Newtonian Thin Films

Thin films provide a rich setup that is of interest from multiple points of view. From application side, these thin films and their instabilities are of relevance to a number of fields that rangefrom glass making to semiconductor and polymer applications, solar cells, to mention just a few fields of applications. From modeling side, thin films are demanding due to the presence of multiple spatial and temporal scales and due to the need to incorporate multi physics that is often of relevance in particular in the setups involving spreading and instabilities of thin films on substrates.

Recent group activities on the topic of Newtonian thin films have focused on modeling and simulations of metal films of nanoscale thickness exposed to laser radiation, in collaboration with the experimental group at U. Tennessee and Oak Ridge National Laboratory. Metal films are relevant to a number of technological fields with applications that include plasmonics, magnetic nanoparticles, control surface optical properties, catalysts for nanowire growth and many others. In many of these fields ordered arrays of nanoparticles are needed. While in liquid phase, the film instabilities lead to such arrays via self- or directed- assembly, and we are interested in the basic mechanisms that are relevant to the instabilities.

At NJIT, we have carried extensive simulations of thin metal films within the long-wave model that reduces the problem to solving a nonlinear 4th order evolution equation of diffusion type, as well as by directly solving Navier-Stokes equations using Volume-of-Fluid method. A number of new and interesting results has been obtained, including much better understanding of the connection between film instability and its geometry, of the influence of fluid/solid interaction forces, heat flow through the film and the substrate, to name just a few aspects.

Earlier works on thin films include a variety of configurations and setups including considerations of stochastic effects, flows down an incline, hanging and evaporating films, among others. Some of the considered configurations were also explored in research projects including undergraduate students in the Capstone Laboratory [link to the capstone page]. The relevant recent publications can be found below, and older works are listed at the publication list or here.

We acknowledge past and current support by NSF and Fulbright Foundation.

External Collaborators:

Rodolfo Cuerno, U. Carlos III (Email)
Javier Diez, UNCPBA (Website)
Francesc Font, McMaster U. (Website)
Jason Fowlkes, ORNL (Website)
Alejandro Gonzalez, UNCPBA (Email)
Esteban Moro, U. Carlos III (Email)
Philip Rack, UT (Website)

Recent Publications

I. Seric, S. Afkhami, and L. Kondic, Influence of thermal effects on stability of nanoscale films and filaments on thermally conductive substrates, Physics of Fluids, 30, 012109, (2018)

C.A. Hartnett, I. Seric, K. Mahady, L. Kondic, S. Afkhami, J.D. Fowlkes, and P.D. Rack, Exploiting the Marangoni Effect To Initiate Instabilities and Direct the Assembly of Liquid Metal Filaments, Langmuir, 33, 8123-8128, (2017)

Utilization of the Marangoni effect in a liquid metal is investigated, focusing on initiating instabilities to direct material assembly via the Rayleigh–Plateau instability. Thin (2 nm) copper (Cu) films are lithographically patterned onto thick (12 nm) nickel (Ni) strips to induce a surface energy gradient at the maximum wavelength of the filament instability predicted by Rayleigh–Plateau instability analysis. The pattern is irradiated with an 18 ns pulsed laser such that the pattern melts and the resultant Ni–Cu surface tension gradient induces Marangoni flows due to the difference in surface energies. The experimental results, supported by extensive direct numerical simulations, demonstrate that the Marangoni flow exceeds the capillary flow induced by the initial geometry, guiding instabilities such that final nanoparticle location is directed toward the regions of higher surface energy (Ni regions). Our work shows a route for manipulation, by means of the Marangoni effect, to direct the evolution of the surface instabilities and the resulting pattern formation.


F. Font, S. Afkhami, and L. Kondic, Substrate melting during laser heating of nanoscale metal films, International Journal of Heat and Mass Transfer, 113, 237-245, (2017)

Abstract We investigate heat transfer mechanisms relevant to metal films of nanoscale thickness deposited on a silicon (Si) substrate coated by a silicon oxide (SiO2) layer and exposed to laser irradiation. Such a setup is commonly used in the experiments exploring self and directed assembly of metal films that melt when irradiated by laser and then evolve on time scale measured in nanoseconds. We show that in a common experimental setting, not only the metal but also the SiO2 layer may melt. Our study of the effect of the laser parameters, including energy density and pulse duration, shows that melting of the substrate occurs on spatial and temporal scales that are of experimental relevance. Furthermore, we discuss how the thicknesses of metal and of the substrate itself influence the maximum depth and liquid lifetime of the melted SiO2 layer. In particular, we find that there is a minimum thickness of SiO2 layer for which this layer melts and furthermore, the melting occurs only for metal films of thickness in a specified range. In the experiments, substrate melting is of practical importance since it may significantly modify the evolution of the deposited nanoscale metal films or other geometries on nanoscale.

N. Dong and L. Kondic, Instability of nanometric fluid films on a thermally conductive substrate, Physical Review Fluids, 1, 063901, (2016)

We consider thin fluid films placed on thermally conductive substrates and exposed to time-dependent spatially uniform heat source. The evolution of the films is considered within the long-wave framework in the regime such that both fluid/substrate interaction, modeled via disjoining pressure, and Marangoni forces, are relevant. We analyze the problem by the means of linear stability analysis as well as by time-dependent nonlinear simulations. The main finding is that when self-consistent computation of the temperature field is performed, a complex interplay of different instability mechanisms results. This includes either monotonous or oscillatory dynamics of the free surface. This oscillatory behavior is absent if the film temperature is assumed to be slaved to the current value of the film thickness. The results are discussed within the context of liquid metal films, but are of relevance to dynamics of any thin film involving variable temperature of the free surface, such that the temperature and the film interface itself evolve on comparable time scales.

K. Mahady, S. Afkhami, and L. Kondic, A numerical approach for the direct computation of flows including fluid-solid interaction: Modeling contact angle, film rupture, and dewetting, Physics of Fluids, 28, 062002, (2016)

In this paper, we present a computationally efficient method for including fluid-solid interactions into direct numerical simulations of the Navier–Stokes equations. This method is found to be as powerful as our earlier formulation [K. Mahady et al., “A volume of fluid method for simulating fluid/fluid interfaces in contact with solid boundaries,” J. Comput. Phys. 294, 243 (2015)], while outperforming the earlier method in terms of computational efficiency. The performance and efficacy of the presented method are demonstrated by computing contact angles of droplets at equilibrium. Furthermore, we study the instability of films due to destabilizing fluid-solid interactions, and discuss the influence of contact angle and inertial effects on film breakup. In particular, direct simulation results show an increase in the final characteristic length scales when compared to the predictions of a linear stability analysis, suggesting significant influence of nonlinear effects. Our results also show that emerging length scales differ, depending on a number of physical dimensions considered.

K. Mahady, S. Afkhami, and L. Kondic, A volume of fluid method for simulating fluid/fluid interfaces in contact with solid boundaries, Journal of Computational Physics, 294, 243-257, (2015)

In this paper, we present a novel approach to model the fluid/solid interaction forces in a direct solver of the Navier-Stokes equations based on the volume of fluid interface tracking method. The key ingredient of the model is the explicit inclusion of the fluid/solid interaction forces into the governing equations. We show that the interaction forces lead to a partial wetting condition and in particular to a natural definition of the equilibrium contact angle. We present two numerical methods to discretize the interaction forces that enter the model; these two approaches differ in complexity and convergence. To validate the computational framework, we consider the application of these models to simulate two-dimensional drops at equilibrium, as well as drop spreading. We demonstrate that the model, by including the underlying physics, captures contact line dynamics for arbitrary contact angles. More generally, the approach permits novel means to study contact lines, as well as a diverse range of phenomena that previously could not be addressed in direct simulations.

K. Mahady, S. Afkhami, and L. Kondic, On the influence of initial geometry on the evolution of fluid filaments, Physics of Fluids, 27, 092104, (2015)

In this work, the influence of the initial geometry on the evolution of a fluid filament deposited on a substrate is studied, with a particular focus on the thin fluid strips of nano-scale thickness. Based on the analogy to the classical Rayleigh-Plateau (R-P) instability of a free-standing fluid jet, an estimate of the minimal distance between the final states (sessile droplets) can be obtained. However, this numerical study shows that while the prediction based on the R-P instability mechanism is highly accurate for an initial perturbation of a sinusoidal shape, it does not hold for a rectangular waveform perturbation. The numerical results are obtained by directly solving fully three-dimensional Navier-Stokes equations, based on a Volume of Fluid interface tracking method. The results show that (i) rectangular-wave perturbations can lead to the formation of patterns characterized by spatial scales that are much smaller than what is expected based on the R-P instability mechanism; (ii) the nonlinear stages of the evolution and end states are not simply related, with a given end state resulting from possibly very different types of evolution; and (iii) a variety of end state shapes may result from a simple initial geometry, including one- and two-dimensional arrays of droplets, a filament with side droplets, and a one-dimensional array of droplets with side filaments. Some features of the numerical results are related to the recent experimental study by Roberts et al. ["Directed assembly of one- and two-dimensional nanoparticle arrays from pulsed laser induced dewetting of square waveforms," ACS Appl. Mater. Interfaces 5, 4450 (2013)].

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S. Nesic, R. Cuerno, E. Moro, and L. Kondic, Fully nonlinear dynamics of stochastic thin-film dewetting, Physical Review E, 92, 061002, (2015)

The spontaneous formation of droplets via dewetting of a thin fluid film from a solid substrate allows materials nanostructuring. Often, it is crucial to be able to control the evolution, and to produce patterns characterized by regularly spaced droplets. While thermal fluctuations are expected to play a role in the dewetting process, their relevance has remained poorly understood, particularly during the nonlinear stages of evolution that involve droplet formation.Within a stochastic lubrication framework, we show that thermal noise substantially influences the process of droplets formation. Stochastic systems feature a smaller number of droplets with a larger variability in size and space distribution, when compared to their deterministic counterparts. Finally, we discuss the influence of stochasticity on droplet coarsening for asymptotically long times.

S. Afkhami and L. Kondic, On the dewetting of liquefied metal nanostructures, Journal of Engineering Mathematics, 94, 5-18, (2015)

Direct numerical simulations of liquefied metal nanostructures dewetting a substrate are carried out. Full three-dimensional Navier–Stokes equations are solved and a volume-of-fluid method is used for tracking and locating the interface. Substratewettability is varied to study the influence of the solid–liquid interaction. The effects of initial geometry on the retraction dynamics is numerically investigated. It is shown that the dewetting velocity increases with increases in the contact angle and that the retraction dynamics is governed by an elaborate interplay of initial geometry, inertial and capillary forces, and the dewetting phenomena. Numerical results are presented for the dewetting of nanoscale Cu and Au liquefied structures on a substrate.

C.A. Hartnett, K. Mahady, J.D. Fowlkes, S. Afkhami, L. Kondic, and P.D. Rack, Instability of Nano-and Microscale Liquid Metal Filaments: Transition from Single Droplet Collapse to Multidroplet Breakup, Langmuir, 31, 13609-13617, (2015)

We carry out experimental and numerical studies to investigate the collapse and breakup of finite size, nano- and microscale, liquid metal filaments supported on a substrate. We find the critical dimensions below which filaments do not break up but rather collapse to a single droplet. The transition from collapse to breakup can be described as a competition between two fluid dynamic phenomena: the capillary driven end retraction and the Rayleigh−Plateau type instability mechanism that drives the breakup. We focus on the unique spatial and temporal transition region between these two phenomena using patterned metallic thin film strips and pulsed-laser-induced dewetting. The experimental results are compared to an analytical model proposed by Driessen et al. and modified to include substrate interactions. In addition, we report the results of numerical simulations based on a volume-of-fluid method to provide additional insight and highlight the importance of liquid metal resolidification, which reduces inertial effects.

L. Kondic, N. Dong, Y. Wu, J.D. Fowlkes, and P.D. Rack, Instabilities of nanoscale patterned metal films, The European Physical Journal Special Topics, 224, 369-378, (2015)

We consider the evolution and related instabilities of thin metal films liquefied by laser pulses. The films are patterned by largescale perturbations and we discuss how these perturbations influence the dynamics. In the experiments, we find that the considered thin films dewet, leading to the formation of primary and secondary drops, with the locations of the primary ones coinciding with the original perturbations. Based on the results of the fully nonlinear time-dependent simulations, we discuss the details of the evolution leading to these patterns. Furthermore, in both experiments and simulations, we discuss the influence of the shape of the initial perturbations on the properties of the final patterns.

J.D. Fowlkes, N.A. Roberts, Y. Wu, J.A. Diez, A.G. Gonzalez, C. Hartnett, K. Mahady, S. Afkhami, L. Kondic, and P.D. Rack, Hierarchical nanoparticle ensembles synthesized by liquid phase directed self-assembly, Nano letters, 14, 774-782, (2014)

A liquid metal filament supported on a dielectric substrate was directed to fragment into an ordered, mesoscale particle ensemble. Imposing an undulated surface perturbation on the filament forced the development of a single unstable mode from the otherwise disperse, multimodal Rayleigh− Plateau instability. The imposed mode paved the way for a hierarchical spatial fragmentation of the filament into particles, previously seen only at much larger scales. Ultimately, nanoparticle radius control is demonstrated using a micrometer scale switch.

N.A. Roberts, J.D. Fowlkes, K. Mahady, S. Afkhami, L. Kondic, and P.D. Rack, Directed assembly of one-and two-dimensional nanoparticle arrays from pulsed laser induced dewetting of square waveforms, ACS applied materials & interfaces, 5, 4450-4456, (2013)

The directed assembly of arrayed nanoparticles is demonstrated by dictating the flow of a liquid phase filament on the nanosecond time scale. Results for the assembly of Ni nanoparticles on SiO2 are presented. Previously, we have implemented a sinusoidal perturbation on the edge of a solid phase Ni, thin film strip to tailor nanoparticle assembly. Here, a nonlinear square waveform is explored. This waveform made it possible to expand the range of nanoparticle spacing−radius combinations attainable, which is otherwise limited by the underlying Rayleigh−Plateau type of instability. Simulations of full Navier−Stokes equations based on volume of fluid method were implemented to gain further insight regarding the nature of instability mechanism leading to particle formation in experiments.

S. Afkhami and L. Kondic, Numerical simulation of ejected molten metal nanoparticles liquified by laser irradiation: Interplay of geometry and dewetting, Physical Review Letters, 111, 034501, (2013)

Metallic nanoparticles, liquified by fast laser irradiation, go through a rapid change of shape attempting to minimize their surface energy. The resulting nanodrops may be ejected from the substrate when the mechanisms leading to dewetting are sufficiently strong, as in the experiments involving gold nanoparticles [Habenicht et al., Science 309, 2043 (2005)]. We use a direct continuum-level approach to accurately model the process of liquid nanodrop formation and the subsequent ejection from the substrate. Our computations show a significant role of inertial effects and an elaborate interplay of initial geometry and wetting properties: e.g., we can control the direction of ejection by prescribing appropriate initial shape and/or wetting properties. The basic insight regarding ejection itself can be reached by considering a simple effective model based on an energy balance. We validate our computations by comparing directly with the experiments specified above involving the length scales measured in hundreds of nanometers and with molecular dynamics simulations on much shorter scales measured in tens of atomic diameters, as by M. Fuentes-Cabrera et al. [Phys. Rev. E 83, 041603 (2011)]. The quantitative agreement, in addition to illustrating how to control particle ejection, shows utility of continuum-based simulation in describing dynamics on nanoscale quantitatively, even in a complex setting as considered here.

K. Mahady, S. Afkhami, J. Diez, and L. Kondic, Comparison of Navier-Stokes simulations with long-wave theory: Study of wetting and dewetting, Physics of Fluids, 25, 112103, (2013)

The classical long-wave theory (also known as lubrication approximation) applied to fluid spreading or retracting on a solid substrate is derived under a set of assumptions, typically including small slopes and negligible inertial effects. In this work, we compare the results obtained by using the long-wave model and by simulating directly the full two-phase Navier-Stokes equations employing a volume-of-fluid method. In order to isolate the influence of the small slope assumption inherent in the longwave theory, we present a quantitative comparison between the two methods in the regime where inertial effects and the influence of gas phase are negligible. The flow geometries that we consider include wetting and dewetting drops within a broad range of equilibrium contact angles in planar and axisymmetric geometries, as well as liquid rings. For perfectly wetting spreading drops we find good quantitative agreement between the models, with both of them following rather closely Tanner’s law. For partially wetting drops, while in general we find good agreement between the two models for small equilibrium contact angles, we also uncover differences which are particularly evident in the initial stages of evolution, for retracting drops, and when additional azimuthal curvature is considered. The contracting rings are also found to evolve differently for the twomodels, with themain difference being that the evolution occurs on the faster time scale when the long-wave model is considered, although the ring shapes are very similar between the two models.

A.G. González, J.A. Diez, Y. Wu, J.D. Fowlkes, P.D. Rack, and L. Kondic, Instability of Liquid Cu Films on a SiO2 Substrate, Langmuir, 29, 9378-9387, (2013)

We study the instability of nanometric Cu thin films on SiO2 substrates. The metal is melted by means of laser pulses for some tens of nanoseconds, and during the liquid lifetime, the free surface destabilizes, leading to the formation of holes at first and then in later stages of the instability to metal drops on the substrate. By analyzing the Fourier transforms of the SEM (scanning electron microscope) images obtained at different stages of the metal film evolution, we determine the emerging length scales at relevant stages of the instability development. The results are then discussed within the framework of a long-wave model. We find that the results may differ whether early or final stages of the instability are considered. On the basis of the interpretation of the experimental results, we discuss the influence of the parameters describing the interaction of the liquid metal with the solid substrate. By considering both the dependence of dominant length scales on the film thickness and the measured contact angle, we isolate a model which predicts well the trends found in the experimental data.

A.G. González, J.A. Diez, and L. Kondic, Stability of a liquid ring on a substrate, Journal of Fluid Mechanics, 718, 246-279, (2013)

We study the stability of a viscous incompressible fluid ring on a partially wetting substrate within the framework of long-wave theory. We discuss the conditions under which a static equilibrium of the ring is possible in the presence of contact angle hysteresis. A linear stability analysis (LSA) of this equilibrium solution is carried out by using a slip model to account for the contact line divergence. The LSA provides specific predictions regarding the evolution of unstable modes. In order to describe the evolution of the ring for longer times, a quasi-static approximation is implemented. This approach assumes a quasi-static evolution and takes into account the concomitant variation of the instantaneous growth rates of the modes responsible for either collapse of the ring into a single central drop or breakup into a number of droplets along the ring periphery. We compare the results of the LSA and the quasi-static model approach with those obtained from nonlinear numerical simulations using a complementary disjoining pressure model. We find remarkably good agreement between the predictions of the two models regarding the expected number of drops forming during the breakup process.

T.-S. Lin, L. Kondic, and A. Filippov, Thin films flowing down inverted substrates: Three-dimensional flow, Physics of Fluids, 24, 022105, (2012)

We study contact line induced instabilities for a thin film of fluid under destabilizing gravitational force in three-dimensional setting. In the previous work [T.-S. Lin and L. Kondic, Phys. Fluids 22, 052105 (2010)], we considered two-dimensional flow, finding formation of surface waves whose properties within the implemented longwave model depend on a single parameter, D = (3Ca)1/3 cot α , where Ca is the capillary number and α is the inclination angle. In the present work we consider fully 3D setting and discuss the influence of the additional dimension on stability properties of the flow. In particular, we concentrate on the coupling between the surface instabilities and the transverse (fingering) instabilities of the film front. We furthermore consider these instabilities in the setting where fluid viscosity varies in the transverse direction. It is found that the flow pattern strongly depends on the inclination angle and the viscosity gradient.

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T.D. Nguyen, M. Fuentes-Cabrera, J.D. Fowlkes, J.A. Diez, A.G. González, L. Kondic, and P.D. Rack, Competition between Collapse and Breakup in Nanometer-Sized Thin Rings Using Molecular Dynamics and Continuum Modeling, Langmuir, 28, 13960-13967, (2012)

We consider nanometer-sized fluid annuli (rings) deposited on a solid substrate and ask whether these rings break up into droplets due to the instability of Rayleigh-Plateau-type modified by the presence of the substrate, or collapse to a central drop due to the presence of azimuthal curvature. The analysis is carried out by a combination of atomistic molecular dynamics simulations and a continuum model based on a long-wave limit of Navier–Stokes equations. We find consistent results between the two approaches, and demonstrate characteristic dimension regimes which dictate the assembly dynamics.

J.D. Fowlkes, L. Kondic, J.A. Diez, A.G. Gonzalez, Y. Wu, N.A. Roberts, C.E. McCold, and P.D. Rack, Parallel assembly of particles and wires on substrates by dictating instability evolution in liquid metal films, Nanoscale, 4, 7376-7382, (2012)

Liquid metal wires supported on substrates destabilize into droplets. The destabilization exhibits many characteristics of the Rayleigh-Plateau model of fluid jet breakup in vacuum. In either case{,} breakup is driven by unstable{,} varicose surface oscillations with wavelengths greater than the critical one ([small lambda]c). Here{,} by controlling the nanosecond liquid lifetime as well as stability of a rivulet as a function of its length by lithography{,} we demonstrate the ability to dictate the parallel assembly of wires and particles with precise placement.

J.A. Diez, A.G. González, and L. Kondic, Instability of a transverse liquid rivulet on an inclined plane, Physics of Fluids, 24, 032104, (2012)

This work concentrates on the stability of a viscous liquid rivulet positioned across an inclined plane under partial wetting conditions. The study is performed within the framework of lubrication approximation by employing a slip model. Both normal and parallel components of gravity are considered.We find the stability regions for given area of the cross section of the rivulet, A, plane inclination angle, α, and static contact angle, θ0, characterizing the wettability of the substrate. For αs smaller than some critical angle, α*, a static solution exists. This solution is characterized by rear/front contact angles given by θ0 ± δ. The linear stability analysis of this solution is performed using an efficient pseudo-spectral Chebyshev method.We analyze the effects of A, θ0, and α on the predictions of the model, such as the dominant wavelength, the maximum growth rate, and the behavior of the most unstable perturbation mode. To verify them, we also carry out experiments with silicone oils spreading on a coated glass substrate for several different fluid volumes and inclination angles. We find very good agreement between the wavelength of maximum growth rate given by the theory and the average distance between the drops after rivulet breakup. An analysis of finite size effects shows that the inclusion of normal gravity effects leads to a better agreement between theoretical and experimental results.