Affine and nonaffine motions in sheared polydisperse emulsions

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Affine and nonaffine motions in sheared polydisperse emulsions

J. Clara-Rahola, T. A. Brzinski, D. Semwogerere, K. Feitosa, J. C. Crocker, J. Sato, V. Breedveld, and Eric R. Weeks

Phys. Rev. E 91, 010301(R) – Published 8 January 2015

Abstract

We study dense and highly polydisperse emulsions at droplet volume fractions $ϕ \geq 0.65$ . We apply oscillatory shear and observe droplet motion using confocal microscopy. The presence of droplets with sizes several times the mean size dramatically changes the motion of smaller droplets. Both affine and nonaffine droplet motions are observed, with the more nonaffine motion exhibited by the smaller droplets which are pushed around by the larger droplets. Droplet motions are correlated over length scales from one to four times the mean droplet diameter, with larger length scales corresponding to higher strain amplitudes (up to strains of about 6%).

Received 25 April 2012
Revised 27 August 2014

DOI:https://doi.org/10.1103/PhysRevE.91.010301

Authors & Affiliations

J. Clara-Rahola^1,*, T. A. Brzinski¹, D. Semwogerere¹, K. Feitosa^2,†, J. C. Crocker², J. Sato³, V. Breedveld³, and Eric R. Weeks^1,‡

¹Department of Physics, Emory University, Atlanta, Georgia 30322, USA
²Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
³School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

^*Present address: Eikhonal Institute, 08750 Barcelona, Spain.
^†Present address: Dept. of Physics and Astronomy, James Madison University, Harrisonburg, VA 22807, USA.
^‡erweeks@emory.edu

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Vol. 91, Iss. 1 — January 2015

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Images

Figure 1
Confocal microscopy image of a polydisperse emulsion. The mean droplet displacement is indicated by the large black arrow at the upper right (2.8 $μ m$ to the left during the half cycle); this is the affine component of the motion. The white arrows indicate the total displacement of each droplet from the peak-to-peak of one-half oscillation cycle, with the mean displacement subtracted off; this is the nonaffine component of the motion. For easier visualization, all arrows are twice their actual length, including the affine displacement arrow. The image corresponds in time to the midpoint of this half cycle. The width of the image is 56 $μ m$ . The depth is $z = 24 μ m$ , the strain amplitude is $γ_{loc} = 0.067$ , and the volume fraction is $ϕ = 0.65$ .
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Figure 2
Probability distribution of droplet sizes in a $ϕ = 0.80$ sample. Droplets with radius $r < 5 μ m$ are fit to an exponential (dashed line) with decay length 0.5 $μ m$ . The data for $r > 11 μ m$ correspond to only three observed droplets, thus the true shape of the distribution is ill defined for larger droplets. The inset shows the same data, with the probability weighted by $r^{3}$ and plotted on a linear scale.
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Figure 3
(a) Probability distributions of the parallel ( $a_{x}$ ) and perpendicular ( $a_{y}$ ) amplitude components. (b) Probability distributions of the phase angles. (c)–(e) Scatter plots of the droplet amplitudes as a function of their radii $r$ . (c) shows $a_{x} - 〈 a_{x} 〉$ , (d) shows $a_{y}$ , and (e) shows $a_{r} \equiv \sqrt{{(a_{x} - 〈 a_{x} 〉)}^{2} + a_{y}^{2}}$ . The horizontal dashed lines in (c) and (d) are at zero, indicating the expected value for purely affine motion. Data are from a $ϕ = 0.65$ emulsion with $γ_{loc} = 0.070$ , $z = 24 μ m$ . The data for $P (a_{x})$ are centered around $〈 a_{x} 〉 = 2.4 μ m$ . $〈 θ_{x} 〉$ is arbitrary as it depends on when we set $t = 0$ , although note that the distribution for $θ_{y}$ is centered around the mean value of $θ_{x}$ . The lack of symmetry in these distributions is due to the finite number of droplets.
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Figure 4
Droplets which move elastically, drawn at their mean position. The color code indicates the parallel (left picture) and perpendicular (right picture) amplitude of each droplet. The color bar denotes the amplitude scale in $μ m$ , where 0 denotes the mean amplitude for these data ( $〈 a_{x} 〉 = 2.4 μ m$ , and using 0 $μ m$ for the $a_{y}$ data). The images correspond to the data shown in Fig. 3.
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Figure 5
(a) Spatial correlation functions of the droplet amplitudes $a_{x}$ (solid symbols) and $a_{y}$ (open symbols). The strain amplitudes are $γ_{loc} = 0.017$ (circles) and $γ_{loc} = 0.085$ (squares). The correlation functions are normalized by their value at $Δ r_{0} = 2 μ m$ . The data are from the same sample as Figs. 3 and 4. (b) Values of the decay lengths for this sample as a function of strain amplitude. The uncertainty of $γ_{loc}$ is 6%. The error bars are the standard deviations found from multiple experiments. The data shown are taken at depth $z = 24 μ m$ . Lines are guides to the eye.
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