Nonlinear frictiophoresis

The simplest case of friction-velocity dependence law is the Stokes's one:

${\displaystyle (1)\qquad F_{dr}(v)=\lambda v,}$

where ${\displaystyle F_{dr}(v)}$ is the friction/drag force applied to a particle moving with velocity ${\displaystyle v}$ in a medium. The friction-velocity law (1) is observed for a slowly moving spherical particle in a Newtonian fluid.

Fig. 1 Linear friction

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, ${\displaystyle d}$, to get the particle moving. This kind of friction-velocity (dry friction) law has a jump discontinuity at ${\displaystyle v=0}$:

${\displaystyle (2)\qquad F_{dr}(v)=\lambda v+d\cdot \mathrm {sign} (v).}$

Fig. 2 Nonlinear friction example

It is nonlinear, see Fig. 2, and is used in this example.

Let ${\displaystyle T>0}$ denote the period of driving force. Chose a time value ${\displaystyle t_{1}}$ such that ${\displaystyle 0<t_{1}<T}$ and two force values, ${\displaystyle F^{+}>0}$, ${\displaystyle F^{-}<0}$ such that the following relations are satisfied:

${\displaystyle \qquad \qquad F^{+}>d,\quad |F^{-}|<d,}$

${\displaystyle (3)\qquad F^{+}t_{1}+F^{-}(T-t_{1})=0.}$

The periodic driving force ${\displaystyle f(t)}$ used in this example is as follows:

${\displaystyle (4)\qquad f(t)={\begin{cases}F^{+},{\text{ if }}0<t\leq t_{1},\\F^{-},{\text{ if }}t_{1}<t\leq T,\quad f(t+T)=f(t).\end{cases}}}$

It is clear that, due to (3), ${\displaystyle f(t)}$ has zero mean:

${\displaystyle \int \limits _{0}^{T}f(t)dt=F^{+}t_{1}+F^{-}(T-t_{1})=0.}$

Fig. 3 Zero mean driving force example

See also Fig. 3.

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 ${\displaystyle \tau \ll t_{1}}$, where ${\displaystyle \tau }$ is the relaxation time. In this situation, the particle driven with force (4) immediately starts moving with constant velocity ${\displaystyle v^{+}={\frac {1}{\lambda }}(F^{+}-d)}$ during interval ${\displaystyle 0<t\leq t_{1}}$ and will immediately stop moving during interval ${\displaystyle t_{1}<t\leq T}$, see Fig. 4.

Fig. 4 Velocity with nonzero mean

This results in the positive mean velocity of unidirectional drift:

${\displaystyle \qquad \qquad {\overline {v(t)}}={\frac {1}{T}}\int \limits _{0}^{T}v(t)dt={\frac {t_{1}}{\lambda T}}(F^{+}-d)>0.}$

In the DNA fragments separation, zero mean periodic electric field is used in zero-integrated-field electrophoresis (ZIFE),[6] 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.

It was observed, [7] 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.

An electric dipole rotating freely around ${\displaystyle z}$-axis in a medium with nonlinear friction can be manipulated by applying electromagnetic wave polarized circularly along ${\displaystyle \pm z}$ and composed of two harmonics. The equation of motion for this system is as follows:

${\displaystyle \qquad \qquad {\dot {\omega }}+\lambda \omega -\epsilon g(\omega )=f(t,\theta (t)),\quad {\dot {\theta }}=\omega ,}$

where ${\displaystyle f(t,\theta (t))}$ is the torque acting on the dipole due to circular wave:

${\displaystyle (6)\qquad f(t,\theta (t))\sim |p|(A_{1}\cos(\omega t\pm \theta )+A_{2}\cos(2\omega t\pm \theta +\psi )),}$

where ${\displaystyle p}$ is the dipole moment component orthogonal to ${\displaystyle z}$-axis and ${\displaystyle \theta }$ defines the dipole direction in the ${\displaystyle XY}$ plane. By choosing proper phase shift ${\displaystyle \psi }$ in (6) it is possible to orient the dipole in any desired direction, ${\displaystyle \theta _{0}}$. The direction ${\displaystyle \theta _{0}}$ is attained due to angular directed drift, which becomes zero when ${\displaystyle \theta =\theta _{0}}$. [8] [9] A small detuning between the first and second harmonic in (6) results in continuous rotational drift.[9]

Fig. 8 Example of modification of potential function ${\displaystyle U(x)}$ due to nonlinear frictiophoresis. (a) initial ${\displaystyle U(x)}$, (b) modified ${\displaystyle U(x)}$.

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 ${\displaystyle U(x)}$ is imposed. Equation of motion in that case is:

${\displaystyle \qquad \qquad {\dot {v}}+\lambda v-\epsilon g(v)=f(t)-\phi (x),\quad {\dot {x}}=v,}$

where ${\displaystyle \phi (x)}$ is the force due to potential field. The drift continues until a steep enough region in the course of ${\displaystyle U(x)}$ is met, which is able to stop the drift. This kind of behavior, as rigorous mathematical analysis shows, [10] results in modification of ${\displaystyle U(x)}$ by adding a linear in ${\displaystyle x}$ term. This may change the ${\displaystyle U(x)}$ 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. [11]

For electrophoresis of colloid particles under a small strength electric field, the force ${\displaystyle f(t)}$ in the right-hand side of Eq. (5) is linearly proportional to the strength ${\displaystyle E(t)}$ 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:

${\displaystyle \qquad \qquad f(t)\sim E(t)+\alpha (E(t))^{3}.}$

In the last expression, even if the applied field, ${\displaystyle E(t)}$ has zero mean, the applied force ${\displaystyle f(t)}$ may happen to have a constant component that can cause a directed drift. [12] As above, for this to happen, ${\displaystyle E(t)}$ 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.[13]