This is a brief article that I wrote together with our VP and which is to appear in International Cement Review - an industry concerned with all things cement-y. It hasn't much to do with oceanography, but it has everything to do with particles!

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The particle size distribution (PSD) of a cement powder is important for its complete characterization. From the PSD it is possible to compute the surface area of the cement powder, which has a close correlation to the rate of cement hydration. However, no international standard for measuring the PSD of cement powder has been agreed upon, and consequently a range of techniques are being used, with laser diffraction being the most common. A major inter-comparison study was completed by the US National Institute of Standards and Technology in 2004 .

Broadly speaking, laser diffraction takes advantage of the fact that forward scattering of light is principally due to diffraction, or in an intuitive way, light that went ’around the particle’. This diffracted light is not affected by the composition of the particles, only its size and shape. A small part of the light is additionally transmitted through the particle and shows up at angles beyond the first two principal diffraction peaks. This ‘transmitted’ fraction of light is affected by composition. Thus the choice of a refractive index affects the ‘large’ angle scattering, whereas the size alone affects the ‘small’ angles region of the scattered light field.

For spherical particles, each particle has its own characteristic scattering pattern, which can be computed using the so-called Mie theory. This is a completely general solution to the scattering problem for spheres, requiring only two inputs: particle diameter and the complex refractive index. Using Mie theory it is possible to compute the scattering pattern for an arbitrary size range of spherical particles.

In a suspension of particles of many sizes, the total scattering pattern is the sum of the scattering patterns that arise from each of the individual particles in the suspension. Mathematically, this can be described as:

E = K * C

Where E is a vector containing the intensity of the diffracted light in a set of solid angle ranges, K is the so-called kernel matrix and C is the volume concentration of the particles creating the diffraction pattern, i.e. the PSD. K contains information on scattering by particles of various sizes into different angles, e.g. from Mie theory. If E is measured and K is known, then C can be obtained via matrix inversion:

C = K-1 * E

A typical laser diffraction instrument consists of a laser beam that illuminates particles, and receive optics that place scattered light on a set of concentric ring detectors. Each ring detector measures scattering into a particular, small angle range. Together, the ring detectors cover a large dynamic range of angles over which the scattered light is sensed; this angle range establishes the range of size of particles that can be analyzed. There are a large number of instruments now on the market. A new version of these is made by the authors’ company, and remains the first and only truly portable laser diffraction particle analyzer (Figure 1). It incorporates 32 detector rings, covering a 200:1 angle range. The size range covered by this device spans 1.25-250 or 2.5-500 microns.

Figure 1: LISST-Portable instrument for laser diffraction of particles in liquid suspension.The LISST-Portable instrument is provided with the option to invert angular scattering data into equivalent sphere size distribution, or into a ‘random shape’ particle size distribution. How and why this was done is explained below.

The scattering kernel, K , can only be computed if it is assumed that the particles are spherical, although lately progress has been made that enables the computation of matrices for non-spherical but regular geometrical shapes, and even irregular shapes but with limitations from computational complexity. Very few particles resulting from natural or industrial process are spheres or regular. Grinding and milling typically produces particles of a general random shape, with numerous pits and edges on their surfaces. Consequently the diffraction pattern arising from these natural, random-shaped particles can be expected to be different from spheres. Since it is not theoretically possible today to properly model the scattering from random-shaped particles we have taken an empirical approach and determined the corresponding kernel matrix, K . We have used this empirical matrix to study how the retrieval of the PSD of random shaped natural sediment particles is affected by the implicit assumption that the scattering particles are spherical, which is the current industry standard.

To construct our empirical kernel matrix K , we used our LISST (Laser In Situ Scattering and Transmissometry) laser technology to measure the scattering pattern from a range of random-shaped particles, pre-sorted into sieve size bins that were equivalent to the 32 size bins covered by the LISST–Portable instrument (2.5-500 µm for the instrument used here). We then compared these scattering patterns to those from spherical particles with the same sieve size. Sieving is only possible for particles larger than approximately 16 µm, so in order to separate smaller particles into size bins all the way down to 2.5 µm we used a density stratified settling column of known viscosity. Stratification kills all turbulence in the column and ensures that the particles settle without being affected by convection currents in the column. Thus, combining sieving and settling techniques, an empirical kernel matrix K was constructed that when used to invert observed scattering, yields size distribution of random shaped particles.

To illustrate the differences between scattering by spheres and random shaped particles, Figure 2 shows comparisons of scattering patterns from spherical and random shaped particles in two narrow size ranges: 25-32 µm and 75-90 µm. It can be seen that the peak of the scattering intensity of the random-shaped particles is displaced one to two detector bins to the left, relative to the spherical particles. Because shifting of diffraction pattern to the left implies larger size, this means that the random shaped particles, with a sieve size equal to that of spherical particles appear as if they are one size bin (18%) coarser than they actually are.

Figure 2. Measured scattering from spheres and random shaped particles, both with a sieve size of 25-32 µm (left) or 75-90 µm (right).The implications for this are that when a laser diffraction measurement is processed, and it is assumed during processing that the particles are spherical, the resulting size distribution becomes too coarse, by approximately one or two size bin, which is 18-36% for the LISST instruments. In other words, shape effect alone implies that non-spherical particles, such as cement powders are actually 18-36% finer than reported using a Mie theory based standard laser diffraction system .

The second effect to note is that of refractive index. Cements have been analyzed with assumptions of absurdly high imaginary parts of the index, ranging from 0.1 to 1. High imaginary index implies blackness. Carbon black has an imaginary index near 0.1 so this is clearly not an appropriate value for cements. High imaginary index kills the part of light transmitted through the particle. In effect, it forces Mie theory to pure diffraction. In reality, the light that is transmitted through particles and is then actually measured, is thus interpreted to originate due to fine particles that did not exist in the suspension; in other words, assumption of high imaginary index invents fine particles.

In figure 3, we have analyzed Portland cement powder standard 114q obtained from NIST on our LISST instruments using both a spherical and a random shape particle matrix. Table 1 summarizes the D10, D50 and D90 for the two size distributions. Our spheres model employs Mie theory with assumed real index of 1.5 and 0 for the imaginary part.

Figure 3: A comparison of the size distribution obtained with an inversion with kernel matrix for spheres and for random shaped particles.From Figure 3 and Table 1, it is clear that using a kernel matrix for random shape particles for the cement powder causes the size distribution to narrow slightly and move to the left, i.e. most of the size distribution becomes finer. We have noted that our research has shown that in general D50 (the median particle size) becomes finer by approximately 25%, which corresponds to one to two size classes for the LISST instruments. This is demonstrated here. Sequoia is the first and only company in the world to have established a laser diffraction processing protocol based on the scattering pattern of naturally random-shaped particles.

D (µm ) | Spherical matrix | Random shape matrix |
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D10 | 5.6 | 6.9 |

D50 | 16.7 | 15.0 |

D90 | 36.9 | 28.3 |

Table 1. Comparison of the 10, 50 and 90 percentiles for Portland cement using the two different matrices for inversion.To conclude, we have revisited the NIST standard 114q with an instrument that uses two advancements: an inversion for random shaped particles, and an implicit refractive index that is not as extreme as past industry practice. The new results suggest, first, that the median sizes of cements are actually a little finer than the equivalent spheres result reported in the past. And, second, that the fines reported in the past may be exaggerated by effects of both, particle shape and refractive index. Further details of this work will shortly be published elsewhere.