1. Introduction

The on-line measurement of the size of powdered minerals would provide a diagnostic for comminution in mineral processing. Grinding in mills is the most energy-intensive phase of the liberation of metal rich ores from gangues. In the absence of real-time particle size information grinding mills are often adjusted to produce finer than necessary particle sizes in order to ensure that ore recovery is high. Over grinding wastes energy, increases the milling time and maintenance costs and lowers the throughput and mean time before failure of the milling plant. An on-line particle size measurement technique would allow better grinding control. Laser speckle decorrelation was investigated as a possible technique for measuring the average size of powders in beds.

Léger et al [1] and Léger and Perrin [2] describe a theory and method for the measurement of the roughness of metallic surfaces by angular speckle decorrelation. A laser of wavelength λ is used to illuminate a surface which has a normally distributed surface roughness (σ) at an angle of incidence (θ₁) and the resulting speckle pattern is recorded on film (camera axis normal to the surface). The film is subsequently exposed to a second speckle field after changing the angle of incidence of the laser beam by a small amount (δθ). They describe the visibility of the resulting Young’s fringe pattern (V(δθ)) from very rough surfaces (σ≫λ) as:

They compared the measured fringe visibility with their theory for three metallic surfaces of known roughness. They concluded that the method could easily determine the roughness and that the method seemed to be independent of the material being measured. They described only roughness measurements on metallic surfaces.

Nitta and Asakura [3] applied the speckle decorrelation method to the measurement of powder beds. They used two different approaches. They compared the Young’s fringe method described by Léger et al [1] with measurements made using a two-dimensional Charge Coupled array Detector (CCD). The fast Fourier transform (FFT) was used to calculate cross-correlations of the recorded speckle patterns. The Young’s fringes and CCD-FFT approaches provided essentially equivalent data. Nitta and Asakura [3] examined alumina, silica, calcite and precipitated chalk powders over a size range of 1.37–13.56µm using a 633nm wavelength HeNe laser. They describe their samples as having approximately log-normal particle size distributions and as having been “carefully smoothed with a glass plate which was then removed”. They found that, for consistently prepared powder beds, the surface roughnesses derived from the speckle decorrelation were highly correlated with particle size and linearly proportional to the mean size of the powder.

More recently variations on the Léger and Perrin [2] technique have been reported [4–7]. Russo et al [6] used a linear diode array detector with simultaneous twin-beam exposure and computed auto-correlation to measure surface roughness standards. They commented that their one-dimensional correlations did not provide a high signal-to-noise ratio. Rebollo et al [7] applied speckle decorrelation to sedimentary rocks using a linear diode array-FFT cross-correlation technique to measure the roughness of oil reservoir rocks and found a linear relationship between the optically determined roughness value of the rock and the porosity. Ohlídal [13,14] examined the performance of the Léger and Perrin theory [1]. He studied metal surfaces with a low ratio of roughness to correlation length (meeting the Fraunhofer criterion) and those surfaces with steeper sloped roughness where the Fresnel approximation applied and found that the Léger and Perrin theory [1] only fitted well when the surface roughness had a large correlation length.

Industrial powders and sedimentary rocks are seldom transparent. To our knowledge there have been no reports examining the effects of sample transparency on the measurement of the particle size or roughness of powder beds or sinters by angular speckle decorrelation although Quintián et al [15] described multiple scattering effects on rough-surfaced transparent diffusers. The work reported here set out to determine whether varying particle transparency has a significant effect on the angular speckle decorrelation rate and whether any such effects could be compensated.

2. Experimental

2.1 Speckle Measurements

All of the speckle measurements reported here were made with apparatus functionally similar to that of Nitta and Asakura [3]. The beam from a 2mW 633nm wavelength HeNe laser was directed at the surface of a powder bed mounted on a θ/2θ goniometer. A monochrome CCD (768×576 pixels) without a lens recorded the speckle pattern scattered from the powder surface. (see Fig. 1.) The CCD was operated with a nominal gamma of 1.0 and with the automatic gain control switched off. The CCD’s exposure time was adjusted to eliminate CCD saturation. Image frames were saved for later processing as 256-level grey-scale bitmaps in the Windows™ 3.0 BMP format. The speckle images collected as the sample was rotated were later cross-correlated by computed two-dimensional fast Fourier transform (2D-FFT). The speckles were approximately 17 pixels wide at half-maximum intensity and there were about 1500 speckles in each CCD frame.

The sample was mounted on the θ arm of the goniometer and the CCD was mounted on the 2θ arm. In this way the CCD was moved at twice the angular rate of the sample. This conveniently eliminated speckle translation across the CCD face allowing larger sample rotations to be used. Russo and Sicre [4] had earlier used a stepper motor arrangement to move their CCD camera, presumably to achieve the same ends.

 

Fig. 1. Schematic diagram of experimental set-up used to make speckle measurements. The bed-CCD spacing was 185mm.

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2.2 Reflection measurements

Reflection measurements were made by illuminating the specimen with chopped light from a 50W quartz-iodide lamp through a planar 0.75m diameter transmissive paper diffuser mounted 0.42m above the specimen. An optical fibre with a field of view smaller than the specimen diameter sampled the diffuse reflection of the specimen from 2.5cm above the powder surface. The sampled light was filtered using a 10nm bandpass 650nm central wavelength interference filter and detected with a reverse-biased silicon photo-diode connected to a lock-in amplifier. Spectralon [8] and compressed MgO powder were used as comparative 100% reflectance standards.

2.3 Powder presentation

The materials examined included calcite, silicon carbide, garnet [9] and glass micro spheres (ballotini [10]). Calcite crystals were crushed and wet sieved [11] into relatively narrow size distributions. The calcite and the ballotini were transparent at 633nm. The silicon carbide was black and opaque at 633nm. The garnet particles were red. The particle shapes of the various materials were quite different. The crushed calcite particles were rhombic, the silicon carbide particles had many flat shiny surfaces, the garnet particles were sub-rounded to sub-angular [9] while the ballotini were approximately spherical.

Specimens were presented in 30mm diameter by 7mm deep aluminium containers. Satisfactory surfaces for measurement could not be prepared by pressing the powder surface down with a glass slide or a polished metal plate because of adhesion of the particles to the slide or plate. Similarly, drawing a straight edge across an overfilled container produced variable quality surfaces depending on the powder size and cohesion. The finer powders did not flow well, even if dried for several days in an oven at 110°C and the surfaces obtained were often variable in appearance. Coarser powders flowed well and levelled out when poured into the container. The specimen levelling technique that produced the most reproducible results was to fill the sample cup with powder and then drop the filled cup five times from a 5cm height onto a hard surface. This drop technique was used for all of the data reported here. The correlation lengths of the surface roughness of the loose prepared powder surfaces were not measured and thus it cannot be said whether the prepared loose powder surfaces met the Fraunhofer criterion [13,14] required by the Léger and Perrin theory [1].

2.4 Data collection and treatment

The usual angles of incidence and measurement used were 2° from the normal. Near-normal incidence was chosen to reduce the decorrelation rate against angular tilt (small sin(θ₁) in Eq. 1.) This allowed coarse powders to be conveniently measured with moderate δθ ′s. Between five and forty CCD frames were captured on each prepared specimen at various incremental tilt angles (δθ). A large spread of tilt angles was used for fine particle samples as the angular decorrelation rate of the speckle was expected to vary slowly with tilt. Typically the total incremental tilt angle was less than one or two degrees but for coarse powders a smaller spread of angles was used. The corresponding square sub-sections of the CCD frames in a set of speckle images were cross-correlated using a computed discrete 2D-FFT method. The best signal-to-noise ratios were obtained when the largest possible sub-matrices of the CCD frames (usually 512×512 pixels) were used. To provide compensation for any stray light reaching the CCD a background frame was collected for each set while the laser beam was blocked. The background frame was subtracted from each of the speckle frames in the set before it was cross-correlated.

The speckles in the individual frames of a set stayed well aligned between frames. The angular shift between the individual frames of a set was estimated from the time between frames and the angular turn rate of the sample on the goniometer. The processed output data set comprised a series of cross-correlation values and the respective angular tilts (referred to the first frame of the set).

Since it was possible that the intensity of the laser could have changed between the CCD frames of a set a normalisation strategy was adopted. Consider the cross correlation of frames “0” and “n” of a set of N frames. The auto-correlation of frame “0” (A₀), the auto-correlation of frame “n” (A_n) and the cross correlation of frames “0” and “n” (C_0,n) were individually calculated. Each of these A₀, A_n, C_0,n values was the respective peak correlation minus the respective correlation average “background” value. The “normalised cross-correlation” used was:

This value was calculated for all values of n from 0 to N. When the normalised cross-correlation was plotted against the sample turn angle the resulting curves were similar in shape to those of Nitta and Asakura [3].

2.5 Monte Carlo modelling

The diffuse optical reflectance (remission) of a powder bed can be described using Kubelka-Munk (KM) theory [12]. KM theory describes remission from the surface of an optically thick powder bed in terms of the optical absorption per unit length (K) and the optical scattering per surface (S). In particular the reflectance of an infinitely thick powder layer (R_∞) is related to the K/S ratio by the KM function:

One-dimensional Monte Carlo model calculations of the scattering in a powder bed were performed for various K/S ratios. In the model the absorption in a particle was linearly proportional to the particle diameter and the scattering occurred at the particle surface (not in the interior of the particle) and was invariant in magnitude with particle size. That is, the scattering modeled was such as would occur at a refractive index discontinuity and the particles were assumed to have homogenous interiors. At each particle each modeled “photon” could either pass through the particle without interacting or else could interact with the particle. If the “photon” interacted it could scatter from the particle or else could be absorbed by the particle. Scattering simply reversed the direction of travel of the “photon”. The sum of the modeled probabilities of scattering, being absorbed and passing through unaffected was unity. The type of each event was selected in the usual Monte Carlo manner using a pseudo-random number sequence generator.

The input parameters to the Monte Carlo model were the particle size, the bed thickness, the absorption and scatter values, the number of “photons” to use and the maximum number of interactions for which the fate of each “photon” would be followed. Each run of the model assumed mono-sized particles all with identical absorption and scattering parameter values. In practice the number of followed interactions was increased until the number of “photons” which had not exited the model or been absorbed was negligible. The bed thickness was, in practice, increased until no “photons” leaked out of the back of the model.

A hundred thousand “photons” were used for each Monte Carlo run of each modeled point in Fig. 5. Each “photon” was followed until it was lost by either exiting the back of the model or by being absorbed or else was remitted out of the entrance of the model. A running total of a “photon’s” absolute travel path length was kept while the “photon” was “alive”. If a particular “photon” was remitted its absolute travel path length was added to the sum total absolute travel path for all previously remitted “photons” for that particular Monte Carlo run. When the supply of “photons” was exhausted the grand total sum of the path lengths of all of the remitted “photons” was divided by the number of remitted “photons”. The calculated quotient was thus the quotient of the grand total sum of the “photon” path lengths divided by the number of remitted “photons”.

3. Results and discussion

3.1 Color and speckle decorrelation

All powders were presented dried. The samples were prepared by the drop method. The water-white calcite was transparent and showed a tight correlation between mean sieve size and angular decorrelation rate. (see Fig. 2.) Sieved silicon carbide (opaque black commercial abrasive) was measured in a similar fashion to calcite. It was very clear that the decorrelation rate for sized opaque silicon carbide was very much lower than that for equivalently sized transparent calcite.

 

Fig. 2. Decorrelation rate as a function of particle size for calcite and silicon carbide.

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The effects of transparency on decorrelation rate were further investigated using heterogeneous mixtures of various proportions of dyed ballotini with similarly sized transparent ballotini and using dyed ballotini with various tints. The 70µm (d₅₀) diameter transparent ballotini flowed freely and it was easy to prepare samples with a surface that looked flat. A quantity of 70µm ballotini was coloured using ethanol-soluble purple dye of the type sold for marking out metal surfaces. The dyed ballotini was dried. Several mixtures were prepared of the natural and dyed ballotini, in various proportions by weight. The decorrelation rates were measured for those samples using a 633nm wavelength HeNe laser. When purple dyed ballotini was mixed in with clear ballotini the decorrelation rate decreased substantially. The mixed samples showed decorrelation rates between those of 100% clear and 100% purple samples with the clear sample showing the highest decorrelation rate. (see Fig. 3.) Presumably the photon path length is more variable in a more transparent powder. Modeling (see Sec. 3.2 below) certainly indicated that the mean path length of remitted photons was longer in transparent powders than in opaque powders.

Further decorrelation measurements with a 633nm HeNe laser on three sizes of garnet abrasive powder (75µm, 80µm and 196 µm) showed that the decorrelation rate varied with particle size in a manner consistent with that seen for calcite and silicon carbide. Léger and Perrin [2] (Eq. 1.) indicate that using a green laser on a metallic surface would produce a faster decorrelation than a red laser because the wavelength is shorter.

 

Fig. 3. Cross-correlation versus incremental tilt angle for mixtures of clear and dyed ballotini. The percentage of purple ballotini in each mixture is shown by weight.

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However when 514nm Ar⁺ laser light was used on the semi-transparent red garnet powder it was found that, in contrast to Léger and Perrin [2], the speckle decorrelation rate was much slower than for red 633nm laser light. (see Fig. 4.) Similarly the decorrelation rate for garnet using 633nm laser light was much slower than for the transparent calcite powder of the same nominal size. These results demonstrate that the application of speckle decorrelation to determine particle size in loosely packed powder beds should take account of the particle transparency at the wavelength used.

 

Fig. 4. Cross-correlation versus incremental tilt angle for coloured and transparent powders of the same nominal size.

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3.2 KM function, path length and decorrelation rate

When the KM transformed Monte Carlo modeled remission was plotted against modelled K/S (not shown) the expected straight-line correlation was obtained. The modelled path length of remitted photons increased with increasing particle size. It was also noticed that, if the logarithm of the path length of remitted photons was plotted against the logarithm of the KM function, a good linear correlation was obtained. (see Fig. 5.) The data is well fitted by Eq.4.

Where a is the scale factor and b is the power factor. See Table 1.

Table 1. Regression fits of Eq. 4. to Monte Carlo data of Fig. 5.

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Fig. 5. MC Calculated mean path lengths of remitted photons in powder beds versus the KM function of the absorption to scatter ratio.

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In each calculated case the power factor was close to -0.5 and the scale factor was proportional to the particle size. In the one-dimensional model used, the scatter could potentially occur at each particle and so, the fits for two powders whose size differed by a factor of two had fitted scaling factors differing by a factor of two. The model indicated that the number of remitted photons (n) travelling a path length (p) in the powder varied with K/S as expected:

It is easy to demonstrate that the mean path length of remitted photons with such an exponential path length distribution varies with K/S as (K/S)^-0.5 which is in essential agreement with regression fits to the MC calculations (see Table 1). The model showed that, when the powder bed was thick and semi-transparent, the relative particle size could be determined from the scale factor if both the path length and the KM function were known simultaneously.

If the KM function is very small and the bed is not infinitely thick the Monte Carlo model predicts that the path length of the remitted photons will asymptotically approach a value of twice the bed thickness as expected. When the KM function is large the model assumption that the surface of the bed is perfectly flat implies a zero mean path length for the remitted photons. In reality at large values of the KM function the mean photon path length would asymptotically approach a value related to the actual surface roughness of the powder bed.

The Monte Carlo trends were investigated experimentally. The surfaces of large numbers of individual particles of ballotini were dyed to various uniform tints of blue to change their K/S. The MC model assumed that each particle was point-like and had scattering and absorption properties. Thus the experiment with superficially dyed particles adequately matched the model in that respect. The experimental data on decorrelation rate and KM function demonstrated similar parallel line behaviour to the Monte Carlo calculations. Both the speckle decorrelation rate and the diffuse reflectance at effectively infinite thickness R_∞ were measured.

The KM function was subsequently calculated from the diffuse reflectance (see Fig. 6). The experimentally fitted scaling factors varied as the particle size. The fitted power factor (b) was approximately -0.5 in each case (see Table 2), suggesting that variations in photon path length with K/S were driving the observed correlations shown in Fig. 6. It would have been preferable to perform the experiment with volume tinted particles (darkened by intense gamma radiation in an irradiation pond perhaps) but no suitable samples were available with well-matched shapes and size distributions.

 

Fig. 6. Experimental data on decorrelation rate and KM function of dyed ballotini of various sizes demonstrated similar parallel line behaviour to the Monte Carlo calculations.

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Table 2. Regression fits of Eq. 4. To the experimental data of Fig. 6.

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The variation in decorrelation rate with angle of incidence was examined. The theory of Léger et al [1] for metal surfaces indicates that the decorrelation rate should increase with increasing incidence angle (θ₁ in Eq. 1.). Opaque SiC powder viewed over a range of θ₁ of 3–13 degrees showed decorrelation rates varying by nearly four fold in accord with the metallic surface theory used to obtain Eq. 1. (see Fig. 7.) In contrast no significant variation in the decorrelation rate with incidence angle was found for calcite powder or red garnet (within the experimental error). This demonstrates the limitations of applying a metal surface theory to semi transparent powders. Further, no variation in decorrelation rate with incidence angle was seen when the red 80µm garnet was examined using a red laser.

 

Fig. 7. Decorrelation versus sin(θ₁) for opaque and transparent powders. Note that the fitted regressions are each forced through zero to hi-light the dependency of the decorrelation technique on the transparency of the material studied.

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The decorrelation behavior of opaque powder with incidence angle was thus in essential agreement with the theory of Léger et al [1]. This is not surprising when it is remembered that their theory was developed for metal surfaces and that photons hitting an opaque surface do not penetrate the sub-surface structure. In contrast, photons may travel a considerable distance into a transparent powder bed before being remitted. Their paths and the variations in their paths could reasonably be assumed to be sensitive to the sub-surface structure of the powder bed and thus comparatively insensitive to the actual surface shape.

4. Conclusions

The angular speckle decorrelation rates of beds of transparent powders of a given particle size are much faster than those of opaque powders of a similar size. Care should be exercised when applying the angular decorrelation theory presented by Léger et al [1] and Léger and Perrin [2] to transparent powders.

An improved experimental estimate of the relative particle size can be derived from combined decorrelation and reflectance measurements. Monte Carlo modelling indicates that it may be possible to compensate for transparency variations and arrive at a better estimate of the particle size by also measuring the diffuse reflectance of the surface at that wavelength. The scaling factor of power law curves fitted to the decorrelation rate and KM function is a more accurate predictor of relative particle size than the raw data. Experimental data demonstrates an approximate -0.5 power law fit for the powders measured, in agreement with the Monte Carlo modelling. Further work is needed to determine whether the best power fit is always -0.5. It may be valuable to study the speckle decorrelation of light scattered from glass porous sinter disks especially since it is theoretically possible to build a three dimensional map of the position of every grain of glass in the sinter by grinding the surface down and photographing each successive surface. The map could then be used with a large computer simulation model to generate the speckle developed by transparent assemblies of particles and to compare the modeled speckle with experimental results.

When speckle decorrelation is used to estimate the surface roughness of powders of varying tint the transparency effect needs to be compensated and the angle of incidence is important. The same considerations might apply when speckle decorrelation is used to estimate the surface roughness of rocks but no measurements were performed to investigate any such possible effects.