Article pubs.acs.org/Langmuir
Spreading of a Droplet over a Nonisothermal Substrate: Multiple Scaling Regimes Kaustav Chaudhury and Suman Chakraborty* Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India S Supporting Information *
ABSTRACT: We envisage the spreading behavior of a twodimensional droplet under a thin-film-based paradigm, under a perfect wetting condition, while the droplet is placed over a nonisothermal substrate. Starting from the onset of thin-film behavior (or equivalently beyond the inertia-dominated initial stage), we identify the existence of mutually contrasting multiple scaling regimes defining the spreading behavior at different time scales. This is attributable to the time-stage-wise upsurge of capillarity or thermocapillarity over the other. In particular, the spreading behavior is characterized by the foot-width (w) evolution with time (t) in a power-law fashion w ∼ tα, with α being the spreading exponent, defining the rate of spreading. Following pertinent thin-film and subsequent similarity analysis, we identify different asymptotes of α over disparate temporal scales, leading to the characterization of different scaling regimes over the entire spreading event starting from the inception of thin-film behavior. Reported literature data are found to correspond well to the present interpretations and estimations.
■
INTRODUCTION The spreading of a droplet over a solid substrate is a longstanding topic of interest, as the scenario mimics several situations of practical importance.1,2 Of late, the potential of droplet spreading has found particular relevance in microscale transport processes, specifically in digital microfluidics, micrototal analysis, and lab-on-a-chip systems.1,2 Despite diverse facets of complexities in the above-mentioned systems, the fundamental principle driving droplet spreading can be organized as follows: the initial stage is inertia-dominated whereas the later stage is governed by viscous and capillary effects.3−12 Beyond the inertial stage, a power-law-based spreading characteristic of the drop width (w ∼ tα) is typical in the thin-film regime. Interestingly, when multiple counteractive forcing parameters are involved, the prevalence of a unique α over the entire thin-film regime of the spreading event is questionable, even under a perfect wetting condition. One such scenario may potentially occur during the spreading of a droplet atop a nonisothermal substrate.3,6,7,10,11 Under this circumstance, capillarity and thermocapillarity (due to the surface tension gradient imposed by the temperature gradient) effects coexist, complicating the spreading characteristics to a considerable extent. Here we focus on the spreading characteristics of a sessile droplet in the thin-film regime (or equivalently beyond the inertial stage) under the perfect wetting condition, taking the combined confluence of capillarity and thermocapillarity into account, while the droplet is placed atop a nonisothermal substrate. Although it is reported7,11 that thermocapillarity can alter α compared to capillarity-driven spreading, the alteration is realized at far later stages of the spreading event and not over © 2015 American Chemical Society
the entire thin-film regime. The fundamental interfacial interactions, leading to the asymptotic attainment of such later-stage dynamics starting from the inception of the thin-film dynamics, however, remain obscure, to the best of our knowledge. The challenges in developing a comprehensive understanding in this regard are intrinsically attributable to a complicated interplay of two distinctive interfacial interactions, namely, the capillary and thermocapillarity phenomena. Under the purview of thin-film dynamics (beyond the inertial stage), here we bring out the dynamics of the contrastingly different scaling regimes of a sessile droplet on a nonisothermal substrate, with a fundamental delineation of the following three regimes, namely, the capillary regime, the linear thermocapillary regime, and the nonlinear thermocapillary regime. We pinpoint the spreading exponents for all of these regimes and characterize the pertinent time scales. These observations are shown to corroborate well with the reported data.
■
PROBLEM DESCRIPTION Figure 1 schematically illustrates the present problem under consideration: A liquid droplet is spreading over a nonisothermal substrate in an ambient air medium. The spreading characteristics of the droplet are given by the foot width w(t) and drop height h(x, t) evolving with time t and/or position x. It is important to mention at the outset that as soon as the droplet comes into contact with the solid substrate, rapid Received: December 8, 2014 Revised: February 19, 2015 Published: March 18, 2015 4169
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir
considering perfectly wetting conditions θ ≤ 5°, resulting in (1 − cos θ)→ 0 without any loss of generality. Thus, in our formulation, we neglect the implications of contact-line dynamics, assuming the perfectly wetting condition. Later on, we discuss the implications of this analysis by bringing in perspectives from full-scale numerical simulation data obtained over a range of contact angles. Next, we justify the use of a linear temperature variation along x for the analysis reported here. Toward that, we note that thermocapillarity is essentially triggered by the surface tension gradient brought about by the temperature gradient. Common practice in introducing the temperature gradient is in maintaining a nonuniform distribution of temperature over the substrate by heating and subsequently cooling two ends of it,6 as schematically represented in Figure 1 by the hot and cold ends. The practical motivation of such studies, for example, may be to mimic a droplet-based cooling system for a microchip having a locally concentrated hotspot. Moreover, silicon is a common choice as a base material for microchips. Accordingly, the high thermal conductivity of silicon allows rapid heat conduction and thereby pertains to the constant temperature gradient across the substrate.6 A possible deviation from the situation can be the perturbation of the thermal field due to the moving and/or spreading droplet. However, the highly conducting nature of silicon is argued to minimize such perturbations.6 Conforming to the above-mentioned physically realistic arguments, here we analyze the implications of thermocapillarity promoted by a substrate having a constant temperature gradient. Following the above-mentioned arguments, here we consider a substrate with temperature Tw(x) such that ∂Tw/∂x = Γ = constant. Now, under the thin-film approximation (h/w ≪ 1), for the liquid droplet region, one can consider the thermal Peclet number to be very small and the dominant mode of heat transfer within the droplet to be governed by the conduction equation of the form ∂2T/∂y2 = 0 (cf. refs 7 and 13 for further details), with T as the temperature within the droplet. This equation is supported by the boundary condition T|y=0 = Tw(x) and interfacial condition −k ∂T/∂y|y=h = ψ(T|y=h − T∞) where k, ψ, and T∞ are the thermal conductivity of the liquid, convective heat transfer coefficient between the liquid/air media, and ambient temperature, respectively. Following these arguments, one can obtain the temperature distribution within the droplet as T = Tw −[(Tw − T∞)(ψy/k)]/[1 + (ψh/k)]. Now, under the thin-film approximation (very small h) it is plausible to assume ψy/k,ψh/k ≪ 1 (cf. refs 7 and 13), resulting in T ≈ Tw(x) from the temperature distribution equation. Thus, the temperature of the substrate can be considered to be “imprinted” across the droplet. Accordingly, we have ∂T/∂x ≈ ∂Tw/∂x = Γ = constant, resulting in T = T0 + Γx with T0 being a reference temperature at x = 0. One of the essential driving factors for the present thermocapillarity-driven free-surface thin-film flow situation is the Marangoni stress τ = ∂σ/∂x = (∂σ/∂T)(∂T/∂x) = (∂σ/ ∂T)Γ, realized at the droplet/air interface. Evidently, the temperature dependence of surface tension coefficient σ(T) is a critical requirement in this regard. Commonly, employed liquid water follows a linear σ(T) pattern. However, an aqueous solution of some long-chain alcohols (such as an aqueous solution of n-heptanol, n-hexanol, etc.) exhibits quadratic σ(T) behavior with well-defined minima.14,15 For both cases, the variation of density and viscosity with the temperature is almost negligible, at least within the temperature range under
Figure 1. Schematic illustration. Cartesian reference frame x − y is fixed to the substrate, and the entire analysis is conducted with respect to this frame.
spreading can be observed. For low-viscosity droplets, this initial stage of spreading behavior can be delineated in a powerlaw fashion where the power-law exponent is much higher than that observed in the later stage of spreading.4,5,8,9 Interestingly, for high-viscosity droplets, instead of a pure power-law-based notion, an essential logarithmic correction needs to be taken into consideration.8 However, for both high- and low-viscosity droplets, the initial stage of spreading is found to be dominated by inertia, by analogy to the inertia-dominated stage of droplet coalescence.8 Following analogies with spreading coalescence, the contribution of inertia becomes negligible when the drop height becomes smaller than the extent of the foot width.8 In reality, condition h/w ≪ 1 pertains to the dominance of viscous and capillary effects. In this situation, it is possible to rationalize the spreading behavior of a droplet by employing a thin-film approximation, neglecting the contribution of inertia, without any loss of generality.12 Here we specifically envisage the spreading behavior in this thin-film regime. The contrasts in the spreading behavior due to the transition from the inertial to the thin-film regime have been aptly pointed out by several researchers.4,5,8,9 Interestingly, even within the thin-film regime, we find the existence of mutually contrasting regimes of droplet spreading, owing to the involvement of multiple physical features, namely, capillarity and thermocapillarity. It is important to mention at this juncture that hereinafter whenever we refer to “early” and “later” stages, for the present paradigm under consideration, it should be considered to be the corresponding stages within the thin-film regime (or equivalently beyond the inertial regime). Interestingly, in the thin-film situation, the film thickness may eventually reach an ultrathin dimension so that the molecularscale interactions start to play an important role. This typically pertains to long-film behavior. The underlying thin-film dynamics can be captured through the introduction of disjoining−conjoining pressure (in addition to the Laplace pressure) of the form7 Π = (2σ0/h*)(1 − cos θ)[(h*/h)3 − (h*/h)2, were θ is the contact angle and h* is the thickness of the energetically favored molecular thin film. However, for Π to be dominant, h has to be comparable to h*, where the latter is typically on the order of 10 nm7. We must note that here we focus our attention on droplets having a microliter volume or equivalently having sizes on the order of millimeters. This consideration is in tandem with the usual practice in microfluidics.1,2 Thus, h in our analysis is typically much greater than h*. Accordingly, we neglect the possible implications of long-film behavior. Moreover, here we are 4170
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir consideration. Needless to say, both water and aqueous solutions of other liquids can be potential choices as liquid droplets, implying the possibility of both monotonic and nonmonotonic σ(T) characteristics. Thus, for the sake of generality, we consider σ(T ) = σ0[1 + a1(T − T0) + a 2(T − T0)2 ]
Equation 3 serve as the primary equation for the present analysis, as discussed subsequently. It is important to mention that the nonlinear thermocapillarity (L4) is actually brought about by quadratic σ(T) characteristics. However, use of the notation “nonlinear” is standard in free-surface thin-film flow situations,11,13 thus we prefer to adhere to it for the sake of convenience. Scaling. Here we analyze the situation with respect to the scaled equation. We begin with the scaling of the form x = lxx̃, h = lhh̃, and t = tct ̃ with lx, lh, and tc being the scales of the corresponding variables where the quantities with tildes represents the corresponding normalized forms. The proper choice of the scales should result in O(x̃), O(h̃), and O(t)̃ = 1 such that O(x) = lx, O(h̃) = lh, and O(t) = tc. Following the geometry of the problem, the choice lx = w0 and lh = h0 satisfy O(x̃) = O(h̃) = 1, where w0 = w|t=0 and h0 = h|x=0,t=0. We also note that h0/w0 = ε < 1. With these considerations of the length scales, the scaled version of eq 3 reads as
(1)
Here, σ0 is the surface tension at a reference temperature T0 with a1,2 being the constants. In Figure 2, we present some
Figure 2. Linear (water) and quadratic (6 × 10−3 m aqueous solution of n-heptanol) σ(T) characteristics following refs 14 and 15. The markers represent the experimental data points of refs 14 and 15 whereas the solid lines are the fittings following eq 1.
with tc remaining to be specified. In fact, eq 5 shows that three different choices of tc are possible, as follows:
typical σ(T) characteristics for water and aqueous solutions of n-heptanol. Accordingly, the Marangoni stress at the droplet/air interface, due to the temperature gradient, can be estimated to be τ=
(a) capillary time scale: O(L1) = O(L 2) ⇒ tc =
(c) nonlinear thermocapillary time scale: O(L1) μ = O(L4) ⇒ tc = εw0σ0a 2 Γ 2
⇒τ = σ0 Γ[a1 + 2a 2(T − T0)] or τ = σ0 Γ[a1 + 2a 2 Γx] (2)
(6)
FORMULATION Thin-Film Model. Under thin-film approximation h/w ≪ 1, the flow field within the droplet is governed by the momentum balance equation 0 = −∂p/∂x + μ∂2u/∂y2 along with flow continuity ∂h/∂t + ∂(∫ h0 u dy)/∂x = 0, where μ, u, and p are the dynamic viscosity of the liquid, velocity along the x direction, and pressure, respectively. At the solid substrate, we consider u|y=0 = 0 (no slip). Across the droplet/air interface, we consider τ = μ ∂u/∂y|y=h (shear stress matching) and p = p∞ − σ0 ∂2h/ ∂x2 (normal stress matching), where p∞ is the pressure in the air medium and ∂2h/∂x2 is the curvature of the droplet/air interface. Note that for imposing the shear and normal stress matching conditions we consider the viscosity of air with respect to μ to be negligible. With these considerations in background, we obtain the thin-film equation
It is worth emphasizing that all the above-mentioned time scales satisfy the basic criterion O(t)̃ = 1. Thus, it is evident that droplet spreading behavior over a nonisothermal substrate can be described by three different time scales, focusing on different interfacial attributes. Accordingly, it is plausible to assume the existence of three regimes of droplet spreading: (i) the capillary regime, (ii) the linear thermocapillary regime, and (iii) the nonlinear thermocapillary regime. The characteristics of each of the regimes remains to be unveiled, if they exist at all, as endeavored in the subsequent section.
■
SPREADING CHARACTERISTICS Here we unveil the spreading characteristics following similarity analysis. Toward this end, we proceed with the stretching transformation of the form x̃ = sxx,̅ h̃ = shh̅, and t ̃ = stt.̅ Note, here sx, sh, and st are the stretching variables, not the characteristic scales. For a similar solution to exist, there must exist a nontrivial (not all unity) relationship among stretching variables sx, sh, and st such that the governing equation remains invariant (similar). It is worth mentioning at this juncture that the volume of the droplet should remain conserved, ∫ h̃ dx̃ = constant. This necessities shsx ∫ h̅ dx̅ = constant. Evidently, for the volume-conservation constraint to remain invariant, we must have sh = sx−1, setting one necessary relationship between sh and sx.
The significance of different terms in eq 3 is as follows: L1 ⇒ transience L 2 ⇒ capillarity L3 ⇒ linear thermocapillarity L4 ⇒ nonlinear thermocapillarity
ε 3σ0
(b) linear thermocapillary time scale: O(L1) = O(L3) ⇒ tc 2μ = εσ0a1Γ
∂σ ∂σ = Γ ∂x ∂T
■
3μw0
(4) 4171
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir Capillary Regime. First, we consider the situation decided by the capillary time scale (eq 6a), resulting in the scaled eq 5 to assume the form
̃ Evidently, at a later stage O(L2) = M1−1t−5/2 ≪ 1. However, the ̃ /M1 unless M2/ same cannot be inferred about O(L4) = M2t1/2 M1 ≪ 1. If σ varies linearly with T (implies M2 = 0 but M1 ≠ 0), then L4 is identically zero. Interestingly, the situation M2/M1 ≪ 1 or M2 ≪ M1 can also be realized when the nonmonotonic part of the σ − T dependence lies far away from the droplet region.11 In effect, the droplet experiences the monotonic σ − T dependence which is equivalent to linear thermocapillary dynamics. Thus, under the strong influence of linear thermocapillarity, the later-stage dynamics can be given as
where M1 = 3w0a1Γ/2ε2 and M2 = 3w02a2Γ2/ε2. Following the stretching transformation and considering sh = sx−1, the stretched version of eq 7 reads as
From eq 8 it is evident that there exist no nontrivial relationship between st and sx that results in the similarity between eq 7 (original equation) and eq 8 (stretched version). Thus, global similarity does not exist for the present problem under consideration. However, if we consider sx = st1/7 and sh = 1/7 st−1/7 (because sh = sx−1) it is possible to define η = x/t ̅ ̅ = x̃/ 1/7 1/7 ̃ ̃t and F(η) = h̅t1/7 ̃ ̅ = ht . Recasting the original eq 7 using η and F(η) we obtain
̃ ) and F(η) = h̃t1/2 ̃ . where η = (x̃/t1/2 Subsequently, the droplet spreading follows the scaling law w ∼ t1/2. Nonlinear Thermocapillary Regime. Finally we consider the situation with the nonlinear thermocapillary time scale (eq 6c) and obtain the scaled equation from eq 5 as
Accordingly, the stretching transformation x̃ = sxx,̅ h̃ = shh̅ and t ̃ = stt ̅ with sh = sx−1 (volume conservation), converts eq 15 to its stretched form
Evidently, eq 9 does not represent a similarity equation owing to the t ̃ dependence. However, in an early stage of spreading, ̃ ≪ 1 and O(L4) = M2t6/7 ̃ ≪1 one can consider O(L3) = M1t5/7 in comparison to L1 and L2. Thus, the early-stage dynamics can be given by an approximate similarity equation
Considering sx = st and sh = st−1 (because sh = sx−1), we can define variables η = x/t ̅ ̅ = x̃/t ̃ and F(η) = h̅t ̅ = h̃t.̃ Accordingly, eq 15 can be recast as
̃ ) and F(η) = h̃t1/7 ̃ . where η = (x̃/t1/7 Following the definition of η, the spreading behavior can be represented as w ∼ t1/7. It is worth mentioning that the similarity form, thus obtained, along with the definitions of similarity variables η and F(η), resembles the popular capillaryspreading dynamics.12 Thus, for a droplet spreading over a nonisothermal substrate, the early-stage dynamics is nothing but the celebrated capillary-spreading dynamics. Linear Thermocapillary Regime. Next, we proceed with the analysis by taking the linear thermocapillary time scale (eq 6b) into consideration. This results in the thin-film eq 5 assuming the normalized form
̃ ≪ 1 and O(L3) Evidently, at a later stage both O(L2) = M2−1t−6 = M1/M2t ̃ ≪ 1. Thus, under the influence of nonlinear thermocapillarity, the later-stage dynamics can be given as
where η = (x̃/t)̃ and F(η) = h̃t.̃ Accordingly, the spreading behavior can be given as w ∼ t.
■
DISCUSSION We first refer to the study of Gomba and Homsy,7 focusing on the spreading characteristics of a droplet under the influence of linear thermocapillarity for different wettability conditions defined by contact angle θ. Their observation (markers) is presented in Figure 3, for foot width evolution at θ = 5° (cf. Figure 3 in Gomba and Homsy7), as it closely matches the perfectly wetting condition and is consistent with our notation. Note that the results are presented in accordance with their normalized form, starting from the onset of droplet spreading in the thin-film regime. They7 have considered lcap = (σ0/ρg)1/2 and tcap = 3 μLcap/σ0 for normalizing w, where ρ is the density of the liquid and g is the acceleration due to gravity. Gomba and
Following the above-mentioned generic stretching transformation x̃ = sxx,̅ h̃ = shh̅ and t ̃ = stt ̅ with sh = sx−1 (volume conservation), eq 11 assumes the stretched form
Here we consider sx = st1/2 and sh = st−1/2 (because sh = sx−1). 1/2 1/2 This pertains to the definition of variables η = x/t ̅ ̅ = x̃/t ̃ 1/2 1/2 ̃ ̃ and F(η) = h̅t ̅ = ht . Accordingly, eq 11 can be recast as 4172
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir Homsy7 have presented their results for t/tcap from 1 to 105. However, a variation of w/lcap is observable only beyond t/tcap = 3 × 102. Thus, here we present their results for t/tcap > 3 × 102. From the figure it is evident that the w(t) characteristics remain within two asymptotes: w ∼ t1/7 in an early stage and w ∼ t1/2 in a later stage of the thin-film regime. It is important to mention that Gomba and Homsy7 have also discussed the emergence of the later-stage asymptote w ∼ t1/2. However, the transition to this later stage from the capillarity-driven scaling asymptote w ∼ t1/7 at an early stage has not been addressed in their study, which we highlight here. One can appreciate that the early stage is characterized by a smaller rate of spreading, compared to the linear-thermocapillarity-driven faster spreading in a later stage. Thus, the variation of foot width in the capillarity-driven early stage is not much as compared to the later stage. The prevalence of the early-stage asymptote is therefore visible while the w/lcap axis is presented on a linear scale. In a log−log plot, otherwise, the capillarity-driven early stage appears as a limiting tangential regime, although the extent of t/tcap is comparable for both asymptotic stages, as can be appreciated from Figure 3.
Figure 4. Evolution of foot width of a 2D water droplet with time, under different wettability conditions. The markers represent the results of our VOF simulations, and the solid and the dashed lines denote the scaling asymptotes.
from θ = 20° onward, earmarking the onset of altered behavior of the droplet at larger contact angles under thermocapillary actuation, consistent with the interpretation of Gomba and Homsy.7 Thus, our estimation of the multiple regime seems to be consistent for the perfect wetting condition. Additionally, according to our notation, the capillarity-driven early stage pertaining to the condition M1[t/(tc)capillary]5/7 ≪ 1. Thus, the capillary stage is expected to be observed for t/(tc)capillary ≪ M1−7/5. Following the data used in our simulations (detailed in Supporting Information), we get M1 = 0.07. Thus, the asymptote w ∼ t1/7 is expected to exist for t/(tc)capillary ≪ 41.4, and an upsurge of the scaling asymptote w ∼ t1/2 is expected thereafter. The effective temporal ranges of the asymptotes, as observed from Figure 3, confirm the notation. A very nice accounting of nonlinear thermocapillarity-driven spreading of a 2D droplet has been given by Karapetsas et al.11 Their primary findings (markers) on the temporal evolution of foot width is shown in Figure 5, according to their
Figure 3. Foot-width evolution with time under the influence of linear thermocapillarity and an estimation of multiple scaling regimes. The markers represent results (normalized form) of Gomba and Homsy,7 and the solid and dashed lines denote the scaling asymptotes.
It is important to mention that wettability may also be of importance in dictating the pertaining transiences. Thus, to assess its implications on the present estimation of the regimes, in Figure 4 we present the spreading characteristics of a water droplet at different wettability conditions in the thin-film regime, as obtained from comprehensive full-scale numerical simulations based on the volume of fluid (VOF) method. (For details, see the Supporting Information.) The results are presented in normalized form, with w0 and capillary time scale (tc)capillary = 3μw0/ε3σ0 for the normalization of w and t, respectively. Note that the w/w0 axis is presented on a linear scale for a better visualization of the early stage. Now, the linear σ(T) characteristics of water are likely to bring the linear thermocapillarity into action, while the temperature of the substrate varies linearly in space. From the comparison presented in Figure 4, it is apparent that the asymptotes w ∼ t1/7 (at early stage) and w ∼ t1/2 (at later stage) can sufficiently describe the spreading characteristics of the droplet in the thinfilm regime, at different wettability conditions. Here we also present the spreading characteristics at larger θ ≥ 5° for a critical assessment of our conjecture. The tendency of the results to departure from our scaling predictions is noteworthy
Figure 5. Foot-width evolution with time under the influence of nonlinear thermocapillarity and estimation of multiple scaling regimes. The markers represent results (normalized form) of Karapetasas et al.11 while the solid and dashed lines denote the scaling asymptotes.
normalization scheme, with lK = w0/2 and tK = μw02/4h0σ0 as the scales for normalizing w and t, respectively. Here we consider only those results which bring nonlinear thermocapillarity into action (cf. Figures 2, 3, 5, and 6 in Karapetsas et al.11). In this situation, also we find the prominent existence of two contrasting asymptotes of spreading. Capillarity-driven early stage w ∼ t1/7 and nonlinear thermocapillarity-driven later 4173
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir stage w ∼ t, as shown in Figure 5, are consistent with the present conjecture on the sequential dominance of capillarity and nonlinear thermocapillarity on the corresponding time scales. For better visualization of the slowly spreading early stage, here we present the w/lk axis on the linear scale. We must mention that Karapetsas et al.11 have also offered a conclusion on the later-stage asymptote, w ∼ t. However, they have arrived at the conclusion based on the analysis of the slope of their simulation data on w(t) in the log−log plane. Here we have arrived at the same conclusion based on similarity analysis, without going into the detailed solution of the thin-film equation. Additionally, the similarity analysis benefits us in identifying the different stages of spreading. Following the present approach, it can be pinpointed that during droplet spreading atop a nonisothermal substrate thermocapillarity does not decide the spreading behavior over all temporal scales. Rather, it is always preceded by the capillarity-driven spreading in an early stage. Specifically, here we are able to show the sequential development of the spreading stages with explicit accounting of the spreading exponents at different stages. Such multiple spreading characteristics at different time scales is of great practical value in real time operations. Prior to concluding our discussion, we must note certain issues with the possible implications of thermal noise, promoted by random molecular motion. This random fluctuation can be realized by a Gaussian white noise stochastic stress S (tensor form) in the governing equations and particularly in the stress matching criteria across the droplet/ air interface.16−18 The implications of the stochastic noise is the prevalence of spreading law w ∼ t1/4, in contrast to the other scaling laws described so far. However, to realize this behavior, one requires16,17 h≪
h0 2 h0 7/6B7/6 where x* = x* (kBT /σ0)1/6
beyond the inertial regime). Although the above-mentioned effects appear altogether in deciding the spreading behavior, their relative dominance is not the same over all temporal regimes. In the early stage of spreading, the capillarity seems to play a dominant role. The dominance of thermocapillarity (linear or nonlinear), on the other hand, is realized in later stages of the spreading event. Thus, there seem to exist multiple scaling regimes during the entire period of the spreading of a droplet atop a nonisothermal substrate. These results may be of immense consequence in designing droplet-based microfluidic systems toward obtaining desired spreading characteristics for achieving multifarious functionalities over disparate spatiotemporal scales.
■
ASSOCIATED CONTENT
S Supporting Information *
Numerical simulation using the volume of fluid (VOF) method. This material is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
■
REFERENCES
(1) Choi, K.; Ng, A. H. C.; Fobel, R.; Wheeler, A. R. Digital Microfluidics. Annu. Rev. Anal. Chem. 2012, 5, 413−440. (2) Darhuber, A. A.; Troian, S. M. Principles of Microfluidic Actuation By Modulation of Surface Stresses. Annu. Rev. Fluid Mech. 2005, 37, 425−455. (3) Mukhopadhyay, S.; Murisic, N.; Behringer, R. P.; Kondic, L. Evolution of Droplets of Perfectly Wetting Liquid under the Influence of Thermocapillary Forces. Phys. Rev. E 2011, 83, 046302. (4) Biance, A.-L.; Clanet, C.; Quéré, D. First Steps in the Spreading of a Liquid Droplet. Phys. Rev. E 2004, 69, 016301. (5) Winkels, K. G.; Weijs, J. H.; Eddi, A.; Snoeijer, J. H. Initial Spreading of Low-Viscosity Drops on Partially Wetting Surfaces. Phys. Rev. E 2012, 85, 055301. (6) Brzoska, J. B.; Brochard-Wyart, F.; Rondelez, F. Motions of Droplets on Hydrophobic Model Surfaces Induced by Thermal Gradients. Langmuir 1993, 9, 2220−2224. (7) Gomba, J. M.; Homsy, G. M. Regimes of Thermocapillary Migration of Droplets under Partial Wetting Conditions. J. Fluid Mech. 2010, 647, 125. (8) Eddi, A.; Winkels, K. G.; Snoeijer, J. H. Short Time Dynamics of Viscous Drop Spreading. Phys. Fluids 2013, 25, 013102. (9) Bird, J.; Mandre, S.; Stone, H. Short-Time Dynamics of Partial Wetting. Phys. Rev. Lett. 2008, 100, 234501. (10) Benintendi, S. W.; Smith, M. K. The Spreading of a NonIsothermal Liquid Droplet. Phys. Fluids 1999, 11, 982. (11) Karapetsas, G.; Sahu, K. C.; Sefiane, K.; Matar, O. K. Thermocapillary-Driven Motion of a Sessile Drop: Effect of NonMonotonic Dependence of Surface Tension on Temperature. Langmuir 2014, 30, 4310−4321. (12) Leal, L. G. Advanced Transport Phenomena; Cambridge University Press, 2007. (13) Oron, A.; Rosenau, P. On a Nonlinear Thermocapillary Effect in Thin Liquid Layers. J. Fluid Mech. 2006, 273, 361. (14) Vochten, R.; Petre, G. Study of the Heat of Reversible Adsorption at the Air-Solution Interface. II. Experimental Determination of the Heat of Reversible Adsorption of Some Alcohols. J. Colloid Interface Sci. 1973, 42, 320−327. (15) Limbourg-Fontaine, M. C.; Petre, G.; Legros, J. C.; Van Ransbeeck, E. Thermocapillary Movements around a Surface Tension
(19)
Parameters include the Boltzmann constant kB and the width of the channel B (measured perpendicular to the plane of the paper). According to the realization of Willis and Freund,17 which they have arrived at through rigorous molecular dynamic simulations, h has to reach an ultrathin dimension down to nanometer-scale resolution to meet the criterion in eq 19. The requirement of nanoscale resolution for the onset of imperative implications of stochastic thermal noise is also appreciated in other studies.19−22 However, here we focus our attention on droplets having a microliter volume or equivalently having sizes on the order of millimeters, as previously mentioned in this article. Thus, stochastic noise seems to have no bearing on the present study. Having said the above, we must mention that an overlap between the contributions of stochastic noise and thermocapillarity may be observed in a later stage down to nanoscale resolution. This is attributable to the relative dominance of the contributing Marangoni and stochastic shear stresses. To resolve this issue, therefore, rigorous molecular dynamics simulations are required to analyze the combined implications of thermocapillarity and thermal noise. This is nevertheless beyond the scope of the present study.
■
CONCLUSIONS We find that during droplet spreading atop a nonisothermal substrate, competing contributions of capillarity and thermocapillarity (linear or nonlinear) emerge in an intriguingly sequential fashion in the thin-film regime (or equivalently 4174
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir Minimum under Microgravity Conditions (part I. Technical Description of the Stem Experiments. D1Mission of Spacelab). Acta Astronaut. 1986, 13, 197−208. (16) Davidovitch, B.; Moro, E.; Stone, H. Spreading of Viscous Fluid Drops on a Solid Substrate Assisted by Thermal Fluctuations. Phys. Rev. Lett. 2005, 95, 244505. (17) Willis, A. M.; Freund, J. B. Enhanced Droplet Spreading due to Thermal Fluctuations. J. Phys.: Condens. Matter 2009, 21, 464128. (18) Grün, G.; Mecke, K.; Rauscher, M. Thin-Film Flow Influenced by Thermal Noise. J. Stat. Phys. 2006, 122, 1261−1291. (19) Moseler, M. Formation, Stability, and Breakup of Nanojets. Science 2000, 289, 1165−1169. (20) Hennequin, Y.; Aarts, D.; van der Wiel, J.; Wegdam, G.; Eggers, J.; Lekkerkerker, H.; Bonn, D. Drop Formation by Thermal Fluctuations at an Ultralow Surface Tension. Phys. Rev. Lett. 2006, 97, 244502. (21) Gopan, N.; Sathian, S. P. Rayleigh Instability at Small Length Scales. Phys. Rev. E 2014, 90, 033001. (22) Willis, A. M.; Freund, J. B. Thermal Capillary Waves Relaxing on Atomically Thin Liquid Films. Phys. Fluids 2010, 22, 022002.
4175
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Spreading of a Droplet over a Nonisothermal Substrate: Multiple Scaling Regimes Kaustav Chaudhury and Suman Chakraborty* Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India S Supporting Information *
ABSTRACT: We envisage the spreading behavior of a twodimensional droplet under a thin-film-based paradigm, under a perfect wetting condition, while the droplet is placed over a nonisothermal substrate. Starting from the onset of thin-film behavior (or equivalently beyond the inertia-dominated initial stage), we identify the existence of mutually contrasting multiple scaling regimes defining the spreading behavior at different time scales. This is attributable to the time-stage-wise upsurge of capillarity or thermocapillarity over the other. In particular, the spreading behavior is characterized by the foot-width (w) evolution with time (t) in a power-law fashion w ∼ tα, with α being the spreading exponent, defining the rate of spreading. Following pertinent thin-film and subsequent similarity analysis, we identify different asymptotes of α over disparate temporal scales, leading to the characterization of different scaling regimes over the entire spreading event starting from the inception of thin-film behavior. Reported literature data are found to correspond well to the present interpretations and estimations.
■
INTRODUCTION The spreading of a droplet over a solid substrate is a longstanding topic of interest, as the scenario mimics several situations of practical importance.1,2 Of late, the potential of droplet spreading has found particular relevance in microscale transport processes, specifically in digital microfluidics, micrototal analysis, and lab-on-a-chip systems.1,2 Despite diverse facets of complexities in the above-mentioned systems, the fundamental principle driving droplet spreading can be organized as follows: the initial stage is inertia-dominated whereas the later stage is governed by viscous and capillary effects.3−12 Beyond the inertial stage, a power-law-based spreading characteristic of the drop width (w ∼ tα) is typical in the thin-film regime. Interestingly, when multiple counteractive forcing parameters are involved, the prevalence of a unique α over the entire thin-film regime of the spreading event is questionable, even under a perfect wetting condition. One such scenario may potentially occur during the spreading of a droplet atop a nonisothermal substrate.3,6,7,10,11 Under this circumstance, capillarity and thermocapillarity (due to the surface tension gradient imposed by the temperature gradient) effects coexist, complicating the spreading characteristics to a considerable extent. Here we focus on the spreading characteristics of a sessile droplet in the thin-film regime (or equivalently beyond the inertial stage) under the perfect wetting condition, taking the combined confluence of capillarity and thermocapillarity into account, while the droplet is placed atop a nonisothermal substrate. Although it is reported7,11 that thermocapillarity can alter α compared to capillarity-driven spreading, the alteration is realized at far later stages of the spreading event and not over © 2015 American Chemical Society
the entire thin-film regime. The fundamental interfacial interactions, leading to the asymptotic attainment of such later-stage dynamics starting from the inception of the thin-film dynamics, however, remain obscure, to the best of our knowledge. The challenges in developing a comprehensive understanding in this regard are intrinsically attributable to a complicated interplay of two distinctive interfacial interactions, namely, the capillary and thermocapillarity phenomena. Under the purview of thin-film dynamics (beyond the inertial stage), here we bring out the dynamics of the contrastingly different scaling regimes of a sessile droplet on a nonisothermal substrate, with a fundamental delineation of the following three regimes, namely, the capillary regime, the linear thermocapillary regime, and the nonlinear thermocapillary regime. We pinpoint the spreading exponents for all of these regimes and characterize the pertinent time scales. These observations are shown to corroborate well with the reported data.
■
PROBLEM DESCRIPTION Figure 1 schematically illustrates the present problem under consideration: A liquid droplet is spreading over a nonisothermal substrate in an ambient air medium. The spreading characteristics of the droplet are given by the foot width w(t) and drop height h(x, t) evolving with time t and/or position x. It is important to mention at the outset that as soon as the droplet comes into contact with the solid substrate, rapid Received: December 8, 2014 Revised: February 19, 2015 Published: March 18, 2015 4169
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir
considering perfectly wetting conditions θ ≤ 5°, resulting in (1 − cos θ)→ 0 without any loss of generality. Thus, in our formulation, we neglect the implications of contact-line dynamics, assuming the perfectly wetting condition. Later on, we discuss the implications of this analysis by bringing in perspectives from full-scale numerical simulation data obtained over a range of contact angles. Next, we justify the use of a linear temperature variation along x for the analysis reported here. Toward that, we note that thermocapillarity is essentially triggered by the surface tension gradient brought about by the temperature gradient. Common practice in introducing the temperature gradient is in maintaining a nonuniform distribution of temperature over the substrate by heating and subsequently cooling two ends of it,6 as schematically represented in Figure 1 by the hot and cold ends. The practical motivation of such studies, for example, may be to mimic a droplet-based cooling system for a microchip having a locally concentrated hotspot. Moreover, silicon is a common choice as a base material for microchips. Accordingly, the high thermal conductivity of silicon allows rapid heat conduction and thereby pertains to the constant temperature gradient across the substrate.6 A possible deviation from the situation can be the perturbation of the thermal field due to the moving and/or spreading droplet. However, the highly conducting nature of silicon is argued to minimize such perturbations.6 Conforming to the above-mentioned physically realistic arguments, here we analyze the implications of thermocapillarity promoted by a substrate having a constant temperature gradient. Following the above-mentioned arguments, here we consider a substrate with temperature Tw(x) such that ∂Tw/∂x = Γ = constant. Now, under the thin-film approximation (h/w ≪ 1), for the liquid droplet region, one can consider the thermal Peclet number to be very small and the dominant mode of heat transfer within the droplet to be governed by the conduction equation of the form ∂2T/∂y2 = 0 (cf. refs 7 and 13 for further details), with T as the temperature within the droplet. This equation is supported by the boundary condition T|y=0 = Tw(x) and interfacial condition −k ∂T/∂y|y=h = ψ(T|y=h − T∞) where k, ψ, and T∞ are the thermal conductivity of the liquid, convective heat transfer coefficient between the liquid/air media, and ambient temperature, respectively. Following these arguments, one can obtain the temperature distribution within the droplet as T = Tw −[(Tw − T∞)(ψy/k)]/[1 + (ψh/k)]. Now, under the thin-film approximation (very small h) it is plausible to assume ψy/k,ψh/k ≪ 1 (cf. refs 7 and 13), resulting in T ≈ Tw(x) from the temperature distribution equation. Thus, the temperature of the substrate can be considered to be “imprinted” across the droplet. Accordingly, we have ∂T/∂x ≈ ∂Tw/∂x = Γ = constant, resulting in T = T0 + Γx with T0 being a reference temperature at x = 0. One of the essential driving factors for the present thermocapillarity-driven free-surface thin-film flow situation is the Marangoni stress τ = ∂σ/∂x = (∂σ/∂T)(∂T/∂x) = (∂σ/ ∂T)Γ, realized at the droplet/air interface. Evidently, the temperature dependence of surface tension coefficient σ(T) is a critical requirement in this regard. Commonly, employed liquid water follows a linear σ(T) pattern. However, an aqueous solution of some long-chain alcohols (such as an aqueous solution of n-heptanol, n-hexanol, etc.) exhibits quadratic σ(T) behavior with well-defined minima.14,15 For both cases, the variation of density and viscosity with the temperature is almost negligible, at least within the temperature range under
Figure 1. Schematic illustration. Cartesian reference frame x − y is fixed to the substrate, and the entire analysis is conducted with respect to this frame.
spreading can be observed. For low-viscosity droplets, this initial stage of spreading behavior can be delineated in a powerlaw fashion where the power-law exponent is much higher than that observed in the later stage of spreading.4,5,8,9 Interestingly, for high-viscosity droplets, instead of a pure power-law-based notion, an essential logarithmic correction needs to be taken into consideration.8 However, for both high- and low-viscosity droplets, the initial stage of spreading is found to be dominated by inertia, by analogy to the inertia-dominated stage of droplet coalescence.8 Following analogies with spreading coalescence, the contribution of inertia becomes negligible when the drop height becomes smaller than the extent of the foot width.8 In reality, condition h/w ≪ 1 pertains to the dominance of viscous and capillary effects. In this situation, it is possible to rationalize the spreading behavior of a droplet by employing a thin-film approximation, neglecting the contribution of inertia, without any loss of generality.12 Here we specifically envisage the spreading behavior in this thin-film regime. The contrasts in the spreading behavior due to the transition from the inertial to the thin-film regime have been aptly pointed out by several researchers.4,5,8,9 Interestingly, even within the thin-film regime, we find the existence of mutually contrasting regimes of droplet spreading, owing to the involvement of multiple physical features, namely, capillarity and thermocapillarity. It is important to mention at this juncture that hereinafter whenever we refer to “early” and “later” stages, for the present paradigm under consideration, it should be considered to be the corresponding stages within the thin-film regime (or equivalently beyond the inertial regime). Interestingly, in the thin-film situation, the film thickness may eventually reach an ultrathin dimension so that the molecularscale interactions start to play an important role. This typically pertains to long-film behavior. The underlying thin-film dynamics can be captured through the introduction of disjoining−conjoining pressure (in addition to the Laplace pressure) of the form7 Π = (2σ0/h*)(1 − cos θ)[(h*/h)3 − (h*/h)2, were θ is the contact angle and h* is the thickness of the energetically favored molecular thin film. However, for Π to be dominant, h has to be comparable to h*, where the latter is typically on the order of 10 nm7. We must note that here we focus our attention on droplets having a microliter volume or equivalently having sizes on the order of millimeters. This consideration is in tandem with the usual practice in microfluidics.1,2 Thus, h in our analysis is typically much greater than h*. Accordingly, we neglect the possible implications of long-film behavior. Moreover, here we are 4170
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir consideration. Needless to say, both water and aqueous solutions of other liquids can be potential choices as liquid droplets, implying the possibility of both monotonic and nonmonotonic σ(T) characteristics. Thus, for the sake of generality, we consider σ(T ) = σ0[1 + a1(T − T0) + a 2(T − T0)2 ]
Equation 3 serve as the primary equation for the present analysis, as discussed subsequently. It is important to mention that the nonlinear thermocapillarity (L4) is actually brought about by quadratic σ(T) characteristics. However, use of the notation “nonlinear” is standard in free-surface thin-film flow situations,11,13 thus we prefer to adhere to it for the sake of convenience. Scaling. Here we analyze the situation with respect to the scaled equation. We begin with the scaling of the form x = lxx̃, h = lhh̃, and t = tct ̃ with lx, lh, and tc being the scales of the corresponding variables where the quantities with tildes represents the corresponding normalized forms. The proper choice of the scales should result in O(x̃), O(h̃), and O(t)̃ = 1 such that O(x) = lx, O(h̃) = lh, and O(t) = tc. Following the geometry of the problem, the choice lx = w0 and lh = h0 satisfy O(x̃) = O(h̃) = 1, where w0 = w|t=0 and h0 = h|x=0,t=0. We also note that h0/w0 = ε < 1. With these considerations of the length scales, the scaled version of eq 3 reads as
(1)
Here, σ0 is the surface tension at a reference temperature T0 with a1,2 being the constants. In Figure 2, we present some
Figure 2. Linear (water) and quadratic (6 × 10−3 m aqueous solution of n-heptanol) σ(T) characteristics following refs 14 and 15. The markers represent the experimental data points of refs 14 and 15 whereas the solid lines are the fittings following eq 1.
with tc remaining to be specified. In fact, eq 5 shows that three different choices of tc are possible, as follows:
typical σ(T) characteristics for water and aqueous solutions of n-heptanol. Accordingly, the Marangoni stress at the droplet/air interface, due to the temperature gradient, can be estimated to be τ=
(a) capillary time scale: O(L1) = O(L 2) ⇒ tc =
(c) nonlinear thermocapillary time scale: O(L1) μ = O(L4) ⇒ tc = εw0σ0a 2 Γ 2
⇒τ = σ0 Γ[a1 + 2a 2(T − T0)] or τ = σ0 Γ[a1 + 2a 2 Γx] (2)
(6)
FORMULATION Thin-Film Model. Under thin-film approximation h/w ≪ 1, the flow field within the droplet is governed by the momentum balance equation 0 = −∂p/∂x + μ∂2u/∂y2 along with flow continuity ∂h/∂t + ∂(∫ h0 u dy)/∂x = 0, where μ, u, and p are the dynamic viscosity of the liquid, velocity along the x direction, and pressure, respectively. At the solid substrate, we consider u|y=0 = 0 (no slip). Across the droplet/air interface, we consider τ = μ ∂u/∂y|y=h (shear stress matching) and p = p∞ − σ0 ∂2h/ ∂x2 (normal stress matching), where p∞ is the pressure in the air medium and ∂2h/∂x2 is the curvature of the droplet/air interface. Note that for imposing the shear and normal stress matching conditions we consider the viscosity of air with respect to μ to be negligible. With these considerations in background, we obtain the thin-film equation
It is worth emphasizing that all the above-mentioned time scales satisfy the basic criterion O(t)̃ = 1. Thus, it is evident that droplet spreading behavior over a nonisothermal substrate can be described by three different time scales, focusing on different interfacial attributes. Accordingly, it is plausible to assume the existence of three regimes of droplet spreading: (i) the capillary regime, (ii) the linear thermocapillary regime, and (iii) the nonlinear thermocapillary regime. The characteristics of each of the regimes remains to be unveiled, if they exist at all, as endeavored in the subsequent section.
■
SPREADING CHARACTERISTICS Here we unveil the spreading characteristics following similarity analysis. Toward this end, we proceed with the stretching transformation of the form x̃ = sxx,̅ h̃ = shh̅, and t ̃ = stt.̅ Note, here sx, sh, and st are the stretching variables, not the characteristic scales. For a similar solution to exist, there must exist a nontrivial (not all unity) relationship among stretching variables sx, sh, and st such that the governing equation remains invariant (similar). It is worth mentioning at this juncture that the volume of the droplet should remain conserved, ∫ h̃ dx̃ = constant. This necessities shsx ∫ h̅ dx̅ = constant. Evidently, for the volume-conservation constraint to remain invariant, we must have sh = sx−1, setting one necessary relationship between sh and sx.
The significance of different terms in eq 3 is as follows: L1 ⇒ transience L 2 ⇒ capillarity L3 ⇒ linear thermocapillarity L4 ⇒ nonlinear thermocapillarity
ε 3σ0
(b) linear thermocapillary time scale: O(L1) = O(L3) ⇒ tc 2μ = εσ0a1Γ
∂σ ∂σ = Γ ∂x ∂T
■
3μw0
(4) 4171
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir Capillary Regime. First, we consider the situation decided by the capillary time scale (eq 6a), resulting in the scaled eq 5 to assume the form
̃ Evidently, at a later stage O(L2) = M1−1t−5/2 ≪ 1. However, the ̃ /M1 unless M2/ same cannot be inferred about O(L4) = M2t1/2 M1 ≪ 1. If σ varies linearly with T (implies M2 = 0 but M1 ≠ 0), then L4 is identically zero. Interestingly, the situation M2/M1 ≪ 1 or M2 ≪ M1 can also be realized when the nonmonotonic part of the σ − T dependence lies far away from the droplet region.11 In effect, the droplet experiences the monotonic σ − T dependence which is equivalent to linear thermocapillary dynamics. Thus, under the strong influence of linear thermocapillarity, the later-stage dynamics can be given as
where M1 = 3w0a1Γ/2ε2 and M2 = 3w02a2Γ2/ε2. Following the stretching transformation and considering sh = sx−1, the stretched version of eq 7 reads as
From eq 8 it is evident that there exist no nontrivial relationship between st and sx that results in the similarity between eq 7 (original equation) and eq 8 (stretched version). Thus, global similarity does not exist for the present problem under consideration. However, if we consider sx = st1/7 and sh = 1/7 st−1/7 (because sh = sx−1) it is possible to define η = x/t ̅ ̅ = x̃/ 1/7 1/7 ̃ ̃t and F(η) = h̅t1/7 ̃ ̅ = ht . Recasting the original eq 7 using η and F(η) we obtain
̃ ) and F(η) = h̃t1/2 ̃ . where η = (x̃/t1/2 Subsequently, the droplet spreading follows the scaling law w ∼ t1/2. Nonlinear Thermocapillary Regime. Finally we consider the situation with the nonlinear thermocapillary time scale (eq 6c) and obtain the scaled equation from eq 5 as
Accordingly, the stretching transformation x̃ = sxx,̅ h̃ = shh̅ and t ̃ = stt ̅ with sh = sx−1 (volume conservation), converts eq 15 to its stretched form
Evidently, eq 9 does not represent a similarity equation owing to the t ̃ dependence. However, in an early stage of spreading, ̃ ≪ 1 and O(L4) = M2t6/7 ̃ ≪1 one can consider O(L3) = M1t5/7 in comparison to L1 and L2. Thus, the early-stage dynamics can be given by an approximate similarity equation
Considering sx = st and sh = st−1 (because sh = sx−1), we can define variables η = x/t ̅ ̅ = x̃/t ̃ and F(η) = h̅t ̅ = h̃t.̃ Accordingly, eq 15 can be recast as
̃ ) and F(η) = h̃t1/7 ̃ . where η = (x̃/t1/7 Following the definition of η, the spreading behavior can be represented as w ∼ t1/7. It is worth mentioning that the similarity form, thus obtained, along with the definitions of similarity variables η and F(η), resembles the popular capillaryspreading dynamics.12 Thus, for a droplet spreading over a nonisothermal substrate, the early-stage dynamics is nothing but the celebrated capillary-spreading dynamics. Linear Thermocapillary Regime. Next, we proceed with the analysis by taking the linear thermocapillary time scale (eq 6b) into consideration. This results in the thin-film eq 5 assuming the normalized form
̃ ≪ 1 and O(L3) Evidently, at a later stage both O(L2) = M2−1t−6 = M1/M2t ̃ ≪ 1. Thus, under the influence of nonlinear thermocapillarity, the later-stage dynamics can be given as
where η = (x̃/t)̃ and F(η) = h̃t.̃ Accordingly, the spreading behavior can be given as w ∼ t.
■
DISCUSSION We first refer to the study of Gomba and Homsy,7 focusing on the spreading characteristics of a droplet under the influence of linear thermocapillarity for different wettability conditions defined by contact angle θ. Their observation (markers) is presented in Figure 3, for foot width evolution at θ = 5° (cf. Figure 3 in Gomba and Homsy7), as it closely matches the perfectly wetting condition and is consistent with our notation. Note that the results are presented in accordance with their normalized form, starting from the onset of droplet spreading in the thin-film regime. They7 have considered lcap = (σ0/ρg)1/2 and tcap = 3 μLcap/σ0 for normalizing w, where ρ is the density of the liquid and g is the acceleration due to gravity. Gomba and
Following the above-mentioned generic stretching transformation x̃ = sxx,̅ h̃ = shh̅ and t ̃ = stt ̅ with sh = sx−1 (volume conservation), eq 11 assumes the stretched form
Here we consider sx = st1/2 and sh = st−1/2 (because sh = sx−1). 1/2 1/2 This pertains to the definition of variables η = x/t ̅ ̅ = x̃/t ̃ 1/2 1/2 ̃ ̃ and F(η) = h̅t ̅ = ht . Accordingly, eq 11 can be recast as 4172
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir Homsy7 have presented their results for t/tcap from 1 to 105. However, a variation of w/lcap is observable only beyond t/tcap = 3 × 102. Thus, here we present their results for t/tcap > 3 × 102. From the figure it is evident that the w(t) characteristics remain within two asymptotes: w ∼ t1/7 in an early stage and w ∼ t1/2 in a later stage of the thin-film regime. It is important to mention that Gomba and Homsy7 have also discussed the emergence of the later-stage asymptote w ∼ t1/2. However, the transition to this later stage from the capillarity-driven scaling asymptote w ∼ t1/7 at an early stage has not been addressed in their study, which we highlight here. One can appreciate that the early stage is characterized by a smaller rate of spreading, compared to the linear-thermocapillarity-driven faster spreading in a later stage. Thus, the variation of foot width in the capillarity-driven early stage is not much as compared to the later stage. The prevalence of the early-stage asymptote is therefore visible while the w/lcap axis is presented on a linear scale. In a log−log plot, otherwise, the capillarity-driven early stage appears as a limiting tangential regime, although the extent of t/tcap is comparable for both asymptotic stages, as can be appreciated from Figure 3.
Figure 4. Evolution of foot width of a 2D water droplet with time, under different wettability conditions. The markers represent the results of our VOF simulations, and the solid and the dashed lines denote the scaling asymptotes.
from θ = 20° onward, earmarking the onset of altered behavior of the droplet at larger contact angles under thermocapillary actuation, consistent with the interpretation of Gomba and Homsy.7 Thus, our estimation of the multiple regime seems to be consistent for the perfect wetting condition. Additionally, according to our notation, the capillarity-driven early stage pertaining to the condition M1[t/(tc)capillary]5/7 ≪ 1. Thus, the capillary stage is expected to be observed for t/(tc)capillary ≪ M1−7/5. Following the data used in our simulations (detailed in Supporting Information), we get M1 = 0.07. Thus, the asymptote w ∼ t1/7 is expected to exist for t/(tc)capillary ≪ 41.4, and an upsurge of the scaling asymptote w ∼ t1/2 is expected thereafter. The effective temporal ranges of the asymptotes, as observed from Figure 3, confirm the notation. A very nice accounting of nonlinear thermocapillarity-driven spreading of a 2D droplet has been given by Karapetsas et al.11 Their primary findings (markers) on the temporal evolution of foot width is shown in Figure 5, according to their
Figure 3. Foot-width evolution with time under the influence of linear thermocapillarity and an estimation of multiple scaling regimes. The markers represent results (normalized form) of Gomba and Homsy,7 and the solid and dashed lines denote the scaling asymptotes.
It is important to mention that wettability may also be of importance in dictating the pertaining transiences. Thus, to assess its implications on the present estimation of the regimes, in Figure 4 we present the spreading characteristics of a water droplet at different wettability conditions in the thin-film regime, as obtained from comprehensive full-scale numerical simulations based on the volume of fluid (VOF) method. (For details, see the Supporting Information.) The results are presented in normalized form, with w0 and capillary time scale (tc)capillary = 3μw0/ε3σ0 for the normalization of w and t, respectively. Note that the w/w0 axis is presented on a linear scale for a better visualization of the early stage. Now, the linear σ(T) characteristics of water are likely to bring the linear thermocapillarity into action, while the temperature of the substrate varies linearly in space. From the comparison presented in Figure 4, it is apparent that the asymptotes w ∼ t1/7 (at early stage) and w ∼ t1/2 (at later stage) can sufficiently describe the spreading characteristics of the droplet in the thinfilm regime, at different wettability conditions. Here we also present the spreading characteristics at larger θ ≥ 5° for a critical assessment of our conjecture. The tendency of the results to departure from our scaling predictions is noteworthy
Figure 5. Foot-width evolution with time under the influence of nonlinear thermocapillarity and estimation of multiple scaling regimes. The markers represent results (normalized form) of Karapetasas et al.11 while the solid and dashed lines denote the scaling asymptotes.
normalization scheme, with lK = w0/2 and tK = μw02/4h0σ0 as the scales for normalizing w and t, respectively. Here we consider only those results which bring nonlinear thermocapillarity into action (cf. Figures 2, 3, 5, and 6 in Karapetsas et al.11). In this situation, also we find the prominent existence of two contrasting asymptotes of spreading. Capillarity-driven early stage w ∼ t1/7 and nonlinear thermocapillarity-driven later 4173
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir stage w ∼ t, as shown in Figure 5, are consistent with the present conjecture on the sequential dominance of capillarity and nonlinear thermocapillarity on the corresponding time scales. For better visualization of the slowly spreading early stage, here we present the w/lk axis on the linear scale. We must mention that Karapetsas et al.11 have also offered a conclusion on the later-stage asymptote, w ∼ t. However, they have arrived at the conclusion based on the analysis of the slope of their simulation data on w(t) in the log−log plane. Here we have arrived at the same conclusion based on similarity analysis, without going into the detailed solution of the thin-film equation. Additionally, the similarity analysis benefits us in identifying the different stages of spreading. Following the present approach, it can be pinpointed that during droplet spreading atop a nonisothermal substrate thermocapillarity does not decide the spreading behavior over all temporal scales. Rather, it is always preceded by the capillarity-driven spreading in an early stage. Specifically, here we are able to show the sequential development of the spreading stages with explicit accounting of the spreading exponents at different stages. Such multiple spreading characteristics at different time scales is of great practical value in real time operations. Prior to concluding our discussion, we must note certain issues with the possible implications of thermal noise, promoted by random molecular motion. This random fluctuation can be realized by a Gaussian white noise stochastic stress S (tensor form) in the governing equations and particularly in the stress matching criteria across the droplet/ air interface.16−18 The implications of the stochastic noise is the prevalence of spreading law w ∼ t1/4, in contrast to the other scaling laws described so far. However, to realize this behavior, one requires16,17 h≪
h0 2 h0 7/6B7/6 where x* = x* (kBT /σ0)1/6
beyond the inertial regime). Although the above-mentioned effects appear altogether in deciding the spreading behavior, their relative dominance is not the same over all temporal regimes. In the early stage of spreading, the capillarity seems to play a dominant role. The dominance of thermocapillarity (linear or nonlinear), on the other hand, is realized in later stages of the spreading event. Thus, there seem to exist multiple scaling regimes during the entire period of the spreading of a droplet atop a nonisothermal substrate. These results may be of immense consequence in designing droplet-based microfluidic systems toward obtaining desired spreading characteristics for achieving multifarious functionalities over disparate spatiotemporal scales.
■
ASSOCIATED CONTENT
S Supporting Information *
Numerical simulation using the volume of fluid (VOF) method. This material is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
■
REFERENCES
(1) Choi, K.; Ng, A. H. C.; Fobel, R.; Wheeler, A. R. Digital Microfluidics. Annu. Rev. Anal. Chem. 2012, 5, 413−440. (2) Darhuber, A. A.; Troian, S. M. Principles of Microfluidic Actuation By Modulation of Surface Stresses. Annu. Rev. Fluid Mech. 2005, 37, 425−455. (3) Mukhopadhyay, S.; Murisic, N.; Behringer, R. P.; Kondic, L. Evolution of Droplets of Perfectly Wetting Liquid under the Influence of Thermocapillary Forces. Phys. Rev. E 2011, 83, 046302. (4) Biance, A.-L.; Clanet, C.; Quéré, D. First Steps in the Spreading of a Liquid Droplet. Phys. Rev. E 2004, 69, 016301. (5) Winkels, K. G.; Weijs, J. H.; Eddi, A.; Snoeijer, J. H. Initial Spreading of Low-Viscosity Drops on Partially Wetting Surfaces. Phys. Rev. E 2012, 85, 055301. (6) Brzoska, J. B.; Brochard-Wyart, F.; Rondelez, F. Motions of Droplets on Hydrophobic Model Surfaces Induced by Thermal Gradients. Langmuir 1993, 9, 2220−2224. (7) Gomba, J. M.; Homsy, G. M. Regimes of Thermocapillary Migration of Droplets under Partial Wetting Conditions. J. Fluid Mech. 2010, 647, 125. (8) Eddi, A.; Winkels, K. G.; Snoeijer, J. H. Short Time Dynamics of Viscous Drop Spreading. Phys. Fluids 2013, 25, 013102. (9) Bird, J.; Mandre, S.; Stone, H. Short-Time Dynamics of Partial Wetting. Phys. Rev. Lett. 2008, 100, 234501. (10) Benintendi, S. W.; Smith, M. K. The Spreading of a NonIsothermal Liquid Droplet. Phys. Fluids 1999, 11, 982. (11) Karapetsas, G.; Sahu, K. C.; Sefiane, K.; Matar, O. K. Thermocapillary-Driven Motion of a Sessile Drop: Effect of NonMonotonic Dependence of Surface Tension on Temperature. Langmuir 2014, 30, 4310−4321. (12) Leal, L. G. Advanced Transport Phenomena; Cambridge University Press, 2007. (13) Oron, A.; Rosenau, P. On a Nonlinear Thermocapillary Effect in Thin Liquid Layers. J. Fluid Mech. 2006, 273, 361. (14) Vochten, R.; Petre, G. Study of the Heat of Reversible Adsorption at the Air-Solution Interface. II. Experimental Determination of the Heat of Reversible Adsorption of Some Alcohols. J. Colloid Interface Sci. 1973, 42, 320−327. (15) Limbourg-Fontaine, M. C.; Petre, G.; Legros, J. C.; Van Ransbeeck, E. Thermocapillary Movements around a Surface Tension
(19)
Parameters include the Boltzmann constant kB and the width of the channel B (measured perpendicular to the plane of the paper). According to the realization of Willis and Freund,17 which they have arrived at through rigorous molecular dynamic simulations, h has to reach an ultrathin dimension down to nanometer-scale resolution to meet the criterion in eq 19. The requirement of nanoscale resolution for the onset of imperative implications of stochastic thermal noise is also appreciated in other studies.19−22 However, here we focus our attention on droplets having a microliter volume or equivalently having sizes on the order of millimeters, as previously mentioned in this article. Thus, stochastic noise seems to have no bearing on the present study. Having said the above, we must mention that an overlap between the contributions of stochastic noise and thermocapillarity may be observed in a later stage down to nanoscale resolution. This is attributable to the relative dominance of the contributing Marangoni and stochastic shear stresses. To resolve this issue, therefore, rigorous molecular dynamics simulations are required to analyze the combined implications of thermocapillarity and thermal noise. This is nevertheless beyond the scope of the present study.
■
CONCLUSIONS We find that during droplet spreading atop a nonisothermal substrate, competing contributions of capillarity and thermocapillarity (linear or nonlinear) emerge in an intriguingly sequential fashion in the thin-film regime (or equivalently 4174
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
Article
Langmuir Minimum under Microgravity Conditions (part I. Technical Description of the Stem Experiments. D1Mission of Spacelab). Acta Astronaut. 1986, 13, 197−208. (16) Davidovitch, B.; Moro, E.; Stone, H. Spreading of Viscous Fluid Drops on a Solid Substrate Assisted by Thermal Fluctuations. Phys. Rev. Lett. 2005, 95, 244505. (17) Willis, A. M.; Freund, J. B. Enhanced Droplet Spreading due to Thermal Fluctuations. J. Phys.: Condens. Matter 2009, 21, 464128. (18) Grün, G.; Mecke, K.; Rauscher, M. Thin-Film Flow Influenced by Thermal Noise. J. Stat. Phys. 2006, 122, 1261−1291. (19) Moseler, M. Formation, Stability, and Breakup of Nanojets. Science 2000, 289, 1165−1169. (20) Hennequin, Y.; Aarts, D.; van der Wiel, J.; Wegdam, G.; Eggers, J.; Lekkerkerker, H.; Bonn, D. Drop Formation by Thermal Fluctuations at an Ultralow Surface Tension. Phys. Rev. Lett. 2006, 97, 244502. (21) Gopan, N.; Sathian, S. P. Rayleigh Instability at Small Length Scales. Phys. Rev. E 2014, 90, 033001. (22) Willis, A. M.; Freund, J. B. Thermal Capillary Waves Relaxing on Atomically Thin Liquid Films. Phys. Fluids 2010, 22, 022002.
4175
DOI: 10.1021/la5047657 Langmuir 2015, 31, 4169−4175
