Anomalous dynamics in tracer-particle motions in an electrohydrodynamically driven oil-in-oil system

Somayeh Khajehpour Tadavani and Anand Yethiraj

Phys. Rev. E 98, 022602 – Published 7 August 2018

Abstract

We characterize the superdiffusive dynamics of tracer particles in an electrohydrodynamically driven emulsion of oil droplets in an immiscible oil medium, where the amplitude and frequency of an external electric field are the control parameters. In the weakly driven electrohydrodynamic regime, the droplets are trapped dielectrophoretically on a patterned electrode, and the driving is therefore spatially varying. We find excellent agreement with a $〈 x^{2} 〉 \sim t^{1.5}$ power law and find that this superdiffusive dynamics arises from an underlying displacement distribution that is distinctly non-Gaussian and exponential for small displacements and short times. While these results are comparable with a random-velocity field model, the tracer particle speeds are in fact spatially varying in two dimensions, arising from a spatially varying electrohydrodynamic driving force. This suggests that the important ingredient for the superdiffusive $t^{1.5}$ behavior observed is a velocity field that is isotropic in the plane and spatially correlated. Finally, we can extract, from the superdiffusive dynamics, a experimental length scale that corresponds to the lateral range of the hydrodynamic flows. This experimental length scale is non zero only above a threshold ion mobility length.

Received 17 April 2018

DOI:https://doi.org/10.1103/PhysRevE.98.022602

Physics Subject Headings (PhySH)

Applications of soft matter Biological fluid dynamics Electric field effects Self-assembly

Hydrodynamics

Polymers & Soft MatterFluid DynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Somayeh Khajehpour Tadavani and Anand Yethiraj

Physics and Physical Oceanography, Memorial University of Newfoundland, St. John's, Newfoundland, A1B 3X7 Canada

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Issue

Vol. 98, Iss. 2 — August 2018

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Images

Figure 1
(a) A Side view of the cell geometry. Three dashed lines indicate three planes at which the experiment is conducted. The top plate is an unpatterned ITO electrode. The bottom electrode is patterned with circular ITO-free regions. Silicone oil is dielectrophoretically trapped in the ITO-free regions and assumes the form of near-hemispherical drops (shown in yellow). The continuous phase is castor oil, seeded with fluorescent PMMA particles (represented by white dots). (b) Snapshots of three planes of interest at (i) $h_{p} = 70$ , (ii) 30, and (iii) $10 μ m$ above the patterned (bottom) electrode at $E = 4.6 V / μ m$ , perpendicular to the page, and $f = 1 Hz$ . The PMMA particles are the bright spots in each frame. Accumulation of PMMA particles at the interface of silicone oil drop and castor oil leads to a bright edge.
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Figure 2
A time lapse series of $5 s$ for three heights of interest, from left to right $h_{p} = 70, 30$ , and $10 μ m$ . Flows tracked by fluorescent PMMA particles are indicated by trails in direction of arrows from sources to sinks. For $h_{p} = 70 μ m$ sources are at $d_{n n} / 2 ≃ 50 μ m$ , and sinks are at the center of the drops. For $h_{p} = 30$ and $10 μ m$ sources and sinks are located at edge of the drops and $d_{n n} / 2 ≃ 50 μ m$ , respectively, indicating an oblate drop deformation. Red circles show position of center of drops in the lower heights. $E = 5.8 V / μ m$ , perpendicular to the page, $f = 3 Hz$ , and scale bar is $50 μ m$ .
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Figure 3
Normalized histogram: panels (a)–(d) show four examples of the normalized distribution histograms along $x$ and $y$ in three successive times and at four different frequencies, $0.5, 1, 2$ , and $6 Hz$ , respectively. $E = 3.5 V / μ m$ , perpendicular to the page and $h_{p} = 10 μ m$ . (e) Variation of $δ$ versus $f$ gives rise to mean value of $1.04 \pm 0.12$ .
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Figure 4
(a) $log (MSD)$ versus $log (t)$ shows a linear dependency with the slope of $γ ≃ 1.5$ consistent with enhanced superdiffusive dynamics in the outer fluid. $γ ≃ 1.5$ is independent of frequency, field amplitude, and the height above the bottom electrode (i.e., the $x − y$ plane) at which the experiment is carried out. (b) MSD versus time for short time, 100 frames recorded at $20 fps$ for $0.5 Hz$ and $10 fps$ for $4 Hz$ equivalent to 5 and $10 s$ , respectively. MSD versus time for long time, 400 frames recorded at $20 fps$ for $0.5 Hz$ and $10 fps$ for $4 Hz$ equivalent to 20 and $40 s$ , respectively. (c) $γ$ versus $h_{p}$ , which is averaged over frequency and amplitude, respectively, shows a mean value of $γ = 1.52 \pm 0.01$ .
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Figure 5
(a) The second moment of the distribution of displacements along $x$ and $y$ depends linearly on $t^{1.5}$ : the slope $K_{γ} = 12.3 \pm 0.1 μ m^{2} / s^{1.5}$ (b) The fourth moment of $x$ and $y$ distribution of displacements depends linearly on $t^{3}$ : the slope $K_{2 γ} = 648.6 \pm 7.1$ . (c) The non-Gaussian parameter has an approximate inverse frequency dependence (shown by the light gray line), with an average value $α_{2} = 0.31 \pm 0.06$ . The inset shows $α_{2}$ calculated from Eq. (5) for different values of the, in principle, unknown $δ$ parameter. A value $α_{2} = 0.31 \pm 0.06$ corresponds to $δ = 1.43 \pm 0.06$ .
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Figure 6
(a) Effective diffusion coefficient, $K_{γ}$ , versus $f$ . (b) An experimental length scale, $λ_{h} = \sqrt{K_{1.5} / f^{1.5}}$ , versus frequency, $f$ , for different field amplitudes, frequencies, and heights, shows a monotonic decrease with increasing $f$ . (c) An expected ionic hydrodynamic length, $l_{h} = μ_{E} E / f$ , plotted for all field amplitudes for comparison with $λ_{h}$ . (d) Experimental length scale, $λ_{h}$ , versus ionic hydrodynamic length scale, $l_{h}$ , shows a linear dependency with an x intercept of $l_{0} ≃ 40 μ m$ .
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