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Surface States of Microdroplet of Suspension D. Jakubczyk,* M. Kolwas, G. Derkachov, and K. Kolwas Institute of Physics of the Polish Academy of Sciences, Al. Lotniko´w 32/46, 02-668 Warsaw, Poland ReceiVed: January 27, 2009; ReVised Manuscript ReceiVed: March 30, 2009
Surface thermodynamic states of single evaporating microdroplets of aqueous suspension of nanospheres were studied by analyzing the surface pressure (surface activity) isotherm. Surface pressure evolution was found from the temporal evolution of the droplet radius. The droplet radius was inferred from the azimuthal angular distribution of the irradiance of s-polarized scattered light. Several thermodynamic states of the surface layer formed of nanospheres were identified: surface gas, surface gas-liquid coexistence, two states of surface liquid, and surface solid. The collapse of the surface monolayer was detected. 1. Introduction Properties of suspensions are of importance to science, technology, and everyday life. These properties (mechanical and optical, etc.) are intrinsically connected with the properties of interfaces present in the suspension and on its surface (e.g., ref 1). An evaporating droplet of suspension makes an even more complicated system. In particular, the state of its surface changes dynamically in an intricate way.2 A drying suspension of nanoparticles can exhibit complex transitory internal structures. This phenomenon has been observed for nanoparticle suspension drying on a substrate.3,4 The evaporation of dispersing liquid can lead to the formation of a monolayer of nanoparticles3 and/ or to the formation of surface assemblies of nanoparticles of various morphologies. By analogy to colloids, it should be possible to distinguish thermodynamic states of the surface layer (e.g., ref 5). We report a study of the evaporation of single, free droplets of aqueous suspension of nanospheres. Individual droplets were levitated in the electrodynamic quadrupole trap.6,7,9 The evaporation of dispersing water led primarily to the gathering of inclusions on the droplet surface and second to the increase of their density in the droplet volume. The process can be perceived as an evaporation-driven assembly of nanostructures under the condition of spherical symmetry imposed by the surface tension of the droplet.2 We followed the evaporation process by analyzing the temporal evolution of the droplet radius. The droplet radius was found from the analysis of the azimuthal angular frequency of the interference pattern of light scattered by the droplet.10,11 The optical properties of a spherical droplet containing inclusions may depend on the inclusion (inclusion structures) morphology (see, e.g., ref 12). This would manifest in patterns of light scattered by such droplet. Some features of the scattering pattern (rainbow angle13 and angular frequency14) practically depend on a limited set of droplet parameters only. Azimuthal angular frequency of s-polarized scattered light, for a certain range of angles, carries information about the radius of a droplet, whereas it is practically independent of the droplet refractive index. It gives opportunity to determine droplet radius even when its refractive index is unknown or the droplet composition is nonuniform. This is a well-established technique used for * To whom correspondence should be addressed. E-mail: jakub@ ifpan.edu.pl.
particle sizing, e.g., in sprays (see, e.g., ref 15 and references therein). There are many variants of this sizing technique under different names: laser imaging for droplet sizing (ILIDS), interferometric particle imaging (IPI), Mie scattering imaging (MSI), and interferometric Mie imaging (IMI), etc. By analyzing the evaporation rate of a droplet (within the framework of a model that we adopted), we were able to determine the effective surface pressure evolution (changes of surface tension). As the droplet of suspension looses water and evolves from liquid to dry aggregate of inclusions, its surface undergoes transitions through various surface thermodynamic states. They can be identified by analyzing the surface pressure. We observed several states: from surface gas of inclusions (first stage of droplet drying), through surface gas-liquid coexistence, surface liquid to the surface solid (2D) followed by its collapse. In other words, the effective surface pressure reflects the formation and evolution of the surface nanostructured layer of inclusions. We believe that the evolution of thermodynamic surface states of a droplet is a fundamental phenomenon influencing evaporation rate of any droplet of broadly defined suspension. We expect that controlled drying of droplets of suspension could also be of technological value. It opens a possibility for engineering of various surface and volume micro- and nanostructures, which could be employed as photonic crystals or meta-materials.
2. Evaporation Model and Surface Phenomena The process of droplet evaporation is associated with the mass and heat transport through the droplet surface. It is driven by the gradients of vapor density and temperature around the droplet. Therefore it depends on the vapor pressure near the droplet surface pa and far from the droplet pcc, as well as on the temperature of the droplet surface Ta and the temperature of the reservoir Tcc. The conventional way of tackling the problem, which can be found, e.g., in ref 16, is based on combining equations of diffusion of mass and heat with the gas kinetic equations. In this manner, convenient expressions describing the dynamics of evaporation of a spherical droplet of pure liquid can be obtained:
10.1021/jp9007812 CCC: $40.75 2009 American Chemical Society Published on Web 05/19/2009
Surface States of Microdroplet of Suspension
a
[
pa(Ta) pcc(Tcc) da ≡ aa˙ ) ΛDa(a, Ta) dt Tcc Ta
J. Phys. Chem. C, Vol. 113, No. 24, 2009 10599
]
where droplet radius is denoted as a,
Da(a, Ta) )
D a/(a + ∆C) + D√2πM/(RTa)/(aR)
(2)
is the effective diffusion coefficient of the vapor surrounding a droplet, accounting for the gas kinetic effects, D is the diffusion constant of vapor in ambient gas, R is the evaporation constant, Λ ) M/(RF), M and F are the molecular weight and density of the liquid, respectively, and R is the universal gas constant. ∆C defines the effective range of ballistic evaporation. It is worth noting that expression 1 is proportional to the droplet surface area change rate d(4πa2)/ dt2. Due to the evaporative cooling, the temperature of the droplet (surface) Ta is slightly lower (for the discussed cases less than 0.1 K) than that of the climatic chamber Tcc.9 Though the evaporation of a droplet of suspension in a humid environment is nearly an isothermal process, accounting for the evaporative cooling is necessary in order to obtain numerically accurate results. The temperature drop at the droplet surface can also be expressed in terms of aa˙:
Ta - Tcc )
qF aa˙ λ
( )
pa ) ps exp
(1)
(4)
where σ is the effective surface tension of the suspension and ps is the saturated vapor pressure of the pure dispersing liquid. It should be stressed that pa is the equilibrium vapor pressure above the surface of a droplet of suspension. Introducing expression 4 into 1 yields
[
aa˙ ) ΛDa(a, Ta)
( )]
ps(Ta) pcc(Tcc) Λ 2σ exp . Tcc Ta Ta a
(5)
The modification of the equilibrium vapor pressure due to the surface curvature is significant only for small droplets. Since for a ) 10 µm the relative modification pa/ps = 0.999 and the initial radius of a droplet in our experiments was a0 > 10 µm, we assumed that pa(t0) = ps, which in view of eq 4 and eq 5 is equivalent to
[
(aa˙)(t0) ) ΛDa(a0, Ta0)
ps(Ta0) pcc(Tcc) Tcc Ta0
]
(6)
where Ta0 ) Ta(t0). Therefore, the surface tension of the droplet of suspension can be expressed as
(3)
where λ is the thermal conductivity of the ambient gas-vapor mixture (gas kinetic effects can be neglected) and q is the enthalpy of vaporization. However, describing the evaporation of a droplet of suspension requires a somewhat more complicated model. The surface of a droplet shrinking due to the evaporation gathers up the inclusions from the evaporated volume. As a result, a nanoparticle film forms at the surface. It can be also seen as the formation of a porous layer of decreasing pore size. It may be justly expected that the value of the evaporation rate should depend on the density and structure of the surface film. To address this issue, first we note that the interaction between totaly submersed inclusions and the evaporating molecules of dispersing liquid is perfectly negligible. The direct interaction between inclusions residing at the interface and the evaporating molecules may also be safely neglected at least for the better part of the droplet evaporation process, as it is known from the analysis of evaporation from wet porous materials (see, e.g., ref 17 and references therein). Nevertheless, inclusions residing at the interface modify the effective surface tension and the (local) curvature of the interface. The effect of surface tension modification seems to prevail. It can be interpreted as arising from the repulsive forces between inclusions coming into contact (surface pressure). All this, in turn, modifies the equilibrium vapor pressure above the interface (see, e.g., refs 17 and 18). Since the surface tension of a suspension is a well-defined measurable quantity,19 we invoke the Laplace equation to describe the equilibrium vapor pressure above the droplet as a whole. In this way it will be possible to characterize the changes of the state of the surface in terms of the effective surface tension of the suspension droplet. From the Laplace equation for a droplet of suspension the corresponding Kelvin equation follows (compare, e.g., refs 17 and 20):
Λ 2σ Ta a
σ)
{
[
]}
Ta Da(a, Ta) aTa 1 ln 1 (aa˙)(t0) aa˙ 2Λ ΛDa(a, Ta) ps(Ta) Da(a0, Ta0)
(7)
At this point, it is convenient to introduce the surface pressure (surface activity) ∆σ, describing the change of surface tension with respect to the surface tension of a pure liquid σw (comparable with a similar procedure for colloids21):
∆σ ) σw - σ
(8)
Combining eq 8 and eq 7 leads to the conclusion that the temporal evolution of the surface pressure ∆σ(t) can be inferred from the temporal evolution of the droplet radius a(t). 3. Experiment The schematic diagram of the experimental setup is presented in Figure 1. The details of the trap construction and of the experimental setup can be found in ref 2 and references therein, while, in what follows, an outline will be sketched. Single droplets of homogeneous aqueous suspension of standardized nanospheres were injected with a piezoelectric injector11 into the quadrupole electrodynamic trap6,7 mounted in a small climatic chamber.2 Our trap resembled the basic arrangement of a Paul trap,6,8 except that cap electrodes were spherical rather than hyperboloidal and the direct current (dc) voltage was applied just between cap electrodes. High resistances in the circuit suppressed discharges and enabled trap operation in humid environment. The experiments were conducted in stationary gas. The climatic chamber provided stable and homogeneous temperature field ((0.15 K relative). The absolute temperature measurement accuracy was (0.3 K. The initial humidity was controlled with 4% accuracy, monitored with 2% accuracy, and
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Figure 2. aa˙(a) (20-point adjacent averaged) for a droplet of aqueous suspension of 200 nm polystyrene spheres. Inset: Corresponding droplet radius temporal evolution a(t): raw data marked with red dots; black dots obtained after 50-point adjacent averaging. aa˙(t0) marked with dashed line.
Figure 1. Experimental setup diagram: equatorial plane sectional view of the climatic chamber.
This method was effective as long as the regular interference pattern was at least partially visible in the scattered light. An example of temporal evolution of the droplet radius a(t), obtained with the described method, is presented in the inset in Figure 2. From the experimentally obtained a(t) we calculated the temporal dependence of aa˙(t). It is convenient to combine these two quantities into aa˙(a) (see Figure 2). Then it was possible to assess the value of (aa˙)(t0), which in the presented example was -7.3 ( 0.1 µm2/s. 4. Surface Pressure Isotherm
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(as we know from our experiments on pure water ) could be regarded as constant during each experimental run (a few tens of seconds). Aqueous suspensions of polystyrene spheres of 200 nm diameter (3200A, Thermo Scientific) and SiO2 spheres of 450 nm diameter (MP-4540, Nissan Chemical Industries) were used. The suspensions obtained from manufacturers were directly diluted with ultrapure (reagent-grade) water (Milli-Q Plus, Millipore) in 100:1 proportion. Stabilizing agents introduced by the manufacturers were not removed. The resulting initial volume concentrations of inclusions were ∼1:1000 and ∼1: 500, respectively. The experiment was conducted at the temperature of 288.2 ( 0.3 K, water vapor relative humidity of 94 ( 2%, and the atmospheric pressure of 1006 ( 2 hPa. The exact values of these thermodynamic parameters were not critical; however, the stability of temperature and humidity was essential. A He-Ne, 632.8 nm, ∼18 mW, continuous wave (CW) laser beam of polarization orthogonal to the scattering plane (spolarized) was used to illuminate the trapped droplet. The azimuthal angular distribution of the intensity of s-polarized scattered light Iss(Θ,t) was recorded with the digital camera (PixelFly, pco.imaging) from the solid angle of ∼0.1 sr around the right angle in the scattering plane. As it has been mentioned in section 1, azimuthal angular frequency of Iss(Θ) (characteristic frequency present in the scattered light azimuthal angular distribution), for the range of observation angles we used, carries information about the radius of a droplet, whereas it is practically independent of the droplet refractive index.14 Carefully applied, the method is precise enough ((100 nm absolute accuracy for ideal data) and especially suitable in our case as independent of the (effective) refractive index of the droplet, which was inevitably changing during the evaporation of dispersing water.
On substitution of experimental a, a˙(a), and (aa˙)(t0) into eq 7, we found the surface pressure as a function of the droplet radius: ∆σ(a). This can be perceived as a surface pressure isotherm (Figure 3). Several regions (separated with vertical dashed lines in Figure 3) can be distinguished in ∆σ(a) dependence. In view of the evolution scenario of the suspension droplet that we proposed in ref 2 (compare with visualization of the scenario22), we associate these regions with various thermodynamic phases of the surface layer of inclusions.23,24 As can be seen in Figure 3 top and bottom, the phenomenon is essentially independent of the precise size and composition of inclusions. Similar surface states and phase transitions can be identified in the evolution of the surface pressure for both suspensions: of 200 nm polystyrene and 450 nm SiO2 nanospheres. We shall discuss this phenomenon for the case of the 200 nm polystyrene nanospheres suspension presented in Figures 3 (top, main) and 4. To identify the phases (states) of the surface correctly, we start from the plateau lying between 7.5 and 5.3 µm. We associate it with a state in which the surface gas of inclusions (G) coexists with the surface liquid of inclusions (Figure 4b). In this state, surface pressure is hardly influenced by the shrinking of the surface since the excess of the surface gas “molecules” condenses. Therefore, this state can actually be interpreted as the phase transition. In a few cases even a slight decrease of ∆σ(a) was observed. This could be associated with spontaneous formation of very large surface aggregatess islets of inclusions. For a > 7.5 µm, there is a surface gas of inclusions on the droplet surface (Figure 4a). Since in this region the experimental noise got amplified due to data processing, the modulation of the isotherm for a > 8.5 µm cannot be unambiguously interpreted.
Surface States of Microdroplet of Suspension
J. Phys. Chem. C, Vol. 113, No. 24, 2009 10601 on the surface. As the fractal dimension of the surface structure increases, the film becomes denser. For a ∼ 4 µm a phase transition to liquid-condensed (L2) is possibly observed (Figure 4d). It manifests as the kink on the isotherm (best seen in Figure 3). Further compression leads to the surface liquid-solid transition for a = 2.7 µm (Figure 4e). Then, after the rapid growth of the surface pressure, a collapse of the surface layer takes place (Figure 4f). The collapse, followed by the formation of a multilayered surface solid, manifests as the rapid decrease in the surface pressure. The values of the surface pressure we obtained for later stages of evaporation (Figure 3) depart significantly from those obtained by other authors for homogeneous aqueous colloids or suspensions (see, e.g., refs 19 and 25). However, as it has been mentioned in section 2, the repulsive forces between inclusions coming into contact become the main source of the surface pressure, which thus can become significant. Therefore, the value of surface pressure can be interpreted as a measure of the surface layer compressibility. The surface gas of inclusions is highly compressible and indeed, at this stage, the surface pressure was in average close to zero. Surface liquid(s) are significantly less compressible which leads to noticeably larger values of surface pressure. The compressibility of the surface solid is obviously even less, and the surface pressure can be perceived as a measure of the Young modulus of the surface layer. Consequently, the collapse of the surface layer corresponds to a yield point on the stress-strain curve. 5. Discussion
Figure 3. Examples of surface pressure isotherms for droplets of suspensions of 200 nm polystyrene spheres (a) and of 450 nm silica spheres (b). Phase transitions are marked with dotted lines. Isotherms from the main panels are repeated for comparison in insets in dotted lines.
Figure 4. Surface pressure isotherm from Figure 3 (main, top) in semilogarithmic scale (red line) together with snapshots from a visualization of simulation presented in refs 22 and 2. The droplet size has been scaled freely for best clarity.
For 5.3 > a > 4 µm we expect the liquid-expanded phase (L1; Figure 4c). The surface film can be described as a structure of fractal dimension below 2. The surface film is composed (as can be inferred from the simulation22) of large loose aggregates. The evaporation of water leads to the compression of surface structures, as well as to the increase of the number of inclusions
We intended to demonstrate a method of studying the surface thermodynamic states of a droplet of suspension rather than to give specific values of measured quantities. However, our model calculations were performed quite rigorously, and the issue of uncertainties can also be addressed. Surface pressure variability was, in practice, a function of a single variable only: the droplet radius (see formulas 7 and 8). Since the values of other variables and constants are known with higher accuracy, the measurement of droplet radius temporal evolution remains the main source of experimental uncertainties. As it has been mentioned in section 3 and is illustrated in the inset in Figure 2, the intrinsic uncertainty of the method used to find the droplet radius at a given moment was (100 nm. However, we know from model considerations as well as from experiments with pure water, where droplet radius was found with (15 nm precision,9 that temporal evolution of the radius can be and should be adequately smoothed in order to remove artifacts introduced by the method itself (see undulation of a 0.4 s period in Figure 2). The persistent uncertainties can be estimated for a large droplet of diluted suspension by local fitting of an evolution of a pure water droplet. We estimated them to be (20 nm. These uncertainties may originate from droplet nonsphericity, inhomogeneity, rotation, and refractive index changes, etc. They manifest distinctly for t > 12 s, when the droplet/particle is nearly dry, as undulation of a 1.5 s period. Calculation of the temporal derivative of the radius evolution at discrete data points a˙(t), apart from amplification of fluctuations present in the data, adds fluctuations dependent on datapoint density. This numerical effect was lifted by further smoothing. Next, aa˙(a) was calculated, aa˙(t0) was found, and its accuracy was estimated as a standard deviation of fluctuations for a > 7.5 µm (compare section 3).
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The final test of achieved accuracy was performed by application of formula 7. This formula had been checked against model-calculated data for pure water. The relative error of σ introduced by the formula itself was found to be ∼1%. As can be seen in Figure 3 the values of surface pressure obtained for the diluted suspension (a > 7.5 µm) fluctuate, as should be, around zero. In top main panel the mean of ∆σ for 7.5 < a < 11.5 µm is 0.1 N/m with a standard deviation of 0.8 N/m. Fluctuations are highest in this region and diminish quickly for smaller droplet radii. 6. Conclusions Extensive knowledge on the evolution of the surface (and internal) structure of an evaporating droplet of suspension can be gained by analyzing the temporal evolution of the droplet radius only. The droplet radius can be found by analyzing the angular distribution of scattered light intensity. This, in turn, by application of the Kelvin equation to the whole droplet, enables construction of the surface pressure isotherm which, in an elegant way, describes the state of the droplet surface. Various structures of inclusions forming on the surface (surface phases) can be distinguished, and the phase transitions can be identified. The phenomenon was found to be independent of the precise size and composition of inclusions. Acknowledgment. This work was supported by Polish Ministry of Education and Science Grant No. 1 P03B 117 29. References and Notes (1) Pogorzelski, S. Colloids Surf., A 2001, 189, 163–176. (2) Derkachov, G.; Kolwas, K.; Jakubczyk, D.; Zientara, M.; Kolwas, M. J. Phys. Chem. C 2008, 112, 16919–16923.
Jakubczyk et al. (3) Bigioni, T.; Lin, X.-M.; Nguyen, T.; Corwin, E.; Witten, T.; Jaeger, H. Nat. Mater. 2006, 5, 265–270. (4) Rabani, E.; Reichman, D.; Geissler, P.; Brus, L. E. Nature 2003, 426, 271–274. (5) Tolnai, G.; Agod, A.; Kabai-Faix, M.; Kova´cs, A.; Ramsden, J. J.; Ho´rvo¨lgyi, Z. J. Phys. Chem. B 2003, 107, 11109–11116. (6) Major, F.; Gheorghe, V.; Werth, G. Charged Particle Traps; Springer: Berlin, 2005. (7) Davis, E.; Buehler, M. F.; Ward, T. L. ReV. Sci. Instrum. 1990, 61, 1281–1288. (8) Paul, W. ReV. Mod. Phys. 1990, 62, 531–540. (9) Zientara, M.; Jakubczyk, D.; Kolwas, K.; Kolwas, M. J. Phys. Chem. A 2008, 112, 5152–5158. (10) Jakubczyk, D.; Derkachov, G.; Zientara, M.; Kolwas, M.; Kolwas, K. J. Opt. Soc. Am. A 2004, 21, 2320–2323. (11) Jakubczyk, D.; Derkachov, G.; Bazhan, W.; Łusakowska, E.; Kolwas, K.; Kolwas, M. J. Phys. D 2004, 37, 2918–2924. (12) Videen, G.; Sun, W.; Fu, Q.; Secker, D.; Greenaway, R.; Kaye, P.; Hirst, E.; Bartley, D. Appl. Opt. 2000, 39, 5031–5039. (13) Saengkaew, S.; Charinpanitkul, T.; Vanisri, H.; Tanthapanichakoon, W.; Mees, L.; Gouesbet, G.; Grehan, G. Opt. Commun. 2006, 259, 7–13. (14) Steiner, B.; Berge, B.; Gausmann, R.; Rohmann, J.; Ru¨hl, E. Appl. Opt. 1999, 38, 1523–1529. (15) Dehaeck, S.; van Beeck, J. Exp. Fluids 2008, 45, 823–831. (16) Pruppacher, H.; Klett, J. Microphysics of Clouds and Precipitation; Kluwer: Dordrecht, The Netherlands, 1997. (17) Chen, Y.; Wetzel, T.; Aranovich, G.; Donohue, M. J. Colloid Interface Sci. 2006, 300, 45–51. (18) Lupis, C. Chemical Thermodynamics of Materials; North Holland: New York, 1983. (19) Okubo, T. J. Colloid Interface Sci. 1995, 171, 55–62. (20) Galvin, K. Chem. Eng. Sci. 2005, 60, 4659–4660. (21) Adamson, A.; Gast, A. P. Physical Chemistry of Surfaces; Wiley: New York, 1997. (22) http://www.ifpan.edu.pl/ON-2/on22/films/dropletevolution.avi. (23) Harkins, W. The Physical Chemistry of Surface Films; Reinhold: New York, 1952. (24) Adam, N. The Physics and Chemistry of Surfaces; Oxford University Press: London, 1941. (25) Kihm, K.; Deignan, P. Fuel 1995, 74, 295–300.
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