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:
 formula_1
where formula_2 is the friction/drag force applied to a particle moving with velocity formula_3 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, formula_4, to get the particle moving. This kind of friction-velocity (dry friction) law has a jump discontinuity at formula_5:
 formula_6
It is nonlinear, see Fig. 2, and is used in this example.
Periodic driving force.
Let formula_7 denote the period of driving force. Chose a time
value formula_8
such that formula_9
and two force values, formula_10, formula_11
such that the following relations are
satisfied:
formula_12
formula_13
The periodic driving force formula_14
used in this example is as follows:
formula_15
It is clear that, due to (3), formula_14 has zero mean:
formula_17
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 formula_18,
where formula_19 is the relaxation time. In this situation, the particle driven with force (4) immediately starts moving with constant velocity 
formula_20 during interval
formula_21
and will immediately
stop moving during interval formula_22, see Fig. 4.
This results in the
positive mean velocity of unidirectional drift:
formula_23
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
formula_14, formula_25, formula_26
is as follows:
formula_27
where the friction/drag force
formula_28 satisfies
the following:
formula_29
It is proven in 
that any solution to (5)
settles down onto periodic regime
formula_30, formula_31, which has nonzero mean:
formula_32
almost certainly, provided formula_14
is not antiperiodic.
For formula_34, two cases of formula_14 have been considered explicitly:
1. Saw-shaped driving force, see Fig. 5:
formula_36
In this case, found in 
first order in formula_37 approximation to formula_30,
formula_39, has the following mean value:
formula_40
This estimate is made expecting formula_41.
2. Two harmonics driving force,
formula_42
In this case, the first order in formula_37 approximation
has the following mean value:
formula_44
This value is maximized in formula_45, formula_46, keeping formula_47 constant. Interesting that the drift value depends on formula_45 and changes its direction twice as formula_45 spans over the interval formula_50. 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 formula_14 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.
Dedicated experiments
have determined formula_52 for a standard DNA ladder up to 1500 bp long in 1.5% agarose gel. The dependence found, see Fig. 6, supports the possibility of nonlinear frictiophoresis in such a system. Based on data in Fig. 6, an optimal time course for driving electric field with zero mean, formula_53, has been found in, which ensures maximal drift for 1500 b.p. long fragment,
see Fig. 7.
The effect of unidirectional drift caused by periodic force with zero integral value has a peculiar dependence on the time course of the force applied. See the previous section for examples. This offers a new dimension to a set of separation problems.
DNA separation with respect to length.
In the DNA fragments separation,
zero mean periodic electric field is used in zero-integrated-field electrophoresis (ZIFE),
where the field time dependence similar to that shown in Fig. 3 is used.
This allows to separate long fragments in agarose gel,
nonseparable by standard constant field electrophoresis.
The long DNA geometry and its manner of movement in a gel,
known as reptation
do not allow to apply directly the consideration based on Eq. (5), above.
Separation with respect to specific mass.
It was observed,
that under certain physical conditions
the mechanism described in Mathematical analysis section, above,
can be used for separation with respect to specific mass,
like particles made of isotopes of the same material.
Extensions.
The idea of organizing directed drift with zero mean periodic drive
have obtained further development for other configurations
and other physical mechanism of nonlinearity.
Rotation by means of circular wave.
An electric dipole
rotating freely around formula_54-axis
in a medium with nonlinear friction can be manipulated by
applying electromagnetic wave polarized circularly along formula_55
and composed of two harmonics. The equation of motion
for this system is as follows:
formula_56
where formula_57 is the torque acting on the dipole due to
circular wave: 
formula_58
where formula_59 is the dipole moment component orthogonal
to formula_54-axis
and formula_61 defines the dipole direction in the formula_62 plane. By choosing proper
phase shift formula_45 in (6) it is possible to
orient the dipole in any desired direction, formula_64.
The direction formula_64 is
attained due to angular directed drift, which becomes zero when
formula_66.
A small detuning between the first and second harmonic in (6) results in continuous rotational drift.
Modification of potential function.
If a particle undergoes a directed drift while moving freely in accordance
with Eq. (5), then it drifts similarly if a shallow enough
potential field formula_67 is imposed. Equation of motion in that case is:
formula_68
where formula_69 is the force due to potential field. The drift continues until a steep enough region in the course of formula_67 is met, which is able to stop the drift. This kind of behavior, as rigorous mathematical analysis shows, 
results in modification of formula_67 by adding a linear in formula_72 term. This may change the formula_67 qualitatively, by, e.g. changing the number of equilibrium points, see Fig. 8. The effect may be essential during high frequency
electric field acting on biopolymers. 
Another nonlinearity.
For electrophoresis of colloid particles under a small strength electric field, the force formula_14
in the right-hand side of Eq. (5) is linearly proportional to the
strength formula_53 of the electric field applied. For a high strength,
the linearity is broken due to nonlinear polarization.
As a result, the force may depend nonlinearly on the applied field:
formula_76
In the last expression, even if
the applied field, formula_53 has zero mean, the applied force formula_14 may happen to have a constant component that can cause a directed drift.
As above, for this to happen, formula_53 must have more than a single sinusoidal harmonic.
This same effect for a liquid in a tube may serve in
electroosmotic pump
driven with zero mean electric field.
References.
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