J. Phys. Chem. C 2010, 114, 4479–4485
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Measuring Flow Profiles and Slip by Single Molecule Tracking Experiments in Thin Liquid Films Arne Schob, Martin Pumpa, Markus Selmke, and Frank Cichos* Molecular Nanophotonics, Institute of Experimental Physics I, UniVersity of Leipzig, 04103 Leipzig, Germany ReceiVed: September 15, 2009; ReVised Manuscript ReceiVed: NoVember 26, 2009
A method for analyzing single molecule tracking data is presented, which allows the measurement of flow profiles in shear flows of ultrathin liquid films. The results show that the velocity profile detected by single molecule tracking is in general different from the analytical solution of the Navier-Stokes equation due to the diffusion of the probe molecules. However, the theory presented allows the extraction of even slip boundary phenomena, which for the first time provides access to boundary conditions on a molecular scale. A verification of the theory and its capabilities is presented by means of tracking single polystyrene particles in a 500 nm thin water film confined in a surface forces apparatus. Introduction
Experimental Setup
The flow of liquids in small channels and in confined geometries has become a very important issue for applications in micro- and nanofluidics1–3 or lubrication.4–10 Since in small dimensions surface effects start to become important for the whole system or even determine the properties of the system, it is of great interest to verify the details of the interaction of liquids at solid boundaries. One of the most discussed properties is the boundary condition for liquid flow, which is termed either “stick” or “slip”. Stick refers to the assumption that liquid molecules next to the solid boundary possess the same velocity as the solid, whereas for slip, liquid molecules slide across the solid surface and have a velocity different from the solid surface. There have been a number of experiments and simulations providing evidence for slip on solid boundaries11–14 and structured surfaces.15 However, none of those experiments is able to provide a molecular picture of this process. Single molecule detection has shown the ability to provide such insight into molecular processes. Single molecule tracking is an especially versatile tool to follow the motion of single molecules in different environments. For example, it has been used to reveal molecular dynamics in lipid membranes.16 Measurements of single molecule motion on random networks of actin filaments17 have demonstrated that directed motion coupled to a random morphology may in the same way lead to apparent diffusive behavior, which cannot be identified by ensemble experiments. Experiments on single molecule diffusion on ultrathin liquid films have provided indications of near order at the solid boundary.18 Single molecule experiments in confined liquid films, where the confinement is controlled by a surface forces apparatus down to nanometer film thickness, further allow access to a new dimension of liquid dynamics.19 Within this report, we present a method that is capable of measuring flow, flow direction, and velocity profiles in ultrathin confined liquids by the use of single molecule tracking. An experimental verification of the method using single fluorescent particles as tracers is presented.
The experiments on liquids under shear have been carried out using a modified surface forces apparatus (SFA) with the ability to track single molecules within the confined liquid as described in detail elsewhere.19 Measurements have been carried out on stained polystyrene beads (d ) 200 nm, Molecular Probes) in water. The particle solution has been diluted to obtain suitable concentrations for single particle tracking. A droplet of the obtained solution has been confined between two glass cover slides mounted on the cylinder surfaces of the SFA. The thickness of the liquid film was about 500 nm, as determined by a capacitive distance sensor. To measure the influence of shear on single particle tracking experiments, different shear velocities between 0.7 and 12.5 µm/s have been applied to the lower glass surface of the SFA. For all measurements an exposure time of ∆t ) 20 ms and a frame rate of f ) 50 Hz was used to record 1000 frames per measurement. The analysis of the recorded images was conducted using a tracking software package developed in the lab.
* To whom correspondence should be addressed,
[email protected].
∆x ) [b(t r + ∆t) - b(t)] r ·b e
Theory and Experimental Results Brownian motion obeys Gaussian statistics, which shows up in a Gaussian step size probability distribution. In three dimensions this is valid for each dimension, and the overall three-dimensional step size distribution is governed by the product of the step size distribution for each dimension given by
p(∆x) )
1
√4πD∆t
(
exp -
∆x2 4D∆t
)
(1)
where D is the diffusion coefficient and ∆t denotes the time. Regardless of the specific system and its dimensionality, one obtains a one-dimensional diffusion problem by projecting the three-dimensional displacement ∆r b ) b(t) r - b(t r + ∆t) onto an arbitrary direction represented by the unit vector b e with
10.1021/jp908915n 2010 American Chemical Society Published on Web 02/23/2010
(2)
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Schob et al. whereas the observed diffusion constant Dobs of the ensemble of tracked particles is
Dobs )
Figure 1. Step size distribution (from about 50000 steps) of diffusing PS spheres in a 500 nm thin sheared water film (V ) 10 µm/s). An offset from ∆x ) 0, which is the result of the applied shear velocity (see eq 4) is clearly visible. The solid line shows a fit with a Gaussian to determine the observed shear speed (offset of the Gaussian) and diffusion constant (width of the Gaussian).
As described in ref 20, this type of analysis can be carried out on any subensemble of molecules, e.g., for certain spatial regions of the sample [r b, b r + dr b] and any desired direction b e and time interval ∆t. This provides spatial and directional sensitivity to the analysis of single molecule dynamics and easily allows the detection of spatial heterogeneities. This analysis can be further extended to the detection of flow in liquids, which is the main topic of this paper. For molecules in a resting liquid the mean 〈∆x〉 of the step size distribution (eq 1) is zero. Moving the liquid at a constant velocity b V results in a constant offset ∆r bflow which adds to the diffusive displacement of a molecule. Projecting this offset onto the same axis b e as used before gives
(3)
Therefore, the step size distribution for diffusing molecules within a flow can be written as31
p(∆x, b V) )
1
√4πD∆t
(
exp -
(∆x - (V b·b e ) · ∆t)2 4D∆t
)
(4)
Following this, the mean of the step size distribution of a diffusing molecule in a flowing liquid is given by 〈∆x〉 ) (V b ·b e) · ∆t. Figure 1 shows as an example the measured step size distribution for polystyrene beads in a thin water film (thickness 500 nm) measured by single particle tracking in a Couette flow with a shear velocity of 10 µm/s. The distribution clearly reveals a Gaussian with a center displaced from the origin ∆x ) 0. The variance σ2 of the Gaussian distribution in eq 1 is still σ2 ) 2D∆t and is independent of the imposed flow velocity. Nevertheless, the displacement of the center is directly related to the flow velocity. Thus, both free parameters of the Gaussian probability distribution allow to separate diffusive and directed motion based on measurements on single particles. Further, the flow direction can be obtained by determining the displacement of the Gaussian (〈∆x〉) as a function of the direction of the vector b e. The observed velocity Vobs of the medium is then given by
Vobs )
〈∆x〉
∆t
(5)
(6)
Note, that the observables in eqs 5 and 6 are denoted as “observed” values, since in general they may be distributed and, therefore, subject to averaging. Furthermore, the relation Dobs/D ) 1 - f∆t/3 ) 2/3 connects the observed apparent Dobs with the true D due to finite exposure time effects in case of equal exposure time ∆t and inverse frame rate f-1.33,34 The amount of averaging depends in detail on the experiment itself. For example, a liquid in a Couette flow between two parallel plates obeys a linear variation of the velocity across the film thickness. The liquid velocity increases linearly from resting to the moving plate. Thus, different particles at different distances from the resting plate will experience different local velocities and, therefore, report different velocities and even diffusion constants. For a simple qualitative picture, one may assume different subensembles of probes which are subject to different shear velocities and, thus, obey a step size distribution, which is shifted by a different amount. If the relative shift between distributions is large enough, the overall step size distribution consisting of all subensembles will be broadened compared to the width of each subensemble and can in general not be described by a Gaussian. The actual step size distribution p(∆x) is then found by convoluting the probability of finding a certain local velocity in the liquid p(V) with the step size distribution p(∆x, b V) as obtained from eq 4. This convolution is given by
p(∆x) )
∆xflow ) ∆b r flow · b e ) (V b·b e ) · ∆t
σ2 2∆t
∫0V
max
p(∆x, b V ·b e ) · p(V) dV
(7)
Thus, in general, a distribution of local velocities will increase the width of the observable step size distribution, which corresponds to an apparent enhancement of the diffusivity similar to the well-known Taylor dispersion.21 This enhancement, however, occurs only in the direction of the imposed flow velocity. In the same way as the velocity distribution enhances the observed diffusivity, the observed velocity itself is modified. In the following, we will develop a more quantitative description of the influence of local velocity distributions and probe diffusivity on the observed velocity and diffusion constant. Choosing the right conditions will allow the determination of velocity profiles even for ultrathin liquid films of a few nanometer thickness and provide access to the hydrodynamic boundary conditions on a molecular scale. Within the following, we will restrict our analysis to the simplest flow profile of a Couette flow. This type of flow is observed, if the liquid is confined by two planar solid substrates, which are moved relative to each other. This applies to all shear experiments carried out in a SFA.22–26 To simplify the calculations and predictions, we introduce the following dimensionless variables. We will use the dimensionless position y* ) y/d within a confined film of thickness d, the relative local velocity V*(y*) ) V(y*)/Vshear where Vshear denotes the velocity applied to the moving surface and a relative slip length δ* ) δ/d for each surface. The slip length δ is defined as the distance from the surface within the solid phase, where the extrapolated flow velocity vanishes.27 A slip length δ ) 0 corresponds to the “stick” boundary condition, whereas
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δ > 0 describes the “slip” boundary condition. The velocity profile of a Couette flow can then be expressed as
V*(y*) )
y* + δ1* 1 + δ1* + δ2*
(8)
where δ1* and δ2* are the corresponding relative slip lengths at the resting and the moving surfaces. For symmetric “stick” boundary conditions (δ1* ) δ2* ) 0), the profile reduces to V*(y*) ) y*. Equation 8 can also be written as
V*(y*) ) [V*(1) - V*(0)](y* - 1/2) + V*(1/2)
(9)
which means that different boundary conditions just provide a linear transformation of the simple profile for symmetric stick. The linearity of the flow profile suggests an equally distributed probability to find a certain velocity within the liquid film. However, this equal distribution is not accessible by single diffusing probes in the film. The step size distribution as determined in an experiment is also the result of a diffusional averaging of different flow velocities. The three spatial coordinates are therefore not anymore independent of each other. Consequently each molecule reports a mean velocity 〈V*〉, which is the average of the local velocities V*(y*) over a certain range of y within the film. This average is given by a convolution of the velocity profile with the probability that a molecule is detected when starting at a certain position y0 ∈ [0, d]. This probability is the Gaussian distribution given in eq 1 and the normalized convolution can then be computed. To take the prohibited diffusion through the boundary into account, a reflective barrier treatment31 is required. The reflecting boundary conditions can also be expressed in terms of a mirrored velocity profile (eq 10) at the interfaces (subscript Θ will denote such an extension), which we introduce with the help of a unit step (Heaviside) function Θ(y).
VΘ*(y) ) Θ(y)Θ(d - y)V*(y) + Θ(-y)V*(-y) + Θ(y - d)V*(d - (y - d))
(10)
Figure 2. Observable velocity profiles within a Couette flow between two parallel plates. The reduced velocity is plotted over the reduced position in the film (see text for details). The dashed line represents the solution of the Navier-Stokes equation for no-slip boundary conditions. The solid lines show the observable velocity profiles as probed by diffusing probes in the liquid. The lower slope is due to the diffusional averaging of different velocities during exposure times (ε: 1.3, 2, 3.3, 10). The observed velocity profiles are further modified due to the hydrodynamic interaction of a finite size probe particle with the solid boundaries (dotted lines). Therefore a reduced probe size R* ) R/d ) 0.01 has been assumed.
as the ratio of film thickness and diffusion length. A large epsilon will refer to a case when diffusional averaging can be neglected and the flow velocity only changes slightly over the diffusion length. With eq 12, eq 11 can be simplified to
〈V*(y0*, ε)〉 )
∫-∞∞ VΘ*(y*) exp[-ε2(y* - y0*)2] dy* ∫-∞∞ exp[-ε2(y* - y0*)2] dy* (13)
According to eq 9, eq 13 can be expressed as a linear transformation of the observable velocity profile obtained for stick boundary conditions
〈V*〉 ) [V*(1) - V*(0)][〈V*〉s - 1/2] + V*(1/2)
(14)
This results in
〈V*(y0)〉 )
[
]
(y - y0)2 V *(y) exp dy -∞ Θ 4Dy∆t
∫
∞
∫-∞ ∞
[
]
(y - y0)2 exp dy 4Dy∆t
where 〈V*〉s is given by eq 13 using eqs 9, 10, and 8 with V*(y*) ) y*
(11)
where Dy denotes the diffusion constant for a vertical motion in the film. In general, Dy differs from the diffusion constant for lateral motion D and depends on the position in the film due to boundary effects.28–30 First, however, we will assume Dy ) D. According to this, the experimentally accessible velocity profiles will also in this case differ from the analytical linear Couette profile. The strength of this deviation depends on the speed of diffusion, characterized by the diffusion length ddiff ) (4D∆t)1/2, which has to be compared to the film thickness d. This is expressed in terms of the dimensionless parameter
ε ) d/ddiff
(12)
e-y0
/2ε2
- e-(y0* - 1) ε √πε (y0* - 1){erf([y0* - 1]ε) + 1} + y0* erf(y0*ε)
〈V*(y0*, ε)〉s )
2 2
(15)
This shows that any observable velocity profile is determined by a linear transformation (eq 14) of eq 15. It is thus sufficient to calculate 〈V*〉s once for different relative film thicknesses ε as depicted in Figure 2 for constant Dy ) D. The observed profile converges toward the analytical profile as obtained from the Navier-Stokes equations for ε f ∞. Thus, the diffusion length has to be small compared to the displacement caused by the shear velocity to be able to measure the real spread of velocities in a particle tracking experiment. If ε is small, however, the velocity profile will merge into the mean reduced velocity V*(1/2). In this case, a particle samples all velocities within the time interval ∆t due to diffusion. Finally, the velocity
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Figure 3. Probability density distribution of observable velocities within a Couette as extracted from the velocity profiles shown in Figure 2. The solid lines correspond to the probability density distributions for diffusional averaging of the velocities in the film. The dotted lines further include the modified diffusivity of probes due to the hydrodynamic interaction of the probe with the solid boundaries.
the previous section is the limited vertical domain in which the particles can move. Rather than probing the entire gap y* ∈ [0, 1] the particle may only acquire positions y* ∈ [R*, 1 R*]. This limits the range of possible reported velocities 〈V*(y*) 〉 to the corresponding true Couette flow velocities within that domain. Furthermore, the close proximity of particles to the wall is expected to modify their mobility as y* f R* and y* f 1 - R*. This expected behavior can be quantified and its implications for the analysis are detailed below. Using the diffusion coefficient for a sphere perpendicular to a single wall, DyI, as given in ref 32, and the linear superposition approximation (LSA35,38) describing the combined interaction of two walls, one may obtain an explicit expression for Dy(y0*, R*)/D. Within this expression, various powers of the ratio R/h (where h is the sphere center height above the boundary) appear which may be expressed as ratios R*/y0*. In order to connect this with the previous formulations of the observable velocity profile, the effect of a variable Dy may be taken into account in the analysis by replacing the constant ε with an effective ε′(y0*, R*) that depends on y0* and R*.
profile can be converted into a probability distribution for finding a certain velocity in the film by
ε′(y0*, R*) ) ε
∂〈V*(y*, ε)〉 p(〈V*(y*)〉) ∝ ∂y*
[
-1
]
y*(〈V*〉)
) [erf(y0*ε) - 1 -
erf((y0* - 1)ε)]-1
|
(
Dy,xI(y0*, R*) D
(
Dy,x(y0*, R*) D
)
) ] [( -1
-1 +
-1/2
( [
)ε1+
Dy,xI(1 - y0*, R*) D
) ]) -1
1/2
-1
(17)
(16)
y*(〈V*〉)
Figure 3 displays the results of numerical calculations using the above-described equations to obtain velocity distributions, which can be observed by single particle tracking in sheared liquid films (using reflecting boundary conditions for the diffusion). The velocity distributions depicted display an enhanced probability at the edges due to the flattening of the velocity profiles close to the solid surfaces. Further, as suggested by Figure 3, the velocity distributions become narrower with decreasing ε. These results demonstrate that the velocity profiles, as determined by single particle tracking, deviate from the analytical solution of a Couette flow. This is due to the fact that the velocity is not probed at a fixed spatial position but rather by a diffusing probe. Thus the particle or molecule probes spatial areas of different velocities within a certain time interval ∆t. The described issues are applicable to any flow type and have to be considered whenever a diffusing probe reports a flow velocity. The results, however, do also suggest that velocity distributions can indeed be observed by single particle or even single molecule tracking under certain conditions. The identification of a velocity distribution works best when the broadening of the step size distribution (7) is considerably larger than the accuracy of measuring the width of the step size distribution at rest. Finite Size Effects of the Probe Particles So far, the diffusion coefficient describing the vertical motion, which is causing the diffusional averaging of the observable flow velocity, has been assumed constant throughout the gap. This simplification turns out to be valid as long as the radius R of the particle as compared to the film thickness d is small, i.e., as long as R* ) R/d , 0.5. In these cases, the following considerations will not alter the picture and the analysis significantly. The effect of finite size probe particles is 2-fold. The first obvious alteration of the line of though presented in
with the influence of a single wall being32
( ) ( ) ( ) ( ) (( ) )
DyI(y0*, R*) 1 R* 3 57 R* 9 R* + )1D 8 y0* 2 y0* 100 y0* R* 11 1 R* 5 +O 5 y0* y0*
4
+
(18)
( ) ( ) ( ) ( ) (( ) )
DxI(y0*, R*) 1 R* 3 9 R* + )1D 16 y0* 8 y0* 1 R* 5 +O 16 y0*
45 R* 256 y0* R* 6 y0*
4
-
(19)
Here, ε ) d/ddiff is relating the limiting bulk diffusional length scale (without boundary effects) ddiff ) (4D∆t)1/2 to the film thickness d. The diffusional averaging got position-dependent in the gap: particles close to the boundaries, y0* ≈ R* or y0* ≈ 1 - R*, have a reduced vertical (as well as lateral) mobility as given by Dy(y0*, R*)/D. The effective ε′ grows large as this ratio decreases along with DyI (eq 18). Analogously to the previous discussion of the effect of increasing constant ε which minimized diffusional averaging and caused a closer correspondence between the reported and the true velocity, the same will hold here locally. A close match between the reported velocity 〈V*(y*, ε, R*) 〉 and the Couette flow velocity profile V*(y*) (or for that matter any other kind of velocity profile) is predicted for particles close to the boundaries. A large particle which is comparable in size to the dimension of the confinement d, i.e., R* ≈ 0.5, is in effect similar to a negligeable small diffusional averaging throughout the major part of the attainable vertical space [R*, 1 - R*]. It resembles a large constant ε for a small particle with R* , 0.5 which only probes this reduced vertical range of possible positions. This equivalence is largely independent of ε > 1. Although the details of the averaging are
Measuring Flow Profiles
Figure 4. Direction dependence of the diffusion constant of particles for a small shear velocity (Vshear ) 0.7 µm/s) as determined from an experiment. Within an experimental accuracy (∆D ) 0.015 µm2/s), isotropic motion is found. The extracted diffusion constant is D ) 0.63 µm2/s.
different (to be discussed in a separate paper in preparation), the result is similar in showing a close match between the reported mean velocity 〈V*(y0*, ε)〉 and the actual local Couette flow velocity V*(y*) over the entire domain probable by the spherical particle. For rather pointlike, i.e., molecular probes, the reported velocities will differ from the results obtained with Dy ) D significantly only close to the boundary, where independently of ε the reported velocity will approximate V*(y*). Thus the curves shown in Figure 2 will approximate the linear Couette flow profile (dashed line) close to R* and 1 - R*. This results in a changed appearance of the probability density p(〈V*(y*)〉) close to the wall (see Figure 3) compared to its form as obtained when Dy ) D is assumed. However, the additional attenuated wing of this new probability density distribution does not change the width of the distribution significantly. Although the lateral diffusion coefficient Dx is affected by the hydrodynamic coupling of particles close to the boundary as well (eq 19), the influence is small in most cases and may be ignored in (4) and (7) for the estimation of the effect on the width of the step-size distribution. This is especially true since one is interested in the directional change of this width as given by the convolution and the effect of the boundary on the lateral mobility is invariant with respect to the direction within the plane. Experimental Verification To verify our description and to demonstrate its applicability, we have analyzed the diffusion of single polystyrene particles (R* ) 0.2) in the confining gap of a surface forces apparatus.19 Measurements at two shear speeds of 0.7 and 12.5 µm/s were carried out. The first measurement at low shear speed (0.7 µm/ s) has been used to determine the accuracy of the measurement. The maximum detected shear velocity in this experiment is Vmax obs ) 0.28 µm/s, which is less than 0.8Vshear as predicted by the above theory. The directional analysis of the step size distribution width as depicted in Figure 4 reveals an isotropic diffusion of particles. No enhanced width of the step size distribution is found. Thus, the influence of the shear velocity is small and the experiment can be used to determine the accuracy of the diffusion constant measurement. The obtained apparent mean diffusion constant is 〈D〉 ) 0.63 µm2/s, and the accuracy of measuring the diffusion constant is ∆D ) 0.015 µm2/s. To compare this result with theoretical predictions, the expression for the single wall lateral diffusion coefficient DxI32 and the
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Figure 5. Direction-dependent analysis of the offset of the step size distribution from the origin. The graph shows the determined velocity for different directions of the analysis vector b e. The results show two lobes with positive and negative shear velocity, which correspond to the forward and backward shear motion of our experimental apparatus. The observed shear direction corresponds well to the applied shear direction.
Figure 6. Variance of the step size distribution (weighted by the exposure time) as a function of the analysis direction b e for a shear velocity of Vshear ) 12 µm/s. A clear enhancement of the variance can be found in the direction of the applied shear flow (y axis). The determined enhancement for this experiment is 0.12 µm2/s while 〈D〉 ) 0.59 µm2/s as obtained from the direction perpendicular to the shear. The solid line shows the expected variance of the step size distribution in the case of zero shear velocity and a diffusion constant of 〈D〉 ) 0.59 µm2/s.
LSA35 are needed and result in a value of 〈D〉 ) 1.2 µm2/s. Although a precise and meaningful weighting incorporating the enhanced dwell time close to the boundary surfaces is nontrivial, this computed mean will be an upper bound for the ensembleaveraged lateral diffusion constant37 that is measured and is thus consistent with the measured absolute value. 〈D〉 may be converted to the film thickness to bulk diffusion length ratio ε ) 1.7. If the enhancement of the width of the step size distribution exceeds this accuracy, ∆D, an influence of the shear velocity distribution in the liquid film can be safely detected and the width or even the shape of the velocity distribution can be extracted. The results of measurements at an imposed shear speed of 12.5 µm/s are displayed in Figure 5 and Figure 6. The directional analysis of the measured velocity clearly shows a directionality along the y axis, which corresponds to the direction of the applied shear velocity. The maximum velocity calculated from the analysis is 3.8 µm/s, which is considerably below half the
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applied shear velocity. As mentioned in the theoretical part above, this points to different boundary conditions at both glass surfaces with different slip lengths. The measured width of the step size distributions also clearly deviates from the circular shape expected for isotropic diffusion. The maximum diffusion constant occurs in the direction of the applied shear flow, while the diffusion perpendicular to the flow is the same as for the low shear speed measurements. This enhanced width of the step size distribution is clearly a sign of the existing velocity distribution within the film. To estimate the width of the velocity distribution measured by single particle tracking, we simplify the shape of the velocity distribution (Figure 3) to correspond to a Gaussian with the same width σ ) B. Thus, the width of the convolution in eq 7 can explicitly be given as the sum of the variances of the Gaussian velocity distribution and the Gaussian step size statistics as
2D*∆t ) 2D∆t + (B Vshear ∆t)2
(20)
where D* is the apparent diffusion constant caused by the applied shear flow. For the case of δ1* ) δ2* ) 0 and large particles as well as very small particles, a crude estimate may be readily obtained from Btheo ) V*(y0* ) 1, ε) - 1/2
Btheo ) 1/2 - R*
{
-ε2
erf(ε) - 1/2 + (e
- 1)/επ
1/2
for R* ≈ 1, ε > 1 for R* , 0.5
(21)
The observed enhancement of the diffusion is then
D* - D )
∆t (BVshear)2 2
(22)
According to this, one obtains the width B for the shown experiment as
B)
1 · 12.5 µm/s
2(0.12 µm2 /s) ) 0.28 20 ms
(23)
This experimentally determined width of the velocity distribution can be directly compared to the expected width as calculated from the theory presented in this paper using a diffusion constant of D ) 0.59 µm2/s and, with that, a relative film thickness of ε ) 1.7. The calculation results in a width of Btheo ) 0.3 (eq 21). The experimental measurement is therefore close to the theoretical limit and demonstrates the capabilities of the presented analysis for the determination of velocity profiles in ultrathin liquid films. As already mentioned, the enhanced diffusion constant can only be observed if the accuracy of measuring the diffusion constant does not exceed the enhancement itself. Using the experimental measurement accuracy ∆D ) 0.015 µm2/s, we obtain the relation ∆D < D* - D, which is being visualized in Figure 7 in terms of a relation between the applied shear velocity Vshear and the parameter ε for the illumination time ∆t used. The shaded area corresponds to parameter pairs for which the measurement accuracy is larger than the shear-induced change in the diffusion constant. The white area represents therefore the area in which the width of the velocity distribution can be safely determined. Plotting the parameter pair Vshear ) 12.5 µm/s and ε ) 1.7 into the diagram reveals the effect of a velocity
Figure 7. The enhancement of the diffusion constant along the axis of the applied shear flow is only observable for certain combinations of the parameters Vshear and ε (white) and not observable for others (gray). The black dot indicates the parameters covered by our experiment. According to that, we can detect the enhancement for shear velocities larger than 4 µm/s only.
distribution should be indeed observed in our experiments. For a fixed ε ) 1.7 (determined by diffusion constant and film thickness), the applied shear velocity must be larger than 4 µm/s to observe an effect of the velocity distribution in our measurements. According to the diagram, the observed effect of the velocity distribution gets largest if the viscosity of the liquid is high (small diffusion coefficient) or the film thickness is large or the shear velocity is high. Thus there are three parameters, which can be adjusted to obtain determine the velocity distribution in a sheared ultrathin liquid film, which finally provides a measure for boundary slip. Conclusion We have developed an analytical method to access velocity distributions and boundary slip in thin liquid films by means of single particle tracking. The method allows the extraction of velocity profiles from a step size analysis of diffusion of single particles in sheared liquids under certain conditions. We have shown that the observable velocity profile using this technique systematically deviates from the analytical profile obtained, e.g., from the Navier-Stokes equations due to the diffusive sampling of different velocity regions in the liquid film. Since the scheme employs single particle tracking algorithms, it is also applicable to single molecule tracking. Further, we have carried out experiments on single particle tracking in thin liquid films under shear, which indeed demonstrate that the applied shear velocity distribution in a Couette flow results in a broadening of the step size distribution of the diffusing particles in the direction of shear. The extracted width of the velocity distribution in the film corresponds well with the theoretical predictions given in the paper. As we strongly believe, the use of this technique in combination with single molecule tracking experiments will give new molecular insight into the slip of liquids at solid boundaries. Acknowledgment. This project was funded by VolkswagenStiftung under Grant Number I/78834. Support of the DFG research unit 877 “From local constraints to macroscopic transport” is acknowledged. References and Notes (1) Ismagilov, R. F. Angew. Chem., Int. Ed. 2003, 42, 4130. (2) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. ReV. Fluid Mech. 2004, 36, 381.
Measuring Flow Profiles (3) Atencia, J.; Beebe, D. J. Nature 2005, 437, 648. (4) Barrena, E.; Kopta, S.; Ogletree, D. F.; Charych, D. H.; Salmeron, M. Phys. ReV. Lett. 1999, 82, 2880. (5) Drummond, C.; Israelachvili, J. N. Phys. ReV. E 2001, 63, 041506. (6) Filippov, A. E.; Klafter, J.; Urbakh, M. Phys. ReV. Lett. 2001, 87, 275506. (7) Mugele, F.; Persson, B.; Zilberman, S.; Nitzan, A.; Salmeron, M. Tribol. Lett. 2002, 12, 123. (8) Gourdon, D.; Israelachvili, J. N. Phys. ReV. E 2003, 68, 021602. (9) Raviv, U.; Giasson, S.; Kampf, N.; Gohy, J.-F.; Jerome, R.; Klein, J. Nature 2003, 425, 163. (10) Tadmor, R.; Janik, J.; Klein, J. Phys. ReV. Lett. 2003, 91, 115503. (11) Craig, V. S. J.; Neto, C.; Williams, D. R. M. Phys. ReV. Lett. 2001, 87, 054504. (12) Bonaccurso, E.; Kappl, M.; Butt, H. J. Phys. ReV. Lett. 2002, 88, 076103. (13) Priezjev, N. V.; Troian, S. M. Phys. ReV. Lett. 2004, 92, 018302. (14) Fetzer, R.; Jacobs, K.; Mu¨nch, A.; Wagner, B.; Witelski, T. P. Phys. ReV. Lett. 2005, 95, 127801. (15) Priezjev, N. V.; Darhuber, A. N.; Troian, S. M. Phys. ReV. E 2005, 71, 041608. (16) Schmidt, T.; Schuetz, G. J.; Baumgartner, W.; Gruber, H. J.; Schindler, H. J. Phys. Chem. 1995, 99, 17662. (17) Vikhorev, P. G.; et al. Langmuir 2008, 24, 13509. (18) Schuster, J.; Cichos, F.; Borzcyskowski, C. v. Eur. Polym. J. 2004, 40, 993. (19) Schob, A.; Cichos, F. J. Phys. Chem. B 2006a, 110, 4354.
J. Phys. Chem. C, Vol. 114, No. 10, 2010 4485 (20) Schob A. Cichos, F. Chem. Phys. Lett., in press. (21) Taylor, G. Proc. R. Soc. London, Ser. A 1953, 219, 186. (22) Gee, M. L.; McGuiggan, P. M.; Israelachvili, J. N. J. Chem. Phys. 1990, 93, 1895. (23) Klein, J.; Kumacheva, E. Science 1995, 269, 16. (24) Demirel, A. L.; Granick, S. J. Chem. Phys. 1998, 109, 6889. (25) Zhu, Y.; Granick, S. Phys. ReV. Lett. 2004, 93, 096101. (26) Mukhopadhyay, A.; Zhao, J.; Bae, S. C.; Granick, S. Phys. ReV. Lett. 2004, 93, 236105. (27) Navier, C. L. M. H. Mem. Acad. R. Sci. Inst. Fr. 1827, 6, 839. (28) Happel, H. Brenner, J. Low Reynolds number hydrodynamics; Kluwer Academic Publisher Group: Boston, MA, 1983. (29) Joly, L.; Ybert, C.; Bocquet, L. Phys. ReV. Lett. 2006, 96, 046101. (30) Lauga, E.; Squires, T. M. Phys. Fluids 2005, 17, 103102. (31) Chandrasekhar, S. ReV. Mod. Phys. 1943, 15, 1. (32) Scha¨ffer, E.; Norrelykke, S. F.; Howard, J. Langmuir 2007, 23, 3654–3665. (33) Savin, T.; Doyle, P. S. Biophys. J. 2005, 88, 623. (34) Montiel, D.; Cang, H.; Yang, H. J. Phys. Chem. B 2006, 110, 19763. (35) Facheux, L. P.; Libchaber, A. J. Phys. ReV. E 1997, 49, 5158. (36) Dufresne, E. R.; Altman, D.; Grier, D. G. Europhys. Lett. 2001, 53 (2), 264. (37) Schnitzer, M. J. Phys. ReV. E 1994, 48 (4), 2553. (38) Benesch, T.; Yiacoumi, S. Phys. ReV. E 2003, 68, 021401.
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