Computer Simulations of Homogeneous Deposition of Liquid Droplets

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Computer Simulations of Homogeneous Deposition of Liquid Droplets Serge Ulrich,† Serge Stoll,*,† and Emile Pefferkorn‡ Analytical and Biophysical Environmental Chemistry (CABE), Department of Inorganic, Analytical and Applied Chemistry, University of Geneva, Sciences II, 30 quai E. Ansermet, CH-1211 Geneva 4, Switzerland, and Institut Charles Sadron - UPR 022, Centre National de la Recherche Scientifique, 6 rue Boussingault, 67083 Strasbourg Cedex, France Received September 2, 2003. In Final Form: December 3, 2003 Understanding the deposition of liquid droplets on surfaces is essential in many environmental and industrial processes. In this paper, we describe computer simulations of homogeneous deposition of liquid droplets on an ideal, smooth, and horizontal solid surface. The statistical evolution of droplet deposition and growth processes are investigated. It is found that three basic events, namely, deposition, incorporation, and coalescence, produce droplets of different sizes and that the droplet size polydispersity is continuously increasing with time leading to a bimodal distribution. By considering the total number of droplets Ntot on the surface versus time, we demonstrate that four growth regimes must be considered. These regimes reflect the relative influence of the three events during the deposition process. Variation with time of the surface coverage Γ as a function of the contact angle θ between the droplets and the surface is also investigated. The effect of a variable contact angle on the value of surface saturation and dynamical growth is calculated. Finally, by considering the total mass of the deposited material and surface coverage, some guidelines to achieve an efficient droplet deposition process and surface coverage versus total droplet mass are proposed.

1. Introduction Wetting of surfaces by liquid droplets constitutes an important field in physical chemistry, statistical physics, and fluid dynamics to describe the formation of films and evolution of the interface of many natural and synthetic systems.1-6 Fundamental understanding of deposition and spreading processes of liquid droplets on surfaces is also essential in many industrial processes as well as engineering applications such as inkjet printing and spray coating. Liquid droplet deposition also plays important roles in environmental science and technology such as in the deposition of pesticides on substrates.7 In all cases, the behavior of such systems depends largely on the ability of the liquid droplets to adsorb and adhere to the surfaces, as well as on the structure of the adsorbed layer. Nonetheless, in view of the complexity of such processes, applications to real systems are often based on empirical and semiempirical observations8-12 and predictions based * To whom correspondence should be addressed. E-mail: [email protected]. † Analytical and Biophysical Environmental Chemistry (CABE), Department of Inorganic, Analytical and Applied Chemistry, University of Geneva. ‡ Institut Charles Sadron - UPR 022, Centre National de la Recherche Scientifique. (1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 828-862. (2) Chebbi, R. J. Colloid Interface Sci. 2000, 229, 155-164. (3) Kalliadasis, S.; Chang, H.-C. Ind. Eng. Chem. Res. 1996, 35, 2860. (4) Shikhmurzaev, Y. D. Phys. Fluids 1997, 9, 266. (5) Varma, T. D.; Sharma, L. K.; Varma, S. Surf. Sci. 1974, 45, 205. (6) Stroem, G.; Fredriksson, M.; Stenius, P.; Radoev, B. J. Colloid Interface Sci. 1990, 134, 107. (7) Basu, S.; Luthra, J.; Nigam, K. D. P. J. Environ. Sci. Health, Part B 2002, 37, 331-344. (8) Sondergard, E.; Kofman, R.; Cheyssac, P.; Stella, A. Surf. Sci. 1996, 364, 467-476. (9) Aste, T.; Botter, R.; Beruto, D. Sens. Actuators, B 1995, B25, 826-829. (10) Carlow, G. R.; Barel, R. J.; Zinke-Allmang, M. Phys. Rev. B: Condens. Matter 1997, 56, 12519-12528. (11) Haderbache, L.; Garrigos, R.; Kofman, R.; Sondergard, E.; Cheyssac, P. Surf. Sci. 1998, 410, L748-L756.

on theoretical or analytical models13-18 which account for the dynamical growth of droplets at the surface are still desirable. To describe irreversible adsorption kinetics and the structure of the layers formed by deposited objects (usually spherical particles), various models can be considered. The most popular model on a 2D surface is the random generalized sequential adsorption (RSA) model,19-27 which has recently received many extensions such as reversible desorption and surface mobile particles.28,29 The RSA model assumes the deposition of monodisperse and hard circles on a surface having periodic boundary conditions. The x,y positions of the circle centers are chosen randomly at the deposition plane. Once adsorbed, it is usually supposed that circles do not diffuse or desorb. If one circle i overlaps during the deposition procedure an already deposited circle j, the deposition trial is not accepted and then a new attempt, independent (12) Zinke-Allmang, M.; Feldman, L. C.; Van Saarloos, W. Phys. Rev. Lett. 1992, 68, 2358-2361. (13) Meakin, P. Rep. Prog. Phys. 1992, 55, 157-240. (14) Family, F.; Meakin, P. Phys. Rev. A 1989, 40, 3836-3854. (15) Family, F.; Meakin, P. Phys. Rev. Lett. 1988, 61, 428-431. (16) Blackman, J. A.; Brochard, S. Phys. Rev. Lett. 2000, 84, 44094412. (17) Meakin, P.; Family, F. J. Phys. A: Math. Gen 1989, 22, L225L230. (18) Aste, T. Phys. Rev. E 1996, 53, 2571-2579. (19) Talbot, J.; Tarjus, G.; Van Tassel, P. R.; Viot, P. Colloids Surf., A 2000, 165, 287-324. (20) Senger, B.; Voegel, J.-C.; Schaaf, P. Colloids Surf., A 2000, 165, 255-285. (21) Meakin, P.; Jullien, R. Physica A (Amsterdam) 1992, 187, 475488. (22) Swendsen, R. H. Phys. Rev. A 1981, 24, 504. (23) Lavalle, P.; Schaaf, P.; Ostafin, M.; Voegel, J.-C.; Senger, B. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 11100-11105. (24) Carl, Ph.; Schaaf, P.; Voegel, J.-C.; Stoltz, J.-F.; Adamczyk, Z.; Senger, B. Langmuir 1998, 14, 7267-7270. (25) Meakin, P.; Jullien, R. Phys. Rev. A 1992, 46, 2029-2038. (26) Loscar, E. S.; Borzi, R. A.; Albano, E. V. Phys. Rev. E 2003, 68, 041106/041101-041106/041109. (27) Filipe, J. A. N.; Rodgers, G. J. Phys. Rev. E 2003, 68, 027102. (28) Jullien, R.; Meakin, P. J. Phys. A 1992, 25, L189-L194. (29) Tarjus, G.; Schaaf, P.; Talbot, J. J. Chem. Phys. 1990, 93, 8352.

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of the previous trial, is performed again. For each trial, either successful or not, simulation time is incremented by one unit. At a high surface coverage, corresponding to long simulation times, it becomes more and more difficult to deposit a new circle. The system has then an asymptotic behavior, and surface coverage is characterized by a jamming limit Γ∞ ≈ 0.547. When monodisperse circles are allowed to diffuse along the surface, they can organize themselves as an hexagonal crystal and a maximum surface coverage equal to Γ∞ ≈ 0.91 is then achieved.19,20 However, understanding the thermodynamic as well as kinetic factors controlling the deposition of 3D liquid droplets impinging and coalescing on a surface is expected to require also the knowledge of parameters involving the surface chemistry, droplet composition and size, and temperature, for example, so as to calculate the resulting contact angle between droplets and surface and take into account possible incorporation and coalescence events between the deposited droplets. A decade ago, models were proposed for liquid coalescence and growth on a ddimensional surface mainly to describe the condensation of droplets on a surface, which is a basic process in a large class of phenomena in material science such as vapor deposition of thin film or the formation of dew. Scaling behaviors were observed in these models and comparisons were made in good agreement with a number of experimental situations.30-32 In particular, the size distribution for droplet sizes, polydispersity, droplet growth, and to a lesser extent surface coverage were considered.15,17,21 It was shown that the behavior of the growth is strongly dependent on the number of deposition sites on the substrate, and two models were proposed: heterogeneous and homogeneous growth processes. Homogeneous deposition means that all x,y coordinates at the surface are possible deposition (or condensation) sites, whereas heterogeneous deposition yields specific places or sites. Focusing on dew formation, which is a phenomena experienced daily, Beysens30-34 demonstrated that general laws can be deduced to describe the evolution of the mean droplet sizes and that the surface coverage saturates at Γ ≈ 0.55 for long times in good agreement with experiments.35 An experimental and theoretical description36 of the early stages of coalescence of two water droplets was also investigated by considering the case of partial wetting. It was demonstrated that the characteristic relaxation time was proportional to the droplet radius. A theoretical study related to the kinetics of relaxation of sessile droplets toward equilibrium under the action of surface tension forces was also presented.37 Theoretical predictions of the early stage of coalescence of two droplets were compared with studies of coalescence of helium drops,38 and a droplet sliding on a tilted solid plane39,40 was considered. Reviews and papers on sessile drops and spreading can also be

found in the literature,41-46 as well as droplet deformation in dispersing flow,47 droplet impact and solidification,48 deposition of tin droplets on a steel plate,49 or numerical investigation of the motion of a growing droplet in a cloud chamber50 just to quote a few. Most of the predictions are based on theoretical or analytical models that contain the essentials of the droplet dynamical growth.13-18 In general, good agreements with model predictions are observed. In this paper, we consider the homogeneous deposition of droplets on an ideal, smooth, and horizontal solid surface of size L × L with periodic boundary conditions. Homogeneous deposition is discussed here by considering different contact angles and their important effects of surface coverage, the droplet growth process, and total number of objects at the surface. The effect of the relative size of deposited droplets versus total accessible surface is also investigated. The statistical evolution of droplet deposition and growth processes is regulated by three dynamical basic events: deposition, incorporation, and coalescence. These events lead to systems having a wide range of drop sizes that we describe by calculating the moments of the size distribution and droplet polydispersity. In particular, and for practical applications so as to control for example the probability of contact between active compounds present in droplets and surface, we focus on the variation with time of the surface coverage Γ and the total number of droplets at the surface. We also discuss the total mass of the deposited material versus surface coverage Γ in order to rationalize the deposition procedure for economic reasons. To our knowledge, no experimental data are presently available to correlate directly with our simulations. The paper is organized as follows. Deposition, incorporation, and coalescence are discussed first, and the model behavior is presented for a given contact angle θ ) π/2. As the mathematical algorithm requires the determination of the largest radius rh of a deposited droplet according to its initial radius r0 prior to adsorption, key mathematical expressions are given. The variation of the effective number of droplets and relative proportion of the three events, deposition, incorporation, and coalescence, versus time are discussed owing to their relative importance in the dynamical growth process. The last part of the discussion addresses the variations of the surface coverage Γ versus some characteristic sizes and the effect of the contact angle modification. Finally, some guidelines are given to achieve an efficient droplet deposition process by considering surface coverage versus total droplet mass.

(30) Beysens, D.; Knobler, C. M. Phys. Rev. Lett. 1986, 57, 14331436. (31) Beysens, D.; Steyer, A.; Guenoun, P.; Fritter, D.; Knobler, C. M. Phase Transitions 1991, 31, 219-246. (32) Beysens, D. Atmos. Res. 1995, 39, 215-237. (33) Fritter, D.; Knobler, C. M.; Roux, D.; Beysens, D. J. Stat. Phys. 1988, 52, 1447-1459. (34) Zhao, H.; Beysens, D. Langmuir 1995, 11, 627-634. (35) Viovy, J. L.; Beysens, D.; Knobler, C. M. Phys. Rev. A 1988, 37, 4965. (36) Andrieu, C.; Beysens, D. A.; Nikolayev, V. S.; Pomeau, Y. J. Fluid Mech. 2002, 453, 427-438. (37) Nikolayev, V. S.; Beysens, D. Phys. Rev. E 2002, 65, 046135. (38) Maris, H. J. Phys. Rev. E 2003, 67, 066309. (39) Kim, H.-Y.; Lee, H. J.; Kang, B. H. J. Colloid Interface Sci. 2002, 247, 372-380. (40) Thiele, U.; Neuffer, K.; Bestehorn, M.; Pomeau, Y.; Velarde, M. G. Colloids Surf., A 2002, 206, 87-104.

(41) Lopez, J.; Miller, C. A. J. Colloid Interface Sci. 1976, 56, 460468. (42) Neogi, P.; Miller, C. A. J. Colloid Interface Sci. 1982, 86, 525538. (43) Gu, Y.; Li, D. Colloids Surf., A 1998, 142, 243-256. (44) Voue´, M.; Semal, S.; De Coninck, J. Langmuir 1999, 15, 78557862. (45) Brochard-Wyart, F.; de Gennes, P. G. Langmuir 1994, 10, 22402443. (46) Seveno, D.; Ledauphin, V.; Martic, G.; Voue´, M.; De Coninck, J. Langmuir 2002, 18, 7971. (47) Feigl, K.; Kaufmann, S. F. M.; Fischer, P.; Windhab, E. J. Chem. Eng. Sci. 2003, 58, 2351-2363. (48) Pasandideh-Fard, M.; Bhola, S.; Chandra, S.; Mostaghimi, J. Int. J. Heat Mass Transfer 2002, 45, 2229-2242. (49) Pasandideh-Fard, M.; Bhola, S.; Chandra, S.; Mostaghimi, J. Int. J. Heat Mass Transfer 1998, 41, 2929-2945. (50) Utheza, F.; Garnier, F. J. Aerosol Sci. 2003, 34, 993-1007.

2. Model Description When a spherical liquid droplet with a volume v0 is deposited onto a solid surface, it does not keep its spherical shape because of the formation of a contact circle whose size is controlled by the contact angle (Figure 1). The

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than or equal to L/2 to avoid finite size effects due to the periodic boundary conditions of the surface plane. b. Largest Radius rh. The largest radius rh of a deposited droplet has to be calculated according to r0 and the contact angle θ. Owing to mass and volume conservation, two droplets coalescing give rise to one new droplet with a volume equal to the sum of the two coalescing droplet volumes. We can relate the volume of the droplet before deposition and the deposited one, which is assumed to be a truncated sphere, according to Figure 1. The liquid droplet before deposition fits a sphere of radius r0 and volume v0. When a liquid droplet is deposited at a solid surface, it generally deforms from its initial spherical shape and spreads. The adsorbed droplet fits in a truncated sphere of radius r2. The surface in contact with the deposition plane is a disk of radius r1. The radii r1 and r2 are strongly dependent on the contact angle θ. The largest radius rh is equal to r1 or r2 depending on the θ value.

surface of the spherical cap in contact with the deposition plane constitutes a circle of radius r1 called the contact radius. Thus, surface coverage is equal to the sum of the areas described by all the contact circles divided by the area of the deposition plane. The spherical cap fits in an equivalent sphere with a radius r2. The angle of contact between the droplet and the surface (contact angle) is referred to as θ. According to the abilities of the droplets to spread on the surface, two distinct equilibrium regimes can be defined: partial wetting, when a finite contact angle is defined, or complete wetting, when θ ) 0. We will consider here the case where θ ∈ ]0; π[ and in particular θ ) π/2, that is, when hemispherical caps are formed at the surface. As previously mentioned, the surface coverage process is controlled by three dynamic events: deposition, incorporation, and coalescence. a. Deposition, Incorporation, and Coalescence Events. At each simulation step, a droplet k of volume v0 is randomly deposited on an horizontal L × L surface with periodic boundary conditions. If the droplet is deposited at a free position (no overlap), then a new “object” is added at the surface with two characteristic radii r1 and r2 (Figure 1). If overlapping occurs with a droplet j which is present at the surface, meaning that the center to center distance is smaller than the sum of the largest radii rh (r1 or r2 depending on the θ value, see Figure 1) of the two considered droplets, then the droplet is incorporated into the droplet j of volume V′. A new droplet is then formed with a volume v0 + V′. On the other hand, coalescence is resulting from an incorporation which provokes an increase of the radius of a droplet j so as to induce overlapping with a droplet i. During the simulations, tests are made to detect possible coalescences after each incorporation. We make the assumption that the droplet resulting from coalescence relaxes instantly to a spherical cap. After coalescence, the new droplet can either grow independently or can still enlarge if overlapping with another droplet is possible. The contact radius rg at the surface plane of the droplet resulting from coalescence of droplets j and i of radius rj and ri is given by 3

rg ) xri3 + rj3

(1)

The x,y coordinates of a new droplet resulting from incorporation or coalescence are calculated by considering the mass center of its relatives. The simulation ends after a given number N0 (usually several millions) of steps or when large droplets are achieved. The simulations are stopped when the radius of the largest droplet is greater

4 1 1 v0 ) πr03 ) πh(3r12 + h2) ) πh2(3r2 - h) (2) 3 6 3 where h represents the height of the deposited droplet and is related to r1 and θ using

h ) r1R

(3)

with R ) 1/sin θ - 1/tan θ. Thus, one can find r1 and r2 as a function of the volume v0 (and thus r0) and the contact angle θ [deg] using 3

r1 )

x

r2 )

6v0

π(3R + R3)

3v0 + h3π 3h2π

(4)

(5)

According to these equations, one observes that when θ f 0, then r1 f ∞ and r2 f ∞, but the radius r2 tends toward the infinite much more quickly than r1. On the other hand, when θ f π, then r1 f 0 and r2 f r0. When θ ) π/2, a particular and important case is obtained; r1 ) r2 ) rh and the deposited droplet of volume V is represented by a hemisphere. Because of its conceptual and mathematical simplicity, this case will be investigated first. c. Contact Surface. If one considers two identical hemispherical droplets of radius r, having a contact surface on the deposition plane equal to 2A, upon coalescence, the new droplet is expected to have a surface A′ which is lower than 2A:

A′ ) 22/3πr2 ) 22/3A

(6)

The surface which is made free by coalescence is due to the difference of dimensionality between the droplets and the deposition plane. However, because of the continuous droplet deposition and incorporation events with time surface coverage Γ is expected to continue to increase globally. Nonetheless, according to eq 1, we note when the difference between the radii rj and ri is important, for instance when rj . ri, the radius rg of the droplet resulting from incorporation or coalescence is close to rj. Hence these events will not change notably surface coverage. Deposition, incorporation, and coalescence will induce a wide range of sizes through complicated dynamical processes. The deposition rate is here not limited by geometric constraints due to particles already present at the surface. Because of coalescence between polydisperse droplets (having different masses and volumes), the position of the droplet centers is expected to change with time. d. Characteristic Sizes. Average droplet sizes are investigated versus time by considering the moments of the droplet size distribution Nj ) ∑inixji where n represents the droplet number and x is either the droplet radius r, contact surface A, or volume V. N1/N0 represents the

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Figure 2. Snapshots of a deposition process versus time with L ) 400, rh ) 1, N0 ) 1.5 × 106, and θ ) π/2. The surface coverages Γ are equal to (a) 0.17, (b) 0.32, (c) 0.53, (d) 0.64, (e) 0.79, and (f) 0.82.

number-average size, and N2/N1 represents the weightaverage size. Surface coverage Γ and the total number of droplets present on the surface Ntot are also reported as a function of time. Using a dynamic-scaling approach, Family and Meakin investigated the kinetics of droplet growth in a homogeneous15 and heterogeneous17 deposition process. The exponents describing the scaling of the droplet size distribution and the growth law for the mean droplet size were calculated exactly. They demonstrated that for homogeneous deposition the weight-average volume S(t) increases with time following the scaling relation

S(t) )

∑i nivi2 ∑i nivi

3. Results and Discussion Simulations were first carried out with L ) 400, r0 ) 1, a total number of deposited droplets N0 equal to 1.5 × 106, and θ ) π/2. Snapshots (Figure 2) were taken at t ) 104, 2.49 × 104, 7 × 104, 1.4 × 105, 106, and 1.5 × 106 with estimated surface coverages Γ equal to 0.17, 0.32, 0.53, 0. 64, 0.79, and 0.82, respectively. As shown in snapshots a and b, continuous droplet deposition increases surface coverage and the droplet radius size polydispersity p which is defined as

p) z

∼t

(7)

where the dynamic exponent z value is related to the droplet dimensionality D and the dimension of the space d (here D ) 3 and d ) 2). Although the dynamic exponent was shown to be theoretically equal to z ) D/(D - d) ) 3, practically z was found to be equal to 2.8 indicating that their simulations were not in the full asymptotic regime.

N2N0 N12

(8)

remains close to 1 (Figure 3). When the surface is saturated with elementary droplets, both incorporation and coalescence events are observed and large droplets are emerging (snapshots c and d). Because of continuous coalescence, larger droplets develop with time and the droplet radius polydispersity increases more because the newly deposited droplets fill the free space between the large droplets as shown in snapshots e and f. a. Variation of the Surface Coverage. In Figure 4 is presented the variation of surface coverage Γ as a


Figure 3. Radius polydispersity versus time with L ) 400, rh ) 1, N0 ) 1.5 × 106, and θ ) π/2 on a double logarithmic scale. Radius polydispersity strongly increases with time when Γ ≈ 0.62 because small droplets are filling the space between the large droplets which have been formed by coalescence.

Figure 4. Surface coverage variation as a function of time for a system with L ) 400, rh ) 1, N0 ) 1.5 × 106, and θ ) π/2. In the first time of simulation, one observes a fast increase of surface coverage, and then surface coverage starts to be limited and reaches a plateau value at an estimated value of Γ ≈ 0.62. The plateau formation is due to coalescence.

function of time. A fast increase of Γ is observed at first, and then saturation of the surface coverage is observed because of the effect of coalescence. The quantitative determination of the value at which the saturation coverage occurs can be deduced from the analysis of the mean droplet size distribution or total number of objects at the surface which are addressed in the next section. The regular deposition of droplets at the surface still produces on average a very slow and continuous increase of surface coverage. In contrast, the RSA model has an asymptotic behavior with a surface coverage characterized by a jamming limit Γ∞ ≈ 0.547. On the other hand, heterogeneous deposition (or condensation) saturates at a value of Γ ≈ 0.55. Shoulders observed in Figure 4 in the saturation regime are mostly due to the statistical variations of the deposition process with time. In particular, when two large droplets (compared to the size of the surface) are coalescing or when multiple coalescences are induced large variations of Γ are observed. The sum of events, that is, simple depositions ∑∆t)100 Ndep, incorporations ∑∆t)100 Ninc, and coalescences ∑∆t)100 Ncoa, was calculated every 100 units of time and represented in Figure 5 to get insight into the relative importance of simple deposition, incorporation, and coalescence versus time. Because droplets can be deposited or incorporated into a droplet already present on the deposition plane

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Figure 5. Sum of events; simple depositions ∑∆t)100 Ndep, incorporations ∑∆t)100 Ninc, and coalescences ∑∆t)100 Ncoa calculated every step of 100 units of time. The effective number of droplets Ntot present on the surface for a simulation with L ) 400, rh ) 1, N0 ) 1.5 × 106, and θ ) π/2 is represented as a function of time. Ntot is continuously decreasing at long times.

Figure 6. Schematic representation of a typical configuration of some droplets on a surface when θ ) π/2. The black disks represent the surface of the droplets in contact with the deposition plan, whereas the dashed zones indicate additional areas where elementary droplets can incorporate a droplet already present on the surface. This size exclusion effect reduces dramatically the available surface for deposition.

prior to the coalescence, ∑∆t)100 Ndep + ∑∆t)100 Ninc is necessarily equal to 100. As the total number of droplets present on the surface Ntot versus time f(t) is also expected to give important information, it was calculated and also presented in Figure 5. Ntot is expected to increase by 1 by a simple deposition, remain constant for one incorporation, and decrease upon coalescence. From Figure 5, clearly four regimes can be defined during the homogeneous deposition process. They are described here. Regime 1. As shown in Figure 5, the first regime is characterized by ∑∆t)100 Ndep > ∑∆t)100 Ninc > ∑∆t)100 Ncoa. Indeed, for a low surface coverage, that is, at short time, the available surface is important and promotes simple deposition. Incorporation is then rare, making coalescence even more anecdotic. However, as Γ increases rapidly, incorporation and coalescence events become less exceptional. As a result, the slope of the variation Ntot which is equal to 1 at short time slightly decreases. Figure 6 illustrates that the available surface for incorporation is equal to the sum of the area of the droplets themselves (black disks) plus the areas defined by rings of thickness

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Figure 7. Number- and weight-average radius, area, and volume variations for a simulation where κ400,1, N0 ) 1.5 × 106, and θ ) π/2. Double logarithmic scale.

equal to r0 (dashed regions in Figure 6). Hence incorporation is expected to become a non-negligible event even at a relative low surface coverage. The slope of the Ntot variation of f(t) is still positive and close to 1. Regime 2. As the available surface for incorporation increases more quickly than direct deposition, we observe at Γ ≈ 0.17 a crossover (also denoted Γ1-2) when ∑∆t)100 Ndep ) ∑∆t)100 Ninc (corresponding to snapshot a in Figure 2). This point corresponds to the situation where the effective available surface for deposition according to the excluded volume effect is less than 0.5. Γ1-2 indicates the end of regime 1 and the beginning of regime 2, in which ∑∆t)100 Ninc > ∑∆t)100 Ndep > ∑∆t)100 Ncoa. Incorporation is thus more important than deposition and coalescence. The slope of the variation of Ntot versus time rapidly decreases and is equal to 0 when Γ ≈ 0.34 (corresponding to snapshot b in Figure 2). Ntot reaches a maximum value at this point. Regime 3. When Γ ≈ 0.34 is reached (also denoted Γ2-3), then coalescence starts to play a non-negligible role and

∑∆t)100 Ndep > ∑∆t)100 Ncoa > ∑∆t)100 Ndep, meaning that coalescence is more frequent than deposition. As a result, the slope of Ntot ) f(t) is now negative as observed in Figure 5. The radius polydispersity of the droplets at the surface is increasing slightly during regimes 1-3, reflecting the fact that the size of the droplets mainly increases by a uniformly distributed incorporation in regime 2. When Γ ≈ 0.34, most of the droplets are adjacent and the incorporation induces more or less systematically a cascade of coalescences. However, radius polydispersity (Figure 3) is still weak, because on average, all the droplets expand at the same time. The point of inflection of Ntot ) f(t) at the beginning of regime 3 (Figure 5) corresponds to the period where coalescence is the most important. Regime 4. Owing to the variation of Ntot in Figure 5, another regime is emerging at Γ ≈ 0.62 (snapshot d in Figure 2). The decrease of Ntot is less significant than in the previous regime, and ∑∆t)100 Ninc > ∑∆t)100 Ncoa > ∑∆t)100 Ndep. On average, the relative importance of the different


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events does not change even if ∑∆t)100 Ndep and ∑∆t)100 Ncoa fluctuate a lot. Despite the fact that droplet coalescence produces free surface, this event continues to be more frequent than deposition because of the probability of inducing multiple coalescence per unit of time. The incorporation of a droplet in much larger droplets which have been obtained in regime 3 induces little changes in the sizes. Thus, coalescence is more frequent for small droplets lately deposited and which are filling space released by coalescence. This explains why the decrease of Ntot ) f(t) is less important in regime 4 than in regime 3. The strong fluctuations observed for ∑∆t)100 Ndep and ∑∆t)100 Ncoa correspond to coalescence of large droplets, leading to multiple coalescences, which makes the surface free for simple deposition. In that regime, droplet radius polydispersity strongly increases owing to the generation, by coalescence, of large droplets and then concomitant deposition of small droplets. Hence, the homogeneous model exhibits a bimodal size distribution, that is, a roughly monodisperse distribution of large droplets superimposed on a polydisperse distribution of smaller droplets. Bimodal distributions have been experimentally observed,9-11 and a mathematical description of a bimodal distribution has been achieved by Blackman and Brochard16 considering the scaling exponent of the mean droplet size which is found to be in good agreement with the results presented here. It should also be noted here that four stages of droplet growth are also usually observed when heterogeneous deposition is considered. These different stages reflect the relative importance of condensation at specific sites and coalescence. In our case, when we consider θ ) π/2, a first and small surface coverage saturation is observed at Γ ≈ 0.34, that is, when incorporation becomes predominant. Then a second, important saturation is provoked by coalescence at Γ ≈ 0.62. This last value is directly extracted from the analysis of the moment of the droplet size distribution which is discussed below. b. Variation of the Droplet Number- and WeightAverage Sizes. In Figure 7 are represented the time variations of the number- and weight-average droplet sizes on a log-log scale. The transition from regime 3 to regime 4 is clearly put in evidence by following the variation of the mean number droplet radius as a function of time which displays a plateau value at Γ ≈ 0.62. A change is also observed here when the variation of the weight droplet average radius as a function of time is considered. That curve moves to lines having slopes equal to 0.65 to 0.49 when Γ ≈ 0.62. Such a slope decrease is due to the importance of the statistical weight of small droplets which are deposited at the surface made free after coalescence. These transitions are also observed at Γ ≈ 0.62 for the number-average droplet area and number-average droplet volume. On the other hand, the weight-average droplet area and weight-average droplet volume are represented by curves with slopes equal to 1.80 and 2.96, respectively, and no change in the slopes is observed. This last value is similar to the prediction (z ) 3) of Family and Meakin for the heterogeneous growth. c. Influence of the Relative Size of Droplets and Cell. To normalize the droplet deposited area versus the total surface area, we defined a parameter κ(L,r1) which is the ratio of the deposition plane area L × L divided by the area described by an elementary droplet contact radius r1:

κ(L,r1) )

L2 πr12

(9)

Figure 8. Three simulation runs with hemispherical droplets (θ ) π/2), with a constant value of κ but different values of L ) 300, 400, and 600 and r0 ) 1, 4/3, and 2. The surface coverage variations versus time could not be differentiated. Variation of surface coverage with time is statistically identical for identical values of surface ratio κ.

Figure 9. Surface coverage as a function of time for simulations with N0 ) 1.5 × 106, θ ) π/2, and different values of κ. When one increases κ through the increase of L or the decrease of r0, the same rate of surface coverage is obtained but with considerable delay.

As shown in Figure 8, for a given contact angle θ ) π/2, variation of surface coverage with time is statistically identical for identical values of κ. To get insight into the role of the relative size between the droplets and the surface area, we carried out several simulations with hemispherical droplets (θ ) π/2) but with different values of L ) 300, 400, 600, and 1000. N0 was fixed at 107, and rmax at L/3. On the other hand, calculations were performed by maintaining L at 400 and adjusting the initial droplet

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Figure 10. Surface coverage as a function of time for simulations with L ) 400, N0 ) 1.5 × 106, r0 ) 1, and different values of θ (10-170° by steps of 20°). Saturation surface coverage which is represented in the inset rapidly decreases with the increase of the contact angle of the droplets.

Figure 11. Total number of effective droplets as a function of time for several simulations of deposition with L ) 400, r0 ) 1, N0 ) 1.5 × 106, and θ ) 10-170° (by steps of 10°).

radius. In Figure 9, when κ increases through the increase of L or the decrease of droplet contact radius r1, surface coverage is considerably delayed. Hence, to achieve a given surface coverage Γ, the corresponding time is expected to be strongly dependent on the initial droplet size; that is, coverage is more efficient for large droplet volumes. Variations of number- and weight-average sizes do not vary when κ changes. d. Influence of the Contact Angle θ. Surface-active agents are added to agrochemicals, paintings, inks, and so forth for optimal surface coverage by modification of the contact angle and so the surface of contact of the deposited droplets (hydrophilic surface). On the other hand, when nonwetting situations are required, the chemistry is adjusted so as to increase θ (hydrophobic surface). We carried out several simulations with L ) 400 and r0 ) 1 for different contact angles. The time dependence of surface coverage Γ and total number of droplets Ntot as a function of various values of θ are presented in Figures 10 and 11, respectively. Figure 10 clearly indicates that, no matter what the contact angle is, the surface coverage always saturates at a value after some time. The increase of the surface coverage is due to the reduction of θ and thus the increase of the droplet radius because of the volume conservation. The time at which the asymptotic value occurs is important, and the hydrophilic

surface saturates faster than the hydrophobic surface because of a lower contact angle. The saturation surface coverage value strongly decreases with the increase of the contact angle as shown in the inset of Figure 10. For instance, saturation surface coverage is equal to 0.75 when θ ) 90° and decreases rapidly to 0.15 when θ ) 150°. Similar trends were obtained by Zhao and Beysens34 where heterogeneous condensation experiments were performed. They demonstrated that even though the dynamics of coalescence and thus the growth morphology are strongly affected by the presence of one hysteretic effect, the regime of temporal self-similar growth is preserved although the saturated coverage is significantly increased depending on the hysteresis strength and contact angle. They also demonstrated that the heterogeneous nucleation theory needs to properly incorporate the effect of surface heterogeneity and subtle influence of the contact angle gradient in order to correctly explain the experimental results. Back to our model and by considering the Ntot variation (Figure 11), when θ increases (the largest radius rh decreases) it is shown that the different regimes are reached more tardily. By further increasing θ to values close to 180°, the Ntot variations display convergence to an asymptotic curve which is representative of the presence of spherical droplets at the surface. Hence, the time corresponding to the transition from one regime to another is strongly related to the contact angle. We demonstrate here that even though the dynamics of homogeneous deposition and thus the growth morphology are strongly affected by the contact angle, the different regimes of the temporal growth are preserved and self-similar although the saturation plateau is significantly increased with decreasing θ. 4. Model Applications In Figure 12 is presented the variation of the total number of deposited droplets Nparticles as a function of surface ratio κ for several Γ values. The angle of contact is set to π/2. When κ is large, “small” droplets are deposited on a large surface, whereas when κ is small, “large” droplets are deposited on a small surface. If the elementary volume of a droplet is known, according to the contact angle, we can easily predict and control the amount of liquid Vtotal required to achieve a specific Γ value. To achieve a given surface coverage, the total number of droplets is a linear function of κ and so a function of the


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Figure 12. Total number of deposited droplets Nparticles as a function of surface ratio κ. For a given surface coverage, the variation of Nparticles is a linear function of κ. Slopes are dependent on the surface coverage (slope values as a function of surface coverage are plotted in the inset).

size of the available surface. The corresponding slopes are strongly dependent on the desired surface coverage. For example, a surface coverage equal to 0.8 will require an important quantity of material. The slopes of the total mass (the total mass is proportional to the number of droplets) versus the surface ratio have been presented as a function of the surface coverage Γ in the inset of the Figure 12. From a practical point of view, important guidelines can be extracted here to achieve a good balance between surface coverage and deposited droplet mass, that is, to control deposition processes. First, surface coverage greater than 0.34 should be avoided because of coalescence which promotes transfer of liquid in the verticality of the large droplets. If an important surface coverage is desired, successive deposition processes at Γ ) 0.34 (after drying) must be promoted. Second, for a given surface area, large droplets should be preferentially used to rapidly increase surface coverage (this could be achieved by lowering the contact angle also). 5. Conclusion We investigated the deposition process of liquid droplets on a smooth and chemically homogeneous flat substrate. When the homogeneous droplet deposition involves coalescence processes, four time-dependent regimes for homogeneous deposition of liquid droplets are clearly demonstrated. These regimes are controlled by the relative importance of the deposition, incorporation, and coalescence events. In particular, the last regime, where the homogeneous model exhibits a bimodal size distribution, is characterized by a strong polydispersity, a slow and progressive increase of the rate of surface coverage, and a continuous decrease of the effective number of droplets present at the surface. By focusing at θ ) π/2, we demonstrated that the surface coverage levels off at Γ ≈ 0.62, which corresponds to the beginning of the last regime, where coalescence dominates. We demonstrated that no matter what the contact angle is, the surface coverage always saturates at the value after some time and that the dynamics of homogeneous deposition is strongly affected by the contact angle. The different regimes of the temporal growth are preserved and self-similar although the saturation plateau is significantly increased with decreasing contact angle. Contrary to the RSA and heterogeneous deposition models where asymptotic values

and saturation are achieved, respectively, here the regular and homogeneous deposition of droplets at the surface still produces on average a slow and continuous increase of the surface coverage. The total amount of deposited liquid, for identical values of the contact angle and relative size of the droplet and surface, can vary enormously as a function of the desired surface coverage. Also, the ratio between the elementary droplet surface and the size of the deposition plane is expected to play a key role. Large droplets have to be preferentially used to limit the effect of coalescence. To cover a surface with a minimum material, it is strongly suggested to stop the deposition of liquid droplets before coalescence starts to play a role (for example at Γ ) 0.34 when θ ) π/2) and then, after drying, start a new deposition process. This paper presents results obtained with an ideal model of homogeneous deposition of liquid droplets, leading to complex processes and results similar to those found in real systems. The results reported here have been obtained by assuming constant contact angle and fast incorporation/coalescence with regard to the droplet deposition rate. However, important factors such as the relaxation of the contact angle, the relaxation of the shape of the droplet during coalescence (hysteresis of the contact angle), or the kinetics of injection rate of the droplets in the system could play important roles. Also, gravity could play a role with the increase of the drop sizes, negating the spherical assumption for droplets on the surface. The simulations reported here are a preliminary step toward a more precise modeling of the problem, and further refinements will be considered in the future by including a “retardation” effect to estimate counterbalancing effects of the rate of deposition and the liquid viscosity. Acknowledgment. The authors are grateful to D. K. Schneider, TECFA (University of Geneva), for providing the computational resources (O2-SGI) and to D. Beysens, J. Buffle, S. Raux, W. Marconi, P. Hewson, and J. E. Sepulveda for useful discussions and comments. This work was supported by the Swiss National Research Project 2100-061750.00. LA030348I

Computer Simulations of Homogeneous Deposition of Liquid Droplets

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