Sizing of Nonspherical Particles and a Method of Representing their Size Distributions P . D. Lark School of Chemistry., Unioersity of New Sourh Wales, Kensington, Australia
Many single meas,ures of particle size have b e e n proposed but these a re not suitable for honspherical particles. S u c h part icles are b e s t c h a r a c t e r i z e d by t h e length a n d breaclith of t h e profiles t h e y p r e s e n t when lying in their stabl e positions. Equipment is d e s c r l b e d which simplifies t h e t a s k of obtaining t h e s e m e a s u r e ments f r o m photi,micrographs. The particle dimensions c a n b e plott e d o n a triangular c h a r t from which both s h a p e a n d t tle more u s u a l n u m b e r , surface, a n d volume o r weighi: distributions a r e obtainable. T h e p r o c e d u r e is illusitrated for two varieties of titanium dioxide pigment. with the microscope, either directly or from phiXomicrographs, the most common practice is to adopt a single measure of size and to characterize the particles with re spect to it without regard to their shape or the kind of distritmtion data required. When dealing wit1h systems containing markedly nonspherical particles, single measures of size (especially Martin’s and Feret’s statistical dirmeters) are inadequate, and at least two measurbs are requiired to characterize individual particles. Alternatively, shape factors must be introduced if accurate volume or surface d istributions are to be obtained. If two measures are to be Iused, these might be one of size and the other of shape; for Iexample, the length of the major axis and the eccentricity of anI ellipse which matches the particle profile in shape and area. An alternative, and the natural choice, is of two dimensions nieasured in mutually perpendicular direo x r . r r r l l Ienoth nnrl hrrslrlth tions which correspc...,\ n A tn the nV.-.I.I Y....Y... a q the particle, it being assumed that the particle is lying in its most stable position. The measurement of the length and breadth of microscopic images is tedious, if not difficult, even after photography or projection. The task is simplified by the use of a diaphragm whose aperture is adjustable in mutually perpendicular directions. Such a diaphragm can be constructed on the parallel rule principle o r by re-entrant, right-angled blades whose edges constitute reference lines for matching with particle images. The latter method lends itself to simple mechanical constructions especially if one blade is fixed and the relative position of the other is defined by scale coordinates translatable into particle dimensions. In addition, the fixing of one blade is of great assistance in lining up a particle for measurement. I N PARTICLE SIZE M e A s u R E m N T
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DESCRIPTION OF INSTRUMENTS
A simple measurer, based on this principle, can be easily made out of a detachable mechanical stage which incorporates the perpendicularly moving elements and scales required for the purpose. Figure 1 illustrates a suitable arrangement. In this, the fixed blade is attached to the framework of the stage and the movable blade is controlled by the longitudinal and transverse adjustments. The blade edges can conveniently take the form of lines scribed on Perspex sheets, one being firmly attached to the underside of the stage and the other held between the stage jaws in the position of a microscope slide. An old Leitz stage was ideal for the purpose as it
Figure 1. A simple measurer had been designed for mounting on the post of a microscope and a similar post (in the form o f a brass disc) could he screwed to the lower Perspex sheet. Furthermore. the knurled knobs which control the stage movements are mounted vertically, which is most desirable. The range of measurement is from 0.2 mm to 3 cm. The absolute precision of measurement of LI particle image dimension is about 0.15 mm so the quality of the photomicrograph and the matching of reference linrs to image arc more important factors in respect to error than the construction o f t h e instrument itself. This device, although easy to construct and use, suffers from the considerable disadvantage that the scale and vernier values must be read, recorded and, because the scales are arbitrary and exclusive--e.g., 0 to 60.61) to 100-brought hack to a common origin by calibration. These are troublesome and time-consuming operations even with the aid of an assistant, and a semiautomatic system by which the observer is relieved of them is desirable. A semiautomatic measurer for use with suJ?iciently enlarged photomicrographs is shown schematically in Figure 2. The fixed blade, whose edges are denoted by ohc in the figure, is painted on the underside of glass ~ 1 3 1 A, ~ . and the movable blade (of thin blackened metal; edge odc), 8, is supported immediately beneath. The rectangular aperture between them IS illuminated from below by a clear low-wattage lamp and is suficiently bnght and sharply edged lo he aligned with particle images when seen through a photographic prl VOL. 39, NO. 1, JANUARY 1967
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. . Figure 2. Schematic diagram of a semiautomatic measurer A , glass plate; B, moving blade; C, two-way slide; D,pantograph; E , marker; F, chart; G, marker stop; H , position of lockable slide; ahcd, adjustable aperture; p , pantograph follower; q, fixed pivot, adjustable along the line q’y” of slide H ; r , sliding pivot; s and t , adjustable pivots; x , y , and i,components of two-way slide C , I being attached to the base. A is set flush in a wooden table about 1 inch above the base The aperture ahcd is illuminated from beneath
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Figure 3. Length-breadth scatter diagrams for two types of TiO, A and B (238 particles of type A and 304 of type 5 are represented)
The metal blade is constrained to move in mutually perpendicular directions by a brass two-way slide, C, to which a pantograph, D,is attached. The follower, p , of the pantograph is free to rotate in a socket in C but has exactly the same travel as any point on the blade. The pantograph magnifies the movement of such a point, say of d, and enables its relative position to be marked on chart F by the free or drawing end, E. The slide is mounted toward one end of a standard drawing board, the other side being used as the plotting table. Plate A is raised about 1 inch above the base board and set flush in a wooden support which serves both as a photomicrograph table and as a cover for the mechanism. In operation, the observer can move a photomicrograph freely over plate A with his left hand and align the particle images. With his right, he controls the pantograph marker pencil, E (and through it, the aperture size and shape), and marks the chart when a particle image is matched. If the photomicrograph is turned so that a particle image lies lengthwise in a particular direction-e.g., across the observer-and is outlined by the aperture edges, abc and adc, respectively, its relative length is plotted in the x direction and breadth in the y direction. In this way, the data can be recorded on a triangular chart. A bar, G, can be pinned to the chart to prevent marker E from being moved across the hypotenuse on which the sizes of equidimensional particles are represented. The chart is calibrated as required for particular enlargement settings of the pantograph. Allowing for positioning the photomicrograph, matching aperture to particle image, and marking off, it is considered possible to make 10 or more pairs of measurements per minute under favorable conditions. In the case of regular particles the operation is so simple that practice is of little account. In the case of very irregular particles an element of judgment enters, but if the observer is content to measure overall dimensions the decisions to be made are usually simple and the rate of measurement is not greatly impeded. At the rate of measurement indicated, the lengths and breadths of 300 particles can be recorded in 30 min or even less, but the total time to process the data depends, of course, on subsequent graphical and numerical operations. With the instrument constructed as described, the practical minimum dimension measurable is abut 2 mm, although 56 *
ANALYTICAL CHEMISTRY
images of smaller size could be matched if the diaphragm blades were to be optically projected and not just shadowed as here. The maximum aperture between the blades is about 4 cm square. This calls for a 10-inch chart when an enlargement factor of 5:1 or 6 : l is set on the pantograph. The latter, a wooden commercial model, can be set for enlargements up to 10:1, but the highest ratios call for a very freemoving two-way slide and an unduly large plotting table h i d base board. (Some freedom in positioning the chart is allowed by mounting the fixed pivot of the pantograph, q , in a lockable slide, H , but if a high ratio is set, the standard sized board must be extended to accommodate the chart and movement of the sliding pivot, r). Because of the restrictions imposed by the use of the pantograph as a mechanical magnifier, photomicrographs must be suitably enlarged before measurements are made with this instrument. The relative simplicity of this instrument involves certain disadvantages. (a) The minimum size of image measurable is rather high at 2 mm; much smaller sizes can be measured with the adapted mechanical stage. (b) The degree of photographic enlargement must be adjusted to obtain an adequate range of image sizes. The linkage is such that (c) particle dimensions can only be recorded on arithmetic scales and (d) some mechanical lack of rigidity is unavoidable. The first disadvantage can be overcome by a much more elaborate optical system but this would lead in turn to greater mechanical complexity. The remaining disadvantages might also be overcome in an instrument of more elaborate design but, in the case of the present instrument, the last can be minimized by careful construction of the two-way slide and by moving the diaphragm blade consistently inward or outward when matching its edges to the image. The representational error is proportional to the pantograph ratio but is more or less independent of the position of the mark on the chart. When the pantograph is set a t 5 : 1 , the uncertainty in the position of a mark is about 1 mm or 1 % of an image dimension of middle size (2 cm on the photomicrograph, 10 cm on the chart) but varies between about 5 and 0.5 for the smallest and largest measurable dimensions, respectively. Taking into account a chart calibration error of similar magnitude due to the uncertainty in the location of scale divisions, the aver-
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Figure 5.
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Linea of constant elongation on the I-b plot
THE TRIANGULAR REPRESENTATION OF LENGTH AND BREADTH
Figure 4. 1;ledronmicro~raphs iif two types of TiOi Original magnification, XZOMH) age overall random error should be roughly 1.5 to 2z. A random error of this magnitude is generally unimportant by comparison with those (both random and systematic) of the various stages of particle sampling and perhaps of photcmicrograph calibration. It is also less than that ofthe measurement of statistical diameters or the projected area diameter.
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Triangular 1-6 diagrams, as produced directly by the semiautomatic instrument or by plotting the data obtained from the simple measurer, are shown in Figure 3 for two samples of titanium dioxide, designated A and B. Sample A is composed of more or less rounded particles and B of rather prismatic particles as can be seen from the photomicrographs in Figure 4. Plots of this form have several convenient features in themselves, and the more conventional frequency distributions can be readily obtained from them. Length, Breadth, and Elongation Ratio Distributions. The sense in which I and b are plotted is shown in Figure 5. It can be seen from this figure that a point representing an equidimensional profile will fall on the hypotenuse while one representing an elongated profile will fall within the triangle. If the elongation ratios, lib, of all the particles in a sample are the same the points representing their sizes will lie on a straight line originating at the left-hand apex and the more elongated the particles the closer will this line lie to the I axis. With fibrous particles from material of constant diameter, the points will lie on a horizontal line at a distance below the I axis corresponding to that diameter. It is not to be expected that the proportions of all the particles in a sample will be the same and consequently the I-b plot will normally reveal a considerable scatter as in Figure 3. Nevertheless, it is an advantage of the graph that it shows (a) the approximate degree of elongation and its variability and (b) the change of elongation ratio with size if this feature is real and not masked by scatter. The latter effect is a feature of potato starch, the smaller grains of which present circular profiles and the larger elliptical; it is not evident in the TiOl examples given here. For the quantitative comparison of samples, frequency distributions with the ratio 116 as variable can be easily ab-
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5
I/b-
(Zn = percentaze of particles per unit of ratio)
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ds > dv for bodies of lower than cubic symmetry, the curves of constant dv are better spread than the dA curves, those for ds (not shown) being intermediate. Number distributions can be found in terms of any of these measures of size by counting the points between suitably spaced lines drawn on the graph. Distributions based on dA and dv for TiO,. sample B are shown in Figures loa, and 6. It can be seen that the dA distribution has greater spread than that related to dv as is required by the inequality,
dA
> dv.
From the point of view of constructing a number distribution, da might be thought the most suitable of the three single measures of size as it alone depends on the dimensions actually observed. However, ds or dv are to be preferred when other distributions are to be derived. Size-Surface and Size-Volume Distributions. Although ds and dv are hypothetical in the sense that they are based on an assumption concerning particle thickness, some such assumption is implicit whenever the surface and volume distributions are derived from a number distribution in terms of a single measure of size. In practice, it is often assumed that the amount of surface or of volume or weight within a given size interval is proportional to the square or cube of the average projected area diameter for that interval, and the surface and volume dis tributions are calculated accordingly. This procedure is unexceptionable in the case of roughly spherical particles. It is illustrated by the British Standards Institution’s method for sizing with the optical microscope (2). However, it is no longer valid if the particles are elongated, as the British Standard specification points out.
The use of ds in calculating the surface distribution and of dv in calculating the volume distribution obviates the difficulty as the particles are now classified into size intervals whose midpoints adequately represent the property concerned when raised to the appropriate power. This is the basis of Comyns and Murley’s (3) method for obtaining volume (weight) distributions by which particle profiles are compared with ellipses representing particular prolate spheroidal volumes. Their method (which is also based on the assumption that b = t ) is designed to obviate the error involved in the comparison of noncircular particle profiles with reference circles. The procedure described here does the same and has a similar basis because it is immaterial whether asymmetrical particles are approximated by a prolate spheroid or a rectangular prism for the purpose of volume calculation. A histogram showing the volume distribution for TiOz sample B is given in Figure 1Oc. As usual, it is shifted toward the right (in the direction of greater size) by Comparison with the corresponding number distribution in Figure lob. A volume distribution based on da and adjusted to the range of dv is superimposed on the histogram in Figure 1Oc. This shows clearly that the use of the projected area diameter for such calculations could convey a misleading impression if the elongation of the particles were considerable. ACKNOWLEDGMENT
The author acknowledges with gratitude the assistance of A. E. Comyns and R. D. Murley of British Titan Products Co. Ltd. who provided the electron micrographs used for illustration and that of B. Clisby of the School of Chemistry Workshop, The University of New South Wales, who constructed the semiautomatic instrument.
RECEIVED for review July 5,1966. Accepted October 17,1966. (3) A. E. Comyns and R. D. Murley, “The Meaning of Particle Size,”
(2) British Standards Inst:itution, B.S. 3406, Part 4, 1963.
Lecture delivered to the Schweizerische Vereinigung der Lackund Farben-Chemiker, 1965.
VOL. 39, NO. 1 , JANUARY 1967
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