+
+
426
Langmuir 1997, 13, 426-431
Evaporation Kinematics of Polystyrene Bead Suspensions James Conway, Heather Korns, and Michael R. Fisch* Department of Physics, John Carroll University, University Heights, Ohio 44118 Received August 22, 1996. In Final Form: November 19, 1996X A study of the evaporation of polystyrene sphere-H2O suspensions placed on nonwetting surfaces was conducted using a video camera and computer-imaging interface. The height, diameter, contact angle, and mass were measured as functions of time for a range of sessile drop sizes, polystyrene sphere diameters, and initial suspension concentrations. For initial concentrations of polystyrene spheres greater than approximately 8%, the drop diameter changed by less than 5%, and universal trends, independent of the sphere and droplet size, in the time dependence of the height, mass, and contact angle were observed. Those situations for which the polystyrene sphere concentration was less than approximately 8% showed a larger variation of diameter with time. For the case where the diameter remained constant, a theoretical model, which is analogous to the Landau theory of phase transitions, successfully predicts the experimental height and mass data over the entire evaporation range.
Introduction The evaporation of spherical drops was first addressed by Maxwell in 1877 when he considered a spherical drop evaporating in a uniform and infinite gaseous medium.1 In 1882, Sreznevsky related Maxwell’s theoretical results to the evaporation of hemispherical drops resting on a plane.2 His findings showed that the rate of evaporation of a drop is approximately proportional to the vapor pressure of the liquid. Since that time, much attention has been given to sessile drops, but mostly in relation to surface tension determinations.3-5 In most cases, the drops of interest are single-component systems. A study along similar lines to our research by Bourge´s-Monnier and Shanahan6 investigated the change in contact angle during the evaporation of water and n-decane on various substrates. Shanahan and Bourge´s7 also studied the effects of different polymer surfaces on the contact angle of an evaporating water drop. These studies observed four different stages in the evaporation kinematics. They also presented a simple model that agrees with their data for the time evolution of the contact angle during the stage of evaporation in which the diameter of the drop is constant. However, what effect does a second component have on the evaporation? Recent research by Kuz shows that a droplet’s evaporation rate can be increased or decreased, dependent on the bulk liquid, by an immiscible layer of nonvolatile liquid.8 The present study includes data which indicate that polystyrene beads increase the evaporation rate of water compared to a clean water surface. The colloidal suspension is an important system in many practical applications, including medicine, surface science, paints, inks, and liquid nutrients.9 A model polystyrene bead suspension can accurately represent the realistic * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, January 15, 1997. (1) Fuchs, N. A. Evaporation and Droplet Growth in Gaseous Media; Pergamon: Oxford, 1959. (2) See ref 1. (3) Coucoulas, L. M.; Dawe, R. A. J. Colloid Interface Sci. 1985, 103, 230. (4) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169. (5) Huh, C.; Reed, R. L. J. Colloid Interface Sci. 1983, 91, 472. (6) Bourge´s-Monnier, C.; Shanahan, M. E. R. Langmuir 1995, 11, 2820. (7) Shanahan, M. E. R.; Bourge´s, C. Int. J. Adhesion Adhesives 1994, 14, 201. (8) Kuz, V. A. Langmuir 1992, 8, 2829. (9) Gelbart, W. M.; Ben-Shaul, A. J. Phys. Chem. 1996, 100, 13169.
situation of a complex fluid, i.e., a liquid containing impurities. Polystyrene particles in solution have been used as model systems in many fields of study. Three noteworthy studies, which may provide some insight into the present observations, address the formation of twodimensional crystals of polystyrene spheres10,11 and the formation of ridges and a striped pattern as a dilute suspension dries on a surface.12 Our research addresses the evaporation of a fluid from a model colloidal suspension and develops a simple approach that describes the kinematics of this system when the diameter of the droplets remains constant. The evaporation of a liquid from a sessile drop containing impurities is fairly common, occurring in solutions such as salt water or coffee and in mixtures as represented by a suspension of latex polystyrene beads. When the impurities are sufficiently dilute, a ridge forms near the perimeter of the drop as it evaporates. The present observations focus on the time dependence of the mass, diameter, height, and contact angle of polystyrene sphere-water drops as the water in the drop evaporates. In this paper, we present and discuss the effects of polystyrene sphere size, concentration, and initial droplet size on the observed time dependence of the above parameters. The evaporation ended with the formation of a ridge near the perimeter of the drop; observations of this phenomena will also be discussed. To help facilitate both the experiments and the subsequent analysis, the drops were placed on substrates of low surface free energy, which promoted beading and allowed the approximation of a spherical cap. For sufficiently high initial sphere concentration, the results show considerable similarity in behavior over a wide range of drop and polystyrene bead sizes, in addition to various concentrations of polystyrene spheres. Using a theoretical model analogous to the Landau theory of phase transitions for the case of fixed diameter droplets, predictions are made for the normalized height and mass of the drops as functions of time which are independent of bead and drop size. We also observe the effect of the polystyrene spheres on the evaporation process, as well as their behavior as a second component in the liquid drop. (10) Dimitrov, A. S.; Dushkin, C. D.; Yoshimura, H.; Nagayama, K. Langmuir 1993, 10, 432. (11) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Inanov, I. B.; Yoshimura, H.; Nagayama, K. Langmuir 1992, 8, 3183. (12) Adachi, E.; Dimitrov, A. S.; Nagayama, K. Langmuir 1995, 11, 1057.
+
+
Evaporation Kinematics of PS Bead Suspensions
Langmuir, Vol. 13, No. 3, 1997 427
between the drops of different initial volumes, we normalized the mass, height, diameter, and time. This was done as follows. The time is normalized such that the normalized time tnorm ≡ 1 when the drop is at half of its original mass. The height and diameter are normalized such that their initial values equal 1, and the mass is normalized by the following formula:
mnorm(t) )
Figure 1. (a) Mass and (b) height data for the 530-nm polystyrene spheres, 8% concentration. Initial drop sizes represented by (b) 100 µL, (2) 75 µL, (9) 60 µL, ([) 45 µL, and (1) 30 µL.
Experimental Section The experimental apparatus is rather simple. The drop evaporates inside an enclosed analytical balance. The profile of the drop is imaged using a long-focal-length microscope connected to a video camera and analyzed by means of image-analysis software. A fluorescent light, placed behind the drop and outside the balance, provides a high-contrast image without increasing the temperature inside the enclosure. The temperature inside the balance is monitored by a thermistor inside the enclosure. The balance platform and camera mount are secured to a metal base and cushioned with rubber sheets between attachments. Glass microscope slides were cleaned with Liqui-nox soap and then rinsed with methanol and purified water. After cleaning, the slide is treated with Rain-X to promote drop beading. The polystyrene suspensions used were surfactant-free, consisting of only latex polystyrene spheres suspended in deionized water. We studied five polystyrene sphere sizes ranging from 110 to 1000 nm in diameter. The original concentration of these samples was 8%; however, lower concentrations were obtained through dilution with filtered deionized water. We obtained higher concentrations by spinning samples in a centrifuge and then removing the lower concentration top portion. Using these methods, we obtained concentrations ranging from 1% to 22%. The drop images were measured to obtain the height, diameter, and contact angles, while the mass was recorded from the balance. Measurements of all five variables were made every 5 min throughout the experiment, which typically lasted for 2-4 h. The dimensions obtained from the images were converted to absolute measurements by calibrating the microscope’s image using a steel rule.
Results and Observations Fixed Initial Concentration of Polystyrene Spheres. We observed five initial drop volumes ranging from 30 to 100 µL for five sphere sizes (110, 200 , 530, 750, and 1000 nm) at 8% initial concentration. Typical results for the mass and height of these initial concentrations of spheres are shown in Figure 1. To facilitate a comparison
m(t) - mfinal minitial - mfinal
(1)
where mnorm(t) is the normalized mass, m(t) is the actual mass at time t, mfinal is the final mass, and minitial is the initial mass of the drop. In this way, the initial normalized mass is 1 and the final normalized mass to 0. The entire 8% initial concentration data set produced universal trends for the normalized height and normalized mass. These results are shown in Figure 2, where all drop and sphere sizes are represented by the same symbol. The solid lines in parts a and b of this figure are a fit to a model developed later in this paper. Figure 2c shows that most of the drops had total diameter changes of approximately 2% or less, which allowed us to treat the drop diameter as essentially fixed. The average contact angle (averaged over both left and right sides of the drop) also followed a general trend, which is shown in Figure 2d. Note that to within the typical uncertainty of the measured angles, which is (5°, there appears to be a universal trend in which the contact angle essentially ceases to change after a normalized time of approximately 1.2. The observation that the contact angle remains constant even though the height continues to decrease indicates that the polystyrene beads are, to some extent, becoming more concentrated near the perimeter. This observation suggests that this is the earliest normalized time at which ridge formation may begin to be observed and is the subject of further experiments. Variable Initial Concentration of Polystyrene Spheres. Various initial concentrations of polystyrene spheres (1-13%) of equal volume (30 µL) and sphere diameter (530 nm) were also studied. The same quantities were measured as before, and the same normalized quantities were studied. The results are shown in Figure 3a-d for six concentrations ranging from 1% to 13%. The important observation is that for this broad range of concentrations, there is no apparent universal behavior in the normalized quantities. This is especially true of the lower initial concentrations. The normalized diameters generally remained constant for initial concentrations of 8% or more. An example of this behavior is shown in Figure 3c. For the lower initial concentrations, a change in diameter was most likely to occur; however, this was not always the case. When the diameter decreased, typically one side of the drop remained pinned, while the other advanced inward. When the diameter changed by 10% or less, the behavior of the height followed the same trend. However, the data show that the diameter of the 1% and 2% samples changed by approximately 30%, and consequently, its height behavior is quite different than the other concentrations. This is shown in parts b and c of Figure 3. For all measurements, as the initial concentration decreased, the data approached that of pure water, as one would expect. The most consistent example of this trend is found in the mass results, shown in Figure 3a. This trend is also found in the height and diameter results, but with some exceptions. For example, the diameter of the 12% drop slipped by 5%, but the 13% and 8% drop diameters remained essentially fixed. We expect that these anomalies are influenced by possible nonuniformity in the surface
+
428
+
Langmuir, Vol. 13, No. 3, 1997
Conway et al.
Figure 2. Composite of normalized 8% concentration data representing (a) normalized mass, (b) normalized height, (c) normalized diameter, and (d) average contact angle as functions of normalized time. Data includes 30-100-µL drop sizes and 100-1000-nm sphere diameters.
Figure 3. Effect of sphere concentration on the measured quantities: (a) normalized mass, (b) normalized height, (c) normalized diameter, and (d) average contact angle. Example of pure water is also included. Specific concentrations are represented by (b) pure H2O, (2) 1%, (9) 2%, ([) 4%, (1) 5.5%, (O) 8%, (4) 12%, and (0) 13%.
treatment. Note also that these data indicate that the polystyrene beads enhance evaporation of the water
compared to pure water. Thus, presumably, the spheres are coated with water and make a rough air-suspension
+
Evaporation Kinematics of PS Bead Suspensions
+
Langmuir, Vol. 13, No. 3, 1997 429
Figure 4. Top view of typical polystyrene drops after evaporation. Final center height is dependent on polystyrene bead diameter. Sphere diameters represented are (a) 200 and (b) 750 nm. Figure 6. Effect of concentration on the width of the polystyrene ridge.
din ) 0.6e-0.1c + 0.3 dout
Figure 5. Effect of sphere diameter on the width of the polystyrene ridge.
surface that has a somewhat larger surface area than that of a similarly sized pure water drop. Ridge Formation. Each evaporation resulted in the formation of a solid ridge on the perimeter of the drop. As the ridge formed, the polystyrene left the bulk of the drop, creating a crater in the drop center. The details of the ridge formation are independent of the drop volume but not the sphere diameter. Smaller sphere diameters resulted in higher ridge heights and deeper centers upon complete evaporation. The 110- and 200-nm sphere trials produced drop centers with only a thin translucent film. The other sphere sizes, however, exhibited opaque drop centers and increasingly higher final center heights with increasing bead diameters. Figure 4 displays these results. We also observed the effect of sphere diameter and concentration on the final width of the ridge. The ratio of the inner diameter to outer diameter provides a good relative comparison between different ridges, provided that one of the diameters remains constant. With this constraint, as this ratio approaces unity, the width of the ridge becomes increasingly thinner. In the following observations, all of the drops observed had equal initial volumes and the outer diameters remained constant throughout the evaporation. We observed the ridge width decrease with increased polystyrene sphere diameter, as shown in Figure 5. Additionally, the width of the ridge increases (the inner diameter decreased) with increasing concentration, as shown in Figure 6, which shows the final diameter ratio (inner diameter divided by outer diameter) for various concentrations of 530-nm spheres. We found that the diameter ratios for a range of concentrations decreased exponentially according to the empirical form
(2)
where din and dout are the inner and outer diameters, respectively, and c is the concentration. This fit is also shown as the solid line in Figure 6. As the ridge width leveled out for increasing concentrations, the excess polystyrene remained in the center of the drop, which consequently increased the final height at the center. Theoretical Model for the Time Dependence of the Height and Mass. A continuing theme in condensed matter science is symmetry, including its role in the static and dynamic behavior of physical systems.13,14 A general conclusion from these works is that a theoretical approach to understanding nature is greatly simplified when the system displays symmetry. In the present case, a reasonable place to look for symmetry is in the configuration of the drops themselves. In particular, do pendent and sessile drops behave in the same manner? If, as in the model that follows, the positive h direction corresponds to sessile drops and the negative h direction to pendent drops and there is no difference in the time dependence of mass and height in these two configurations, we may conclude that positive h and negative h are equivalent. In order to verify the existence of this symmetry, the apparatus was modified to allow measurements of a drop of polystyrene spheres placed on the bottom of a glass slide. When treated with Rain-X, however, the drop would not affix to the slide’s underside. Without the Rain-X treatment, the larger free-energy difference caused the drop to cling to the slide while inverted, allowing observation of its evaporation. In order to compare like systems, we studied both inverted and upright drops placed on untreated slides. The resulting time dependence of the mass and height is shown in Figure 7. We observed identical behavior in the time dependence of the height and mass in the two different configurations. Hence, we assume that although different slide treatments may result in different height time dependencies, a symmetry with respect to positive and negative height exists which is self-consistent for a particular surface treatment. At this point, we follow an analogy to the Landau theory of phase transitions15 with the intention of predicting the normalized height and normalized mass as functions of (13) Anderson, P. W. Basic Notions of Condensed Matter Physics; Addison-Wesley: Reading, MA, 1984. (14) De Gennes, P. G.; Prost, J. The Physics of Liquid Crystals, 2nd Ed.; Clarenden: Oxford, England, 1993. (15) The following references provide more details: Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry; John Wiley and Sons: New York, 1980; pp 884 ff. Chandler, D. Introduction to Modern Statistical Mechanics; Oxford University: New York, 1987. Prigogine, I. The
+
430
+
Langmuir, Vol. 13, No. 3, 1997
Conway et al.
To lowest order in time, we assume the following time dependence for R2′(t) and R4′(t): R2′(t) ) R2(t - tc) and R4′ ) R4, i.e., a constant, where R2 and R4 are phenomenological constants and tc is the normalized time at which h ) 0. R2′(t) must have the form shown in order to obtain a situation where for t < tc, h * 0. Since in equilibrium the Gibbs free energy is a minimum, the first derivative of eq 3 with respect to h may be set equal to zero and we obtain
h)
(
)
-R2(t - tc) 2R4
1/2
(4)
Based on this expression, we see that there are two equally likely non-zero, symmetric values for the normalized height at a given normalized time of less than tc and that as t approaches tc, h becomes smaller. This is in agreement with our experimental results. The susceptibility, χ, describes the change in the order parameter as a result of a variation in the conjugate field. The second derivative of the free energy can be shown to equal the inverse susceptibility14,15 such that
∂2G 1 ) ) 2R2(t - tc) + 12R4h2 ∂h2 χ
(5)
Inserting the previous expression for the normalized height, eq 4, we obtain
χ-1 ) -4R2(t - tc) Figure 7. Time dependence of mass and height for sessile (2) and pendent (4) drops.
time. Landau theory begins with a power series expansion of the appropriate free energy in terms of a parameter that is non-zero at some point and zero at another and varies between these values. This parameter is called the order parameter. In the present model, the order parameter is the normalized height, h. What is the appropriate free energy in this case? The drops evaporate sufficiently slowly so that the system is very near equilibrium, and hence, we treat the drop as being in an equilibrium state at constant atmospheric pressure and temperature.16 Furthermore, we assume that the cooling of the sample due to evaporation is small compared to the absolute temperature and may be ignored. Thus, the drop is in a minimum Gibbs free-energy state. The free energy is dependent on time and the physical characteristics of the system. Since the diameters of the drops under observation remained nearly constant, it follows that G ) G(h,t), where G is the Gibbs free energy, h is the normalized height of the drop, and t is the normalized time. Following these premises, we expand the Gibbs free energy in powers of the normalized height. Since the system is at its minimum free energy, there is no linear term in h. Furthermore, since h and -h have been shown to be equivalent, there can be no odd powers of h in the expansion. If such terms were included, the energy would be different for h > 0 and h < 0. Thus, to leading order, the power series is
G ) G0 + R2′(t)h + R4′(t)h + ... 2
4
(3)
where G0 is the energy of an evaporated drop of zero height. Molecular Theory of Solutions; North Holland: Amsterdam, 1957. The equivalence of mean field and Landau theory is shown in: Plischke, M.; Bergersen, B. Equilibrium Statistical Physics; Prentice Hall: Englewood Cliffs, NJ, 1989; pp 52ff. (16) Conway, James. M.S. Thesis (unpublished), John Carroll University, 1996.
t < tc
(6)
The inverse susceptibility, defined by the second derivative of the free energy, has dimensions of energy divided by length squared, or force per unit length. Since the diameter of the drop is constant and the mass varies with time, on dimensional grounds, an inverse susceptibility is
χ-1 )
gm(t) ∝ m(t) d
(7)
where g is acceleration due to gravity, m(t) is the normalized mass, and d is the diameter of the drop. Using eq 6 for the susceptibility, we obtain the following expression for the time dependence of the normalized mass:
m(t) ∝ χ-1 ) 4R2(tc - t)
(8)
We have obtained expressions, eqs 4 and 8, for the normalized height and mass which are functions of the normalized time. Inserting the initial value of 1 at t ) 0 in both eqs 4 and 8 results in the following equations:
h(t) )
( ) tc - t tc
1/2
(9a)
and
m(t) )
( ) tc - t tc
(9b)
This model predicts that the mass should approach zero with the time dependence of h(t)2. In practice, a mean field theory like this does not usually predict the correct exponents.15 Experimental values of the exponents are obtained for a particular system by fitting the data to the above models, eqs 9a and 9b, replacing the exponents 1/2 and 1 by variables β and γ. Nevertheless, we still expect that the time dependence of the mass will be that of h(t)2.
+
+
Evaporation Kinematics of PS Bead Suspensions
Langmuir, Vol. 13, No. 3, 1997 431
Table 1. Least-Squares Fit Results for the Normalized Height and Mass Data
h)
(
m)
(
model
)
tc - t tc t* - t t*
intercept time
critical exponent
β
γ
)
tc ) 1.75 ( 0.01
β ) 0.6 ( 0.02
t* ) 2.04 ( 0.02
γ ) 1.18 ( 0.02
Figure 8. Residuals resulting from the least-squares fits to the normalized mass and normalized height denoted by b and 4, respectively.
Table 1 shows the results of the nonlinear fits performed on the normalized height and mass data. The critical exponents and intercept times are left as free parameters. The results of the fits are shown as the solid lines in Figure 2a and b.17 The residuals of both fits, which are equally dispersed about the mean, are displayed in Figure 8. The order parameter, which corresponds to the normalized height, approaches zero at a critical time tc. Furthermore, in the present model, the inverse susceptibility should have the same critical time as the order parameter. This is not what we observe here as the normalized mass approaches zero at a normalized time t*, which is 0.3 longer than the tc. This deviation from the Landau analogy might be explained by adsorption behavior, which implies a difference in concentration at the surface compared to the bulk of the drop. Experimental results indicate that soaps and detergents accumulate on the surface and decrease the surface tension of the water.18 Conversely, inorganic salts accumulate in the bulk of the drop and increase the surface tension. Although the present mixtures are neither of the above, similar adsorptive behavior could be expected. In support of this assumption, we have observed colored reflections from the surfaces of some of the drops; this is presumably due to interference in a surface layer. We may then assume that this surface layer slows the evaporation rate, prolonging the change in the mass past the critical time as defined by the order parameter. This results in a normalized time t* that is approximately 0.3 greater than (17) Another test of the absence of the h3 term in eq 3 was found by including it in eq 3 and solving for the equations, analogous to eqs 9a and 9b. When the coefficient of the h3 term was an independent variable, the resulting least-squares fits obtained a value of this coefficient much smaller than either R2 or R4. Thus, any departure from the equivalence of h and -h is rather small. (18) Vemulapalli, G. K. Physical Chemistry; Prentice Hall: Englewood Cliffs, NJ, 1993.
[ [
error matrix
0.0189 0.0144
0.0144 0.0136
0.0223 0.0191
0.0191 0.0169
] ]
the critical time, which is consistent over the entire 8% concentration data set. These results indicate the following successes of the model: (a) both sessile and pendent drops are predicted and observed to have the same time dependence of the mass and height; (b) universal behavior is both predicted and observed; (c) within the uncertainty of the fits, m(t) ∝ h(t)2, as predicted by this mean field theory; and (d) although this system will ultimately be described solely by statistical mechanics, we have found that the phase transition analogy serves well as an initial step toward developing a phenomenological theory of colloidal drop evaporation in those situations where the diameter remains constant. These results may be compared to the study of Shanahan and Bourge´s,7 who predict that under the assumption of spherical cap drops and the condition of constant diameter and for pure solvent evaporation, the height should decrease linearly with time. Such behavior is in fact observed in their experiment during one stage of the evaporation. Even though both models are predictions for constant diameter drops, attempts to apply this result to the present work are difficult at best. First, the present drops systematically deviate from the spherical caps. For spherical cap drops, the following relationship exists between the drop radius, r, height, h, and contact angle, θ: (h/r)/(tan(θ/2)) ) 1. Analysis of the present experiment at times well before ridge formation yields an average value of 0.9 ( 0.1. Moreover, and more importantly, the observed height is nonlinear over the whole time range. Its time dependence is extremely well fit to a 0.6 power in the time difference for the whole time of evaporation. Conclusion In summary, the research presented here provides detailed observations of the time dependence of the mass, height, diameter, and contact angle during evaporation of colloidal suspensions. We have observed the shift in the kinematic behavior toward that of pure water with decreasing initial suspension concentration. Also, preliminary observations were made concerning the effects of sphere concentration and diameter on perimeter ridge formation. Finally, for the case of constant diameter, the universal trends in height and mass were predicted through a theoretical model analogous to the Landau theory. Future research will investigate other surfaces as well as a more complete study of ridge formation. Acknowledgment. We thank the Research Corporation for their generous support and the Clare Luce Booth Scholarship committee for their financial support of Ms. Korns. We express our gratitude to Mr. Daniel Harrison for extensive insight and discussion. We also thank the referees for their comments. This research was financially supported by the NSF under Grant No. DMR-9321924. LA960833W
