Lennard-Jones potential diagram

Figure 4.4 shows a section of a perfect single crystal surface [such as Pt(lll)] which is approached by a diatomic molecule (say 02) undergoing dissociative chemisorption. The progress of this process is illustrated by a contour plot of the energetics as a function of the distance x of the molecule from the surface and of the separation y between the two atoms, together with the well-known one-dimensional Lennard-Jones potential diagram. (The molecular axis is assumed to be parallel to the surface... 

The Lennard-Jones potential diagram reproduced in Fig. 1.4 illustrates the progress of dissociative adsorption only in a simplified one-dimensional manner. In reality, this process is... 

The interaction of hydrogen (deuterium) molecules with a transition metal surface c an be conveniently described in terms of a Lennard--Jones potential energy diagram (Pig. 1). It cxxislsts of a shallcw molecular precursor well followed by a deep atomic chemisorption potential. Depending on their relative depths and positions the wells m or may not be separated by an activation energy barrier E as schematically Indicated by the dotted cur e in Fig. 1. 

The classical approach for discussing adsorption states was through Lennard-Jones potential energy diagrams and for their desorption through the application of transition state theory. The essential assumption of this is that the reactants follow a potential energy surface where the products are separated from the reactants by a transition state. The concentration of the activated complex associated with the transition state is assumed to be in equilibrium... 

Lennard-Jones potential energy diagram, JABLONSKI DIAGRAM Leucine,... 

Fig. 12. Equation of state (a) and phase diagram (b) of a bead-spring polymer model. Monomers interact via a truncated and shifted Lennard-Jones potential as in Fig. 6 and neighboring monomers along a molecule are bonded together via a finitely extensible non-linear elastic potential of the form iJpENE(r) = — 15e(iJo/

Figure 25.3 Lennard-jones potential energy diagram of a Hj molecule interacting with an active (full line) and an inactive metal surface (dotted line) as a schematic one-dimensional description of the activated (non-activated) hydrogen adsorption. The dashed line indicates the potential energy U(z) for a pre-dis-sociated Hj molecule (shifted by the dissociation energy E, , with respect to energy zero)...

Figure 3.2. Lennard-Jones potential energy diagram for the interaction of hydrogen with the surface of a metal of Groups 8-10 (see text for description). The lower part of the diagram shows possible configuration at three points in the chemisorption process.

Selected theoretical values of y and a Broughton and Gilmer [ 14] used molecular dynamics to calculate the surface energy and surface stress for a unary system with a P-T diagram of the type in Figure 1.4 for which the atoms were assumed to interact according to a Lennard-Jones potential. The solid phase was assumed to take the fee structure. The results from this simulation for temperatures and pressures near... 

Obviously anisotropic systems are not only more complicated but the addition of other degrees of freedom result in a richer phase-diagram. For instance diatomic particles interacting through a Lennard-Jones potential (Kriebel Winkelmann, 1996 Sumi et al.. 

Figure 3.3. Potential energy diagrams for dissociative adsorption of a diatomic molecule approaching a surface (a) Two-dimensional contour plot with variation of x and y. (b) One-dimensional plot, showing the potentials for the interaction of Aj and 2A, respectively, with the surface (Lennard-Jones potential), (c) Variation of the potential along the reaction coordinate (see a).

Fig. 7.11 Phase diagrams for a symmetrical off-lattice mixture with Na = Ng = N = 20, where both components are modeled as bead-rod chains, and all nonbonded beads interact with standard Lennard-Jones potentials which are truncated at 2.5b), and data are shown for a bulk system (full dots) and thin films with repulsive walls for thickness IO.Sct (squares) and 5cr (triangles). Lines represent a fit according to xi - xic oc (T- (From Kumar et...

Figure 6 Electrostatic and steric fields in CoMFA studies are calculated from Coulomb and Lennard-Jones potentials, respectively. Because of the steep slopes of these functions, cut-off values define the upper limits (and lower limits of the Coulomb potential not shown in the diagram) of individual grid values (redrawn from Ref. 3 with kind permission from Spektrum Akademischer Verlag, Heidelbeig)...

In atomic and molecular systems the range and strength (inverse temperature) of the attraction is set by quantum mechanics. In many cases a Lennard-Jones potential describes the pair interaction quite well [7]. The phase behavior of atomic and molecular systems is often represented in a pressure versus temperature diagram. Quite often the distance between triple point (tp) and critical point (cp) is significant so that there is a wide region where a liquid exists the liquid window is then wide. 

The density functional approach has also been used to study capillary condensation in slit-like pores [148,149]. As in the previous section, a simple model of the Lennard-Jones associating fluid with a single associative site is considered. All the parameters of the interparticle potentials are chosen the same as in the previous section. Our attention has been focused on the influence of association on capillary condensation and the evaluation of the phase diagram [42]. 

Fig. 7. Potential energy diagram for van der Waals (aa) and chemisorbed hydrogen (66) [J. E. Lennard Jones, Trana. Faraday Soc. 28, 333 (1932)]. aQ and A5 represent heats of chemisorption and van der Waals adsorption. aF represents activation energy for chemisorption.

The use of Eq. (8b) is also intrinsic for the BOC-MP treatment of AB dissociation. Consider the activation barrier AE (Bgfor dissociation ABg - As + Bs when AB approaches a surface from the gas phase. The traditional one-dimensional Lennard-Jones (LJ) potential diagram refers A B,gto the intersection point of the molecular AB and atomic A + B curves, as shown in Fig. 2a.Thus, the transition state (TS) where the configuration switch occurs corresponds to jcXI = 0, reducing Eq. (8b) to... 

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