Report Francesco Marino Contents 1 Introduction 1 2 Theoretical backgroung 2 2.1 Quasi-elastic neutron scattering . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Simulation details 6 4 Results and discussion 7 5 Summary and conclusion 9 References 9 Abstract This report details the work I have undertaken at Institut Laue Langevin (ILL) as a participant of the XRay and Neutron Science Summer Programme 2017 over the period September 4 - 29. I worked under the supervision of Peter Fouquet, helping him set up a molecu- lar dynamics (MD) simulation of the diffusion of ammonia molecules on a graphite substrate. MD results show that adsorbate molecules follow a very fast jump diffusion mech- anism, characterized by a diffusion coefficient of the order of 10 − 8 m 2 /s , in agreement with time-of-flight neutron spectroscopy measurements. These encouraging preliminary studies suggest tha MD can be a valid tool for future investigation of surface diffusion. 1 Introduction The diffusion of ammonia on graphite is particularly interesting for potential applications of graphene and graphitic material surfaces. These include the use of graphene as sensor for ammonia groups and the n-doping of graphene by means of ammonia gas. 1
In this project, the properties of the adsorption of ammonia on an exfoliated graphite substrate were studied combining experimental and computational methods [2]. Experi- ments were carried out with neutron time-of-flight (TOF) technique at IN6 beamline at Institut Laue Langevin (ILL). Quasi-elastic neutron scattering (QENS) is well suited for studying fast diffusion processes as it can resolve the motion of atoms and molecules on a timescale from ns to ps. Furthermore, models used for describing different types of diffusion make distinct predictions on the properties of the dynamical scattering function, namely about the shape and momentum transfer dependence of the quasi-elastic broadening; this makes it possible to compare theory and QENS data. As for the computational side, we set up a molecular dynamics (MD) simulation. MD allows to follow the evolution of the system in time and is therefore useful to have a more intuitive feeling of the mechanism that rules the diffusion process. Moreover, since MD is an ab initio method, no additional assumptions are needed: the observed behaviour is the consequence of the interaction between atomic constituents. COMPASS II was chosen as a force field for our simulations. Whilst it has been extensively validated for bulk studies, there is few evidence concerning its accuracy as far as surface processes are concerned. Thus, one further motivation for our work on MD is carrying a preliminary test on whether this kind of simulations may be reliable tools for studying adsorption. Experimental and molecular dynamics results show a good agreement, at least in the general features of the adsorption mechanism. In fact, they both suggest that ammonia molecules follow a very fast jump motion. 2 Theoretical backgroung 2.1 Quasi-elastic neutron scattering The time-of-flight (TOF) spectroscopy technique is used to measure the dynamic structure factor, or dynamic scattering function, S ( Q , ∆ E ) of neutrons scattered by a sample. ∆ E = E f − E i and Q = k f − k i are the energy transfer and the momentum transfer, respectively. The basic experimental procedure is the following (Figure 1). A beam of neutrons is sent through a chopper or a velocity selector. Only neutrons with a defined velocity are allowed to pass: as a result, a monochromatic beam of known wavelength and kinetic energy is produced. The selected neutrons are then directed against the sample, where they are scattered by nuclei. The interaction leads to a variation in the energy as well as the momentum of probes. Scattered neutrons are collected by the detectors, which cover a wide angle around the sample. The time interval between scattering and detection is recorded. As the flight distance is known, this allows to calculate the kinetic energy of scattered neutrons. Thus, the energy transfer ∆ E can be easily obtained. 2
The momentum transfer Q , instead, is related to the scattering angle by simple formula: Q = 4 π λ sin( θ ), where λ is the wavelength and θ is the angle between the incident and the scattered beams. Figure 1: Schematic representation of IN6 beamline at ILL, where TOF measurements have been performed In our experiment, TOF was used to carry out quasi-elastic neutron scattering (QENS) measurements. QENS [3] is characterized by small energy transfers (of the order of neV to meV ) between neutrons and target particles, leading to a broadening of the elastic peak in the energy distribution. This so called quasi-elastic broadening is the signature of diffusive processes. Theoretical models describing diffusion mechanisms make prediction on the quasi-elastic broadening, namely about the shape of the quasi-elastic profile and the dependence of its width on the momentum transfer . We will refer to the FWHM of the energy distribution as Γ( Q ). Generally, there are three standard cases for diffusion: ballistic diffusion, continuous diffusion and jump diffusion. Ballistic diffusion describes processes when there is no sig- nificant friction between adsorbate and substrate particles. The shape of the quasi-elastic profile as a function of energy transfer is a Gaussian curve. The quasi-elastic broadening is proportional to the magnitude of the momentum transfer: � 2 ln(2) k B T Γ( Q ) = 2 � Q (1) m Continuous diffusion is described by the Brownian model. Adsorbate particles are 3
assumed to follow a random walk, i.e. molecules move uniformely and, when two of them collide, their velocities change randomly. Brownian motion is charaterized by a high friction coefficient between substrate and adsorbates. The quasi elastic profile, in this case, is a Lorentzian distribution, whose FWHM is proportional to Q 2 . Γ( Q ) = 2 � DQ 2 (2) Here, D is the diffusion coefficient, related to the temperature T and the friction coef- fcient η by the Einstein’s relation: D = k B T (3) mη Finally, jump diffusion model describes processes where there is little friction, but activation energy is high. Adsorbate molecules follow a stepwise motion: they spend on average a time τ on an equilibrium position, then make an istantaneous jump to another (random) site, as a consequence of thermal fluctuations. The quasi-elastic profile is still Lorentzian-shaped, but the dependence on the momentum transfer is different. In fact, if we call l the jump vector connecting two adiacent sites, the expression of the quasi-elastic broadening is a sinusoidal function of Q · l , namely: sin 2 ( l · Q Γ( Q ) = � � ) (4) τ 2 l Figure 2: 2D plot of the dynamic scattering function S ( Q, δE ) extracted from the neutron TOF data relative to a 0.9 ML ammonia coverage sample at 94 K. 4
2.2 Molecular dynamics We briefly present the fundamental ideas of molecular dynamics. For a more detailed introduction to the subject, see [5] and [4]. For COMPASS force field, read [1], [6] and [7]. Molecular dynamics (MD) is a simulation method in which a system made of atoms and molecules is allowed to evolve from an initial configuration according to the laws of classical physics. Positions and velocities of the constituents of the system at each time step are calculated numerically solving Newton’s equation: m i ¨ r i = −∇ i V ( r 1 , ..., r N ) (5) Here, r i denotes the position vector of the i-th particle of the system, V is the potential energy and ∇ i = ( ∂ ∂ ∂ ∂ x i , ∂ y i , ∂ z i ). The potential energy function has to be carefully designed in order to achieve a realistic output. In COMPASS II force field, V is splitted in bonded, non bonded and cross- terms: V = V bond + V non − bond + V cross (6) Bonded terms deal with strong chemical bonds (covalent, ionic). Bonds are modelled as springs (with additional degrees of freedom) connecting atoms. V bond is the sum of polinomial (2nd to 4th order) and sinusoidal contributions used to describe stretch, bending and torsion of the bonds as perturbations around equilibrium. Non-bonded terms, instead, concern weaker inter-atomic pair interactions, i.e. electro- static and van der Waals interactions: V non − bond = V vdW + V elec (7) As for the former, the Coulomb potential is used: q i q j � V elec = (8) r ij i>j The sum runs over all pairs of atoms in the system. q i and q j denote the partial electric charges on the i-th and j-th atoms and r ij is the distance between nuclei. Van der Waals forces are described by the so called Lennard-Jones pair potential, whose general expression is: ǫ ( A ij − B ij � V vdW = ) ( n > 6) (9) r n r 6 ij ij i>j While the term proportional to − r − 6 is motivated by theoretical calculations, the ex- ij ponent n and the values of the coefficients are empirically chosen. In COMPASS II, n = 9. 5
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