2786
Langmuir 1993,9, 2786-2788
Surface Melting of Ice Induced by Hydrocarbon Films R. Bar-Ziv and S. A. Safran* Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel Received July 20, 1993. In Final Form: September 20, 199P
We apply the continuum theory of van der Waals interactions to a four-component,layered system, composed of a bulk vapor phase, a thin hydrocarbon film, and a water film which is in equilibrium with bulk ice, at the triple point. We find that for thin hydrocarbon films, these interactions result in a finite film of water at the ice surface with a thickness which increases with the hydrocarbon film thickness, d, reaching a value of about loo0 8, for d is: 300 A. This corresponds to incomplete surface melting of ice but with a relativelythick wetting layer. However, for larger d , we predict a discontinuouswetting transition as the water film thickness jumps to infinity, indicating complete surface melting of ice. The phenomenon of surface melting of solids which is an important component in the understanding of lubication, friction, and wear has recently received considerable experimental and theoretical attention.' Surface melting is the process in which a liquid film completely wets the solid-vapor interface with a thickness which grows indefinitely as the triple point is approached from below. As aresult, the solid melts from its surface. Recently, Elbaum, Lipson, and Dash2 observed that water films existing on the surfaces of isolated single H2O crystals were of nonzero, but finite, thickness at all temperatures below the melting temperature, indicating partial wetting of water on ice and hence incomplete surface melting of ice. These findings were in reasonable agreement with theoretical predictions by Elbaum and Schick3 who considered the contribution of van der Waals (vdW) interactions to the problem of the surface melting of ice. They found that the vdW interactions among ice, water, and vapor at the triple point do not favor a macroscopically thick film of water at the ice-vapor interface; thus, there is only partial wetting of the solid-vapor interface by the fluid, and surface melting is incomplete. They estimated the film thickness Lminwhich minimizes the free energy of the vdW interaction to be about 36 A; excess liquid coexists with the film in the form of droplets with an estimated contact angle of about 0 . 2 O . In this paper we predict that the addition of an insoluble hydrocarbon (e.g., tetradecane) film, of thickness d , between the ice-water4 and vapor interface, as shown5in the inset of Figure 1,induces an increase in the equilibrium thickness L,i, of the wetting layer of water, reaching a limiting value of Lminis: 1000 8,for d i= 300 A. For larger values of d there is a discontinuous transition from incomplete to complete surface melting; bulk ice is wet by a macroscopic film of water, and Lminjumps to infinity. Abstract published in Advance ACSAbstracts, October 15,1993. (1) For recent reviews where surface melting is discussed, see: Dosch, H. Critical Phenomena at Surfaces and Interfaces, Evanescent X-Ray and Neutron Scattering; Springer-Verlag: Berlin, Heidelberg, 1992. Forgacs, G.; Lipowsky,R.; Nieuwenhuizen, Th. M.; In Phase Transitions and critical phenomena; Domb, C., Lebowitz, J., Eds.; Academic Press: New York, 1991;Vol. 14 and references therein. (2)Elbaum, M.; Lipson, S. G.; Dash, J. G . J.Cryst. Growth 1993,129, 491-505. (3) Elbaum, M.; Schick, M. Phys. Reu. Lett. 1991, 66, 1713. (4)Using datagiven by R. J. Hunter (Foundations of Colloid Science; Clarendon Press: Oxford, 1986, Vol. 1) for the surface tensions u m = 72.75,u m = 21.8,and u w = 50.8 erg/cm* of water-vapor, hydrocarbon (CeHd-water, and hydrocarbon-vapor, respectively, we conclude that hydrocarbons on water should have small or zero contact angles. Even if the wetting is not complete, in the presence of a large enough quantity of hydrocarbon, our predictions will apply to the region under the macroscopic hydrocarbon droplet. (5)We imagine the ice connected to a substrate so that it does not displace water. Also, we limit our calculations to thicknesses of the fluid films where the effects of gravity are not important. @
1000
vapor
z=-d
hydrocarbon water
0 -
500
0 0
140
280
hydrocarbon film thickness d [Ao]
Figure 1. Calculated equilibrium water film thickness L d as a function of hydrocarbon film thickness d. Inset: illustration of the interface configuration. For larger values of d the minimum free energy state is 1 with L-.
-
This suggests a discontinuous wetting transition from partial wetting (incomplete surface melting) to complete wetting (complete surface melting), which may be observed at the triple point of water and ice by tuning the thickness of the hydrocarbon layer. The physical origin of this effect is directly related to the fact that, in a wide range of frequencies, the hydrocarbon is more polarizable than the water which is more polarizable than the ice, thus enhancing those components of the van der Waals interactions which promote thickening of the water film. The calculation is based on the approach of Dzyaloshinskii, Lifshitz, and Pitaevskiie (DLP) who first calculated the excess free energy per unit area of a film of a given medium between two other media by utilizing the full frequency dependent dielectric function. Since their work, the theory has been generalized and applied to diverse s y ~ t e m s . ~The - l ~ general scheme for calculating the vdW free energy F in layered geometry is
The function W(q,o)determines the dispersion relation (6)Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. Adu. Phys. 1961, 10, 165.
(7)Ninham, B. W.; Parsegian, V. A.; Weiss, G. H.;J. Stat.Phys. 1970,
n "no
k , JLJ.
(8)Barash, Yu. S.;Ginzburg, V. L. In The Dielectric Function of Condensed Systems; Keldysh, L. V., Kirzhnitz, D. A,, Maradudin, A. A,,
Eds.;North Holland Amsterdam, 1989).
(9)Parsegian, V. A. In Annual Reuiew of Biophysics and Bioengineering; Mullins, L. J., Hagins, W. A., Stryer, L., Eds.; Annual Reviews Inc.: Palo Alto, 1973. (10)Safran, S. A. Statistical Thermodynomics ofsurfaces,Interfaces and Membranes; Addison-Wesley: Reading, MA; to be published.
Q743-7463/93/2~Q9-2786$04.00/00 1993 American Chemical Society
Letters of the electromagnetic normal modes which satisfy the correspondingboundary conditions and Maxwell's equations. For planar systems, with the surface normal in the z direction, these modes are the surface E and H waves8 which depend on the distances between the bodies and their dielectric functions. In eq 1, q, and q, are the components of the wave vector which are parallel to the surfaces and tn = (2?rk~T/h)n.The prime on the sum indicates the n = 0 term is multiplied by a factor of 1/2. We now consider the calculation of the dispersion relation. Any arbitrarily polarized plane wave may be resolved into two waves, one of which is linearly polarized with its electric field perpendicular to the plane of incidence and one of which is linearly polarized and the magnetic field is perpendicular to the plane of incidence. Therefore, one needs to consider both cases independently with the corresponding boundary conditions, which are the continuity of DI, Ell, and HII,The temporal Fourier transforms of Maxwell's equations are
( i w / c ) k , = a x E,
(2)
(iw/c)c(o)E, = -9 x R,
(3)
For the case where the electric field is erpendicularso = EyS,and H, the plane of incidence (E wave) one has = H,2 + Hz2. From eqs 2 and 3 one obtains
5,
(4) Setting E, = f(z)eiqxone obtains the general solution for f(z): f ( z ) = Ae-Pz Bep,, with p2 = q2 - (w2/c2) e(w). The magnetic field components are given by H, = -@/io)(an/ az)eiqx,and H,= (c/w)qf(z)eiqx.The function f is written in the form
+
AeP+ ifzI-d BePhz + Ce-P@ if -d Iz I0 f(z) = DePG + Ee-Pd if 0 Iz IL Fe-P" ifLIz where L is the thickness of the water film and d is the thickness of the hydrocarbon layer. The continuity of E, and H, at the interfaces yields the condition for the existence of a nontrivial solution from which one obtains a dispersion relation:
{
-
with R(xj, ~ k = ) ( ~ -j X k ) / ( x j + x k ) and p: = q2 - (w2/ c2)cj(w). This form is normalized such that A(q,o) 1for L m. When d = 0, this reduces to the familiar DLP expression for a water film between the vapor and ice halfplane^.^ For the case where the magnetic field is perpendicular = f(z)eiqxS, to the plane of incidence (H wave) one takes and obta@s, by a similar analysis, another dispersion celation, A(q,w) = 0, using the continuity o_fE, and H,.In A, R(pj,pk) is replaced by R ( P j , P k ) , where R ( X j , X k ) = (XjCjk - x k e j ) / ( x j e k + x k e j ) . The complete dispersion relation is then given by
-
a,
W(q,w)= A(q,w) &q,w) = 0 (6) Equation 1 is the excess free energy per unit area as a function of the water film thickness, L, for a fixed thickness, d, of the hydrocarbon layer. With the following changes
2.2
,
Langmuir, Vol. 9, No.11, 1993 2787 I
water
/
hydrocarbon
1.8
1 .o 2
0 1 log,,, ( %,,lev 1
-1
Figure 2. Dielectric functions of water, ice, and hydrocarbon tetradecane at the imaginary frequencies it,,.
of variables, p = ~pw[t,(if'n)]-'/~/&,x = rnp,and rn = 2&,1/2 (i[n)L/c, one obtains after some algebra
with
g ( R ) = 1~ ( x ~ , x R(Xh,X,,,) ,)
(
+ ~ ( x ~ , x R(x,,x,)e-2Thd/L ,)
1- R(x,,Xh) R(x,,x,)e-hhd/L
e-' (8)
with X u = X ; X k = [ X 2 + rn2(ek/t, - 1)I1I2. To calculate the free energy, eq 1,one needs the complex dielectric functions t(w) which can be parameterized as €(a)= 1
+
c
fj
(9)
!e - iltogj - (hw12 Their approximation from spectroscopic data has been described e l s e ~ h e r e . ~ JThe ~ * ~static constants are not included in expression 9; e,(O) = 88.2, eh(0) = 2, and ei(0) = 91.5 at 0 "C. The data for water and ice are taken from ref 3 while for hydrocarbon tetradecane we used the comprehensive data given by Parsegian and Weiss.12 For is apmost hydrocarbons, the dielectric function proximately constant (e = 2) for t = 0 to the ultraviolet and varies little from one material to another.13J4 We also used the less comprehensive data given by Hough and White15 for alkanes with carbon numbers n = 8 to n =14 and found essentially the same results. The dielectric functions as a function of imaginary frequencies are shown in Figure 2. As discussed in ref 3, the fact that surface melting of ice is incomplete is due to a crossover frequencywo = 2 X 10l6rad/s, in the ultraviolet, above which ice is more polarizable than water. This can be understood from an approximate expressiongJ0 for the free energy of a water film between ice and another J
€(in
(11) Russel, W. B.;Saville, D. A.; Schowalter, W. R.; Colloidal Dispersions; Cambridge Univerisity Press: Cambridge, 1989. (12) Parsegian,V. A.;Weiss, G. H. J. Colloid Interface Sci. 1981,81, 285. (13) It is known that vdW forces predict that various hydrocarbons have different wetting properties at the water-air interface; Bee: Richmond, P.; Ninham, B. W.; Ottewill, R. H. J. Colloid Interface Sci. 1973, 4.569. However, we are looking at the effect of hydrocarbon layers on the water-ice system, and therefore the use of one hydrocarbon over another is not expected to alter our results significantly. (14) Strickly speaking, hydrocarbontetradecane solidifies at about 6 O C . However,mixingitwithasmallamountofashorter chainhydrocarbon may prevent it from solidifying at 0 OC, without alteringsignificantlythe dielectric function. In any case, we have checked that the results are unchanged for carbon numbers 8-12 which are fluid at the triple point. (15) Hough, D.B.;White, L. R. Adu. Colloid Interface Sci. 1980,14, 3-41.
Letters
2788 Langmuir,Vol. 9,No.11,1993 1 105 0
- 4 10-5
I
0
I
400
800
0.25
0
0.5
water film thickness L [km]
water film thickness L [Ao]
Figure 3. Contribution F(L) to the surface free energy per unit area as a function of water f i b thickness for hydrocarbon tetradecane film thickness d = 50 A.
Figure 4. Free energy F(L) for hydrocarbon film thickness d = 280,290, and 400 A, indicating a discontinuouswetting transition from incomplete to complete surface melting.
medium, which could be either vapor or hydrocarbon (for the case where the hydrocarbon completely displaces the 01): vapor, d
tension and the vdW contributions, is expanded about the minimum:
-
G(L) = 2 where x = u or h. Consider first the vapor-waterhe system where ew - cu > 0 for all frequencies. Since ei - e, < 0 for f n < wo and ti - e, > 0 for &, > wg, the low-frequency contributions tend to thin the film while the high-frequency contributions tend to thicken the film. For thin water films the overall effect is a tendency for the film to thicken. However, for thicker films the high frequencies experience retardation damping due to the exponential factor in eq 10 and the tendency to thicken the film weakens. The result3 is a minimum free energy at finite thickness Lmi, = 36 A. When the vapor is completely replaced by hydrocarbon, this delicate effect is reversed. Since e, - Ch < 0 for almost all frequencies (except for In = 0 and f,, >> wo), the low frequencies (&, < woj) tend to thicken the film while the high frequencies ( f n > wo) tend to thin it. For thin films the overall effect is again to thicken the film. However, as the film thickens, retardation suppresses the high frequencies, just the ones tending to thin the film; hence, the film continues to grow. We now consider a finite layer of hydrocarbon of thickness d. The calculation shows that the free energy F(L) attains a minimum at a finite value Lmi,, corresponding to incomplete surface melting, as in the d = 0 resultS(see Figure 3 for d = 50 A). Lmh is found to increase continuously with d , reaching a value of L- = 1019 A for d E 287 A (see Figure 1 for tetradecane). When d 288 A, a discontinuous wetting transition occurs as the jumps to infinity, indicating equilibrium thickness complete surface melting (see Figure 4). Since the free energy barrier for the transition is small, one needs to evaluate the mean square fluctuations about the equilibrium water film thickness, L m h , to check that the incomplete wetting is stable to thermal fluctuations. Assume the system is at the estimated transition thickness with d = 287 A and L,i, = 1019 A. One can write the local thickness of the water film as
-
(11)
The free energy, G(L), which includes both the surface
(
(”) IT( + (”)
d i u(VLI2+ aL2 ,L (L - Lmin12)=
)hGh4 (12) aL2 L,, where u is the oil-water surface tension. Then, uq2
2
where
[-2
= (a2F/aL2)L,/u, so that
Here a E 3 A is a molecular cutoff length. Inserting Lmi, lo00 A, u = 70 erg/cm2,and (a2F/dL2)min= 4 X 1@16kBT one finds f E 0.22 cm and thus
( ( L ( i )- Lmin12) 13 A2
