J. Phys. Chem. 1989, 93, 3102-3703
3702
Therefore, applying the definitions of eq 36 and 38 to eq A5, we obtain a(E)
+ a ( - E + 2Ee) = 1
('46)
where eE, = eEe
+ PA, + Y2 k T In {(l + (3)/@) < 1 in eq 42, we see &E) < g ( E )
Setting E = Ee in eq A6, we finally obtain a ( P )=
72
Appendix B Using eq 42, we prove here that &Ee) < 'I2and a N ( E e ) > hold for a finite value of @. W e define a monotonically increasing function of E as
(B2)
Using the property tanh x
(B3)
Then, we obtain a'(Ee)
< g(Ee) < g(Ee + ( k T / 2 e ) In ((1 + @ ) / P I ) = -P + (P(1 + P ) ) 1 / 2< Y2
034)
That is, we have proved a'(Ee) Since &Ee)
< Y2
(B5)
+ a N ( E e ) = 1 holds, we obtain a N ( E e )>
72
(B6)
A Study of the Evaporation Rates of Small Water Drops Placed on a Solid Surface K. S. Birdi,* D. T. Vu, Fysisk- Kemisk Institut, Technical University, Building 206, 2800- Lyngby, Denmark
and A. Winter Geological Survey of Denmark (D.C.U),Thoravej-8, Copenhagen, Denmark (Received: July 6, 1988; In Final Form: November 9, 1988)
The evaporation rate of a sessile drop of water resting on a smooth solid surface (such as glass) has been studied at very high sensitivity. The rate of evaporation of sessile drops is found to depend on the radius of the liquidsolid interface. The radius of the liquidsolid interface was found to remain constant. Since drop volume decreases, the contact angle must decrease. The shape of a sessile drop during evaporation was monitored by photography.
Introduction
Rates of evaporation from liquid drops, sessile or suspended in air, have been studied by various investigator^.]-^ These investigations have been found to be useful in understanding other complex systems, such as clouds, fogs, oil burners, or engines, etc. In a recent study,1° it was shown that all sessile drops formed by condensation of their own vapor have a positive free energy of formation which can be expressed as a function of the equilibrium contact angle and drop size. This expression included the free energy of both the solid-liquid and the liquid-vapor interface. If there is no contact and the mechanical equilibrium of the drop is rapidly established during drop growth, the expression for the vapor pressure is the same as that given by the Kelvin equation applied to the liquid-vapor interface alone.I0 However, although ( I ) Houghton, H. G. Physics 1933, 4 , 419. (2) Ranz, W. E.; Marshall, W. R. Chem. Eng. Prog. 1952, 48(3), 141; 1952, 48(4), 173. (3) Fuchs, N. A. Evaporation and Droplet Growth in Gaseous Media; Pergamon: London, 1959. (4) Ivchenko, 1. N.; Muradyan, S. M. Izu. Akad. Nauk. SSSR, Mekh. Zhidk. Gaza 1982, 112. (5) Ivchenko, I. N. Sou. Phys. Dokl. 1984, 29(1), 59. (6) Yang, W.-J.; Nouri, A. A. Lett. Heat Mass Transfer 1981, 8, 115. (7) Lou, Y. S . J . Appl. Phys. 1978, 49(4), 2350. (8) OBrien, R. N.; Saville, P. Langmuir 1987, 3, 41. (9) Ray, A. K.; Lee, J.; Tilley, H . L. Langmuir 1988, 4 , 631. (IO) Schrader. M . E.; Weiss, G. H. J . Phys. Chem. 1987, 91, 353.
0022-3654/89/2093-3102$01.50/O
a number of investigations on this subject have been published, there is still a need for accurate experimental measurements of the evaporation of liquid sessile droplets resting on solid surfaces. This study reports the evaporation rate of a sessile drop (water) resting on a solid (glass) surface. Results and Discussion When a liquid is carefully placed on the surface of a given solid, it remains as a drop with the formation of a contact angle between the liquid and solid phases, provided the liquid does not wet the solid. The magnitude of the contact angle depends on the physical characteristics of both the liquid and the solid phase.11,'2 It is of importance to understand the physical forces that stabilize such systems from the determination of the dimensions of the sessile drop and the curvature of the liquid-vapor interface. The higher value of the contact angle of a given volume of a liquid always gives a thicker sessile drop with a smaller base radius. The contact angle (and hence the solid substrate) plays therefore an important role in the rate of evaporation of the sessile drop. The rate of evaporation of droplet a t a given moment can be expressed by rate = I = -dm/dt = -p(dV/dt) (1) ( 1 1) Chattoraj, D. K.; Birdi, K. S. Adsorption & the Gibbs Surface Excess; Plenum Press: New York, 1984. (12) Birdi, K. S.; Winter, A,; Vu, D. T. Colloid Polym. Sci. 1988, 266, 470.
$2 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3103
Evaporation Rates of Small Water Drops
.250. E-6
cn
I
1
I
I
C
1 0 0 . E-6
-
1
2
2.2
I
2.4
1
2.6
I I 2.8 3 Radius [ m m l
Figure 2. Plot of evaporation rate against radius of liquidsolid interface.
Ttme [minl
Figure 1. Weight of sessile drop (water/glass) as a function of time. Drop weights are 5 X 1O-’g (X), 10 X g (A),and 15 X lo-’ g (+). TABLE I: Evaporation Rate of a Water Drop on a Glass Surface with Different Drop Sizes at 22 “C drop volumz, uL 5
10 15
1.6 d m i n 0.000 1 17 5 0.000 162 7 0.000 202 8
“Liquid drop volumes were measured by a syringe. *The average value of two experiments. where m is the mass of the drop, Vis the volume of the drop, p is the density of liquid, and t is the time. Sessile drops of water with various volumes (5, 10, and 15 pL), which were placed on a clean glass surface ( 2 X 2 cm2 microscope glass slide), were investigated. Glass slides were cleaned by rinsing in ethanol, followed by acetone (and left overnight before use), resulting in a mildly hydrophobic surface. The rate of evaporation of a sessile drop was investigated by weighing. The experimental measurements were conducted in a chamber with controlled temperature and humidity. The weight of a liquid drop was measured with a sensitivity of f 5 p g (at a rate of 0.1 s). The plots of the drop weight of different initial drop sizes as a function of time are given in Figure 1. These data showed that the rate of evaporation of water on glass is a linear function of time. The magnitude of the contact angle, 0, was measured a t time = 0 and found to be 41°, a mildly hydrophobic condition (independent of drop volume). The more common hydrophilic condition gives a contact angle of very nearly zero. When a very small liquid drop is placed on a horizontal plane, it is safe to assume that the drop has the shape of a spherical segment. Accordingly, the volume of a sessile drop, V, can be expressed by lo I/ = ar,3(2 - 3 cos
e + cos3 @/(3
sin3 e)
(2)
where r b is the radius of the base (Le., radius of liquid-solid interface) and 0 is the contact angle. From photographic measurements, it was found that the magnitude of the radius, rb, remained constant during the evaporation process. On the other hand, the magnitude of 6 decreased. By using eq 2, we can estimate the magnitude of the radius of the liquidsolid interface. The dependence of the rate of evaporation on the radius, r, was estimated from the data in Figure 1 (Table I). It is seen that if the rate is dependent on rb, then the plots of (I = -dm/dt) versus rb should be linear. This is indeed seen in Figure 2, where the rate is plotted versus radius for drops of radius greater than 2 mm and less than 3 mm. The data in Figure 2 fit the following relation: rate = -p(dV/dt) = a ,
+ u2(rb)
(3)
From linear regression analysis the magnitude of the slope of the
plot is found to be equal to 9.2119 X g/(min mm). The correlation coefficient for the plot is 0.999983. These observations thus indicate that even when drops of diameter ca. 3 mm were studied, the degree of nonsphericity is much lower than the experimental accuracy. This shows that the rate of evaporation of a sessile drop in the present case is linearly proportional to the radius of the liquidsolid interface. These data can be explained by considering the evaporation as being essentially a gas diffusion The case of a simple system, Le., stationary evaporation of a spherical droplet, and motionless relative to an infinite (uniform) medium, has been described in the literature. It is assumed that the vapor concentration at the interface of the liquid drop is equal to its equilibrium concentration, co (which is the saturation vapor a t the temperature of the drop). The rate of diffusion of the vapor of the droplet across a spherical interface with radius r was given as3 rate = I = -4nr2(dc/dr)D g / s
(4)
where D is the diffusion constant of the vapor and c its concentration (g/cm3). At infinite distance from the liquid interface, the following boundary conditions exist:
c = c,
03
(5)
c = co when r = rd
(6)
when r =
and where rd is the radius of the liquid drop. From the above equations one obtains3
I = hrdD(C0 - C m )
(7)
It is thus seen that under these conditions the rate of evaporation, I , is completely determined by the rate of diffusion of the vapor. Further, it is also observed that the rate of evaporation should be proportional to the diameter (2r) of the sphere rather than the surface area. This model thus describes the present data satisfactorily. Although, it must be stressed that any nonhorizontal liquid surface at which evaporation is taking place would generate some density gradient, and hence some convection motion must be present to a varying degree. Under the present experimental conditions, however, it seems that these effects are negligible.
Conclusions The evaporation of a water drop resting on a glass surface is a stationary process, where the rate of evaporation is constant. The rate of evaporation of small spherical sector drops is proportional to their radii.3 The evaporation of a sessile drop in the present case is linearly proportional to the radius of the liquidsolid interface. The radius of the liquidsolid interface remains constant, but the contact angle decreases during evaporation. Acknowledgment. We thank the Ministry of Energy (Denmark) and the Nordic Ministry (“Nordisk Ministerraad”) for support of this research. Registry No. H,O, 7732-18-5.
