Attraction and Repulsion. in Biaxial Molecular Interactions. Interaction Potential

Transcription

1 Attraction and Repulsion Interaction Potential Slide in Biaxial Molecular Interactions Epifanio G. Virga SMMM Soft Matter Mathematical Modelling Department of Mathematics University of Pavia, Italy Slide The most general quadratic pair-potential was introduced by Straley (974) V = U { ξq q + ( q b + b q ) + b b } U typical interaction energy ξ,, dimensionless parameters Summary alternative representation Molecular Biaxiality Interaction Potential Stability Symmetry V = U { ( + ξ) + (ξ )(m m ) +( + )(e e ) + ( )(e e ) } Romano (4), Longa (5) Molecular Biaxiality Stability Slide molecular tensors We think of biaxial molecules as being described by a biaxial tensor that can be decomposed into two traceless, irreducible orthogonal components. q := m m I m Slide 4 The local stability of the ground state of V, where all three molecular axes are equally oriented, is guaranteed by the following conditions ξ = > + > ξ = >.8 b := e e e e m long molecular axis.6.4. m,e,e tensor axes of any molecular polarizability e e

2 potential extrema symmetric attraction For ξ = the stability region enjoys further properties that can be phrased in terms of the extrema of V For ξ = and = the interaction potential V can be given the symmetric form V attains its absolute minimum at (q,b) = (q,b ) (q,b) and (q,b ) have one and the same eigenframe at all maxima of V V = U (q + b) (q + b ) Slide 5 there are extrema of V at which (q,b) and (q,b ) do not share the same eigenframe, but they are neither minima nor maxima. Gartland (5) Slide 7 symmetric superposition The pair-potential V can uniquely be written as superposition of two orthogonal symmetric components { V = U a + q + q + + a q q } with q + q = maxima chart Precisely, q + = q + + b q = q + b Slide 6 Slide 8 with and ± = ± ( ) + 6 a + = + a = + + strong attraction The inner triangle, where V attains its maxima when all corresponding molecular axes are mutually orthogonal, is interpreted as the region of strongest molecular attraction. 4

3 graphical construction full attraction.5 Both a + and a are positive whenever >. All potentials represented by points within the dispersion parabola are fully attractive, as they are superpositions of attractive, symmetric potentials. mild repulsion Slide 9.5 Slide Either a + or a is negative whenever < within the stability region. The potentials represented by these points are mildly repulsive..5 / All excluded-volume potentials so far studied seem to fall within this category Slide Each point (,) represents an interaction potential which can be written as a linear superposition of two orthogonal, purely quadratic (symmetric) potentials represented by points ( +, + ) and (, ) on the dispersion parabola =. Each point (,) on a straight line through the point (, ) is associated with the same pair ( +, + ) and (, ), but with different coefficients a + and a. Slide Symmetry V -invariant transformations V = U {ξ q q + (q b + q b ) + b b } e = e ξ = = + = + e = e ξ = = = m = m.5.5 / ξ = = = Longa(5), De Matteis (5) 5 6

4 rescaling conjugation charts Provided that ξ, we can set either ξ = or ξ =, depending on whether ξ > or ξ <. Correspondingly, the pairs (, ) and (, ) become = = Slide = = Slide ξ = symmetry properties We denote by τ, τ, and τ the scaled transformations. They enjoy the following properties:.8.6 τ i τ i =.4 Slide 4 τ i τ j τ k = for i j k lines + + =, + =, and = are mutually conjugated parabola = is self-conjugated De Matteis (5) Slide ξ = 7 8

5 Acknowledgements Slide 7 Co-Authors Fulvio Bisi Giovanni De Matteis Georges Durand Chuck Gartland Silvano Romano André M. Sonnet Institutions Royal Society of London Sothampton-Pavia Collaborative Project Italian MIUR PRIN Grant No More information Slide 8 Soft Matter Mathematical Modelling Department of Mathematics University of Pavia, Italy 9