Instability Mechanism for Micrarray Protrusions in Nanoscale Films

Theoretical studies

Experiments by several groups during the past decade have demonstrated that a liquid nanofilm subject to a large transverse thermal gradient can undergo spontaneous formation of periodic microarrays resembling a forest of protrusions. The arrays self-assemble with lamellar, square or hexagonal symmetry although the mechanism for pattern generation remains unknown. The formation of these arrays provides an interesting new approach to 3D non-contact lithography for photonic and optoelectronic application. In particular, the molten structures can be solidified in-situ thereby resulting in specular surfaces suitable for the most demanding optical applications. There are two prevailing models in the literature to help explain this behavior. The first model suggests that coherent reflections of low frequency acoustic phonons within the film cause periodic modulation of the radiation pressure which thereby enhances the growth of protrusions. This model represents an acoustic analog of the Casimir effect in ultraconfined systems. The second model suggests that the protrusions are caused by an electrostatic attraction between surface charges embedded along the polymer interface and surface induced charge on a nearby substrate. The first proposal relies on the dynamics of glassy behavior in entangled polymer films - the second proposal on the poor electrical conductivity of polymer films.

We have recently developed a different model, applicable to any liquefiable nanofilm, based on a hydrodynamic instability caused by interfacial thermocapillary forces. These forces, which normally play a minor role in macroscopic systems, assume a critical role in nanoscale films subject to large thermal gradients. A perturbation analysis provides an expression for the wavelength and time scale of the fastest growth mode, which can be directly compared to experiment. The Lyapunov functional developed for this system helps differentiate between states of different symmetry. Numerical simulations of the evolution equation for the moving boundary (highly non-linear 4th order PDE) are being used to elicit the response of the system to larger amplitude disturbances and initial film profiles with non-uniform shape.
 

Experimental Studies

Researchers during the past decade have been exploring the use of large transverse thermal gradients in liquid nanofilms for directed growth of nanopillar microarrays. When the free surface of a molten polymer nanofilm is exposed to a large transverse thermal gradient, periodic protrusions resulting from an instability are observed to grow toward the cooler target until contact is achieved. There has been significant excitement over these findings since application of a distribution of thermal gradients can be used to generate arbitrary, three dimensional nanostructures directly from the melt. Once solidified, these structures exhibit ultra smooth interfaces suitable for the most demanding optical applications. For controlled growth, it is necessary to identify the predominant growth mechanism, to determine the variables governing the growth rate, and to minimize the instability wavelength for enhancing pattern density and spatial resolution.

We are currently conducting optical microscopy and white light interferometric measurements of the characteristic lateral spacing and growth rate for microarray protrusions in polymer nanofilms. The resulting images are dynamically characterized by Fourier analysis, pair correlation functions and Voronoi tessellation as a function of time. The early time studies are challenging to implement since the initial fluctuations in film thickness are only of the order of a few tens of nanometers. The late time studies indicate a shift in symmetry and wavelength caused by contact with the cooler substrate. Tests are underway to validate which of the models proposed in the literature provides the dominant mechanism for pillar formation.

We are also conducting studies with cooled prepatterned templates to selectively shape nanofilms for constructing optical lens microarrays and thin film optical waveguides. These studies reveal ways in which to use the competition in wavelengths between the hydrodynamic instability and the external patterned template to produce 3D patterns that are either in-phase or out-of-phase with the templated structures. Our hope is to use a combination of numerical modeling and experimental efforts to design waveguide and optical resonator shapes not currently accessible to conventional photolithographic techniques.
 

Phase Transitions in Freely Suspended Nanoscale Films

Freely suspended bilayer films consisting of mobile, charged self-assembling "walls" of surfactant monomers, which can undergo exchange with micelles and polymer molecules in the bulk, are ubiquitous in nature and form the essential ingredients of soap films, tear films in the eye and cell membranes. At wall separation distances in the nanometer range, these films become subject to an oscillatory disjoining pressure which causes thinning transitions between metastable states called layering transitions. Studies of 2- or 3- component liquid films containing micelle-polyelectrolyte or micelle-polymer complexes reveal that the discrete jumps in film thickness correlate closely with the characteristic size of micelles or the micelle-polymer coil size, respectively. Measurements by other groups by a thin film pressure balance or laser light scattering have confirmed that discrete layers of fluid can be successively expelled due to an interaction potential stemming from van der Waals, electrostatic and hydration forces.

Our investigations of the dynamics of freely suspended, bilayer nanoscale films containing micelle-polymer complexes (whose characteristic size is larger than the confining dimension) have revealed two novel instabilities. Video microscopy of film thinning toward the final metastable state has revealed spontaneous nucleation of a microdomain phase that rapidly permeates the surrounding thicker film. The rapidly expanding perimeter of these microdomains, however, is not circular or elliptical as common in similar systems without complexation, but develops a highly ramified front whose fractal dimension correlates strongly with the liquid viscosity. Despite that the surfaces of the freely suspended film are highly mobile and not confined between rigid walls, the lateral growth of this distinct phase shows striking resemblance to the Saffman-Taylor instability in macroscale systems. For sufficiently high polymer molecular weight, the permeating phase undergoes a secondary instability that generates a densely packed array of flattened nanodroplets whose packing density within the fractal perimeter exhibits 4-fold symmetry. This reduced symmetry stands in sharp contrast to the usual hexagonal packing structure observed in conventional simple fluid or colloidal systems. We are investigating these and similar systems in order to understand how nanoconfinement between deformable or soft interfaces can trigger phase transitions whose moving boundaries undergo rapid shape transformations toward dynamic fractal structures.
 

Tunable Optofluidic Components Based on Modulation of Interfacial Flow

An ever growing number of studies involving bacterial, cell or genomic assays rely on miniaturized diagnostic platforms known as labs-on-a-chip. These devices are being used both for fundamental research and commerical applications. Microfluidic chips, which typically require much smaller sample sizes than conventional diagnostic systems, allow for precise control over the spatial and temporal distribution of flow speeds as well as the concentration of nutrients, catalysts, competitor colonies, etc. The majority of such devices use pressure or electric field gradients to generate internal flow within enclosed microchannels. By contrast, our laboratory is exploring a number of alternative driving mechanisms for generating free surface flow in micro- and nanoscale films and droplets along flat and curved surfaces. Surface directed or "open architecture" flows can be achieved by activated gradients in temperature, concentration, electric, magnetic or acoustic fields or by selective actuation of adjacent boundary motion. In this way, surface stresses generated at air-liquid, liquid-liquid or liquid/solid interfaces can be used to tune the shape and speed of stationary or moving liquid elements. The capability to tailor the shape of micro- or nanoscale liquid structures introduces a wealth of applications for biofluidic and optofluidic applications including the design of various nanowell channels as well as tunable filters, gratings, waveguides and photonic structures.
 

Slip Boundary Condition for Liquid-on-Solid Flow

The celebrated no-slip condition used to calculate velocity and stress fields for all hydrodynamic systems dictates that a liquid element adjacent to a solid surface must equal the velocity of that surface. This boundary condition has proven remarkably successful in reproducing most macroscopic flows. There exist, however, notable examples for which this condition leads to a divergence in the viscous shear stress. Examples include the dynamics of moving triple lines (i.e. line separating gas/liquid/solid or liquid/liquid/solid interfaces) and the flow dynamics of macromolecular sytems consisting of long molecular chains.

It is now well accepted that suitably constructed boundary conditions which allow fluid elements to slip past the adjacent solid surface can regularize such flows and reproduce realistic behavior. Unfortunately, these slip models are phenomenological in origin and provide no universal understanding of the nature of momentum transport at liquid/solid interfaces. Using molecular dynamics simulations of liquids in planar shear, we have shown there exists a general non-linear function relating the slip length to the local shear rate along smooth or roughened liquid-solid interfaces. This lengthscale is governed by the in-plane structure factor and 2D diffusion coefficient, the density of contact points and the excluded volume in the vicinity of the solid boundary. We are currently investigating observed deviations between the predictions of molecular dynamics simulations and continuum (hydrodynamic) studies of nanoscale liquid films in planar shear along energetically patterned surfaces to establish how the local shear rate, liquid structure factor, wall roughness and variations in wall surface energy affect the degree of slip in non-inertial flows. These deviations offer insight useful to the development of multiscale models.
 

Evolution of Digitated Structures in Marangoni Driven Flows

Theoretical Studies

The spreading of surface active molecules, such as many organics like surfactants, proteins, dye molecules and liquid crystals, on a very thin liquid film of higher surface tension is known to produce a dynamic instability. The spreading front undergoes repeated branching and tip-splitting forming arterial patterns characterized by a fractal dimension similar to other processes governed by Laplacian growth. Calculations based on linear stability and transient growth analysis suggest that the coupling of Marangoni and capillary stresses causes dramatic variations in film thickness and surfactant concentration which rapidly destabilize the advancing front. The coupled evolution equations for the film thickness and surface concentration (highly nonlinear 4th order PDEs) exhibit rich behavior including self-similarity, algebraic growth, non-modal transient phenomena and front sharpening. We are conducting non-modal stability analysis to identify the conditions leading to instability in microscale films. The analogous problem of surfactant monolayers spreading on a thin liquid film is of relevance to the study of respiratory diseases known to be alleviated by exogenous delivery of lung surfactant. While many studies have focused on the equilibrium properties of the formulations used in clinical applications, our laboratory focuses instead on the dynamics and transport along to gas/liquid interface which hinders uniform surface coverage. A mapping to other Laplacian growth driven systems suggests that the fractal behavior of this system arises from a nonlinear process.

Experimental Studies

We use techniques such as refracted Moire topography to reconstruct the spatial and time dependent waveforms associated with the spreading dynamics of a model lipid monolayer. This system closely mimics the behavior of an insoluble surfactant driven to spread on a thin viscous layer under the action of Marangoni stresses induced by variations in surfactant concentration. The film thickness profiles exhibit a strong surface depression ahead of the surfactant source capped by an elevated rim at the surfactant leading edge. The surface slope and shape as well as the propagation speed of the advancing rim are directly compared with numerical solutions of a lubrication model based on Marangoni driven spreading of a surfactant monolayer. Comparison between the theoretical and measured profles reveals the importance of the initial shear stress in determining the evolution in the film thickness and surfactant distribution. This initial stress appears to thin the underlying liquid support so drastically that the surfactant droplet behaves as a finite and not an infinite source even though there is always present an excess of surfactant at the origin.

Hot and Cold Chips for Microfluidic Applications

We have developed a microfluidic chip which combines thermocapillary stresses with selective substrate patterning to guide and tune the flow of liquid samples along the surface of a substrate. By controlling the voltage applied to embedded thin metal film heater arrays, we can selectively apply on demand specific temperature distributions with high spatial resolution. The local thermal gradients alter the surface tension of the adjacent liquid sample thereby inducing thermocapillary stresses which propel the liquid away from warm regions and toward cooler regions of the substrate. The liquid surface temperature can also be tuned by varying the intensity of radiative heating of the air-liquid interface using an overhead laser and programmable mirror array. With either method, the liquid flow speed and direction of liquid trajectories can be electronically tuned.

We have used this device as a miniature automated platform for such functions as a droplet router for polar and organic liquids, droplet trapping and release, droplet scission, controlled sample mixing, and monitoring of chemical reactions. What makes this technique particularly attractive for microfluidic applications is the variety of tasks possible solely by control of the liquid surface temperature. These experimental studies are also complemented by an extensive theoretical program which includes hydrodynamic modeling as well as molecular dynamics simulations.
 

Microdetection and Analysis by Integrated Evanescent Wave Sensing

Our laboratory has previously demonstrated two methods for droplet detection and sensing for microfluidic devices which makes use of the coplanar microelectrode arrays used for thermocapillary actuation. The first method monitors the thermal rise time of embedded microheaters, from which can be extracted droplet location, volume or composition due to changes in thermal conductivity induced by the presence of overlying liquid film. The second method records the capacitance change induced by an overlying droplet. Rapid response for electrode widths which are comparable to the liquid film thickness allows for accurate detection of droplet position, volume, composition and evaporative loss even for nanoliter liquid samples. These sensing techniques, which probe samples by thermal or electric fields, however, are not suitable for all applications. In this respect, integrated optical and spectroscopic probes, such as evanescent sensing, offer significant advantages over these diagnostic techniques.

In a recent set of experiments, we have demonstrated a non-intrusive optical method for microfluidic detection and analysis based on evanescent wave sensing. The device consists of a planar thin film waveguide integrated with a microfluidic chip based on thermocapillary flow. Microliter droplets are electronically transported and positioned over the waveguide surface by actuation of a glass-embedded microelectrode array. The attenuated intensity of propagating modes is used to detect droplet location, to monitor dye concentration in aqueous solutions, and to measure the increase in chemical reaction rates as a function of increasing substrate temperature for a chromogenic biochemical assay. This study illustrates just a few of the capabilities possible by direct integration of optical sensing with surface directed fluidic devices. Our design also offers high sensitivity with few additional fabrication steps and is especially well suited to any fluidic device based on droplet manipulation by modulation of surface tension. We are currently investigating alternate substrate structures which offer the possibility of increasing the evanescent field intensity by one to two orders of magnitude useful for increasing the sensitivity and specificity of this device.
 

Fabrication of Amorphous Si Thin Film Transistors by Offset and Letterpress Printing

Large area electronics such as light emitting displays require far less stringent resolution limits than conventional photolithography. The demand for applications with lower resolution has triggered the development of patterning and fabrication methods which are large area, high throughput and low cost. We have demonstrated that contact printing methods like offset and letterpress methods when scaled down to the microscale can be used to produce wet and dry etch resist masks of arbitrary shape and thickness on flat or spherical surfaces. These structures can be fabricated cheaply and rapidly yet accommodate wide area formats containing disparate sizes and shapes. Recently, our lab has demonstrated that polymer etch masks printed with a microscale letterpress stamp can be used to fabricate amorphous thin film transistors with I-V curves comparable to those fabricated by conventional photolithography. This study, which focuses on flexible electronics, also includes a significant modeling effort based on energy minimization and fluid dynamical studies designed to parametrize and optimize the flow and surface conditions required for various stages of the printing process.
 

Evolution of Digitated Structures in Marangoni Driven Flows

Theoretical Studies

The spreading of a surfactant coated droplet on a thin liquid film of higher surface tension is known to produce an unusual fingering instability near the initial deposition edge. The spreading front undergoes repeated branching and tip-splitting forming arterial or dendritic patterns characterized by a fractal dimension similar to processes governed by Laplacian growth. Calculations based on linear stability and transient growth analysis suggest that the coupling of Marangoni and capillary stresses causes dramatic variations in film thickness and surfactant concentration which rapidly destabilize the advancing front. Interestingly, this system of coupled PDEs is unique in that the corresponding disturbance equations harbor the potential for large transient growth despite that the flow is well characterized as a lubrication type flow for which the Reynolds number plays no role. We are currently conducting experimental and theoretical studies to identify the characteristics of the base state profiles and conditions leading to instability to identify the source of unstable flow in sufficiently thin films. This problem of a surfactant monolayer spreading on a thin liquid film is of relevance to the study of certain respiratory diseases in newborn infants, which are alleviated by the exogenous delivery of lung surfactant. While many studies have focused on the equilibrium properties of the formulations used in clinical applications, our laboratory focuses instead on the dynamics of the spreading process in order to prevent non-uniform coverage of the surfactant concentration.

Experimental Studies

Most recently, we have used refracted Moire topography to reconstruct the spatial and time dependent waveforms associated with the spreading dynamics of a model lipid monolayer. This system closely mimics the behavior of an insoluble surfactant driven to spread on a thin viscous layer under the action of Marangoni stresses induced by variations in surfactant concentration. The film thickness profiles exhibit a strong surface depression ahead of the surfactant source capped by an elevated rim at the surfactant leading edge. The surface slope and shape as well as the propagation speed of the advancing rim are directly compared with numerical solutions of a lubrication model based on Marangoni driven spreading of a surfactant monolayer. Comparison between the theoretical and measured profles reveals the importance of the initial shear stress in determining the evolution in the film thickness and surfactant distribution. This initial stress appears to thin the underlying liquid support so drastically that the surfactant droplet behaves as a finite and not an infinite source even though there is always present an excess of surfactant at the origin.
 

Moving Front Instabilities in Microscale Films

When a thin liquid film is forced to coat a solid substrate by application of a body force like centrifugal or gravitational forces or surface force like thermocapillary or Maxwell stresses, the advancing front can develop a narrow and highly curved capillary ridge at the leading edge. This ridge is highly unstable and typically breaks into numerous parallel rivulets which act as channels which selectively direct the flow. Examples of this instability include liquid paint streaming down a vertical wall, the spin coating of a liquid film on a rotating substrate, the thermocapillary migration of a liquid film on a differentially heated substrate, or the forced spreading of a viscous film by an overhead gas stream. We have investigated these types of fingering instabilities via modal and transient growth analysis. For the thermocapillary driven system, the optimal perturbations which give rise to the fingering behavior rapidly asymptote to the most unstable modes determined from linear stability of the base state traveling wave solution. This holds true even if the base state equation includes van der Waals interactions. Our work provides the basis for understanding the excellent agreement between experiment and linear stability theory despite the non-normal character of the relevant disturbance matrix. We are extending our studies to other coating flows and to surfaces of mixed wettability obtained by chemical micropatterning.