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Prediction of Self-Assembled Dewetted Nanostructures for Photonics Applications via a Continuum Mechanics Framework

When a liquid film lies on a non-wettable substrate, the configuration is unstable and the film then retracts from a solid substrate to form droplets. This phenomenon, known as dewetting, commonly leads to undesirable morphological changes. Nevertheless, recent works have demonstrated the possibility to harness dewetting by employing templated substrates with a degree of precision on par with advanced lithographic processes for high-performance nanophotonic applications. Since resonant behavior is highly sensitive to geometrical changes, predicting quantitatively dewetting dynamics is of high interest. In this work, we develop a continuum model that predicts the evolution of a thin film on a patterned substrate, from the initial reflow to the nucleation and growth of holes. We provide an operative framework based on macroscopic measurements to model the intermolecular interactions at the origin of the dewetting process, involving length scales that span from sub-nanometer to micron range. A comparison of experimental and simulated results shows that the model can accurately predict the final distributions, thereby offering novel predictive tools to tailor the optical response of dewetted nanostructures.


Introduction
Flows of thin films over substrates are of central interest owing to their ubiquity in natural as well as industrial environments. Depending on the interaction between film, substrate and surrounding environment, thin films may dewet and re-arrange into droplets. 1 The miniaturization of modern devices involving ultra-thin layers has brought a novel focus on the question of dewetting. This ubiquitous phenomenon threatens the integrity of thin films, typically yielding semi-ordered tessellation patterns. This has often been an undesirable phenomenon, jeopardizing the film morphology with little practical use. Nevertheless, a few research endeavors have investigated the potential of dewetting as an efficient self-assembly process. Several approaches have been proposed based on selective wetting 2,3 or spatially-patterned electric fields. 4,5 These investigations have demonstrated interesting possibilities but remained limited in terms of materials, scalability, geometry or resolution. By combining engineered substrates with functional materials, thermal dewetting has emerged as a viable alternative, as illustrated in FIG. 1. 6,7,8,9 In opposition to mono-or polycrystalline thin films that dewet according to surface diffusion mechanisms, amorphous thin films dewet following bulky viscous flow mechanisms. While surface diffusion over templates has been the object of several investigations, 6,10,11 the flow of amorphous films over pre-patterned substrates have thus far only made the object of linear stability analyses, which fail to predict the complete dewetting dynamics. 12,13 This is surprising, as isotropic material properties associated with amorphous materials allow for improved control of the rearrangement mechanism and higher complexity in the final microstructures. Resorting to highindex dielectric glasses bears relevance in nanophotonics, 14 where the control of the nanoresonator geometry allows for wide engineering of scattering and resonant properties. To increase the method's accuracy and further extend its realm of application, the prediction in the most precise manner of the final microstructure based on the exact substrate shape and film-substrate interactions is of paramount importance, which motivates the present study.
Herein, we propose a continuum model predicting the evolution of a templated film evolving over pre-patterned substrates, based on the modeling of intermolecular interactions occurring on various substrates, for contact angles less than 90°. By comparing experimental and simulated thickness profiles over various patterns, we show that the proposed model is suitable for the accurate prediction of the final morphology, and over several length scales (from nm to m scale). We thereby offer improved control of the dewetting patterns, allowing for the realization of precise architectures relevant to nanophotonics.

Dewetting on flat substrates: validation
An accurate dynamic description of dewetting constitutes a particularly challenging problem. The theoretical framework for the description of fluid flow is based on approaches from continuum mechanics. We initially consider a flat horizontal substrate and introduce a coordinate system x, y, z , where the z direction, along which the film thickness is measured, coincides with the vertical one, as sketched in FIG. 2(a). The integration of the Navier-Stokes equations along the z direction, under the classical assumptions of the long-wavelength approximation, 15,16 leads to an evolution equation for the thin film thickness h in the x, y directions, so-called lubrication or thin film equation: 16 ∂h ∂t Where κ is the curvature, ∇ operates in the x, y plane, μ is the fluid dynamic viscosity, γ is the surface tension coefficient between the fluid and the air, and is the so-called disjoining pressure. In the classical framework of the long-wavelength approximation, the curvature is implemented with its linearized version, i.e. ϰ ∇ h, that holds for small slopes of the film. As discussed further on, we resort here to the complete curvature to properly account for arbitrary height profiles. Note that, since the thickness is a single-valued function of the position, contact angles greater than 90° cannot be described by this model.
The disjoining pressure term is assumed to stem from a classical Lennard-Jones type potential: 17,18,19 where A=A123 is the so-called Hamaker constant of the system substrate (1) -film (2) -air (3) and B is the Born coefficient, employed to model respectively the molecular long-range attractive and short-range repulsive forces. The combination of a repulsive and an attractive term defines a minimum of the potential for an equilibrium "precursor" film thicknessh 48πB/A / , obtained by imposing φ′ h 0 (FIG. 2 (b)).
The force derived from the Lennard-Jones potential stems from an imbalance in the interactions between the various constituent molecules. This imbalance is classically embedded in the Hamaker constant A123, which establishes the influence of constituent materials in long-range interactions, in the presence of multiple bodies according to Lifschitz theory. 20,21,22,23 The previously introduced Lennard-Jones potential is linked to the disjoining pressure through: A positive Hamaker constant induces destabilizing pressure gradients for films larger than the equilibrium thickness h . When a region of the film reaches the precursor film thickness h , the local equilibrium at the interface between the precursor film and the thicker regions defines an apparent contact angle (FIG. 2(a)) given by: 17,24 Considering solely angles between 0° to 90°, relation (4) provides a bijective relationship between the contact angle and the precursor film h .
To validate this approach, we now proceed to simulate the evolution of a thin film and evaluate the resulting contact angle. The thin film equation is implemented with the full expression of the interface curvature: 25,26,27,28 ϰ ∇ ⃗ . n (5) n 1 In the rare previous works that developed a model based on the Lennard-Jones potential, 16,17,19 the contact angle was inferred from accurate Hamaker constant and Born coefficient data, with a good agreement between theoretically derived and experimentally measured contact angles. This approach assumes the prior knowledge of the Born coefficient, significantly harder to quantify than the Hamaker constant, and thus constitutes a significant limitation for broader use of such modeling scheme. Moreover, this requires the knowledge of the precursor film thickness, a challenging quantity to experimentally measure (typically in the Angström range). In this work, we propose to use the Born coefficient, and thus the equilibrium thickness given by (4) is sufficient to effectively model the thin film dynamics.

Templated Dewetting
We now turn to the evolution of a film with fixed contact angle over a pre-patterned (or "templated") substrate, of height profile h measured along the z direction starting from the horizontal reference previously introduced (see FIG. 3(a)). In this work, we consider two different types of templates, (i) lines or two-dimensional templates, characterized by triangular trenches of base W and period P along the direction, invariant along the direction orthogonal to the periodicity one ( direction), and (ii) pyramids or three-dimensional templates, characterized by pyramidal trenches with periodicity P along both and directions. We resort to amorphous We adapt the numerical scheme with the latter thickness definition and ensure that it yields the desired contact angle over substrates with arbitrary inclination (See FIG. 4).
To validate the proposed scheme, numerical simulations using experimental atomic scanning microscope profiles of nanoimprinted substrates are calculated. We take as initial

Application in Photonics: Phase Control
We now turn to exploit this in-depth understanding and control of template dewetting to realize advanced optical metasurfaces. Ordered high-index nanoparticles bear particular importance for metasurfaces or meta-gratings, which enable to tailor the phase, amplitude and polarization of light over reduced thicknesses, in stark contrast with current bulky optical components. 30 together define a relationship between evaporated film thickness and line width. Since the system is highly sensitive to slight geometrical changes as low as 10 nm, experimental and simulated reflection spectra appear relatively well in line. We now turn to the evolution in phase for a range of periods ranging from 1200 nm to 1500 nm. Interestingly, interference between the various individual particle Mie mode cumulate to yield a cumulative phase shift over the complete 2π phase range at λ = 784 nm. The phase shift is gradual, spanning from 1200 nm to 1500 nm, hence allowing for phase control with experimentally attainable accuracies. One can also note a

Sample Fabrication
Chalcogenide thin films (Se, As2Se3) are first thermally evaporated (UNIVEX 350， Oerlikon, Germany) onto three types of substrate: two UV-curable polymers (Ormocomp®, Ormostamp®) well suited for nanoimprint lithography and a pure Silica texture obtained by solgel process. We evaporate the films both on textured and non-textured regions to later compare these two relative situations. The film thickness is monitored during evaporation using a quartz crystal (Inficon, Switzerland). Film viscosity is dramatically reduced upon annealing above their glass transition temperature, and enhanced chain mobility allows for dewetting to occur.

Contact Angle Measurements
To experimentally determine contact angles, we proceed to dewet evaporated thin films. This is achieved by thermal annealing above the glass transition temperature of the film for extended durations, e.g. twice the time required to observe stable microstructure based on top-view observations using optical microscopy. Cross-sections of the obtained samples are then prepared using liquid nitrogen. All SEM samples were coated with a 10 nm carbon film. The SEM images were taken with a Zeiss Merlin field emission SEM equipped with a GEMINI II column operating at 1.0 kV with a probe current of 70 pA. Contact angles are further measured by Scanning Electron Microscopy cross-sectional images and processed using image analysis to accurately extract the contact angles (Image J, Contact Angle Module).

Numerical Simulations
The numerical implementation of the lubrication equation (1) with complete curvature (5), and together with the interface potential expressions (2), is performed in the finite-element solver COMSOL Multiphysics. The equations are discretized for the variables ℎ, . We consider quadratic Lagrangian elements for the spatial discretization, with a triangular non-structured grid for the two-dimensional case. We exploit the built-in Backward Differentiation Formula algorithm for the time marching, setting a tolerance of 10 -5 . The numerical convergence is achieved by performing several simulations with ℎ 0 and verifying the convergence of the contact angle to the desired value.
As outlined above, the approach for the simulation of experimental conditions is based on the choice of the contact angle and the retrieval of the equilibrium thickness and Born coefficient. The Lennard-Jones potential reads: where the Hamaker constant A is estimated based on the Lifschitz theory. Following reference 24, the macroscopic contact angle at the equilibrium is given by: where φ(heq) is the equilibrium potential, obtained imposing φ'(heq)=0, where heq is the equilibrium thickness (i.e. precursor film thickness). Once the contact angle is fixed, the previous relation gives a unique value of the equilibrium potential in the range [0°, 90°], with A>0. The value of the equilibrium potential can be used to evaluate the equilibrium thickness. Deriving expression (2) with respect to h and evaluating at the equilibrium thickness ℎ yields: The equilibrium potential at the equilibrium thickness can be written as (c.f. equation (2)): Where ℎ is associated to a unique contact angle between 0° to 90° according to equation (4).
We therefore obtained the Born coefficient that satisfies the imposed contact angle.

Evaluation Of The Hamaker Constant Using The Lifschitz Theory And Typical Values
The

Roughness Measurements
Atomic force microscopy images ( To limit contact angle hysteresis, we proceed in successive steps. The first step consists of a preparation of substrates with low surface roughness. We resort to several substrates, including: (i) sol-gels based on acid-catalyzed Methy (triethoxysilane), followed by a pyrolysis step at 400°C, or (ii) UV curable commercial resins, in particular Ormocomp® and Ormostamp® from Microresist, Germany. The pyrolysis step for sol-gels helps to densify the resulting silica structure while removing residual organic components. To control the resulting surface roughness of silicabased on sol-gel processes as well as the commercial UV resins, we proceed to measure by atomic force microscopy the surface roughness (FIG. SI 1). All root mean squared (RMS) roughness values are inferior to 1 nm, which, although not competitive with typical Si wafer roughness, compares favorably with most other surfaces.
A second step aims at cleaning thoroughly the substrates to remove any chemical inhomogeneities. As reported in the literature, this is a critical step to ensure a homogeneous substrate surface energy, and consequently homogeneous dewetting patterns. The extensive use of polymers largely restricts the use of commonly-used aggressive solvents to clean wafers such as H2SO4 or HF. Nevertheless, common cleaning procedures help to wash away nano-imprinting residues such as silicone oil traces from the PDMS or other contaminants, All substrates are subsequently washed systematically using isopropanol (degreasing agent), ethanol, and water. The wash cycle is commonly repeated three times, followed by a gentle nitrogen or air gun to dry the substrate surface and blow away eventual debris or particles remaining at the surface.
The hysteresis caused by roughness and chemical heterogeneity is further assessed by evaluating the standard deviation of contact angles measurement for a given film/substrate couple.
The error bar stem from a combination of imaging analysis related uncertainties and substrate surface roughness, which induce local triple line pinning and deviation for equilibrium contact angle. As apparent in FIG. SI 1, limited contact angle hysteresis does occur in all systems studied.
Further techniques to decrease the surface roughness of nanoimprinted substrate as well as improve the chemical washing procedure would be of considerable help to reduce contact angle standard deviation and thereby offer better control over templated dewetting. deposited on a plasma-treated silica textured sample with increasing spacing-to-period ratio. The structural transition from the absence of mesa (e.g. particle in between pits) to single mesa and then double mesa is visible with an initial film thickness of 60 nm. In the 100 nm thickness case, the film only shows a single structural transition from the absence of mesa to single mesa.
FIG. SI 6 -All-dielectric Huygens Meta-Gratings for linear features according to the geometry shown in FIG. 3(a).
The simulated dash curves are modeled purely enforcing the volume conservation hypothesis, with respective periods of 1620 nm and 1520 nm and an initial film thickness of 85 nm. The experimental data is obtained for a film thickness of 80 nm based on microbalance measurements in-situ during evaporation and respective periods of 1600 nm and 1500 nm. The colormap shows the phase map in reflection, with a sharp transition around 690 nm. The phase shift is plotted for λ=684 nm, showing a clear 2π phase shift for a range of periods spanning from 1400 nm to 1900 nm.