Dynamics of gas–surface interactions: 3D generalized Langevin model applied to fcc and bcc surfaces

A theory of phonon-induced friction on molecular adsorbates

In this manuscript, we provide a general theory for how surface phonons couple to molecular adsorbates. Our theory maps the extended dynamics of a surface's atomic vibrational motions to a generalized Langevin equation, and by doing so captures these dynamics in a single quantity: the non-Markovian friction. The different frequency components of this friction are the phonon modes of the surface slab weighted by their coupling to the adsorbate degrees of freedom. Using this formalism, we demonstrate that physisorbed species couple primarily to acoustic phonons while chemisorbed species couple to dispersionless local vibrations. We subsequently derive equations for phonon-adjusted reaction rates using transition state theory and demonstrate that these corrections improve agreement with experimental results for CO desorption rates from Pt(111).

On the quantum dynamical treatment of surface vibrational modes for reactive scattering of H2 from Cu(111) at 925 K

We construct the effective Hartree potential for H2 on Cu(111) as introduced in our earlier work [Dutta et al., J. Chem. Phys. 154, 104103 (2021), and Dutta et al., J. Chem. Phys. 157, 194112 (2022)] starting from the same gas-metal interaction potential obtained for 0 K. Unlike in that work, we now explicitly account for surface expansion at 925 K and investigate different models to describe the surface vibrational modes: (i) a cluster model yielding harmonic normal modes at 0 K and (ii) slab models resulting in phonons at 0 and 925 K according to the quasi-harmonic approximation-all consistently calculated at the density functional theory level with the same exchange-correlation potential. While performing dynamical calculations for the H2(v = 0, j = 0)-Cu(111) system employing Hartree potential constructed with 925 K phonons and surface temperature, (i) the calculated chemisorption probabilities are the highest compared to the other approaches over the energy domain and (ii) the threshold for the reaction probability is the lowest, in close agreement with the experiment. Although the survival probabilities (v' = 0) depict the expected trend (lower in magnitude), the excitation probabilities (v' = 1) display a higher magnitude since the 925 K phonons and surface temperature are more effective for the excitation process compared to the phonons/normal modes obtained from the other approaches investigated to describe the surface.

Modeling Surface Vibrations and Their Role in Molecular Adsorption: A Generalized Langevin Approach

The atomic vibrations of a solid surface can play a significant role in the reactions of surface-bound molecules, as well as their adsorption and desorption. Relevant phonon modes can involve the collective motion of atoms over a wide array of length scales. In this paper, we demonstrate how the generalized Langevin equation can be utilized to describe these collective motions weighted by their coupling to individual sites. Our approach builds upon the generalized Langevin oscillator (GLO) model originally developed by Tully. We extend the GLO by deriving parameters from atomistic simulation data. We apply this approach to study the memory kernel of a model platinum surface and demonstrate that the memory kernel has a bimodal form due to coupling to both low-energy acoustic modes and high-energy modes near the Debye frequency. The same bimodal form was observed across a wide variety of solids of different elemental compositions, surface structures, and solvation states. By studying how these dominant modes depend on the simulation size, we argue that the acoustic modes are frozen in the limit of macroscopic lattices. By simulating periodically replicated slabs of various sizes, we quantify the influence of phonon confinement effects in the memory kernel and their concomitant effect on simulated sticking coefficients.

H atom scattering from W(110): A benchmark for molecular dynamics with electronic friction

Molecular dynamics with electronic friction (MDEF) at the level of the local density friction approximation (LDFA) has been applied to describe electronically non-adiabatic energy transfer accompanying H atom collisions with many solid metal surfaces. When implemented with full dimensional potential energy and electron density functions, excellent agreement with experiment is found. Here, we compare the performance of a reduced dimensional MDEF approach involving a simplified description of H atom coupling to phonons to that of full dimensional MDEF calculations known to yield accurate results. Both approaches give remarkably similar results for H atom energy loss distributions with a 300 K W(110) surface. At low surface temperature differences are seen; but, quantities like average energy loss are still accurately reproduced. Both models predict similar conditions under which H atoms that have penetrated into the subsurface regions could be observed in scattering experiments.

Molecular dynamics with electronic friction (MDEF) at the level of the local density friction approximation (LDFA) has been applied to describe electronically non-adiabatic energy transfer accompanying H atom collisions with many solid metal surfaces.

First-Principles Insights into Adiabatic and Nonadiabatic Vibrational Energy-Transfer Dynamics during Molecular Scattering from Metal Surfaces: The Importance of Surface Reactivity

Energy transfer is ubiquitous during molecular collisions and reactions at gas-surface interfaces. Of particular importance is vibrational energy transfer because of its relevance to bond forming and breaking. In this Perspective, we review recent first-principles studies on vibrational energy-transfer dynamics during molecular scattering from metal surfaces at the state-to-state level. Taking several representative systems as examples, we highlight the intrinsic correlation between vibrational energy transfer in nonreactive scattering and surface reactivity and how it operates in both electronically adiabatic and nonadiabatic pathways. Adiabatically, the presence of a dissociation barrier softens a bond in the impinging molecule and increases its couplings with other molecular modes and surface phonons. In the meantime, the stronger interaction between the molecule and the surface also changes the electronic structure at the barrier, resulting in an increase of nonadiabatic effects. We further discuss future prospects toward a more quantitative understanding of this important surface dynamical process.

Dynamics of Initial Hydrogen Spillover from a Single Atom Platinum Active Site to the Cu(111) Host Surface: The Impact of Substrate Electron-Hole Pairs

The initial impulsive diffusion of hot hydrogen atoms resulted from the dissociative chemisorption of H₂ at atomically dispersed Pt atoms embedded in Cu(111) is investigated using ab initio molecular dynamics. Upon dissociation, one of the two hydrogen atoms tends to roam away from the dissociation site while the other remains trapped. It is shown that the fraction of diffusion and the average diffusion length increase with the incident energy and H₂ vibrational excitation, due apparently to the increased initial kinetic energy of the hot atoms. Most importantly, the strong interaction with surface electron-hole pairs, modeled using an electronic friction model, is shown to play an important role in rapid energy dissipation and significant retardation of the impulsive diffusion.

Random Force in Molecular Dynamics with Electronic Friction

Originally conceived to describe thermal diffusion, the Langevin equation includes both a frictional drag and a random force, the latter representing thermal fluctuations first seen as Brownian motion. The random force is crucial for the diffusion problem as it explains why friction does not simply bring the system to a standstill. When using the Langevin equation to describe ballistic motion, the importance of the random force is less obvious and it is often omitted, for example, in theoretical treatments of hot ions and atoms interacting with metals. Here, friction results from electronic nonadiabaticity (electronic friction), and the random force arises from thermal electron-hole pairs. We show the consequences of omitting the random force in the dynamics of H-atom scattering from metals. We compare molecular dynamics simulations based on the Langevin equation to experimentally derived energy loss distributions. Despite the fact that the incidence energy is much larger than the thermal energy and the scattering time is only about 25 fs, the energy loss distribution fails to reproduce the experiment if the random force is neglected. Neglecting the random force is an even more severe approximation than freezing the positions of the metal atoms or modelling the lattice vibrations as a generalized Langevin oscillator. This behavior can be understood by considering analytic solutions to the Ornstein-Uhlenbeck process, where a ballistic particle experiencing friction decelerates under the influence of thermal fluctuations.

Computational approaches to dissociative chemisorption on metals: towards chemical accuracy

We review the state-of-the-art in the theory of dissociative chemisorption (DC) of small gas phase molecules on metal surfaces, which is important to modeling heterogeneous catalysis for practical reasons, and for achieving an understanding of the wealth of experimental information that exists for this topic, for fundamental reasons. We first give a quick overview of the experimental state of the field. Turning to the theory, we address the challenge that barrier heights (E_b, which are not observables) for DC on metals cannot yet be calculated with chemical accuracy, although embedded correlated wave function theory and diffusion Monte-Carlo are moving in this direction. For benchmarking, at present chemically accurate E_b can only be derived from dynamics calculations based on a semi-empirically derived density functional (DF), by computing a sticking curve and demonstrating that it is shifted from the curve measured in a supersonic beam experiment by no more than 1 kcal mol^-1. The approach capable of delivering this accuracy is called the specific reaction parameter (SRP) approach to density functional theory (DFT). SRP-DFT relies on DFT and on dynamics calculations, which are most efficiently performed if a potential energy surface (PES) is available. We therefore present a brief review of the DFs that now exist, also considering their performance on databases for E_b for gas phase reactions and DC on metals, and for adsorption to metals. We also consider expressions for SRP-DFs and briefly discuss other electronic structure methods that have addressed the interaction of molecules with metal surfaces. An overview is presented of dynamical models, which make a distinction as to whether or not, and which dissipative channels are modeled, the dissipative channels being surface phonons and electronically non-adiabatic channels such as electron-hole pair excitation. We also discuss the dynamical methods that have been used, such as the quasi-classical trajectory method and quantum dynamical methods like the time-dependent wave packet method and the reaction path Hamiltonian method. Limits on the accuracy of these methods are discussed for DC of diatomic and polyatomic molecules on metal surfaces, paying particular attention to reduced dimensionality approximations that still have to be invoked in wave packet calculations on polyatomic molecules like CH₄. We also address the accuracy of fitting methods, such as recent machine learning methods (like neural network methods) and the corrugation reducing procedure. In discussing the calculation of observables we emphasize the importance of modeling the properties of the supersonic beams in simulating the sticking probability curves measured in the associated experiments. We show that chemically accurate barrier heights have now been extracted for DC in 11 molecule-metal surface systems, some of which form the most accurate core of the only existing database of E_b for DC reactions on metal surfaces (SBH10). The SRP-DFs (or candidate SRP-DFs) that have been derived show transferability in many cases, i.e., they have been shown also to yield chemically accurate E_b for chemically related systems. This can in principle be exploited in simulating rates of catalyzed reactions on nano-particles containing facets and edges, as SRP-DFs may be transferable among systems in which a molecule dissociates on low index and stepped surfaces of the same metal. In many instances SRP-DFs have allowed important conclusions regarding the mechanisms underlying observed experimental trends. An important recent observation is that SRP-DFT based on semi-local exchange DFs has so far only been successful for systems for which the difference of the metal work function and the molecule's electron affinity exceeds 7 eV. A main challenge to SRP-DFT is to extend its applicability to the other systems, which involve a range of important DC reactions of e.g. O₂, H₂O, NH₃, CO₂, and CH₃OH. Recent calculations employing a PES based on a screened hybrid exchange functional suggest that the road to success may be based on using exchange functionals of this category.

Atomic-scale thermal manipulation with adsorbed atoms on a solid surface at a liquid-solid interface

Modulating thermal transport through interfaces is one of the central issues in nanoscience and nanotechnology. This study examined thermal transport between atoms adsorbed on a solid surface and a liquid phase based on non-equilibrium molecular dynamics. The heat flux was detected at sub-atomic spatial resolution, yielding a two-dimensional map of local heat flux in the vicinity of the adsorbed atoms on the surface. Based on the detected heat flux, the possibility of atomic-scale thermal manipulation with the adsorbed atoms was examined by varying the interaction strengths between the liquid molecules and atoms adsorbed on the surface. The results of the local heat flux at the single-atom scale clearly showed effects of the adsorbed atoms on the thermal transport through the liquid-solid interface; they can significantly enhance the heat flux at the single-atom scale using degrees of freedom normal to the macroscopic temperature gradient. The effect was especially evident for a low wettability surface, which provides key information on local enhancement at the single-atom scale of the thermal transport through a liquid-solid interface.

Dynamics in reactions on metal surfaces: A theoretical perspective

Recent advances in theoretical characterization of reaction dynamics on metal surfaces are reviewed. It is shown that the widely available density functional theory of metals and their interactions with molecules have enabled first principles theoretical models for treating surface reaction dynamics. The new theoretical tools include methods to construct high-dimensional adiabatic potential energy surfaces, to characterize nonadiabatic processes within the electronic friction models, and to describe dynamics both quantum mechanically and classically. Three prototypical surface reactions, namely, dissociative chemisorption, Eley-Rideal reactions, and recombinative desorption, are surveyed with a focus on some representative examples. While principles governing gas phase reaction dynamics may still be applicable, the presence of the surface introduces a higher level of complexity due to strong interaction between the molecular species and metal substrate. Furthermore, most of these reactive processes are impacted by energy exchange with surface phonons and/or electron-hole pair excitations. These theoretical studies help to interpret and rationalize experimental observations and, in some cases, guide experimental explorations. Knowledge acquired in these fundamental studies is expected to impact many practical problems in a wide range of interfacial processes.

Stochastic wave packet approach to nonadiabatic scattering of diatomic molecules from metals

In this contribution, we present a quantum dynamical approach to study inelastic scattering of diatomic molecules from metal surfaces at normal incidence. The dissipative dynamics obeys a stochastic Schrödinger equation describing the time-evolution of the system as a piecewise deterministic process. Energy exchange between the molecular vibrational degrees of freedom and the metal electrons is represented using operators in tensor product form, which are coupled via anharmonic transition rates calculated from first-order perturbation theory. Full dimensional observables are obtained by averaging over simulations in 4D-including the internal stretch, the distance to the surface, and the orientation angles-at different surface sites. The method is applied to the state-resolved scattering of vibrationally excited NO from Au(111), revealing important channels for quantized energy relaxation.

Dynamics of N2 sticking on W(100): the decisive role of van der Waals interactions

The reactive dynamics of N₂ on W(100) has been investigated by means of quasi-classical trajectory calculations using an interpolated six-dimensional potential energy surface (PES) based on density functional theory energies obtained employing the vdW-DF2 functional.

Strong Anisotropic Interaction Controls Unusual Sticking and Scattering of CO at Ru(0001)

Complete sticking at low incidence energies and broad angular scattering distributions at higher energies are often observed in molecular beam experiments on gas-surface systems which feature a deep chemisorption well and lack early reaction barriers. Although CO binds strongly on Ru(0001), scattering is characterized by rather narrow angular distributions and sticking is incomplete even at low incidence energies. We perform molecular dynamics simulations, accounting for phononic (and electronic) energy loss channels, on a potential energy surface based on first-principles electronic structure calculations that reproduce the molecular beam experiments. We demonstrate that the mentioned unusual behavior is a consequence of a very strong rotational anisotropy in the molecule-surface interaction potential. Beyond the interpretation of scattering phenomena, we also discuss implications of our results for the recently proposed role of a precursor state for the desorption and scattering of CO from ruthenium.

Vibrational Excitation of H2 Scattering from Cu(111): Effects of Surface Temperature and of Allowing Energy Exchange with the Surface

In scattering of H₂ from Cu(111), vibrational excitation has so far defied an accurate theoretical description. To expose the causes of the large discrepancies with experiment, we investigate how the feature due to vibrational excitation (the "gain peak") in the simulated time-of-flight spectrum of (v = 1, j = 3) H₂ scattering from Cu(111) depends on the surface temperature (T_s) and the possibility of energy exchange with surface phonons and electron-hole pairs (ehp's). Quasi-classical dynamics calculations are performed on the basis of accurate semiempirical density functionals for the interaction with H₂ + Cu(111). The methods used include the quasi-classical trajectory method within the Born-Oppenheimer static surface model, the generalized Langevin oscillator (GLO) method incorporating energy transfer to surface phonons, the GLO + friction (GLO+F) method also incorporating energy exchange with ehp's, and ab initio molecular dynamics with electronic friction (AIMDEF). Of the quasi-classical methods tested, comparison with AIMDEF suggests that the GLO+F method is accurate enough to describe vibrational excitation as measured in the experiments. The GLO+F calculations also suggest that the promoting effect of raising T_s on the measured vibrational excitation is due to an electronically nonadiabatic mechanism. However, by itself, enabling energy exchange with the surface by modeling surface phonons and ehp's leads to reduced vibrational excitation, further decreasing the agreement with experiment. The simulated gain peak is quite sensitive to energy shifts in calculated vibrational excitation probabilities and to shifts in a specific experimental parameter (the chopper opening time). While the GLO+F calculations allow important qualitative conclusions, comparison to quantum dynamics results suggests that, with the quasi-classical way of describing nuclear motion and the present box quantization method for assigning the final vibrational state, the gain peak is not yet described with quantitative accuracy. Ways in which this problem might be resolved in the future are discussed.

Dissipation of the excess energy of the adsorbate-thermalization via electron transfer

A new scenario for the thermalization process of adsorbates at solid surfaces is proposed. The scenario is based on the existence of an electric dipole layer in which the electron wavefunctions extend over the positive ions, creating a strong local electric field which drags the electrons into the solid interior and repels the positive ions. During adsorption the electrons tunnel into the solid interior, conveying the excess energy. The positive ions are retarded by the field, losing the excess kinetic energy, and are located smoothly into the adsorption sites. In such a scheme, the excess energy is not dissipated locally, avoiding melting or the creation of defects which is in accordance with experiments. The scenario is supported by ab initio calculation results, including density function theory of the slabs representing the AlN surface and the Schrodinger equation for the time evolution of hydrogen-like atoms at the solid surface.

Dissociative adsorption dynamics of nitrogen on a Fe(111) surface

The dissociative adsorption dynamics of N₂ on clean Fe(111) surfaces is theoretically investigated by means of quasi-classical trajectory calculations based on a multidimensional potential energy surface built from density functional theory.

Adsorption dynamics of molecular nitrogen at an Fe(111) surface

We present an extensive theoretical study of N₂ adsorption mechanisms on an Fe(111) surface.

Dissociative and non-dissociative adsorption of O2 on Cu(111) and CuML/Ru(0001) surfaces: adiabaticity takes over

Adiabatic molecular spin-quenching during the approach of O₂ to Cu(111) and Cu_ML/Ru(0001) surfaces.

Green's function nonequilibrium molecular dynamics method for solid surfaces and interfaces

This study presents a comprehensive procedure to calculate the exact dynamic Green's function of a harmonic semi-infinite solid and the time trajectories of the atoms, in the framework of the Green's function molecular dynamics. This Green's function properly describes the energy dissipation caused by excitations of the surface phonons, and the simulated atoms serve as well-defined thermo- and barostats for the nonequilibrium surface and interface systems. Moreover, the use of the exact dynamic Green's function coupled with a fast convolution algorithm significantly improves both the accuracy and the computing speed. The presented method is applied to a diamond (001) surface, and the results demonstrate that the properties of the nonreflecting boundary, the thermal fluctuations, and the energy dissipations involving long-wavelength phonons are correctly reproduced. These distinctive performances potentially allow us to reveal the nonequilibrium phenomena in a wide spectrum of applications such as catalysis, thermal transport, fracture mechanics, mechanochemistry, and tribology.

Electron-Hole Pair Effects in Polyatomic Dissociative Chemisorption: Water on Ni(111)

The influence of electron-hole pairs in dissociative chemisorption of a polyatomic molecule (water) on metal surfaces is assessed for the first time using a friction approach. The atomic local density dependent friction coefficients computed based on a free electron gas embedding model are employed in classical molecular dynamics simulations of the water dissociation dynamics on rigid Ni(111) using a recently developed nine dimensional interaction potential energy surface for the system. The results indicate that nonadiabatic effects are relatively small and they do not qualitatively alter the mode specificity in the dissociation.

Quantum and classical dynamics of reactive scattering of H2 from metal surfaces

State-of-the-art theoretical models allow nowadays an accurate description of H₂/metal surface systems and phenomena relative to heterogeneous catalysis. Here we review the most relevant ones investigated during the last 10 years.

Molecular dynamics simulation of O2 adsorption on Ag(110) from first principles electronic structure calculations

State of the art simulations show that the physisorption state could be important for O₂/Ag(110) adsorption.

Frictional effects near a metal surface

When a classical master equation (CME) is used to describe the nonadiabatic dynamics of a molecule at metal surfaces, we show that in the regime of reasonably strong molecule-metal couplings, the CME can be reduced to a Fokker-Planck equation with an explicit form of electronic friction. For a single metal substrate at thermal equilibrium, the electronic friction and random force satisfy the fluctuation-dissipation theorem. When we investigate the time scale for an electron transfer (ET) event between the molecule and metal surface, we find that the ET rates show a turnover effect (just as in Kramer's theory) as a function of frictional damping.

Local pressure components and interfacial tensions of a liquid film in the vicinity of a solid surface with a nanometer-scale slit pore obtained by the perturbative method

A classical molecular dynamics simulation was conducted for a liquid-solid interfacial system with a nanometer-scale slit pore in order to reveal local thermodynamic states: local pressure components and interfacial tensions of a liquid film in the vicinity of the slit. The simulation also examined the transition mechanism between the two states of the liquid film: (a) liquid film on the slit and (b) liquid film in the slit, based on the local thermodynamic quantities from a molecular point of view. An instantaneous expression of the local pressure components and interfacial tensions, which is based on a volume perturbation, was presented to investigate time-dependent phenomena in molecular dynamics simulations. The interactions between the particles were described by the 12-6 Lennard-Jones potential, and effects of the fluid-solid interaction intensity on the local pressure components and interfacial tensions of the fluid in the vicinity of the slit were examined in detail by the presented perturbative method. The results revealed that the local pressure components tangential to the solid surface in the vicinity of the 1st fluid layer from the solid surface are different in a two dimensional plane, and the difference became pronounced in the vicinity of the corner of the slit, for cases where the fluid-solid interaction intensities are relatively strong. The results for the local interfacial tensions of the fluid inside the slit suggested that the local interfacial tensions in the vicinity of the 2nd and 3rd layers of the solid atoms from the entrance of the slit act as a trigger for the transition between the two states under the influence of a varying fluid-solid interaction.

The dynamics of adsorption and dissociation of N2 in a monolayer of iron on W(110)

Does N₂ adsorption increase on strained Fe monolayers?

Surface strain improves molecular adsorption but hampers dissociation for N2 on the Fe/W(110) surface

We compare the adsorption dynamics of N(2) on the unstrained Fe(110) and on a 10% expanded Fe monolayer grown on W(110) by performing classical molecular dynamics simulations that use potential energy surfaces calculated with density functional theory. Our results allow us to understand why, experimentally, the molecular adsorption of N(2) is observed on the strained layer but not on Fe(110). Surprisingly, we also find that while surface strain favors the molecular adsorption of N(2) it seems, on the contrary, to impede the dissociative adsorption. This result contrasts with previous examples for which strain is found to modify equally the energetics of chemisorption and dissociation.

Role of physisorption states in molecular scattering: a semilocal density-functional theory study on O2/Ag(111)

We simulate the scattering of O2 from Ag(111) with classical dynamics simulations performed on a six-dimensional potential energy surface calculated within semilocal density-functional theory. The enigmatic experimental trends that originally required the conjecture of two types of repulsive walls, arising from a physisorption and chemisorption part of the interaction potential, are fully reproduced. Given the inadequate description of the physisorption properties in semilocal density-functional theory, our work casts severe doubts on the prevalent notion to use molecular scattering data as indirect evidence for the existence of such states.

ReaxFF molecular dynamics simulations of CO collisions on an O-preadsorbed silica surface

A quasiclassical trajectory dynamics study was performed for carbon monoxide collisions over an oxygen preadsorbed β-cristobalite (001) surface. A reactive molecular force field (ReaxFF) was used to model the potential energy surface. The collisions were performed fixing several initial conditions: CO rovibrational states (v = 0-5 and j = 0, 20, 35), collision energies (0.05 ≤ E(col) ≤ 2.5 eV), incident angles (θ(v) = 0°, 45°) and surface temperatures (T(surf) = 300 K, 900 K). The principal elementary processes were the molecular reflection and the non-dissociative molecular adsorption. CO₂ molecules were also formed in minor extension via an Eley-Rideal reaction although some of them were finally retained on the surface. The scattered CO molecules tend to be translationally colder and internally hotter (rotationally and vibrationally) than the initial ones. The present study supports that CO + O(ad) reaction should be less important than O + O(ad) reaction over silica for similar initial conditions of reactants, in agreement with experimental data.

Dynamic wetting at the nanoscale

Although the capillary spreading of a drop on a dry substrate is well studied, understanding and describing the physical mechanisms that govern the dynamics remain challenging. Here we study the dynamics of spreading of partially wetting nanodroplets by combining molecular dynamics simulations and continuum phase field simulations. The phase field simulations account for all the relevant hydrodynamics, i.e., capillarity, inertia, and viscous stresses. By coordinated continuum and molecular dynamics simulations, the macroscopic model parameters are extracted. For a Lennard-Jones fluid spreading on a planar surface, the liquid slip at the solid substrate is found to be significant, in fact crucial for the motion of the contact line. Evaluation of the different contributions to the energy transfer shows that the liquid slip generates dissipation of the same order as the bulk viscous dissipation or the energy transfer to kinetic energy. We also study the dynamics of spreading on a substrate with a periodic nanostructure. Here it is found that a nanostructure with a length scale commensurate with molecular size completely inhibits the liquid slip. The dynamic spreading is thus about 30% slower on a nanostructured surface compared to one that is atomically smooth.