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A method for measuring the size distribution of latex particles by scanning force microscopy
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Ultramicroscopy 100 (2004) 319–329
A method for measuring the size distribution of latex particles by scanning force microscopy G. Roea, L. McDonnellb,*, A. Ghanemc a School of Science, Galway-Mayo Institute of Technology, Galway, Ireland Department of Applied Physics and Instrumentation, Centre for Surface and Interface Analysis, Cork Institute of Technology, Bishopstown, Cork, Ireland c Analytical Technologies, Solvay Research and Technology, Rue de Ransbeek 310, 1120 Brussels, Belgium b
Received 1 July 2003; received in revised form 5 December 2003; accepted 7 January 2004
Abstract A methodology has been developed to accurately determine the size distribution of latex particles using the scanning force microscope (SFM). Unlike other workers, who have generally measured the lateral dimensions of monolayers of latex particles using a global quantification method, we have measured the heights of individual latex particles located at the edges of latex monolayers that were immobilised onto mica substrates. In agreement with other work, we noted that the edges of monolayers of latex particles provided stable and reproducible scanning force imaging. Whilst SFM imaging noise, image processing artifacts, tip/sample forces and variations in the mica substrate are sources of measurement error that should not be overlooked, our experience has been that the variation over time of the sensitivity of the Z actuator is the greatest potential uncertainty in determining the heights of latex particles. The methodology that we used requires frequent calibration of the Z actuator of the SFM, typically before and after two or three images, in order to ensure that the uncertainties in the Z sensitivity are known and minimised. This methodology was developed for an SFM instrument that was equipped with open loop piezoelectric actuators following a careful study of the behaviour of those actuators. Using this methodology, we have measured the size distributions of populations of 300–400 latex particles from each of several different latex samples, with the maximum variation in the Z-actuator calibration experienced during the measurement of a sample being less than 2%, often about 1% and occasionally better still. In so doing, we have demonstrated that SFMs equipped with open loop actuators can be used for high confidence quantitative measurements of step heights. r 2004 Elsevier B.V. All rights reserved. PACS: 06.30.Bp; 68.37.Ps Keywords: Latex; Scanning force microscopy; Size distribution
1. Introduction *Corresponding author. Tel.: +353-21-4326230; fax: +35321-4345355. E-mail address: [email protected] (L. McDonnell).
Following its invention in 1986 [1], the scanning force microscope (SFM) was soon deployed to visualise surface topography for a wide range of
0304-3991/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2004.01.019
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sample types, materials and applications. Contrast in SFM images is derived from direct measurements of surface topography and thus threedimensional measurements can be readily extracted from SFM images. This is unlike most optical and scanning electron microscopies where metrology is limited to lateral dimensions, image contrast in these cases being derived from slopes in surface topography rather than from heights. Despite this intrinsic quantitative ability, most SFMs have been designed to visualise rather than measure surface topography and therefore considerable care is needed if valid measurements are to be derived from SFM images. Quantification of SFM images can be divided into two categories: the measurement of global parameters, such as surface roughness, that ‘‘average’’ the entire image and the measurement of the dimensions of discrete image features, the latter being the category of measurement that this paper specifically relates to. Whilst the SFM can be used as a miniature coordinate measurement machine, it has some limitations as illustrated in Fig. 1(a). Firstly, with the exception of those few instruments equipped with tips that probe vertically and laterally
SFM tip Object
Re-entrant zone
Tip convolution
(a)
SFM tip Object
Tip convolution
Re-entrant zone
(b) Fig. 1. Relationships between the SFM tip, the object being imaged, the imaged shape and the imaged dimensions when the tip has (a) sloped sides and (b) vertical sides. Neither tip is able to image the re-entrant zone of the object.
(Dimension X3D, Veeco Instruments, Santa Barbara, CA, USA), SFMs cannot access re-entrant regions such as the lower part of the object shown in Fig. 1(a). Another issue arises when the SFM tip interacts with the sample, the point of interaction on the tip being referred to as the proximal point. During imaging, the proximal point moves across the tip to an extent that depends on the relative sizes and shapes of both the tip and the object of interest, thereby convoluting the object’s shape with the proximal point’s trajectory over the tip. As the object in Fig. 1(a) is imaged from left to right, the proximal point moves down the leading edge of the tip to the tip apex and then up the trailing edge of the tip, thereby adding a sloped skirt to the spherical object with clear implications for subsequent quantification [2]. As indicated in Fig. 1(b), using a tip with a vertical side wall will eliminate the shape change although a dimensional increase in the width will remain. Fortunately, SFM images can be corrected for tip imaging artifacts by extracting the tip profile from the image. The tip profile can be determined directly, using either a tip characteriser or a scanning electron microscope image of the tip, or virtually, using a blind reconstruction algorithm [3] that identifies the motif in the image that is due to the tip. However, it should be noted that the SFM tip acts as a filter and the above correction/ reconstruction processes cannot recover information that the tip has rejected, such as topographic features that are finer than the tip shape can resolve. Crucially, regardless of tip dimensions or shape, heights will always be measured correctly by the SFM provided that (i) the tip has unrestricted access to at least one side of the immobilised object and (ii) there is no distortion of the sample by the tip (or vice versa) during imaging. SFM was applied to latex particles quite early in its history with latex particles being used, e.g., as a calibration artefact [4]. The SFM has also been used to study the behaviour of latex film formation [5], latex film defects [6,7] and the effects of crosslinking [8]. The scientific literature also contains several references (e.g. Refs. [9–13]) to different methods by which latex particle sizes, and their distribution in size, can be determined by the SFM. Generally, workers have favoured measuring lateral
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dimensions [14]. In this paper, we present an alternative method by which the size distribution of latex particles can be accurately determined by measuring the heights of particles at the edges of monolayers with the SFM.
2. Methodologies to measure the size distribution of latex particles 2.1. Lateral measurements of latex particle diameters One of the more common methods (e.g. Refs. [9,10,13]) of determining latex particle sizes is to obtain the particle diameters from the lateral line profiles taken along the points of contact between nearest neighbours as illustrated in Fig. 2. There are a number of potential errors associated with this approach, the most fundamental being those introduced by tip imaging effects at the interstices between nearest neighbours of differing diameters. Fig. 3 shows schematically how tip shape, tip asymmetry and the size ratios of the nearest neighbours affect the location of the imaged interstice and hence the perceived diameter of the latex particles. Using a tip with a vertical side wall (left-hand side of tip A in Fig. 3), the calculated maximum error in the smaller particle’s diameter is
Fig. 2. SFM line profile through a row of latex particles with a vacancy.
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1% for a nearest neighbour size ratio of 75% and 11% for a ratio of 50%. For sloped tips (tips B and C in Fig. 3) and a given nearest neighbour size ratio, the error increases as the tip angle increases. Fig. 4 shows the effect of tip imaging on the lateral dimensions of latex particles bordering a vacancy within a layer (a) and at the edge of a layer (b) and demonstrates qualitatively the importance of omitting such particles from lateral measurements. It is also possible to use Fourier transforms [11], radial distribution functions [12] and power spectral density analysis to quantify the diameters of large areas of close-packed latex particles. These methods emphasise the spatial position of the (higher) centres of the particles rather than the (lower) interstices and so are less sensitive to tip imaging effects. However, the accuracy of these methods is undermined by the contributions of particles that are not close-packed.
A
B
C
Fig. 3. Schematic diagram showing how the perceived diameters of latex particles in SFM images are influenced by the shape and asymmetry of the SFM tip and the relative sizes of the nearest neighbours. Tip A is asymmetric with one vertical side wall. Tip B is symmetric. Tip C is also symmetric but with a larger tip angle than tip B.
Fig. 4. (a) 3 mm 3 mm SFM image of a vacancy in a closepacked latex layer and (b) 5 mm 5 mm SFM image of the edge of a close-packed latex monolayer.
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2.2. Vertical measurements of latex particle diameters The method chosen in this work to determine the size distribution was to measure the heights of individual particles at the edges of single layers of latex particles that were dried onto flat mica substrates. This method has two advantages: (i) it eliminates tip imaging effects from the measurement as there is unrestricted access from one side of the particle of interest; and (ii) it enables the datum level of the substrate to be established at a point that is very close to the particle of interest. The validity of height measurements extracted from an SFM image is most likely to be affected by three factors: (i) tip/sample forces; (ii) errors introduced by the image processing algorithms used to remove tilt and curvature; and (iii) uncertainties in the position of the tip. These issues are discussed in Section 3. Two other factors, imaging noise and variations in the height datum also contribute to uncertainty in the height measurement. Care has been taken to minimise such effects and their contribution to the uncertainty is cited later.
mode to minimise the force applied to the latex particles. Although no particle height variations were observed over a range of cantilever oscillation amplitudes, non-contact SFM imaging was carried out at the smallest possible oscillation amplitudes. 3.2. Position of the tip The piezoelectric materials used in SFM actuators suffer from creep, ageing, non-linearity and hysteresis and hence present significant challenges if valid measurements are to be derived from SFM images. The first generation of SFMs used piezoelectric actuators in open loop and these instruments are fully exposed to the inadequacies of the actuators. Second generation SFMs use closed loop positional control of the X, Y and Z movements to minimise the deficiencies of the piezoelectric actuators and this has helped the SFM to evolve into a more quantitative instrument. Nevertheless, the behaviour of both types of instrument needs careful characterisation so that appropriate calibration procedures are developed. 3.3. Image processing algorithms
3. Uncertainties in height measurements 3.1. Tip/sample forces There are three general modes of SFM operation: contact mode, intermittent contact mode (or Tapping ModeTM) and non-contact mode. In contact mode the tip is in permanent repulsive contact with the sample and the vertical and lateral forces applied during imaging can distort or damage the sample [15]. Intermittent contact mode has proved to be very effective at reducing the distortion arising from lateral interactions, but for very soft samples may still apply significant vertical force. The lowest forces are applied in non-contact mode provided that the tip always operates in the attractive regime, thereby avoiding any contact [16–18]. Although it was possible to image the latex particles in contact mode without apparent damage, it was deemed prudent to use non-contact
Because SFM image contrast is generated by surface form, sample tilt, the curvature of the piezoelectric scanner and surface topography, SFM images generally contain unwanted backgrounds that require removal by image processing algorithms. Sample tilt can affect SFM images in two ways. Firstly, the tilt alters the angle at which the sample’s topographic features are presented to the SFM tip and introduces a geometric distortion that is further compounded by tip imaging as discussed earlier. Secondly, many image processing or ‘‘levelling’’ algorithms remove tilt incorrectly. Fig. 5 shows the effect of sample tilt on the SFM measurement of a step height. The solid black line shows the actual tilted surface and the grey line shows the path of an ideal SFM tip of negligible diameter and vertical side walls. The dotted line shows the ‘‘levelled’’ surface that is generated by commonly used levelling algorithms that simply rotate the flat portions AB and CD about points B and C, respectively, as indicated by
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D C
h
E
θ
F
B A Fig. 5. Schematic diagram illustrating the effect of tilt angle y on the measurement of a step edge of height h: The grey line (ABCD) shows the profile followed by an ideal SFM tip. The curved arrows and the dashed line (EBCF) show the effect of incorrect planar levelling.
the curved arrows. It is clear from Fig. 5 that there is now an error in the measured step height as given by the relationship: error in height ¼ h½ð1=cos yÞ 1; where h is the true step height and y is the angle of tilt. For a 5 tilt the calculated maximum error is +1.5%, for a 3 tilt the error is +0.6% whilst for a 1 tilt the error is +0.1%. Clearly, for the highest accuracy, care needs to be taken to ensure that the tilt is minimised. Fu et al. [19] have shown that levelling algorithms should remove tilt by carrying out a rotational transformation of the co-ordinates for the entire image. In Fig. 5, this would be achieved by rotating the entire line profile about either point A or D. They also demonstrated that the error introduced by sample tilt can be severely compounded by the algorithm used to calculate the step height, with errors of more than 31% being experienced for single atomic step height measurements on the Si(1 1 1) surface when using one commercial algorithm. Most SFMs use either tube or tripod piezoelectric actuators to scan either the sample or the tip over trajectories that are curved rather than planar. This is demonstrated in Fig. 6(a) which shows a longrange SFM image of a super-smooth polished glass sample1 obtained by the ExplorerTM SFM (Veeco Instruments, Santa Barbara, CA, USA) used in this work. Fig. 6(b) also shows a line profile through the
Fig. 6. 3D view SFM image (a) of a super-smooth polished glass surface obtained with the ExplorerTM SFM and a line profile (b) from the centre of this image.
centre of this image. The flatness of the glass sample is such that the curvature observed in Fig. 6 can be fully attributed to the curved trajectory of the scanner. Whilst this curvature can be removed by software algorithms, care needs to be taken to ensure that other errors are not introduced during the levelling process [20,21]. Fortunately, because of the relatively large length of the tripod scanner in the ExplorerTM SFM, the curvature introduced into the full scan is much smaller than many other SFMs and is quantified later in Section 4.4.
4. Experimental 4.1. Sample preparation
1
European Commission BCR project. Characterisation and development of surfaces with roughnesses in the 0.1 nm range. Contract No. 3425/1/0/185/91/4-BCR-D(30).
The latex particles were immobilised onto mica substrates (Agar Scientific, Stansted, England)
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using the general method described by Van Cleef et al. [12]. Firstly, the stock suspension (nominal dilution of 0.1%) was sonicated for 30 min to remove agglomerates. Then, a 0.15 ml aliquot of the suspension was immediately deposited, without dilution, onto a piece of freshly cleaved mica. The substrate was then placed in a dessicator for 24 h, after which the dried sample was ready for imaging. 4.2. Scanning force microscope imaging conditions Non-contact SFM imaging was carried out in air with an ExplorerTM SFM equipped with a long lateral range (130 mm) tripod type scanner. Silicon non-contact cantilevers (type SCF11 UltraSharpTM, Silicon-MDT Co., Moscow) with a nominal force constant of 5 Nm1 were used. Before use, the shape of the cantilever tip was determined by imaging a tip characteriser (type TGT01, Silicon-MDT Co., Moscow). As discussed earlier, in principle the SFM tip radius is not crucial to either the accuracy or the repeatability of a step height measurement, provided that the tip has unrestricted access to the step. Nevertheless, SFM tips were rejected if the images did not show sufficiently deep interstices between adjacent closely packed latex particles. Sharp interstices are not indicative of a sharp tip as a very blunt tip will also produce a sharp turning point as it crosses the interstice. SFM tips were also rejected if image noise, or other imaging instabilities, were experienced as such effects are indicative of tip contamination and could affect the height measurements by increasing noise and altering the tip– sample interactions on the top of the latex particle vis-a" -vis the mica substrate. SEM images of tips that degraded during imaging revealed that in many such cases latex particles from the sample had attached to the tip. The samples were placed for SFM analysis onto a computer controlled XY translator. Sample tilt was measured from line profiles obtained in the X and Y directions and reduced where necessary using the manual adjustments on the ExplorerTM SFM. In all cases, tilt of the samples (and of the step height standards used in calibration) was reduced to less than 1 so that the resulting
systematic error in step height measurement was insignificant (+0.1%) and could be neglected. Suitable general locations on the mica substrate were identified using the internal video camera on the ExplorerTM SFM. These locations were then inspected by SFM using large scan ranges (50 mm 50 mm or 100 mm 100 mm). Suitable locations were then centred using the XY translator for subsequent zoom imaging by the SFM. A high-resolution SFM image, of either 10 mm 10 mm area (500 pixels 500 pixels) or 5 mm 5 mm area (400 pixels 400 pixels), was taken for the larger and smaller latex sample sizes, respectively. The pixel resolution was selected to maximise the accuracy of the height measurement and to minimise the image acquisition time. The calculated maximum error in height measurement arising from image pixellation has been calculated to be 0.06% for a 600 nm diameter particle (20 nm 20 nm pixels) and 0.2% for a 200 nm diameter particle (12.5 nm 12.5 nm pixels). This error is determined from the height error that is introduced if the top of the latex particle is not imaged due to image pixellation. The error becomes more significant for a given pixel size as the latex particle diameter decreases. Images were typically acquired at a line scan rate of 0.67 Hz. 4.3. Calibration of the Z actuator In order to have confidence in the height measurements made by the ExplorerTM, the behaviour of its independent Z actuator was studied. Heyde et al. [22] have shown that a relatively low-cost fibre-optic displacement sensor can be used for such purposes. In the work reported here, the performance of the Z actuator was studied using such a displacement sensor (Philtec, Annapolis, MD, USA) and discrete step height standards. A crucial observation that was made during this study concerned the effect of offset voltages on the behaviour of the Z actuator. This actuator, used when the instrument is configured for long lateral ranges, is provided with a voltage offset that allows feedback to be engaged when the Z actuator is off-centre. Unfortunately, invoking the voltage offset invalidates the calibration as shown in Fig. 7. The data
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540 530 520 510 500 490 30
108 107 106 105 104 103 0
40
50 Z offset voltage (V)
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Fig. 7. The effect of varying the voltage offset of the independent Z piezoelectric actuator of the ExplorerTM SFM on measurements of a 512 nm step height standard.
in Fig. 7 demonstrate that the measurement of a 512 nm step height varied by almost +5% when voltage offsets corresponding to about +20% of the actuator range were used. Consequently, all images of step height standards and latex samples were obtained with a zero offset voltage, i.e. with 50 V applied to the Z actuator, so that the instrument remained in calibration. Furthermore, the reproducibility of the Z actuator for a given step height was studied and it was found that the step height measurement varied over time. In order to minimise this effect, the Z actuator was typically calibrated before and after two or three images of the latex particles using the same step height standard and the average step height was used to retrospectively calibrate the ‘‘bracketed’’ set of images. Fig. 8 shows the variation in the measurements of a 106 nm step height artefact for the calibrations performed during several SFM imaging sessions of a latex sample that took place over 5 days, the longest period during which we obtained data from a given sample. The maximum uncertainty in the step height measurement for any pair of calibrations was 0.8% and is somewhat less than that for the overall set of calibrations (2%). Where ‘‘bracketed’’ images included an overlap between all successive images, the overlapped particles in the image (see e.g. Fig. 9) were used as internal standards to determine the calibration correction for that particular image. The most stable imaging performance of the ExplorerTM SFM was obtained by ‘‘warming up’’ its XY piezoelectric actuators by continuously scanning in free space for a period of 1 h, the scan
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100 Time (hours)
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Fig. 8. The variation with time of measurements of a 106 nm step height standard made with the independent Z piezoelectric actuator of the ExplorerTM SFM during four imaging sessions.
speed being the same as that used for subsequent imaging. During this warm-up period the Z actuator was maintained at its mid-point voltage. Piezoelectric actuators perform best if they are operated over the same voltage range at the same scan speed. Changing scan speed and, more importantly, changing the scan range degraded the stability of the piezoelectric actuators. It was also found to be beneficial to leave the SFM electronics switched on continuously, although it should be noted that, in order to avoid damage, the high voltages were only applied to the piezoelectric actuators during the warm-up period and whilst imaging. The ExplorerTM SFM is a scanned cantilever instrument of the ‘‘stand-alone’’ type [23] and is lifted off the sample stage in order to change samples and/or step height artefacts. This physical disturbance can introduce instabilities in the subsequent performance of the instrument. Instabilities can also be introduced whenever feedback is disengaged as the actuator is exercised over a much larger range during retraction of the SFM tip than when imaging. This was demonstrated by continuously imaging a step height artefact and obtaining a superior repeatability of step height measurements compared to discontinuous, sequential imaging where the feedback was disengaged and engaged between consecutive images. 4.4. Measurement of latex particle heights Prior to quantification, the tilt in the SFM images was removed using the three point planar levelling algorithm within the software provided
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326 10 µm
5 µm
10 µm
0 µm 0 µm
5 µm
10 µm
5 µm
0 µm 0 µm
5µm
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Fig. 9. Example of two overlapped SFM images.
with the ExplorerTM SFM. Using this algorithm, three locations on the (planar) mica substrate were defined and used to determine the planar levelling correction. As discussed earlier (Section 3.3), planar levelling algorithms can introduce step height measurement errors. Therefore, the efficacy of this commercial algorithm was determined using simulated images of step height artefacts with tilts similar to the experimental SFM images. This algorithm was found to introduce maximum calculated height errors that were less than 0.05%. We did not remove the scanner curvature described earlier (Section 3.3) as the line profile in Fig. 6(b) shows that the maximum calculated height error associated with a 1 mm lateral distance (the relevant scale for the step height measure-
ments made in this work) is less than 0.39 nm. The latex particle heights were measured from the levelled SFM images using software written in Matlab (Release 12, The Mathworks, Natick, MA, USA). With this software the user selects with a cross-hairs cursor the centre of the latex particle of interest. The line profile is displayed to enable the user to identify the top of the latex particle and the position on the neighbouring mica substrate that is to be used as the height datum. The latex particle height is then determined as the difference between these two heights. In all cases the lateral distance between the two height measurements is less than 1 mm to ensure that the error due to the residual curvature is less than 0.39 nm, as discussed above. In this method it is presumed that the latex datum
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continues without variation underneath the latex particle. Indeed, it cannot be determined whether the latex particles are sitting on a different datum arising from variations in the mica or from contamination underneath the particle. However, the mica substrate is extremely flat and over distances of 1 mm height variations due to atomic steps are negligible. A particular feature of the software is its ability to search three dimensionally about the selected line and points therein to determine that the top of the latex particle has been found and that the mica substrate is flat. To test the repeatability of the measurement procedure, 30 measurements were made of the height of the same particle from a given image from a range of samples. These measurements involved opening the image, selecting the top of the latex particle and its mica datum. In all cases the standard deviation of the repeated measurements was better than 0.15%. This uncertainty includes contributions arising from imaging noise and variations in the mica datum. 4.5. Example of size distribution of latex particles
65 3 66 6. 5 m or e
55 9 57 2. 5 58 5. 9 59 9. 3 61 2. 7 62 6. 2 63 9. 6
70 60 50 40 30 20 10 0 54 5. 6
Frequency
Fig. 10 shows the size distribution obtained for one of several latex samples that have been measured using the methodology described above. The mean diameter for the 421 latex particles measured for this sample (from 25 images) was 635 nm with a standard deviation of 19 nm. The Z calibrations for these data were obtained using a 512 nm step height standard and the maximum calibration uncertainty experienced in any imaging
Height (nm)
Fig. 10. Distribution of heights for a latex sample. In all, 421 particles were measured and found to have a mean height of 635 nm with a standard deviation of 19 nm.
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Table 1 Table of uncertainties and their sources for the latex particle measurements presented in Fig. 10 Source of uncertainty a
Image pixellation Step height standardb Planar levelling algorithma Scanner curvaturea Particle height algorithm (includes imaging noise and datum variation)b Scanner calibrationa Combined uncertainty
Uncertainty (%) 0.06 1.00 0.05 0.07 0.08 0.56 1.06
Note: In calculating the combined uncertainty it is assumed that the root-mean-square sum of all uncertainties is applicable. Uncertainties that are expressed as maximum errors are divided by the square root of three prior to the latter calculation. a Refers to an uncertainty expressed as a maximum error. b Refers to an uncertainty expressed as a standard deviation.
session was 2.9 nm, i.e. 0.56% (detailed data not shown here). Table 1 lists all the uncertainties associated with the height measurements of Fig. 10 together with the combined uncertainty (rootmean-square sum of all uncertainties).
5. Conclusion Latex particles are important in many industrial applications and the measurement of their size and their size distribution is crucial to quality assurance and research. Several techniques are available for such measurements. They include turbidity, light diffraction, dynamic light scattering, ultraand disc centrifugation, field flow fractionation, capillary hydrodynamic fractionation, scanning and transmission electron microscopy (SEM/ TEM) and SFM. These techniques are complementary, each having its own advantages and disadvantages, and comparative studies can be found in the literature. The main advantages of microscopy are high resolution and ease of data interpretation as there is no need for a model or for a priori knowledge of material properties such as density or refractive index. One disadvantage of SEM and TEM is the latex particle shrinkage that may occur with such techniques. As discussed in the introduction the SEM does not provide actual
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height data and furthermore there is insufficient contrast for automated or semi-automated image analysis of lateral dimensions. A recent comparative study concluded that SFM was more suitable than SEM and TEM for the quantification of latex particles [24]. In this work, a methodology has been developed to accurately determine the size distribution of latex particles using scanning force microscopy. Unlike other workers, who have generally measured the lateral dimensions of monolayers of latex particles using a global quantification method, we have measured the heights of individual latex particles at the edges of latex monolayers immobilised onto mica substrates. In agreement with other work (e.g. Ref. [4]) we noted that the edges of monolayers of latex particles provided stable and reproducible SFM imaging. Whilst SFM imaging noise, image processing, tip/sample interactions and variations in the mica substrate are also sources of measurement error, our experience is that variations over time of the sensitivity of the Z actuator is the greatest uncertainty in determining the heights of latex particles by this method. The methodology that we used requires frequent calibration, before and after every two or three (typically) images, of the Z actuator of the SFM, thereby ensuring that the uncertainties in the Z sensitivity are minimised and known. We have measured the size distributions of populations of 300–400 latex particles from several different latex samples with the maximum variation in the Z-actuator calibration experienced during the measurement of a sample being less than 2%, often about 1% and occasionally better still. This methodology should only be used to quantify latex particles with narrow size distributions as there will be no size related segregation effects and thus the latex particles at the edges of monolayers will be representative of the sample. Furthermore, SFM imaging is a relatively slow process and the calibration procedure described here is time consuming and thus this method is probably best reserved for quantifying standard samples which will serve to calibrate faster techniques (e.g., disc centrifugation and light scattering).
The methodology described here was developed for an instrument equipped with open loop piezoelectric actuators after a careful study of the behaviour of those actuators. We have demonstrated that scanning force microscopes equipped with open loop actuators can be used for high confidence quantitative measurements. Indeed, a recent inter-laboratory comparison of nanometric step heights measured by scanning force microscopy confirms that open loop actuator (first generation) instruments can perform as well as closed loop actuator (second generation) instruments [25]. Whilst this methodology was specifically developed to measure latex particles, it is applicable to ‘‘step height’’ measurement in general.
Acknowledgements This work was carried out at Cork Institute of Technology with support from SolVin and Solvay Research and Technology. The authors are grateful to Greta Vanmarcke (Solvay Research and Technology) for fruitful contributions and discussions on error analysis. The authors also thank the referees as their comments during the review process have helped to improve the manuscript.
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binding events, Ph.D. Thesis, Cork Institute of Technology, 1999. J. O’Mahony, L. McDonnell, Poster Presentation at STM ’01 Conference, Vancouver, Canada, July 2001. Extended manuscript submitted to Ultramicroscopy. M. Heyde, H. Sturm, K. Rademann, Surf. Interface Anal. 27 (1999) 291. M. Hipp, H. Bielefeldt, J. Colchero, O. Marti, J. Mlynek, Ultramicroscopy 42–44 (1992) 1498. C. Serra Rodriguez, J. M!endez Fernandez, M.A. CorreaDuarte, L.M. Liz-Marzan, Eur. Microsc. Anal. 65 (2000) 11. M. Senoner, L. Koenders, W. Unger, G. Wilkening, Proceedings of the Fifth Seminar on Quantitative Microscopy and First Seminar on Nanoscale Calibration Standards and Methods (NanoScale 2001); PTB-Bericht F-44, Physikalisch-Technische Bundesanstalt, Braunschweig, 2002, pp. 70–77.
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A method for measuring the size distribution of latex particles by scanning force microscopy G. Roea, L. McDonnellb,*, A. Ghanemc a School of Science, Galway-Mayo Institute of Technology, Galway, Ireland Department of Applied Physics and Instrumentation, Centre for Surface and Interface Analysis, Cork Institute of Technology, Bishopstown, Cork, Ireland c Analytical Technologies, Solvay Research and Technology, Rue de Ransbeek 310, 1120 Brussels, Belgium b
Received 1 July 2003; received in revised form 5 December 2003; accepted 7 January 2004
Abstract A methodology has been developed to accurately determine the size distribution of latex particles using the scanning force microscope (SFM). Unlike other workers, who have generally measured the lateral dimensions of monolayers of latex particles using a global quantification method, we have measured the heights of individual latex particles located at the edges of latex monolayers that were immobilised onto mica substrates. In agreement with other work, we noted that the edges of monolayers of latex particles provided stable and reproducible scanning force imaging. Whilst SFM imaging noise, image processing artifacts, tip/sample forces and variations in the mica substrate are sources of measurement error that should not be overlooked, our experience has been that the variation over time of the sensitivity of the Z actuator is the greatest potential uncertainty in determining the heights of latex particles. The methodology that we used requires frequent calibration of the Z actuator of the SFM, typically before and after two or three images, in order to ensure that the uncertainties in the Z sensitivity are known and minimised. This methodology was developed for an SFM instrument that was equipped with open loop piezoelectric actuators following a careful study of the behaviour of those actuators. Using this methodology, we have measured the size distributions of populations of 300–400 latex particles from each of several different latex samples, with the maximum variation in the Z-actuator calibration experienced during the measurement of a sample being less than 2%, often about 1% and occasionally better still. In so doing, we have demonstrated that SFMs equipped with open loop actuators can be used for high confidence quantitative measurements of step heights. r 2004 Elsevier B.V. All rights reserved. PACS: 06.30.Bp; 68.37.Ps Keywords: Latex; Scanning force microscopy; Size distribution
1. Introduction *Corresponding author. Tel.: +353-21-4326230; fax: +35321-4345355. E-mail address: [email protected] (L. McDonnell).
Following its invention in 1986 [1], the scanning force microscope (SFM) was soon deployed to visualise surface topography for a wide range of
0304-3991/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2004.01.019
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sample types, materials and applications. Contrast in SFM images is derived from direct measurements of surface topography and thus threedimensional measurements can be readily extracted from SFM images. This is unlike most optical and scanning electron microscopies where metrology is limited to lateral dimensions, image contrast in these cases being derived from slopes in surface topography rather than from heights. Despite this intrinsic quantitative ability, most SFMs have been designed to visualise rather than measure surface topography and therefore considerable care is needed if valid measurements are to be derived from SFM images. Quantification of SFM images can be divided into two categories: the measurement of global parameters, such as surface roughness, that ‘‘average’’ the entire image and the measurement of the dimensions of discrete image features, the latter being the category of measurement that this paper specifically relates to. Whilst the SFM can be used as a miniature coordinate measurement machine, it has some limitations as illustrated in Fig. 1(a). Firstly, with the exception of those few instruments equipped with tips that probe vertically and laterally
SFM tip Object
Re-entrant zone
Tip convolution
(a)
SFM tip Object
Tip convolution
Re-entrant zone
(b) Fig. 1. Relationships between the SFM tip, the object being imaged, the imaged shape and the imaged dimensions when the tip has (a) sloped sides and (b) vertical sides. Neither tip is able to image the re-entrant zone of the object.
(Dimension X3D, Veeco Instruments, Santa Barbara, CA, USA), SFMs cannot access re-entrant regions such as the lower part of the object shown in Fig. 1(a). Another issue arises when the SFM tip interacts with the sample, the point of interaction on the tip being referred to as the proximal point. During imaging, the proximal point moves across the tip to an extent that depends on the relative sizes and shapes of both the tip and the object of interest, thereby convoluting the object’s shape with the proximal point’s trajectory over the tip. As the object in Fig. 1(a) is imaged from left to right, the proximal point moves down the leading edge of the tip to the tip apex and then up the trailing edge of the tip, thereby adding a sloped skirt to the spherical object with clear implications for subsequent quantification [2]. As indicated in Fig. 1(b), using a tip with a vertical side wall will eliminate the shape change although a dimensional increase in the width will remain. Fortunately, SFM images can be corrected for tip imaging artifacts by extracting the tip profile from the image. The tip profile can be determined directly, using either a tip characteriser or a scanning electron microscope image of the tip, or virtually, using a blind reconstruction algorithm [3] that identifies the motif in the image that is due to the tip. However, it should be noted that the SFM tip acts as a filter and the above correction/ reconstruction processes cannot recover information that the tip has rejected, such as topographic features that are finer than the tip shape can resolve. Crucially, regardless of tip dimensions or shape, heights will always be measured correctly by the SFM provided that (i) the tip has unrestricted access to at least one side of the immobilised object and (ii) there is no distortion of the sample by the tip (or vice versa) during imaging. SFM was applied to latex particles quite early in its history with latex particles being used, e.g., as a calibration artefact [4]. The SFM has also been used to study the behaviour of latex film formation [5], latex film defects [6,7] and the effects of crosslinking [8]. The scientific literature also contains several references (e.g. Refs. [9–13]) to different methods by which latex particle sizes, and their distribution in size, can be determined by the SFM. Generally, workers have favoured measuring lateral
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dimensions [14]. In this paper, we present an alternative method by which the size distribution of latex particles can be accurately determined by measuring the heights of particles at the edges of monolayers with the SFM.
2. Methodologies to measure the size distribution of latex particles 2.1. Lateral measurements of latex particle diameters One of the more common methods (e.g. Refs. [9,10,13]) of determining latex particle sizes is to obtain the particle diameters from the lateral line profiles taken along the points of contact between nearest neighbours as illustrated in Fig. 2. There are a number of potential errors associated with this approach, the most fundamental being those introduced by tip imaging effects at the interstices between nearest neighbours of differing diameters. Fig. 3 shows schematically how tip shape, tip asymmetry and the size ratios of the nearest neighbours affect the location of the imaged interstice and hence the perceived diameter of the latex particles. Using a tip with a vertical side wall (left-hand side of tip A in Fig. 3), the calculated maximum error in the smaller particle’s diameter is
Fig. 2. SFM line profile through a row of latex particles with a vacancy.
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1% for a nearest neighbour size ratio of 75% and 11% for a ratio of 50%. For sloped tips (tips B and C in Fig. 3) and a given nearest neighbour size ratio, the error increases as the tip angle increases. Fig. 4 shows the effect of tip imaging on the lateral dimensions of latex particles bordering a vacancy within a layer (a) and at the edge of a layer (b) and demonstrates qualitatively the importance of omitting such particles from lateral measurements. It is also possible to use Fourier transforms [11], radial distribution functions [12] and power spectral density analysis to quantify the diameters of large areas of close-packed latex particles. These methods emphasise the spatial position of the (higher) centres of the particles rather than the (lower) interstices and so are less sensitive to tip imaging effects. However, the accuracy of these methods is undermined by the contributions of particles that are not close-packed.
A
B
C
Fig. 3. Schematic diagram showing how the perceived diameters of latex particles in SFM images are influenced by the shape and asymmetry of the SFM tip and the relative sizes of the nearest neighbours. Tip A is asymmetric with one vertical side wall. Tip B is symmetric. Tip C is also symmetric but with a larger tip angle than tip B.
Fig. 4. (a) 3 mm 3 mm SFM image of a vacancy in a closepacked latex layer and (b) 5 mm 5 mm SFM image of the edge of a close-packed latex monolayer.
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2.2. Vertical measurements of latex particle diameters The method chosen in this work to determine the size distribution was to measure the heights of individual particles at the edges of single layers of latex particles that were dried onto flat mica substrates. This method has two advantages: (i) it eliminates tip imaging effects from the measurement as there is unrestricted access from one side of the particle of interest; and (ii) it enables the datum level of the substrate to be established at a point that is very close to the particle of interest. The validity of height measurements extracted from an SFM image is most likely to be affected by three factors: (i) tip/sample forces; (ii) errors introduced by the image processing algorithms used to remove tilt and curvature; and (iii) uncertainties in the position of the tip. These issues are discussed in Section 3. Two other factors, imaging noise and variations in the height datum also contribute to uncertainty in the height measurement. Care has been taken to minimise such effects and their contribution to the uncertainty is cited later.
mode to minimise the force applied to the latex particles. Although no particle height variations were observed over a range of cantilever oscillation amplitudes, non-contact SFM imaging was carried out at the smallest possible oscillation amplitudes. 3.2. Position of the tip The piezoelectric materials used in SFM actuators suffer from creep, ageing, non-linearity and hysteresis and hence present significant challenges if valid measurements are to be derived from SFM images. The first generation of SFMs used piezoelectric actuators in open loop and these instruments are fully exposed to the inadequacies of the actuators. Second generation SFMs use closed loop positional control of the X, Y and Z movements to minimise the deficiencies of the piezoelectric actuators and this has helped the SFM to evolve into a more quantitative instrument. Nevertheless, the behaviour of both types of instrument needs careful characterisation so that appropriate calibration procedures are developed. 3.3. Image processing algorithms
3. Uncertainties in height measurements 3.1. Tip/sample forces There are three general modes of SFM operation: contact mode, intermittent contact mode (or Tapping ModeTM) and non-contact mode. In contact mode the tip is in permanent repulsive contact with the sample and the vertical and lateral forces applied during imaging can distort or damage the sample [15]. Intermittent contact mode has proved to be very effective at reducing the distortion arising from lateral interactions, but for very soft samples may still apply significant vertical force. The lowest forces are applied in non-contact mode provided that the tip always operates in the attractive regime, thereby avoiding any contact [16–18]. Although it was possible to image the latex particles in contact mode without apparent damage, it was deemed prudent to use non-contact
Because SFM image contrast is generated by surface form, sample tilt, the curvature of the piezoelectric scanner and surface topography, SFM images generally contain unwanted backgrounds that require removal by image processing algorithms. Sample tilt can affect SFM images in two ways. Firstly, the tilt alters the angle at which the sample’s topographic features are presented to the SFM tip and introduces a geometric distortion that is further compounded by tip imaging as discussed earlier. Secondly, many image processing or ‘‘levelling’’ algorithms remove tilt incorrectly. Fig. 5 shows the effect of sample tilt on the SFM measurement of a step height. The solid black line shows the actual tilted surface and the grey line shows the path of an ideal SFM tip of negligible diameter and vertical side walls. The dotted line shows the ‘‘levelled’’ surface that is generated by commonly used levelling algorithms that simply rotate the flat portions AB and CD about points B and C, respectively, as indicated by
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D C
h
E
θ
F
B A Fig. 5. Schematic diagram illustrating the effect of tilt angle y on the measurement of a step edge of height h: The grey line (ABCD) shows the profile followed by an ideal SFM tip. The curved arrows and the dashed line (EBCF) show the effect of incorrect planar levelling.
the curved arrows. It is clear from Fig. 5 that there is now an error in the measured step height as given by the relationship: error in height ¼ h½ð1=cos yÞ 1; where h is the true step height and y is the angle of tilt. For a 5 tilt the calculated maximum error is +1.5%, for a 3 tilt the error is +0.6% whilst for a 1 tilt the error is +0.1%. Clearly, for the highest accuracy, care needs to be taken to ensure that the tilt is minimised. Fu et al. [19] have shown that levelling algorithms should remove tilt by carrying out a rotational transformation of the co-ordinates for the entire image. In Fig. 5, this would be achieved by rotating the entire line profile about either point A or D. They also demonstrated that the error introduced by sample tilt can be severely compounded by the algorithm used to calculate the step height, with errors of more than 31% being experienced for single atomic step height measurements on the Si(1 1 1) surface when using one commercial algorithm. Most SFMs use either tube or tripod piezoelectric actuators to scan either the sample or the tip over trajectories that are curved rather than planar. This is demonstrated in Fig. 6(a) which shows a longrange SFM image of a super-smooth polished glass sample1 obtained by the ExplorerTM SFM (Veeco Instruments, Santa Barbara, CA, USA) used in this work. Fig. 6(b) also shows a line profile through the
Fig. 6. 3D view SFM image (a) of a super-smooth polished glass surface obtained with the ExplorerTM SFM and a line profile (b) from the centre of this image.
centre of this image. The flatness of the glass sample is such that the curvature observed in Fig. 6 can be fully attributed to the curved trajectory of the scanner. Whilst this curvature can be removed by software algorithms, care needs to be taken to ensure that other errors are not introduced during the levelling process [20,21]. Fortunately, because of the relatively large length of the tripod scanner in the ExplorerTM SFM, the curvature introduced into the full scan is much smaller than many other SFMs and is quantified later in Section 4.4.
4. Experimental 4.1. Sample preparation
1
European Commission BCR project. Characterisation and development of surfaces with roughnesses in the 0.1 nm range. Contract No. 3425/1/0/185/91/4-BCR-D(30).
The latex particles were immobilised onto mica substrates (Agar Scientific, Stansted, England)
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using the general method described by Van Cleef et al. [12]. Firstly, the stock suspension (nominal dilution of 0.1%) was sonicated for 30 min to remove agglomerates. Then, a 0.15 ml aliquot of the suspension was immediately deposited, without dilution, onto a piece of freshly cleaved mica. The substrate was then placed in a dessicator for 24 h, after which the dried sample was ready for imaging. 4.2. Scanning force microscope imaging conditions Non-contact SFM imaging was carried out in air with an ExplorerTM SFM equipped with a long lateral range (130 mm) tripod type scanner. Silicon non-contact cantilevers (type SCF11 UltraSharpTM, Silicon-MDT Co., Moscow) with a nominal force constant of 5 Nm1 were used. Before use, the shape of the cantilever tip was determined by imaging a tip characteriser (type TGT01, Silicon-MDT Co., Moscow). As discussed earlier, in principle the SFM tip radius is not crucial to either the accuracy or the repeatability of a step height measurement, provided that the tip has unrestricted access to the step. Nevertheless, SFM tips were rejected if the images did not show sufficiently deep interstices between adjacent closely packed latex particles. Sharp interstices are not indicative of a sharp tip as a very blunt tip will also produce a sharp turning point as it crosses the interstice. SFM tips were also rejected if image noise, or other imaging instabilities, were experienced as such effects are indicative of tip contamination and could affect the height measurements by increasing noise and altering the tip– sample interactions on the top of the latex particle vis-a" -vis the mica substrate. SEM images of tips that degraded during imaging revealed that in many such cases latex particles from the sample had attached to the tip. The samples were placed for SFM analysis onto a computer controlled XY translator. Sample tilt was measured from line profiles obtained in the X and Y directions and reduced where necessary using the manual adjustments on the ExplorerTM SFM. In all cases, tilt of the samples (and of the step height standards used in calibration) was reduced to less than 1 so that the resulting
systematic error in step height measurement was insignificant (+0.1%) and could be neglected. Suitable general locations on the mica substrate were identified using the internal video camera on the ExplorerTM SFM. These locations were then inspected by SFM using large scan ranges (50 mm 50 mm or 100 mm 100 mm). Suitable locations were then centred using the XY translator for subsequent zoom imaging by the SFM. A high-resolution SFM image, of either 10 mm 10 mm area (500 pixels 500 pixels) or 5 mm 5 mm area (400 pixels 400 pixels), was taken for the larger and smaller latex sample sizes, respectively. The pixel resolution was selected to maximise the accuracy of the height measurement and to minimise the image acquisition time. The calculated maximum error in height measurement arising from image pixellation has been calculated to be 0.06% for a 600 nm diameter particle (20 nm 20 nm pixels) and 0.2% for a 200 nm diameter particle (12.5 nm 12.5 nm pixels). This error is determined from the height error that is introduced if the top of the latex particle is not imaged due to image pixellation. The error becomes more significant for a given pixel size as the latex particle diameter decreases. Images were typically acquired at a line scan rate of 0.67 Hz. 4.3. Calibration of the Z actuator In order to have confidence in the height measurements made by the ExplorerTM, the behaviour of its independent Z actuator was studied. Heyde et al. [22] have shown that a relatively low-cost fibre-optic displacement sensor can be used for such purposes. In the work reported here, the performance of the Z actuator was studied using such a displacement sensor (Philtec, Annapolis, MD, USA) and discrete step height standards. A crucial observation that was made during this study concerned the effect of offset voltages on the behaviour of the Z actuator. This actuator, used when the instrument is configured for long lateral ranges, is provided with a voltage offset that allows feedback to be engaged when the Z actuator is off-centre. Unfortunately, invoking the voltage offset invalidates the calibration as shown in Fig. 7. The data
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540 530 520 510 500 490 30
108 107 106 105 104 103 0
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Fig. 7. The effect of varying the voltage offset of the independent Z piezoelectric actuator of the ExplorerTM SFM on measurements of a 512 nm step height standard.
in Fig. 7 demonstrate that the measurement of a 512 nm step height varied by almost +5% when voltage offsets corresponding to about +20% of the actuator range were used. Consequently, all images of step height standards and latex samples were obtained with a zero offset voltage, i.e. with 50 V applied to the Z actuator, so that the instrument remained in calibration. Furthermore, the reproducibility of the Z actuator for a given step height was studied and it was found that the step height measurement varied over time. In order to minimise this effect, the Z actuator was typically calibrated before and after two or three images of the latex particles using the same step height standard and the average step height was used to retrospectively calibrate the ‘‘bracketed’’ set of images. Fig. 8 shows the variation in the measurements of a 106 nm step height artefact for the calibrations performed during several SFM imaging sessions of a latex sample that took place over 5 days, the longest period during which we obtained data from a given sample. The maximum uncertainty in the step height measurement for any pair of calibrations was 0.8% and is somewhat less than that for the overall set of calibrations (2%). Where ‘‘bracketed’’ images included an overlap between all successive images, the overlapped particles in the image (see e.g. Fig. 9) were used as internal standards to determine the calibration correction for that particular image. The most stable imaging performance of the ExplorerTM SFM was obtained by ‘‘warming up’’ its XY piezoelectric actuators by continuously scanning in free space for a period of 1 h, the scan
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Fig. 8. The variation with time of measurements of a 106 nm step height standard made with the independent Z piezoelectric actuator of the ExplorerTM SFM during four imaging sessions.
speed being the same as that used for subsequent imaging. During this warm-up period the Z actuator was maintained at its mid-point voltage. Piezoelectric actuators perform best if they are operated over the same voltage range at the same scan speed. Changing scan speed and, more importantly, changing the scan range degraded the stability of the piezoelectric actuators. It was also found to be beneficial to leave the SFM electronics switched on continuously, although it should be noted that, in order to avoid damage, the high voltages were only applied to the piezoelectric actuators during the warm-up period and whilst imaging. The ExplorerTM SFM is a scanned cantilever instrument of the ‘‘stand-alone’’ type [23] and is lifted off the sample stage in order to change samples and/or step height artefacts. This physical disturbance can introduce instabilities in the subsequent performance of the instrument. Instabilities can also be introduced whenever feedback is disengaged as the actuator is exercised over a much larger range during retraction of the SFM tip than when imaging. This was demonstrated by continuously imaging a step height artefact and obtaining a superior repeatability of step height measurements compared to discontinuous, sequential imaging where the feedback was disengaged and engaged between consecutive images. 4.4. Measurement of latex particle heights Prior to quantification, the tilt in the SFM images was removed using the three point planar levelling algorithm within the software provided
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326 10 µm
5 µm
10 µm
0 µm 0 µm
5 µm
10 µm
5 µm
0 µm 0 µm
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Fig. 9. Example of two overlapped SFM images.
with the ExplorerTM SFM. Using this algorithm, three locations on the (planar) mica substrate were defined and used to determine the planar levelling correction. As discussed earlier (Section 3.3), planar levelling algorithms can introduce step height measurement errors. Therefore, the efficacy of this commercial algorithm was determined using simulated images of step height artefacts with tilts similar to the experimental SFM images. This algorithm was found to introduce maximum calculated height errors that were less than 0.05%. We did not remove the scanner curvature described earlier (Section 3.3) as the line profile in Fig. 6(b) shows that the maximum calculated height error associated with a 1 mm lateral distance (the relevant scale for the step height measure-
ments made in this work) is less than 0.39 nm. The latex particle heights were measured from the levelled SFM images using software written in Matlab (Release 12, The Mathworks, Natick, MA, USA). With this software the user selects with a cross-hairs cursor the centre of the latex particle of interest. The line profile is displayed to enable the user to identify the top of the latex particle and the position on the neighbouring mica substrate that is to be used as the height datum. The latex particle height is then determined as the difference between these two heights. In all cases the lateral distance between the two height measurements is less than 1 mm to ensure that the error due to the residual curvature is less than 0.39 nm, as discussed above. In this method it is presumed that the latex datum
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continues without variation underneath the latex particle. Indeed, it cannot be determined whether the latex particles are sitting on a different datum arising from variations in the mica or from contamination underneath the particle. However, the mica substrate is extremely flat and over distances of 1 mm height variations due to atomic steps are negligible. A particular feature of the software is its ability to search three dimensionally about the selected line and points therein to determine that the top of the latex particle has been found and that the mica substrate is flat. To test the repeatability of the measurement procedure, 30 measurements were made of the height of the same particle from a given image from a range of samples. These measurements involved opening the image, selecting the top of the latex particle and its mica datum. In all cases the standard deviation of the repeated measurements was better than 0.15%. This uncertainty includes contributions arising from imaging noise and variations in the mica datum. 4.5. Example of size distribution of latex particles
65 3 66 6. 5 m or e
55 9 57 2. 5 58 5. 9 59 9. 3 61 2. 7 62 6. 2 63 9. 6
70 60 50 40 30 20 10 0 54 5. 6
Frequency
Fig. 10 shows the size distribution obtained for one of several latex samples that have been measured using the methodology described above. The mean diameter for the 421 latex particles measured for this sample (from 25 images) was 635 nm with a standard deviation of 19 nm. The Z calibrations for these data were obtained using a 512 nm step height standard and the maximum calibration uncertainty experienced in any imaging
Height (nm)
Fig. 10. Distribution of heights for a latex sample. In all, 421 particles were measured and found to have a mean height of 635 nm with a standard deviation of 19 nm.
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Table 1 Table of uncertainties and their sources for the latex particle measurements presented in Fig. 10 Source of uncertainty a
Image pixellation Step height standardb Planar levelling algorithma Scanner curvaturea Particle height algorithm (includes imaging noise and datum variation)b Scanner calibrationa Combined uncertainty
Uncertainty (%) 0.06 1.00 0.05 0.07 0.08 0.56 1.06
Note: In calculating the combined uncertainty it is assumed that the root-mean-square sum of all uncertainties is applicable. Uncertainties that are expressed as maximum errors are divided by the square root of three prior to the latter calculation. a Refers to an uncertainty expressed as a maximum error. b Refers to an uncertainty expressed as a standard deviation.
session was 2.9 nm, i.e. 0.56% (detailed data not shown here). Table 1 lists all the uncertainties associated with the height measurements of Fig. 10 together with the combined uncertainty (rootmean-square sum of all uncertainties).
5. Conclusion Latex particles are important in many industrial applications and the measurement of their size and their size distribution is crucial to quality assurance and research. Several techniques are available for such measurements. They include turbidity, light diffraction, dynamic light scattering, ultraand disc centrifugation, field flow fractionation, capillary hydrodynamic fractionation, scanning and transmission electron microscopy (SEM/ TEM) and SFM. These techniques are complementary, each having its own advantages and disadvantages, and comparative studies can be found in the literature. The main advantages of microscopy are high resolution and ease of data interpretation as there is no need for a model or for a priori knowledge of material properties such as density or refractive index. One disadvantage of SEM and TEM is the latex particle shrinkage that may occur with such techniques. As discussed in the introduction the SEM does not provide actual
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height data and furthermore there is insufficient contrast for automated or semi-automated image analysis of lateral dimensions. A recent comparative study concluded that SFM was more suitable than SEM and TEM for the quantification of latex particles [24]. In this work, a methodology has been developed to accurately determine the size distribution of latex particles using scanning force microscopy. Unlike other workers, who have generally measured the lateral dimensions of monolayers of latex particles using a global quantification method, we have measured the heights of individual latex particles at the edges of latex monolayers immobilised onto mica substrates. In agreement with other work (e.g. Ref. [4]) we noted that the edges of monolayers of latex particles provided stable and reproducible SFM imaging. Whilst SFM imaging noise, image processing, tip/sample interactions and variations in the mica substrate are also sources of measurement error, our experience is that variations over time of the sensitivity of the Z actuator is the greatest uncertainty in determining the heights of latex particles by this method. The methodology that we used requires frequent calibration, before and after every two or three (typically) images, of the Z actuator of the SFM, thereby ensuring that the uncertainties in the Z sensitivity are minimised and known. We have measured the size distributions of populations of 300–400 latex particles from several different latex samples with the maximum variation in the Z-actuator calibration experienced during the measurement of a sample being less than 2%, often about 1% and occasionally better still. This methodology should only be used to quantify latex particles with narrow size distributions as there will be no size related segregation effects and thus the latex particles at the edges of monolayers will be representative of the sample. Furthermore, SFM imaging is a relatively slow process and the calibration procedure described here is time consuming and thus this method is probably best reserved for quantifying standard samples which will serve to calibrate faster techniques (e.g., disc centrifugation and light scattering).
The methodology described here was developed for an instrument equipped with open loop piezoelectric actuators after a careful study of the behaviour of those actuators. We have demonstrated that scanning force microscopes equipped with open loop actuators can be used for high confidence quantitative measurements. Indeed, a recent inter-laboratory comparison of nanometric step heights measured by scanning force microscopy confirms that open loop actuator (first generation) instruments can perform as well as closed loop actuator (second generation) instruments [25]. Whilst this methodology was specifically developed to measure latex particles, it is applicable to ‘‘step height’’ measurement in general.
Acknowledgements This work was carried out at Cork Institute of Technology with support from SolVin and Solvay Research and Technology. The authors are grateful to Greta Vanmarcke (Solvay Research and Technology) for fruitful contributions and discussions on error analysis. The authors also thank the referees as their comments during the review process have helped to improve the manuscript.
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