1 (2000) 4, 299-305 research paper diffusion of single

299

Single Mol. 1 (2000) 4, 299-305 RESEARCH PAPER MoleculesSingle

Diffusion of Single Molecules Close to Interfaces

Jörg Schuster 1), Frank Cichos 1), Jörg Wrachtrup 2) and Christian vonBorczyskowski 1)

1) TU Chemnitz, Institut für Physik,D-09107 Chemnitz

2) University Stuttgart, 3. Physikalisches Institut,D-70550 Stuttgart

Correspondence toFrank CichosTU Chemnitz, Institut für Physik,D-09107 ChemnitzPhone +49 371/531 3066Fax +49 371/531 3060E-mail [email protected]

submitted 03 Nov 2000published 12 Jan 2001

Abstract

The diffusion of individual rhodamine 6G molecules inethylene glycol close to a glass interface has been studied.Diffusion coefficients are analyzed by photon burst analysis.It is shown that the diffusion of dye molecules becomesslower near the interface as compared to the bulk value. Weattribute this to anomalous diffusion of molecules next tothe interface, due to attachment and detachment ofmolecules caused by molecule surface interaction.

The analysis of the measurements is tested bycomputer simulations of fluorescence bursts.

Introduction

Diffusion of single molecules in liquids has been extensivelystudied by confocal microscopy using fluorescencecorrelation spectroscopy [1, 2, 3] or fluorescence burstanalysis [4, 5, 6]. Both techniques allow the determinationof diffusion constants of dye molecules in a solventaveraging over a huge number of individual moleculescrossing the focus of the confocal microscope. In contrast,wide field imaging of individual diffusing molecules is amethod where the fluorescence from a region of interest isimaged to a CCD camera allowing the reconstruction ofdiffusion trajectories following one individual molecule overmany frames. Single dye tracking by wide field imaging wasdemonstrated in different environments such as lipids [7,8], gels [9], pure solvents [10] or within complexbiomolecules [11, 12].

In the present work fluorescence burst analysis isapplied to studies of single molecule diffusion in liquiddroplets, wetting a surface. Wetting of liquids on solidsurfaces plays an important role in a variety of technologicalapplications. The ability to control wetting properties leadsto new challenging applications [13, 14, 15]. Even thoughwetting research has a long tradition, established theoriesexist only for the macroscopic description of wettingphenomena [16, 17, 18] whereas little is known about themicroscopic origin of the macroscopic phenomena (contactangles, evolution of droplet profiles etc.).

So far only molecular dynamics simulations have thepotential to analyze microscopic processes leading tocertain wetting dynamics. For example the evolution of thinprecursor layers [19 - 23] wetting the surface in front of amacroscopic droplet, has been reproduced by differentsimulations [24, 25]. From experiments it is known that theradius of the precursor layer is proportional to the squareroot of time t. Molecular dynamics simulations reproduce

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the same dynamics, however starting from differentmicroscopic models, assuming fluids of chainlike molecules[24] or atomic fluids [25]. Obviously, the spreadingdynamics of the precursor layer is rather general andindependent from molecular details.

In contrast, in a recent experiment [23] a drastic changeof spreading rates of macromolecular liquids(polydimethylsiloxane) by variation of surface properties(concentration of free silanol sites) was observed. Theauthors attribute this to frequent attachment anddetachment of the macromolecules on the silanol sites ofthe surface, hindering the diffusion of other molecules. Thiscan be understood as a kind of diffusion in the presence ofobstacles [23], generating additional friction in the fluid.The concentration of the silanol sites on the surfaces thusinfluences the attachment frequency of the macromoleculesand thus also the spreading rates.

Although in [23] and other papers [26, 27] molecularpictures are developed, experiments have only beenperformed on macroscopic, indirect quantities (contactangle, droplet profiles, collective diffusion coefficients) tostudy wetting dynamics. Thus, there is an obvious need touse experimental techniques which give direct insight in thedynamics of individual molecules in wetting liquids on amicroscopic scale. We will demonstrate first results in thepresent paper.

Experimental

Sample Preparation

R h o d a m i n e 6 G w a s d i s s o l v e d i n s p e c t r o s c o p i c g r a d e e t h y l e n e g l y c o l (A l d r i c h ) an d d i l u t e d t o a c o n c e n t r a t i o n of 10-12 mol/l. A small droplet with a diameter of about onemillimeter was deposited on a conventional glass cover slip.The cover slips were cleaned first in spectroscopic gradeethanol (Aldrich) followed by spectroscopic grade water(Aldrich) and dried in a stream of hot air (200 °C).

For comparison of the bulk diffusion constants, sampleswith water (spectroscopic grade, Aldrich) instead of theethylene glycol were prepared the same way.

Confocal Setup

We use a home built confocal microscope, mounted on anoptical table. The samples are illuminated by the 514 nmline of an argon ion laser which is additionally filtered by ainterference band pass filter. Fluorescence is collected by amicroscope objective (ZEISS, 100 x, 1.3 NA, oil immersion),filtered by a holographic notch filter (Kaiser Optics) andimaged by a lense (250 mm focal length) to a pinhole (100µm diameter) in front of a photo multiplier tube (S20, EMI).

Reflected light from the sample is collected in a differentdetection channel.The overall detection efficiency of the setup is about 5 %.The excitation intensity used in the confocal arrangementwas about 1 mW on the sample. No excitation energydependence (up to 1 mW) was found in our experiments,ensuring that optical trapping of molecules is negligible.

The radius (1/e value) of the focal volume wasdetermined by imaging single molecules under theconditions described above to be w0 = 320 nm.

The sample (Fig. 1) is mounted on a piezo scanner. Therepositioning accuracy of the scanner (without opticalfeedback) is about 0.5 micron, due to nonlinearities of thepiezo.

A personal computer controls the scanner and countsthe detected photons to allow image acquisition. A multichannel analyzer (Canberra) was used to record time tracesof the fluorescence from a given point of the sample.Commonly a few thousand single molecule bursts have tobe recorded which requires a number of 105 data points toobtain good statistics.

Fig. 1. Illustration of the confocal sample arrangement -illumination and fluorescence detection are realized throughthe glass slide, the liquid droplet is located directly on thebottom of the glass slide.

Analysis of Fluorescence Time Traces

Fluorescence traces recorded with the confocal setup areanalyzed using burst width analysis [5]. This techniquemeasures the width in time of the fluorescence bursts(emitted photons) originating from molecules while diffusingthrough the focal volume of the confocal excitation spot.The width of the fluorescence bursts is related to thediffusion constant of the dye in the solvent. Although photobleaching limits the burst width it can be neglected for low

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Diffusion of Single Molecules Close to InterfacesSingleMolecules

excitation powers if the survival time of the molecules untilphoto bleaching occurs is much larger than the meandiffusion time through the spot. For data analysis we set athreshold to define begin and end of the fluorescencebursts and thus the burst width tB. The threshold value wasset to twice the background luminescence intensity whichwas proven to separate bursts from the background noisewithout changing the long time tail of the burst widthdistribution.

Diffusion studies by confocal microscopy are restrictedto the analysis of the diffusion components perpendicular tothe optical axis of the confocal spot [1]. Since the spotdimensions along the optical axis are much larger (by afactor of about 5) than perpendicular to it, the probe spacecan be modeled as a long cylinder and the diffusion of thedye molecules through the spot can be treated as 2-dimensional [1, 5]. Thus the experimental situation issimilar to the so called "first passage problem", whichdescribes the diffusion of a particle starting from the centerof a given circular area. The exact solution of the firstpassage problem is equation (1):

pD

JD

Bn

n

n

nBt

a

w a

a

wt( ) = å -

æ

èç

ö

ø÷

-

¥

1 02

1

2

02

2( )

exp (1)

It describes the probability distribution p(tB) of the burstwidths tB needed to diffuse from the center to the edge ofthe droplet [5, 28] (J1 is a first kind Bessel function of order1, an are the roots of J0(an) = 0, w0 denotes the radius ofthe focal volume and D the diffusion constant of themolecule). In the limit of large tB eq. (1) can be reduced to asimple Poissonian distribution since a an

2 >> 12 for n > 1.

Replacing p(tB) by burst frequencies N(tB) instead ofprobabilities thus leads to

N(tB) = N(0) exp(-tB/tD). (2)

The characteristic diffusion time tD is then given by

tD = wo2/(a1

2 D) (3)

where a1 = 2.4. It can be easily extracted from eq. (2) byplotting the logarithm of eq. (2) and calculating the inverseof the slope of the long time tail. In contrast to fluorescencecorrelation spectroscopy [29] only a linear fit to the data isnecessary and no further assumptions are needed for dataanalysis.

Results and Discussion

Simulations

Despite the similarity of the "first passage problem" to theexperimental situation a main difference lays in the initialconditions. The particle in the first passage problem starts

its trajectory in the center of the area (center to edgediffusion, CED), while it starts at the edge of the area in theexperiment (edge to edge diffusion, EED). The traveleddistance of a particle diffusing through the focal volume inthe EED can thus be much smaller than the radius of thefocal cylinder, while the particle in the CED has to travel atleast the cylinder radius. Consequently the distribution N(tB)is dominated by very short bursts belonging to moleculesentering and leaving the focal volume not exploring thecomplete volume of the focal cylinder [30].

Fig. 2. Comparison of burst width distribution for CED(center to edge diffusion, (1)) and for EED (edge to edgediffusion, (2)) from computer simulations (see text). Thesolid line represents the long term slope of bothdistributions.

The EED problem cannot be treated analytically. Thereforecomputer simulations have been carried out, to study thedifferences between EED and CED in detail. For simplicitydiffusion of a single molecule was modeled as a threedimensional random walk [31] while more sophisticatedmodels as i.e. a Levy walk might be more adequate for thesimulation of diffusing polymers or macromolecules onsurfaces. The intensity distribution in the focal volume isgiven by the confocal point spread function [32]

I(r,z) = 16 J14(r)/r4 sin4(z/4)/(z/4)4, (4)

with I(0,0)=1, J1(r) being a Bessel function and z, r beingcoordinates along and perpendicular to the optical axis. Thepoint spread function is set to zero for all points (r,z) withI(r,z) > 1/e and thus defines the focal volume. The randomwalk was started at an arbitrary point out of the focalvolume for the EED problem or in the center (r = 0, z = 0) ofthe focal volume for the CED problem respectively. Thefluorescence is assumed to be proportional to the intensitydistribution I(x,y,z) and is recorded during the random walk.The resulting time traces of the fluorescence intensity are



Fig. 4. Fluorescence (left) andreflection mode (right) imageof a droplet of silicone oil(PDMS) on a glass surface.Interference fringes clearlyvisible in the reflected lightallow the reconstruction ofdroplet profiles.

analyzed as described above with a threshold of zero (zeronoise). Fig. 2 shows the resulting burst width distributions.While the CED burst width distribution approaches zero forshort burst width, the EED solution is as expecteddominated by short bursts which is also observed inexperiment [4, 6, 30]. The deviation from a Poissoniandistribution is thus caused by the initial conditions of theexperimental situation and does not involve any trapping ofmolecules in the laser focus as suggested in reference [4,6]. Despite the differences in the CED and EED distributionsfor small tB, eq. (2) can be used to analyze the experimentaldata since the slopes of both distributions are the same forlarge tB as already assumed in ref. [5]. These resultsshould also apply to diffusion in a very thin liquid film.

Measurement of Bulk Diffusion Constants

To check the data analysis, bulk diffusion constants ofrhodamine 6G in water and ethylene glycol were recorded.Fluorescence traces have been measured in large dropletsof the solvent far from any interface. As shown in Fig. 3 thesignal to noise ratio is about 5. Fluorescence burst widthanalysis results in characteristic diffusion times of tD = 1.1(± 0.2) ms for ethylene glycol and tD = 70 (± 50) ms forwater. According to the Stokes-Einstein relation

D = kT/(6pha), (5)

h being the solvent viscosity and a the spherical diameter ofthe molecules, the value of tD for water should be by afactor of 19 smaller than for ethylene glycol, consideringviscosities of h = 0.89 mPas for water and h = 16.79 mPasfor ethylene glycol [33]. Within the error of tD this is in goodagreement with the experimental results shown above. Fromequation (2) diffusion constants of rhodamine 6G can becalculated to be D = 2.5 (± 1.7) 10-6 cm2/s in water and D= 1.6 (± 0.3) 10-7 cm2/s in ethylene glycol which is in goodagreement with measurements reported in literature [5].

Fig. 3. Fluorescence trace for rhodamine 6G in ethyleneglycol (0.5 ms per bin). See text for more details.

Determination of Droplet Profiles

Cross sections of the sample can be imaged with theconfocal microscope in the reflected light or in thefluorescence mode. Figure 4 shows images of a silicon oildroplet heavily doped with rhodamine 6G as recordedsimultaneously in the fluorescence and reflection channel ofthe confocal microscope. The interference fringes visible inFig. 4 allow a precise determination of the droplet shapewith a thickness resolution of 100 nm.

For liquids with a higher contact angle on glass such asethylene glycol the distance of the interference fringes isbelow the optical resolution. Thus the droplet shape wasdetermined with a lower accuracy (1 mm) in this case (seeFig. 5). Cover slip, droplet and air can be identified in theimages by differences in the background luminescence or inreflectivity. By recording time series of such images it wasverified that shape and position of droplets of ethyleneglycol remain stable over tens of minutes, the time requiredto record sufficient data for fluorescence burst analysis.

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Fig. 5. Reflection mode image of an ethylene glycol dropleton a glass cover slip. Shown is a cross sectionperpendicular to the sample surface, as illustrated by theinset. The reflection at the solid/air interface is clearlyvisible. The liquid/air interface is only partly visible,depending on the local curvature. The reflection from thesolid/liquid interface is too weak to be visible.

Diffusion Constants Close to an Interface

Fluorescence time traces were recorded at differentpositions in the droplet (see Fig. 6), in the bulk droplet (A)or near the interface (B far from the droplet edge, C at thedroplet edge). Figure 6 shows the resulting burst widthdistributions N(tB) at these different positions.

Note that the curves presented in Fig. 6 show negativedeviations from a Poissonian distribution for small values ofthe burst width while computer simulations predict positivedeviations due to multiple recrossings of the focus. Thisdiscrepancy can be explained by the fact, that very shortand weak bursts (which contribute to signals in the rangeof the noise level) are discriminated by the threshold set inthe data analysis and thus cannot enter the burst widthstatistics.

The determined characteristic diffusion times and thecorresponding diffusion coefficients are summarized intable 1.

Table 1.

Sample point tD [ms] D [cm2/s]

A 1.1 ± 0.2 1.6 (± 0.3) 10-7

B 1.3 ± 0.2 1.4 (± 0.3) 10-7

C 2.0 ± 0.4 8.8 (± 1.8) 10-8

The measurements point out that the characteristicdiffusion times tD become longer compared to the bulkvalues when approaching the liquid-solid interface. Thusthe diffusion is slowed down at the edge of the droplet (C)while the slow down it is not significant at (B). The reasonfor the slow down of the diffusion is the strong interaction ofthe surface with the liquid molecules which also leads tothe well know effect of molecular layering [34]. Theobserved slow down is however not as drastic as expectedsince the experimental technique applied here alwayssamples the emission from a layer more than 100 nm thick,thus containing molecules not interacting with the surface.This is the case for (B) and also (C) since ethylene glycolhas a fairly high contact angle on glass surfaces.

Fig. 6. Measured burst width distributionsfrom different regions of the droplet. Theinset (cross section through the droplet)shows a schematic view of the sample,including points A, B and C where data havebeen recorded.



The experimental data contains therefore a mean value ofthe diffusion constant close to the interface and in the bulk.Nevertheless, the effect is stronger in point C, because ofthe smaller film thickness.

We pronounce that the burst width statistics in (C) isclearly described by a single exponential (the observeddeviations for very long bursts widths have no statisticalsignificance). This can only be interpreted by a broaddistribution of diffusion constants of the individualmolecules. A simple model with two dominant diffusionconstants (i.e. a bulk and an interface value) wouldconsequently lead to bi-exponential burst width statisticswhich is obviously not the case in our measurements.

We thus conclude that the real interface diffusioncoefficients are even smaller than the ones measured here.Correct measurement of the surface diffusion coefficientswould require a liquid film thickness of a few nm over therange of the confocal spot which was not the case in ourexperiments. We also note that the technique used is notcapable of distinguishing between a normal diffusion and adiffusion process which is influenced by attachmentperiods. Results from wide field imaging of diffusiontrajectories of individual molecules which are currently inprogress [35] suggest however, that molecules close tointerfaces show anomalous diffusion, i.e. the diffusion isstrongly influenced by attachment periods of molecules onthe glass surface. This will be the issue of a forthcomingpaper using direct observation of diffusion trajectories [35].

Conclusions

We have performed computer simulations to prove that themeasurement of diffusion constants using the long timebehavior of burst width statistics gives the same resultswhen comparing it with the analytical solution of the ”firstpassage problem”.

Measurements of droplet profiles with optical methodsallows us to perform burst width measurements in distinctregions of a droplet with respect to the substrate.Depending on the droplet shape, profiles down to a liquidfilm thickness of about 130 nm (l/4) can be determined.

A decrease of the diffusion constants of rhodamine 6Gin ethylene glycol in presence of a glass interface wasobserved, which is attributed to a strong surface interaction.The decrease of the diffusion constants near the interfacemight also be influenced short attachments of dyemolecules on the glass surface. However, such attachmentperiods can not be identified with the experimentaltechnique applied in this report. Therefore we started torecord diffusion trajectories in wetting films with a wide fieldimaging setup. This will allow a detailed study of attachmentand detachment processes of individual molecules in aliquid spreading on a solid surface and will provide a keylink to understand wetting dynamics even in very thin liquid

films where the dynamics is clearly dominated byinteractions with the interface and far from purehydrodynamics.

The present paper demonstrates the ability of singlemolecule studies in liquid environments near interfacesbeing a powerful tool to study details of the moleculardynamics in wetting structures.

Acknowledgment: Financial support of the DeutscheForschungsgemeinschaft via the Schwerpunkt ”Benetzungund Strukturbildung an Grenzflächen” is acknowledged.

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