge, B.; Vovelle, L. Polymer 1996, 37, 2465. (5) Welters, W. J. J.; Fokkink, L. G. J. Langmuir 1998, 14, 1535. (6) Ivosevic, N.; Zutic, V. Langmuir 1998, 14, 231. (7) Verheijen, H. J. J.; Prins, M. W. J. Rev. Sci. Instrum. 1999, 70, 3668. (8) Verheijen, H. J. J.; Prins, M. W. J. Langmuir 1999, 15, 6616.
Figure 1. Sessile drop on a smooth, inert, nondeformed solid surface. Young definition of an equilibrium contact angle from the mechanical equilibrium condition on the three-phase contact line eq 1.
of charges. Assuming further Csl ) 0/τ ( being the dielectric constant of the double layer and τ its thickness) to be potential independent and performing double integration with respect to φsl, results in the relationship between the cosine of the contact angle and the potential5,6
cosθq ) cosθ +
0(φsl - φ0sl)2 2τγlv
(4)
Equation 4 suggests that electrowetting is the direct consequence of interface polarization and formally it is no different from the classical electrocapillary effect. Although eq 4 qualitatively predicts the observed nearparabolic behavior of cosθq vs φ, an experimental test of the mechanical balance between the surface tensions at the contact line in the horizontal direction does not hold.1,2 This means that the modification of the contact angle under an applied potential cannot be attributed to the Lippman phenomenon, and combination of eqs 1 and 2 cannot solve the problem of the three-phase equilibrium in the presence of electric charges. It is necessary to minimize the total energy of the three-phase system, with an account of electrostatic terms. A second serious shortcoming of the above approach is its disregard of the line tension effect, a property characteristic of three-phase equilibrium, and of possible redistribution of the charges on the three-phase boundaries, which can have considerable influence on the wetting behavior. In this paper, we attempt to show that the change in wettability due to the presence of surface and line charges
10.1021/la991308a CCC: $19.00 © 2000 American Chemical Society Published on Web 07/12/2000
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Digilov
is governed by the redistribution of the charges on the three-phase contact line. Equations of Capillarity in the Presence of Charges. Consider a droplet of a conducting liquid with a charge density σlv distributed uniformly on its surface placed on a smooth surface of the dielectric solid in equilibrium with own vapor, as shown in Figure 1. The solid is assumed to be inert, homogeneous, and nondeformed with surface charge σsv. The surface charge of the solid-liquid interface is σsl and gravity is neglected. The system thus consists of three bulk homogeneous phases: solid (s), liquid (l), and vapor (v), which following Gibbsian thermodynamics, are considered as uniform and isotropic phases up to the dividing liquid-vapor (lv), solidvapor (sv), and solid-liquid (sl) interfaces. The dividing interfaces are considered as uniform and isotropic twodimensional surface phases up to the dividing solidliquid-vapor (slv) contact line, which is considered as a uniform and isotropic one-dimensional line phase with excess line charge χslv. Gibbs9 and more recently Johnson10 carried out a thermodynamic analysis of a similar capillary system in the absence of electric charges. Later on Neimann11 took the line tension effect into consideration. The GibbsJohnson-Neimann (GJN) approach assumes that a free liquid drop automatically takes the shape that minimizes the total system energy. The classical Laplace and Young equations of capillarity are the consequence of the mechanical equilibrium of the system. We shall use the GJN approach to determine the influence of electric charges, brought to the interfaces and to the three phase contact line on the condition of thermodynamic equilibrium of the capillary system. To establish equilibrium conditions, we minimize the system total energy. In additional to bulk, interfacial, and linear energies, we involve the energy of the electric charges in the treatment. As a consequence of the mechanical equilibrium of the system we expect to obtain the generalized Young and Laplace equations in the presence of charges. We shall consider the following: 1. The system is isolated thermally from its surroundings. This means that its total entropy is maximum, St ) max, i.e. 3
∫V dSs + ∫V dSl + ∫V dSv + ∑∫
δSt ) δ(
s
l
v
j)1
dS(j) ω ωj
+
∫LdSslv) ) 0
(5)
where δ denotes variation and d differential. The integration is over the volume occupied by solid Vs, liquid Vl, and vapor Vv, interfaces are ωj, and three-pase contact line L. 3 Ss,(l),(v) is the entropy of the bulk phases, Sslv ) Sω - ∑j)1 S(j) ω is the excess entropy of the three phase contact line. Sω ) St - Ss - Sl - Sv is the excess surface entropy, Sω(j) is the excess entropy of the jth boundary interface far away from the three phase contact line. The subscript ω denotes the excess surface values and the summation index j indicates summation over the sv, lv, and sl boundary interfaces. Similar notations are used for the other equations.
2. The system is isolated mechanically. This means that its total volume is constant, Vt ) constant, i.e.
∫
δVt ) δ (dVs + dVl + dVv) ) 0
3. The number of molecules in the system is constant, Nt ) constant, i.e.
δNt ) δ(
∫
dNs + V s
∫
dNl + V l
∫
3
dNv + V v
∫ω dN(j)ω + ∑ j)1 j
∫LdNslv) ) 0
(7)
where Ns,(l),(v) is the number of particles in the bulk phases, 3 N(j) Nslv ) Nω - ∑j)1 ω is the excess number of particles at the three phase contact line, Nω ) Nt - Ns - Nl - Nv is the excess number of particles at the interface, and N(j) ω is the excess number of particles at the jth boundary interface far away from the three phase contact line. The total energy of the heterogeneous system Et is conserved at constant St, Vt, and Nt. Therefore δ(Et)St,Vt,Nt ) 0, which reads
∫V dEs + ∫V dEl + ∫V dEv +
δ(Et)S,Vt,Nt ) δ(
s
l
v
3
∫ω dE(j)ω + ∫LdEslv)S ,V ,N ) 0 ∑ j)1 j
t
t
t
(8)
where Es,(l),(v) is the energy of the bulk phases, Eω(j) is the 3 E(j) excess energy of the jth interface, Eslv ) Eω - ∑j)1 ω is the excess energy of the three-phase contact line, and Eω ) Et - Es - El - Ev is the total excess surface energy of the system. Since the solid is assumed rigid, its bulk energy may be considered to be constant, that is δ(∫VsEs)St,Vt,Nt ) 0 and there is no necessity to involve this contribution in the variational procedure (eq 8). Now if we include the charged components in the system, the energy of the system may be considered as in the absence of an external field, and the ordinary thermodynamic equation may be applied
dE ) TdS - PdV + µ˜ dN where T is the temperature, P is the pressure, N is the number of molecules or ions, µ˜ is electrochemical potential, for which the standard expression may be used: µ˜ ) µ + U, where U(r) ) qφ(r) is the energy of a particle in the electric field of potential φ(r), q the particle charge, and µ the chemical potential. With this expression we have
dE ) TdS - PdV + µdN + UdN Now, the following equations can be written for the terms of eq 8:
∫V (TldSl - PldVl + µldNl + UldNl)
(9)
∫V (TvdSv - PvdVv + µvdNv + UvdNv)
(10)
∫ω (T(j)ω dS(j)ω + γjdωj + µ(j)ω dN(j)ω + U(j)ω dN(j)ω )
(11)
∫L(TslvdSslv + ηdl + µslvdNslv + UslvdNslv)
(12)
El ) Ev ) E(j) ω )
Eslv ) (9) Gibbs, J. W. Thermodynamics. The Collected Works of J. Willard Gibbs. Yale University Press: New Haven, CT, 1928; Vol. 1, pp 314331. (10) Johnson, R. E., Jr. J. Chem. Phys. 1959, 63, 1665. (11) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977, 66, 5464.
(6)
l
v
j
where Ul,(v) is the energy of the electrostatic interaction per particle in the liquid and vapor bulk phases; Uω(j) is the excess interface energy of the electrostatic interaction
Wettability/Charge Distribution at 3-Phase Contact Line
per particle at the jth boundary interface; Uslv ) Uω 3 U(j) ∑j)1 ω is the excess electrostatic energy per particle of the three-phase contact line; Uω ) Ut - Ul - Uv is the surface excess electrostatic energy of the particle; η is the line tension, the excess energy per unit length of the threephase contact line; dl is a length element of the threephase contact line. We shall treat the electric field of the present charges as a function of the coordinates only. The energy of the entire system including the electrostatic terms is conserved. However, there may be variations between phases, associated with independent variations of the entropy, geometric elements, external forces, and the number of particles in the phases, under which the thermodynamic equilibrium of the system is conserved. Variation of the geometrical parameters δdV, δdω, δdl in the presence of charges necessitates variation of the pressure P, interface tensions γj and line tension η. The geometric variation parameters are shown in Figure 2.Variation of the elements in eqs 9-12 at constant temperature T and chemical potentials µ reads
δEl )
∫V (TlδdSl - δ(PldVl) + µlδdNl + l
UlδdNl + δUldNl) (13) δEv )
∫V (TvδdSv - δ(PvdVv) + µvδdNv + v
UvδdNv + δUvdNv) (14) δE(j) ω )
∫ω [T(j)ω δdS(j)ω + δ(γjdωj) + µ(j)ω δdN(j)ω + j
(j) (j) (j) U(j) ω δdNω + δUω dNω ] (15)
δEslv )
∫
(TslvδdSslv + δ(ηdl) + µslvδdNslv + L
3
∫V δdSl + Tv∫V δdSv + ∑ T(j)ω ∫ω δdS(j)ω + l
v
j
j)1
∫L δdSslv ) 0
Tslv (µl + Ul)
Figure 2. Illustration of the virtual displacement of the threephase contact line at the fixed total volume of the system. For definitions see text.
Here µ˜ k ) µk + Uk ) µk + qφk is the electrochemical potential. Thus, under thermodynamic equilibrium the temperature and total chemical potential in the phases, on the boundary interfaces and three-phase contact line, are equal. Equation 19 expresses the conditions for mechanical equilibrium of the system in the presence of charges. Now, it is useful to consider some trivial relationships. The total volume of the system is constant dVt ) dVs + dVl + dVv ) 0. Since the solid is assumed nondeformed dVs ) 0 and we have dVv ) -dVl. For the differentials of the number of particles, one can write:
dNl ) nldVl, dNv ) nvdVv ) -nvdVl, dN(j) ω ) n(j) ω dω, dNslv ) nslvdl
UslvδdNslv + δUslvdNslv) (16)
As the variations of variables are independent, the sum of eqs 13-16 can be separated into three independent equations to satisfy the necessary condition δ(Et)T,µ ) 0.
Tl
Langmuir, Vol. 16, No. 16, 2000 6721
∫V
δdNl + (µv + Uv)
v
∫V
v
(17)
δdNv +
3
3
(j) (j) (µ(j) ∑ ω + Uω ) ∫ω δdNω + (µslv + Uslv) ∫L δdNslv ) 0 j)1 j
(18)
∫V [-δ(PldVl) + dNlδUl] + ∫V l
v
[-δ(PvdVv) +
3
dNvδUv] +
∫ω (δγjdωj + γjδdωj + ∑ j)1 j
dN(j) ω
δU(j) ω)
∫L (δηdl + ηδdl + dNslvδUslv) ) 0
+ (19)
Equation 17 agrees with eq 5 if the conditions for thermal equilibrium in the system hold:
Tl ) Tv ) Tsv ) Tlv ) Tsl ) Tslv
(20)
Similarly, eq 18 agrees with eq 7 if the conditions for electrochemical equilibrium hold
µ˜ l ) µ˜ v ) µ˜ sv ) µ˜ lv ) µ˜ sl ) µ˜ slv
where nl,(v) is the particle density in the bulk liquid (l) and vapor (v) phases, nω(j) is the surface density of the particles at the jth boundary interface and nslv is the line density of the particles at the three phase contact line. Furthermore, the total surface area of the nondeformed solid as a sum of the free solid-vapor, ωsv and solid-liquid, ωsl area is constant, ωsv + ωsl ) constant, that is d(ωsv + ωsl) ) 0, so dωsv ) -dωsl. Taking into account these relationships, eq 19 may be rewritten
(21)
∫∑ (γjδdωj + δγjdωj + n(j)ω dωjδU(j)ω ) j)1
∫[(Pl - Pv)δdVl + δ(Pl - Pv)dVl] + ∫(ηδdl + δηdl) + ∫nslvδUslvdl + ∫(nlδUl - nvδUv)dVl ) 0 (22) Variation of the liquid-vapor interface area has two components: one due to the curvature of the liquid vapor interface, and the other due to the position of the 3 δdωj ) (1/r1 + three-phase contact line. Therefore, ∑j)1 3 1/r2)δndωlv + ∑j)1dlδtj, where δtj is the virtual displacement of the contact line normal to dl along jth boundary interface (Figure 2), and δn is the variation of the normal to the liquid-vapor interface. Variation of the (sv) and (sl) interfaces has only one component because of the position of the contact line. Furthermore δdVl ) dωlvδn. Variation of the differential element dl of the three-phase contact line due to variation of curvature 1/r in the horizontal plane is δdl ) dl/r δr. Using these geometric relationships in eq 22, after grouping similar terms, one obtains
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Langmuir, Vol. 16, No. 16, 2000
Digilov
∫[δ(γsv - γsl) + nsvω δUsvω - nslω δUslω]dωsv +
[( ) ] ∫(∑ ) ∫ 1
∫ γlv r
1
+
1
δn + δγlv - (Pl - Pv)δn +
r2
3
nlv ω
δUlv ω
dωlv +
γjδtj +
j)1
nslvδUslv dl -
(
η r
+
)
∂η ∂r
δr +
[δ(Pl - Pv) - (nlδUl - nvδUv)]dVl ) 0 (23)
Since the increments dωj, dl, and dV are independent and the variations are arbitrary, every integrand of eq 23 must equal zero. The first surface integrand yields sv sl sl δ(γsv - γsl) ) -(nsv ω δUω - nω δUω )
(24)
(j) (j) (j) (j) (j) (j) Since n(j) ω δUω ) nω qδφω ) σ δφω , where σ is the surface density of the charges at the jth interface, integration of eq 24 yields the Lippman equation
∫0
(γsv - γsl)q - (γsv - γsl) ) -
φ
(σsvdφsv - σsldφsl) (25)
The Laplace generalized equation of capillarity in the presence of charges derives from the second surface integrand
(
∆P ) γlv
)
dγlv ∂Ulv 1 1 ω + nlv + + ω r1 r2 dn ∂n
(26)
where the first term is the Laplace pressure, dγlv/dn measures the change in surface tension with the curvature, and the last term is the electrostatic contribution. Let us lv lv lv transform it: nlv ω ∂Uω /∂n ) nω q(∂φω /∂n) ) -Elvσlv, where lv σlv ) nω q is the surface density of the charges at the (lv) interface; Elv ) - ∂φlv ω /∂n is the normal to the liquidvapor interface component of the electrostatic field. Thus, we have
(
)
dγlv 1 1 - Elvσlv + ∆P ) γlv + r1 r2 dn
(27)
Here, the electrostatic term is related to the Lippman effect of the change γlv due to the presence of the charges at the curved liquid-vapor interface, which coincides with the electric contribution obtained in ref 13 for the pressure difference of a spherical layer between two phases, in the limit when the layer thickness approaches zero. The third integrand yields the generalization of the Young equation in the presence of charges
∑j γjδtj +
(
η r
+
)
∂η ∂r
δr + nslvδUslv ) 0
(28)
Here ∑jγjδtj is the Young term, η/r is the line tension contribution, ∂η/∂r defines the curvature dependence of the line tension, and the last term is the electrostatic (12) Gaydos, J.; Boruvka, L.; Rotenberg, Y.; Chen, P.; Neumann, A. W. The Generalized Theory of Capillary. In Applied Surface Thermodynamics, Neumann, A. W., Spelt, J. K., Eds. Marcel Dekker: New York, 1996; Chapter 1. (13) Sanfeld, A. Introduction to the Thermodynamics of Charged and Polarized Layers; Wiley; London, 1968.
Figure 3. The three-phase contact line equilibrium in the presence of electric charges. On a level with the three surface tensions forces γlv, γsv, γsl and line tension contribution η/r, the electric driving force of the wetting, fslv is affected by the triple line polarization.
contribution due to the presence of line charges. Thus, the mechanical equilibrium between the surface tensions and the line tension at the three-phase contact line in the presence of surface and line charges involves the electrostatic contribution from the line charges only. The last integrand gives the equation for distribution of the hydrostatic pressure in the bulk phases in the presence of bulk charges
δ(Pl - Pv) - (nlδUl - nvδUv) ) 0
(29)
Thus, eqs 25-29 express conditions for mechanical equilibrium of the heterogeneous system in the presence of electric charges. Let us consider eq 28 in detail. The Young Equation in the Presence of Charges. The excess electrostatic energy per unit length of the threephase contact line reads
nslvδUslv ) nslvqδφslv ) χslvδφslv
(30)
where χslv ) nslvq is the line density of the electric charges and δφslv is the change of the electrostatic potential on the three-phase contact line. Substituting eq 30 in eq 28 and opening the summation sign, one has
γlvδtlv + γsvδtsv + γslδtsl +
(ηr + ∂η∂r)δr + χ
slvδφslv
)0 (31)
Rearrangement with δtsl ) -δtsv and δtlv/δtsl ) cosθq (Figure 2), where θq is the contact angle in the presence of charges, yields:
γsv - γsl ) γlvcosθq +
(ηr + ∂η∂r)δtδr + χ
∂φslv (32) ∂tsl
slv
sl
Here, δr/δtsl ) cosβ, where β is the angle of inclination. For a flat substrate (our case) δr ) δtsl and cosβ ) 1. The derivative δφslv/δtsl ) Eslv is the strength of the electrostatic field at the wetting line. Substituting these relationshnsips in eq 32 and disregarding ∂η/∂r, one obtains Young’s equation in the presence of charges
γsv - γsl ) γlvcosθq +
η - χslvEslv r
(33)
where fslv ) χslvEslv is the specific electric driving force of the wetting (Figure 3). In the absence of charges on the triple line, eq 33 becomes Young’s equation with the line tension correction12
γsv - γsl ) γlvcosθ +
η r
(34)
where θ is the equilibrium contact angle in the absence
Wettability/Charge Distribution at 3-Phase Contact Line
Langmuir, Vol. 16, No. 16, 2000 6723
of charges. From the difference between eqs 34 and 33 it follows that
an applied potential is governed by the electric charges on the three-phase contact line.
χslvEslv cosθq ) cosθ + γlv
(35)
Denote by ηq the line tension in the presence of line charges at fixed radius of the triple line r0 and contact angle θq. Then Young’s equation reads
γsv - γsl ) γlvcosθq +
ηq r0
(36)
The difference between eq 36 and eq 33 results in
η - ηq ) χslvEslvr0
(37)
which shows that the electric driving force of the wetting is governed by reduction of the line tension. Combining eq 37 with the line analogue of the Lipmann equation:
η - ηq )
∫φφχslvdφslv 0
where φslv is the potential at the three-phase contact line, eq 35 one writes
( ) ∂cosθ ∂φslv
T,µ
)
1 χ r0γlv slv
(38)
Thus, the change in the cosine of the contact angle under
Conclusions The main conclusions of the present study can be summarized as follows: Variation treatment of a capillary system in the presence of surface and the line charges is provided. As one of the natural boundary conditions for minimization of the system internal energy, a novel relationship is proposed for the generalized form of the Young equation in the presence of charges. The generalized Young equation shows that the change in the contact angle is caused by the reduction of the line tension due to the redistribution of charges on the threephase contact line. Application of this result to analysis of the potentialinduced change in the contact angle enables us to conclude that the potential applied to the interface causes polarization of the interface and the three-phase contact line. The first phenomenon leads to reduction of interface tension (primary electrocapillary effect), whereas the second leads to change in the line tension and as thereby in the contact angle (secondary electrocapillary effect). Thus, the electrowetting phenomenon is a consequence of the secondary electrocapillary effect and operates through redistribution of the charges on the three-phase contact line. Acknowledgment. The author would like to thank Dr. Reiner for her help and the Israel Ministry of Immigration and Absorption for its financial support. LA991308A
