Understanding the static structure factor

The structure factor \(S(q)\) is an important quantity for characterizing disordered systems of particles, like colloids. Its significance comes from the fact that it can be directly measured in a light scattering experiment and is related to other quantities that characterize a system's microscopic arrangement and inter-particle interactions. However, it's difficult to learn about it the context of disorder because it's primarily used in crystallography. Crystals are far from being disordered.

In this post, I'll explore the nature of the static structure factor, which is something of an average structure factor over many microscopic configurations of a material. A dynamic structure factor describes the statistics of a material in time as well as space.

The structure factor of a disordered material can be measured by illuminating the material with a beam of some type of radiation (usually X-rays, neutrons, or light). The choice of radiation depends on the material. It is also important that the material should not scatter the incident beam too strongly because the structure function is typically found in singly-scattered radiation (see the first Born approximation for a discussion about a related concept). If the material is multiply scattering, the information about the material's structure, which is largely carried by the singly scattered light, is washed out.

In a measurement, the sample is usually placed at the center of rotation of a long rotating arm. A detector for the radiation is placed at the opposite end of the arm. The arm is rotated about this axis and the intensity of the scattered radiation as determined by the detector is recorded as a function of the angle. This data set essentially contains the structure factor, but must be transformed and corrected for, accordingly.

First, the structure factor is usefully represented as a function of the scattering wave number, \(q\), and not as a function of the scattering angle. In optics, \(q\) is usually given by

\[ q = \frac{4 \pi n}{\lambda} \sin( \theta / 2) \]

where \(n\) is the refractive index of the background material (usually a solvent like water) and \(\lambda\) is the wavelength of the light. \(\theta\) is the scattering angle.

Additionally, the structure factor must be corrected for a large number of confounding factors, such as scattering from the sample cell and radiation frequency-dependent detectors. A classic paper that details all these corrections to find \(S(q)\) in a neutron scattering experiment is given here.

Once the structure factor is found in an experiment, it may be Fourier transformed numerically to give the radial distribution function, \(g(r)\) (see Ziman for the proper conditions for which this applies) of particles. This function gives the probability of finding a particle at a radial distance from another particle in the system. Many important thermodynamic properties are related to \(g(r)\). Importantly, the pair-wise interaction potential between any two particles is related to \(g(r)\), and the pair-wise interaction determines many macroscopic system properties.

The structure factor as \(q\) (or equivalently the scattering angle) goes to zero is also an important quantity in itself. \(S(0)\) is equal to the macroscopic density fluctuations of particles in the medium (see Ziman, Section 4.4, p. 130). But density fluctuations can be calculated from thermodynamics and leads to the isothermal compressibility of a material.

In the language of optics, which I'll stick to for the rest of this post, the density fluctuations would correspond to large regions of refractive index variations across the sample.

This leads to an interesting problem, the resolution of which reminds me of the fallibility in taking some models too literally: for a homogeneous and non-scattering optical material, like a very nice piece of glass, the value of the density fluctuations in the refractive index are essentially zero (this is true because the disorder in a glass is at a length scale that is much smaller than the wavelength of light). This means that \(S(0) = 0\). At the same time, the scattered intensity in the type of experiment measured above is directly proportional to the structure factor:

\[I(q) \sim S(q).\]

So, if I illuminate a nice piece of glass with a laser beam, and I know that \(S(0)\) is equal to zero, the above expression means that there should be no scattered intensity in the forward direction. But this is a silly conclusion, because when I do this experiment in the lab I see the laser beam shining straight through the glass! In other words, \(I(0)\) is not zero.

The problem is that this expression is for the scattered intensity. In random media, we often talk about the scattered light and the ballistic light. The latter of these two is not scattered but directly transmitted through the material as if the material were not there. So, even though no light is scattered into the forward direction, there is still the ballistic, unscattered beam, that is passing straight through the sample.

Most small angle light scattering experiments measure as close as they can to \(q=0\) and extrapolate to the structure function's limiting value. \(S(0)\) can't actually be measured. But, it's determination is important for materials with significant long-range order, such as those near a phase transition, because the small angles correspond to large distances, due to their inverse Fourier relationship.

One can also engineer a material to not transmit any light into the forward direction. To do this, \(S(q)\) must be zero AND there must be no ballistic light passing through the material. This can be achieved with a crystal that diffracts all the light into directions other than the forward direction, such as a blazed grating.

On a final note, the structure factor can sometimes be related to important material properties beyond the radial distribution function. Ziman says in section 4.1, pg. 126 that the direct correlation function (which measures interactions between pairs of particles) can be derived directly from the structure factor. This correlation function is related to the Percus-Yevick model for liquids.

A beautiful experiment on a nonequilibrium thermodynamic system

I just finished reading an impressive article from 2003 entitled "Observing Brownian motion in vibration-fluidized granular matter". In the article's beginning, the authors established a simple question: can linear response theory describe a nonequilibrium thermodynamic system? This question is important because systems that are not in thermodynamic equilibrium are both difficult to analyze and serve as appropriate models for most natural phenomena. However, very powerful mathematical tools exist for systems that are in equilibrium, so it would be very convenient if their mathematical formalism could be extended to nonequilibrium cases.

In particular, the authors explore whether a torsion oscillator driven by a "heat bath" of randomly vibrating glass beads can be described by the fluctuation-dissipation theorem (FDT). The FDT is arguably the hallmark of linear response theory and describes the return to equilibrium of a many-body system subjected to a small perturbation. (A small perturbation means that the response is linearly proportional to the perturbation.)

A canonical example where the FDT finds use is in describing the motion of ions in a fluid between two plates of a capacitor after a voltage difference has been applied to the plates. Prior to the voltage being applied, the motion of the ions is erratic and Brownian. A long time after the constant voltage is applied, they move with an average velocity that is proportional to the electric field between the plates, the proportionality constant being called the mobility. The FDT describes the very short times immediately after the field is applied. It also links the noise (the random movement of the ions in equilibrium) to the mobility of the ions.

Returning to the article, the authors find that the motion of the oscillator is described by the FDT so long as an "effective" temperature is adopted. Effective temperatures are very appealing as analytical tools for describing nonequilibrium systems because they are very, very simple modifications to the FDT. Simply replace T with T_eff and you're done.

As Cugliandolo points out, to be a good thermodynamic descriptor, an effective temperature should be measurable by a thermometer. I'm not sure what the thermometer is in this system, but I suspect that it's the torsion oscillator itself. Furthermore, she stresses that not all nonequilibrium phenomena are describable by effective temperatures. It seems that one requires coupling between fast processes and slower observables, among other requirements. The beauty of the Nature article is that the authors not only confirmed this point (which seems to currently be an area of contention), but did so convincingly by measuring the relevant quantities directly and under a number of different conditions.

I'm not sure whether the effective temperature is a universal property of nonequilibrium systems; I'm inclined to say it is not. Hopefully more experiments like this one will be done that may further elucidate the current maze of theoretical papers on the topic.

Energies and fields in statistical mechanics

I've recently encountered a failure of understanding in my attempts at a statistical mechanics treatment for systems that are best described by fields rather than energies.

The problem lies with constructing the partition function of a system whose degrees of freedom are field quantities, rather than energetic quantities. In the classical Boltzmann formulation, the partition function is a sum over exponentials whose arguments are functions of the energies of the corresponding microstates:

The partition function ultimately goes into the calculation of how likely it is that a system will be in a given state subject to constraints and may determine important system parameters such as total energy and entropy. 

Importantly, in the equation above, the energy Es of each microstate is a sum of the energies of each degree of freedom in the system. For example, the energy associated with a microstate of a container of gas is the sum of all the energies of the gas molecules in that microstate. However, if the degrees of freedom of the system are best described by field quantities, then the energy of a given microstate becomes nontrivial. Fields are represented by two numbers (real and imaginary parts) and the energy carried by the total field is not simply the sum of the energies corresponding to each degree of freedom taken alone.

This result is commonly known in optics. Irradiance (which intuitively may thought of as how much energy a beam carries) is the time-averaged square of the electric field. Two beams that are coherent and allowed to interfere with one another will produce a different irradiance than two incoherent beams. As another example, electrical engineers know that the power delivered to a device is a product of current and voltage, which are also field quantities. The power thus becomes a function of the phase lag between the two.

The conclusion is that the energy of a microstate now depends on the sum of field quantities, which depends on quantities like their relative phase and state of coherence. This makes the Es appearing in the equation above much more difficult to calculate.

I have just discovered the topic of statistical field theory, which apparently deals with the statistical mechanics of fields, but it looks a bit difficult and may require a large amount of time to grasp its concepts.

The Effective Temperature as a Description for Non-equilibrium Systems

Cugliandolo wrote a survey last year of recent work involving the effective temperature. This is a macroscopic quantity that characterizes a system driven out of equilibrium. The review states that the effective temperature was initially used as an intuitive description of glassy and slowly relaxing systems, but only recently have theoreticians placed it on firmer ground by linking it to the fluctuation-dissipation theorem (FDT).

In practice, the effective temperature is the negative inverse slope of a system's dc susceptibility (a.k.a. its time-integrated impulse response) vs. the time-correlation function of some observable (a.k.a. the description of its thermal fluctuations). Importantly, a departure from the straight line joining the points (1,0) and (0, 1/temperature) on a properly normalized plot may signify a system that is not at equilibrium with its bath. In the paper, Cugliandolo assumes a canonical ensemble, or a system coupled to an equilibriated thermal bath. Also, because it is based on the FDT, this treatment is only valid for extremely small perturbations to the system such that an impulse response is an appropriate description.

Most recent work has been focused on determining whether the effective temperature meets our intuitive requirements for a temperature, like being measurable by a thermometer, and whether it is an appropriate thermodynamic description, i.e. it is a single number that summarizes the state of a large ensemble of random system parts. It seems that very slow relaxations, either forced or natural, must be present for this quantity to be useful.

What is nonequilibrium thermodynamics?

An important topic in thermodynamics and statistical mechanics is the description of systems that are not in equilibrium. It is important because most systems are not in thermodynamic equilibrium and routinely exchange energy and matter with their surroundings. Somewhat surprisingly, the equilibrium thermodynamics of pioneers such as Boltzmann, Gibbs, and Carnot has sufficed for many years, in part, I think, because of its success at guiding the design of heat engines and describing chemical reactions. A theoretical description of nonequilibrium systems, though, still remains a challenge and active area of research.

So what is a nonequilibrium thermodynamic system? I am seeking an intuitive answer, not the unenlightened, yet all-too-common statement "a system that is not in equilibrium."

Unfortunately I cannot find the answer in any one place. I've read several research articles, particularly on active matter, which provide about zero insight to the question. This is probably because journal articles typically assume some familiarity with a topic. Wikipedia's page on nonequilibrium thermodynamics, which I linked to above, seems to provide a good answer in the form of a long description. However, I usually run into problems when trying to identify what about a particular system drives it out of equilibrium or why classical thermodynamics fails to describe the system. For example, on the Wikipedia page noted above under basic concepts, a system between two thermostats at different temperatures is described as a nonequilibrium system, even though basic heat engines from (equilibrium) thermodynamics texts are modeled in this way.

I suspect that a general definition of a nonequilibrium system is elusive because we usually must appeal to specialized statements about the system at hand, such as what spatial and temporal scales we are interested in, and whether the system is in a steady state.

My intuitive understandings about nonequilibrium steady states are given below:

1. Thermal, pressure, or particle density gradients are present, resulting in fluxes.
2. The behavior of the system changes with spatial and temporal scales.
3. There are many ways to drive a system out of equilibrium, so general descriptions must include the nature of the driving processes.
4. Macroscopic properties are not easily defined. Microscopic properties, however, appear easier to describe.

Addendum
The link about nonequilibrium steady states contains one important quality about these systems: work must continuously be performed on a system to maintain its state.

What is a good description for entropy?

"Insight into Entropy," by Daniel F. Styer, is a nice paper that appeared in the American Journal of Physics in 2000. In the paper, he argues for a qualitative explanation of entropy that involves two ideas: disorder and freedom.

Entropy as disorder is a common analogy given to students who are learning about thermodynamics, but Styer provides several arguments for why this qualitative description fails to adequately explain the idea. One such argument involves a glass of shredded and broken ice. Despite the fact that the ice has been shattered into many pieces, the entropy of the bowl of ice is less than that of an identical bowl filled with water. The water may seem to be more ordered because it is homogeneous, but it does not possess a lower entropy.

Styer's idea of entropy as freedom attempts to explain how systems can possess multiple classes of states (commonly known as macrostates) and how entropy limits the microscopic details of each class. In the game of poker, the probability of getting a royal flush is identical to any other five-card selection without replacement. However, the number of configurations that form a royal flush is extremely small, so the entropy of the class of hands forming a royal flush is low. This very low entropy class of poker hands restricts the possible configurations of the microstate—the description of what five cards are in one's hand—and completes the analogy with freedom. High entropy macrostates have greater freedom in choosing their microstate by having a larger number of microstates to choose from; low entropy macrostates (royal flushes, for example) have less freedom.

Styer does propose retaining the "entropy as disorder" description by suggesting that both the freedom and disorder analogies be presented simultaneously to negate any emotions commonly associated with either word. His example of such an analogy goes as "For macrostates of high entropy, the system has the freedom to choose one of a large number of microstates, and the bulk of such microstates are microscopically disordered."

Finally, on a different train of though: teaching ideas by analogy apparently must be done with sensitivity to the common emotions associated with a word. I've never considered this idea before, but will surely be mindful of it in the future.

A point about negative temperatures

Negative temperatures occur when the derivative of entropy with respect to a system's energy is negative. In other words, the entropy decreases with added energy. As Daniel Schroeder points out in his Introduction to Thermal Physics, this may only occur when the total energy that a system may take is limited, such as a two-state paramagnet. In other, more common systems, such as a gas in a container, the total energy that the system may absorb is practically unlimited. This is why negative temperatures are not observed.

I think a lot of confusion in learning thermodynamics is that the common sense notion of temperature is very different from its thermodynamic definition. Other ideas, such as force or work, do not contradict common sense quite so much and are more readily adopted.

Finally, Schroeder references this article for an experiment in which negative temperature was observed.

Osmotic fun

In my readings on how to properly maintain cell cultures underneath a microscope, I came across the entry for osmotic pressure on Wikipedia. In the introduction, a thought experiment is described as such:

In order to visualize this effect, imagine a U shaped clear tube with equal amounts of water on each side, separated by a membrane at its base that is impermeable to the sugar molecules (made from dialysis tubing). Sugar has been added to the water on one side. The height of the water on each side will change proportional to the pressure of the solutions.

Just like my post on a proposed experiment dealing with entropic elasticity in rubber bands, this looks like an interesting and easy demonstration to perform. That is, if I have time to devise the setup (I'm also still working on the entropic elasticity experiment and a demonstration of Schlieren photography for CREOL's student group's outreach program, CAOS).

Fun with thermodynamics

Admittedly, the thought experiment I'm about to tell you about is simply explained by thermodynamics. Despite this, I puzzled over it for a while since it is very counter-intuitive, at least to me.

Suspend a weight from an elastic band so it is stretched (only slightly) beyond its equilibrium point. Now, heat the band with a hair dryer. Does the weight move up or down?

Give up? It moves upward. Do you know why?

I plan to verify this experimentally at some point.