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Investigating Diffusing on Highly Curved Water-Oil Interface Using Three-Dimensional Single Particle Tracking Yaning Zhong, Luyang Zhao, Paul M. Tyrlik, and Gufeng Wang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b01721 • Publication Date (Web): 23 Mar 2017 Downloaded from http://pubs.acs.org on March 30, 2017

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The Journal of Physical Chemistry

Investigating Diffusing on Highly Curved Water-Oil Interface using Three-Dimensional Single Particle Tracking

Yaning Zhong, Luyang Zhao, Paul M. Tyrlik, and Gufeng Wang* Department of Chemistry, North Carolina State University, Raleigh, NC 27695-8204

Address all correspondence to: Gufeng Wang, Department of Chemistry, North Carolina State University, Raleigh, NC 276958204. Tel: (919) 515-1819; E-mail: [email protected]

1 ACS Paragon Plus Environment


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ABSTRACT Diffusion on highly curved surfaces is important to many industrial and biological processes. Despite the progress made in theoretical studies, how diffusion is affected by the curvature is unclear due to experimental challenges. Here, we measured the trajectories of polystyrene nanoparticles diffusing on highly curved water-silicone oil interface, where the oil droplet diameter ranges from several µm to as small as ~400 nm. To analyze the diffusion coefficients on curved surface, an analytical solution developed by Castro-Villarreal containing an infinite series can be used. Through Monte Carlo simulations, we simplified Castro-Villarreal Equation and defined the conditions that satisfy corresponding approximations. For the experiments, unexpectedly, we found that the diffusion slows down significantly when the oil droplet becomes smaller. Possible reasons were discussed and a diffusion-induced droplet deformation and interface fluctuation model is consistent with the experimental results. This study reveals an unexpected decrease of particle diffusion on small oil droplet surface and sheds new light on understanding diffusion on highly curved interface.


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INTRODUCTION Diffusing of biomolecules and nano-objects on highly curved liquid-liquid interface is important to many biological and industrial processes.1-10 For example, biomolecules diffusion on bilipid vesicle surfaces is tightly related to the vesicular functions in the whole spectrum of cellular metabolism and signaling such as endocytosis, exocytosis, active transport in live cells, etc.11-13 In industries, e.g., fine chemical, mining and paint industries, nanoparticles are added to stabilize or activate the emulsions.2,14,15 However, how the nanoparticle interacts with the emulsion is not well understood. Despite its importance, the experimental study of diffusion on highly curved surfaces falls far behind the theoretical studies.2,16-18 The theoretical solution on how to determine the diffusion coefficient (D), as well as the curvature effect on the D, have been given in the literature. Experimentally, single particle tracking technique has been used to track nano-objects in congested environments with a dimension of micrometers such as in porous media and cytoplasm.19-21 However, to the best of our knowledge, there is no report about the diffusion on small bilipid vesicle or droplet surfaces that have a diameter below 1 µm. One major reason is the lack of proper experimental techniques to track the particles’ motion in all three dimensions (3D) with sufficiently high precision and time resolution. As a consequence, how the diffusion is affected by the large curvature is largely unclear due to limited studies. Recently, there have been efforts toward tracking particles or molecules in the full 3D space. These methods can be categorized into three types: intensity-based tracking,22-26 multi-zimage scanning,2,27-31 and imaging using z-sensitive point spread functions (PSF).11,32-37 Intensity-based tracking techniques, e.g. total internal reflection fluorescence (TIRF) microscopy, suffer from very limited tracking range in the z-direction.22-26 In multi-z-image based methods,



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collecting a stack of z-images allows one to track the particle in the 3D space with high accuracy and precision. In addition, multiple particles in the field of view can be tracked simultaneously. However, it is very time consuming to collect a whole stack of z-images, which results in an insufficient time resolution for many applications. Recently, it has been realized that in order to determine the particle’s 3D position, only few data points in the whole image stack are needed. Thus, several groups have developed confocal fluorescence microscopy-based 3D tracking techniques that selectively image a limited number of spots in the object space. This greatly improves the temporal resolution to be below ~1 ms time scale.2,28-30 The shortcomings of this type of techniques are expensive instrumentation, tracking limited to one particle at a time, and the high tendency of losing the particle during the tracking. In z-sensitive PSF imaging, instead of scanning in the z-direction in the object space, the spatial intensity distribution function in the z-direction is altered using a special technique so that a point object’s z-position can be resolved in a single camera exposure.11,32-37 In these methods, several different approaches are used, including wedged prism-based imaging,33 astigmatismbased imaging where vertical rays and horizontal rays are focused to different planes,11,34 achromatic aberration-based bifocal plane imaging,35,36 and double helix point spread functionbased imaging.37 Because of its wide-field imaging nature and only one exposure is required to recover its full 3D information, the temporal resolution can be greatly improved and is only limited by the signal strength and the camera speed. Multiple particles can be tracked simultaneously using this approach. We recently developed a diamond PSF-based 3D single tracking technique. It belongs to the astigmatism-based 3D imaging but the z-working range is greatly extended to be as large as ~10 µm!34 This method is simple and can be easily adopted on most of the commercial


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fluorescence microscopes. By simply inserting a weak cylindrical lens to the optical path, different image patterns of a point source can be generated depending on its axial position. From the image pattern, its z-position can be recovered. With this method, we can localize and track 100 nm nanoparticles in solution with a precision of 8~20 nm in all 3 axes and a time resolution of 30 ms. In this study, we applied this technique to study 100 nm carboxylated polystyrene nanoparticles diffusing on the surfaces of silicone oil droplets with a diameter ranging from ~400 nm to 5 µm. Unexpected results were observed and its possible origins were discussed.

EXPERIMENTAL Chemicals and sample preparation. Silicone oil was from Fisher Scientific (CAS 63148-62-9). Its density was 0.97 g/cm3 and kinematic viscosity 500 cSt. The 100 nm (diameter) nanoparticles were carboxylated polystyrene nanoparticles doped with fluorescent dyes from Life Technology. The 100 nm particles in aqueous suspensions were packaged as 1% solids. The particles were diluted 500,000-fold in 25 mM pH 10.0 CHES buffer. Oil emulsion was prepared by adding 10 µL silicone oil into 100 µL aqueous nanoparticles suspensions and then sonicated for 1-5 seconds. The droplet size can be controlled by changing the sonication time. After preparation, the emulsion was sealed in a solution chamber composed of a glass slide and a coverglass with Scotch tapes as the spacers. The sample was then aged for ~10 min to let the oil droplets settle and attach onto the coverglass surface. The sample was then observed on an upright fluorescence microscope. The oil droplets have diameters ranging from several hundred nanometers to several micrometers, which were estimated using dark field microscopy. The actual diameters of each individual droplets used in the figures were determined from the 3D trajectories. From the top view (xy-plane projection)



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and the side view (xz-plane projection) of the trajectory, the lengths in x, y and z directions were read, averaged, and used as the diameter of the droplet. Epi-fluorescence microscopy and CCD camera. We used an upright Nikon Eclipse 80i microscope to fulfill the 3D single particle tracking experiments.34 To generate astigmatism and obtain different image patterns at different z-positions, we inserted a cylindrical lens (f = 1.0 m) between the microscope objective and the tube lens (~7.5 cm from the tube lens). The signal was collected by a 100× Apo TIRF/1.49 oil immersion objective. A P-725 PIFOC long-travel objective scanner from Physik Instrument (model no. P725.2CD) was used to control the axial distance from the sample to the objective. The images and movies were acquired by an Andor iXon 897 camera (512 × 512 imaging array, 16 µm pixel size). 30 ms integration time was used in all dynamic tracking unless specified. MATLAB and NIH ImageJ were used to analyze and process the collected images and videos.

RESULTS AND DISCUSSION Theory of diffusion on curved surfaces. On a flat, infinite surface, the diffusion equation (Fick’s second law) in Cartesian coordinates is:

∂2P ∂2P 1 ∂2P + = ∂x 2 ∂y 2 D ∂t 2

(1)

where P (x, y, t) is the probability of finding a diffusing entity in the small vicinity near of the point (x, y) at time t; D is the diffusion coefficient. Given that the diffusing entity is at the origin at t = 0, the solution for Equation 1 is: r2 v P ( r , t ) = ( 4πDt ) −n / 2 exp( − ) 4 Dt

(2)


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v where r is the position vector in polar coordinates; n is the number of dimensions and here it

equals to 2. The first moment of the probability distribution (or the mean of r) is zero, which tells that the average position of the diffusing entity is the origin. The second moment of the probability distribution gives the mean squared displacement (MSD) ~ t relationship: r 2 = 2nDt

(3)

Thus, the diffusion coefficient D can be calculated from the MSD using Equation 3, where MSD is defined as: r 2 = ∆x 2 + ∆y 2 + ∆z 2

(4)

On a curved surface, however, the relationship in Equation 3 is no longer valid. Starting from the diffusion equation (Fick’s second law), Castro-Villarreal derived a general polynomial form for the mean-squared geodesic distance (MSGD) ~ t relationship on a n-dimensional curved surface with a constant curvature.38 For a spherical surface in the 3D space:

s 2 = 4 Dt −

2 ( −2) 2 Rg ( Dt ) 2 + Rg ( Dt ) 3 + ...... 3 45

(5)

where s is the geodesic distance, i.e., the shortest distance on the curved surface between two points; Rg is the scalar curvature of the surface (Rg = 2/R for spheres where R is the radius of the sphere). This equation predicts that the MSGD will be linear at the short time limit and level off at the long time limit. This trend is consistent with the experimentally acquired MSD curves as discussed in the experimental 3D trajectory section in this manuscript. While Equation 5, or the Castro-Villarreal Equation, gives the analytical solution on how to determine the diffusion coefficient, it is impractical to use it because of the infinite number of the polynomial terms in the equation. Thus, we must make reasonable approximations to



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simplify Equation 5 by truncating high order polynomials in determining experimental diffusion coefficients.

Monte Carlo simulation of diffusion on spherical surface. To find out how to simplify Equation 5, we used Monte Carlo simulation to model particle diffusing on spherical surfaces with a known diffusion coefficient. Then, the recovered diffusion coefficients using Equation 5 simplified to different extents were compared. A successfully simplified Equation 5 shall be simple yet give sufficiently accurate recovered diffusion coefficients. In the simulation, we assumed that the particle randomly walked on the spherical surface with a diffusion coefficient D. The rational was that when the time interval is sufficiently small, the diffusion on a curved surface can be approximated as the diffusion on a flat surface within the time interval. As the particles diffused on the surface, their steps were projected to the sphere surface and an arbitrarily long trajectory in the 3D can be constructed. In practice, the simulation was started by assuming a particle’s initial position. The tangential plane was determined and the particle then randomly walked in the 2D tangential plane for a step. The random step was generated by assuming two independent variables: a normally distributed step size r and a uniformly distributed angle φ between 0 and π. The step size r for each step was randomly generated using Matlab program so that it has a Gaussian distribution with a standard deviation of σ:

σ 2= 4 Dt

(6)

where t is the time interval between steps. After each step, the new position of the particle was determined in the tangential plane and then projected onto the spherical surface. The new position of the particle on the sphere surface then served as the starting point of the next step.


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The particle was kept diffusing on the spherical surface and a trajectory with fine time resolution was constructed. By skipping data points on the trajectory with fine resolution, we then built trajectories with lower time resolution. We simulated the trajectories of particle diffusing on differently sized droplet surface with different diffusion coefficients. The presented simulation data shows D’s ranging from 0.10 to 0.40 µm2/s, which are similar to the D’s observed in the experiments as discussed later. The droplet size varied from 300 nm to 3 micrometers. In each case, 400 trajectories were simulated and the reported average and standard deviation of the D’s were from the 400 simulations. A time interval of 10 µs was used. Figures 1abc show the simulated trajectories of a point object diffusing on droplets with a size of 2 µm, 1 µm, and 500 nm, respectively. The diffusion coefficient was 0.40 µm2/s and the trajectory time resolution was 30 ms to approximate the experimental conditions. Corresponding movies can be found in the Supporting Information (Movies 1-3, respectively). The moving speed of the particle on the droplet surface is very similar to those of experimentally collected trajectories (Movies 4b and 5b).

Recovering D in the linear region of Castro-Villarreal Equation (Equation 5) from simulated data. We then attempted to recover the D’s form the simulated data. By using the straight line distance r, we plotted the MSD curves as a function of time (for examples, see Figures 1def). As has been discussed earlier, when the time resolution is sufficiently small so that each diffusion step is much smaller than the droplet size, the diffusion on a curved surface can be approximated as 2D diffusion within a few steps. Then, the diffusion coefficient can be recovered using conventional 2D MSD method using Equation 3, or, the Castro-Villarreal



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equation containing only the first order term. To satisfy this approximation, it requires that either the droplet size is large, or the time interval is small, or both. This requirement is clearly disclosed from the MSD curves: when the droplet is large (2 µm), the MSD curve is fairly linear for the first a few data points (Figure 1d). However, as the droplet size becomes smaller, the MSD curve deviates from a straight line and levels off quickly (Figures 1def). This is because the diffusion distance in each step is large compared to the droplet size so that the curvature can no longer be ignored. Nonetheless, we recovered the D’s using linearized Castro-Villarreal Equation, which is equivalent to Equation 3 with a dimensionality n of 2. In the linear fitting, the first 3 data points in the MSD curve were used for consistency even though the apparent curvature can be identified for those of small droplets. Figures 2ab show the recovered diffusion coefficient as a function of the droplet size and the assumed collection time resolution of the trajectory. The assumed D’s were 0.40 and 0.10 µm2/s, respectively, for Figures 2a and 2b. The low time resolution trajectories were obtained by skipping data points from high time resolution trajectories produced in the simulation. Each data point in Figure 2 is the average from 400 simulated trajectories and the error bar stands for the standard deviation. The difference between the assumed and recovered D’s is apparent. First, when the assumed collection time resolution is high, e.g., 10 µs, the recovered D’s are accurate and identical to the assumed D’s: 0.40 ± 0.01 µm2/s and 0.10 ± 0.01 µm2/s, respectively, even for the smallest droplet with a diameter of 300 nm. Second, as the observation interval becomes longer, the recovered D’s are significantly underestimated for small droplets, which is caused by the deviation of the MSD curves from a straight line (e.g., Figures 1def). Especially, when the time resolution is low, e.g., 30 ms, the


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recovered D drops quickly as the droplet size decreases (Figure 2a). The recovered D is only 12.5% of the actual D for 300 nm droplets. Figure 2b shows that using a smaller assumed D (0.10 µm2/s) gives the same trend but less extent of underestimation. These results show that the small time interval assumption is no longer valid as the droplet size decreases and/or the observation time interval increases. The contributing factors for the error are two-fold: (1) substituting the geodesic distance with the straight line distance; and (2) the higher order polynomials in Equation 5 become significant when the curvature becomes larger. Third, under what conditions can we use linearized Castro-Villarreal Equation (Equation 3) in measuring diffusion coefficient on curved surfaces? Apparently, the underestimation error is related to all 3 factors: the diffusion coefficient D, the droplet diameter d, and the time interval t in measurements. There does not seem to be a simple relationship to relate them with errors. However, the error must be only related to the ratio between the average step size r within time t and the droplet size d, regardless of their absolute sizes. Empirically, we found that as the average step size r (estimated from Equation 3) is smaller than 22% of the droplet size d, the underestimation error is smaller than ~17.5%. For example, when D is 0.40 µm2/s and droplet size is 1 µm, using a 30 ms integration time will yield a D ~17.5% (0.33 µm2/s) smaller than the true value. We deem that at this condition, the D can still be satisfactorily estimated. To summarize, the condition that the conventional MSD method (Equation 3) can be used on a curved surface is: 4 Dt / d < 0.22

(7)

which will give an error in D 0.98), the D can be accurately recovered from the linearized Equation 5. For the MSGD curves showing apparent leveling off, higher order polynomials of Equation 5 must be used. Practically, we found that the MSGD curve need to be truncated before it reaches ~90% of the maximum value in order to use simplified Equation 5 with lower order of polynomials. For example, Figure 3c shows the second order fitting results using different number of data points in the MSGD curve for differently sized droplets. For large droplets, e.g., 1500 nm and 3000 nm droplets, even when the MSGD curvature is small and the leveling off of the MSGD curve is not significant, a second order fitting using 3-8 data points



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still gave the expected D values irrespective of the number of data points used. However, for 400 nm droplets, the MSGD curve approaches the maximum value after a total of 4 data points (Figure 3d). The second order fitting using 3 or 4 data points on MSGD curve gave the expected value while incorporating more data points in the fitting will underestimate the D. This is because once the curve levels off, it can no longer be approximated by a lower order polynomial function. Thus, we must truncate the MSGD curve before it reaches ~90% of the maximum value. With this criterion, using how many data points has little effect on the recovered D value; however, we usually use a minimum of 4 data points for the fitting in Figure 3a. As an alternative, if we use the full MSGD curve, we must incorporate more polynomial terms in Equation 5 for fitting. Practically, we found that a minimum of ~10 terms are needed to obtain a good fitting of the curve, possibly because the polynomial terms converge to zero slowly. Moreover, more variations in the recovered D’s were encountered. We deem that using the full MSGD curve for recovering the D is impractical and unnecessary. (5) For too small droplets, there will not be enough data points collected before the MSGD curve reaches maximum. We found that the low limit of the droplet can be determined by 4 Dt / d > 0.55

(9)

Or, for 400 nm droplet, the 30 ms time resolution is too poor to resolve a D of 0.40 µm2/s. To summarize, we found that to resolve diffusion coefficient on highly curved surfaces, the quadratic form of Equation 5 (i.e., Equation 5 truncated to the second order) can be used to recover accurate D’s. The applicable conditions are given in Equations 7 and 9.

Experimental 3D trajectories. With the knowledge obtained from the simulations, we measured the 3D trajectories of 53 particles diffusing on silicone oil droplet surface using our astigmatism-


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based 3D imaging technique. Under the fluorescence microscope with artificially induced astigmatism, the nanoparticles gave diamond-like imaging pattern (for examples, see Movies 4a and 5a). When they diffused on oil droplet surface, the particle images moved laterally depending on the droplet size. Meanwhile, the image pattern changed, indicating that they were moving along the axial direction simultaneously. Using our 3D tracking technique, we recovered the 3D trajectories of 53 diffusing 100nm polystyrene nanoparticles on silicone oil droplet-water interface. Figures 4 and 5 show typical trajectories (1000 frames, or 33 s) of 100 nm nanoparticles diffusing on a large and a small oil droplet, respectively. The large droplet is slightly deformed (Figure 4; Movie 4a: the original fluorescence images; Movie 4b: the recovered 3D trajectory). The top view shows a circular shape, with the diameter in the y-direction slightly longer than that in the x-direction. The side view shows that the droplet is round on the sides but flat on the top, suggesting the droplet is attached to the glass surface on the top. The middle portion of the top view shows little particle activity, confirming that the flat region is caused by the contact of the oil and glass surface so that the particle did not diffuse in there. The average size (diameter) is 2.6 µm from the 3 directions. Figure 5 shows the 3D trajectory of a smaller oil droplet (Movie 5a: the original fluorescence images; Movie 5b: the recovered 3D trajectory). The average size from the 3 dimensions is 690 nm. From the 3D trajectories, one can tell that the oil droplet is attached to the coverglass by showing a flat top (Figure 5b). It is clear that the nanoparticle is only diffusing on the surface of the oil droplet. Figures 5c and 5d show the top views of the particle trajectories at selected z-sections. In the middle part of the droplet, the projection of the particle trajectory forms a ring shape, indicating that the particle is only diffusion on the surface of the droplet



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(Figure 5d). At the droplet-glass contact region, the particle trajectory has a smaller footprint, with the middle portion open, again indicating that this open part is inaccessible to the particle because of the oil-glass contact. From the 3D trajectories, we can tell that the localization precision is excellent. The localization error is negligible as compared to the droplet dimension as small as several hundred nanometers in diameter. Our 3D trajectories also show that the oil droplets are not perfectly spherical, possibly because the contact of the droplet with glass surface distorted the droplet, leading to a deformed spherical trajectory.

Experimentally recovered D on oil droplet surface. Form the 3D trajectories, we were able to estimate the diffusion coefficient D of the particles on oil droplet surfaces. The time resolution is crucial in obtaining the diffusion coefficient especially on small oil droplets. However, increasing the time resolution sacrifices the signal to noise ratio thus the localization precision. To compromise, we used 30 ms integration time throughout this manuscript. From earlier studies, we know that this time resolution is sufficient to recover D’s (~0.40 µm2/s) on 400 nm droplets. It is important to point out here that for the Brownian motion of a massive particle under the influence of friction (Ornstein–Uhlenbeck process), the MSD scales ballistically with respect to time at a short time scale,39 i.e., r 2 ∝ Dt 2 . The inertial effect could affect the diffusion process in our system. We plotted the MSD curves for these diffusing particles on oil droplet surfaces. They show similar trends as those from the simulated trajectories: deviating from a straight line and leveling off with respect to time (Figures 4d and 5f). Thus, we did not observe the inertial effect, possibly because: (1) the time resolution is poor (30 ms), longer than the typical


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momentum relaxation time; and (2) the MSD curve is dominated by the curvature effect on the spherical droplet surface. Using the knowledge obtained from the simulations and the second order CastroVillarreal Equation, we recovered the 3D trajectories of the 53 particles diffusing on silicone oil droplet surface, plotted their MSGD curves as a function of time, and determined their diffusion coefficients using the quadratic from of Equation 5. The droplet size ranged from 390 nm to 4.9 µm. Figure 6a shows their D’s as a function of the oil droplet size. The diffusion coefficients have values ranging from 0.10 ~ 0.55 µm2/s, with the largest measured value ~1 order of magnitude smaller than those in aqueous solution (4.4 µm2/s).34 The smaller D’s on oil droplet surface is expected because the dynamic viscosity of the silicon oil (kinematic viscosity 500 cSt reported by the vendor) is ~500 times larger than that of water. Surprisingly, the recovered diffusion coefficient shows a decreasing trend as the droplet size decreases. To better show this trend, we averaged the D’s for each 5 neighboring data points with similar droplet size (Figure 6b). There is a significant difference between D’s from large droplets and from small droplets. The diffusion coefficient tends to level off, giving a Langmuir type of curve. To understand this trend, we did a nonlinear least squares (NLLS) fitting using an empirical equation according to the Langmuir shape of the curve: D=

A 1+ B / d β

(10)

where A, B and β are all empirical parameters. It shows that a first order d-dependence (β = 1) gave the best fit for all integer β values (Figure 6b). The fitting results are A = 0.42 µm2/s and B = 0.63 µm, respectively. This fitting shows that as the diameter of the droplet becomes larger (or, approaching a flat interface), the limiting diffusion coefficient is ~0.42 µm2/s. 17 ACS Paragon Plus Environment


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Unexpected slowing down of diffusion on small oil droplet surface. However, the D ~ d relationship is not expected and does not seem to be supported by conventional theories. Under current hydrodynamic theory,14,40-42 the dragging force for the particle diffusing on the interface of two immiscible phases is contributed from both phases. When the dragging force from one of the phase is dominant, e.g. the oil phase, the diffusion coefficient can be approximated according to the modified Stokes-Einstein equation:15,43,44 D=

k BT f

(11)

where kB is Boltzmann constant; T is the temperature; f is the friction coefficient, and is a function of water-oil-particle three-phase contact angle α according to Young’s equation:15,43,44

f = 6πηRP tanh

32(1 + cos α ) 9π 2

(12)

where η is the viscosity of the silicone oil; RP is the radius of the particle; and cos a =

γ po − γ pw γ ow

(13)

γ po , γ pw , and γ ow are the interfacial tensions between particle-oil, particle-water, and oil-water,

respectively (Figure 7a). Since the interfacial tension only changes significantly when the droplet size is reduced to a few nanometers,45 the contact angle should not change for particles on a water-oil droplet interface for droplets have a size of several hundred micrometers. So, the diffusion coefficient should not change significantly as the oil droplet size changes in the hundreds of nm to several µm scale. One possible source for the particle diffusion slowing down on small droplet surface is the increased hydrodynamic friction caused by the increased velocity gradient inside the small 18 ACS Paragon Plus Environment

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droplets due to the droplet boundary confinement. This is also one of the reasons for the particle slowing down when they diffuse near a solid surface23 or in a hollow tube.46 To the best of our knowledge, the solution for the hydrodynamic friction in such a droplet system is complicated and has not been solved in the literature. However, this hydrodynamic friction source is less likely as the increased friction only becomes significant when the gap between the particle surface and the confinement surface is reduced to several nanometers.23,46 In our experiments, the smallest droplet is ~400 nm. Thus, the decrease in diffusion coefficient is less likely caused by the increased friction inside the droplets.

Diffusion-induced droplet surface fluctuation and deformation: fluctuation-dissipation theorem. The increased fluctuation and deformation of the interface induced by particle diffusion on small droplet surface is also a possible reason.47-51 Specifically, the tangential movement of particle will deform the highly curved interface on small droplets (Figure 7b), causing more fluctuations of the interface. Unlike an equilibrium state, the random fluctuation of the interface produces transient, random force F(t) on the particle.52-54 Although the average of the force over time is zero (i.e. F (t ) = 0 ), their effect on the particle is not zero but is equivalent to an additional resistance according to the fluctuation-dissipation theorem:47,55 f =

∞ k BT 1 = k B T ∫ F ( 0 ) F ( t ) dt − ∞ D 2

(14)

where f is the friction coefficient; F is the total random force the particle is experiencing. On the water-small oil droplet interface, the total fluctuation force should include at least two sources – the force from thermal activity of the solvent molecules FH as that on a flat interface, and the force from the fluctuation of interface FI induced by the particle diffusion: F = FH + FI

(15) 19 ACS Paragon Plus Environment


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These two can be viewed as uncorrelated.56 The integration of Equation 14 results in: f = fH + fI

(16)

where fH is the hydrodynamic friction coefficient as one would expect in the absence of the curvature in Equation 12. fI is a new friction coefficient due the interface fluctuation and deformation. The successful prediction of fI requires the knowledge about the characteristic length of the interface fluctuation and the characteristic correlation time of the fluctuation force,56 which are both dependent on the droplet size. At this stage, we are unable to predict fI. However, if we assume that fI is inversely proportional to the droplet size to the β’s power: fI = k / d β

(17)

we have: D=

kT kT kT = = f f H + f I γ (1 + k H

γH

1 ) dβ

D0

= 1+

γH 1

(18)

k dβ

where k is a coefficient; D0 is the diffusion coefficient in the absence of the curvature. Thus, Equation 18 is consistent with the experimentally acquired empirical Equation 10, which suggests that the increased surface fluctuation and deformation may be a reasonable cause for the observed diffusion slowing down on small droplet surface. Now, the parameter A in Equation 10 has the meaning of the diffusion coefficient in the absence of curvature. Parameter B is a characteristic length for the droplet size, at which the diffusion coefficient will decrease by half. We realize that this is not a strict proof. However, this hypothesized mechanism provides a plausible explanation of unexpected experimental observation and shows consistency with the experimental results. It shows that the diffusion of particles on highly curved surface starts to


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significantly slow down when the droplet size decreases to ~ 630 nm. Further studies are needed to fully understand this abnormal slowing down of particles diffusing on small droplets.

CONCLUSIONS To summarize, we measured for the first time the diffusion coefficient for nanoparticles diffusing on oil droplet surface with a diameter from several µm to as small as ~ 400 nm using the 3D single particle tracking technique. A second order Castro-Villarreal Equation can satisfactorily fit the curved mean squared geodesic distance and recover the diffusion coefficient on these small droplets. Unexpectedly, we found that when the oil droplet becomes smaller, the particle diffusion slows down significantly as compared to that on a large droplet surface. This new phenomenon is explained by the particle diffusion-induced interfacial deformation and fluctuation: the particle diffusing on a highly curved surface will lead to the deformation of the interface, which causes more fluctuation of the interface thus an additional large friction for the particle to move according to the fluctuation-dissipation theorem. Such an understanding will help us understand better of the multi-phase nanoparticle-droplet systems.

Acknowledgments This work was supported by the North Carolina State University start-up funds and a FRPD award to G.W.

Supporting Information Available: 7 Movies and movie descriptions. This material is available free of charge via the Internet at http://pubs.acs.org.



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FIGURE CAPTIONS

Figure 1. Simulated trajectories of particle diffusing on droplet surface. (a-c) Top view of the trajectories. (d-f) Droplet size 2000 nm. (d-f) MSD as a function of time. (a) and (d) Droplet size 2000 nm. (b) and (e) 1000 nm. (c) and (f) 500 nm. Diffusion coefficient was assumed to be 0.40 µm2/s. The time interval was assumed to be 30 ms.



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Figure 2. Recovered diffusion coefficient using linearized Castro-Villarreal Equation for simulated data as a function of the droplet size and the collection time resolution. Diffusion coefficient was assumed to be 0.40 µm2/s for (a) and 0.10 µm2/s for (b), respectively.


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Figure 3. Recovering diffusion coefficient using Castro-Villarreal Equation truncated to different orders of polynomials. (a) Recovered diffusion coefficient as a function of droplet size and the highest order of the polynomials using simulated trajectories. The collection time was assumed to be 30 ms. Diffusion coefficient was assumed to be 0.40 µm2/s. (b) The average standard deviation of recovered D’s as a function of the order of the polynomials in the fitting. (c) Fitting results with quadratic form of Equation 5 using different number of data points in the MSGD. (d) A sample fitting of the MSGD using the first order and the second order polynomials for a particle diffusing on 400 nm droplet surface.



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Figure 4. 3D trajectory of a 100 nm nanoparticle diffusing on a 2600 nm oil droplet surface. (a) Top view of the trajectory. (b) Side view. (c) 3D view. (d) Mean squared displacement as a function of the time.


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Figure 5. 3D trajectory of a 100 nm nanoparticle diffusing on a 690 nm oil droplet surface. (a) Top view of the trajectory. (b) Side view. (c) Top view of the particle trajectory with a z-range above 900 nm. (d) Top view of the particle trajectory with a z-range of 400-600 nm. (e) 3D view. (f) Mean squared displacement as a function of the time.



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Figure 6. Recovered diffusion coefficients on silicone oil droplet surface. (a) The D for all 53 particles vs. droplet size. The droplet size was determined from the experimentally determined 3D trajectories. (b) Averaged D from each 5 neighboring data points with similar droplet size. The red curve is a theoretical fitting using Equation 10.


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Figure 7. Schematics of particle diffusion at oil-water interface. (a) Three-phase contact angle. (b) Additional deformation and fluctuation of the droplet surface caused by the diffusion of the particle. The graph was drawn with a particle-to-droplet diameter ratio of 1:6.2.



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