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.