Nonlinear frictiophoresis is the unidirectional drift of a particle in a medium caused by periodic driving force with zero mean. The effect is possible due to nonlinear dependence of the friction-drag force on the particle's velocity. It was discovered theoretically., and is mainly known as nonlinear electrofrictiophoresis
. At first glance, a periodic driving force with zero mean is able to entrain a particle into an oscillating movement without unidirectional drift, because integral momentum provided to the particle by the force is zero. The possibility of unidirectional drift can be recognized if one takes into account that the particle itself loses momentum through transferring it further to the medium it moves in/at. If the friction is nonlinear, then it may so happen that the momentum loss during movement in one direction does not equal to that in the opposite direction and this causes unidirectional drift. For this to happen, the driving force time-dependence must be more complicated than it is in a single sinusoidal harmonic.
A simple example - Bingham plastic
Nonlinear friction
The simplest case of friction-velocity dependence law is the
Stokes's one:
where is the friction/drag force applied to a particle moving with velocity in a medium. The friction-velocity law (1) is observed for a slowly moving spherical particle in a Newtonian fluid. It is linear, see Fig. 1, and is not suitable for nonlinear frictiophoresis to take place. The characteristic property of the law (1) is that any, even a very small driving force is able to get particle moving. This is not the case for such media as Bingham plastic. For those media, it is necessary to apply some threshold force, , to get the particle moving. This kind of friction-velocity (dry friction) law has a jump discontinuity at :
It is nonlinear, see Fig. 2, and is used in this example.
Periodic driving force
Let denote the period of driving force. Chose a time
value
such that
and two force values, , such that the following relations are satisfied:
The periodic driving force
used in this example is as follows:
It is clear that, due to (3), has zero mean:
See also Fig. 3.
Unidirectional drift
For the sake of simplicity, we consider here the physical situation when inertia may be neglected. The latter can be achieved if particle's mass is small, velocity is low and friction is high. This conditions have to ensure that , where is the relaxation time. In this situation, the particle driven with force (4) immediately starts moving with constant velocity during interval
and will immediately
stop moving during interval , see Fig. 4.
This results in the
positive mean velocity of unidirectional drift:
Mathematical analysis
Analysis of possibility to get a nonzero drift by periodic force with
zero integral has been made in
. The dimensionless equation of motion for a particle driven by periodic force , , is as follows:
where the friction/drag force
satisfies
the following:
It is proven in
that any solution to (5) settles down onto periodic regime , , which has nonzero mean:
almost certainly, provided
is not antiperiodic.
For , two cases of have been considered explicitly:
1. Saw-shaped driving force, see Fig. 5:
In this case, found in
first order in approximation to , , has the following mean value:
This estimate is made expecting .
2. Two harmonics driving force,
In this case, the first order in approximation
has the following mean value:
This value is maximized in , , keeping constant. Interesting that the drift value depends on and changes its direction twice as spans over the interval . Another type of analysis, based on symmetry breaking suggests as well that a zero mean driving force is able to generate a directed drift.
Applications
In applications, the nature of force in (5), is usually electric, similar to forces acting during standard electrophoresis. The only differences are that the force is periodic and without constant component. For the effect to show up, the dependence of friction/drag force on velocity must be nonlinear. This is the case for numerous substances known as non-Newtonian fluids. Among these are gels, and dilatant fluids, pseudoplastic fluids, liquid crystals.