Applied fractal geometry and powder technology

Pergamon 0960-0779(94)00265-7

Chaos, Solitons & Fractals Vol. 6, pp. 245-253, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960-0779/95 $9.50 + .00

Applied Fractal Geometry and Powder Technology B.H. K A Y E Department of Physicsand Astronomy,LaurentianUniversity, Sudbury,Ontario, Canada, P3E 2C6

Abstract - Powder technology is important to many industries including mining, food processing, paint technology, powder metallurgy and space research. Applied fractal geometry is being used in these industries to describe and comprehend the complex interactions of many causes. The topics to be discussed in this review include, the study of important fractally structured pigments, the fractal structure of hazardous fumes, and the role of fractal soot in world climate studies. We shall discuss in particular the way in which fractal dimensions derived from the study of the avalanching behaviour of powder heaps are creating a revolution in powder rheology.

INTRODUCTION In the years since the publication of Mandelbrot's seminal book on Fractal Geometry it has become apparent that it is useful to distinguish between theoretical fractal geometry and applied fractal geometry. Theoretical fractal geometry deals with the structure and properties of the exotic patterns such as the Mandelbrot set, Julia sets and Fatu dusts. These systems can properly be described as ideal fractal structures. They have the property of being self-similar at all scales of scrutiny. Applied fractal geometry deals with the application of the concepts of fractal geometry to rugged systems and other phenomena which exhibit statistically self-similarity over a finite range of inspections (1,2,3). As has been discussed at length elsewhere systems which manifest fractal structure and or behaviour are generated by the random interaction of many causes and it is not surprising that powder technology with its complex generative and processing variables has proved to be an area where the concepts of fractal geometry have found many applications.

F R A C T A L DIMENSIONS A TERM W I T H MANY MEANINGS. One of the consequences of the very rapid development of applied fractal geometry is that there is some confusion over the exact meaning of various terms. Thus if we consider a carbonblack profile of Figure l(a) the properties of such an agglomerate have been described with various parameters which have been described as 245

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Figure 1. The structure of an agglomeratedcarbonblackprofile can be described by various parametersbased on the concepts of fmctal geometry. (a) (b) (c)

Profile of a highly magnified carbonblackagglomerate. Typical polygons generated by structuredwalk explorations. Richardson plot of a series of explorationsof the profile of (a)

fractal dimensions. First of all if one looks at the ruggedness of the projected boundary of the agglomerate one can describe the structure of the boundary in terms of a boundary fractal dimension. This parameter is a measure of the ruggedness of the boundary. Such boundary fractal dimensions have been studied by various image analysis techniques with the most familiar technique being described as the structured walk or yardstick method. In this technique one estimates the perimeter of the profile at a given resolution by striding around the profile with a pair of compasses set at a magnitude ~, as shown in Figure l(b). The symbol )~ represents the resolution of inspection parameter and the polygon perimeter, P, generated by striding around the profile represents the estimate of the perimeter at resolution )~. It is common practise to plot the data for an exploration of a profile at various values of 1 by normalizing the estimates of the perimeter and the step size using the maximum projected length of the profile known as the Feret Diameter, FD. Data from a set of explorations of this kind plotted on log-log graph paper are summarized in Figure l(c). A graph of this kind is known for historic reasons as a Richardson plot. An operational view of fractal systems is that if a set of explorations of the kind described in the

Applied fractal geometry and powder technology

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forgoing section generates linear relationship on a Richardson plot then one can usefully regard a slope of a linear region as defining the fractal addendum to the topological dimension of the system to generate a descriptive fractal dimension of the system. It can be seen in the graph of Figure l(c) that there are two linear relationships and it can be shown that it is useful to describe the data line at coarse resolution as defining the structural boundary fractal dimension of the profile. The structural boundary fractal dimension is useful in describing the aerodynamic behaviour of the agglomerated soot fineparticle and if one is using a carbonblack, or other pigment, a structural fractal dimension probably describes the way in which the profile physically interacts with its surroundings (2,4). The fractal dimension deduced from the data generated at high resolution inspection of the profile can usefully be described as the textural fractal dimension of the profile. It has recently been suggested that the magnitude of the textural and structural boundary fractal dimensions found by examining a series of agglomerates generated by fuming in a turbulent flame contains frozen information on the formation dynamics of the profile (5). The study of the formation of agglomerates in the flame is a branch of aerosol physics. Currently an active area of research in aerosol physics is to model the generation of the fractal structure of an agglomerate on a computer using randomwalk - Monte Carlo methods and then comparing the structures of real soot or fume agglomerates with the structure of simulated fumed agglomerates to see if one can deduce formation dynamics by discovering the algorithm which generates fractal structures similar to those observed in real systems (6,7,8,9). Boundary fractal dimensions have been used to study the flow properties of powders (10,11) and to study the viscosity of suspensions (2). The search for correlations between the fractal structure of systems such as fumes and fragmented material is being hampered by lack of specialized automated image analysis. Significant developments in the use of fractal parameters to describe aerosol systems will be dependent on the development of low cost fast systems for looking at images of fractally structured systems. An alternate technique for studying the structure of a cloud of aerosol fineparticles is to probe the system with x-rays, light rays or other energy beams. When one explores the structure of the agglomerates by such a technique one is not measuring the configuration of the profile in space rather one is exploring the way that the subunits of the agglomerate are packed to occupy three-dimensional space. Thus fractal dimensions measured by light scattering studies should properly be called mass or density fractal dimensions. Many workers have used such fractal dimensions to study fumes and pigments (12,13). The mass fractal dimension of soot and other fineparticles produced by combustion processes are becoming an important element in any models for use in predicting future climate changes (14). It is likely that the people attempting to calculate the radiation balance of the earth in an atmosphere disturbed by the presence aerosol flneparticles will need to know both the mass fractal dimension, to understand the scattering behaviour of the fineparticles, and the boundary fractal dimensions, which will govern the rate at which particles settle out of the atmosphere and/or the way in which they interact with atmospheric species such as acid rain drops. Currently there is very little instrumentation available for direct measurement of'the fractal structure of air borne agglomerates. (See however reference 6). Because of the importance of the topic we can expect to see an intensive development of techniques for measuring the boundary fractal structures of airborne fineparticles. (See also comments later in this review on fractal studies in occupational hygiene.)

FRACTAL DIMENSIONS IN DATA SPACE.

The fractal dimensions discussed so far have been directly connected with physical structures. The term fractal dimension is also used to describe self-similar behaviour in time and space domains. Consider for

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example recent studies of the fracture of material by ballistic impact. If one considers the failure of the body subject to stress, if pressure is applied to the system by slow compression, failure occurs by means of crack propagation through the flaws present in the body. In ballistic fracture, failure occurs because shock waves created by the impact of an item, such as a piece of rock on a surface, are reflected in a way that creates tension zones in the body leading to ballistic fracture. Recent studies have established that if one takes a piece of material and fractures it by a single ballistic impact the fragment size distribution can be plotted to generate a straight line on log-log graph paper. It is useful to describe the slope of this line as being a fractal dimension in data space. Thus in Figure 2 some data reported recently by Gareth Brown of the Mining Department of Nottingham University is shown. Brown is exploring the possibility that the fractal dimension in data space of the fragments is related to the fractal structure of the fragments themselves which is in turn related to the physical properties of the materials such as hardness and elasticity (15,16,17). The concept of using fractal dimensions to describe self-similar systems in data space can also be applied to behaviour and frequency of events and time series data. Consider for example the equipment shown in Figure 3. This is being used to study the dynamic avalanching of a powder system such as that which would be generated if powder were flowing out of a storage hopper. Using the equipment of Figure 3(a) the data of graphs (b) and (c) were generated. The powder studied was what is known as rock tailings, the material left after the valuable constituents of an ore body have been taken out of the crushed ore. When this powder was allowed to form an avalanching powder heap the frequencies of avalanches of a given size for the untreated powder was as illustrated by the data plotted in the graphs of Figure 3 indicated as being without Cab-o-Sil (11). We have already noted that when we were looking at physical fractals that they were not infinitely self-similar. In the case of a real powder system forming avalanches in a dynamic system one can have more than one data line. The physical significance of the three data lines for the untreated powder of Figure 3(b) are as follows. For the majority of the avalanches the fractal dimension in data space of 1.35 described the behaviour of a

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Figure 3. The dynamic behaviour of an avalanching powder heap can be described using the concepts of fractal dimensionsin data space. (a) (b)

Simplifiedsketch of the equipmentused to study powder avalanching. Weight distribution for avalanchingof mine tailings.

(c)

Distribution for time between avalanches for the avalanches of (b).

system. However the data line of the smaller avalanches showed that some process in the avalanching system was suppressing the frequency of the smaller avalanches. At the other end of the scale larger avalanches were much less frequent that anticipated from the bulk of the behaviour of the powder system indicating that when the powder build up reached a certain value the heaps became more unstable than anticipated in a study of the average behaviour of the heap. In Figure 3(c) the same type of data summary indicates that fractal dimensions in data space can be used to describe the time between avalanches of a powder heap. In powder technology a strategy often used to influence the flowability of a powder is to add so called flow agents to the powder. One type of flow agent is a very finely divided amorphous silica (which itself is fractally structured). In Figure 3(b) and (c) the effect of the flow agent on the avalanching behaviour of the powder is illustrated with the symbol (Q). This is currently a very active area of powder rheology research (16,17,3).

250

B.H. KAYE I N D U S T R I A L A P P L I C A T I O N S O F F R A C T A L DESCRIPTORS. Now that we have discussed the various meanings of the term fractal dimension in applied fractal

geometry we can now explore industrial applications of the fractal concepts in specific areas of powder technology and aerosol physics. Industrial workers are often exposed to fumes which constitute an inhalation hazard. Currently some of the technology used to measure the property of dust that form respirable hazards is simple and generates information on what is known as the aerodynamic diameter. The aerodynamic diameter is the size of a unit density sphere which has the same dynamic properties of the dust. The aerodynamic diameter of a profile is useful for calculating the entry into the lung but gives limited information on the actual health hazard constituted by the dust. Devices are available which can isolate isoaerodynamic fineparticles from a respirable dust and in Figure 4 a set of isoaerodynamic dusts for several dangerous dusts are shown. The first thing to notice is that the aerodynamic diameter grossly underestimates the surface area of the dust and this means that for many of the present studies of occupation health dust hazards grossly underrate the health hazard particularly if the dust contains adsorbed carcinogenic chemicals from cigarette smoke and or diesel fumes. Moreover the possibility that a dust fineparticle will be captured by a filter is not usefully related to the

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Isoaerodynamiccoal fragments. Isoaerodynamicuraniumdioxide fineparticles.

(c)

Isoaerodynamic thorium dioxide fmeparticles.

Applied fractal geometry and powder technology

251

aerodynamic diameter of the profile. The structure of filters for respirators and the structure of deposits formed in filters by the capture of dust is an active area of fractal research (18,19,20,21,22). A subject closely allied to the problems of assessing the health hazards of respirable dust is that of treating patients with therapeutic aerosols. Again if one generates an aerosol and measures its structure from diffraction pattern analysis or aerodynamic investigation one is not able to assess the biological activity and the behaviour of the fineparticles in the lung because the chemical reactivity will be determined by the fractal structure of the individual fineparticle and the lodgability of the dusts in the lung is a function of the physical structure of the fineparticle. The study of the efficiency of systems for delivery of aerosols to the lung is a very active area of research and one in which fractal geometry is assuming increasing importance in the search for a better understanding of the efficiency of drug delivery systems. The pharmaceutical industry is also interested in the pore structure of the tablets that they produce. Recently it has been shown that one can study the pore structure of a tablet by placing a drop of a suitable

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Figure 5. The fractal boundaries generated by the spread of fluid on porous substances is proving to be useful in the quality control of the manufacture of items as different as pharmaceutical tablets and paper. It is also being used to look at the quality of letter boundaries in printing technology. (a) (b) (c)

Boundary produced by a drop of coloured liquid in a pharmaceutical tablet. Data for the quantitative study of the boundary of a letter O printed on newsprint (2,10). Data for the boundary of a letter O printed on high quality glossy magazine paper.

252

B.H. KAYE

coloured fluid onto the surface of the tablet. The fractal front produced by the spreading of the liquid as shown in Figure 5(a) is related to the pore structure of the tablet which in turn is related to its physical properties when used to deliver drug systems to the patient (16). The pattern produced by the spreading fluid in the tablet is similar to that generated when a fluid such as an ink is fractionated on a filter paper to produce the familiar patterns of liquid chromatography. The fact that an ink spreads on a piece of paper to produce a fractal boundary is of interest to the people who study print quality in the printing industry as illustrated by the data of Figure 5(b) and (c). CONCLUSIONS In this brief review of the use of fractal concepts in powder technology and aerosol physics it has not been possible to describe all the studies underway in many laboratories in detail. However the case histories reviewed should indicate the great interest and activity in the use of fractal geometry to study powder systems. It is an area of research where Monte Carlo simulations of systems and the examination of real physical systems are intertwined in the search for understanding.

Future research is likely to focus on better methods of

measuring the fractal structure of complex systems so that the usefulness of Monte Carlo simulation of the systems can be established.

Acknowledgement - The writer wishes to acknowledge the help given by G.G. Clark in the preparation of the

diagrams and text of this communication.

REFERENCES Mandelbrot first described his theories of Fractal Geometry in a book entitled: Fractals, Form, Chance and Dimension published in (1977). In 1983 Mandelbrot published an updated and expanded version of the book under the fl0e: The Fractal Geometry of Nature. W. Freeman, San Francisco, (1983). This book is considered by Mandelbrot to be the definitive book on the subject. (Personal Communication). B.H. Kaye, A Randomwalk Through Fractal Dimensions. VCH Publishers, Weinheim, Germany, (1989). B.H. Kaye, Chaos and Complexity; Discovering the Surprising Patterns of Science and Technology. VCH Publishers, Weirdaeim, Germany, (1993). B.H. Kaye, Characterizing the Structure of Famed Pigments Using the Concepts of Fractal Geometry, Part. Part. Syst., Charact. 9, pp. 63-71 (1991). B.H. Kaye and G.G. Clark, Chapter 24, Formation Dynamics Information: Can it Be Derived from the Fractal Structure of Famed Fineparticles? in Particle Size Distribution II, Assessment and Characterization, Edited by Theodore Provder, (ACS Symposium Series, 472), American Chemical Society, Washington, D.C., (1991). 6.

I. Colbeck, Dynamic Shape Factors of Fractal Clusters of Carbonacous Material. Journal of Aerosol Science, Vol. 21, supplement 1, pages $43-46.

7.

P.C. Reist, M.T. Hsieh, P.A. Lawless, Fractal Characterization of the Structure of Aerosol Agglomerates Grown at Reduced Pressures. Aerosol Science and Technology, Vol. 12, pp. 91-99, (1989).

8.

T. Cleary, R. Samson, J. Gentry, Methodology for Fractal Analysis of Combustion Aerosols in Particle Clusters. Aerosol Science and Technology, Vol. 12, pp. 518-525 (1990).

9.

P. Meakin, Section 3.12 in The FractalApproach to Heterogeneous Chemistry. D. Avnir, (ed). John Wiley & Sons, (1989).

10.

M. Peleg, M.D. Normand, Characterization of the Ruggedness of Instant Coffee Particle Shape by Natural Fractals. J. Food Sci., 50, pp. 829-831 (1985).

I1.

M. Peleg, M.D. Normand, Mechanical Stability as the Limit to the Fractal Dimension of Solid Particle Silhouettes. Powder TechnoL, 43 (2) July 15, pp. 187-188 (1985).

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12.

D.W. Schaeffer, Fractal Models and the Structure of Materials in The Materials Research Society Bulletin, Volume 13, Number 2, pp. 22-27, February (1988).

13.

D.W. Schaeffer, Polymers, Fractal and Ceramic Materials, Science, Volume 243, pp. 1023-1027, February 24 (1989).

14.

J. Nelson, Nature. 339 pp. 661 (1989).

15.

Personal communication with Gareth Brown.

16.

B.H. Kaye, Applied Fractal Geometry and the Fineparticle Specialist. Part I: Rugged Boundaries and Rough Surfaces. Part. Part. Syst. Charact., Volume 10(3), pp. 99-110 (1993).

17.

B.H. Kaye, Fractal Dimensions in Data Space; New Descriptors for Fineparticle Systems. Part. Part. Syst. Charact., Volume 10 (4), pp. 191-200 (1993).

18.

D.S. Ensor, M.E. Mullins, The Fractal Nature of Dendrites formed by the Collection of Particles on Fibers. Part. Charact. 2, pp. 77-78 (1985).

19.

B.H. Kaye, Describing Filtration Dynamics from the Perspective of Fractal Geometry. KONA (1991) published by Hosokawa Foundation, 780 Third Avenue, New York, N.Y. 10017.

20.

R.A. Trottier, I. Stenhouse, B.H. Kaye, Possible Links Between the Fractal Structure of Dust Capture Tree Deposits In A Fibrous Filter and Loading Effects. Extended Abstract proceedings of the 5th Annual Conference of the Aerosol Society, Loughborough University of Technology, Loughborough, England, March 26-27 (1991).

21.

C. Kanoaka, H. Emi, S. Hiragi, T. Myojo, Morphology of Particulate Agglomerates on a Cylindrical Fiber and a Collection Efficiency of a Dust Loaded Fiber. Aerosols Formation and Reactivity. Proceedings of the Second International Aerosol Conference, West Berlin, 22-26, September (1986), Pergamon Journals Ltd., Printed in Great Britain.

22.

B.H. Kaye, R. Trottier, Effect of Shape, Structure and Texture on the Accuracy of Size Characterization of Fineparticles by Light Scattering. Lasers in Industry Conference, June (1988), Portugal.