January 19731 PARTICLE SIZE ANALYSIS 13 Particle Size Analysis The following are summaries of the two papers presented a t a meeting of the Particle Size =Inalysis Group held on March lst 1972 and reported in the April issue of Proceedi?zgs (11. 7 8 ) . A Suggestion for the Exact Determination of Particle Size and Shape BY H. 11. SUTTOK ASD N. BTXDALLI (Wavren Spvzng Laborntoiy Stet enage He? tjovdshwe) PRIMARY Standards for the absolute measurement of particle size and shape are difficult to define. In practice particles are assessed by Secondary Standards such as sieving sediment- ation permeability adsorption light scattering and Coulter counting whereas the Primary Standards which can be obtained only by direct measurement on the particle image require a prohibitive amount of labour unless automatic scanning devices are available.DEFISITIONS- a point in space. or in three dimensions as where 4 is the azimuthal angular co-ordinate. be expanded as a Fourier series1- The size or shape of a particle is defined as the locus of points lying on its surface about In two dimensions this can be expressed in polar co-ordinates as * . .. .. . * (1) Y = r(0) a . . . . . y = r(@,4) .. . . .. .. . * I . (2) ,4t present only the two-dimensional case will be considered. The function r ( 0 ) can m r(0) = a + 2 (a cos 120 + b sin 120) . . .. .. * (3) n=l The coefficients ao a, a2,. . . . . . . . and b, b2,. . . . . . . . can be calculated and the parameters conveniently represent the size and shape of the particle represented by r(0). It can be shown that if a point in the particle is used as the origin such that a and b are identically zero then in a quantitative way the n = 2 terms describe the “elongation” of 14 PARTICLE SIZE ANALYSIS CProc.SOL. Analyf. Chew. the particle. The n = 3 terms are large if the particle has a “triangular” appearance (strictly if it has apexes separated by 2 ~ / 3 radians) the u = 4 terms are large for a “square” appearance and so on. The relative proportions of a and b determine the orientation of the particle relative to 0 = 0. In fact equation (3) can be written as y ( e ) = a +- C cos (ne + En) . . .. .. ‘ (4) 2 n = 2 where Cn2 = an2 + bn2 and En = arctan (-Cn/bn). If the zero is chosen so that E = 0 then the convention is rendered unique by the criterion I E I <n/3 because the n = 2 terms have a two-fold symmetry.The parameter a is the mean length of all the radii taken since i The other parameters are given by and a = - s2 Y cos no d8 x 0 0 .. .. * * ( 6 ) .. . . .. .. .. .. . . * * ( 7 ) .. 0 where M = 1 2 3 . . . . . . . . . etc. The coefficient a. can be taken as a size parameter while the others a and b,, are shape parameters. However this is not absolute; for example a diameter defined in terms of area immediately involves coefficients other than a, thus showing the explicit influence of size on shape. The “size” and “shape” of a particle are thus both logically and mathema- tically inseparable since the exact geometry can only be expressed in terms of all the Fourier coefficients. USE OF COXPUTERS- A computer program has been written to evaluate particle size and shape in terms of Fourier series.The program can process a digitised form of data of the kind which might be obtained from an automatic scanning microscope or a flying spot microdensitometer. Initially the data are scanned and the positions of points on the perimeter of particles are stored in the computer in a main storage array. I t is assumed that the original scanning instrument scans the specimen from left to right registering respectively zeros outside and units inside the particle profiles in the form of a grid. Preliminary rectangular co-ordinates are initially taken to cover the total scan area. These Cartesian co-ordinates of grid points on the perimeters of the particles are stored in the computer. The data [in a form similar to Fig. 1 (a) are scanned row by row left t o right starting from the top row of the data grid.Thus K and K [Fig. l,(b)] will be detected as two points on the outline of a particle in row R. The program incorporates a procedure for determining if K and K are perimeter points of a new particle that has not been scanned or of a particle that already has perimeter points in the store. The process of scanning and storing co-ordinates continues until all the data have been scanned. If the storage limit of the computer is reached the scanning of data stops and the Fourier parameters of each particle are calculated. Scanning of the data is then resumed because the data in the store are now redundant and can be overwritten. January 19731 PARTICLE SIZE ANALYSIS 15 Thus from the stored co-ordinates each particle is identified in turn and computations performed on the particle.The centroid of the particle is first calculated from x = - Xi and y = - Y i where 72 is the total number of scan points in a particle and X and Y refer to the preliminarj. Cartesian co-ordinate system. Then with the centroid as the origin the polar co-ordinates of the perimeter of the particle are evaluated. The radii of these points are interpolated a t uniform angle intervals in preparation for the calculation of Fourier parameters by \T'hittaker and Kobinson's method where the integrations in equations (ti) (6) and (7) are performed by using the trapezoidal rule. The parameters of each particle are computed before the com- putation of the next particle begins. - n lls 3 = 1 "fl. l2 a = 1 ! '*5 10 15 20 25 30 <5 Y 4 5- 10- 15- 20- 251 I 30.35; I 40. 45 50- 55- 60- 65; 75 I! 0 x- Y 4 R -1 R Fig. 1 ( a ) Test data processed by computer; and ( b ) scanning of data for particle end-point5 I< and K The Fourier series method for particle shape determination cannot deal with particles which have more than one value of r for any value of 8. The present computerised method cannot deal with particles that are orientated in such a way that the scan line crosses and re- crosses the particle outline (see Fig. 2). However as well as the advantages of identification of shape features in terms of large values of parameters mentioned earlier the whole of a scanned particle can be reconstructed by using the stored data (Fig. 3). 16 PARTICLE SIZE .4NALYSIS [Proc. Soc. Afzaljd. Chewi. nth row ( a ) (bi Fig. 2 (u) Re-entrant particle shape and ( b ) particle scanned as two separate particles in first row of scan In principle the Fourier coefficients can be used separately in any ad hoc correlation.For example in industrial quality control it may be desirable to have spherical particles and upper limits on the ratios @,/a and b,/a can be set. For elongated particles the ratios bz/aq and cgfao should be large compared with the other coefficients. Furthermore it is possible to make use of computers in simulating random packing of a collection of particles of known size or shape Similarly the tensile strength of a computer-generated compact can be determined by another averaging process that allows for the nature of the inter- particle forces. The number of parameters calculated for each particle does not substantially affect the compu- tation time.The computation time can be reduced by reducing the number of interpol- ations. For example if only two parameters are required for each particle ten interpol- ations would be sufficient. The computation time depends mainly on the amount of data being analysed. using parameters > O la0 > O 04ao > O 03ao 32 -016 a2 -016 61 - 0 0 3 a4 -010 a4 -0.10 a2 -016 a7 - 0 0 5 a9 -0 0 4 all -0 03 a7 - 0 0 5 a3 - 0 0 3 5 a9 - 0 0 4 63 - 0 0 3 611 -0 03 ( d ) Parameters (e) First 8 ( f ) First 22 (g) Actual shape > O 03ao parameters parameters of particle from first 14 61 - 0 0 3 h3 -0.03 67 -0.03 a3 -0.035 a7 - 0 0 5 Fig. 3. Reconstruction of particle shapes from Fourier para- meters with the Hewlett-Packard calculator/plotter January 19731 PARTICLE SIZE ATi-4LYSIS 17 The first part of the program i.e.scanning and storing can be put to a variety of other uses. For example by setting threshold levels to the intensity values to be processed different coloured areas of the specimen can be selectively analysed. Hence the program can be applied in biological work metallurgy and mineralogy. Finally to obtain data for three-dimensional particles Y(f?,q5) a sample of particles could be set in an appropriate resin and sectioned with a moving microtome although this would be difficult with hard or fine particles. The r(6) data and the thickness of the sections could then be converted to the r(B,+) form by computer. The corresponding Fourier series could be evaluated if required. Arrangements can be made for the computer program described in this paper to be made available.REFERENCES 1. 2. Cheng D C. H. and Sutton H. AT. S a t u r e (Plzyszcal Sczence) 1971 232 S o . 35 192. LVhittaker E. T. and Robinson G. "The Calculus of Observation Blackie London 1946 266; pp. The Effect of Particle Size on the Tensile Strength of Powders BY D. C.-H. CHENG (Warren Spring Laboratwy Stevenage Hevifoovdsltzvr) THIS note outlines a theory which relates the tensile strength of a compact of powders or granular material to the particle-size distribution. Details of the theory and experimental verification have been discussed elsewhere,l but some new modifications are presented here. The theory is based on the assumption that in a powder compact individual particles are packed against each other in a more or less random fashion.On average each particle is in contact with a number (the co-ordination number c) of other particles. The co-ordination number depends on the density of the compact p by the relationship .. . I .. * (1) .. 3 C = 1 - PIPS where p s is the density of the solids. When the compact is split under tension a number of pairs of particles originally in contact will be separated. In an ideal situation these pairs of particles are grouped about a plane normal to the direction of the tensile force and the tensile strength T can be defined as the force per unit area of this fracture plane required to break the compact. The tensile strength depends on the number of pairs of particles associated with the unit fracture plane Nppla which in turn depends on the sizes of the particles involved.Con- sidgr a volume marked out by tlLe centres of the pairs of particles. On average this volume is d per unit area of split (where d is the mean effective diameter which will be defined below) and the number of particles contained therein is where ArPBIV is the number of pairs of particles per unit volume of the compact which is related to the number of particles per unit volume of the compact ATPIv by - N,,, = dXDP,V . . * . .. * . * . (2) The latter in turn is related to the density of the compact by Therefore N*/vCp'ps = p . . * . .. .. . * (4) N D D / & = - - - .. .. .. . . ' ' (5) c Z p 2 Cpps 18 PARTICLE SIZE ANALYSIS [Proc. Soc. Analyt. Chem. The second factor determining the tensile strength is the force F which acts between the particles in a pair of particles.In general F will be inclined a t an angle 0 to T and it is the resolved component F cos 8 that co3tributes to T . Assuming that all 8 are equally probable the mean effective value of F F can be found by averaging over 6 - . . * . (6) F = + F . . .. * . Since the particles are not smooth when they come into contact it is the surface pro- An area of contact Aij can be defined within trusions that touch and form micro-contacts. which the micro-contacts lie and then F can be expressed in terms of this area where 12 is the interparticle force per unit area of contact. I t also depends on the property of h which arises because of the presence of van der IYaals electrochemical and other interatomic forces that depend on distance between the interacting atoms. In addition the number of micro-contacts at which the interatomic forces act also varies as the surface separation t between the two particles varies.Therefore h is a function of 1 The exact mathematical relation between h and t is unknown but in general it can be expected that h is large when t tends to zero and it falls to zero when t attains some charac- teristic value 1,. The distance I between particle centres in a pair of particles is . . . . ( 7 ) F = hAij . . . . . . The area of contact depends on the sizes of the two particles involved. . . . . (8) h = h(t) * . . . The surface separation can be related to the compact density. .. . . ’ . (9) I = +(di + dj) + t . . . . If it is assumed that there is no change in the type of particle packing as density changes- 1 P E p .. I . .. . . (10) Then for small changes to - t .... . Po which rearranges into In these equations po is the density at which t = to and d has been substituted for 1. The area of contact can now be defined as the area over which the separation between any two points one on the surface of one particle and the other on the second particle is less than to. For two spherical particles of diameters di and d j respectively . . (13) Summarising the factors that contribute to the tensile strengtli- T = C f .. . I . . . . . . . . (14) for all pairs of particles in unit area of split. From equations (6) and (7)- T = I ] dij .. . . .. . . . . (15) for all pairs of particles in unit area of split. On grouping the pairs of particles according to the particle sizes involved- xijSPP a -4ij . . . . . . . (16) January 1973 PARTICLE SIZE AKALTSIS 19 where xij is the proportion of pairs of particles out of all possible pairs that are composed of particles of sizes di and dj.On using equations (5) and (13)- . . .. . . . . (17) where .. .. * . .. . . (18) di + dj S = i ? i < j Equations (15) and (12) are the theoretical relations linking T to experimental quantities. I t is shown in the earlier paper1 how pa can be deduced from experimental results. The particle-size parameters d S and 5 can be calculated if the particle-size distribution is known. For example if the distribution consists of number fractions ni nj . . . . . . . . of spheres of diameters di dj . . . . . . . . . then i < j I . .. .. . . (20) i < j and ; = 2 nidi3 .. .. .. .. . . (21) 6 i and S is given by equation (18) and from the use of equation (19).fractions in the form The theory can be tested by plotting data obtained by using different particle-size T d~ pips against (i - 1) when the data for different size fractions fall on the same curve. From this curve the value of to can be deduced and hence the function k ( t ) can be constructed in the form of a master curve which can then be used to generate the tensile strength of any other compacts for which the particle-size distributions are known. By invoking the principle of corresponding states it is possible to arrive a t a useful method for comparing the interparticle forces1 and shear properties3 of different powders or granular materials. The effect of particle shape can be introduced by modifying equations (13) (18) and (20) and this aspect of the problem is currently being studied a t li’arren Spring Laboratory. 4 77 1 - PIPS Further extension of the theory to cover moist powders is possible.2 REFEREXES 1. -. 3 3. Chcng D. C.-H. Chew. Engng Sci. 1968 23 1403. __ J . Adhesion 1970 2 82. Chcng D. C.-H. Farley R. and Valetitin F. H. H. Insin Chew E n g ~ s Synzp. Sw. S o . 29 1968 14.