Ballastic and Diffusive motions

The impacts of heat bath on a small system embedded in it are clear on macroscopic and stationary scale, but they remain a challenging subject from the microscopic and dynamic point of view. A simple example is the Brownian motion of a single particle placed in a air or other medium. The equilibrium statistical theory was forwarded by Einstein a century ago, yet what actually take place over very short periods are still under intensive study (see previous entries). Here comes a new report [Science, 332:802(2011)]:

For many years after Einstein's contributions, it was expected that the transition from ballistic to diffusive motion would be quite sharp, corresponding to an exponential decay of the particle's memory of its earlier velocity. However, about 50 years ago, hints from computer simulations and theory started to suggest a more complex scenario. In particular, hydrodynamic vortices in the liquid created by the particle's motion lead to memory effects, and the particle's velocity decays much more slowly than exponentially, exhibiting a t^−3/2 “long-time-tail” (12). Detailed analysis by Huang et al. of data like that shown in the second figure, panel B, where the ballistic-to-diffusive transition spans more than three decades in time, has now provided a thorough verification of the full, complicated hydrodynamic theory (13, 14). Although several previous experiments had observed the breakdown of the simple diffusion picture [e.g., (15)], the present studies extend into the ballistic regime.

What next? Li et al. mention the fascinating prospect of laser cooling a trapped particle to a temperature at which quantization of the energy of this mesoscopic object could be observed (16). Huang et al. suggest extending their measurements to Brownian motion in confined regions and heterogeneous media. Here, understanding the details of prediffusive motion over subnanometer distances could well be relevant to some biological processes, such as the lock-and-key mechanism of enzyme action.

Cellular networks

Cellular networks are common in nature, such as seen in the cells, bubbles, and polycrystals. In all these problems, a central question is, how do the cell boundaries distribute in time in their geometric structures and textures ? A key phenomena is that, at large times, the evolution of the distribution ubiquitously leads to a steady state that follows Boltzman's law. Here is an attempt to address this problem. The approach is traditional and phenomenological.

Mesoscale experiment and simulation permit harvesting information about both geometric features and texture in polycrystals. The grain boundary character distribution (GBCD) is an empirical distribution of the relative length [in two dimensions (2D)] or area (in 3D) of an interface with a given lattice misorientation and normal. During the growth process, an initially random distribution of boundary types reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, it is found that if the given energy density depends only
on lattice misorientation, then the steady-state GBCD and the energy are related by a Boltzmann distribution. This is among the simplest nonrandom distributions, corresponding to independent trials with respect to the energy. In this paper, we derive an entropy-based theory that suggests that the evolution of the GBCD satisfies a Fokker-Planck equation, an equation whose stationary state is a Boltzmann distribution. Cellular structures coarsen according to a local evolution law, curvature-driven growth, and are limited by space-filling constraints. The interaction between the evolution law and the constraints is governed primarily by the force balance at triple junctions, the natural boundary condition associated with curvature-driven growth, and determines a dissipation relation. A simplified coarsening model is introduced that is driven by the boundary conditions and reflects the network level dissipation relation of the grain growth system. It resembles an ensemble of inertia-free spring-mass dash pots. Application is made of the recent characterization of Fokker-Planck kinetics as a gradient flow for a free energy in deriving the theory. The theory predicts the results of large-scale two-dimensional simulations and
is consistent with experiment. [PHYSICAL REVIEW B 83, 134117 (2011)]

Metastable states are important in reality

Metastable states are common in nature: supercooled or superheated liquid are daily examples. These states are not the lowest-energy state, and, by thermodynamic principles, one should not expect them to be long-lived in nature. Indeed, they exist only under very stringent conditions, and very little disturbance can push the system off to a stable state around. Thermodynamically stable states dwell on one of the global minima of the free energy, around which, however, local minima might exist that are separated from the global minima by energy barriers. When a system has not reached the stable states, it will be constantly kicked by its surroundings and eventually transits from where it is in to a stable state after some period (the life of the metastable state). The interesting point is that, the life time can sometimes be very long and real transitions can hardly be seen, a situation similar to ergodicity breaking. For example, diamond has a higher energy than graphite, but it can exist for ever. The reason is because the time to make the transition is cosmologically long, due to the hugely high barrier. Another material is graphene, which should not be stable according to Wagner-Mermin-Honberg theorem. Yet, it was produced in 2004. Glasses are the third examples, in which case, transition has been frustrated by its structure. In the case of supercooled water, the transition is suppressed by distilling process.

Remarks on phase transitions

It is usually said that, phase transitions are associated with singular, non-analytic and discontinuous behaviors of physical functions such as the thermodynamic potential or other non-equilibrium ones. But this is so only for infinite systems, in which any small difference in energy density can be infinitely magnified due the infinity of volumn. In infinite systems, phase transitions are sharp and abrupt, and ergodicity is completely lost when certain state is selected under symmetry breaking. This means infinite life time of the selected state, and thus very sharp transition. For finite systems, which represent the reality, phase transitions are never as sharp as that in infinite systems, since in this case the life time, though long, but finite. Also, no genuine singularities exist. Only strong crossovers can be observed, provided sufficient resolution. A crucial feature of finite systems might be that, configurations with small energy density differences could have very strong mixing and fluctuations and may not be distinguishable for certain resolutions.

Why does it grow in the observed way ?

Crystal growth has been all the time an intriguing but complicated problem. The macroscopic shape depends in a subtle way on several factors, such as the properties of the surroundings. Indebted to the increasing power of computers, physicists are enabled to simulate a real growth. On the other hand, experiments also provide important insights. "Despite the many parameters involved, theorists have predicted that icicles, as well as other natural features like stalactites, should all converge to the same shape as they grow. In a paper appearing in Physical Review E, Antony Chen and Stephen Morris at the University of Toronto, Canada, describe an experimental setup that allows them to image icicles as they grow under controlled conditions, and test these predictions. They mounted a camera through a slot in the side of a refrigerator, within which icicles formed as water dripped from a nozzle and onto a rotating wooden support. Rotating the support helps even out the effects of drafts and temperature gradients." [http://physics.aps.org/synopsis-for/10.1103/PhysRevE.83.026307]

snowflakes

Oh, man, I love this diagram:

http://www.its.caltech.edu/~atomic/publist/rpp5_4_R03.pdf

Reversible order-disorder transition

Usually order-disorder transitions are not exactly reversible. Nevertheless, here [1] comes a report just on this seldom reversibility. It occurs to an organic overlayer on a metal surface upon cooling.
[1] Science 329, 303 (2010)

Inverse melting or disordering, in which the disordered phase forms upon cooling, is known for a few cases in bulk systems under high pressure. We show that inverse disordering also occurs in two dimensions: For a monolayer of 1,4,5,8-naphthalene-tetracarboxylic dianhydride on Ag(111), a completely reversible order-disorder transition appears upon cooling. The transition is driven by strongly anisotropic interactions within the layer versus with the metal substrate. Spectroscopic
data reveal changes in the electronic structure of the system corresponding to a strengthening of the interface bonding at low temperatures. We demonstrate that the delicate, temperature dependent balance between the vertical and lateral forces is the key to understanding this unconventional phase transition.

glass transition dynamics and surface layer mobility in unentangled polystyrene films

Science 25 June 2010: Vol. 328. no. 5986, pp. 1676 - 1679 DOI: 10.1126/science.1184394

Prev | Table of Contents | Next

Reports

Glass Transition Dynamics and Surface Layer Mobility in Unentangled Polystyrene Films

Zhaohui Yang,¹ Yoshihisa Fujii,¹ Fuk Kay Lee,^1,2 Chi-Hang Lam,² Ophelia K. C. Tsui^1,*

Most polymers solidify into a glassy amorphous state, accompanied by a rapid increase in the viscosity when cooled below the glass transition temperature (T_g). There is an ongoing debate on whether the T_g changes with decreasing polymer film thickness and on the origin of the changes. We measured the viscosity of unentangled, short-chain polystyrene films on silicon at different temperatures and found that the transition temperature for the viscosity decreases with decreasing film thickness, consistent with the changes in the T_g of the films observed before. By applying the hydrodynamic equations to the films, the data can be explained by the presence of a highly mobile surface liquid layer, which follows an Arrhenius dynamic and is able to dominate the flow in the thinnest films studied.

¹ Department of Physics, Boston University, Boston, MA 02215, USA.
² Department of Applied Physics, Hong Kong Polytechnic University, Hong Kong.

^* To whom correspondence should be addressed. E-mail: okctsui@bu.edu.

eletron emisson delay experimetally seen

Look at a hydrogen atom made of a proton and an electron. Now imagine a photon impinges upon it, and the electron may be kicked out. The probability for such an event can be easily evaluated with quantum mechanics. And, according to quantum mechanics, the physics here is simple: the moment the photon disappears, the elctron is emitted. However, this is not so for a multi-electron atom. For such atoms, how an electron is thrust out by a photon is still being investigated, due to electron interactions. In this case, a senseful scenario might be like this: the photon is aborbed by the electron cloud of this atom, meanwhile the electron cloud changes to a higher energy state, and later on, this cloud relaxes to a lower energy state in the accompany of electron emission. Thus, there is a delay between the photon vanishing and electron escape. This delay is of the order of a thousandth of femtosecond, difficult to be spotted. A recent study [1] just attacks this barrier.

[1] M. Schultze et al., Science 328, 1658 (2010).

A perspective and a useful list of references on this work is found here.

A grand unification

In science, there is one thing that always makes me curious and awe. Roughly, I'd like to call it 'grand unification', which means a simple concept that connects various seemingly unrelated and independent phenomena which occur in distant disciplines. Such unification corroborates the conviction that, the universe has a common underlying mathematical structure. Here I talk of one more example of this.

This example is about fast symmetry breaking phase transitions (FSPT), in contrast with the usual cases where one talks of phase transitions near equilibrium states which may be described within hydrodynamic jargon. Up to my knowledge, no general effective theory for the moment has been established for FSPT. Despite this, it is possible to find out on some quantities very generic constraints which derive from the basic structure (such as symmetry and causality) of the first principles theory. Suppose a physical system is in equilibrium. Now one changes an experimental knob (e.g., pressure). This system shall then evolve away from its initial equilibrium state according to the corresponding dynamic theory, and shall eventually reach another new equilibrium state. However, what this final state might be should depend on the evolution and how the knobs are changed. An interesting thing is that, during this evolution some topological defects (which are textures that break the overall symmetry) shall form. Generally, it is expected that, as constrained by causality, which means no physical signal can travel faster than a typical velocity special to the system, two textures separated by a typical distance, say x, could not develop significant correlations during the evolution. Assume that a texture be characterized by 'direction vector'. Then, the causality indicates that, two distant textures should choose their directions independently in statistical sense. Therefore, all direction vectors may be found with certain textures because of the symmetry respected by the first principles model. Now, an interesting question is, how does x scale with the phase transition rate ?

In 1976, Kibble made an estimation [1], which is based on a cosmological model. It deals with cosmic strings, which are inherently stable topological objects that are expected to survive to today. Experimental verification of his idea is however quite difficult, since it is about the whole universe, which is so vast. A breakthrough came by Zurek [2], who tried Kibble's idea on Helium, which undergoes super-fluid transition at much low temperatures. And in such system, topological defects, which are known as fluxons and antifluxons by their winding numbers, could form as the transition is passed. Therefore, the helium system poses a remarkable realization of the big universe in this respect. To obtain the relation between x and transition rate, Zurek employed the dynamic Landau's theory, which was supposed to govern the temporary evolution. It turns out that, the relation follows a simple power law. Verification of this law came afterward. The latest one was done on superconductor [3]. For a good review see Ref.[4].

It is exciting to see that, the thing speculated about cosmos has its image on earth. I'd like to pose another question concerning FSPT, how does x scale with the dimension of the system ?

It may be worth pointing out that, as was emphasized by Anderson in his 'more is different' address, to deal with emergent phenomena, it is not enough to know just the first principles model (e.g., BCS model), which are usually not much useful in obtaining intuition and understanding. It is more useful to come up with an effective model (e.g., Landau's model), which concerns the quantities under imminent interest instead of those in the first principles model.

[1]J.Phys.A, 9:1387(1976)
[2]Nature, 317:505(1985)
[3]PRB, 80:180501(R), 2009
[4]Phys.Today, 60:47(2007)