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'''Electrophoretic light scattering''' is based on [[dynamic light scattering]]. The [[frequency]] shift or [[phase (waves)|phase]] shift of an incident [[laser]] beam depends on the [[dispersed particles]] mobility. In the case of ''dynamic light scattering'', Brownian motion causes particle motion. In the case of ''electrophoretic light scattering'', [[oscillating]] [[electric field]] performs the same function.

This method is used for measuring [[electrophoretic mobility]] and then calculating [[zeta potential]]. Instruments to apply the method are commercially available from several manufacturers. The last set of calculations requires information on [[viscosity]] and [[dielectric permittivity]] of the [[dispersion medium]]. Appropriate [[electrophoresis]] theory is also required. Sample dilution is often necessary in order to eliminate particle interactions.

==Instrumentation of Electrophretic light scattering==
[[Image:Optics 0002.jpg|thumb|right|200px|Fig.3. Commercial heterodyne optical system of an electrophoretic light scattering instrument with a modulator (from Ref. 11).]]

A laser beam passes through the electrophoresis cell, irradiates the particles dispersed in it, and is scattered by the particles. The scattered light is detected by a photo-multiplier after passing through two pinholes. There are two types of optical systems: heterodyne and fringe.
Ware and Flygare (9) developed a heterodyne-type ELS instrument, that was the first instrument of this type. In a fringe optics ELS instrument (10), a laser beam is divided into two beams. Those cross inside the electrophresis cell at a fixed angle to produce a fringe pattern. The scattered light from the particles, which migrates inside the fringe, is amplitude-modulated. The frequency shifts from both types of optics obey the same equations. The observed spectra resemble each other.
Oka et al. developed an ELS instrument of heterodyne-type optics (11) that is now available commercially. Its optics is shown in Fig. 3.

A modulator is a translationally moving mirror. A frequency of reference light shifts by the movement. Scattered light from dispersed particle without electrophoretic movement is line broadened. That with electrophoretic movement is line broadened and Doppler shifted. The moving mirror moves to shorten optical pass length, then the frequency shift toward higher frequency.

==Heterodyne light scattering==
The frequency of light scattered by particles undergoing electrophoresis is shifted by the amount of the Doppler effect, <math>\upsilon_D\,</math>  from that of the incident light, :<math>\upsilon\,</math> .
The shift can be detected by means of heterodyne optics in which the scattering light is mixed with the reference light.
The autocorrelation function of intensity of the mixed light, <math>g(\tau) \,</math>, can be approximately described by the following damped cosine function [7].

: <math>g(\tau)=A+B \exp(-\Gamma\tau)\cos(2\pi\upsilon_o)+C \exp(-2\Gamma \tau)\,\qquad (1)</math>

where <math>\Gamma\,</math>    is a decay constant and A, B, and C are positive constants dependent on the optical system.

Damping frequency <math>\upsilon_o\,</math>  is an observed frequency, and is the frequency difference between scattered and reference light.

:    <math>\upsilon_o =| \upsilon_s - \upsilon_r   | = | (\upsilon_i+\upsilon_D)-(\upsilon_i+\upsilon_M ) |\qquad(2)</math>

where  <math>\upsilon_s\,</math>  is the frequency of scattered light, <math>\upsilon_r\,</math>  the frequency of the reference light,
<math>\upsilon_i\,</math>  the frequency of incident light (laser light),
and <math>\upsilon_M\,</math>  the modulation frequency.

The power spectrum of mixed light, namely the Fourier transform of <math>g(\tau) \,</math>, gives a couple of Lorenz functions at <math>\pm\Delta\upsilon \,</math>  having a half-width of  <math>\Gamma/2\pi\,</math> at the half maximum.

In addition to these two, the last term in equation (1) gives another Lorenz function at
<math>\upsilon = 0\,</math>

The Doppler shift frequency and the decay constant are dependent on the geometry of the optical system and are expressed by the equations.

:    <math>\upsilon_D = \frac{Vq}{2\pi} \qquad(3)
</math>
and

:<math>\Gamma = D|q|^2 \qquad(4)</math>

where <math>\upsilon_D \,</math>  is velocity of the particles and <math>\ D \,</math> is the [[diffusion constant|translational diffusion constant]] of particles.

The amplitude of the scattering vector <math>\ q \,</math>   is given by the equation

:     <math>\  |q|  = \frac{4 \pi n}{\lambda_0 }\sin\left( \frac{\theta}{2}\right) \qquad(5)</math>

Since velocity <math>\  V \,</math>      is proportional to the applied electric field, <math>\  E \,</math>, the apparent electrophoretic mobility <math>\  \mu_{obs}  \,</math>  is define by the equation

:     <math>\  \vec{V}  =  \mu_{obs} \vec{E}  \qquad(6)</math>

Finally, the relation between the Doppler shift frequency and mobility is given for the case of the optical configuration of Fig. 3 by the equation

:    <math>\upsilon_D = \mu_{obs} \frac{n E}{\lambda_0} \sin \theta \qquad(7)</math>

where  <math>\  E \,</math>   is the strength of the electric field, <math>\ n\,</math> the refractive index of the medium, <math>\ \lambda_0  \,</math>, the wavelength of the incident light in vacuum, and <math>\  \theta \,</math>    the scattering angle.
The sign of  <math>\  v_D  \,</math> is a result of vector calculation and depends on the geometry of the optics.

The spectral frequency can be obtained according to Eq. (2).
When <math>\ | \upsilon_M | > | \upsilon _D  | \,</math>, Eq. (2) is modified and expressed as

:    <math>\upsilon_p = \upsilon_o = \pm( \upsilon _D -| \upsilon _M | ) \qquad(8)</math>

The modulation frequency <math> \upsilon _M \,</math>  can be obtained as the damping frequency without an electric field applied.

The particle diameter is obtained by assuming that the particle is spherical. This is called the hydrodynamic diameter,  <math>\  d_H\,</math>   .

:    <math>\ d_H =  \frac{k_BT}{3 \pi  \eta D} \qquad(10)</math>

where <math>\  k_B  \,</math>   is [[Boltzmann constant|Boltzmann coefficient]], <math>\  T  \,</math>  is the absolute temperature, and <math>\  \eta  \,</math>  the [[dynamic viscosity]] of the surrounding fluid.

==Profile of Electro-Osmotic Flow==
Figure 4 shows two examples of heterodyne autocorrelation functions of scattered light from sodium polystyrene sulfate solution (NaPSS; MW 400,000; 4&nbsp;mg/mL in 10 mM NaCl). The oscillating correlation function shown by Fig. 4a is a result of interference between the scattered light and the modulated reference light.
The beat of Fig. 4b includes additionally the contribution from the frequency changes of light scattered by PSS molecules under an electrical field of 40 V/cm.

Figure 5 shows heterodyne power spectra obtained by Fourier transform of the autocorrelation functions shown in Fig. 4.

Figure 6 shows plots of Doppler shift frequencies measured at various cell depth and electric field strengths, where a sample is the NaPSS solution.
These parabolic curves are called profiles of electro-osmotic flow and indicate that the velocity of the particles changed at different depth.
The surface potential of the cell wall produces electro-osmotic flow.
Since the electrophoresis chamber is a closed system, backward flow is produced at the center of the cell.
Then the observed mobility or velocity from Eq. (7) is a result of the combination of osmotic flow and electrophoretic movement.

Electrophoretic mobility analysis has been studied by Mori and Okamoto [16], who have taken into account the effect of electro-osmotic flow at the side wall.

The profile of velocity or mobility at the center of the cell is given approximately by Eq. (11) for the case where k>5.
:    <math>\ U_a (z) = AU_0(z/b)^2 + \Delta U_0(z/b)+(1-A) U_0 + U_p \qquad(11)</math>
where
: <math>\ z  = \,</math>  cell depth
: <math>\ U_a (z) = \,</math>  apparent electrophoretic velocity of particle at position z.

: <math>\ U_p = \,</math>  true electrophoretic velocity of the particles.

: <math>\ z/b = \,</math> thickness of the cell

:   <Math>\ U_0 = \,</math> average velocity of osmotic flow at upper and lower cell wall.

:  <Math>\Delta U_0 = \,</math> difference between velocities of osmotic flow at upper and lower cell wall.

:  <Math>\ A = \frac{1}{( 2/3) - (0.420166/k)} \qquad(12)</math>
:  <math> \ k = a/b \,</math> , a ratio between two side lengths of the rectangular cross section.

The parabolic curve of frequency shift caused by electro-osmotic flow shown in Fig. 6 fits with Eq. (11) with application of the least squares method.

Since the mobility is proportional to a frequency shift of the light scattered by a particle and the migrating velocity of a particle as indicated by Eq. (7), all the velocity, mobility, and frequency shifts are expressed by parabolic equations.
Then the true electrophoretic mobility of a particle, the electro-osmotic mobility at the upper and lower cell walls, ware obtained.
The frequency shift caused only by the electrophoresis of particles is equal to the apparent mobility at the stationary layer.

The velocity of the electrophoretic migration thus obtained is proportional to the electric field as shown in Fig. 7.
The frequency shift increases with increase of the scattering angle as shown in Fig. 8.
This result is in agreement with the theoretical Eq. (7).

<gallery>
Image:CorrelationFuncs_0012.gif|Fig.4 a and b; Correlation function with and without electric field. Sample: NaPSS solution (MW: 400,000) of 4&nbsp;mg/ml in 10 mM NaCl. Electric field applied: (a) 0 V/cm; (b) 40 V/cm. Scattering angle 7.0 degree, temperature 25 +-0.3 
Image:Spectra_0004-2.gif|Fig. 5. Heterodyne power spectra obtained by FFT of the correlation functions. 
Image:ParabolicProfile_0018.gif|Fig. 6. Frequency shifts observed at various cell depths.  
Image:ElectricFieldDependence_0016.gif|Fig. 7. Electric field dependence of the velocity at the stationary layer. 
Image:ScatteredAngleDependence_0014.gif|Fig. 8. Frequency shift as a function of scattering angle. 
</gallery>

==References==
(1) Surfactant Science Series, Consulting Editor Martin J. Schick Consultant New York, Vol. 76 Electrical Phenomena at Interfaces Second Edition, Fundamentals, Measurements and Applications, Second Edition, Revised and Expanded. Ed by Hiroyuki Ohshima, Kunio Furusawa. 1998. K. Oka and K. Furusawa, Chapter 8　Electrophresis, p.&nbsp;152 - 223. Marcel Dekker, Inc,

(7) B.R. Ware and D.D. Haas, in Fast Method in Physical Biochemistry and Cell Biology. (R.I. Sha'afi and S.M. Fernandez, Eds), Elsevier, New York, 1983, Chap. 8.

(9) R. Ware and W.H. Flygare, J.Colloid Interface Sci. 39: 670 (1972).

(10) J. Josefwiicz and F.R. Hallett, Appl. Opt. 14: 740 ( (1975).

(11) K. Oka, W. Otani, K. Kameyama, M. Kidai, and T. Takagi, Appl. Theor. Electrophor. 1: 273-278 (1990).

(12) K. Oka, W. Otani, Y. Kubo, Y. Zasu, and M. Akagi, U.S. Patent Appl. 465, 186: Jpn. Patent H7-5227 (1995).

(16) S. Mori and H. Okamoto, Flotation 28: 1 (1980). (in Japanese): Fusen 28(3): 117 (1980).

(17) M. Smoluchowski, in Handbuch der Electrizitat und des Magnetismus. (L. Greatz. Ed). Barth, Leripzig, 1921, pp.&nbsp;379.

(18) P. White, Phil. Mag. S 7, 23, No. 155 (1937).

(19) S. Komagat, Res. Electrotech. Lab. (Jpn) 348, March 1933.

(20) Y. Fukui, S. Yuu and K. Ushiki, Power Technol. 54: 165 (1988).

{{DEFAULTSORT:Electrophoretic Light Scattering}}
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[[Category:Colloidal chemistry]]