Dr. Astrid S. de Wijn

*How square ice helps lubrication*

Astrid S. de Wijn and Lars G. M. Pettersson

Phys. Rev. B**95**, 165433 (2017).

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In the context of friction we use atomistic molecular-dynamics simulations to investigate water confined between graphene sheets over a wide range of pressures. We find that thermal equilibration of the confined water is hindered at high pressures. We demonstrate that, under the right conditions, square ice can form in an asperity, and that it is similar to cubic ice VII and ice X. We simulate sliding of atomically flat graphite on the square ice and find extremely low friction due to structural superlubricity. The conditions needed for square ice to form correspond to low sliding speeds, and we suggest that the ice observed in experiments of friction on wet graphite is of this type.*News and Views | Nanoscience: Flexible graphene strengthens friction*

Astrid S. de Wijn

Nature**539**, 502–503 (2016).

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Previous observations showed that friction on graphene increases gradually when a probe starts to slide across the material's surface. Simulations now reveal that this effect is related to bending of the graphene sheet. See Letter Nature**539**, page 541*Imaging high-speed friction at the nanometer scale*

Per-Anders Thorén, Astrid S. de Wijn, Riccardo Borgani, Daniel Forchheimer, and David B. Haviland

Nature Communications**7**, 13836 (2016).

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Friction is a complicated phenomenon involving nonlinear dynamics at different length and time scales. The microscopic origin of friction is poorly understood, due in part to a lack of methods for measuring the force on a nanometer-scale asperity sliding at velocity of order cm/s. Dispite enormous advances in experimental technique this combination of small length scale and high velocity remained illusive. Here we present a technique for rapidly measuring the frictional forces on a single asperity (an AFM tip) over a velocity range from zero to several cm/s. At each image pixel we obtain the velocity dependence of both conservative and dissipative forces, revealing the transition from stick-slip to a smooth sliding friction. We explain measurements on graphite using a modified Prandtl-Tomlinson model that takes into account the damped elastic deformation of the asperity. With its significant improvement in force sensitivity and very small sliding amplitude, our method enables rapid and detailed surface mapping of the full velocity-dependence of frictional forces to sub 10 nm spatial resolution.*Emergent friction in two-dimensional Frenkel-Kontorova models*

Jesper Norell, Annalisa Fasolino, Astrid S. de Wijn

Phys. Rev. E**94**, 023001 (2016).

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Simple models for friction are typically one-dimensional, but real interfaces are two-dimensional. We investigate the effects of the second dimension on static and dynamic friction by using the Frenkel-Kontorova (FK) model. We study the two most straightforward extensions of the FK model to two dimensions and simulate both the static and dynamic properties. We show that the behavior of the static friction is robust and remains similar in two dimensions for physically reasonable parameter values. The dynamic friction, however, is strongly influenced by the second dimension and the accompanying additional dynamics and parameters introduced into the models. We discuss our results in terms of the thermal equilibration and phonon dispersion relations of the lattices, establishing a physically realistic and suitable two-dimensional extension of the FK model. We find that the presence of additional dissipation channels can increase the friction and produces significantly different temperature dependence when compared to the one-dimensional case. We also briefly study the anisotropy of the dynamic friction and show highly nontrivial effects, including that the friction anisotropy can lead to motion in different directions depending on the value of the initial velocity.*Collective superlubricity of graphene flakes*

Merel M. van Wijk, Astrid S. de Wijn, and Annalisa Fasolino

J. Phys.: Condens. Matter**28**134007 (2016).

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We investigate solid lubrication of graphene and graphene flakes using atomistic molecular-dynamics simulations. We find that graphene flakes yield lower friction than graphene as a result of a collective mechanism that emerges from the independent behaviour of the flakes. By freezing out different degrees of freedom of the flakes, we are able to attribute the low friction to non-simultaneous slipping of the individual flakes. We also compare the results of the atomistic simulations to those of a simplified two-dimensional model and find that the behaviour of the latter is strongly dependent on parameters, which emerge naturally from the atomistic simulations.*Preface to the special section on nano- and mesoscale friction*

Eran Bouchbinder, Adam S. Foster, Oğuzhan Gürlü, Ernst Meyer, Susan Perkin, Andre Schirmeisen and Astrid S. de Wijn

J. Phys.: Condens. Matter**28**130301 (2016).

pdf*Effects of molecule anchoring and dispersion on nanoscopic friction under electrochemical control*

A. S. de Wijn, A. Fasolino, A. E. Filippov, and M. Urbakh

J. Phys.: Condens. Matter**28**, 105001 (2016).

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The application of electric fields is a promising strategy for in situ control of friction. While there have recently been many experimental studies on friction under the influence of electric fields, theoretical understanding is very limited. Recently, we introduced a simple theoretical model for friction under electrochemical conditions that focused on the interaction of a force microscope tip with adsorbed molecules whose orientation was dependent on the applied electric field. Here we focus on the effects of anchoring of the molecules on friction. We show that anchoring affects the intensity and width of the peak in the friction that occurs near a reorientation transition of adsorbed molecules, and explain this by comparing the strength of molecule–molecule and molecule–tip interactions. We derive a dispersion relation for phonons in the layer of adsorbed molecules and demonstrate that it can be used to understand important features of the frictional response.*Chaotic properties of spin lattices near second-order phase transitions*

A. S. de Wijn, B. Hess, and B. V. Fine

Phys. Rev. E**92**, 062929 (2015).

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We perform a numerical investigation of the Lyapunov spectra of chaotic dynamics in lattices of classical spins in the vicinity of second-order ferromagnetic and antiferromagnetic phase transitions. On the basis of this investigation, we identify a characteristic of the shape of the Lyapunov spectra, the "G-index", which exhibits a sharp peak as a function of temperature at the phase transition, provided the order parameter is capable of sufficiently strong dynamic fluctuations. As a part of this work, we also propose a general numerical algorithm for determining the temperature in many-particle systems, where kinetic energy is not defined.*Understanding and controlling regime switching in molecular diffusion*

S. Hallerberg and A. S. de Wijn

Phys. Rev. E**90**, 062901 (2014).

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Diffusion can be strongly affected by ballistic flights (long jumps) as well as long-lived sticking trajectories (long sticks). Using statistical inference techniques in the spirit of Granger causality, we investigate the appearance of long jumps and sticks in molecular-dynamics simulations of diffusion in a prototype system, a benzene molecule on a graphite substrate. We find that specific fluctuations in certain, but not all, internal degrees of freedom of the molecule can be linked to either long jumps or sticks. Furthermore, by changing the prevalence of these predictors with an outside influence, the diffusion of the molecule can be controlled. The approach presented in this proof of concept study is very generic, and can be applied to larger and more complex molecules. Additionally, the predictor variables can be chosen in a general way so as to be accessible in experiments, making the method feasible for control of diffusion in applications. Our results also demonstrate that data-mining techniques can be used to investigate the phase-space structure of high-dimensional nonlinear dynamical systems.*Preferential sliding directions on graphite*

Balakrishna S.G., Astrid S. de Wijn, and Roland Bennewitz

Phys. Rev. B**89**, 245440 (2014).

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The anisotropy of friction on graphitic surfaces is investigated by a combined friction force microscopy and modeling study. Friction vectors deviate up to 15° from pulling directions. The strongest deviations are found for pulling directions which lie almost along one zigzag direction of the honeycomb structure, the preferred sliding direction on graphite surfaces and epitaxial graphene grown on SiC(0001). Atomic stick-slip events along and across molecular rows determine direction and magnitude of friction. Simulation and modeling reveal the role of temperature and of the two-dimensional character of the surface potential for the friction anisotropy.*A criterion for condensation in kinetically constrained one-dimensional transport models*

D. M. Miedema, Astrid S. de Wijn, and Peter Schall

Phys. Rev. E**89**, 062812 (2014).

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We study condensation in one-dimensional transport models with a kinetic constraint. The kinetic constraint results in clustering of immobile vehicles; these clusters can grow to macroscopic condensates, indicating the onset of dynamic phase separation between free flowing and arrested traffic. We investigate analytically the conditions under which this occurs, and derive a necessary and sufficient criterion for phase separation. This criterion is applied to the well-known Nagel-Schreckenberg model of traffic flow to analytically investigate the existence of dynamic condensates. We find that true condensates occur only when acceleration out of jammed traffic happens in a single time step, in the limit of strong overbraking. Our predictions are further verified with simulation results on the growth of arrested clusters. These results provide analytic understanding of dynamic arrest and dynamic phase separation in one-dimensional traffic and transport models.*Absence of exponential sensitivity to small perturbations in nonintegrable systems of spins 1/2*

Boris V. Fine, Tarek A. Elsayed, Chahan M. Kropf, and Astrid S. de Wijn

Phys. Rev. E**89**, 012923 (2014).

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We show that nonintegrable lattices of spins 1/2, which are often considered to be chaotic, do not exhibit the basic property expected for classical chaotic systems, namely, exponential sensitivity to small perturbations. Specifically, we compare the responses of chaotic lattices of classical spins and nonintegrable lattices of spins 1/2 to imperfect reversal of spin dynamics known as Loschmidt echo. In the classical case, Loschmidt echoes exhibit exponential sensitivity to small perturbations. This sensitivity is controlled by twice the value of the largest Lyapunov exponent of the underlying chaotic dynamics. In the case of spins 1/2, Loschmidt echoes are only power-law sensitive to small perturbations. The above findings imply that it is impossible to define Lyapunov exponents for translationally invariant lattices of spins 1/2 even in the macroscopic limit. The power-law sensitivity of spin 1/2 lattices to small perturbations is predicted to be measurable in nuclear magnetic resonance experiments. The above findings are encouraging for the efforts to create quantum simulators.*Nanoscopic friction under electrochemical control*

A. S. de Wijn, A. Fasolino, A. Filippov, and M. Urbakh

Phys. Rev. Lett.**112**, 055502 (2014).

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We propose a theoretical model of friction under electrochemical conditions focusing on the interaction of a force microscope tip with adsorbed polar molecules of which the orientation depends on the applied electric field. We demonstrate that the dependence of friction force on the electric field is determined by the interplay of two channels of energy dissipation: (i) the rotation of dipoles and (ii) slips of the tip over potential barriers. We suggest a promising strategy to achieve a strong dependence of nanoscopic friction on the external field based on the competition between long range electrostatic interactions and short range chemical interactions between tip and adsorbed polar molecules.

supplemental material: pdf movie 1 movie 2*Reconstruction of tip-surface interactions in multimodal intermodulation atomic force microscopy*

Stanislav S. Borysov, Daniel Platz, Astrid S. de Wijn, Daniel Forchheimer, Eric A. Tolen, Alexander V. Balatsky, and David B. Haviland

Phys. Rev. B**88**, 115405 (2013).

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We propose a theoretical framework for reconstructing tip-surface interactions using the intermodulation technique when more than one eigenmode is required to describe the cantilever motion. Two particular cases of bimodal motion are studied numerically: one bending and one torsional mode, and two bending modes. We demonstrate the possibility of accurate reconstruction of a two-dimensional conservative force field for the former case, while dissipative forces are studied for the latter.*Lyapunov instabilities in lattices of interacting classical spins at infinite temperature*

A. S. de Wijn, B. Hess, and B. V. Fine

J. Phys. A: Math. Theor.**46**, 254012 (2013).

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We numerically investigate Lyapunov instabilities for one-, two- and three-dimensional lattices of interacting classical spins at infinite temperature. We obtain the largest Lyapunov exponents for a very large variety of nearest-neighbor spin-spin interactions and complete Lyapunov spectra in a few selected cases. All Lyapunov spectra have weakly convex shapes and do not exhibit the Lyapunov modes observed for gases of hard-core particles. We investigate the dependence of the largest Lyapunov exponents and whole Lyapunov spectra on the lattice size and find that both quickly become size-independent. Finally, we analyze the dependence of the largest Lyapunov exponents on the anisotropy of spin-spin interaction with the particular focus on the difference between bipartite and nonbipartite lattices.*Atomic scale friction of molecular adsorbates during diffusion*

B. A. J. Lechner¹, A. S. de Wijn¹, H. Hedgeland, A. P. Jardine, B. J. Hinch, W. Allison, and J. Ellis [¹joint first authors]

J. Chem. Phys.**138**, 194710 (2013).

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Experimental observations suggest that molecular adsorbates exhibit a larger friction coefficient than atomic species of comparable mass, yet the origin of this increased friction is not well understood. We present a study of the microscopic origins of friction experienced by molecular adsorbates during surface diffusion. Helium spin-echo measurements of a range of five-membered aromatic molecules, cyclopentadienyl (Cp), pyrrole and thiophene, on a copper(111) surface are compared with molecular dynamics simulations of the respective systems. The adsorbates have different chemical interactions with the surface and differ in bonding geometry, yet the measurements show that the friction is greater than 2/ps for all these molecules. We demonstrate that the internal and external degrees of freedom of these adsorbate species are a key factor in the underlying microscopic processes and identify the rotation modes as the ones contributing most to the total measured friction coefficient.*Criticality in Dynamic Arrest: Correspondence between Glasses and Traffic*

A. S. de Wijn, D. M. Miedema, B. Nienhuis, and P. Schall

Phys. Rev. Lett.**109**, 228001 (2012).

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Dynamic arrest is a general phenomenon across a wide range of dynamic systems including glasses, traffic flow, and dynamics in cells, but the universality of dynamic arrest phenomena remains unclear. We connect the emergence of traffic jams in a simple traffic flow model directly to the dynamic slowing down in kinetically constrained models for glasses. In kinetically constrained models, the formation of glass becomes a true (singular) phase transition in the limit T→0. Similarly, using the Nagel-Schreckenberg model to simulate traffic flow, we show that the emergence of jammed traffic acquires the signature of a sharp transition in the deterministic limit p→1, corresponding to overcautious driving. We identify a true dynamic critical point marking the onset of coexistence between free flowing and jammed traffic, and demonstrate its analogy to the kinetically constrained glass models. We find diverging correlations analogous to those at a critical point of thermodynamic phase transitions.*The effect of temperature and velocity on superlubricity*

Joost A. van den Ende, Astrid S. de Wijn, and Annalisa Fasolino

J. Phys.: Condens. Matter**24**, 445009 (2012).

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We study the effects of temperature and sliding velocity on superlubricity in numerical simulations of the Frenkel–Kontorova model. We show that resonant excitations of the phonons in an incommensurate sliding body lead to an effective friction and to thermal equilibrium with energy distributed over the internal degrees of freedom. For finite temperature, the effective friction can be described well in terms of a viscous damping force, with a damping coefficient that emerges naturally from the microscopic dynamics. This damping coefficient is a non-monotonic function of the sliding velocity which peaks around resonant velocities and increases with temperature. At low velocities, it remains finite and nonzero, indicating the preservation of superlubricity in the zero-velocity limit. Finally, we propose experimental systems in which our results could be verified.*(In)commensurability, scaling, and multiplicity of friction in nanocrystals and application to gold nanocrystals on graphite*

Astrid S. de Wijn

Phys. Rev. B**86**, 085429 (2012).

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The scaling of friction with the contact size A and (in)commensurabilty of nanoscopic and mesoscopic crystals on a regular substrate are investigated analytically for triangular nanocrystals on hexagonal substrates. The crystals are assumed to be stiff, but not completely rigid. Commensurate and incommensurate configurations are identified systematically. It is shown that three distinct friction branches coexist, an incommensurate one that does not scale with the contact size (*A*^0) and two commensurate ones which scale differently (with*A*^(1/2) and*A*) and are associated with various combinations of commensurate and incommensurate lattice parameters and orientations. This coexistence is a direct consequence of the two-dimensional nature of the contact layer, and such multiplicity exists in all geometries consisting of regular lattices. To demonstrate this, the procedure is repeated for rectangular geometry. The scaling of irregularly shaped crystals is also considered, and again three branches are found (*A*^(1/4),*A*^(3/4),*A*). Based on the scaling properties, a quantity is defined which can be used to classify commensurability in infinite as well as finite contacts. Finally, the consequences for friction experiments on gold nanocrystals on graphite are discussed.*Largest Lyapunov Exponents for Lattices of Interacting Classical Spins*

A. S. de Wijn, B. Hess, and B. V. Fine

Phys. Rev. Lett.**109**, 034101 (2012).

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We investigate how generic the onset of chaos in interacting many-body classical systems is in the context of lattices of classical spins with nearest-neighbor anisotropic couplings. Seven large lattices in different spatial dimensions were considered. For each lattice, more than 2000 largest Lyapunov exponents for randomly sampled Hamiltonians were numerically computed. Our results strongly suggest the absence of integrable nearest-neighbor Hamiltonians for the infinite lattices except for the trivial Ising case. In the vicinity of the Ising case, the largest Lyapunov exponents exhibit a power-law growth, while further away they become rather weakly sensitive to the Hamiltonian anisotropy. We also provide an analytical derivation of these results.*Viscosity of liquid mixtures: The Vesovic-Wakeham method for chain molecules*

Astrid S. de Wijn, Nicolas Riesco, George Jackson, J. P. Martin Trusler, and Velisa Vesovic

J. Chem. Phys.**136**, 074514 (2012).

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New expressions for the viscosity of liquid mixtures, consisting of chain-like molecules, are derived by means of Enskog-type analysis. The molecules of the fluid are modelled as chains of equally sized, tangentially joined, and rigid spheres. It is assumed that the collision dynamics in such a fluid can be approximated by instantaneous collisions. We determine the molecular size parameters from the viscosity of each pure species and show how the different effective parameters can be evaluated by extending the Vesovic-Wakeham (VW) method. We propose and implement a number of thermodynamically consistent mixing rules, taking advantage of SAFT-type analysis, in order to develop the VW method for chain molecules. The predictions of the VW-chain model have been compared in the first instance with experimental viscosity data for octane-dodecane and methane-decane mixtures, thus, illustrating that the resulting VW–chain model is capable of accurately representing the viscosity of real liquid mixtures.*Spectral Narrowing in Coherent Rayleigh-Brillouin Scattering*

A. Manteghi, N. J. Dam, A. S. Meijer, A. S. de Wijn, and W. van de Water

Phys. Rev. Lett.**107**, 173903 (2011).

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Coherent Rayleigh-Brillouin scattering is a four-wave mixing technique that provides information on various physical properties of the scattering medium in the spectral domain. Being based on density gratings generated by dipole forces, the method requires two pump beams of sufficient spectral width to span the full response bandwidth of the scattering medium. We provide experimental data on the scattered spectrum as a function of the coherence between the two pump beams and derive the corresponding pump beam spectrum. We argue that all experiments on coherent Rayleigh-Brillouin scattering to date, have, in fact, been performed in the incoherent regime and show that orders of magnitude in scattering efficiency are to be expected when the experiments are performed with bandwidth-limited picosecond laser pulses.*Low friction and rotational dynamics of crystalline flakes in solid lubrication*

Astrid S. de Wijn, Annalisa Fasolino, A. E. Filippov, and M. Urbakh

EPL**95**, 66002 (2011).

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Solids at incommensurate contact display low-friction, "superlubric", sliding. For graphene flakes on a graphite surface, superlubric sliding is only temporary due to rotation of the flakes from incommensurate to commensurate contact with the substrate. We examine this rotational channel of friction in a prototype geometry of meso- and macroscopic solid lubrication. By molecular-dynamics simulations and theoretical arguments we find that two surfaces lubricated by mobile, rotating graphene flakes exhibit stable superlubric sliding as for ideally incommensurate contacts also when they are covered by randomly oriented pinned graphene patches. For commensurate surfaces, we find a low-friction state at low temperature where incommensurate states are not destroyed by thermal fluctuations.*A radius of curvature approach to the Kolmogorov–Sinai entropy of dilute hard particles in equilibrium*

Astrid S. de Wijn and Henk van Beijeren

J. Stat. Mech. P08012 (2011).

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We consider the Kolmogorov–Sinai entropy for dilute gases of N hard disks or spheres. This can be expanded in density as*h*ₖₛ ∝*nN*[ln(*na^d*)+*B*+O(*na^d*)+O(1/*N*)] , with a the diameter of the sphere or disk, n the density, and d the dimensionality of the system. We estimate the constant B by solving a linear differential equation for the approximate distribution of eigenvalues of the inverse radius of curvature tensor. We compare the resulting values of B both to previous estimates and to existing simulation results, finding very good agreement with the latter. Also, we compare the distribution of eigenvalues of the inverse radius of curvature tensor resulting from our calculations to new simulation results. For most of the spectrum the agreement between our calculations and the simulations again is very good.*Internal degrees of freedom and transport of benzene on graphite*

Astrid S. de Wijn

Phys. Rev. E**84**, 011610 (2011).

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In this paper, the chaotic internal degrees of freedom of a benzene molecule adsorbed on a graphite substrate, their interplay with thermal noise, and their effects on the diffusion and drift are investigated analytically by making use of the presence of two different time scales as well as by molecular-dynamics simulations. The effects of thermal noise are investigated, and it is found that noise does not significantly alter the dynamics of the internal degrees of freedom yet does affect the friction and diffusion of the center of mass. Qualitative and quantitative theoretical predictions for the friction and diffusion of the molecule on the substrate are made and are compared to molecular-dynamics simulations. Contributions to the friction and diffusion from the finite heat bath as well as the slow dynamics of the center of mass are formally identified. It is shown that the torsion in benzene, which dominates the nonlinear coupling, significantly affects the friction of the molecule on the surface. The results compare favorably with recent results from He-neutron spin echo experiments on this system. Based on the analytical and numerical results, some suggestions are made for experimental conditions under which the effects of internal degrees of freedom might be observable.*Coherent and spontaneous Rayleigh-Brillouin scattering in atomic and molecular gases and gas mixtures*

M. O. Vieitez, E. J. van Duijn, W. Ubachs, B. Witschas, A. Meijer, A. S. de Wijn, N. J. Dam, and W. van de Water

Phys. Rev. A**82**, 043836 (2010).

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We study Rayleigh-Brillouin scattering in gases of N₂, O₂, and SF₆ molecules, Kr atoms, and He-Xe and He-CO₂ mixtures at pressures ranging from 1 to 3 bar and using two different experimental setups. In one setup, we measure spectra of light scattered by thermal density fluctuations (spontaneous Rayleigh-Brillouin scattering); in the second setup density waves are induced in the overlap region of two counterpropagating laser beams (coherent Rayleigh-Brillouin scattering). We compare measured spectra to the Tenti models and to a recent model for mixtures. We find new values of the bulk viscosity, which is a parameter in line-shape models that allows for internal degrees of freedom. Both experiments agree on the value of the bulk viscosity. Our results indicate a need for new line-shape models for mixtures of molecules with internal degrees of freedom.*Coherent Rayleigh–Brillouin scattering measurements of bulk viscosity of polar and nonpolar gases, and kinetic theory*

A. S. Meijer, A. S. de Wijn, M. F. E. Peters, N. J. Dam, and W. van de Water

J. Chem. Phys.**133**, 164315 (2010).

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We investigate coherent Rayleigh–Brillouin spectroscopy as an efficient process to measure the bulk viscosity of gases at gigahertz frequencies. Scattered spectral distributions are measured using a Fizeau spectrometer. We discuss the statistical error due to the fluctuating mode structure of the used pump laser. Experiments were done for both polar and nonpolar gases and the bulk viscosity was obtained from the spectra using the Tenti S6 model. Results are compared to simple classical kinetic models of molecules with internal degrees of freedom. At the extremely high (gigahertz) frequencies of our experiment, most internal vibrational modes remain frozen and the bulk viscosity is dominated by the rotational degrees of freedom. Our measurements show that the molecular dipole moments have unexpectedly little influence on the bulk viscosity at room temperature and moderate pressure.*Equivalence of kinetic-theory and random-matrix approaches to Lyapunov spectra of hard-sphere systems*

Astrid S. de Wijn

Physica D: Nonlinear Phenomena**239**, 1834–1841 (2010).

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In the study of chaotic behaviour of systems of many hard spheres, Lyapunov exponents of small absolute values exhibit interesting characteristics leading to speculations about connections to non-equilibrium statistical mechanics. Analytical approaches to these exponents so far can be divided into two groups, macroscopically oriented approaches, using kinetic theory or hydrodynamics, and more microscopically oriented random-matrix approaches in quasi-one-dimensional systems. In this paper, I present an approach using random matrices and weak-disorder expansion in an arbitrary number of dimensions. Correlations between subsequent collisions of a particle are taken into account. It is shown that the results are identical to those of a previous approach based on an extended Enskog equation. I conclude that each approach has its merits, and provides different insights into the approximations made, which include the Stoßzahlansatz, the continuum limit, and the long wavelength approximation. The comparison also gives insight into possible connections between Lyapunov exponents and fluctuations.*Stability of Low-Friction Surface Sliding of Nanocrystals with Rectangular Symmetry and Application to W on NaF(001)*

Astrid S. de Wijn and Annalisa Fasolino

Tribology Letters**39**, 91-99 (2010).

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We investigate the stability of low-friction sliding of nanocrystal with rectangular atomic arrangement on rectangular lattices, for which analytical results can be obtained. We find that several incommensurate periodic orbits exist and are stable against thermal fluctuations and other perturbations. As incommensurate orientations lead to low corrugation, and therefore low friction, such incommensurate periodic orbits are interesting for the study of nanotribology. The analytical results compare very well with simulations of W nanocrystals on NaF(001). The geometry and high typical corrugation of substrates with square lattices increase the robustness when compared to typical hexagonal lattices, such as graphite.*Stability of superlubric sliding on graphite*

Astrid S. de Wijn, Claudio Fusco, and Annalisa Fasolino

Phys. Rev. E**81**, 046105 (2010).

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Recent atomic force microscope (AFM) experiments have shown that the low-friction sliding of incommensurate graphite flakes on graphite can be destroyed by torque-induced rotations. Here we theoretically investigate the stability of superlubric sliding against rotations of the flake. We find that the occurrence of superlubric motion critically depends on the physical parameters and on the experimental conditions: particular scan lines, thermal fluctuations, and high loading forces can destroy the stability of superlubric orbits. We find that the optimal conditions to achieve superlubric sliding are given by large flakes, low temperature, and low loads, as well as scanning velocities higher than those used in AFM experiments.*Relating chaos to deterministic diffusion of a molecule adsorbed on a surface*

Astrid S. de Wijn and A. Fasolino

J. Phys.: Condens. Matter**21**264002 (2009).

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Chaotic internal degrees of freedom of a molecule can act as noise and affect the diffusion of the molecule on a substrate. A separation of timescales between the fast internal dynamics and the slow motion of the centre of mass on the substrate makes it possible to directly link chaos to diffusion. We discuss the conditions under which this is possible, and show that in simple atomistic models with pair-wise harmonic potentials, strong chaos can arise through the geometry. Using molecular dynamics simulations, we demonstrate that a realistic model of benzene is indeed chaotic, and that the internal chaos affects the diffusion on a graphite substrate.*A kinetic theory description of the viscosity of dense fluids consisting of chain molecules*

Astrid S. de Wijn, Velisa Vesovic, George Jackson, and J. P. Martin Trusler

J. Chem. Phys.**128**, 204901 (2008).

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An expression for the viscosity of a dense fluid is presented that includes the effect of molecular shape. The molecules of the fluid are approximated by chains of equal-sized, tangentially jointed, rigid spheres. It is assumed that the collision dynamics in such a fluid can be approximated by instantaneous collisions between two rigid spheres belonging to different chains. The approach is thus analogous to that of Enskog for a fluid consisting of rigid spheres. The description is developed in terms of two molecular parameters, the diameter σ of the spherical segment and the chain length (number of segments)*m*. It is demonstrated that an analysis of viscosity data of a particular pure fluid alone cannot be used to obtain independently effective values of both σ and*m*. Nevertheless, the chain lengths of*n*-alkanes are determined by assuming that the diameter of each rigid sphere making up the chain can be represented by the diameter of a methane molecule. The effective chain lengths of n-alkanes are found to increase linearly with the number*C*of carbon atoms present. The dependence can be approximated by a simple relationship*m*= 1+(*C*−1)/3. The same relationship was reported within the context of a statistical associating fluid theory equation of state treatment of the fluid, indicating that both the equilibrium thermodynamic properties and viscosity yield the same value for the chain lengths of*n*-alkanes.*Strong-field ionization in time-dependent density functional theory*

A. S. de Wijn, M. Lein, and S. Kümmel

EPL**84**43001 (2008).

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Ionization in a strong laser field is a prime example of non-perturbative, correlated electron dynamics. Simulating such processes on a first-principles basis is a major theoretical challenge. We demonstrate that time-dependent density functional theory (TDDFT) on the basis of a newly developed functional for the correlation potential incorporates the relevant electron interaction effects. The central idea of our approach is to exploit information about exact ground-state correlation in approximate time-dependent calculations. The new functional provides an accurate description of the paradigm problem of the double ionization of the Helium atom, showing that TDDFT is a viable tool for strong-field calculations.*Vertical chaos and horizontal diffusion in the bouncing-ball billiard*

Astrid S. de Wijn and Holger Kantz

Phys. Rev. E**75**, 046214 (2007).

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The bouncing-ball billiard is a low-dimensional system with which transport properties of real physical systems can be studied theoretically. We study the bouncing-ball billiard with nonconvex scatterers and small slopes. We show that between the horizontal and vertical motion there is a separation of time scales, which is controlled by the slope of the billiard. We apply the theory of time-scale separation developed by Kantz et al. Physica D**187**200 (2004) If the vertical motion is chaotic, the horizontal motion is diffusive, but if the vertical motion is (quasi)periodic, there is no diffusion. We confirm the results with numerical simulations. Hence, the order-chaos transition in the vertical degrees of freedom translates into a localization-delocalization transition for the horizontal motion.*Numerical aspects of real-space approaches to strong-field electron dynamics*

Astrid S. de Wijn, Stephan Kümmel, and Manfred Lein

J. Comp. Phys**226**, 89-103 (2007).

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Numerical methods for calculating strong-field, nonperturbative electron dynamics are investigated. Two different quantum–mechanical approaches are discussed: the time-dependent Schrödinger equation and time-dependent density functional theory. We show that when solving the time-dependent Schrödinger equation, small errors in the initial ground-state wave function can be magnified considerably during propagation. A scheme is presented to efficiently obtain the ground state with high accuracy. We further demonstrate that the commonly-used absorbing boundary conditions can severely influence the results. The requirements on the boundary conditions are somewhat less stringent in effective single-particle approaches such as time-dependent density functional theory. We point out how results from accurate wave-function based calculations can be used to improve the density functional description of long-ranged, nonlinear electron dynamics. We present details of a method to reconstruct, numerically, the full, unapproximated, Kohn–Sham potential from the density and current of the exact system.*Lyapunov spectra of billiards with cylindrical scatterers: Comparison with many-particle systems*

Astrid S. de Wijn

Phys. Rev. E**72**, 026216 (2005).

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The dynamics of a system consisting of many spherical hard particles can be described as a single point particle moving in a high-dimensional space with fixed hypercylindrical scatterers with specific orientations and positions. In this paper, the similarities in the Lyapunov exponents are investigated between systems of many particles and high-dimensional billiards. The billiards contain cylindrical scatterers which have isotropically distributed orientations and homogeneously distributed positions. The dynamics of the isotropic billiards are calculated using a Monte Carlo simulation, and a reorthogonalization process is used to find the Lyapunov exponents. The results are compared to numerical results for systems of many hard particles as well as the analytical results for the high-dimensional Lorentz gas. The smallest three-quarters of the positive exponents behave more like the exponents of hard-disk systems than the exponents of the Lorentz gas. This similarity shows that the hard-disk systems may be approximated by a spatially homogeneous and isotropic system of scatterers for a calculation of the smaller Lyapunov exponents, apart from the exponent associated with localization. The method of the partial stretching factor is used to calculate these exponents analytically, with results that compare well with simulation results of hard disks and hard spheres.*Kolmogorov-Sinai entropy for dilute systems of hard particles in equilibrium*

Astrid S. de Wijn

Phys. Rev. E**71**, 046211 (2005).

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In an equilibrium system, the Kolmogorov-Sinai entropy*h*ₖₛ equals the sum of the positive Lyapunov exponents, the exponential rates of divergence of infinitesimal perturbations. Kinetic theory may be used to calculate the Kolmogorov-Sinai entropy for dilute gases of many hard disks or spheres in equilibrium at low number density n. The density expansion of*h*ₖₛ is*N*ν̅*A*[ln*n*+*B*+O(*n*)], where ν̅ is the single-particle collision frequency. Previous calculations of A were successful. Calculations of B, however, were unsatisfactory. In this paper, I show how the probability distribution of the stretching factor can be determined from a nonlinear differential equation by an iterative method. From this the Kolmogorov-Sinai entropy follows as the average of the logarithm of the stretching factor per unit time. I calculate approximate values of B and compare these to results from existing simulations. The agreement is good.*Chaos in systems with many degrees of freedom*

Astrid S. de Wijn

PhD thesis, Utrecht University (2004), ISBN 90-393-3868-X.

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In this thesis I discuss some of the chaotic properties specific to systems of many particles and other systems with many degrees of freedom. A dynamical system is called chaotic if a typical infinite perturbation of initial conditions grows exponentially with time. The chaoticity of a system is characterised by the Lyapunov exponents, which indicate the possible rates at which an infinitesimal perturbation of initial conditions may grow or decrease. A system has as many Lyapunov exponents as it's phase space has dimensions, and so a system with many degrees of freedom has many Lyapunov exponents. A system is chaotic if it has at least one positive exponent. The sum of the positive Lyapunov exponents equals the maximal rate of information increase of the system and is referred to as the Kolmogorov-Sinai entropy. The dynamical properties of a system are thought to have bearing on the non-equilibrium behaviour. This thesis contains calculations of Lyapunov exponents of three different systems; namely, systems consisting of many, freely moving hard disks and hard spheres are discussed, as well as the Lorentz gas, which is a system consisting of fixed spherical scatterers with a point particle moving between and colliding elastically with them, and a similar system in which the scatterers are cylindrical. Chapter 3 contains calculations of the smallest positive and negative exponents of hard disks. These are known from simulations to have interesting behaviour, if the system is large enough. They are explained as belonging to Goldstone modes associated with the symmetries of the tangent space, and the Lyapunov exponents are calculated. This allows their calculation by formulating a generalised Boltzmann equation and solving this perturbatively. There is a slight discrepancy between the predicted values and simulation results, which may be attributed to the neglect of ring-collision terms in the Boltzmann equation. In chapter 4 the Kolmogorov-Sinai entropy is calculated for the same system. It is known to be proportional to the collision frequency multiplied by a factor that equals the logarithm of the density plus a constant. Previous calculations of this constant have produced unsatisfactory results. I discuss the cause of this and how to calculate the constant correctly. A system of freely moving hard particles can be described as a point particle in a high-dimensional space, colliding with cylindrical scatterers with very specific positions and orientations. Because this is very similar to the Lorentz gas, where the scatterers are hyperspheres, in chapter 5 I examine the Lyapunov exponents of the Lorentz gas in an arbitrary number of dimensions. The full spectrum is calculated analytically and the similarities and differences between the high-dimensional Lorentz gas and systems of hard disks or spheres are discussed. In chapter 6 I describe numerical calculations of the Lyapunov exponents of systems consisting of isotropically oriented, homogeneously distributed, cylindrical scatterers. Because of the similarity in the shape of the scatterers, the Lyapunov spectrum of this system is much more similar to the spectrum of hard disks than the spectrum of the Lorentz gas. Using the techniques developed in chapters 4 and 5, I estimate the smallest positive and negative Lyapunov exponents of hard disks and hard spheres which are not associated with Goldstone modes.*Lyapunov spectrum of the many-dimensional dilute random Lorentz gas*

Astrid S. de Wijn and Henk van Beijeren

Phys. Rev. E**70**, 036209 (2004).

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For a better understanding of the chaotic behavior of systems of many moving particles, it is useful to look at other systems with many degrees of freedom. An interesting example is the high-dimensional Lorentz gas, which, just like a system of moving hard spheres, may be interpreted as a dynamical system consisting of a point particle in a high-dimensional phase space, moving among fixed scatterers. In this paper, we calculate the full spectrum of Lyapunov exponents for the dilute random Lorentz gas in an arbitrary number of dimensions. We find that the spectrum becomes flatter with increasing dimensionality. Furthermore, for fixed collision frequency the separation between the largest Lyapunov exponent and the second largest one increases logarithmically with dimensionality, whereas the separations between Lyapunov exponents of given indices not involving the largest one go to fixed limits.*Goldstone modes in Lyapunov spectra of hard sphere systems*

Astrid S. de Wijn and Henk van Beijeren

Phys. Rev. E**70**, 016207 (2004).

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In the study of chaotic behavior, Lyapunov exponents play an important part. In this paper, we demonstrate how the Lyapunov exponents close to zero of a system of many hard spheres can be described as Goldstone modes, by using a Boltzmann type of approach. At low densities, the correct form is found for the wave number dependence of the exponents as well as for the corresponding eigenvectors in tangent space. The predicted values for the Lyapunov exponents belonging to the transverse mode are within a few percent of the values found in recent simulations, the propagation velocity for the longitudinal mode is within 1%, but the value for the Lyapunov exponent belonging to the longitudinal mode deviates from the simulations by 30%. For higher densities, the predicted values deviate more from the values calculated in the simulations. These deviations may be due to contributions from ring collisions and similar terms, which, even at low densities, can contribute to the leading order.