Nanomaterials synthesis by a novel phenomenon: The nanoscale Rayleigh-Taylor instability

The Rayleigh-Taylor (RT) interfacial instability has been attributed to physical phenomenon in a wide variety of macroscopic systems, including black holes, laser generated plasmas, and thick fluids. However, evidence for its existence in the nanoscale is lacking. Here we first show theoretically that this instability can occur in films with thickness negligible compared to the capillary length when they are heated rapidly inside a bulk fluid. Pressure gradients developed in the evaporated fluid region produce large forces causing the instability. Experiments were performed by melting Au films inside glycerol fluid by nanosecond laser pulses. The ensuing nanoparticles had highly monomodal size distributions. Importantly, the spacing of the nanoparticles was independent of the film thickness and could be tuned by the magnitude of the pressure gradients. Therefore, this instability can overcome some of the limitations of conventional thin self-organization techniques that rely on film thickness to control l...

The Rayleigh-Taylor (RT) interfacial instability has been attributed to physical phenomenon in a wide variety of macroscopic systems, including black holes, laser generated plasmas, and thick fluids. However, evidence for its existence in the nanoscale is lacking. Here we first show theoretically that this instability can occur in films with thickness negligible compared to the capillary length when they are heated rapidly inside a bulk fluid. Pressure gradients developed in the evaporated fluid region produce large forces causing the instability. Experiments were performed by melting Au films inside glycerol fluid by nanosecond laser pulses. The ensuing nanoparticles had highly monomodal size distributions. Importantly, the spacing of the nanoparticles was independent of the film thickness and could be tuned by the magnitude of the pressure gradients. Therefore, this instability can overcome some of the limitations of conventional thin self-organization techniques that rely on film thickness to control length scales. C

I. INTRODUCTION
There are many applications of nanostructures, such as nanoparticles (NPs), of any variety of hard matter, including metals, ceramics and semiconductors, that require their arrangement on surfaces with precise control over size and spacing. Some broad categories of applications include solar energy harvesting, biosensing, and photocatalysis. [1][2][3][4][5][6][7][8][9] In recent years the spontaneous deformation of a thin film leading to nanoscale patterns governed by self-organizing principles have generated great interest. The reason is that by controlling the choice of materials, such as film and substrate, intrinsic forces of surface tension and dispersion can be changed to modify the time and length scales of pattern formation. [10][11][12] However, while the reliance of intrinsic forces leads to robustness and predictability in pattern formation, such thin film self-organization invariable limits the control over pattern length scale to the form λ ∝ h n , where λ is the self-organized length, h is the film thickness and n varies depending on the conditions used. For instance, in the well studied liquid-phase spinodal dewetting instability, the pattern length scale λ varies with film thickness h as λ ∝ h 211, [13][14][15] and could be modified somewhat by introducing thermal gradients, 16 or by relying on solid state mass transport. 17 While some other ways to overcome this constraint have been demonstrated for polymer liquid films, such as by chemical or morphological modifications to the substrate surface, 18-20 similar flexibility has not been shown for the vast majority of high temperature fluids, such as metals and semiconductors.
Here we propose a novel way to decouple the self-organized length scale from film thickness by using the Rayleigh-Taylor (RT) instability. From a historical perspective, the length scale in theclassical RT pattern-forming instability, such as the beading up of a water film on the ceiling, does not depend on the thickness of the film. 21 But, the classical RT instability cannot be observed in nanoscale thin films because the acceleration forces acting on the fluid come from gravity, thus 047116-2 S. Yadavali and R. Kalyanaraman AIP Advances 4, 047116 (2014) requiring thick films (thickness greater then the capillary length of order 0.1 mm). On a fundamental level, the primary requirement for the RT instability is that a less dense fluid should be accelerated into a more dense fluid. 22,23 This interface destabilization between two fluids can be achieved in a number of different ways, 21 as evident by the large number of examples from an extremely diverse range of systems, including in astronomical structures such as black holes and supernova, [24][25][26] geophysical phenomenon, 27, 28 fusion, [29][30][31][32] and bulk fluid-fluid interfaces. [33][34][35] The new idea proposed in this work is based on the interesting thermal behavior found when a film-fluid interface is heated rapidly by short laser pulses, with the fluid being converted into gas. Since the temperature falls rapidly on going away from the film/gas interface, strong gas pressure gradients are generated. This leads to a local, film-thickness-dependent, pressure that can overcome the stabilizing effects of surface tension and also dominate the destabilizing effects of intermolecular attractions found in nanoscale thickness films. When this result is incorporated into mass transport models, linear analysis predicts a classical RT instability scaling behavior. We performed experiments involving nanosecond (ns) pulsed laser melting of nanometer thick Au films immersed in glycerol liquid. The ensuing nanoparticle size and spacing was studied as a function of laser energy and film thickness. The results were in excellent agreement with the theoretical predictions of the RT instability, including the independence of nanoparticle spacing on film thickness. To the best of our knowledge this is the first observation of RT instability at the nanoscale and its application towards synthesizing materials in a controllable manner.

II. BACKGROUND
In the macroscopic or classical RT instability, a liquid film hanging on a substrate is destabilized when gravitational effects overcome the pressure exerted by the ambient atmosphere. 21 In this work we are primarily concerned with films whose thickness is substantially smaller then capillary lengths, which, for example, is of the order of 0.1 mm for the metal films we will investigate here. In this situation, gravity cannot play a role in stabilizing/destabilizing the film. Instead, other mechanisms to destabilize the films exist. One widely studied case pertains to the scenario when intermolecular attractive forces, such as dispersion forces, can overcome surface tension and result in spontaneous dewetting. This phenomenon has been widely investigated in organic and inorganic films in the context of spinodal dewetting and is typically exhibited by ultrathin films (normally <100 nm). 12,[36][37][38] The characteristic feature of this instability is the strong dependence of the pattern length scale on the film thickness, i.e. λ dewet ∝ h 2 o , where h o is the initial film thickness. 14, 15 First, we describe a general scenario in which an RT-like instability can arise in nanometer thick films lying on a substrate, as shown in Fig. 1(a). Picture a surface perturbation of the liquid film shown in Fig. 1(a) occurring into a hydrostatic pressure field where the pressure acting on the film is dependent on the local height of the film at any position x along the perturbed film, i.e. p(h) = P(x, z). Two possible situations can be envisioned. In the first case, let the local pressure increase with increasing film height h. In this situation, the increased pressure will counter the growth of the perturbation (adding to the effect of surface tension) and therefore, increase the overall stability of the film to deformations. On the other hand, consider the case when the local p(h) decreases with increasing h. In this case, the pressure field opposes the stabilizing effect of surface tension and, if it is large enough, can foster the growth of the perturbation, leading eventually to spatial and temporal behavior dependent on the various forces in play. Recently, two physical processes to achieve this type of scenario have been postulated. Bestehorn et al. 39 suggested the theoretical possibility that a thin film on the underside of a substrate (i.e. RT unstable film) in equilibrium with its vapor phase under a steady-state thermal gradient can lead to patterns influenced by the evaporative process. A second possibility, a non-gravitationally generated RT instability, was reported as an experimental observation by Chen et al. 40 in which the pulsed laser melting of a bulk Si surface immersed in water produced patterns that resemble the early-stage hexagonal structures predicted by the Cahn-Hillard form of the RT evolution. The physical mechanism invoked was the explosive boiling of water in contact with the Si surface, which could generate sufficient pressures to produce the resulting patterns. However, the physical origin of the local p(h) behavior described above that is essential to destabilize the surface was not discussed. In addition, an important quantitative hallmark of the RT instability, which is its characteristic length scale λ was not presented. The length scale of RT instability patterns should behave as λ ∝ 2γ ρa , where γ is the surface tension of the film (or bulk) surface and ρa characterizes the effective pressure per unit length (force per unit volume) arising under the acceleration a. 21 It should be emphasized that such non-gravitationally induced RT instability behavior has not been experimentally verified in reported works for either bulk or thin fluid films, including in situations producing patterns with characteristic length scales in the nanometers.
We propose an alternate way to foster RT instabilities based on the thermal profile that occurs at a substrate/film/fluid interface that is rapidly heated by a short laser pulse. The rapid heating of the film by absorption of short [such as nanosecond (ns)] pulse laser light is well known to lead to boiling of a low temperature fluid in contact with the film. [41][42][43] Because of the rapid heating rates, a large drop in temperature occurs on going away from the absorbing film, i.e. a large magnitude of thermal gradient |T Z | = | dT d Z | with a negative sign (i.e. T Z < 0, indicating decrease in temperature in going from the interface into the vapor/fluid region) exists. When these thermal gradients are incorporated into the pressure dependence of temperature of the gas phase, large transient gas pressures, that fall with distance away from the film, develop. This pressure gradient P Z = d P d Z , with sign P Z < 0, can produce a local film thickness-dependent gas pressure P[x, T(z)] [ Fig. 1(a)] such that it drives the fluid to flow from regions of locally lower film thickness to higher ones, i.e. grows perturbations and produces the RT instability, as we demonstrate in the remainder of this manuscript.

III. THEORY AND MODELING
The estimate of pressure gradients P Z perpendicular to a film/gas boundary was evaluated with specific reference to the ns pulsed laser heating of a quartz/Au/Glycerol system irradiated by 266 nm wavelength laser light at normal incidence from the quartz side. At this wavelength, the photons are absorbed only by the Au film layer (and are transparent to quartz and Glycerol). 44 The ensuing physical situation for laser energies of relevance here resulted in the melting of the Au film, as well as the formation of a gas phase region of Glycerol bounded by its fluid on one side and the molten Au film on the other, as shown in the schematic, Fig. 1(a). First, a simple order of magnitude estimate (and sign) of the pressure gradient P Z in the vapor was made by estimating a temperature gradient T Z away from the film/gas interface through a thermal diffusion analysis as follows. For the T Z in glycerol, we assumed that the spatial extent of the vaporized glycerol L v gl can be approximated as one thermal diffusion length, i.e. L gl = D T h gl τ p , where D T h gl and τ p are the thermal diffusivity of liquid glycerol and the laser pulse time respectively. Using D T h gl of 10 −9 m 2 .s -1 from Ref. 45, and using a laser pulse width of 9 ns, we get L gl of ∼30 nm. The simplest approximation for the gradient is a linear one between the melting point of Au (1337.3 K) on the film side to boiling point of glycerol (563 K) on the glycerol vapor/fluid boundary side. This gives an order of magnitude for the thermal gradient, T Z , as ∼−25 K/nm, with the negative sign indicating that the temperature decreases in going from the interface into the Glycerol vapor/fluid region. Now, the P T = dP/dT for glycerol from P vs T empirical data available from standard references 45 varies with T and is ∼5 × 10 6 Pa/K at the melting point of Au. In other words the transient pressure gradient P Z is estimated to be ∼−1.25 × 10 17 Pa/m, and is a negative quantity, consistent with the sign of T Z and the positive sign of P T typically seen in gases. Therefore, for a metal like Au with density of ∼20 × 10 −3 Kg.m -3 , by using the dimensional equality of |P Z | = |ρa| = |P T × T Z |, we get an extremely high value for the acceleration a of ∼6 × 10 19 m 2 /s, substantially larger then the estimates of Chen et al. 40 To better quantify this thermal effect, we performed finite element calculations to estimate the T Z (and thereby P Z ) based on Fourier heat transport models for multilayer heating that incorporate phase change effects. We have previously developed these models in Refs. 12 and 46, where appropriate use of thermally varying materials parameters allowed us to successfully explain the thermal transport behavior in such nanoscale systems. The Fourier heat transport form of thermal transport was expressed as follows: This equation was solved in each material layer, with ρ being the mass density of the layer, C p the isobaric heat capacity, k the thermal conductivity, z the axis normal to the planar, fixed surface pointing from the substrate, as shown in Fig. 1(a), and Q laser (z, t) being the laser heating contribution given by is the thickness dependent reflectivity, and A(z) is the thickness dependent absorption coefficient. R(z) and A(z) were calculated using the rigorous method described in Ref. 47. The temperature dependent ρ, C p and k for glycerol liquid and gas phase were calculated from Ref. 48 and for gold from. 49 The incident laser power was modeled as a Gaussian distribution with 9 ns pulse width, consistent with the characteristics of the laser used in the experiments to be described. Further details of the calculation are provided in the supplemental information Sec. II and in supplement Fig. 1. 50 In Fig. 1(b), temperature vs distance behavior at the time corresponding to the peak temperature rise in a 8 nm thick Au film for various laser energy densities is shown. The large change in temperature with distance away from the film/gas interface (as well as film/substrate interface) is clearly evident and was calculated to be of the order of magnitude of 30 K/nm at the time of the peak temperature T * . The gradients were estimated from these T vs distance plots as follows. A linear interpolation was applied between the position of the peak temperature at the film/gas interface to the position in the gas phase when the temperature dropped to the glycerol phase change temperature (T = 563 K). Next, by using the empirically known P vs T for glycerol, 45 the effective pressure force, P Z , was estimated for films of various thickness by selecting the laser energy density to achieve the same maximum temperature rise (of T * = 1750 K). The magnitude of the obtained P Z is shown by the dashed line in Fig. 1(c). The two important features are its large magnitude, i.e. O(10 16 ) N/m 3 , and its relative independence to the thickness of the film (for the same peak temperature T * ). As we discuss shortly, the magnitude of this P Z force (in Pa/m or N/m 3 ) is substantially larger then other destabilizing forces, such as the intermolecular dispersion forces estimated for the quartz/Au system 49 and shown by the dotted line in Fig. 1(c).
To theoretically evaluate the role of this large P vs z on film behavior, we solved the mass transport dynamics using the thin-film hydrodynamic equation in the lubrication approximation of the Navier-Stokes equations. 51 This form allowed us to include the gas pressure in addition to the surface tension and dispersion forces due to intermolecular interactions that are already present in such ultrathin film on substrate systems, as depicted in Fig. 1(a). 13,52 A similar approach to solving for the hydrodynamics has been used by us previously in Ref. 12 and 46 to explain the linear and non-linear dynamics of ultrathin fluid films. The total pressure P(x, z) acting on the film at position x was expressed as the superposition of a disjoining pressure ( ), Laplace pressure P L and pressure due to the gas layer P[T(z(x)], as: We solved for the evolution of the instantaneous film height h in 1-dimensions (1D) by the mass conservation equation of the form Where γ is the surface tension of the film-gas interface [γ GF in Fig. 1(a)], η is the viscosity of the film, and the disjoining pressure for a thin film on substrate is given by , where A is the Hamaker coefficient. Since the gas pressure acting at any point along the film surface is dependent on the temperature through its dependence on height, i.e. P(x, z) = P[T(z(x)], it's spatial derivative can be expressed as follows.
Now since z is the coordinate measuring the height h(x), we can express the 1 st and 2 nd order spatial derivatives of pressure as P Z ∂h(x) ∂ x and P Z ∂ 2 h(x) ∂ x 2 , respectively, yielding an evolution equation of the form: Here, we have ignored any contribution to film evolution resulting from the evaporation of the film material during the heating process. As shown in the supplement Fig. 2, 50 actual estimates of Au evaporation based on time and temperatures achieved here suggested negligible contributions. We also chose to ignore the dispersion force contributions, A V F in Fig. 1(a), acting across the glycerol fluid/glycerol gas and glycerol gas/metal film interfaces. This can be justified due to the large thickness of the gas layer (˜150 nm based on thermal model estimates) and partly due to the small values of A that typically exist across a gas medium. Therefore, the primary contribution came from the A V S term in the dispersion forces. Equation (4) was solved using linear stability analysis under boundary conditions of no slip at the film/substrate surface v z = 0 at z = 0, and no shear stress ( ∂v z (z=h) ∂ x = 0) at the film/gas interface with sinusoidal perturbations of form h(x) = h o + εsin(kx)cos(ωt). This yielded a dispersion relation for deformation rate ω to wavevector k as: Where h = − A 2π h 4 , with the negative value signifying an attractive term (with the Hamaker coefficient taken as a positive quantitiy), and the pressure gradient P Z also having a negative sign, as explained earlier. From this, the characteristic wavelength λ RT = 2π /k m was obtained from the maximum growth rate of perturbations k m (i.e. k 2 m = − h −P Z 2γ , when dω dk = 0) as: Also, the time scale τ RT = 2π ω m associated with the growth of perturbations with wavelength λ was obtained by letting k = k m in Eq. (5), giving:

IV. EXPERIMENTS
To analyze the implications of this model, experiments were performed for Au films (with thickness ranging from 4 to 12 nm) deposited on transparent bulk quartz (SiO 2 ) substrates, with this system immersed in a bulk glycerol liquid solution. The Au films were deposited in high vacuum (∼1 × 10 −8 Torr) by electron beam deposition at room temperature onto commercially obtained, optical quality, SiO 2 substrates. 15 The deposition rate used here was typically 1.5 nm/min as confirmed by atomic force microscope (AFM) step height measurements. Energy dispersive Xray spectrometry (EDS) was used to measure the Au counts of the deposited films in a scanning electron microscope (SEM). The EDS counts were converted into an equivalent thickness value by using calibration based on step-height measurements of the film thickness. Surface roughness of the deposited films were measured by AFM and established an upper limit of 0.5 ± 0.2 nm for the average root mean square (RMS) roughness over the entire thickness range. The Au/SiO 2 samples were placed on the bottom a transparent glass beaker (with substrate side of the sample in contact with the beaker) filled with pure glycerol. Laser irradiation was performed at normal incidence from the quartz side, as depicted in Fig. 1(a), with an unfocused laser beam of area 1 × 1 mm 2 at a repetition rate of 50 Hz for a total of approximately 5000 pulses (∼100 seconds). Following the irradiation, the samples were removed from the fluid, dried in air, and the resulting nanostructures were investigated in a scanning electron microscope (SEM) for various values of initial film thickness h o , and the laser energy density E. Fig. 2(a) shows an SEM image of the prototypical nanostructure obtained following the laser irradiation experiment, in this case for a 4 nm Au film (at 80 mJ/cm 2 energy density). Nanoparticles with a well-defined inter-particle spacing and highly monomodal size distribution were obtained. The lower inset in the figure shows the fast fourier transform (FFT) of the contrast in the SEM images. The annular form of the FFT captures the spatial short range ordering typically prevalent in such thin film instabilities. The FFT analysis also allowed accurate calculations of the characteristic nanoparticle spacing. The particle size histogram (upper inset), showed that the standard deviation from the average size was less then 20%, consistent with the definition of a monomodal distribution used in chemical synthesis routes. 53 Such histograms allowed accurate determination of particle diameter.

V. RESULTS AND DISCUSSION
One of the key implications of the theory (Eq. (6)) is that when the magnitude of the thermal forces given by P Z are substantially larger then the dispersion forces (A/2π h 4 ) then the RT instability length scale should have no explicit dependence upon the film thickness. Indeed, as found by our thermal modeling calculations earlier, and depicted in Fig. 1(c), our experimental situation should be dominated by the P Z forces, and hence the observed length scale should be explicitly independent of the film thickness and be given by: To test this result, we performed experiments as a function of various Au film thickness by maintaining a similar magnitude of the P Z forces. This was achieved by first obtaining a laser energy density that resulted in identical maximum temperature of T * = 1750 K for the different thickness via thermal model calculations. This temperature was chosen for ease of experimental purposes. The modeling result to determine the laser energy density is shown in Fig. 2(b) (this result was, in turn, used previously in Fig. 1(b) to generate the P Z force values). In Fig. 2(c) the experiment (closed symbols) and theory predictions (dashed line) for the nanoparticle spacing versus film thickness are shown for similar P Z . Excellent agreement in trend and magnitude between theory prediction and experimental measurement is evident, strongly implying that the RT instability was responsible for the observed nanoparticles. In addition, the length scale was virtually independent of the film thickness, supporting the hypothesis of a classical RT-like instability.
In Fig. 2(c), the experiment (open symbols) and theory prediction (dotted line) for the nanoparticle diameter versus film thickness is also shown. If the initial volume of material is conserved during the laser irradiation process, the final diameter can be predicted from the characteristic length scales to be D = c(θ )(λ 2 RT h o ) 1/3 , 12 where the constant c(θ ) will be a function of the contact angle of the droplets. Since our thermal model calculations and the known temperature-dependent evaporation rate of Au predicted extremely small material loss due to evaporation effects (result provided in the supplement Fig. 2 50 ), we applied volume conservation analysis to the measured nanoparticle size and spacing. Using a hemispherical particle size, which gives a diameter of D = ( 3 2 λ 2 h o ) 1/3 , very good agreement in experiment and theory was found.
One of the well-known features of the classical RT instability is the appearance of hexagonal patterns during the early stages of deformation. In our experimental results, we are unable to capture these early stage features due to the extreme rapidity of the process. We estimated the time scales from Eq. (7) and found them to be of the order of ∼O(10 −10 − 10 −11 ) sec for the various laser energy and film thickness conditions explored here, as shown in Fig. 3(a). Since this time scale is substantially smaller then the laser pulse width of 9 ns, therefore, it was only possibly to observe the final ripened structures following destabilization of the film by the gas pressure driven RT instability process. From Fig. 3(a), it can also be noted that the RT instability time scale is much smaller then the spinodal dewetting time scale for all conditions investigated here, consistent with the much larger forces driving the RT instability effect. One of the interesting behaviors of the RT instability is the linear stability analysis prediction of a decrease in time scale with increasing film thickness. By assuming a negligible dispersion force, the RT instability time scale can be expressed as τ RT = 24γ πη , which is in stark contrast to spinodal dewetting in which the time scale varies as τ ∝ h 5 o . Equation (8) suggested that the RT instability length scale will be independent of the film thickness, and its confirmation was presented in Fig. 2(c). That figure also showed that the particle size could be changed for a fixed spacing. However, practical implementation of this pressure-driven instability to potential applications also requires the flexibility in tuning spacing of the nanoparticles. This can be achieved by noting that varying the magnitude of P Z can change the patterning length scales. From the thermal model result, Fig. 1(b), increasing laser energy density E increases the magnitude of the P Z force due to the increased T Z and this in turn should lead to a decrease in length scale through Eq. (8). This result is depicted in Fig. 3(b) for measurements done on Au films of two different thickness. Clearly, increasing the laser energy density leads to substantial decrease in the nanoparticle spacings. In addition, the decreasing length scale predicted by theory calculations (dashed line) is in very good agreement with the experimental measurements (symbols).

VI. CONCLUSION
In this work we show a new way to synthesize nanoparticle arrays on a substrate with independent control of size and spacing, which overcomes some limitations of spinodal dewetting length scales. The approach is based on applying the Rayleigh Taylor instability to the nanoscale, a first such demonstration, since this instability, thus far, has only been observed in macroscopic systems. Theoretical arguments show that when thin films are rapidly melted inside a bulk fluid, the ensuing gas pressure gradients can produce huge acceleration forces that can destabilize the film leading to the RT instability and predictable nanoscale sizes. Experimental studies performed for nanosecond pulsed laser melting of Au films in glycerol agreed with the theory. Nanoparticles with a highly monomodal size could be synthesized. Importantly, the spacings between nanoparticles was independent of the film thickness and depended primarily on the pressure gradients, which could be controlled by the laser energy. Since the theory underlying this nanoscale Rayleigh-Taylor instability appears independent of the type of thin film material and/or substrate, this technique could be widely applied to synthesis of nanoparticles from many different materials classes i.e. polymers, ceramics, or metals.