Coarsening Dynamics of True Morphological Phase Separation in

Mar 10, 2017 - Unstable thin films can spontaneously phase separate into two equilibrium flat-film morphologies of different thicknesses under the inf...
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Coarsening Dynamics of True Morphological Phase Separation in Unstable Thin Films Chaitanya Narayanam,† Avanish Kumar,‡ Sanjay Puri,‡ and Rajesh Khanna*,† †

Department of Chemical Engineering, Indian Institute of Technology Delhi, Delhi 110016, India School of Physical Sciences, Jawaharlal Nehru University, Delhi 110067, India

‡

S Supporting Information *

ABSTRACT: Unstable thin films can spontaneously phase separate into two equilibrium flat-film morphologies of different thicknesses under the influence of gravity or favorable combinations of apolar and polar excess intermolecular forces. Two distinct pathways and associated dynamics of this morphological phase separation are presented based on numerical simulations of the 2-D thin film equation. The two pathways are (1) the “direct pathway” whereby the thicker phase forms directly and concurrently with the thinner phase and (2) the “defect pathway” whereby the thicker phase forms by the coarsening of defects of intermediate thickness and appears much later than the thinner phase. The defect pathway is favored by films whose initial thickness lies closer to the thickness of the thinner phase. Both pathways show an initial power law decay with exponent −1/4 in time followed by a plateau in the number density of domains/defects. Thereafter, the defect pathway shows another universal power law decay with exponent −1/3 and ends with a logarithmic decay, which is specific to the d = 2 case as there is no interfacial curvature in d = 2. The direct pathway skips the second power law decay and goes directly to the logarithmic decay.

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INTRODUCTION

droplet is governed by the equilibrium contact angle and conservation of mass. Three distinct stages, namely, (1) early, (2) intermediate, and (3) late, have been identified in MPS.3 The early stage is characterized by the emergence of the spinodal wave from the random surface fluctuations and its further amplification to form the flat-film phase and droplets. The late stage refers to coarsening of the droplets as larger droplets feed off smaller droplets because of the Laplace pressure deficit. The intermediate stage corresponds to the interval between the first instance of appearance of the flat-film phase anywhere on the substrate, that is, at the end of the early stage, and the emergence of the flat-film phase across the whole substrate, that is, at the start of the late stage. In general, the flat-film phase is a true equilibrium phase of a definite thickness, whereas the droplet phase keeps getting thicker or thinner. Hence, it is more apt to treat the droplet as a “defect”3 rather than as a true phase. In some cases, gravity also becomes important as the thickness of the droplets increases beyond a few hundred nanometers because of coarsening. Balance of excess intermolecular forces and gravitational attraction imposes a maximum ceiling on the thickness in the liquid morphology. As a result, growing spherical droplets gradually convert to flattopped12 or mesa droplets of the given maximum thickness. Further coarsening of such droplets also does not result in any

Morphological phase separation (MPS) and subsequent pattern formation in thermodynamically unstable thin liquid films on solid surfaces1−3 are important in many technological and scientific applications,4−9 such as polymeric and optical coatings, ordered microstructures and nanolithography, microfluidics, and so forth. MPS can be thought of as a redistribution of the liquid contained in an almost flat film into an array of droplets connected by a thinner flat film. MPS has to be contrasted with dewetting, which is an alternative redistribution of the film liquid around isolated dry spots.10,11 These dry spots or holes grow laterally on the underlying substrate and force the surrounding liquid to gather into long cylindrical threads that break to form isolated droplets. Both MPS and dewetting are initiated by a spontaneous surface instability whenever the second derivative of the excess free energy (per unit area) due to excess intermolecular forces is negative. An excess intermolecular force field comprising a long-range attraction that starts the redistribution of the film liquid and a shorterrange repulsion that prevents dewetting is essential for MPS.3 Here, attraction and repulsion refer to the interaction between the film surface and the underlying solid. The droplets formed in MPS coarsen in time to form the thermodynamically stable configuration of a single droplet surrounded by the equilibrium flat-film phase. The thickness of the equilibrium film is given by the balance of repulsive and attractive components of the excess intermolecular force field. The thickness of the equilibrium © XXXX American Chemical Society

Received: March 6, 2017 Published: March 10, 2017 A

DOI: 10.1021/acs.langmuir.7b00752 Langmuir XXXX, XXX, XXX−XXX

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Langmuir increase in the maximum thickness. Thus, the film separates into two morphological flat-film phases of distinct equilibrium thicknesses under the influence of excess intermolecular forces and gravity. We refer to this as “true MPS” as opposed to “MPS”, which is the separation into spherical defects of variable and continually changing thicknesses and a film phase of equilibrium thickness. True MPS implies the coexistence of two equilibrium phases, which in turn is contingent to Maxwell’s double-tangent construction in the excess free-energy diagram. Now, the excess free energy for a generic coated-apolar system,13 ΔGa, which exhibits MPS due to ubiquitous Lifshitz−van der Waals forces,14 is given by ΔGa = −

Ac 2

12πh

−

A s − Ac 12π (h + δ)2

(1)

Here, h and δ are film and coating thicknesses, respectively. As > 0 (attractive interactions) and Ac < 0 (repulsive interactions) are the effective Hamaker constants of the film liquid corresponding to the substrate and coating, respectively. Figure 1 shows the excess free-energy diagram of this coated-apolar

Figure 2. Variation in the excess free energy per unit area (ΔG) and spinodal parameter (∂2ΔG/∂h2) with film thickness (h) for a coatedapolar-gravity system. The parameters are δ = 128 nm, As = 1.4 × 10−20, and Ac = −1.4 × 10−21.

modeled as an exponential decay to give the free energy of the polar−apolar system, ΔGp, as ΔGp = −

system. It is clear from the figure that the double-tangent construction is not possible in the case of coated-apolar system for thinner films. Thicker films require the inclusion of gravitational attraction15−18 and the excess free energy for a coated-apolar-gravity system, ΔGg, becomes Ac 12πh2

−

A s − Ac 12π (h + δ)2

+ ρg

h2 2

2

h⎞ ⎟⎟ ⎠

(3)

The prefactor, Sp, can be considered to be the spreading coefficient due to polar forces. lp is the decay length of the polar forces, and d0(=0.158 nm) is the minimum cut-off thickness to avoid nonphysical divergence during simulations. The variation in ΔGp with thickness for a polar system with long-range repulsive interactions due to the substrate (As < 0) and shortrange attractive polar interactions (Sp < 0) is shown in the top panel of Figure 3. ΔGp decreases continuously from a high positive value (not shown in the figure) and moves toward zero asymptotically with increasing thickness. The ΔGp curve can

Figure 1. Variation in the excess free energy per unit area (ΔG) and spinodal parameter (∂2ΔG/∂h2) with film thickness (h) for a coatedapolar system. The parameters are δ = 10 nm, As = 1.4 × 10−20, and Ac = −1.4 × 10−21.

ΔGg = −

⎛d − + Sp exp⎜⎜ 0 12πh ⎝ lp As

(2)

where ρ is the density of the liquid film and g is the acceleration due to gravity. Top panel of Figure 2 shows the excess freeenergy diagram of the coated-apolar-gravity system. Apolar attractive and repulsive interactions contribute predominantly to ΔGg at smaller thicknesses. The gravitational attraction dominates completely at higher thicknesses. The relative strength of the three interactions controls the shape of the curve during the intermediate thicknesses. The two thicknesses where all three forces balance each other (P1 and P2) facilitate a double-tangent construction. These two thicknesses are at the same chemical potential and can coexist with each other at equilibrium. Thinner films subjected to a generic polar−apolar excess intermolecular force field can also enforce a ceiling on the maximum thickness for a particular range of parameters.19 The polar component of the excess free energy can be satisfactorily

Figure 3. Variation in the excess free energy per unit area (ΔG) and spinodal parameter (∂2ΔG/∂h2) with film thickness (h) for a polar− apolar system. The parameters are As = −47 × 10−21, Sp = −3.75 × 10−4 J/m2, d0 = 0.158 nm, and lp = 0.6 nm. B

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Langmuir ⎡ ⎛ 2πh 2 ∂ΔG ⎞⎤ ∂H(X⃗ , T ) 0 = ∇·⎢H3∇⎜ − ∇2 H ⎟⎥ ∂T ⎝ |A s| ∂H ⎠⎥⎦ ⎣⎢

assume many forms based on the relative magnitudes of As and Sp. The ΔGp curve takes the form shown in Figure 2 only when |As/Sp| lies between 48πlp2 exp(d0/lp − 2) and 324πlp2 exp(d0/lp − 3). The excess free-energy curve (Figure 2) allows for a double-tangent construction for the above-mentioned restricted range of parameter values. The points of contact of the tangent denote the thicknesses of two true equilibrium phases, one thinner and the other thicker. Thus, polar−apolar systems can also exhibit true MPS in thin films. Recent efforts3,20 to compare and contrast MPS to conventional spinodal phase separation (SPS) processes such as unmixing of binary liquid mixtures showed that the absence of the second true equilibrium phase in MPS leads to important differences between the two. It changes the late-stage kinetics from a sluggish logarithmic in SPS21 to a much faster power law kinetics in 2-D MPS. Also in MPS, quenched disorder does not affect the late-stage dynamics unlike slowing down in SPS. This difference between MPS and SPS does not allow the highly developed understanding and huge reservoir of results of SPS5,22,23 to be directly applied to MPS. The seemingly minor difference between true MPS and MPS, that is, the formation of two and only one true equilibrium phases, respectively, results in major changes in the kinetics of thin film systems, which are of great experimental significance. We show that the saturation of evolving thicknesses to these equilibrium thicknesses imposes a slow logarithmic kinetics as opposed to the much faster power law kinetics. Similarly, distinctions in morphologies of true MPS and MPS can have strong ramifications in pattern formation also in terms of generating well-defined nanostructures. In this article, we identify new pathways and corresponding scaling regimes in the true MPS of thin films. Further, our article identifies new pathways and corresponding scaling regimes in the true MPS of thin films. We also establish the universality of the kinetics of true MPS by investigating two physically realizable but vastly different systems.

The excess free energy for coated-apolar-gravity system (ΔGg) in its nondimensional form is given by ⎤ 2πh0 2 ∂ΔGg 1⎡ 1 − R R = ⎢ + 3 + QH ⎥ 3 |A s| ∂H 3 ⎣ (H + D) ⎦ H

2πh0 2 ∂ΔGp ⎛D − H⎞ 1 R ⎟ =− 3 + exp⎜ ⎝ L ⎠ |A s| ∂H 6L 3H

(7)

12πSph02/As

Here, R = and D = d0/h0 and L = lp/h0 are the nondimensional minimum cutoff thickness and the decay length of hydrophobic interactions, respectively. The linear stability analysis of eq 5 for fluctuations about H = 1 predicts a dominant spinodal wave of wavelength, LM, given by the expression 4π

LM =

2

−

2πh0 ∂ 2ΔG |A s| ∂H2

H=1

(8)

Large-scale simulations were carried out to numerically solve eq 5 in d = 2, starting with an initial small amplitude (≃0.01) random perturbations about H = 1. The system sizes were chosen in multiples of dominant wavelength, that is, nLM (n ranges from 64 to 16384). A typical value of LM is 34.11, which is for a 214 nm thick film. Periodic boundary conditions were applied on both sides of the computational domain. Central differencing in space with half-node interpolation was used for discretization and Gear’s algorithm, which is highly efficient for solving stiff equations was used for time marching. A grid density of 64-points per LM was found to be adequate for the current numerical scheme. All simulations mentioned in the article have been carried out with at least one more grid density and at an acceptable tolerance limit. The codes have also been checked by solving for the asymptotic case of zero gravity,3 and the results match with the earlier published results.

MODEL AND SIMULATION The system of study is an incompressible and Newtonian thin liquid film supported on a solid substrate and bounded on the top by a fluid. The spatiotemporal evolution of the thin film surface is modeled by the thin film equation derived by applying lubrication approximation24 to equations of motion. The total free energy is given by

■

∫ [ΔG(h) + γ(∇⃗ h)2 /2] dx ⃗

RESULTS AND DISCUSSION Figure 4 presents a typical morphological evolution of the film’s surface in true MPS. A coated-apolar-gravity system as described in Figure 2 and eq 2 is chosen for illustration. Figure 4a shows the spinodal wave (solid line) that has emerged from the random initial surface perturbations (not shown). The wave amplifies in time (dashed line) and 30 welldefined and equispaced undulations can be seen in a domain of size 30LM. We observe that the linear theory (eq 8) satisfactorily predicts the spinodal length-scale in true MPS as it does in the case of MPS. Some of the troughs of the spinodal wave have already amplified to the thinner film phase (P1), whereas all crests (defects) are still much thinner than the thicker film phase (P2). Morphologically, this is characteristic of the intermediate stage of MPS. Thus, the evolution so far has been similar to the early and intermediate stages of the MPS. Figure 4b,c presents the coarsening of defects whereby some

Here, Fe and Fi denote the net excess and interfacial free energies, respectively. The mobility (M(h) = h3/(3μ)) is thickness-dependent corresponding to Stokes flow with noslip.25 The resultant equation is ⎡ h3 ⎛ ∂ΔG ⎞⎤ ∂h(x ⃗ , t ) = ∇⃗ ·⎢ ∇⃗⎜ − γ ∇2 h⎟⎥ ⎠⎦ ∂t ⎣ 3μ ⎝ ∂h

(6)

Here, D = δ/h0 is the nondimensional thickness of the nanocoating and R = Ac/As. Q = 2πρgh04/|As| is the representative coefficient of the gravity term. Similarly, the nondimensional form of excess free energy for the polar−apolar system (ΔGp) is

■

Fs[h] = Fe + Fi =

(5)

(4)

where all gradients are taken in the plane of the substrate. The quantities γ and μ refer to surface tension and viscosity of the liquid, respectively. To reduce the parameters, mean film thickness (h0), characteristic length-scale for van der Waals case ((2πγ/|As|)1/2h02), and time-scale (12π2μγh05/As2) are used to scale the corresponding variables (h, x, and t) to their nondimensional values (H, X, and T). The thin film equation in its nondimensional form is given by C

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Figure 5. Evolution of a 400 nm thin film on a 192 nm coating into two equilibrium phases through the direct pathway. Panels a−d present the morphology at their corresponding nondimensional times (T). H is nondimensional film thickness. System size is 8192LM, and simulation parameters are R = −0.1, D = 0.48, and Q = 0.105.

again, a coated-apolar-gravity system is chosen to highlight the differences with the defect pathway. The free-energy diagram of the system is available in the Supporting Information. We start by presenting the amplified spinodal wave in Figure 5a. The evolution is similar to that of the defect pathway until the appearance of the low amplitude spinodal wave (not shown) from the random surface perturbations. The amplification of the crests and troughs in this case is markedly different from that in the defect pathway (Figure 4a). First, the amplification toward thicker or thinner equilibrium thicknesses is comparable for any pair of crest and trough. This is expected as the initial film thickness is roughly midway between the two equilibrium thicknesses, P1 and P2, whereas it was much closer to the thinner phase in the defect pathway. Second, the number of crests and troughs in the amplified wave (solid and dashed lines) is much less than that predicted by the linear theory. Third, there is marked variation in the amplitude of the wave at different locations. The last two observations show that the spinodal wave coarsens also while it amplifies. This additional coarsening that happens even before the formation of phases is due to “surface diffusion” as opposed to “bulk diffusion”, which happens during coarsening of defects through the thinner equilibrium film phase. Figure 5a−c clearly illustrates this coarsening at two locations. The first defect from the left and the third defect from the right in Figure 5c are results of this type of coarsening only. Figure 5b,c clearly shows that thicker and thinner phases appear at almost the same time. We denote this pathway of true MPS as “direct pathway” as both phases are formed directly without any coarsening of well-formed defects. Further coarsening of these domains is similar to that in the case of defect pathway. Figure 5d presents two flat-topped and much wider domains of the thicker phase, which form because of the coarsening of the equilibrium domains shown in Figure 5c. Both direct and defect pathways involve coarsening to form the thicker and thinner phases. We observe that direct pathway involves coarsening during amplification of the spinodal wave, whereas the defect pathway requires significant coarsening after the amplification. The coarsening in the direct pathway is mostly surface diffusion based, whereas in defect pathway, it is bulk diffusion based. The initial number of thicker domains in

Figure 4. Evolution of a 214 nm thin film on a 128 nm coating into two equilibrium phases through the defect pathway. Panels a−f present the morphology at their corresponding nondimensional times (T). H is the nondimensional film thickness. System size is 8192LM, and the simulation parameters are R = −0.1, D = 0.6, and Q = 8.6 × 10−3.

defects grow in thickness and width at the expense of some others. The height of the defects still remains much less than that of the thicker film phase. This evolution is similar to the late stage in the MPS. Figure 4d presents the situation when the thickness of the defects reaches that of the thicker film phase because of coarsening. This marks the end of similarity in the morphological evolution between MPS and true MPS. Further coarsening leads to widening and flattening of the defects marking the appearance of the thicker film phase as shown in Figure 4e. As expected, the defects do not thicken beyond the height of the thicker film phase. The appearance of the second film phase as a distinct domain characterizes the true MPS. The domain coarsens further through material transfers from other defects and domains that are situated beyond the limits of the substrate shown in the figure. The figure shows only 30LM out of the total length of 8192LM to focus on the key morphological changes. We denote this pathway of true MPS as the defect pathway as the thicker phase forms only after the coarsening of defects. It should be noticed that there is a vast delay between the time of formation of thinner phase, T ≈ 105, and the first appearance of the thicker phase, T ≈ 109. It is interesting to see that only 2 out of initial 30 defects could become the thicker equilibrium domains. The sequence of the morphological changes in the defect pathway of true MPS can be summarized as follows: random surface perturbations → spinodal wave → amplification of spinodal wave → formation of the first film phase and defects → increase in the thickness of defects due to coarsening → formation of domains of the second film phase → coarsening of the second film phase → single domains of both phases. The thicker film phase can form directly also by amplification of the spinodal wave without coarsening of defects. Figure 5 presents the morphological evolution of such a case. Once D

DOI: 10.1021/acs.langmuir.7b00752 Langmuir XXXX, XXX, XXX−XXX

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N). This results in a correspondingly longer time to access the thicker phase, starting from the original spinodal instability. One should realize that the choice of the defect pathway requires existence of defects, which in turn is contingent to the presence of a thinner film phase. If thicker film phase forms around the same time as the thinner one, then there is no possibility of formation of defects and associated defect pathway. Thus, evolution of minimum and maximum thicknesses (Hn and Hx) during true MPS can be a good marker of the morphological pathway. The top panel of Figure 7 presents typical evolution of Hx and Hn for defect and direct

both pathways is also very different. The direct pathway of true MPS results in approximately 10 true thick phase domains (Figure 5), whereas the defect pathway resulted in only two true thick phase domains in 30LM size (Figure 4). The corresponding numbers for a much larger size of 8192LM are 178 and 1295 for the defect and direct pathways, respectively. Whether true MPS happens via the defect pathway or direct pathway depends on the mean thickness of the film. The spinodal parameter curve for the coated-apolar-gravity system (bottom panel of Figure 2) shows the range of unstable film thicknesses (between h1 and h2), which can exhibit true MPS via spinodal decomposition. Simulations for a large range of parameter values confirm that unstable films with thicknesses closer to the thinner phase are likely to choose the defect pathway, whereas films with thicknesses closer to the thicker phase are likely to follow the direct pathway. The thickness at which a crossover from one pathway to the other will happen is hard to establish because of the asymmetry of the free-energy curve. However, the thickness halfway to both the thinner and thicker phases serves as an excellent first estimate. If this estimate is well within the unstable region, then the system can exhibit both defect and direct pathways depending on the thickness of the film. If this estimate lies outside of the unstable region, then the system is likely to exhibit true MPS through the defect pathway only. Figure 6 presents a typical phase

Figure 6. Phase diagram showing the binodal and spinodal boundaries for coated-apolar-gravity systems. The system parameters are As = 1.4 × 10−20 and Ac = −1.4 × 10−21.

Figure 7. Variation in maximum thickness [Hx] and minimum thickness [Hn] (top panel), number of equilibrium domains (middle panel), and number density of local maxima (bottom panel) with time for a 214 nm film on 128 nm coating and a 400 nm film on 192 nm coating undergoing true MPS. System size is 8192LM, and simulation parameters are R = −0.1, D = 0.6, Q = 8.6 × 10−3 for 214 nm film and R = −0.1, D = 0.48, Q = 0.105 for 400 nm film.

diagram for the coated-apolar-gravity systems. This phase diagram that shows the binodal and spinodal boundaries for true MPS can be compared and contrasted with the typical phase diagrams of SPS. The binodals (solid lines) depict the final equilibrium thicknesses, P1 and P2, and the corresponding spinodals (dashed lines) present the range of unstable thicknesses. The spinodal lines are symmetric as expected for a phase separation process. However, the asymmetric nature of the binodal lines arising from the asymmetricity in the freeenergy diagram distinguishes true MPS from SPS. It is clear that, for a coating thickness of 192 nm, true MPS can be observed through either defect pathway (A) or direct pathway (B). However, for coating thickness of 84 nm, any unstable film will undergo true MPS only through the defect pathway as the thicker phase (P2) is always farther than the thinner phase (P1) for all possible thicknesses in the unstable region (e.g., M and

pathways. Once again, a apolar-coated-gravity system is chosen for illustration. The emergence of thinner and thicker phases can be readily observed in the figure. For 214 nm film (dashed lines), there is a time delay of approximately 5 decades between the emergence of the thinner phase and the thicker phase. By contrast, the two phases appear almost at the same time for the 400 nm film (solid lines). This behavior by itself indicates that 214 and 400 nm films will exhibit defect and direct pathways, respectively. Various morphological events related to the appearance of thinner and thicker phases can be marked to describe true MPS. Let T1 refer to the time of first appearance of the thinner E

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thinner flat-film phase (T = T1) is necessarily characterized by this decay. This decay leads to T1 through a plateau because of the absence of any coarsening during the amplification of the spinodal wave in the defect pathway of 400 nm. The number density corresponding to this plateau of the defect pathway is accurately predicted by the linear theory. The decay continues a little longer in the case of direct pathway and then reaches T1 through a plateau of a much lower number density. This is because the surface coarsening continues even during the amplification of the spinodal wave in the case of direct pathway as shown in figure (use morphological evolution picture) and discussed earlier. The decay during the intermediate stage, which ends with T2 has no characteristic behavior as the morphology is a mix of thinner equilibrium phase, defects, and local minima. The decay in the post-T3 stage is logarithmic because of the coarsening among equilibrium domains as shown in the inset of the bottom panel. The decay in the defect stage, between T2 and T3, starts with another universal power law exponent of −1/3 because of the coarsening of the defects.3,13 This slowly changes to a logarithmic decay as some of the defects grow to become equilibrium phases as corroborated by the rising curve in the top panel. The direct pathway has no defect stage and hence does not show any −1/ 3 power law decay before the logarithmic decay of the last stage (post-T3). The kinetics of various stages of true MPS can be summarized as follows: • The early stage in both pathways is characterized by a power law (−1/4) decay followed by a plateau. • The intermediate stage shows no characteristic kinetics. • The defect stage is characterized by a power law (−1/3) decay, which gradually changes to a logarithmic decay. The defect stage is absent in the direct pathway. • The final stage of both pathways is characterized by a logarithmic decay, corresponding to the 1-d coarsening of coexisting phases. This growth is driven by the weak interactions of defect tails and should be contracted with curvature-driven growth which arrives for d ≥ 2. The curvature-driven growth regime is characterized by t1/3 growth.5 We now present the typical variation in interfacial (Fi), excess (Fe), and the total free energies (Fs = Fi + Fe) with time for defect (Figure 8) and direct (Figure 9) pathways for the coatedapolar-gravity system under discussion. The times T1, T2, and T3 have been marked based on tracking the morphological evolution. Both Fi and Fe do not

phase at any location on the substrate or the onset of phase separation. Let T2 and T3 refer to the time of appearance of the thinner and thicker phases at all possible locations, respectively. Therefore, at T2 all local minima would have thinned to P1, and at T3 all local maxima would have thickened to P2. By T3, the system is supposed to have phase separated completely into the two equilibrium phases. True MPS can then be described as a succession of following four stages based on T1, T2, and T3. 1 An initial or pre-T1 stage, which is the same as initial stage of MPS. 2 An intermediate stage, which spans from T1 to T2 and is the same as the intermediate stage of MPS. 3 A defect stage, which spans from T2 to T3 and involves coarsening of defects. This stage is similar to the late stage of MPS as long as all defects have thicknesses below that of the thicker equilibrium phase. Coarsening of equilibrium domains is also possible later on when some of the defects thicken to become the thicker equilibrium domains. 4 A final or post-T3 stage, which involves coarsening of the equilibrium domains only. The defect pathway comprises of all of these stages, whereas the direct pathway is characterized by the absence of the defect stage or T2−T3. It is clear that the variation in Hn helps in estimating T1 only. We now explore the kinetics of number density of local maxima/defects/domains, the most efficient and robust marker of MPS, to estimate T2 and T3. This will uncover the kinetic signatures, if any, of the different stages as well. The variation in the number of equilibrium domains and the local maxima with time is presented in the middle and bottom panels of Figure 7, respectively, for the 214 and 400 nm films undergoing true MPS. The representative times T1, T2, and T3 have been marked in the figure based on the detailed tracking of the morphological evolution. The number of local maxima is scaled with n, size of the substrate in units of LM, to present the comparison with the linear theory and remove the effect of the substrate size. It should be noted that local maxima represent all different morphological features that lead to the thicker phase, namely, crests of the spinodal wave, defects, and equilibrium domains. The number of equilibrium domains is presented as a fraction of total number of local maxima. Its value remains zero until the first appearance of the thicker equilibrium phase and becomes unity when the thicker phase has formed at all possible locations, namely, at T = T3. It should be noticed that these curves themselves are a result of detailed tracking of the local maxima and hence can be used to mark T3 with precision. T1 and T2 can be estimated from the figure for the case of direct pathway only as the first appearance of the thicker phase (first nonzero value of the curve) coincides with the first appearance of the thinner phase (T = T1) and T2−T3 for the direct pathway. It should be kept in mind that the thicker phase appears much later than the thinner phase in the defect pathway. Both films show an initial power law decay in the number of local maxima with a universal exponent of −1/4. A universal curve can represent this part for films under all force fields if one corrects the time for the different initial conditions.3 This decay characterizes surface diffusion effected coarsening in the absence of any connecting flat films such as coarsening of undulations of the surface fluctuations shown in Figure 5b,c. Naturally, all coarsening before the first appearance of the

Figure 8. Variation in the interfacial (Fi), excess (Fe), and total (Fs) free energies for a 214 nm film on 128 nm coating undergoing true MPS through the defect pathway. System size is 8192LM, and simulation parameters are R = −0.1, D = 0.6, and Q = 8.6 × 10−3. F

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Figure 9. Variation in the interfacial (Fi), excess (Fe), and total (Fs) free energies for a 400 nm film on 192 nm coating undergoing true MPS direct pathway. System size is 8192LM, and simulation parameters are R = −0.1, D = 0.48, and Q = 0.105.

change significantly during the early stage but change dramatically just after T1 for both pathways. Fi increases because of the generation of more interface, whereas Fe decreases because of the thinning of the film. Fi keeps on increasing as more and more interface is created but reaches a maxima at T2 and decreases after that as the coarsening reduces the amount of interface in both pathways. Fe keeps on decreasing continuously as the system evolves toward the thinner and thicker equilibrium phases. The decrease in both energies is sharper in the defect stage than in the final stage as defects coarsen much faster (power law decay) than the equilibrium domains (logarithmic decay). The variation in Fs, which is a summation of Fi and Fe, is dominated by the variation in Fe because of the vast difference in their magnitudes. This is not surprising as true MPS is driven mostly by excess intermolecular forces. The characteristic features of true MPS, its pathways and associated kinetics, are not limited to gravity-affected thicker films only. As discussed earlier, any thin film system that allows construction of Maxwell’s double tangent in its free-energy curve such as the polar−apolar system shown in Figure 3 will exhibit true MPS. The thickness of the films in such systems can be much less than that of the gravity systems. Extensive numerical simulations show that apolar−polar system with freeenergy curves similar to Figure 3, indeed undergo true MPS via the defect and direct pathways. We present briefly the kinetics of true MPS in apolar−polar systems here to avoid unnecessary repetition. The morphological evolution and the variation in free energies, which closely resemble their counterparts for the coated-apolar-gravity system, are presented in the Supporting Information. We find that a 2.2 nm film undergoes true MPS through the defect pathway and a 2.7 nm film takes the direct pathway. We will now see that the proposed description of the true MPS in terms of stages and representative times is universal in the sense that it does not depend on the force field. Figure 10 presents the typical kinetics of true MPS in polar− apolar systems. The top panel shows the evolution of Hn and Hx for both 2.2 and 2.7 nm films. The simultaneous appearance of the thicker and the thinner phases indicates that the 2.7 nm film (solid lines) undergoes true MPS though the direct pathway. Similarly, the long delay between their appearances suggests that the 2.2 nm film (dashed lines) follows the defect pathway. It should be noticed that 2.2 and 2.7 nm are closer to the thinner and thicker phases, respectively. This confirms our earlier assertion that thinner films are likely to follow the defect pathway. We also see that the representative time events of T1,

Figure 10. Variation in maximum thickness [Hx] and minimum thickness [Hn] (top panel), number of equilibrium domains (middle panel), and number density of local maxima (bottom panel) with time for 2.2 and 2.7 nm aqueous films undergoing true MPS. System size is 8192LM, and simulation parameters are D = 0.071, L = 0.272, R = −14.87 for the 2.2 nm film and D = 0.059, L = 0.22, R = −21.54 for the 2.7 nm film.

T2, and T3 can describe the true MPS for both pathways in this case also. The middle and bottom panels of Figure 10 present the variation in the equilibrium domains and number density of local maxima with time, respectively. The equilibrium domains and the local maxima are scaled as explained for the case of apolar-coated-gravity system. The times T1, T2, and T3 are marked based on the morphological evolution. The figure clearly marks out all four stages of true MPS and their associated kinetics, which shows their similarity with that of the coated-apolar-gravity system.

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CONCLUSIONS We conclude that unstable thin films can morphologically phase separate into two equilibrium flat-film phases of different thicknesses whenever Maxwell’s double-tangent construction is possible in the free-energy diagram. This phase separation takes place via defect or direct pathway depending on the initial thickness of the film. In both pathways, the asymptotic coarsening regime of true MPS is analogous to that in conventional SPS. This is because the late stages of both phenomena are characterized by the elimination of interfacial defects between two coexisting phases. This aspect is absent in MPS, where there is only one proper phase, and the height of the other “phase” diverges to infinity.3,13,20 G

DOI: 10.1021/acs.langmuir.7b00752 Langmuir XXXX, XXX, XXX−XXX

Article

Langmuir

(10) de Gennes, P.-G.; Brochard-Wyart, F.; Quéré, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves; Springer-Verlag: New York, 2004. (11) Rauscher, M.; Dietrich, S. Wetting phenomena in nanofluidics. Annu. Rev. Mater. Res. 2008, 38, 143−172. (12) Gratton, M. B.; Witelski, T. P. Coarsening of unstable thin films subject to gravity. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2008, 77, 016301. (13) Vashishtha, M.; Jaiswal, P. K.; Khanna, R.; Puri, S.; Sharma, A. Spinodal phase separation in liquid films with quenched disorder. Phys. Chem. Chem. Phys. 2010, 12, 12964−12968. (14) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1985. (15) Lister, J. R.; Rallison, J. M.; Rees, S. J. The nonlinear dynamics of pendent drops on a thin film coating the underside of a ceiling. J. Fluid Mech. 2010, 647, 239−264. (16) Gomba, J. M.; Perazzo, C. A. Closed-form expression for the profile of partially wetting two-dimensional droplets under gravity. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2012, 86, 056310. (17) Mayo, L. C.; McCue, S. W.; Moroney, T. J. Gravity-driven fingering simulations for a thin liquid film flowing down the outside of a vertical cylinder. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2013, 87, 053018. (18) Narendranath, A. D.; Hermanson, J. C.; Kolkka, R. W.; Struthers, A. A.; Allen, J. S. The Effect of Gravity on the Stability of an Evaporating Liquid Film. Microgravity Sci. Technol. 2014, 26, 189−199. (19) Sharma, A.; Verma, R. Pattern Formation and Dewetting in Thin Films of Liquids Showing Complete Macroscale Wetting: From “Pancakes” to “Swiss Cheese”. Langmuir 2004, 20, 10337−10345. (20) Jaiswal, P. K.; Vashishtha, M.; Puri, S.; Khanna, R. Morphological phase separation in unstable thin films: Pattern formation and growth. Phys. Chem. Chem. Phys. 2011, 13, 13598− 13603. (21) Lifshitz, I. M.; Slyozov, V. V. The kinetics of precipitation from supersaturated solid solutions. J. Phys. Chem. Solids 1961, 19, 35−50. (22) Kostorz, G. Phase Transformations in Materials, 1st ed.; WileyVCH: Weinheim, New York, 2001. (23) Onuki, A. Phase Transition Dynamics, 1st ed.; Cambridge University Press: Cambridge, U.K., 2002. (24) Ruckenstein, E.; Jain, R. K. Spontaneous rupture of thin liquid films. J. Chem. Soc., Faraday Trans. 2 1974, 70, 132−147. (25) Mitlin, V. S. Dewetting of solid surface: Analogy with spinodal decomposition. J. Colloid Interface Sci. 1993, 156, 491−497.

The pathways have been described as a sequence of initial, intermediate, defect, and final stages based on representative times, which are related to the formation of the two phases. We have been able to establish the major differences between the kinetics of the two pathways. The defect pathway is characterized by a delayed formation of the thicker phase as it forms only as a result of the coarsening of the defects. Last, we highlight the universality in the kinetics of different stages by showing that decay characteristics do not depend on the precise intermolecular force field but rather only on the coexistence of two well-defined equilibrium phases. The simulation results presented in this paper have important experimental implications. It is our hope that these numerical results will be tested in relevant experimental scenarios.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b00752. Free energy and spinodal parameter curves for the 192 nm coated-apolar-gravity system discussed in Figures 5− 7, morphological evolution during true MPS of the 2.2 nm aqueous film discussed in Figure 10, morphological evolution during true MPS of the 2.7 nm aqueous film discussed in Figure 10, variation in interfacial, excess and total free energies with time during true MPS for the 2.2 nm aqueous film discussed in Figure 10, and variation in interfacial, excess and total free energies with time during true MPS for the 2.7 nm aqueous film discussed in Figure 10 (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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DOI: 10.1021/acs.langmuir.7b00752 Langmuir XXXX, XXX, XXX−XXX