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Spatial cross-correlated diffusion of colloids under a shear flow Na Li, Wei Zhang, Zehui Jiang, and Wei Chen Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01803 • Publication Date (Web): 17 Aug 2018 Downloaded from http://pubs.acs.org on August 22, 2018
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Spatial cross-correlated diffusion of colloids under a shear flow Na Li1 , Wei Zhang2, Zehui Jiang3, and Wei Chen1 *
1 State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200438, China
2 School of Physical Science and Technology, China University of Mining and Technology, Xuzhou 221116, China
3 Department of Physics, Harbin Institute of Technology, Harbin 150001, China
Corresponding Author
*E-mail:
[email protected].
KEYWORDS: EYWORDS: Colloidal monolayer, Spatial cross-correlated diffusion, Brownian motion,
Shear flow
ABSTRACT: The spatial cross-correlated diffusion of colloidal particles is often used as an essential tool to study the dynamic properties of a fluid because it directly describes the hydrodynamic interaction between two particles in a fluid. However, the experimental measurement of cross-correlated diffusion can be substantially modified by even a weak shear flow. In this work, the effect of a shear flow on spatial cross-correlated diffusion is demonstrated using experimental measurements that show a clear dependence on pair angles. An analytical solution is proposed to explain the experimental observations. A numerical 1 ACS Paragon Plus Environment
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simulation is performed to systemically demonstrate the influence of a shear flow on spatial cross-correlated diffusion. The results of the experiment, theoretical analysis and numerical simulation agree well with each other. Therefore, this research provides a sensitive experimental method to determine the weak shear flow in any quasi-two-dimensional fluid systems.
INTRODUCTION In a static fluid without any shear flow, the colloidal particles move in Brownian motion. The coupling motion of colloidal particles occurs due to hydrodynamic interactions1-3 , which are dependent on the dynamic properties of the fluid4-7 and the boundary conditions of the system8-13. Usually, the spatial cross-correlated diffusion (SCCD) of colloidal particles is measured to understand the behavior of the hydrodynamic interactions between the particles, which is a function of the distance between the particle pairs. The SCCD is a method base on two-particle measurement, which established by Crocker et al. to investigate the microrheology of the fluid2, 7. Comparing the traditional single particles microrheology, the method of SCCD can be applied in the inhomogeneous fluid because it is driven by the long-wavelength modes in the system and cannot be influenced by the local environment of the particles. In the 3D bulk of a Newtonian fluid, the SCCD of the colloids decays with distance as
(r)~ 1⁄ 7, whereas it follows the logarithmic form of (r)~ ln ( ⁄) in a 2D system8-9, 14-17
. In quasi-two-dimensional static fluid systems, such as colloids near a liquid-liquid
interface, where Oseen tensor for the semi-infinite space should be applied, the SCCD of the colloids in the transverse direction decays as (r)~ 1⁄ when the distance between the 2 ACS Paragon Plus Environment
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colloidal particles is much larger than the characteristic length of the system2,
8-9
. The
tendency of SCCD plays a key role in elucidating the dynamic properties of a liquid. However, the results of SCCD measurements in some experiments often slightly deviate from the theoretical expectation, i.e., the SCCD decays more quickly than that of ~ 1⁄ 18. In this paper, we demonstrate that this deviation is explained by the presence of a weak shear flow in the system. Shear flows exist in many artificial microfluidic systems19-23 , such as in fluids containing vesicles and red blood cells in biological systems, and play an important role in the hydrodynamic behavior of the vesicles in the fluid 19, 24-28. A. Ziehl et al. found that shear flow affects the time correlation between the movement of the colloids: the non-diagonal elements in the temporal cross-correlated diffusion matrix between colloidal pairs becomes negative instead of zero under a shear flow29. Shani et al. found that the SCCD between the motions of the colloids have a directional dependence under a shear flow3. In the present paper, two types of mechanisms are shown to contribute to the SCCD of colloids under a shear flow. One mechanism is caused by the coupling of the particles under Brownian motion due to hydrodynamic interactions, and the other mechanism is caused by the shear flow in the fluid. The strength of the former depends on the distance between the particle pairs, and the strength of the latter depends on both the particle pair angle θ and . In fact, the
contribution of the shear flow to the SCCD increases with increasing distance , which causes even a weak shear flow to substantially influence experimental measurements. In particular, the influence of shear flow is pronounced at the large limit of , where the tendency of SCCD is often regarded as a fingerprint feature to identify the dynamic properties of a fluid system. An experimental method could be developed based on the 3 ACS Paragon Plus Environment
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measurement of SCCD to determine a weak shear flow. In fact, the average additional displacements of colloidal particles smaller than 20 nm are detected under a spatial resolution of 90 nm/pixel in our experimental setup. In many systems with restricted time and space for experiments, researchers cannot obtain adequate data in a limited time to determine the flow field. Therefore, the method proposed in this paper can be widely applied in studies on, for instance, the dynamics of microorganisms22, 30-34 or the movement of living cells in fluids35, using only a common microscope-CCD device. This paper is organized as follows: First, the experimental setup is introduced. The experimental SCCD results of the colloidal monolayer in the transverse direction, () and (θ), are plotted and discussed. Second, the analytical formula of (θ) under a shear
flow is derived and used to fit the experimental data. Third, a numerical simulation is performed to calculate the SCCD under a shear flow. The results of the numerical simulation (θ) are plotted systematically at different distances , and are fitted by the analytical formula. A summary is presented at the end. EXPERIMENTAL SETUP The silica particles used in this experiment have a diameter = 3.1 μm (purchased from Bangs Laboratories), and were centrifuged in deionized water 8 times to remove surfactants in the original sample solution. The cleaned particles were suspended in deionized water (18.2МΩ.cm) in a stainless steel sample cell with an internal diameter of 1.0 cm. A cover slip adhered to the bottom of the sample cell served as the observation window. The separation between the cover slip and the water interface was 1 mm. The sample cell was settled upside down. The surface tension of water will hold the water in the sample cell. The particles sink onto the water-air interface under the gravity. The area fraction of the prepared colloidal 4 ACS Paragon Plus Environment
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monolayer prepared was 0.30 ± 0.02. An inverted microscope (Leica X71, 60X) with a camera (Prosilica GE1050, 1024×1024 pixels) was used to capture the image sequences of the colloids at a frame rate of 17 fps. The spatial resolution of the images was 0.09 µm/pix. The images obtained by the microscope are shown in Figure 1A. The trajectories of every particle were calculated with a homemade tracking software program. The area fraction of 2D closest packing is 84%36-37. The area fraction 30% of the particle monolayer is of a relatively low concentration. The particles monolayer could be regarded as a liquid phase, and the monolayer could response the shear flow of the fluid totally. For a high area fraction, the particle monolayer may go to the glass region, and the possible plasticity of the monolayer will affect the response of the shear flow.
EXPERIMENTAL RESULTS The separation between the particle monolayer and the water-air interface can be estimated by the relationship
=1+ (
( )
! " # !$ #
)( & ), where '( (0) is the single particle %
diffusion constant of particles in the monolayer under the dilute limit of particle concentration (n >0)，' is the single particle diffusion constant when the particles locates in the infinite
3D bulk of water, )* is the viscosity of bulk of water, ) % the viscosity of bulk of air, and a
the radius of particles38. The separation was estimated to be 1.9 ±0.2µm in our experiment. Such a separation is so small that the particle monolayer can be regarded as being attached to the interface. Thereby, the particles can response to the flow on the water-air interface totally. The SCCD between particles pairs in the transverse direction () is defined as () =
2
. 〈∆((/,1)∆(- (/,1)3(4"5 .,2 (/)) 〉.72
1
.
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Here, as shown in Figure 1B, ∆8 9 and ∆8 represent the components of the displacements :
∆8;9 and ∆8;: of the particles < and = in the transverse direction during lag time >. To
eliminate possible global drift flow in the system, the average displacement of all particles in the observation field was subtracted from the displacements of each particle before equation (1) was applied. The measured () is plotted as black solid dots in Figure 2A. The decay
of transverse () of a liquid film on the water interface converts from ln(?⁄) (at the very short limit of r) to 1⁄ ( at the large limit of )2. In the intermediate distance ,
() will show a cross-over dependency on . In our measurements, it looks close to a ~1⁄ decay phenomenally when < 30 μm.
The curves reflect a faster decay than the theoretical expectation of ~1⁄ at large limit
of . Moreover, (r) can have negative values when > 80 µm, where the black solid
dots that represent experimental data points do not appear in the log-log coordinate diagram shown in Figure 2A. The negative SCCD at large implies that the particle pairs have
anti-correlated motions when the distance is sufficiently large. In a further investigation,
three curves for (θ) were calculated for the particle pairs with pair distances = 60, 80
and 90 µm. Here, θ is the angle between the line that connects the center of two particles in a pair and the horizontal axis of the image in the lab frame, as shown in Figure 1A. The results are plotted as black solid dots in Figure 2B. All of the curves show a clear dependence on the pair angle θ, which implies that the rotational symmetry of the fluid field is broken.
Moreover, shows an anti-correlational behavior at large distances =80 and 90 µm,
whereas this anti-correlation occurs only near the angles of 0.8 or 3.9 rad for = 60 µm. These results suggest that a central symmetric shear flow is embedded in the data3 because
the global drift flow has already been numerically removed. 6 ACS Paragon Plus Environment
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When a shear flow exists in a system, colloids still exhibit Brownian motion39. Particle pairs with a large distance are likely to be separated by the symmetric axis of the shear flow. The particles in such pairs have opposite velocities under the central symmetric shear flow. The larger the distance between the particle pairs, the more negative the rate of
SCCD for those pairs. This relationship can explain why the curve of (r) decays more
quickly than expected by the theoretical expression ~1⁄ for large . The quantitative analysis of the contribution of the shear flow on (r) is presented in the next section. THEORETICAL ANALYSIS
Suppose a shear flow, as shown in Figure 3A, exists in a system. The symmetric axis of the shear flow goes through the center of the circular image field. The velocities along the symmetric axis are zero in the field. The shear angle θ is defined as the angle between the symmetric axis of the flow and the horizontal axis of the image in the lab frame. The shear rate is denoted as C. The motion of particles in a fluid with shear flow are independently driven by two mechanisms 39: Brownian motion and shear flow. The displacement of each particle ∆8; can
be written as∆8; = ∆8;DEFGHI + ∆8;KLMED . Here, ∆8;DEFGHI is the component of the displacement introduced by Brownian motion and ∆8;KLMED is the component introduced by the shear flow.
The SCCD of the particle pairs
= =
〈 〈
2
2
. . (∆(;NOPQRS $∆(;TUVON )(∆(;NOPQRS $∆(;TUVON ) 2
. ∆(;NOPQRS ∆(;NOPQRS
+〈
1
2
〉+〈
. ∆(;TUVON ∆(;NOPQRS
1
1
2
〉
. ∆(;NOPQRS ∆(;TUVON
1
〉+〈
2
. ∆(;TUVON ∆(;TUVON
1
〉.
〉
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The first term 〈
() ≡ 〈
2
. ∆(;NOPQRS ∆(;NOPQRS
1
2
. ∆(;NOPQRS ∆(;NOPQRS
1
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〉 is the SCCD due to Brownian motion, denoted as
〉. The fourth term 〈
2
. ∆(;TUVON ∆(;TUVON
1
〉 is the SCCD due to the shear
flow, which is a function of , C, θ and θ and is denoted as KLMED(, C, θ, θ ) ≡
〈
2
. ∆(;TUVON ∆(;TUVON
1
〉. The values of the second and third terms are zero, because ∆8;KLMED and
∆8;DEFGHI are independent of each other and 〈∆8;DEFGHI 〉 = 0. Therefore, the entire expression for SCCD is linearly superposed by two parts: 〈
∆(;. ∆(;2 1
〉 = (, C, θ, θ ) = () + KLMED(, C, θ, θ ).
(3)
The formula for KLMED (, C, θ, θ ) in the transverse direction is derived as follows. The
particle pair
tuv "z{ v
v
"tuv "z { v
v
v
"s4"tuv "wx. y
v "tuv "wx.
g 9 g x9[ y9[ . :
The averaged SCCD of the particle pairs
