Correlated diffusion of colloidal particles near a liquid-liquid interface
aa r X i v : . [ c ond - m a t . s o f t ] A p r Correlated diffusion of colloidal particles near a liquid-liquid interface
Wei Zhang , , † , Song Chen , Na Li , Jiazheng Zhang , and Wei Chen ∗ State Key Laboratory of Surface Physics and Department of Physics,Fudan University, Shanghai 200433, China Department of Physics, Jinan University,Guangzhou 510632, China (Dated: April 12, 2013)Abstract - Optical microscopy and multi-particle tracking are used to investigate the cross-correlated diffusion of quasi two-dimensional (2D) colloidal particles near an oil-water interface.It is shown that the effect of the interface on correlated diffusion is asymmetric. Along the linejoining the centers of particles, the amplitude of correlated diffusion coefficient D k ( r ) is enhancedby the interface, while the decay rate of D k ( r ) is hardly affected. At the direction perpendicular tothe line, the decay rate of D ⊥ ( r ) is enhanced at short inter-particle separation r . This enhancingeffect fades at the long r . In addition, both D k ( r ) and D ⊥ ( r ) are independent of the colloidal areafraction n at long r , which indicates that the hydrodynamic interactions (HIs) among the particlesare dominated by that through the surrounding fluid at this region. However, at short r , D ⊥ ( r ) isdependent on n , which suggests the HIs are more contributed from the 2D particle monolayer self. PACS numbers: 82.70.Dd, 68.05.Gh, 05.40.-a, 83.85.Jn
I. INTRODUCTION
Much attention has been attracted to the dynamic be-havior of confined colloidal suspensions recently [1–6]. Inreal circumstances, particles are usually spatially con-fined in special environments, such as microfluidic de-vices, porous media, fluid interface and cell membrane[7–11]. The dynamical behaviors of colloids in spatiallyconfined environments are more complicated than thatin unbounded three-dimensional (3D) fluid bulk. In theunbounded 3D bulk suspensions, the longitudinal andtransverse correlated diffusion coefficient D k and D ⊥ arewell known as D k ∝ /r , D ⊥ ∝ /r and D k = 2 D ⊥ [12]. For the particles confined near the solid wall, thestrength of hydrodynamic interactions (HIs) decay withinter-particle separation r as 1 /r [4, 13, 14]. In addi-tion, spatial symmetry break due to the boundary con-ditions brings asymmetry feature to cross correlated dif-fusion of particles [2, 12–15]. For the particles on theair-water interface or a viscous membrane, both experi-mental [16] and theoretical [15] studies show D k ∝ /r and D ⊥ ∝ /r . Oppenheimer’s et al. calculation [17]showed that at the membrane adjacent to a solid wall,the correlated diffusion of particles is a function of thedistance between the membrane and solid wall.The influence of different boundary conditions could bedirectly reflected on diffusion behavior of particles [18–22]. Many studies base on solid wall condition, a non-slipboundary, which could cut off the fluid field in its vicin-ity. For a fluid-fluid interface (say a soft wall), there isa slip boundary which partially transforms the flow field ∗ Corresponding author. E-mail: [email protected]; † E-mail: [email protected] around. Compared with extensive investigations on thesolid wall’s effects, there are few experimental studiesabout the influences of the fluid-fluid interface on parti-cles’ correlated diffusion.In this paper, we report the experimental measurementof the cross-correlated diffusion of colloidal particles nearthe water-decahydronaphthalene (decalin) interface. Theparticle monolayer is similar to a viscous membrane. Thecorrelated diffusion of particle pair shows asymmetric be-havior in two normal directions. Along the line joiningthe centers of particles, the decay rate of correlated dif-fusion is constant. At the perpendicular direction, thedecay rate increases with the particle separation. In bothdirections, the correlated diffusion is independent of thearea fraction of particles for long particles separation.The influence of fluid-fluid interface on the dynamic be-havior of the colloidal monolayer tends to be saturated,when the particle separation is much larger than the dis-tance between the interface and monolayer. These phe-nomena reveal weights of HIs through different paths.The remainder of the paper is organized as follows.Section II is devoted to the description of the experi-mental setup. Experimental results and discussions arepresented in Sec. III . Finally, the work is summarizedin Sec. IV.
II. EXPERIMENTAL SETUP
Two kinds of colloidal particle with the same diameterd =2.0v µm are used in the experiment. The first one ispolystyrene (PS) latex spheres purchased from Invitro-gen with sulfate and high density of carboxyl functionalgroups on the surface. The other one is silica spherespurchased from Bangs, which have anionic SiO − groupson the surface when dispersed in water. The density ofthe silica spheres are about 1.9 times larger than that ofPS. Both the polystyrene and silica spheres are the repre-sentative of charged particles commonly used in colloidalscience.The experimental setup is similar to that described inRef. [23]. The sample cell is made from a stainless steeldisk with an inner container in which there is a hole withthe diameter 8.3 mm . The bottom of the hole is sealedwith a 0.1 mm thick glass cover slip, which also serves asan optical window. We first fill the hole to the top edgewith deionized water of 18.2 M Ω · cm in which the par-ticles cleaned 7 times by centrifugation were suspended.And then we add decalin (a mixture of cis and trans withdensity 0.896 g/cm ) to the top of the water, filling theentire height of the inner container. The solvent decalinwas purchased from Sigma-Aldrich. Another cover slipis used to cover the top of the inner container with thewater-decalin interface sandwiched between the top andbottom cover slips. Then we overturn the cell. Particlesfall onto the water-decalin interface caused by the gravityfor one or two hours and form a monolayer, as shown inFig. 1. The colloidal particles are hydrophilic and willprefer to stay in water rather than in decalin. In addition,particles keep a certain distance away from the interfacebecause of image charge in decalin. The image chargeforce is repulsive for particles approach the decalin-waterinterface from the medium (water) with the higher di-electric constant [24]. The water and decalin layers areboth 0.8 mm high. The distance between the cover slipand colloidal monolayer is long enough to eliminate theinfluence of cover slips.With an inverted microscope Olympus Cool-SNAPIX71, the motion of colloidal particles wasrecorded by a digital camera (Prosilica GE1050) of14 frames per second. Each image sequence includes500 consecutive frames ( ≃ . ∗ µm . From these image sequences, usinghomemade software, we obtain the particle positionsand construct trajectories with a spatial resolution of60-100nm. III. EXPERIMENTAL RESULTS ANDDISCUSSIONS
Interaction potential U ( r ) of particles was estimatedfrom pair correlated function g ( r ) of particles. r is theparticle separation. At dilute limit, g ( r ) is related to U ( r ) through the function of U ( r ) /k B T = − ln ( g ( r )).In our experiment, g ( r ) was calculated by averaging 10 particle positions under area fraction n = 0 .
04 for PS andsilica spheres. For even higher area fractions n = 0 . g ( r ) of PS and silica spheres could overlap withthe formers respectively. That indicates the area fractionused could be regarded as the dilute limit. Curves of U ( r )are shown in Fig. 2. Both particles look like hard sphereswhile r > . d . When r is shorter than 2 . d , an attrac-tive potential well appears. For silica particles, the depth of attractive potential well is around 0 . k B T at r = 1 . d .For PS particles, the depth is around 0 . k B T at r = 1 . d .What we found accords with the results in Ref. [25, 26]:like charge particles could attract each other near an in-terface. PS particle shows a wider and deeper attractivepotential than silica, because it has more surface chargethan silica [26].PS monolayer is located at higher position above theoil-water interface than silica because PS sphere has lessmass density and more surface charge. The average dis-tance z from particle monolayer to the interface couldbe estimated by measuring the single diffusion coeffi-cient of particles. From the particle trajectories s ( t ) ,we calculated the single-particle mean square displace-ment (MSD) h ∆ s ( τ ) i = h| s ( t + τ ) − s ( t ) | i . The self-diffusion coefficient D S is obtained according to the equa-tion h ∆ s ( τ ) i = 4 D S ∗ τ for different particle area frac-tion n . Figure 3 shows the dependence of the diffu-sion coefficient D S /D on n for PS and silica spheres. D = k B T / πη w d is the diffusion coefficient of a sin-gle particle in water. The solid lines in Fig. 3 are thesecond-order polynomial fitted results, D S /D = α (1 − βn − γn ) . (1)The parameter α stands for the local viscosity felt by asingle sphere at dilute limit: the larger α is , the smallerthe viscosity is. The parameter β reflects the strengthof two body HIs between the spheres: the smaller β , thesmaller two-body HIs. The fitted values of α , β and γ are given in Table I.The value of α can be used to estimate the distance be-tween the interface and the sphere monolayer z . Accord-ing to the classical prediction given by Lee et al. [22, 27],the distance from particle’s center to the interface z canbe written as: za = 3(2 η w − η o )16( η w + η o )( α − , (2)where a is particle radius, η w is the viscosity of waterand η o = 2 . cP is the viscosity of decalin at 22.5 C .Substituting the fitted values of α into Eq. 2, we ob-tain z = 2 . µm ± . µm (1 . µm ± . µm ) for PS (silica)sphere. The calculation is consistent well with the ex-perimental observation under the microscope: particleskeeps 1 − µm distance away from the interface.The particles slightly fluctuate in vertical direction un-der k B T . The fluctuation amplitude of silica particles issmaller than PS because the former has heavier massdensity. Within the area fraction region we measured,particles near the interface could be viewed as a mono-layer. We focus our investigation on the lateral motion ofparticles, parallel to the oil-water interface, in followingpart of the paper.Following the trajectory s i ( t ) of individual particles i ,particles’ cross-correlated motion is obtained via the en-semble averaged tensor product of the particle displace-ments [12]: M xy ( r, τ ) = < ∆ s ix ( t, τ )∆ s jy ( t, τ ) > i = j,t . (3)Here ∆ s ix ( t, τ ) = s ix ( t + τ ) − s ix ( t ). i, j are particle in-dicates, x and y represent different coordinates, and r is the separation between particle i and j (shown as theinset of Fig. 4). The off-diagonal elements are uncor-related. We focus on the diagonal elements of this ten-sor product: M rr which indicates the correlated motionalong the line joining the centers of particles (called par-allel direction), and M θθ which represents the correlatedmotion perpendicular to this line (called perpendiculardirection). We found that the measured correlated mo-tion M rr and M θθ are linear functions of τ for small lagtime τ . Thus, the cross-correlated diffusion coefficientsare defined as D k = M rr / τ and D ⊥ = M θθ / τ .Figure 4 exhibits D k and D ⊥ of PS and silica spheresas a function of r for different area fraction n . Each curveof an arbitrary n , whose deviation was given in the legendof Fig.4, was obtained by averaging at least 10 particlepositions. The behaviors of D k and D ⊥ of PS and silicamonolayer are qualitatively similar. With the increase of r , D k and D ⊥ of PS and silica sphere decrease followingthe power law. The specific form of the power law willbe discussed in the following section. When r > . d ,the curves of D k ( r ) and D ⊥ ( r ) for different n almost col-lapse onto a single curve respectively, i.e., particle areafraction hardly affects the particles’ cross-correlated mo-tion. When r < . d , however, D ⊥ ( r ) is dependent onthe area fraction n . With the increase of n , the decay rateof D ⊥ ( r ) decreases, i.e., the curves become more flat.The mechanism of n independence effect mentionedabove is in line with Oppenheimer et al. ’s theoreticaldescription in Ref. [17]. In the particle monolayer, thefar-field response of the correlated particle motion mainlyarises from the momentum diffusion through the 3D sur-rounding fluid. Usually the viscosity of 3D fluid suspen-sion is the function of volume fraction φ of particles. Inour system φ could be regarded as zero almost, no matterhow the area fraction n of monolayer changes. The vis-cosity of 3D fluid almost keeps as a constant. Thus thefar-field 3D HIs between the particles are independent ofthe area fraction n , so does the cross-correlated diffusion D k ( r ) and D ⊥ ( r ). Cui et al. [13] also observed the similarconcentration independent effects for particles confinedbetween two plates, but the reasons to their phenom-ena are different. In the confined particle monolayer, thefluid momentum is absorbed by the solid boundaries, andthe far-field fluid response arises solely from mass-dipoleperturbation, which is not influenced by the presence ofneighbor particles.The situation will be more complicated for short r ( < . d ), because the particles reveal an attractive poten-tial in the range (Fig. 2). Both HIs and thermodynamicinteraction involve in particle’s correlated motion now.For the potential is attractive, the correlation of particlemotion will become stronger in the range. As a result,the curves become more flat. For HIs, the near-field re-sponse of the correlated motion mainly arises from themomentum diffusion through the two-dimensional (2D)colloidal layer itself. The 2D monolayer viscosity η m in- creases with the area fraction n of the particles. As aresult, the correlated motion is a function of n .To focus on the influences of the oil-water interface onthe correlated diffusion and eliminate the n effects, weaverage the data of different n in Fig. 4. Figure 5 showsthe averaged correlated diffusion coefficient D k ( r ) and D ⊥ ( r ) normalized by αD , which is the single particlediffusion coefficient obtained by Eq. 1, as a function ofparticle separation r . At the parallel direction, there is D k ( r ) /αD = A k /r λ k . Here, A k is the amplitude coef-ficient and λ k = 1 is the decay rate of cross-correlatedmotion. The interface affects only the amplitude A k butnot the decay rate λ k . The amplitude A k of the silicaspheres is larger than that of PS, which is in accordancewith the results from single diffusion measurement: thefitted value of β of silica is greater than that of PS inTable I. The phenomenon indicates that the strength ofHIs between the particle pair is more strong in the silicamonolayer than in PS monolayer. The amplitude of A k of the silica spheres is found almost 1.2 times larger thanthat of PS, which number is just equal to the ratio ofthe β (0 . / .
62 = 1 . λ k = 1 forboth monolayers indicates that the form of HIs is hardlyaffected by the interface in the parallel direction.At the perpendicular direction, D ⊥ ( r ) /αD = A ⊥ /r λ ⊥ . Both the amplitude A ⊥ and decay rate λ ⊥ are functions of the distance z . The decay rates λ ⊥ ofthe silica spheres is larger than that of PS at short r (1.8 vs. 1.5, as labeled in Fig. 4). Different from theconstant λ k , the decay rate λ ⊥ increases with r . At thelong r limit, both λ ⊥ in PS and silica monolayers mergeinto a value of around 2, as shown in Fig. 5. The in-set of Fig. 5 shows that the scaled diffusion coefficient e D ⊥ ( e r ) = D ⊥ ( e r ) / ( αβD ) for both PS and silica particles,where e r = r/z . Both curves are overlapped at long regionof e r >
15, which indicates the effect of the liquid-liquidinterface tends to be saturated.We compare our results with the studies before fordifferent boundary conditions: the colloidal spheres dis-persed in unbounded 3D bulk, confined by the solid wallor at the air-water interface [13–16]. We find the crosscorrelated motion of colloidal spheres near oil-water in-terface is similar to that of a viscous membrane [15, 16]: λ k = 1 and λ ⊥ = 2 at long r limit. For a viscousmembrane, long r limit means that r ≫ L s , where L s ≡ η m / η b is Saffman length [28]. η m is the 2D mem-brane viscosity and η b is the 3D bulk viscosity of sur-rounding fluid. For our system, limit of long r meansthat r ≫ M ax { z, L s } . M ax { z, L s } is the larger one be-tween z and L s .Theory in [15] shows that the D k ( r ) /αD does notdepend on membrane viscosity η m for large r , but D ⊥ ( r ) /αD is dependent on η m . This prediction accordswith our results of λ k and λ ⊥ . Our results suggest thatthe colloidal monolayer near the the interface could be re-garded as a ’membrane’. Even though the precise valueof 2D viscosity η m of colloidal monolayer is hard to beobtained from our present data, the value of λ ⊥ (1.8 or1.5) indicates the η m usually is very small. For a ’mem-brane’ with viscosity η m , the value of λ ⊥ will approachto 2 at low η m limit and approach to 0 for high η m limit[16]. Precisely, D ⊥ ( r ) will tend to a logarithmic form athigh η m limit. Hence L s should be very small also, andthe limit of long r usually means r ≫ z in our system.At first glance, the result that λ ⊥ = 1 . λ ⊥ = 1 . λ shouldcorresponds to a smaller monolayer viscosity. Since thesilica spheres are closer to the interface, it is located ina more viscous environment comparing with PS. Thus, λ ⊥ of the silica spheres seems should be less than that ofPS. The fact is that the viscosity η ( z ) felt by the spheresis the 3D viscosity of the local surrounding fluid, not the2D viscosity η m of the membrane. The influence of thissurrounding viscosity η ( z ) has been removed by 1 / ( αD )scaling, where α stands for the effect of local viscosity η ( z ). The result that λ ⊥ of silica is larger than λ ⊥ ofPS stems from HIs modified by the interface. Oil is moreviscous than water. The flow field induced by the particlein water is suppressed by the oil-water interface. HIs be-tween particles decay faster with their separation r whenthe monolayer is closer to the interface, which leads to alarger measured λ ⊥ for a smaller z .Similar to the mechanism of a membrane near a solidwall [15], HIs among the particles in our system trans-mit through three paths: I) The 2D flow through themonolayer. II) The flow through the fluid layer sand-wiched between the monolayer and interface. III) Theflow through the upper water bulk or the below oil bulk.HIs through path I is dependent on 2D monolayer viscos-ity η m , while HIs through path II and III are dependenton 3D viscosity η o and η w of surrounding fluid.The cross-correlated motion caused by HIs throughPath I is a function of n , because η m is a function of n usually. HIs through Path I only contributes to corre-lated motion when the separation r is within the orderof Saffman length L s = η m / ( η o + η w ). In our system η m of the monolayer is very small. With r increasing, theweight of HIs through path I decreases quickly. Mean-while, HIs through Path II contributes to correlated mo- tion all the time. So, correlated motion is a function of z (see Fig. 5). At long separation limit r ≫ M ax { z, L s } ,the HIs through path III dominate. Since η o and η w are hardly changed with area fraction n , the correlatedmotion should be independent of n (see Fig. 4). Thecorrelated motion is also independent of z for r ≫ z . Atmiddle range of r , HIs through all three paths contribute,their relative weights change gradually with r . The de-cay rate of the cross-correlated motion shows a crossovertendency. IV. CONCLUSION
We investigated experimentally the cross-correlateddiffusion of colloidal particles near the oil-water inter-face. Correlated diffusion coefficient D k ( r ) and D ⊥ ( r )are independent of area fraction n unless at the shortpair separation r . The interface affects correlated diffu-sion coefficient D k ( r ) and D ⊥ ( r ) distinctly. The interfaceenhances the amplitude of D k ( r ), but does not affect itsdecay rate. The results indicate: along the line joiningthe centers of particle pair, the interface modifies the am-plitude of the HIs between the particle pairs, but not theform of HIs. At the direction perpendicular to the line,the influence of the interface changes with increasing r .The interface enhances the decay rate at short r , whilesuch influence tends to be saturated at the long r limit( r & z , z is the distance between the interface and themonolayer). The dependence of the decay rate on r ev-idences that the weights of HIs through different pathschange gradually. Acknowledgments
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04 for PS and silicaparticles.Fig. 3 (color online) Measured self diffusion coefficient D S scaled by D as a function of particles’ area fraction n . Different symbols represent the data for different par-ticles. The solid lines show the second-order polynomialfitting D S /D = α (1 − βn − γn ).Fig. 4 (color online) Measured correlated diffusion co-efficient D k (solid symbols) and D ⊥ (open symbols) as afunction of inter-particle distance r with various valuesof the area fraction n for (a) PS and (b) silica spheres.The solid lines with the slope − − . − . TABLE I: The distance z from particle’s center to the in-terface and the fitted values of α , β and γ in Fig. 3. Here a = 1 µm is the radius of particles.Sample z/a α β γ PS 2.3 ± ± ± ± ± ± ± ± Fig. 5 (color online) The mean correlated diffusioncoefficient D k /αD (solid symbols) and D ⊥ /αD (opensymbols) obtained by averaging the data for various val-ues of the area fraction n in Fig. 4. The square andcircle represent silica and PS particles, respectively. Theinset shows the correlated diffusion coefficient e D ⊥ (opensymbols) as a function of scaled inter-particle distance e r (= r/z ) for PS and silica spheres. The solid lines withthe slope − − Figure 2 r/d U (r) / k B T Silica PS
Figure 3 D S / D n Silica PS -4 -2 -4 -2 -4 -2 D
1/ r r (=r/z) Silica PS D // / D , D / D r ( m)r ( m)