Unified, Ideal FFF

Field-Flow Fractionation (FFF) is an elegant separation technique because it is a simple concept that is extremely broad in practice:

An external transverse force is applied across a channel to a solution of solutes (maybe colloids, maybe polysaccharides maybe cells, whatever). The perpendicular field pushes the ensemble of solutes against the bottom wall but diffusion disperses the solutes and resists the inhomogeneity. The competition between thermal and potential energy results in an exponential concentration distribution in equilibrium. Importantly, each species of solute has a different concentration profile (perhaps because they have different masses if gravity or centerfugation is being used or because they have different charge if an electric field is being used, etc).

If nonuniform flow profile carries these solutes through the channel, their elution times will differ. Since each species has a different concentration distribution, each species samples the velocity distribution with a different weighting and so has a different mean velocity. Solute particles near the wall are subject to slower flow than particles near the centre; therefore, samples with a mean height close to the wall are carried along with a slower average velocity than samples with a larger mean height.

This seems simple enough but interestingly this simple system possesses surprisingly rich elution behaviour.

Hydrodynamic Chromatography

Imagine that the external field is turned off or negligible. In this case the solutes can diffuse across the entire extent of the channel and uniformly sample the parabolic velocity distribution. EXCEPT that there is an excluded region near the walls. If the solutes are hard spheres then there is a stericly excluded film that is exactly equal to their radii. This excluded zone changes with particle size and so the average solute velocity is different. This is the basis of Hydrodynamic Chromatography.

Normal-Mode FFF

If instead we imagine that the field is strong and the particles are small (so small that we might model them as point particles). Then we only need to average the parabolic fluid velocity weighted by an exponential probability distribution. The particle velocity wouldn’t have an explicit size dependence (since it would be modelled as a point particle) but only a external force dependence.

Steric-Mode FFF

As the particles get larger and larger, the point-particle approximation breaks down and one needs to include both the external force on the colloids and the steric force when they interact with the wall.

We were able to combine all three of these “operational-modes” of FFF into a single unified theory by simply writing the retention theory explicitly in terms of particle size. When this is done FFF retention theory can predict all three modes and the transitions between these modes.

Faxén-Mode FFF

We were happy to have a unified theory but found that when we let the particle size approach the channel height the particle was predicted to go 50% faster than the fluid — a highly unlikely thing. So we went back and added a bit more hydrodynamics: We integrated the stress on the particle due to the fluid flow. Because the fluid profile is parabolic, the particle moves slightly slower than the carrier fluid, which of course means that the particles elute later than if this effect were neglected.

We mapped out the transitions between these ideal modes of operation.

Find more details or the citation for this in our article in the Journal of Chromatography A.

Multi-Particle Collision Dynamics

Physics is a science of models. The equations that we write down and the schematics that we draw are necessarily simplifications of reality but they capture some aspect of existence that we consider to be most important. The best models describe many situations while remaining simple. At times, our models can be stunningly simple and yet surprisingly ubiquitous (like the simple harmonic oscillator). At other times our models can be simple in concept yet require computers to solve (like a Lennard-Jones fluid).

Molecular Dynamics is a computational technique that considers spherical fluid molecules that interact through pair-wise Lennard-Jones potentials (check out our review for more details). But calculating the force between each pair of particles for each time step is a costly procedure. At times, more coarse-grained models of fluids will suffice.

One such model is Multi-Particle Collision Dynamics (MPCD), also called Stochastic Rotation Dynamics (SRD) (which, if you keep reading, you’ll realize is a specific MPCD realizations) or Real-coded Lattice Gas (RLG).

In MPCD collisions between fluid particles are replaced by multi-particle collision events that omit the molecular details and eliminate the need to calculate long-ranged forces between the fluid particles. These collisions are not physically insightful but rather are just constructed to conserve mass, momentum, and energy such that the hydrodynamic equations of motion are obeyed on sufficiently long length and time scales.

Details

MPCD simulations occur in two steps. During the first, or streaming step, the particles move ballistically, and their positions are updated in discrete time intervals δt. During this portion of the algorithm the particles are simply an ideal gas. The second, or collision step transfers momentum between particles. The simulation domain is partitioned into cells. The number of particles in each cell may vary from one cell to another but the total number is conserved. Each cell has a centre of mass velocity, which corresponds to the local macroscopic velocity. Again, the collision step is a simple non-physical scheme that is constructed to conserve momentum. There are many different choices for this collision operator:

Stochastic Rotation Dynamics – The collision operator is a rotation through a given angle (which sets the fluid viscosity) about a randomly chosen axis. The randomly chosen axis can either be a random choice between a given set of orthogonal axes (say x,y,z)) or a random axis can be generated for every collision event.

There is a nice version of MPCD that includes an Andersen thermostat as an integrated parts of the simulation technique. I’m particularly fond of this collision operator. It simply generates random velocities from a Maxwell-Boltzmann distribution. It sets the velocity of each particle in a cell to be the cell’s centre of mass velocity plus a random velocity minus the average of all the random velocities generated for that cell.

Another thermostatted MPCD algorithm uses a Langevin thermostat with noise and a friction terms. My code has this version implemented as well but I tend to use the Andersen-MPCD version most.
A few of these collision operators can be extended to conserve angular momentum within collision events but this is usually not an issue for the systems I am concerned with.

A sharp reader may have noticed that Galilean invariance is broken by the discretization of space into cells in the MPCD algorithm. If the mean-free path of the molecules is comparable or larger than the cell dimension then this is a relatively insignificant effect. However, this can be completely remedied by performing the collision operation in a cell grid which is shifted each time step by a random vector. Galilean invariance is then restored.

A wonderful interactive can be found on the Institute of Complex Systems’ webpage.

More information about MPCD and other simulation techniques that we often use in Dr. Gary Slater’s research group can be found in our review of computational methods.

Uses

The MPCD code has proven a useful tool for revealing the behavior of solutes eluting through microfluidic devices while subject to an external, normal field. In this case, MPCD simulations corroborated experimental findings (with Dr. Michel Godin’s research group) and led to a numerical model for the retention of particles as a function of size.

We have also used it to study electrophoresis of polyelectrolytes and polyampoholytes, to look at electro-osmosis around a tethered polymer and to study polymer brushes.

Polyampholytes

An interesting application of our mean field MPCD-MD Debye-Hückel algorithm is the determination of the electrophoretic mobility of net neutral polyampholytes.

One might think that since the chains have a net neutral charge that an electric field can’t induce them to electrophorese but this isn’t true. We investigated net neutral diblock ring polymers cut to create a net neutral linear triblock polyampholyte with a negatively charged centre. The ends were both positively charged but had different lengths.

If the cut makes a diblock polyampholyte then the mobility is indeed zero. But if the cut is at any other position they have a non-zero mobility. The mobility is largest for symmetric triblock polyampholyte (one little detail here – non-monoticity can occur if the field is large enough to fold the chain over and polarize).

Why is there a non-zero electrophoretic mobility? The monomers at the ends matter more than those in the centre. Less counterion condensation occurs at the ends and their effective charge is significantly higher. For this difference to be apparent, the Debye screening length must be significantly smaller than the length scale of the charge heterogeneity of the polyampholyte. This means that the electrophoretic mobility is large for thin Debye layers and disappears as the Debye layer thickens.

Find more details or the citation for this in our article in the PRL.

Brush FFF

A soft-matter technique for nanoengineering surfaces is to graft or otherwise associate polymers to a hard surface. These polymer brushes are fascinating in their own right (you can read about our work on polymers grafted to the inside of capillaries or polyelectrolyte brushes subject to normal electric fields on this site) but they can also be used as components in microfluidic devices.

An idea of ours is to use a polymer brush to improve Field-Flow Fractionation (FFF). We have developed an ideal theory that accounts for retardation of flow within the brush and the free energy cost of solutes to entering the brush (which depends on the difference between particle size and grafting density (or brush height if the particle is large enough)), which suggests that there is a size range in which separation can be greatly improved by the presence of the brush and are currently running MPCD simulations on this system.

More to come on this project once we publish our work.

FFF in Microfluidic Channels

The ideal, unified Field-Flow Fractionation (FFF) analysis, suggested that a novel operational-mode exists at the largest particle sizes. Therefore, our collaborators in Dr. Michel Godin’s laboratory constructed a 18micron microfluidic channel to to test this prediction for polystyrene colloids subject to a gravitational field and to determine how well the retention theory would match experimental data across multiple operational-modes.

Our work has recently been submitted for publication. The manuscript presents experimental data, MPCD simulations and a modified, numerical retention theory to build up a consistent picture of FFF of large colloids eluting through microfluidic devices.

EOF from Single Tethered Polyelectrolytes

You can read in the polyelectrolyte project page of this site or in our review paper about how the counterion sheath surrounding a polyelectrolyte causes hydrodynamic interactions to be screened, which makes polyelectrolytes “free-draining”. In brief, this occurs because the electric field acts on the polymer, which moves through the surrounding fluid shearing it. However, the total force on the counterion sheath is equal and opposite to the force on the polymer and the counterions also shear the fluid. The shear virtually cancels out on length scales longer than the thickness of the counterion sheath (the Debye length).

But imagine what happens if both an electric field and also a mechanical force act on the polyelectrolyte chain. The counterions do not form a connected object and so the mechanical force doesn’t act on them – only the electric field acts on the counterions. Therefore, the shear doesn’t cancel out and there is a net flow of counterions relative to the polyelectrolyte’s reference frame.

When the polymer is subject to a mechanical force (such as a tethering tension) there is (EOF) at the surface of the chain generated by the electrophoresis of the counterion sheath relative to the stationary chain. This is all very well described by the principle of Electro-Hydrodynamic Equivalence Principle.
The Equivalence Principle states that when polyelectrolytes with a thin counterion sheath are acted on by an electric and mechanical force simultaneously, one can replace the electrostatic and hydrodynamic equations of motion with an effective local flow equal to the translational velocity that the polyelectrolyte would have during in free-solution electrophoresis. It can not be overstated how significant this is for electrophoresis of polyelectrolytes. According to the Equivalence Principle, researchers who are able to design devices that apply any mechanical force in concert with an electric field may achieve length-dependent size separation.

It is well established that the Equivalence Principle can be used to replace the complicated electrohydrodynamics with an equivalent incident flow with respect to chain conformation. But what is more difficult to demonstrate is how approximate the equivalence is for the generated electro-osmotic flow (EOF) of the internal and surrounding fluid.

MPCD (with our mean field MPCD-MD Debye-Hückel algorithm) is the ideal computational method for testing the accuracy of the Equivalence Principle from the perspective of the fluid.