Turbulent flows in active matter

The power of toy models and effective theories in describing and understanding emergent phenomena is illustrated by a 2012 study of the turbulence in the fluid flow of swimming bacteria.

Meso-scale turbulence in living fluids

Henricus H. Wensink, Jörn Dunkel, Sebastian Heidenreich, Knut Drescher, Raymond E. Goldstein, Hartmut Löwen, and Julia M. Yeomans

They found that a qualitative and quantitative description of observations of flow patterns, energy spectra, and velocity structure functions was given by a toy model of self-propelled rods (similar to that proposed for flocking of birds) and a minimal continuum model for incompressible flow. For the toy model, they presented a phase diagram (shown below) as a function of the volume fraction of the fluid occupied by rods and the aspect ratio of the rods. There were six distinct phases: dilute state (D), jamming (J), swarming (S), bionematic (B), turbulent (T), and laned (L). The turbulent state occurs for high filling fractions and intermediate aspect ratios, covering typical values for bacteria.

The horizontal axis is the volume fraction, going from 0 to 1.

The figure below compares the experimental data (top right) for the vorticity and the toy model (lower left) and the continuum model (lower right).

Regarding this work, Tom McLeish highlighted the importance of the identification of the relevant mesoscopic scale and the power of toy models and effective theories in the following beautiful commentary taken from his book, The Poetry and Music of Science

“Individual ‘bacteria’ are represented in this simulation by simple rod-like structures that possess just the two properties of mutual repulsion, and the exertion of a constant swimming force along their own length. The rest is simply calculation of the consequences. No more detailed account than this is taken of the complexities within a bacterium. It is somewhat astonishing that a model of the intermediate elemental structures, on such parsimonious lines, is able to reproduce the complex features of the emergent flow structure. 

Impossible to deduce inductively the salient features of the underlying physics from the fluid flow alone—creative imagination and a theoretical scalpel are required: the first to create a sufficient model of reality at the underlying and unseen scale; the second to whittle away at its rough and over-ornate edges until what is left is the streamlined and necessary model. To ‘understand’ the turbulent fluid is to have identified the scale and structure of its origins. To look too closely is to be confused with unnecessary small detail, too coarsely and there is simply an echo of unexplained patterns.”

Confusing bottom-up and top-down approaches to emergence

Due to emergence, reality is stratified. This is reflected in the existence of semi-autonomous scientific disciplines and subdisciplines. A major goal is to understand the relationship between different strata. For example, how is chemistry related to physics? How is genetics related to cell biology?

Before describing two alternative approaches —top-down and bottom-up —I need to point out that in different fields, these terms are used in opposite senses. That can be confusing!

In the latest version of my review article on emergence, I employ the same terminology traditionally used in condensed matter physics, chemistry, and biology. It is also consistent with the use of the term “downward causation” in philosophy. 

Top-down means going from long-distance scales to short-distance scales, i.e., going down in the diagrams shown in the figure above. In contrast, in the quantum field theory of elementary particles and fields (high-energy physics), “top-down” means the opposite, i.e., going from short to long distance length scales. This is because practitioners in that field tend to draw diagrams with high energies at the top and low energies at the bottom.

Bottom-up approaches aim to answer the question: how do properties observed at the macroscale emerge from the microscopic properties of the system? 

History suggests that this question may often be best addressed by identifying the relevant mesoscale at which modularity is observed and connecting the micro- to the meso- and connecting the meso- to the macro. For example, high-energy degrees of freedom can be "integrated out" to give an effective theory for the low-energy degrees of freedom.

Top-down approaches try to surmise something about the microscopic from the macroscopic. This has a long and fruitful history, albeit probably with many false starts that we may not hear about, unless we live through them or read history books. Kepler's snowflakes are an early example. Before people were completely convinced of the existence of atoms, the study of crystal facets and of Brownian motion provided hints of the atomic structure of matter. Planck deduced the existence of the quantum from the thermodynamics of black-body radiation, i.e. from macroscopic properties. Arguably, the first definitive determination of Avogadro's number was from Perrin's experiments on Brownian motion, which involved mesoscopic measurements. Comparing classical statistical mechanics to bulk thermodynamic properties gave hints of an underlying quantum structure to reality. The Sackur-Tetrode equation for the entropy of an ideal gas hinted at the quantisation of phase space. The Gibbs paradox hinted that fundamental particles are indistinguishable. The third law of thermodynamics hints at quantum degeneracy. Pauling’s proposal for the structure of ice was based on macroscopic measurements of its residual entropy. Pasteur deduced the chirality of molecules from observations of the facets in crystals of tartaric acid. Sometimes a “top-down” approach means one that focuses on the meso-scale and ignores microscopic details.

The top-down and bottom-up approaches should not be seen as exclusive or competitive, but rather complementary. Their relative priority or feasibility depends on the system of interest and the amount of information and techniques available to an investigator. Coleman has discussed the interplay of emergence and reductionism in condensed matter. In biology, Mayr advocated a “dual level of analysis” for organisms. In social science, Schelling discussed the interplay of the behaviour of individuals and the properties of social aggregates. In a classic study of complex organisations in business, understanding this interplay was termed differentiation and integration.

I thank Jeremy Schmit for requesting clarification of this terminology.

The role of superconductivity in development of the Standard Model

In 1986, Steven Weinberg published an article, Superconductivity for Particular Theorists, in which he stated

"No one did more than Nambu to bring the idea of spontaneously broken symmetries to the attention of elementary particle physicists. And, as he acknowledged in his ground-breaking 1960 article  "Axial Current Conservation in Weak Interactions'', Nambu was guided in this work by an analogy with the theory of superconductivity,..."

In the 1960 PRL, referenced by Weinberg, Nambu states that in the BCS theory, as refined by Bogoliubov, [and Anderson]

"gauge invariance, the energy gap, and the collective excitations are logically related to each other as was shown by the author. [Y. Nambu, Phys. Rev. 117, 648 (1960)] In the present case we have only to replace them by (chiral) (gamma_5) invariance, baryon mass, and the mesons." 

This connection is worked out explicitly in two papers in 1961. The first is

Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I

Y. Nambu and G. Jona-Lasinio

They acknowledge, 

"that the model treated here is not realistic enough to be compared with the actual nucleon problem. Our purpose was to show that a new possibility exists for field theory to be richer and more complex than has been hitherto envisaged,"

Hence, I consider this to be a toy model for an emergent phenomena.

The model consists of a massless fermion field with a quartic interaction that has chiral invariance, i.e., unchanged by global gauge transformations associated with the gamma_5 matrix. (The Lagrangian is given above.) At the mean-field level, this symmetry is broken. Excitations include massless bosons (associated with the symmetry breaking and similar to those found earlier by Goldstone) and bound fermion pairs. It was conjectured that these could be analogues of mesons and baryons, respectively. The model was proposed before quarks and QCD. Now, the fermion degrees of freedom would be identified with quarks, and the model illustrates the dynamical generation of quark masses. When generalised to include SU(2) or SU(3) symmetry the model is considered to be an effective field theory for QCD, such as chiral effective theory.

Multi-step spin-state transitions in organometallics and frustrated antiferromagnetic Ising models

In previous posts, I discussed how "spin-crossover" material is a misnomer because many of these materials do not undergo crossovers but phase transitions due to collective effects. Furthermore, they exhibit rich behaviours, including hysteresis, incomplete transitions, and multiple-step transitions. Ising models can capture some of these effects.

Here, I discuss how an antiferromagnetic Ising model with frustrated interactions can give multi-step transitions. This has been studied previously by Paez-Espejo, Sy and Boukheddaden, and my UQ colleagues Jace Cruddas and Ben Powell. In their case, they start with a lattice "balls and spring" model and derive Ising models with an infinite-range ferromagnetic interaction and short-range antiferromagnetic interactions. They show that when the range of these interactions (and thus the frustration) is increased, more and more steps are observed.

Here, I do something simpler to illustrate some key physics and some subtleties and cautions.

fcc lattice

Consider the antiferromagnetic Ising model on the face-centred-cubic lattice in a magnetic field. 

[Historical trivia: the model was studied by William Shockley back in 1938, in the context of understanding alloys of gold and copper.]

The picture below shows a tetrahedron of four nearest neighbours in the fcc lattice.

Even with just nearest-neighbour interactions, the lattice is frustrated. On a tetrahedron, you cannot satisfy all six AFM interactions. Four bonds are satisfied, and two are unsatisfied.

The phase diagram of the model was studied using Monte Carlo by Kammerer et al. in 1996. It is shown above as a function of temperature and field. All the transition lines are (weakly) first-order.

The AB phase has AFM order within the [100] planes. It has an equal number of up and down spins.

The A3B phase has alternating FM and AFM order between neighbouring planes. Thus, 3/4 of the spins have the same direction as the magnetic field.

The stability of these ordered states is subtle. At zero temperature, both the AB and A3B states are massively degenerate. For a system of 4 x L^3 spins, there are 3 x 2^2L AB states, and 6 x 2^L   A3B states. At finite temperature, the system exhibits “order by disorder”.

On the phase diagram, I have shown three straight lines (blue, red, and dashed-black) representing a temperature sweep for three different spin-crossover systems. The "field" is given by h=1/2(Delta H - T Delta S). In the lower panel, I have shown the temperature dependence of the High Spin (HS) population for the three different systems. For clarity, I have not shown the effects of the hysteresis associated with the first-order transitions.

If Delta H is smaller than the values shown in the figure, then at low temperatures, the spin-crossover system will never reach the complete low-spin state.

Main points.

Multiple steps are possible even in a simple model. This is because frustration stabilises new phases in a magnetic field. Similar phenomena occur in other frustrated models, such as the triangular lattice, the J1-J2 model on a chain or a square lattice.

The number of steps may change depending on Delta S. This is because a temperature sweep traverses the field-temperature phase diagram asymmetrically.

Caution.

Fluctuations matter.

The mean-field theory phase diagram was studied by Beath and Ryan. Their phase diagram is below. Clearly, there are significant qualitative differences, particularly in the stability of the A3B phase.

The transition temperature at zero field is 3.5 J, compared to the value of 1.4J from Monte Carlo.

Monte Carlo simulations may be fraught.

Because of the many competing ordered states associated with frustration, Kammerer et al. note that “in a Monte Carlo simulation one needs unusually large systems in order observe the correct asymptotic behaviour, and that the effect gets worse with decreasing temperature because of the proximity of the phase transition to the less ordered phase at T=0”. 

Open questions.

The example above hints at what the essential physics may be how frustrated Ising models may capture it. However, to definitively establish the connection with real materials, several issues need to be resolved.

1. Show definitively how elastic interactions can produce the necessary Ising interactions. In particular, derive a formula for the interactions in terms of elastic properties of the high-spin and low-spin states. How do their structural differences, and the associated bond stretches or compressions, affect the elastic energy? What is the magnitude, range, and direction of the interactions?

[n.b. Different authors have different expressions for the Ising interactions for a range of toy models, using a range of approximations. It also needs to be done for a general atomic "force field".]

2. For specific materials, calculate the Ising interactions from a DFT-based method. Then show that the relevant Ising model does produce the steps and hysteresis observed experimentally.

"Ferromagnetic" Ising models for spin-state transitions in organometallics

In recent posts, I discussed how "spin crossover" is a misnomer for the plethora of organometallic compounds that undergo spin-state phase transitions (abrupt, first-order, hysteretic, multi-step,...)

In theory development, it is best to start with the simplest possible model and then gradually add new features to the model until (hopefully) arriving at a minimal model that can describe (almost) everything. Hence, I described how the two-state model can describe spin crossover. An Ising "spin" has values of +1 or -1, corresponding to high spin (HS) and low spin (LS) states. The "magnetic" field is half of the difference in Gibbs free energy between the two states. 

The model predicts equal numbers of HS and LS at a temperature

The two-state model is modified by adding Ising-type interactions between the “spins” (molecules). The Hamiltonian is then of the form

 The temperature dependence in the field arises because this is an effective Hamiltonian.

The Ising-type interactions are due to elastic effects. The spin-state transition in the iron atom leads to changes in the Fe-N bond lengths (an increase of about 10 per cent in going from LS to HS), changing the size of the metal-ligand (ML6 ) complex. This affects the interactions (ionic, pi-pi, H-bond, van der Waals) between the complexes. The volume of the ML6 complex changes by about 30 per cent, but typically the volume of the crystal unit cell changes by only a few per cent. The associated relaxation energies are related to the J’s. Calculating them is non-trivial and will be discussed elsewhere. There are many competing and contradictory models for the elastic origin of the J’s.

In this post, I only consider nearest-neighbour ferromagnetic interactions. Later, I will consider antiferromagnetic interactions and further-neighbour interactions that lead to frustration. 

Slichter-Drickamer model

This model was introduced in 1972 is beloved by experimentalists, especially chemists, because it provides a simple analytic formula that can be fit to experimental data.

The system is assumed to be a thermodynamic mixture of HS and LS. x=n_HS(T) is the fraction of HS. The Gibbs free energy is given by

This is minimised as a function of x to give the temperature dependence of the HS population.

The model is a natural extension of the two-state model, by adding a single parameter, Gamma, which is sometimes referred to as the cooperativity parameter.

The model is equivalent to the mean-field treatment of a ferromagnetic Ising model, with Gamma=2zJ, where z is the number of nearest neighbours. Some chemists do not seem to be aware of this connection to Ising. The model is also identical to the theory of binary mixtures, such as discussed in Thermal Physics by Schroeder, Section 5.4.

Successes of the model.

good quantitative agreement with experiments on many materials.

a first-order transition with hysteresis for T_1/2 < Tc =z J.

a steep and continuous (abrupt) transition for T_1/2 slightly larger than Tc.

Values of Gamma are in the range 1-10 kJ/mol. Corresponding vaules of J are in the range 10-200 K, depending on what value of z is assumed.

Weaknesses of the model.

It cannot explain multi-step transitions.

Mean-field theory is quantitatively, and sometimes qualitatively, wrong, especially in one and two dimensions.

The description of hysteresis is an artefact of the mean-field theory, as discussed below.

Figure. Phase diagram of a ferromagnetic Ising model in a magnetic field. (Fig. 8.7.1, Chaikin and Lubensky). Vertical axis is the magnetic field, and the horizontal axis is temperature. Tc denotes the critical temperature, and the double-line denotes a first-order phase transition between paramagnetic phases where the magnetisation is parallel to the direction of the applied field.

Curves show the free energy as a function of the order parameter (magnetisation) in mean-field theory. The dashed lines are the lines of metastability deduced from these free-energy curves. Inside these lines, the free energy has two minima: the equilibrium one and a metastable one. The lines are sometimes referred to as spinodal curves.

The consequences of the metastability for a field sweep at constant temperature are shown in the Figure below, taken from Banerjee and Bar.

How does this relate to thermally induced spin-state transitions?

Consider the phase diagram shown above of a ferromagnetic Ising model in a magnetic field. The red and blue lines correspond to temperature scans for two SCO materials that have different values of the parameters Delta H and DeltaS.

The occurrence of qualitatively different behaviour is determined by where the lines intercept the temperature and field axes, i.e. the values of T_1/2 /J and Delta H/J. If the former is larger than Tc/J, as it is for the blue line, then no phase transition is observed. 

The parameter Delta H/J determines whether at low temperatures, the complete HS state is formed.

The figure below is a sketch of the temperature dependence of the population of HS for the red and blue cases.

Note that because of the non-zero slope of the red line, the temperature  T_1/2 is not the average of the temperatures at which the transition occurs on the up and down temperature sweeps.

Deconstructing hysteresis.

The physical picture above of metastability is an artefact (oversimplification) of mean-field theory. It predicts that an infinite system would take an infinite time to reach the equilibrium state from the metastable state.

(Aside: In the context of the corresponding discrete-choice models in economics, this has important and amusing consequences, as discussed by Bouchaud.)

In reality, the transition to the equilibrium state can occur via nucleation of finite domains or in some regimes via a perturbation with a non-zero wavevector. This is discussed in detail by Chaikin and Lubensky, chapter 4.

The consequence of this “metastability” for a first-order transition in an SCO system is that the width of the hysteresis region (in temperature) may depend on the rate at which the temperature is swept and whether the system is allowed to relax before the magnetisation (fraction of HS) is measured at any temperature. Emprically, this is observed and has been highlighted by Brooker, albeit without reference to the theoretical subtleties I am highlighting here. She points out that up to 2014, chemists seemed to have been oblivious to these issues and reported results without testing whether their observations depended on the sweep rate or whether they waited for relaxation.

(Aside. The dynamics are different for conserved and non-conserved order parameters. In a binary liquid mixture, the order parameter is conserved, i.e., the number of A and B atoms is fixed. In an SCO material, the number of HS and LS is not conserved.)

In the next post, I will discuss how an antiferromagnetic Ising model can give a two-step transition and models with frustrated interactions can give multi-step transitions.