Diffusion in supercooled liquids via normal mode analysis

The instantaneous normal mode (INM) approach to liquid state dynamics is presented. INM is put in historical context, and the underlying physical ideas, including the importance of the potential energy landscape, are explained. It is shown that INM can be the basis of a general starting point for dynamical calculations in liquids, and the theoretical developments necessary for future development are indicated. New results are given for the general INM formalism, as well as for depolarized light scattering, the “Boson peak” in supercooled liquids, friction on a vibrating bond, nonadiabatic solvent induced transitions of a quantum system coupled to a liquid, and diffusion in supercooled liquids.

The investigations repor ted at this sympos ium are in the process of publication elsewhere [1, 2]; since the la t te r papers are r a the r detailed, we will content ourselves here with a ra ther brief survey of the work, with only a few addit ional results. The t e rm (~ Monte Carlo ,) has come into use to designate numerical methods in which specifically stochastic elements are introduced, in contras t to the whole body of classical numerical techniques which consist of numerical evaluations of complete ly de te rminan t algebraic expressions. The par t icular Monte Carlo me thod used here was devised by METROPOHS et al. [3]; its essential fea ture is t ha t i t produces a Markov chain [4] in which the individual 5 Ia rkov s tates are points in the usual configuration space of s tat is t ical mechanics, for a sys tem of N molecules confined a t a t e m p e r a t u r e T in ~ volume Y. The t ransi t ion probabil i t ies character izing the Markov chain are de termined in such a way tha t the va lue of any funct ion of configuration state, averaged over all s tates developed in the Markov cha.in, approaches the pe t i t canonical ensemble average value of the same function. I t should be noticed t ha t the present me thod does no t consist in an evaluat ion of the configurational phase in tegral b y means of a random, selection of points in configuration space. A me thod of the l a t t e r sort, which m a y be compared with a sequence of independent throws of a die, is indeed

Under normal liquid conditions (at equilibrium or when moderately supercooled), molecular rotational and translational diffusion adhere closely to the simple Stokes-Einstein-Debye hydrodynamic model. Deeply supercooled "fragile" liquids present a striking exception, with translational diffusion alone occurring up to 102 — 103 times "too fast". This anomaly rests upon the statistics of fluctuating local viscosity in the fragile glass-forming medium. The simplest version of this picture identifies large fluidized domains that spontaneously appear and disappear in a dynamically inert matrix. Plausible and self-consistent values characterizing these fluctuations are cited for orthoterphenyl (OTP) at its glass transition.

The problem of the global Lagrangian description of a dynamical system which has a global Hamiltonian description is considered; the motion of a charged particle in the field of a static magnetic monopole is a prototype of such dynamical problems. The symplectic structure on the (cotangent bundle of the) manifold is associated with a closed but not necessarily exact two-form. In conventional language this means that the tensor field inverse to the symplectic metric is divergence free but not necessarily expressible globally as the curl of a vector field; and hence the usual passage to Hamilton's principle and the Lagrangian can be carried out in patches but not necessarily globally. Generalizing techniques developed elsewhere, we show that the fundamental expression for the action integral is like the surface integral of the closed but not necessarily exact antisymmetric tensor field (two-form) rather than like the line integral of a vector field (one-form). This naturally leads to the path-space formalism in which the action functional depends not only on a point in configuration space but also a path from a chosen fixed point terminating in the relevant point in configuration space. We then show how this is a universal unfolding and how the topological obstructions in the configuration-space description are circumvented in the path-space description. The path-space description contains redundancies and they can be reduced; locally the new manifold is obtained by associating an angle of rotation [a U(1) fiber] with each configuration-space point, but globally there would be a nontrivial fiber-bundle structure. These questions naturally carry over to the quantum theory of such systems and it is shown how quantization conditions arise. The structure of the quantum-theory formalism for such systems is analyzed. The considerations are applied to the monopole problem where the enlarged space in which the Lagrangian description is possible is explicitly identified as the manifold of the group SU(2).

We describe a global optimization technique using `basin-hopping' in which the potential energy surface is transformed into a collection of interpenetrating staircases. This method has been designed to exploit the features which recent work suggests must be present in an energy landscape for efficient relaxation to the global minimum. The transformation associates any point in configuration space with the local minimum obtained by a geometry optimization started from that point, effectively removing transition state regions from the problem. However, unlike other methods based upon hypersurface deformation, this transformation does not change the global minimum. The lowest known structures are located for all Lennard-Jones clusters up to 110 atoms, including a number that have never been found before in unbiased searches.

The mechanism of acoustic wave propagation in supercooled liquids is not yet\nfully understood since the vibrational dynamics of supercooled liquids are\nstrongly affected by their amorphous inherent structures. In this paper, the\nacoustic wave propagation in a supercooled model liquid is studied by using\nnormal mode analysis. Due to the highly disordered inherent structure, a single\nacoustic wave is decomposed into many normal modes in broad frequency range.\nThis causes the rapid decay of the acoustic wave and results in anomalous\nwavenumber dependency of the dispersion relation and the rate of attenuation.\n

Robotic manipulators designed for home assistance and new surgical procedures often have significant uncertainty in their actuation due to compliance requirements, cost constraints, and size limits. We introduce a new integrated motion planning and control algorithm for robotic manipulators that makes safety a priority by explicitly considering the probability of unwanted collisions. We first present a fast method for estimating the probability of collision of a motion plan for a robotic manipulator under the assumptions of Gaussian motion and sensing uncertainty. Our approach quickly computes distances to obstacles in the workspace and appropriately transforms this information into the configuration space using a Newton method to estimate the most relevant collision points in configuration space. We then present a sampling-based motion planner based on executing multiple independent rapidly exploring random trees that returns a plan that, under reasonable assumptions, asymptotically converges to a plan that minimizes the estimated collision probability. We demonstrate the speed and safety of our plans in simulation for (1) a 3-D manipulator with 6 DOF, and (2) a concentric tube robot, a tentacle-like robot designed for surgical applications.

Definitions Symmetry operations leave a set of objects in indistinguishable configurations which are said to be equivalent . A set of symmetry operators always contains at least one element, the identity operator E . When operating with E the final configuration is not only indistinguishable from the initial one, it is identical to it. A proper rotation , or simply rotation , is effected by the operator R ( φ n ), which means “carry out a rotation of configuration space with respect to fixed axes through an angle φ about an axis along some unit vector n .” The range of φ is − π φ ≤ π . Configuration space is the three-dimensional (3-D) space ℛ 3 of real vectors in which physical objects such as atoms, molecules, and crystals may be represented. Points in configuration space are described with respect to a system of three space-fixed right-handed orthonormal axes x , y , z , which are collinear with OX, OY, OZ (Figure 2.1(a)). (A right-handed system of axes means that a right-handed screw advancing from the origin along OX would rotate OY into OZ, or advancing along OY would rotate OZ into OX, or along OZ would rotate OX into OY.) The convention in which the axes x , y , z remain fixed, while the whole of configuration space is rotated with respect to fixed axes, is called the active representation .

It is a commonly observed phenomenon that spherical particles with inertia in an incompressible fluid do not behave as ideal tracers. Due to the inertia of the particle, the planar dynamics are described in a four-dimensional phase space and thus can differ considerably from the ideal tracer dynamics. Using finite-time Lyapunov exponents, we compute the sensitivity of the final position of a particle with respect to its initial velocity, relative to the fluid, and thus partition the relative velocity subspace at each point in configuration space. The computations are done at every point in the relative velocity subspace, thus giving a sensitivity field. The Stokes number, being a measure of the independence of the particle from the underlying fluid flow, acts as a parameter in determining the variation in these partitions. We demonstrate how this partition framework can be used to segregate particles by Stokes number in a fluid. The fluid model used for demonstration is a two-dimensional cellular flow.

By combining a well‐known theorem of Longuet‐Higgins on the phase changes of electronic wave functions along loops encircling a conical intersection of potential surfaces, and a recently proven “vertical symmetry theorem” on the relative distributions of point symmetry groups and potential surface critical points in configuration space, a new result is obtained, suitable for the prediction of the presence of a critical point within a domain of configurations.

Computer methods for sampling statistical ensembles generate chains of configurations in which subsequent members differ only slightly. Statistical errors are determined by the number of independent configurations contained in the sample. A quantitative treatment of the persistence of correlation shows how the first two moments of the autocorrelation function of a variablef along the chain are connected with the expected variance of its mean. The variance of the potential energy in the canonical ensemble is shown be to larger than that in the microcanonical one by a factor which is the ratio of the system heat capacity to that of an ideal gas. A comparison has been made of the efficiency of Metropolis Monte Carlo (MC), the usual microcanonical molecular dynamic (MDM) and a modification of molecular dynamics for canonical ensemble sampling (MDC). The analysis is focused on three aspects: the mean square displacement of a representative point in configuration space, the persistence of correlation in the potential energyV and also in a function of interest in free-energy-difference calculations. In MDM simulations of crystalline solids, it was found thatV behaves as an «oscillatory» variable and that the variance of its mean is reduced by antithetic variations of its values.

Given an eigenstate of a time-independent Hamiltonian, an equation is derived which generates the corresponding energy by means of a pointwise application of only the kinetic operator. Satisfaction of the equation at all points in configuration space, for arbitrary values of a scale factor, constitutes a necessary eigenstate condition. Explicit knowledge of the potential operator, other than knowledge of its homogeneity, is not required. In addition, integration of the pointwise equation yields a generalization of the familiar virial theorem. 2 references.

It is shown that it is possible to define ‘true local entropy’ in velocity space, in an approximate version of the two-fluid formulation of quantum theory introduced by the present author in earlier papers. Using this definition, it is then shown that it is possible to define finite forms for total entropy at all points in configuration space. This important step is achieved by the introduction of a ‘responding’ velocity space. The use of a basis system which responds to occupation number density, makes possible a clear separation of the statistics and the dynamics of the underlying quantum process, and also makes possible the unambiguous use of certain divergent and oscillatory integrals.

The dynamic specific heat Cp(ν) and thermal conductivity, κ, of the simple glass-forming liquid di-n-butylphthalate are measured over a wide frequency range (4 mHz–8 kHz) by a nonadiabatic ac technique due to Birge and Nagel. The relaxation frequency obtained from the specific heat is found to be non-Arrhenius down to the lowest temperatures attained in this measurement. The temperature dependence of the relaxation frequency is in close agreement with that obtained from shear modulus and dielectric susceptibility measurements indicating that all motions in the supercooled liquid are strongly coupled to one another. The shape of the relaxation spectrum changes with temperature. The width of the relaxation is similar to that of the dynamic specific heat of other supercooled liquids. The thermal conductivity is frequency independent through the calorimetric glass transition. The static specific heat is reported over a wide range in temperature extending up to 100 K above the melting point. No evidence is found for thermodynamic anomalies accompanying the proposed formation of spatial inhomogeneities in the liquid. The high frequency relaxation shape exhibits a power law which crosses over to a weaker frequency dependence as seen in the dielectric response.

We measured the thermal diffusivity of amorphous solid and supercooled liquid in a Zr41Ti14Cu12Ni10Be23 bulk metallic glass (BMG) and its crystalline counterpart alloy at temperatures ranging from room temperature to 700 K, using a laser flash method. The thermal diffusivity and conductivity of the amorphous solid were weakly temperature dependent and increased with increasing temperature up to the glass transition temperature. The thermal diffusivity of the supercooled liquid was approximately 3.5×10−6 m2 s−1 and quite constant with temperature. The amorphous solid and supercooled liquid of Zr41Ti14Cu12Ni10Be23 BMG showed lower thermal diffusivity and conductivity than the crystalline counterpart in the range from room temperature to crystallization temperature.