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High-resolution particle sizing by optical tracking of single colloidal particles
PHYSICA£ ELSEVIER
Physica A 235 (19971 291-305
High-resolution particle sizing by optical tracking of single colloidal particles Norbert Garbow*, Jtirgen Miiller, Klaus Schnitzel, Thomas Palberg Johannes-Gutenberg-Universiti~t Mainz, Institut fiir Physik, Staudinger Weg 7, D-55099 Mainz, Germany
Abstract
The motion of individual Brownian particles is observed using the Confocal Tracking Microscope recently introduced by Schnitzel (K. Schnitzel, W. G. Neumann, J. M/iller and B. Materzok, App. Opt. 31 (1992) 770-778). Particles are laterally trapped in a strongly focused laser beam. By evaluating the light-pressure-induced drift velocity and the backscattered intensity we are able to determine particle size histograms with a resolution better than 2%. This is demonstrated on a mixture of seven species of polystyrene latex spheres in the diameter range between 300 and 450 nm, where six classes of diameters are identified. We discuss the scope of the method and potential applications. Keywords: Particle sizing; Multimodal distributions; Polydispersity; Confocal microscopy; Colloidal dispersions
1. Introduction
The synthesis of colloidal dispersions is a well-understood industrial process which yields particles of controlled mean diameters from say 10nm for semiconducter particles up to some microns for latex spheres. Standard deviations below some 5% are achievable and usually referred to as monodisperse. For many applications standard sizing methods like e.g. dynamic light scattering are sufficient, Since these techniques simultaneously average over a large ensemble of particles, the mean diameter may be determined with high accuracy, while an analysis of the size distribution, or the resolution of bi- and multimodal mixtures, is severely limited. Typically, bimodal distributions may be resolved, if the component diameters differ by a factor of two or more. In some cases, however, a better resolution is of vital importance. Colloidal suspensions have, for example, proven to be valuable model systems in the study of
* Corresponding author. 0378-4371/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0378-43 7 1 (96)00349-4
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statistical mechanics and condensed matter physics. Recent theoretical descriptions explicitly include effects of polydispersity, and describe the structure and dynamics of bimodal mixtures [2, 3]. There is a strong evidence, that already the narrow polydispersities observed in standard latex particles, and even more so slight changes in the size ratio of bimodal mixtures, singificantly influence the glass and the freezing transition in these systems [4, 5]. Quantitative experimental studies therefore require careful characterisation of the size distribution. In this paper, we show a new approach different from standard light scattering techniques. Using a specially-built laser tracking confocal microscope, single colloidal particles in a dilute suspension (108-109 cm 3) are observed. A size histogram is collected from typically 103 individual particles. Single particles are illuminated by a strongly focused laser beam. While absorption may be neglected for most common model colloids, particle motion is strongly influenced by the illuminating beam. Perpendicular to the beam optical gradient forces form a potential well. Particles having a refractive index higher than the suspension are bound in the region of highest intensity. This results in a laterally restricted diffusive motion of the bound Brownian particle [6]. In contrast to an optical tweezer the axial gradient forces are not strong enough to dominate the light pressure, hence the particle is driven out of focus. By automatically adjusting the position of the focussing optics, it is possible to follow the particle motion for several seconds, corresponding to some 500 particle diameters. Evaluation of lateral diffusion results in absolute size information, but with a large uncertainty of 10-15%. Alternatively, we measure the Mie backscattered intensity and the light-pressure-induced drift velocity, with accuracies better than 5% and 1%, respectively. While individual measurements of the latter quantities provide only relative size information, their combination can be calibrated to yield absolute particle diameters with a resolution of better than 2%. This paper is organised as follows: After a short introduction to the theoretical background, we give a detailed description of the apparatus and the measurement procedures. Results obtained on a multimodal mixture are presented and discussed in the final section.
2. Bound diffusion (lateral particle motion) Of the three quantities accessible in single particle tracking, the lateral diffusion provides the most direct measure of particle size. We determine displacements in the x and y direction independently. The lateral particle motion is restricted by optical gradient forces (see below), hence they can be considered harmonically bound. For such a Brownian particle, the mean square displacement (MSD) over a time t in one direction is given by [6]
~(x(O)--x(t)2))=2~Z~x2)(1--exp( ( ADtx 2 ) ) ) .
(1)
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the limiting value for long times, (Ax 2 ), is determined by the width of the potential well. Do is the Stokes-Einstein diffusion coefficient for non-interacting particles given by
kT Do = 3n~ld'
(2)
with a thermal energy kT, the viscosity q and the particle diameter d. This coefficient determines the slope of the MSD for small delay times t. As will be briefly discussed in the last section, it is possible to concurrently measure the electrophoretic mobility # = v/E by applying an additional sinusoidai electrical field E, e.g. in the x direction. The induced velocity v changes the mean particle position with the frequency o9/27t and the amplitude A and is in equilibrium with the Stokes friction Fs = - v3rcqd: A-
/~Eo
(3)
(O
The mean square displacement of Eq. (1) is therefore modified to ((x(0) - x(t)2)) = 2 ( A x 2)
(
I -exp
(Ax2)
+ A2(1 -cos(e)t)).
(4)
Measured mean square displacements are shown in Fig. 4. Note that the MSD saturates after some 80 ms. The 'ponderomotive forces' acting on a particle in a highly inhomogeneous intensity profile can be interpreted as force on a dielectric medium in an inhomogeneous mean absolute electric field. In the direction perpendicular to the beam the potential well built up by this force is deep enough for trapping the particles. Typically, they can move in an area of about 100 nm radius around the point of highest intensity. For an optical dipole the depth W of this potential is given by [7-9]
W
= _
2re
R3 np
~Np + 2n 2'
(5)
where I is the peak intensity, R the radius of the particle, n and np the refractive indices of the suspension and the particle and c the vacuum speed of the light. According to calculations by Barton [10], this estimate should hold for particles up to 800 nm in diameter in a focus of 1.2 gm diameter with an accuracy of about 10%. From the known focus width of 1.4 lam and the 1 kT width of approximately 190 nm measured in the long-time limit we estimate the total depth of the potential well to be of the order of 56 kT for the example given in Fig. 4. Depending on the particle size, focal radius, and the laser power, values in the range 30-200 kT are realised. An alternative, intuitive picture for these forces is the momentum transfer of photons passing the particle with np > n. For homogeneous illumination, this results in light pressure only, as the lateral momentum components cancel. In the case of a strong intensity gradient, as it is found close to the focus, the asymmetric momentum transfer leads to a lateral force driving the particle towards the region of highest intensity.
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3. Light pressure and intensity Determining the individual particle sizes m a y alternatively proceed via a c o m b i n e d m e a s u r e m e n t of the axial drift velocity and the backscattered intensity. While the particle is laterally bound, it is axially driven forward by the light pressure with typical velocities of several microns per second. F o r Rayleigh scatterers, the light pressure scales with their cross-sectional area, whereas the friction described in Eq. (3) grows linearly with the diameter. Both forces cancel in the quickly reached equilibrium and a constant (axial) drift velocity is obtained, which linearly depends on the particle size. The relation is still m o n o t o n o u s for particle sizes up to a micron, but corrections for the light pressure using the Mie theory becomes necessary. Also the backscattered intensity m a y be calculated from Mie theory, since our particles are significantly smaller than the focus diameter and the incident light in the 200
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6. velocity [a.u.]
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o 6-
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4-
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d i a m e t e r [nm]
Fig. 1. Axial drift velocity and backscattered intensity for polystyrene lattices (n = 1.59) in a water/glycerol solvent (n = 1.385) and ,~ = 532 nm, calculated using Mie-theory. In contrast to the backscattered intensity the drift velocity grows monotonically with particle size (for particles significantly smaller than the focal diameter).
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centre of the focus may be approximated as a plane wave. In contrast to the monotonous relation between particle diameter and drift velocity, the backscattered intensity for particles of more than 100 nm in diameter has a strong, size-dependent modulation. Calculations for both, the drift velocity and the backscattered intensity, are shown in Fig. 1. The intensity inside the focus is not precisely known. Hence, neither the drift velocity nor the intensity alone is sufficient to determine the absolute size of the particle. However, the combination of measured drift velocity and scattered intensity provides an elegant way to distinguish different sizes and types of particles. For each particle additional data, like mobility, diffusive constant or depth of the potential are determined independently. We note that the statistical accuracy of this approach is increasing linearly in time, since the axial displacements of the particle increase linearly during the measurement. This has to be compared to the short-time lateral displacements due to diffusion, which increase as the square root of time and so does their statistical accuracy. While from the determination of the lateral diffusion the size is directly available, the velocity measurement yields a higher resolution of size differences. lateral position position sensitive diode cuvette
_7
objective polarising beamsplitter
axial position
photo diodes
tilting mirror
Nd:YAG-Laser, 100mW, Z=532nm
~
Lens r-
• [~
Focus Detector
Fig. 2. Simplified experimental set-up. The microscope objective images the first intermediate focus into the cuvette and thus defines the observation volume. A 2/4-plate on the front lens of the objective rotates the polarisation of the backscattered light by 90 °. A particle under observation is imaged onto the detectors, where the axial and lateral position as well as the backscattered intensity are determined.
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4. Set-up In this section we present the experimental realisation of a confocal optical tracking microscope, which allows for the concurrent measurement of the MSD, the backscattered intensity, the light-pressure-induced drift velocity and the electrophoretic motion of an harmonically bound particle (see Fig. 2). A confocal backscattering geometry is chosen to obtain a sufficiently intense illumination and a small observation volume to enable extended measurements on individual particles. Both points are easily fulfilled by focusing the beam of a 100 mW Nd : YAG frequency doubled cw laser through a microscope objective into a highly diluted sample. The microscope objective has the focal length in the ~tm range. The same microscope objective is employed to collect the backscattered light from a solid angle of some 40 °. A 2/4 plate glued to the front lense of the objective and the polarising beamsplitter suppresses reflections not originating from the sample. The lateral position of the particle with respect to the beam centre and its backscattered intensity is determined by imaging it onto a position-sensitive photodiode. The axial position of the particle is measured independently, using two photodiodes combined with pinholes positioned some cm before and behind the image plane, respectively. Both detectors receive the same intensity ifa particle is exactly in focus. As the particle is axially driven out of focus by the light pressure, the difference signal between these diodes is used to control the positioning of the objective via a piezo-driven translation stage. Therefore, our set-up realises a tracking geometry in the axial direction, with typical deviations between the axial particle position and centre of the focus of about one particle diameter and a residual uncertainty of the objective position of about 50 nm. This has to be compared to a focus length (l/e intensity profile) of 13 I.tm. A piezo-tilting mirror is used to scan quickly a larger suspension volume until a particle is caught and the scanning motion is aborted. Successful catches are readily visible in a high and comparably constant backscattered intensity, while rare double catches show up in high-intensity fluctuations. An example is given in Fig. 3. Data taken after the appearance of a second particle are discarded. An important condition for the sample preparation is the use of highly diluted suspensions of typically l0 3 pro-3. This compromises between the necessary scanning time to find a particle (10 20s) and the possible tracking time without double catches (10 15 s). Some further 20 s are needed for data evaluation. After a short delay time to centre and trap the particle laterally, the measuring and axial tracking starts with a sampling clock rate of approximately 800 s-1. The observation time for a single particle is restricted either by the maximum possible tracking displacement of the microscope objective or the available memory of the computer. The current limit in our set-up is approximately 15 s. Size limits for the observation are the optical contrast for small, the maximum tracking velocity for large particles. Currently, the contrast limit is caused by residual internal reflections of the microscope objective rather than detector noise. Interference of this backscattered light with the signal disturbs the autofocus mechanism and
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time [s] Fig. 3. Measured axial position and intensity for a 301 nm particle. The slope in the first graph gives the drift velocity induced by light pressure. The mean value of the second graph leads to the mean backscattered intensity. The intensity variations during the first 12 s result mainly from motion of the particle in the intensity profile due to diffusion and electrophoresis.
prevents valid tracking of particles significantly fainter than the used 300 nm latices. The upper size limit for the measurement is of the order of 500 nm, corresponding to a maximum tracking velocity of about 90 I~m s- 1 However, as the trapping improves with increasing particle size, the laser power may then be reduced leading to lower drift velocities and hence to a shift in the range of accessible particle sizes. The MSD in x and y direction are directly calculated from the signal of the position-sensitive photodiode. A measurement in the presence of a sinusoidal electric field of about 50 Hz and 16.2 V/cm is given as curve A in Fig. 4. The true MSD (curve B) is derived by subtracting a periodic signal of constant amplitude. An initial linear increase and a saturation behaviour at long times are observed, consistent with qualitative theoretical considerations for bound diffusion. In an intermediate time regime a quantitative fit of Eq. (1) is not perfectly possible, since the true beam profile
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25000'
oOOOO: [nm2115ooo-
A
~
looo0-/f
0
AA
~
B
50
1O0 delay [ms]
150
200
Fig. 4. Mean square displacement characterizes the diffusive motion of a 301 nm particle in the optical potential well. The points give the measured values with electrophoresis(A) and corrected for it (B). The drawn line representsEq. (1) for a harmonicallybound particle with the lmiting values of incident slope and saturation. The applied sinusoidal electric field had an amplitude of 16.2 V/cm, and the resulting position amplitude of the particle is 49 nm.
is Gaussian rather than parabolic. Nevertheless, a diffusion coefficient Do may be derived from the initial slope, in this case corresponding to 323 nm diameter. A closer inspection of the data reveals some inherent limitations of statistical accuracy. Only for short times the M S D is dominated by free diffusion and only few data points can be used to obtain Do. In addition, displacements have to be calculated as differences of lateral position data. While the detector noise limits the positional accuracy to some 20 nm, displacements are of the order of 50 nm; hence their uncertainty may easily reach 50%. The alternative route to particle sizing proceeds via the combined measurement of backscattered intensity and drift velocity, which is derived directly from the timedependent microscope objective position. D a t a for a particle with 301 nm diameter are shown in the Figs. 3(a) and (b). The axial position is seen to increase linearly in time; from the slope of the curve in Fig. 3(a) we obtain a constant drift velocity of 16.6 p.m/s. The diffusion in this direction is not visible within the resolution of our axial position detectors. During the main time of observation the backscattered intensity (second part of Fig. 3) shows a more or less constant mean value of 550 AD-units with a significant noise. The observed fluctuations correspond mainly to lateral displacements of the particle due to diffusion and electrophoretic motion in the applied electrical field. Using Mie theory the particle sizes are inferred from a combination of backscattered intensity and drift velocity, if the intensity of the illumination and the viscosity of
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n1=1.580/
I
375or I
/
o
8OO
n =1.590
/
/ /,0nm
intensity [AD-UnitS]oo
400
200 -
32
4Onto
y 110
I
15
210
215
I
30
axial drift velocity [pm/s] Fig. 5. Scatterplot of backscattered intensity versus light-pressure induced for a set of measurements taken on polystyrene particles of diameter 375 nm (triangles). The full lines represent Mie results for two values of the particle refractive index, with the particle diameter as parameter; the theoretical curve for n~ = 1.590 was scaled to fit the data. The dotted lines connect points of the same diameter on the different Mie curves; the error bars included for one particle represent the long time variation of mean intensity and drift velocity. These errors were corrected during the evaluation.
the m e d i u m are known. The former m a y be calibrated using particles of a b r o a d size distribution and a k n o w n refractive index in a well-characterised suspending medium. In Fig. 5 we plot the data for a suspension of polystyrene latex spheres (n = 1.590) of nominal diameter d N = 378 n m suspended in a 1:2 mixture of glycerol and water at 22°C. The data have been corrected for systematic drifts (see below). This sample shows a strongly peaked size distribution with a standard deviation of 2% and a weak tail towards smaller particle sizes clearly visible. The Mie curve for polystyrene particles is fitted to the cluster of raw data to yield a illuminating intensity of 17.7 mW/Ixm 2 with an independently measured viscosity value of r / = 2.6 cP. The projection of the cluster m a x i m u m onto the Mie curve corresponds to a particle size of 377 nm, the mean particle diameter (cf. Table 1) being somewhat smaller due to the presence of the tail in the distribution. We note that the symbol size of the scatterplot corresponds to the uncertainty of each single measurement and is m u c h smaller than the variation between particles, since both mean backscattered intensity and drift velocity m a y be determined with extremely high accuracy. D u r i n g the measurements the optical set-up was thermostatically maintained only by controlling the ambient temperature. The intensity
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Table 1 Particle diameters obtained using different methods. All methods except SPT use only one kind of particle at time, SPT data are corrected for systematic drifts. Standard deviation and residual polydispersity is also corrected for statistical uncertainty Single particle tracking (nm)
Standard deviation A(nm)
Residual polydispersity a (%)
Refractive Static light index scattering (nm)
Dynamic. light scattering (nm)
Nominal diameter (TEM) (nm)
Nominal polydispersity 2a (%)
320 301 375 401 454 401 438
2.4 1.4 6.8 < 4.5 0.9
0.7 0.4 1.8 < 1.3 0.1 < 1
1.585 1.59 1.59 1.59 1.585 1.585 1.585
332 308 386 426 488 448 454
301 306 378 401 403 412 415
2.0 1.07 5.9 1.85 1.6 1.9 2.1
<5
311 301 362 395 434 407 412
t u r n e d out to be fairly r e p r o d u c i b l e if the a m b i e n t t e m p e r a t u r e was k e p t constant, but was subject to larger u n s y s t e m a t i c drifts, if fluctuations in t e m p e r a t u r e exceeded I°C. Also the drift velocity s h o w e d m a r k a b l e v a r i a t i o n s as the optical p a t h length on the d e t e c t o r side fluctuates with t e m p e r a t u r e , but to a much lesser extent. T a k i n g into a c c o u n t the t e m p e r a t u r e fluctuations we estimate the r e p r o d u c i b i l i t y of single m e a s u r e m e n t s to be better than 7 % in the intensity a n d better t h a n 2 % in the velocity, as is i n d i c a t e d by the e r r o r b a r in Fig. 5. As the t e m p e r a t u r e - d e p e n d e n t drifts occur on time scales larger t h a n s o m e 50 single m e a s u r e m e n t s a n d m a y be c o r r e c t e d for, this c o m b i n e s to an u n c e r t a i n t y in the d i a m e t e r of less t h a n 2 % for each i n d i v i d u a l particle. F o r clusters larger t h a n say Ni = 100 particles situated a w a y from the M i e e x t r e m a the m e a n d i a m e t e r m a y therefore be d e t e r m i n e d with a statistical e r r o r smaller t h a n 0.2% a n d a systematic u n c e r t a i n t y of a b o u t 0.5% in the present case c o r r e s p o n d i n g to dspx = 375 _+ 7 nm. F u r t h e r this yields a residual p o l y d i s p e r s i t y PsPv defined as the relative s t a n d a r d d e v i a t i o n from the m e a n d i a m e t e r dsvr in a h i s t o g r a m of drift corrected data. F o r the particles investigated in Fig. 5, Psvr = 1.8%. Also higher m o m e n t s of the size d i s t r i b u t i o n m a y easily be resolved with high accuracy, too, a n d the tail to small d i a m e t e r s is clearly visible in Fig. 5. F o r clusters in the vicinity of M i e e x t r e m a an a d d i t i o n a l a m b i g u i t y m a y arise due to the choice of the index of refraction. The statistical accuracy, however, is not affected.
5. Polydispersity and muitimodal distributions T o explore the range of a p p l i c a b i l i t y a n d the resolution of the a p p a r a t u s , a suspension m i x e d from 7 different types of p o l y s t y r e n e lattices in a n o m i n a l d i a m e t e r range 301-415 n m was analysed. All samples h a d been a n a l y s e d individually by single
N. Garbow et al. IPhysica A 235 (1997) 291-305
10oo-
800" intensity [AD-units] 600-
400"
301nm 320nm , 375nm ,lol°m
o
n1=1.585 n1=1580/ / / n1=1590 ~ / / , 460n
~
~~
301
Iit'"
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/ /."7
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438nr,
.~...~.~~~j .~..~~ - 440nm
329nrn "" /'///
200
0
1'o
l's
2'0
2'5
3'0
axial drift velocity [pm/s] Fig. 6. Scatterplot for a multimodal suspension. As in Fig. 5 the full lines are theoretical curves for particles with the indicated refractive indices and diameters between approximately 250 and 480 nm; the dotted lines connect (calculated) values for particles of the same size and different refractive indices. Each kind of symbols represent a different type of particles. Six of the seven particle types in the suspension are resolved, while the nominal sizes 412 and 401 nm are combined in the 401 nm cluster (triangles downwards).
particle tracking (SPT) before. Their number concentrations ranged from 10-s to 2.2 × 10 -4 Ixm-3. Results of this measurement are shown in Fig. 6. Included are four theoretical curves for increasing particle size corresponding to four different refractive indices. Again, the intensity in the focus was rescaled to fit the dN= 378 nm particles discussed before. Six clusters corresponding to the seven kinds of particles are clearly resolved in the scatterplot and marked with different symbols. This high resolution is emphasized in Fig. 7, where a histogram of the mixture is shown. For comparison the size distribution resulting from a CONTIN-fit [11] to a dynamic light scattering experiment is indicated by the dashed line. Table 1 compares the mean particle sizes and standard deviations with values derived from dynamic and static light scattering and to the manufacturer's data, measured by electron microscopy. We would like to highlight the performance and limitations of our instrument by discussing two examples: (a)the discrimination between the two smallest kinds of particles and (b) between the three kinds of particles positioned in the Mie curve minimum. The main feature of the measurement is a clear discrimination between particles of same refractive index and, in the case of 301/320 nm particles, a difference in mean size
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140
,
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°'Selativ e number 0,6
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301nm
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-
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2 0 - J .
438nm
,
o-
[--.300
0,4
.-"
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400
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Diameter [nm] Fig. 7. Size histogram of the multimodal suspension, derived from the data in Fig. 6 by projecting the individual particle observations on the Mie curves for the respective cluster• The histogram resolution is 1 nm, the total number of valid observations is approximately 1550. The dashed line indicates the size distribution obtained by a CONTIN-analysis of a conventional DLS measurement.
of only 6%. Note that the dashed lines in Fig. 6 correspond to lines of constant radius, and that the discrimination of radii proceeds along the Mie curves• In fact, the limit of resolution between different monodisperse clusters is given by the combined uncertainty due to the temperature stability of the present set-up and the statistical accuracy. It is of the order of 2% size difference. Our particles show a small but significant polydispersity. Taking the systematic uncertainties to be independent of the particle size, the residual polydispersities (relative standard deviations of the diameters corrected for both systematic drifts and statistical accuracy) of the 301 and 320 nm particles are estimated from the second moments of the clusters to be about 0.4% and 0.7%, respectively. While the resolution in the former case was excellent, it is less favourable in the case of the particles with diameters between 400 and 440 nm. As was already known from the reference measurements, the data points of particles with nominal diameters 401 and 412 nm coincide, i.e. they are measured to have the same diameter of 401 nm. In addition, a loose heap of only a few observations is formed by the particles with 415 nm nominal diameter. Although it is clearly discernable from the 401 nm cluster, the catch efficiency is fairly low. This is even more clearly visible in Fig. 7, where these three sorts appear to be the least present species• Note that the data of this histogram are corrected for systematic drifts only. The shape and number of data points belonging to the cluster are strongly influenced by the minimum intensity necessary
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for tracking a particle which provides a cut-off at about 200 AD-units due to the residual reflections. In contrast to the other sorts, no well-defined analysis of polydispersity or of the third moment of the distribution are possible here. These difficulties may, however, be circumvented by using different wavelengths leading to a significant shift of the clusters along the shape of the Mie curves and away from the minimum. This would also solve the ambiguities resulting from uncertainties in refractive index in the vicinity of Mie curve extrema. A comparison between the results of single particle tracking (SPT) and the values obtained by static light scattering (SLS), dynamic light scattering (DLS) and transmission electron microscopy (TEM) shows two systematic trends in the obtained results (Table 1). First, TEM usually gives the smallest size, followed by SLS, SPT and DLS. A good agreement between all methods, and in particular between SPT and TEM, is obtained for particles with refractive index 1.590, while deviations in mean size are significantly larger for particles with lower refractive index. This points to swelling processes in solution or to shrinking in the vacuum conditions of the electron microscope used by the manufacturer. Homogeneous swelling of particles may be caused by residual monomer (of refractive index 1.550) or by solute. Additional sources of deviations would be the presence of an extended or rough boundary region only formed in solution. Second, the SPT diameters range between the data of static and dynamic light scattering (SLS, DLS). SLS yields the mean geometric size. In SPT the light pressure and the backscattered intensity depend on the geometric size, but the Stokes friction depend on the hydrodynamic radius. The DLS data were taken on diluted samples with millimolar salt concentration to minimize electrolyte friction and thus to obtain the true hydrodynamic radius 1-12]. The good agreement obtained for all experiments on particles with n = 1.590 supports the assumption of negligable electrolyte friction. While for particles of refractive index 1.585 SPT and SLS yield rather close results, the deviations between dynamic and static light scattering are as large as 20%. Different degrees of swelling may not be the cause here, since both latter experiments were performed on the same samples. Another source of an extended hydrodynamic radius as compared to the geometric radius may be the presence of surface roughness, which would be differently detected by the different experiments. The amount of surface roughness may, however be connected to an already completed swelling. Apart from polymer surface roughness the hydrodynamic radius may also be influenced by the structure of the boundary region. This may include layers of adsorbed molecules or condensed counter ions, which are in the nm range 1-13]. It may also be due to hairy structures. While no detailed knowledge about the surface of the particles exists, our data seem to be indicative of an enhanced surface roughness for swollen particles. The most extreme deviations are observed for the sample with nominal diameter 403 nm. It is interesting to note further that this sample shows an unexpectedly low
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polydispersity in the SPT measurement, which is smaller by more than a factor of five as compared to the TEM results. This behaviour is reminiscent of the swelling behaviour of network colloids synthesized in emulsion polymerisation. There, particle sizes are controlled by the emulsion droplet size but individual droplets may differ in their polymer content and degree of crosslinking. After synthesis the emulsifier is removed e.g. by dialysis. At large crosslinking densities the maximum size in solution is still given by the original droplet size, while the minimum size under vacuum conditions depends on the polymer content. Unfortunately, we do not have detailed informations about the manufacturing process of this sample and its peculiar properties remain a challenge.
6. Further applications SPT allows an access to further information on single particles, and in particular to their surface potential via the electrophoretic mobility /~. This is an important property for several reasons. First, it gives a criterion of the stability against aggregation. Second, it allows an estimate of adequate handling procedures like in filtering or viscometry. Finally, one may here measure the electrophoretic mobility under conditions of large double layer extensions as compared to the particle radius practically without any interference by interaction with neighbouring particles. Since the surface potential may depend on the particle number density, the mobility measured at high dilutions may not always be used to estimate particle interactions. However, informations about stability and handling usually remain valid and useful also at higher concentrations. The MSD at the lateral position is shown in Fig. 4. The points on the oscillating curve are the measured data including electrophoresis. A line represents Eq.(1) with the limiting value for long times reached after some lOOms. The maximum lateral displacement caused by diffusion is about 94 nm. If a DC electric field of sufficient strength (here: 16.2 V/cm) is applied, the particle is moved out of the beam trap. We use here an AC-field to avoid excursions larger than the particle radius. In addition, a correction for electroosmotic effects is not necessary at field frequencies higher than 20 s-1 [14]. In Fig. 4 the field-induced position amplitude with respect to Eq. (4) is A = 49 + 1.5 nm. From this we calculate the electrophoretic mobility to be /~ = 2.23 + 0.07 (lam/s)/(V/cm). This accuracy of 3% is well comparable to the one reached in e.g. Doppler velocimetry. Following the theory of O'Brian and White [15] the corresponding surface potential is 140 mV. Detailed electrokinetic characterisations for our particle mixture and for the mobility of single particles will be the subject of a forthcoming paper. In principle, additional information may also be gained about the swelling properties of colloidal particles from an analysis of the cluster shape and orientation with respect to the Mie curves. The ellipticity of e.g. the particles with nominal diameter 378 nm due to pure size variation was oriented along the Mie curves. In an experiment
N. Garbow et al./Physica A 235 (1997) 291-305
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investigating particles subjected to a swelling agent at different times of say lower refractive index, the swollen particles will show a lower refractive index. In Fig. 6 the corresponding cluster will again show an elliptical form; however, the main axis of ellipticity will now be tilted against the Mie curves. Only in a high precision experiment like S P T the accuracy of size determination is sufficient to perform such a sensitive study.
7. Conclusions A novel a p p r o a c h to the characterisation of colloidal particles, single particle tracking (SPT), using a conmfocal tracking microscope, has been presented and t h o r o u g h l y tested. It allows for a high-resolution determination of first- and higherorder m o m e n t s of particle size distributions. In mixtures this allows for the discrimination of kinds of particles with extremely small size differences. Systematic c o m p a r i s o n to other methods gives access to information on surface roughness. Further applications like characterisation of stability and swelling properties were d e m o n s t r a t e d and discussed.
Acknowledgements Financial support of the Deutsche Forschungsgemeinschaft (Scha 389/2) is gratefully acknowledged.
References [1] K. Schnitzel, W.G. Neumann, J. Miiller and B. Materzok, App. Opt. 31 (1992) 770 778. [2] R. Klein, in: Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution, eds. Chen, Huang and Tartaglia, NATO ASI Series C, vol. 369 (1992) pp. 39 110. [-3] P. Bartlett, R.H. Onewill and P.N. Pusey, J. Chem. Phys. 93 (1990) 1299-1312. [-4] H. L6wen, J.P. Hansen and J.N. Roux, Phys. Rev. A 44 (1991) 1169. [5] A. Meller and J. Stavans, Phys. Rev. Lett. 68 (1992) 3646. [-6] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1 89. [-7] G. Joos, Theoretical Physics, 3rd Ed. (Dover, New York, 1986). [-8] M.M. Bertolotti and C. Sibilia, Opt. Acta 31 (1984) 567 577. [9] M.M. Burns, J.-M. Fournier and J.A. Golovchenko, Science 249 (1990) 749-754. [10] J.P. Barton, University of Nebraska, Lincoln, Neb. 68508, personal communication (1991). [-11] S.W. Provencher, Comput. Phys. Commun 27 (1982) 229-242. [12] N.D. Denkov and D.N. Petsev, Physica A 183 (1992) 462-489. [13] S. Alexander, P.M. Chaikin, P. Grant, G.J. Morales, P. Pincus and D. Hone, J. Chem. Phys. 80 (1984) 5776. [14] W. Weise, Ph.D. thesis, Kiel, (1993).
Physica A 235 (19971 291-305
High-resolution particle sizing by optical tracking of single colloidal particles Norbert Garbow*, Jtirgen Miiller, Klaus Schnitzel, Thomas Palberg Johannes-Gutenberg-Universiti~t Mainz, Institut fiir Physik, Staudinger Weg 7, D-55099 Mainz, Germany
Abstract
The motion of individual Brownian particles is observed using the Confocal Tracking Microscope recently introduced by Schnitzel (K. Schnitzel, W. G. Neumann, J. M/iller and B. Materzok, App. Opt. 31 (1992) 770-778). Particles are laterally trapped in a strongly focused laser beam. By evaluating the light-pressure-induced drift velocity and the backscattered intensity we are able to determine particle size histograms with a resolution better than 2%. This is demonstrated on a mixture of seven species of polystyrene latex spheres in the diameter range between 300 and 450 nm, where six classes of diameters are identified. We discuss the scope of the method and potential applications. Keywords: Particle sizing; Multimodal distributions; Polydispersity; Confocal microscopy; Colloidal dispersions
1. Introduction
The synthesis of colloidal dispersions is a well-understood industrial process which yields particles of controlled mean diameters from say 10nm for semiconducter particles up to some microns for latex spheres. Standard deviations below some 5% are achievable and usually referred to as monodisperse. For many applications standard sizing methods like e.g. dynamic light scattering are sufficient, Since these techniques simultaneously average over a large ensemble of particles, the mean diameter may be determined with high accuracy, while an analysis of the size distribution, or the resolution of bi- and multimodal mixtures, is severely limited. Typically, bimodal distributions may be resolved, if the component diameters differ by a factor of two or more. In some cases, however, a better resolution is of vital importance. Colloidal suspensions have, for example, proven to be valuable model systems in the study of
* Corresponding author. 0378-4371/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0378-43 7 1 (96)00349-4
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statistical mechanics and condensed matter physics. Recent theoretical descriptions explicitly include effects of polydispersity, and describe the structure and dynamics of bimodal mixtures [2, 3]. There is a strong evidence, that already the narrow polydispersities observed in standard latex particles, and even more so slight changes in the size ratio of bimodal mixtures, singificantly influence the glass and the freezing transition in these systems [4, 5]. Quantitative experimental studies therefore require careful characterisation of the size distribution. In this paper, we show a new approach different from standard light scattering techniques. Using a specially-built laser tracking confocal microscope, single colloidal particles in a dilute suspension (108-109 cm 3) are observed. A size histogram is collected from typically 103 individual particles. Single particles are illuminated by a strongly focused laser beam. While absorption may be neglected for most common model colloids, particle motion is strongly influenced by the illuminating beam. Perpendicular to the beam optical gradient forces form a potential well. Particles having a refractive index higher than the suspension are bound in the region of highest intensity. This results in a laterally restricted diffusive motion of the bound Brownian particle [6]. In contrast to an optical tweezer the axial gradient forces are not strong enough to dominate the light pressure, hence the particle is driven out of focus. By automatically adjusting the position of the focussing optics, it is possible to follow the particle motion for several seconds, corresponding to some 500 particle diameters. Evaluation of lateral diffusion results in absolute size information, but with a large uncertainty of 10-15%. Alternatively, we measure the Mie backscattered intensity and the light-pressure-induced drift velocity, with accuracies better than 5% and 1%, respectively. While individual measurements of the latter quantities provide only relative size information, their combination can be calibrated to yield absolute particle diameters with a resolution of better than 2%. This paper is organised as follows: After a short introduction to the theoretical background, we give a detailed description of the apparatus and the measurement procedures. Results obtained on a multimodal mixture are presented and discussed in the final section.
2. Bound diffusion (lateral particle motion) Of the three quantities accessible in single particle tracking, the lateral diffusion provides the most direct measure of particle size. We determine displacements in the x and y direction independently. The lateral particle motion is restricted by optical gradient forces (see below), hence they can be considered harmonically bound. For such a Brownian particle, the mean square displacement (MSD) over a time t in one direction is given by [6]
~(x(O)--x(t)2))=2~Z~x2)(1--exp( ( ADtx 2 ) ) ) .
(1)
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the limiting value for long times, (Ax 2 ), is determined by the width of the potential well. Do is the Stokes-Einstein diffusion coefficient for non-interacting particles given by
kT Do = 3n~ld'
(2)
with a thermal energy kT, the viscosity q and the particle diameter d. This coefficient determines the slope of the MSD for small delay times t. As will be briefly discussed in the last section, it is possible to concurrently measure the electrophoretic mobility # = v/E by applying an additional sinusoidai electrical field E, e.g. in the x direction. The induced velocity v changes the mean particle position with the frequency o9/27t and the amplitude A and is in equilibrium with the Stokes friction Fs = - v3rcqd: A-
/~Eo
(3)
(O
The mean square displacement of Eq. (1) is therefore modified to ((x(0) - x(t)2)) = 2 ( A x 2)
(
I -exp
(Ax2)
+ A2(1 -cos(e)t)).
(4)
Measured mean square displacements are shown in Fig. 4. Note that the MSD saturates after some 80 ms. The 'ponderomotive forces' acting on a particle in a highly inhomogeneous intensity profile can be interpreted as force on a dielectric medium in an inhomogeneous mean absolute electric field. In the direction perpendicular to the beam the potential well built up by this force is deep enough for trapping the particles. Typically, they can move in an area of about 100 nm radius around the point of highest intensity. For an optical dipole the depth W of this potential is given by [7-9]
W
= _
2re
R3 np
~Np + 2n 2'
(5)
where I is the peak intensity, R the radius of the particle, n and np the refractive indices of the suspension and the particle and c the vacuum speed of the light. According to calculations by Barton [10], this estimate should hold for particles up to 800 nm in diameter in a focus of 1.2 gm diameter with an accuracy of about 10%. From the known focus width of 1.4 lam and the 1 kT width of approximately 190 nm measured in the long-time limit we estimate the total depth of the potential well to be of the order of 56 kT for the example given in Fig. 4. Depending on the particle size, focal radius, and the laser power, values in the range 30-200 kT are realised. An alternative, intuitive picture for these forces is the momentum transfer of photons passing the particle with np > n. For homogeneous illumination, this results in light pressure only, as the lateral momentum components cancel. In the case of a strong intensity gradient, as it is found close to the focus, the asymmetric momentum transfer leads to a lateral force driving the particle towards the region of highest intensity.
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3. Light pressure and intensity Determining the individual particle sizes m a y alternatively proceed via a c o m b i n e d m e a s u r e m e n t of the axial drift velocity and the backscattered intensity. While the particle is laterally bound, it is axially driven forward by the light pressure with typical velocities of several microns per second. F o r Rayleigh scatterers, the light pressure scales with their cross-sectional area, whereas the friction described in Eq. (3) grows linearly with the diameter. Both forces cancel in the quickly reached equilibrium and a constant (axial) drift velocity is obtained, which linearly depends on the particle size. The relation is still m o n o t o n o u s for particle sizes up to a micron, but corrections for the light pressure using the Mie theory becomes necessary. Also the backscattered intensity m a y be calculated from Mie theory, since our particles are significantly smaller than the focus diameter and the incident light in the 200
250
300
350
400
450
I
I
I
I
I
500
8-
6. velocity [a.u.]
4.
2-
J
o 6-
'
I
I
I
'
I
'
I
'
I
'
intensity [a.u.]
4-
0
200
I
I
!
I
I
250
300
350
400
450
500
d i a m e t e r [nm]
Fig. 1. Axial drift velocity and backscattered intensity for polystyrene lattices (n = 1.59) in a water/glycerol solvent (n = 1.385) and ,~ = 532 nm, calculated using Mie-theory. In contrast to the backscattered intensity the drift velocity grows monotonically with particle size (for particles significantly smaller than the focal diameter).
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centre of the focus may be approximated as a plane wave. In contrast to the monotonous relation between particle diameter and drift velocity, the backscattered intensity for particles of more than 100 nm in diameter has a strong, size-dependent modulation. Calculations for both, the drift velocity and the backscattered intensity, are shown in Fig. 1. The intensity inside the focus is not precisely known. Hence, neither the drift velocity nor the intensity alone is sufficient to determine the absolute size of the particle. However, the combination of measured drift velocity and scattered intensity provides an elegant way to distinguish different sizes and types of particles. For each particle additional data, like mobility, diffusive constant or depth of the potential are determined independently. We note that the statistical accuracy of this approach is increasing linearly in time, since the axial displacements of the particle increase linearly during the measurement. This has to be compared to the short-time lateral displacements due to diffusion, which increase as the square root of time and so does their statistical accuracy. While from the determination of the lateral diffusion the size is directly available, the velocity measurement yields a higher resolution of size differences. lateral position position sensitive diode cuvette
_7
objective polarising beamsplitter
axial position
photo diodes
tilting mirror
Nd:YAG-Laser, 100mW, Z=532nm
~
Lens r-
• [~
Focus Detector
Fig. 2. Simplified experimental set-up. The microscope objective images the first intermediate focus into the cuvette and thus defines the observation volume. A 2/4-plate on the front lens of the objective rotates the polarisation of the backscattered light by 90 °. A particle under observation is imaged onto the detectors, where the axial and lateral position as well as the backscattered intensity are determined.
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4. Set-up In this section we present the experimental realisation of a confocal optical tracking microscope, which allows for the concurrent measurement of the MSD, the backscattered intensity, the light-pressure-induced drift velocity and the electrophoretic motion of an harmonically bound particle (see Fig. 2). A confocal backscattering geometry is chosen to obtain a sufficiently intense illumination and a small observation volume to enable extended measurements on individual particles. Both points are easily fulfilled by focusing the beam of a 100 mW Nd : YAG frequency doubled cw laser through a microscope objective into a highly diluted sample. The microscope objective has the focal length in the ~tm range. The same microscope objective is employed to collect the backscattered light from a solid angle of some 40 °. A 2/4 plate glued to the front lense of the objective and the polarising beamsplitter suppresses reflections not originating from the sample. The lateral position of the particle with respect to the beam centre and its backscattered intensity is determined by imaging it onto a position-sensitive photodiode. The axial position of the particle is measured independently, using two photodiodes combined with pinholes positioned some cm before and behind the image plane, respectively. Both detectors receive the same intensity ifa particle is exactly in focus. As the particle is axially driven out of focus by the light pressure, the difference signal between these diodes is used to control the positioning of the objective via a piezo-driven translation stage. Therefore, our set-up realises a tracking geometry in the axial direction, with typical deviations between the axial particle position and centre of the focus of about one particle diameter and a residual uncertainty of the objective position of about 50 nm. This has to be compared to a focus length (l/e intensity profile) of 13 I.tm. A piezo-tilting mirror is used to scan quickly a larger suspension volume until a particle is caught and the scanning motion is aborted. Successful catches are readily visible in a high and comparably constant backscattered intensity, while rare double catches show up in high-intensity fluctuations. An example is given in Fig. 3. Data taken after the appearance of a second particle are discarded. An important condition for the sample preparation is the use of highly diluted suspensions of typically l0 3 pro-3. This compromises between the necessary scanning time to find a particle (10 20s) and the possible tracking time without double catches (10 15 s). Some further 20 s are needed for data evaluation. After a short delay time to centre and trap the particle laterally, the measuring and axial tracking starts with a sampling clock rate of approximately 800 s-1. The observation time for a single particle is restricted either by the maximum possible tracking displacement of the microscope objective or the available memory of the computer. The current limit in our set-up is approximately 15 s. Size limits for the observation are the optical contrast for small, the maximum tracking velocity for large particles. Currently, the contrast limit is caused by residual internal reflections of the microscope objective rather than detector noise. Interference of this backscattered light with the signal disturbs the autofocus mechanism and
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0
2
4
6
8
10
12
I
I
I
I
I
I
I
14
250
200
axial position [pm] 15o 100
50 2500
I
I
I
I
I
I
Ig
2OOO
intensity
[ADU] 1500
1000
500
2
4
6
8
10
12
14
time [s] Fig. 3. Measured axial position and intensity for a 301 nm particle. The slope in the first graph gives the drift velocity induced by light pressure. The mean value of the second graph leads to the mean backscattered intensity. The intensity variations during the first 12 s result mainly from motion of the particle in the intensity profile due to diffusion and electrophoresis.
prevents valid tracking of particles significantly fainter than the used 300 nm latices. The upper size limit for the measurement is of the order of 500 nm, corresponding to a maximum tracking velocity of about 90 I~m s- 1 However, as the trapping improves with increasing particle size, the laser power may then be reduced leading to lower drift velocities and hence to a shift in the range of accessible particle sizes. The MSD in x and y direction are directly calculated from the signal of the position-sensitive photodiode. A measurement in the presence of a sinusoidal electric field of about 50 Hz and 16.2 V/cm is given as curve A in Fig. 4. The true MSD (curve B) is derived by subtracting a periodic signal of constant amplitude. An initial linear increase and a saturation behaviour at long times are observed, consistent with qualitative theoretical considerations for bound diffusion. In an intermediate time regime a quantitative fit of Eq. (1) is not perfectly possible, since the true beam profile
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25000'
oOOOO: [nm2115ooo-
A
~
looo0-/f
0
AA
~
B
50
1O0 delay [ms]
150
200
Fig. 4. Mean square displacement characterizes the diffusive motion of a 301 nm particle in the optical potential well. The points give the measured values with electrophoresis(A) and corrected for it (B). The drawn line representsEq. (1) for a harmonicallybound particle with the lmiting values of incident slope and saturation. The applied sinusoidal electric field had an amplitude of 16.2 V/cm, and the resulting position amplitude of the particle is 49 nm.
is Gaussian rather than parabolic. Nevertheless, a diffusion coefficient Do may be derived from the initial slope, in this case corresponding to 323 nm diameter. A closer inspection of the data reveals some inherent limitations of statistical accuracy. Only for short times the M S D is dominated by free diffusion and only few data points can be used to obtain Do. In addition, displacements have to be calculated as differences of lateral position data. While the detector noise limits the positional accuracy to some 20 nm, displacements are of the order of 50 nm; hence their uncertainty may easily reach 50%. The alternative route to particle sizing proceeds via the combined measurement of backscattered intensity and drift velocity, which is derived directly from the timedependent microscope objective position. D a t a for a particle with 301 nm diameter are shown in the Figs. 3(a) and (b). The axial position is seen to increase linearly in time; from the slope of the curve in Fig. 3(a) we obtain a constant drift velocity of 16.6 p.m/s. The diffusion in this direction is not visible within the resolution of our axial position detectors. During the main time of observation the backscattered intensity (second part of Fig. 3) shows a more or less constant mean value of 550 AD-units with a significant noise. The observed fluctuations correspond mainly to lateral displacements of the particle due to diffusion and electrophoretic motion in the applied electrical field. Using Mie theory the particle sizes are inferred from a combination of backscattered intensity and drift velocity, if the intensity of the illumination and the viscosity of
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1000
n1=1.580/
I
375or I
/
o
8OO
n =1.590
/
/ /,0nm
intensity [AD-UnitS]oo
400
200 -
32
4Onto
y 110
I
15
210
215
I
30
axial drift velocity [pm/s] Fig. 5. Scatterplot of backscattered intensity versus light-pressure induced for a set of measurements taken on polystyrene particles of diameter 375 nm (triangles). The full lines represent Mie results for two values of the particle refractive index, with the particle diameter as parameter; the theoretical curve for n~ = 1.590 was scaled to fit the data. The dotted lines connect points of the same diameter on the different Mie curves; the error bars included for one particle represent the long time variation of mean intensity and drift velocity. These errors were corrected during the evaluation.
the m e d i u m are known. The former m a y be calibrated using particles of a b r o a d size distribution and a k n o w n refractive index in a well-characterised suspending medium. In Fig. 5 we plot the data for a suspension of polystyrene latex spheres (n = 1.590) of nominal diameter d N = 378 n m suspended in a 1:2 mixture of glycerol and water at 22°C. The data have been corrected for systematic drifts (see below). This sample shows a strongly peaked size distribution with a standard deviation of 2% and a weak tail towards smaller particle sizes clearly visible. The Mie curve for polystyrene particles is fitted to the cluster of raw data to yield a illuminating intensity of 17.7 mW/Ixm 2 with an independently measured viscosity value of r / = 2.6 cP. The projection of the cluster m a x i m u m onto the Mie curve corresponds to a particle size of 377 nm, the mean particle diameter (cf. Table 1) being somewhat smaller due to the presence of the tail in the distribution. We note that the symbol size of the scatterplot corresponds to the uncertainty of each single measurement and is m u c h smaller than the variation between particles, since both mean backscattered intensity and drift velocity m a y be determined with extremely high accuracy. D u r i n g the measurements the optical set-up was thermostatically maintained only by controlling the ambient temperature. The intensity
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Table 1 Particle diameters obtained using different methods. All methods except SPT use only one kind of particle at time, SPT data are corrected for systematic drifts. Standard deviation and residual polydispersity is also corrected for statistical uncertainty Single particle tracking (nm)
Standard deviation A(nm)
Residual polydispersity a (%)
Refractive Static light index scattering (nm)
Dynamic. light scattering (nm)
Nominal diameter (TEM) (nm)
Nominal polydispersity 2a (%)
320 301 375 401 454 401 438
2.4 1.4 6.8 < 4.5 0.9
0.7 0.4 1.8 < 1.3 0.1 < 1
1.585 1.59 1.59 1.59 1.585 1.585 1.585
332 308 386 426 488 448 454
301 306 378 401 403 412 415
2.0 1.07 5.9 1.85 1.6 1.9 2.1
<5
311 301 362 395 434 407 412
t u r n e d out to be fairly r e p r o d u c i b l e if the a m b i e n t t e m p e r a t u r e was k e p t constant, but was subject to larger u n s y s t e m a t i c drifts, if fluctuations in t e m p e r a t u r e exceeded I°C. Also the drift velocity s h o w e d m a r k a b l e v a r i a t i o n s as the optical p a t h length on the d e t e c t o r side fluctuates with t e m p e r a t u r e , but to a much lesser extent. T a k i n g into a c c o u n t the t e m p e r a t u r e fluctuations we estimate the r e p r o d u c i b i l i t y of single m e a s u r e m e n t s to be better than 7 % in the intensity a n d better t h a n 2 % in the velocity, as is i n d i c a t e d by the e r r o r b a r in Fig. 5. As the t e m p e r a t u r e - d e p e n d e n t drifts occur on time scales larger t h a n s o m e 50 single m e a s u r e m e n t s a n d m a y be c o r r e c t e d for, this c o m b i n e s to an u n c e r t a i n t y in the d i a m e t e r of less t h a n 2 % for each i n d i v i d u a l particle. F o r clusters larger t h a n say Ni = 100 particles situated a w a y from the M i e e x t r e m a the m e a n d i a m e t e r m a y therefore be d e t e r m i n e d with a statistical e r r o r smaller t h a n 0.2% a n d a systematic u n c e r t a i n t y of a b o u t 0.5% in the present case c o r r e s p o n d i n g to dspx = 375 _+ 7 nm. F u r t h e r this yields a residual p o l y d i s p e r s i t y PsPv defined as the relative s t a n d a r d d e v i a t i o n from the m e a n d i a m e t e r dsvr in a h i s t o g r a m of drift corrected data. F o r the particles investigated in Fig. 5, Psvr = 1.8%. Also higher m o m e n t s of the size d i s t r i b u t i o n m a y easily be resolved with high accuracy, too, a n d the tail to small d i a m e t e r s is clearly visible in Fig. 5. F o r clusters in the vicinity of M i e e x t r e m a an a d d i t i o n a l a m b i g u i t y m a y arise due to the choice of the index of refraction. The statistical accuracy, however, is not affected.
5. Polydispersity and muitimodal distributions T o explore the range of a p p l i c a b i l i t y a n d the resolution of the a p p a r a t u s , a suspension m i x e d from 7 different types of p o l y s t y r e n e lattices in a n o m i n a l d i a m e t e r range 301-415 n m was analysed. All samples h a d been a n a l y s e d individually by single
N. Garbow et al. IPhysica A 235 (1997) 291-305
10oo-
800" intensity [AD-units] 600-
400"
301nm 320nm , 375nm ,lol°m
o
n1=1.585 n1=1580/ / / n1=1590 ~ / / , 460n
~
~~
301
Iit'"
\
/ /."7
I."I I
438nr,
.~...~.~~~j .~..~~ - 440nm
329nrn "" /'///
200
0
1'o
l's
2'0
2'5
3'0
axial drift velocity [pm/s] Fig. 6. Scatterplot for a multimodal suspension. As in Fig. 5 the full lines are theoretical curves for particles with the indicated refractive indices and diameters between approximately 250 and 480 nm; the dotted lines connect (calculated) values for particles of the same size and different refractive indices. Each kind of symbols represent a different type of particles. Six of the seven particle types in the suspension are resolved, while the nominal sizes 412 and 401 nm are combined in the 401 nm cluster (triangles downwards).
particle tracking (SPT) before. Their number concentrations ranged from 10-s to 2.2 × 10 -4 Ixm-3. Results of this measurement are shown in Fig. 6. Included are four theoretical curves for increasing particle size corresponding to four different refractive indices. Again, the intensity in the focus was rescaled to fit the dN= 378 nm particles discussed before. Six clusters corresponding to the seven kinds of particles are clearly resolved in the scatterplot and marked with different symbols. This high resolution is emphasized in Fig. 7, where a histogram of the mixture is shown. For comparison the size distribution resulting from a CONTIN-fit [11] to a dynamic light scattering experiment is indicated by the dashed line. Table 1 compares the mean particle sizes and standard deviations with values derived from dynamic and static light scattering and to the manufacturer's data, measured by electron microscopy. We would like to highlight the performance and limitations of our instrument by discussing two examples: (a)the discrimination between the two smallest kinds of particles and (b) between the three kinds of particles positioned in the Mie curve minimum. The main feature of the measurement is a clear discrimination between particles of same refractive index and, in the case of 301/320 nm particles, a difference in mean size
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140
,
1,o
Contin Analysis, 90 ° . I
120 t
Counts
"-..
100 1
8o-]
.-
320nm
°'Selativ e number 0,6
454nm
301nm
60 •. . . 40
i"
401nm
-
375nm
2 0 - J .
438nm
,
o-
[--.300
0,4
.-"
350
400
0,2
0,0
450
Diameter [nm] Fig. 7. Size histogram of the multimodal suspension, derived from the data in Fig. 6 by projecting the individual particle observations on the Mie curves for the respective cluster• The histogram resolution is 1 nm, the total number of valid observations is approximately 1550. The dashed line indicates the size distribution obtained by a CONTIN-analysis of a conventional DLS measurement.
of only 6%. Note that the dashed lines in Fig. 6 correspond to lines of constant radius, and that the discrimination of radii proceeds along the Mie curves• In fact, the limit of resolution between different monodisperse clusters is given by the combined uncertainty due to the temperature stability of the present set-up and the statistical accuracy. It is of the order of 2% size difference. Our particles show a small but significant polydispersity. Taking the systematic uncertainties to be independent of the particle size, the residual polydispersities (relative standard deviations of the diameters corrected for both systematic drifts and statistical accuracy) of the 301 and 320 nm particles are estimated from the second moments of the clusters to be about 0.4% and 0.7%, respectively. While the resolution in the former case was excellent, it is less favourable in the case of the particles with diameters between 400 and 440 nm. As was already known from the reference measurements, the data points of particles with nominal diameters 401 and 412 nm coincide, i.e. they are measured to have the same diameter of 401 nm. In addition, a loose heap of only a few observations is formed by the particles with 415 nm nominal diameter. Although it is clearly discernable from the 401 nm cluster, the catch efficiency is fairly low. This is even more clearly visible in Fig. 7, where these three sorts appear to be the least present species• Note that the data of this histogram are corrected for systematic drifts only. The shape and number of data points belonging to the cluster are strongly influenced by the minimum intensity necessary
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for tracking a particle which provides a cut-off at about 200 AD-units due to the residual reflections. In contrast to the other sorts, no well-defined analysis of polydispersity or of the third moment of the distribution are possible here. These difficulties may, however, be circumvented by using different wavelengths leading to a significant shift of the clusters along the shape of the Mie curves and away from the minimum. This would also solve the ambiguities resulting from uncertainties in refractive index in the vicinity of Mie curve extrema. A comparison between the results of single particle tracking (SPT) and the values obtained by static light scattering (SLS), dynamic light scattering (DLS) and transmission electron microscopy (TEM) shows two systematic trends in the obtained results (Table 1). First, TEM usually gives the smallest size, followed by SLS, SPT and DLS. A good agreement between all methods, and in particular between SPT and TEM, is obtained for particles with refractive index 1.590, while deviations in mean size are significantly larger for particles with lower refractive index. This points to swelling processes in solution or to shrinking in the vacuum conditions of the electron microscope used by the manufacturer. Homogeneous swelling of particles may be caused by residual monomer (of refractive index 1.550) or by solute. Additional sources of deviations would be the presence of an extended or rough boundary region only formed in solution. Second, the SPT diameters range between the data of static and dynamic light scattering (SLS, DLS). SLS yields the mean geometric size. In SPT the light pressure and the backscattered intensity depend on the geometric size, but the Stokes friction depend on the hydrodynamic radius. The DLS data were taken on diluted samples with millimolar salt concentration to minimize electrolyte friction and thus to obtain the true hydrodynamic radius 1-12]. The good agreement obtained for all experiments on particles with n = 1.590 supports the assumption of negligable electrolyte friction. While for particles of refractive index 1.585 SPT and SLS yield rather close results, the deviations between dynamic and static light scattering are as large as 20%. Different degrees of swelling may not be the cause here, since both latter experiments were performed on the same samples. Another source of an extended hydrodynamic radius as compared to the geometric radius may be the presence of surface roughness, which would be differently detected by the different experiments. The amount of surface roughness may, however be connected to an already completed swelling. Apart from polymer surface roughness the hydrodynamic radius may also be influenced by the structure of the boundary region. This may include layers of adsorbed molecules or condensed counter ions, which are in the nm range 1-13]. It may also be due to hairy structures. While no detailed knowledge about the surface of the particles exists, our data seem to be indicative of an enhanced surface roughness for swollen particles. The most extreme deviations are observed for the sample with nominal diameter 403 nm. It is interesting to note further that this sample shows an unexpectedly low
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polydispersity in the SPT measurement, which is smaller by more than a factor of five as compared to the TEM results. This behaviour is reminiscent of the swelling behaviour of network colloids synthesized in emulsion polymerisation. There, particle sizes are controlled by the emulsion droplet size but individual droplets may differ in their polymer content and degree of crosslinking. After synthesis the emulsifier is removed e.g. by dialysis. At large crosslinking densities the maximum size in solution is still given by the original droplet size, while the minimum size under vacuum conditions depends on the polymer content. Unfortunately, we do not have detailed informations about the manufacturing process of this sample and its peculiar properties remain a challenge.
6. Further applications SPT allows an access to further information on single particles, and in particular to their surface potential via the electrophoretic mobility /~. This is an important property for several reasons. First, it gives a criterion of the stability against aggregation. Second, it allows an estimate of adequate handling procedures like in filtering or viscometry. Finally, one may here measure the electrophoretic mobility under conditions of large double layer extensions as compared to the particle radius practically without any interference by interaction with neighbouring particles. Since the surface potential may depend on the particle number density, the mobility measured at high dilutions may not always be used to estimate particle interactions. However, informations about stability and handling usually remain valid and useful also at higher concentrations. The MSD at the lateral position is shown in Fig. 4. The points on the oscillating curve are the measured data including electrophoresis. A line represents Eq.(1) with the limiting value for long times reached after some lOOms. The maximum lateral displacement caused by diffusion is about 94 nm. If a DC electric field of sufficient strength (here: 16.2 V/cm) is applied, the particle is moved out of the beam trap. We use here an AC-field to avoid excursions larger than the particle radius. In addition, a correction for electroosmotic effects is not necessary at field frequencies higher than 20 s-1 [14]. In Fig. 4 the field-induced position amplitude with respect to Eq. (4) is A = 49 + 1.5 nm. From this we calculate the electrophoretic mobility to be /~ = 2.23 + 0.07 (lam/s)/(V/cm). This accuracy of 3% is well comparable to the one reached in e.g. Doppler velocimetry. Following the theory of O'Brian and White [15] the corresponding surface potential is 140 mV. Detailed electrokinetic characterisations for our particle mixture and for the mobility of single particles will be the subject of a forthcoming paper. In principle, additional information may also be gained about the swelling properties of colloidal particles from an analysis of the cluster shape and orientation with respect to the Mie curves. The ellipticity of e.g. the particles with nominal diameter 378 nm due to pure size variation was oriented along the Mie curves. In an experiment
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investigating particles subjected to a swelling agent at different times of say lower refractive index, the swollen particles will show a lower refractive index. In Fig. 6 the corresponding cluster will again show an elliptical form; however, the main axis of ellipticity will now be tilted against the Mie curves. Only in a high precision experiment like S P T the accuracy of size determination is sufficient to perform such a sensitive study.
7. Conclusions A novel a p p r o a c h to the characterisation of colloidal particles, single particle tracking (SPT), using a conmfocal tracking microscope, has been presented and t h o r o u g h l y tested. It allows for a high-resolution determination of first- and higherorder m o m e n t s of particle size distributions. In mixtures this allows for the discrimination of kinds of particles with extremely small size differences. Systematic c o m p a r i s o n to other methods gives access to information on surface roughness. Further applications like characterisation of stability and swelling properties were d e m o n s t r a t e d and discussed.
Acknowledgements Financial support of the Deutsche Forschungsgemeinschaft (Scha 389/2) is gratefully acknowledged.
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