Differential dynamic microscopy of bidisperse colloidal suspensions

Abstract

Research tasks in microgravity include monitoring the dynamics of constituents of varying size and mobility in processes such as aggregation, phase separation, or self-assembly. We use differential dynamic microscopy, a method readily implemented with equipment available on the International Space Station, to simultaneously resolve the dynamics of particles of radius 50 nm and 1 μm in bidisperse aqueous suspensions. Whereas traditional dynamic light scattering fails to detect a signal from the larger particles at low concentrations, differential dynamic microscopy exhibits enhanced sensitivity in these conditions by accessing smaller wavevectors where scattering from the large particles is stronger. Interference patterns due to scattering from the large particles induce non-monotonic decay of the amplitude of the dynamic correlation function with the wavevector. We show that the position of the resulting minimum contains information on the vertical position of the particles. Together with the simple instrumental requirements, the enhanced sensitivity of differential dynamic microscopy makes it an appealing alternative to dynamic light scattering to characterize samples with complex dynamics.

Fig. 1

a–c Intermediate scattering function f(q,t) as a function of lag time t measured for bidisperse mixtures of particles of radius 50 nm and 1 μm formulated at a large-to-small volume fraction ratio r of a 0.03, b 0.01, and c 0.003 at wavevectors of q = 6.8 μm⁻¹ (30°, squares), 11.2 μm⁻¹ (50°, diamonds), and 18.7 μm⁻¹ (90°, triangles). Red lines indicate fitting functions: Eq. 2 for r = 0.03 and q = 6.8 μm⁻¹ and Eq. 1 otherwise. d Predicted scattering intensity I(q) for small particles at ϕ = 10⁻³ and large particles at volume fraction ratios of r = 0.03, 0.01, and 0.003 as a function of wavevector q using standard equations for hard spheres. The range of wavevectors probed by DLS and DDM are indicated by dashed and dash-dotted lines, respectively. e–g Intermediate scattering function f(q,t), extracted from DDM measurements, as a function of lag time t measured for bidisperse mixtures of particles of radius 50 nm and 1 μm formulated at large-to-small volume fraction ratios r of a 0.03, b 0.01, and c 0.003. For each ratio, data were analyzed over the wavevector range 0.98 μm⁻¹ < q < 3.01 μm⁻¹; the figure shows representative correlation functions obtained for wavevectors q = 1.08 μm⁻¹ (squares), 2.05 μm⁻¹ (diamonds), or 2.92 μm⁻¹ (triangles). Red lines indicate fits to Eq. 4

Fig. 2

a Inverse of the large-particle time scale ${\tau }_L^{-1}$ as a function of the square of the wavevector q ² for bidisperse mixtures of particles of radius 50 nm and 1 μm formulated at large-to-small volume fraction ratios r = 0.03 (squares), 0.01 (diamonds), and 0.003 (triangles). Data at r = 0.01 and r = 0.003 are offset by one and two unit increments on the y axis, respectively, for clarity. b Comparison of DLS and DDM inverse time scales for the small particles as a function of q ². Data at low wavevectors are acquired in a bidisperse mixture using DDM; data at higher wavevectors are acquired in unary solutions using DLS. Dashed red lines in a and b indicate linear fits

Fig. 3

a DDM signal amplitude A(q) as a function of wavevector squared q ² for bidisperse mixtures of particles of radius 50 nm and 1 μm at varying volume fraction ratios r. b and c describe the contributions to signal intensity from small and large particles, A _S(q) and A _L(q), respectively

Fig. 4

a Relative contribution to the DDM signal from the large particles f _L(q) as a function of the square of the wavevector q ². Arrows indicate predicted minima from the diameter of the interference rings. The incident light intensity and sample thickness were kept constant between all samples but the condenser numerical aperture was set to ~0.1 for r = 0.03 and 0.01 and to ~0.15 for r = 0.003 to improve the signal-to-noise ratio. b Examples of interference rings seen in microscope images (top) and image differences (bottom) at a lag time τ = 1.6 s for each sample. Dashed circle indicates the diameter of the interference ring. c DDM signal amplitude A(q) for large particles segregated via sedimentation to the bottom of a glass sample chamber and imaged at various heights z above the plane of the segregated particles. Arrows indicate the predicted minima from the diameter of interference rings. d Examples of interference rings in microscope images for segregated particles imaged at the same heights. Dashed circles indicate the diameters of the interference rings corresponding to arrows in c. Microscope images and image differences were modified to increase brightness and contrast for clarity