Nonlinear frictiophoresis explained

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.,^[1] and is mainly known as nonlinear electrofrictiophoresis^[1] .^[2] 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 theStokes's one:

( 1 ) F _{dr} (v) =λ v, where F _{dr} (v) is the friction/drag force applied to a particle moving with velocity v 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, d , to get the particle moving. This kind of friction-velocity (dry friction) law has a jump discontinuity at v =0 :( 2 ) F _{dr} (v) =λ v + d ⋅ sign (v). It is nonlinear, see Fig. 2, and is used in this example.

Periodic driving force

Let

T> 0 denote the period of driving force. Chose a timevalue t _{1} such that 0 <t _{1<T} and two force values, F ^{+>0} , F ^{-<0} such that the following relations aresatisfied:F ^{+>d,} |F ^{-|<d,} ( 3 ) F ^{+} t _{1} + F ^{- (T - t} _{1 ) =0 .} The periodic driving force f(t) used in this example is as follows:( 4 ) f(t) = \begin{cases} F ^{+,} if 0 <t\le t ^{- ,} _{1 , \\ F} if t _{1<t\le} T, f(t + T) = f(t). \end{cases} It is clear that, due to (3), f(t) has zero mean:\int\limits ^{T} _{0} f(t)dt = F ^{+} t _{1} + F ^{- (T - t} _{1 ) =0 .} 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 ifparticle's mass is small, velocity is low and friction is high. This conditions have to ensure that

\tau\ll t _{1} ,where \tau is the relaxation time. In this situation, the particle driven with force (4) immediately starts moving with constant velocity v ^{+= (1)/(λ)} (F ^{+-d)} during interval0 <t\le t _{1} and will immediatelystop moving during interval t _{1<t\le} T , see Fig. 4.This results in thepositive mean velocity of unidirectional drift:\overline{v(t)} = (1)/(T) \int\limits ^{T} _{0} v(t)dt = (t _{1})/(λ T) (F ^{+-d)>0.}

Mathematical analysis

Analysis of possibility to get a nonzero drift by periodic force withzero integral has been made in.^[1] The dimensionlessequation of motion for a particle driven by periodic force

f(t) , f(t +1 ) = f(t) , \int ^{1 f(t)dt =0} _{0} is as follows:( 5 ) • v +λ v - \epsilon g(v) = f(t), where the friction/drag forceF _{dr} (v) =λ v - \epsilon g(v) satisfiesthe following:F _{dr} ( - v) =- F _{dr} (v), (d)/(dv) F _{dr} (v)\ge 0 . It is proven in ^[1] that any solution to (5)settles down onto periodic regimev ^{*(t)} , v ^{*(t+1)=v} ^{*(t)} , which has nonzero mean:\overline{v ^{1 v} _{0} ^{*(t)dt\ne} 0 , almost certainly, provided f(t) is not antiperiodic.^[3]

For

g(v) = v ^{3} , two cases of f(t) have been considered explicitly:

1. Saw-shaped driving force, see Fig. 5:

f(t) = at, t\in[ -1 / 2 ; 1 / 2 ], f(t +1 ) = f(t), t\in ] -inf ty; inf ty[. In this case, found in ^[1] first order in \epsilon approximation to v ^{*(t)} ,v ^{*(t)} _{1} , has the following mean value:| \overline{v ^{*} _{1 (t)} |} \equiv | \int\limits ^{*(t)dt | \ge (2)/(3)} _{1} \epsilon a ^{3/λ} ^{5.} This estimate is made expecting λ \gg 1 .

2. Two harmonics driving force,

f(t) = a\cos( 2 \pi t) + b\cos( 4 \pi t + \psi). In this case, the first order in \epsilon approximationhas the following mean value:| \overline{v ^{*} _{1 (t)} | = | \int\limits} ^{*(t)dt | \ge (2)/(27)} ^{1} (\epsilon)/(\left( λ) (M)/(λ) \right) ^{3 (1)/(( 1+ (4 \pi ^{2})/()( 1+ (16 \pi ^{2})/(λ ^{2}) ) ^{1/2} λ ^{2}))} . This value is maximized in \psi , a,b , keeping a + b = M constant. Interesting that the drift value depends on \psi and changes its direction twice as \psi spans over the interval [ 0 ; 2 \pi] . Another type of analysis,^[4] 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

f(t) 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.^[5] Dedicated experiments^[2] have determined F _{dr} (v) 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, E(t) , has been found in,^[2] 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),^[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 reptationdo not allow to apply directly the consideration based on Eq. (5), above.

Separation with respect to specific mass

It was observed,^[7] that under certain physical conditionsthe 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 drivehave obtained further development for other configurationsand other physical mechanism of nonlinearity.

Rotation by means of circular wave

An electric dipolerotating freely around

z -axisin a medium with nonlinear friction can be manipulated byapplying electromagnetic wave polarized circularly along \pm z and composed of two harmonics. The equation of motionfor this system is as follows:• \omega +λ \omega - \epsilon g(\omega) = f(t,\theta(t)), • \theta = \omega, where f(t,\theta(t)) is the torque acting on the dipole due tocircular wave: ( 6 ) f(t,\theta(t))\sim |p|(A _{1\cos(\omega} t\pm\theta) + A _{2\cos(2\omega} t\pm\theta + \psi)), where p is the dipole moment component orthogonalto z -axisand \theta defines the dipole direction in the XY plane. By choosing properphase shift \psi in (6) it is possible toorient the dipole in any desired direction, \theta _{0} .The direction \theta _{0} isattained due to angular directed drift, which becomes zero when\theta = \theta _{0} .^{[8] [9]} 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 accordancewith Eq. (5), then it drifts similarly if a shallow enoughpotential field

U(x) is imposed. Equation of motion in that case is:• v +λ v - \epsilon g(v) = f(t) - \phi(x), • x = v, where \phi(x) is the force due to potential field. The drift continues until a steep enough region in the course of 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 U(x) by adding a linear in x term. This may change the U(x) qualitatively, by, e.g. changing the number of equilibrium points, see Fig. 8. The effect may be essential during high frequencyelectric field acting on biopolymers. ^[11]

Another nonlinearity

For electrophoresis of colloid particles under a small strength electric field, the force

f(t) in the right-hand side of Eq. (5) is linearly proportional to thestrength 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:f(t)\sim E(t) + \alpha (E(t)) ^{3.} In the last expression, even ifthe applied field, E(t) has zero mean, the applied force f(t) may happen to have a constant component that can cause a directed drift.^[12] As above, for this to happen, E(t) must have more than a single sinusoidal harmonic.This same effect for a liquid in a tube may serve inelectroosmotic pumpdriven with zero mean electric field.^[13]

Notes and References

1. Vidybida. Alexander. Serikov. Alexander. Electrophoresis by alternating fields in a non-Newtonian fluid. Physics Letters. 108. 3. 1985. 170–172. 10.1016/0375-9601(85)90853-9. 1985PhLA..108..170V.
2. Vidybida. A.K.. Erkızan. V.. Altungöz. O.. Çelebi. G.. Testing of nonlinear electrofrictiophoresis in agarose gel. Bioelectrochemistry. 92. 2000. 1. 91–101. 10.1016/S0302-4598(00)00088-X. 11059582.
3. Function f(t) is antiperiodic if for some A the following relation f(t + A) =- f(t) holds for all t . Individual harmonics, like \sin(\omega t) , are antiperiodic.
4. Nordén. B.. Zolotaryuk. Y.. Christiansen. P.L.. Zolotaryuk. A.V.. Ratchet due to broken friction symmetry. Physical Review E. 65. 1. 2001. 011110. 10.1103/PhysRevE.65.011110. 11800680. 2001PhRvE..65a1110N. 19482936. Nordén. B.. Zolotaryuk. Y.. Christiansen. P.L.. Zolotaryuk. A.V.. Ratchet device with broken friction symmetry. Applied Physics Letters. 80. 14. 2002. 2601–2603. 10.1063/1.1468900. 2002ApPhL..80.2601N. 53695277.
5. Simoff. D.A.. Porter. R.S.. Rheology and Properties of Thermotropic Liquid Crystalline Poly (Bisphenol E Isophthalate-Co-Naphthalate). Molecular Crystals and Liquid Crystals. 110. 1–4. 2011. 1–26. 10.1080/00268948408074493.
6. Noolandi. J.. Turmel. C.. Preparation, manipulation, and pulse strategy for one-dimensional pulsed-field gel electrophoresis (ODPFGE). Mol Biotechnol. 4. 1. 1995. 25–43. 10.1007/BF02907469. 8521038. 9625993.
7. Vidybida. A.. Unpublished observation.
8. Andruschenko. A.P.. Vidybida. A.K.. Control of internal rotation in molecular-scale objects by alternating space-homogeneous electric field. Automation. 5. 1988. 58–61.
9. Doktor thesis. Vidybida. A.K.. 1999. Dynamical mechanisms of influence of alternating electromagnetic fields on macromolecular and cooperative systems. Ch.3 in. Bogolyubov Institute for Theoretical Physics. http://bitp.kiev.ua/files/doc/thesis/1999/diser_Vidybida.pdf.
10. Vidybida. A.K.. Modification of the Potential Function of a Mechanical System Caused by Periodic Action. Soviet Physics Doklady. 32. 1987. 113–115. Vidybida. A.K.. Modification of the Potential Function of a Mechanical System Caused by Periodic Action. Acta Mechanica. 67. 1–4. 1987. 183–190. 10.1007/BF01182131. 120582314.
11. Vidybida. A.K.. Periodic electric field as a biopolymer conformation switch: a possible mechanism. Eur Biophys J. 16. 6. 1989. 357–61. 10.1007/BF00257884. 2924736. 8672314. Doktor thesis. Vidybida. A.K.. 1999. Dynamical mechanisms of influence of alternating electromagnetic fields on macromolecular and cooperative systems. Ch.2 in. Bogolyubov Institute for Theoretical Physics. http://bitp.kiev.ua/files/doc/thesis/1999/diser_Vidybida.pdf.
12. Dukhin. S.S.. Dukhin. A.S.. Serikov. A.A.. Vidybida. A.K.. Aperiodic Electrophoresis. Directed drift of dispersed particles in a uniform anharmonic alternating electric field. Kolloidnyi Zh. 49. 5. 1988. 752–755. Mishchuk. N.A.. Barinova. N.O.. Theoretical and experimental study of nonlinear electrophoresis. Colloid J. 73. 2011. 88–96. 10.1134/S1061933X11010133. 95565907.
13. US. 20050129526. application. Method of using unbalanced alternating electric field in microfluidic devices . 2005-06-16 . 2003-12-10. DUKHIN ANDREI S. . DUKHIN STANISLAV .