Introduction
Some physical properties of any granular material depend on the average coordination number of grains in the material (Reference GublerGubler, 1978) but coordination number has proven a difficult quantity to measure. (Here, coordination number is defined as the number of grains in direct contact with a given grain; the surface of contact between grains is a grain bond.) The most accurate way to measure coordination number is by careful examination of closely spaced serial sections of a material. This method is so time-consuming, however, that it is generally impractical. It is faster but less accurate to estimate coordination numbers from measurements taken on a single section plane. Several methods have been proposed for doing this but all suffer from serious flaws. Here, we briefly review these previously published methods and then present a new method for estimating three-dimensional coordination number from measurements on a single plane of section.
Symbols Used
Previous Methods
All of the methods for estimating three-dimensional coordination number, n 3, from measurements on a planar cross-section involve the use of certain simple counting measurements and certain assumptions regarding the size, derived quantities used in such methods are given above. The counting measurements and simple derived quantities above (A m, β, E, n 2, f 2(n 2j), N A, N Ab, N Lb, N Lf) require only that analysis be conducted on a random plane in an isotropic material or that averages be computed over all directions in an anisotropic material. All other quantities given above depend on assumptions regarding geometry in the material, and these assumptions generally cannot be tested rigorously. It is thus important that the method selected for calculating n3 be insensitive to deviations from assumed geometry. Further discussion of both basic measurements and derived quantities can be found in a number of sources, including Reference UnderwoodUnderwood (1970), Reference KryKry (1975), and Reference Alley and BentleyAlley and others (1982). The measurement of n 2 and its use as a qualitative indicator of firn densification has been discussed byReference Fuchs Fuchs (1959) and Reference Ebinuma and MaenoEbinuma and Maeno (1985). We will present only a sketchy development of this material here.
Calculation of n 3 for planar, circular grain bonds of uniform size between mono-sized spheres was considered by Reference UnderwoodUnderwood (1970, p. 102). He showed that
where N Lb is the number of intersections per unit length between randomly oriented test lines and grain bonds on the plane of section, N Ab is the number of grain bonds per unit area intersected by the plane of section, Nvb is the number of grain bonds per unit volume, and N v is the number of grains per unit volume. The quantity N v is strongly dependent on the grain shape. For spherical grains,Reference Underwood Underwood (1970, p. 96) showed that
where N A is the number of grains per unit area intersected by the plane of section and R is the average grain radius. The variation of N v with grain shape has been discussed by Reference Dehoff and RhinesDeHoff and Rhines (1961), who showed that relatively small shape variations from sphericity can lead to errors in N v in excess of 100% if Equation (2) is used. Although Reference Dehoff and RhinesDeHoff and Rhines (1961) derived equations equivalent to Equation (2) for a variety of shapes including prolate and oblate spheroids of arbitrary axial ratio, they assumed that all grains in a material have the same shape. This requirement of shape constancy is not met in many real materials, so that estimates of N v should be considered inaccurate. We thus followReference Kry Kry (1975) in considering that any estimate of nS based on Nv should not be used to draw quantitative conclusions.
The assumption of uniform bond size used in Equation (1) was relaxed by Reference FullmanFullman (1953) and Reference Kry Kry (1975), who showed that
where E is the harmonic mean of the lengths of intersections of bonds with the plane of section. Again, this method yields excellent results if and only if N v can be evaluated accurately. Considering the uncertainty in N v, Reference KryKry (1975) went so far as to suggest that this method may yield little better than order-of-magnitude estimates of n3. Natural variation of n3 is little more than an order of magnitude, so this is not too useful.
A slightly different approach was adopted by Reference AlleyAlley and others (1982), who estimated that
where C’ is a shape factor (discussed below), A m is the average area of grains observed on a plane of section, β is the fraction of grain surface occupied by bonds, N Lf and N Lb are respectively the number of intersections per unit length of randomly oriented test lines with ice–air surfaces and with grain bonds, a is the average area of bonds, and N vb is given in Equation (3). (Note that the equation for β inReference Alley Alley and others (1982, p. 9) is in error; the correct expression is given above.) The estimate of a is from Reference FullmanFullman (1953) and assumes circular, planar bonds; this assumption of bond shape was tested by Reference KryKry (1975) and Reference Dehoff and RhinesAlley (paper in preparation), and seems to be an accurate approximation. The factor (6C’ A m) in Equations (4) is the average surface area per grain, and the shape factor C’ corrects for deviations of grains from spherical form and for grain-size distributions. The shape factor was taken to be identically 1 by Reference AlleyAlley and others (1982) but it can vary significantly with grain shape (Reference UnderwoodUnderwood, 1970, p. 90-93; Table I ). The shape factor is not known a priori for any real material and is thus the major source of error in this method. This method may be more accurate than those relying on N v but less accurate than we would like.
Values of Shape Factor Equation (4), from Underwood C’ in(1970, P 90 -93)
An interesting approach developed by Reference GublerGubler (1978) for study of seasonal snow leads to an estimate of the distribution of coordination numbers f 3(n 3j). It was shown by Reference GublerGubler (1978) that any assumed distribution f 3(n 3j) can be used to predict a distribution function of coordination numbers in a plane of section, f2(n 2j, p, i), that depends on a tunable parameter, i, and on the probability, p, that a random cut through a single grain with one bond will intersect that bond. The probability, p, depends on average grain radius, R, average bond radius, r, the number of grains per unit volume, N V, an empirically evaluated constant, and a shape factor that must be estimated. To calculate three-dimensional coordination, p is calculated, trial values of f 3(n 3j) and i are selected, and the two-dimensional distribution function f 2(n 2j,p, i) consistent with the trial values is calculated. Then f3(n 3j) and i are adjusted until the predicted f 2(n 2j, p, i) matches the observed two-dimensional distribution f3(n3j) as closely as possible. The trial distribution f 3(n 3j) that produces the closest match is then the best estimate of the actual three-dimensional distribution of coordination numbers. This method requires that (r/R) « 1, which is realized in the snow studied by Reference GublerGubler (1978) but not in most firn (Reference AlleyAlley and others, 1982; paper in preparation by R.B. Alley). Although this method may prove valuable in the study of seasonal snow, the use of a tunable parameter, a shape factor, an empirical constant, and the uncertain quantity N v, and the requirement of small bonds, render it suspect for application in firn. The use of observed two-dimensional coordination numbers to estimate three-dimensional coordination is an excellent idea, however, and we adopt it in developing our new model.
Model
Stated briefly, we have developed a transfer function from the average two-dimensional coordination number on a plane of section, n 2, to the average three-dimensional coordination number in the material, n3, based on the average probability that a plane of section will intersect a circular, planar bond on the surface of a spherical grain. No solution is available for the exact geometry of bonds on grains, so we construct limiting cases and choose their average value for our transfer function. The transfer function is
where Γ, the average fraction of bonds intersected by a plane of section through a sphere, depends only on α, the ratio of average bond radius, r, to average grain radius, R.
Consider a spherical grain of radius R with a planar, circular grain bond of radius r. Clearly, for r < 0 the grain cannot be a true sphere. The actual geometry is limited by the cases of inscribed bonds (Fig. 1a) and tangential bonds (Fig. 1b). We model each case by calculating rnl for inscribed bonds and Γtan for tangential bonds, and then take r to be the average of the two.
Limiting geometries for grain bonds, (a) Inscribed bonds, (b) Tangential bonds.
Geometry for inscribed grain bonds, with plane of section near center of grain.
We begin with the inscribed case, which is shown in detail in Figure 2. Bonds are assumed to be circles of radius r distributed randomly over the grain. A section cut at position z will intersect all bonds with centers falling between z’ and z” (see Fig. 2). The region between z’ and z” contains surface area of the sphere 2πR(z’ –z”) out of total surface area 4 πR 2; thus, the cut at z intersects fraction F1 of the total bonds on the grain, where
(The spatial distribution on a grain of bonds of non-zero size cannot be truly random; however, we require only that, on average, fraction F 1 of the surface area of a grain contains fraction F 1 of the bond centers on that grain, which is realized if bonds lack a preferred orientation.)
From the geometry of Figure 2
Equations (7) can be re-written in terms of sines and cosines of ψ> and ϕ using standard trigonometric identities, which can then be re-written in terms of r,α, and z. This leads to
We have ignored one complication thus far, however. When ψ‹ ϕ a cut at z will not intersect the grain at all if there is a grain bond centered between (ψ – ϕ) and (ϕ – ψ), as shown in Figure 3. This region has area 2πR(R– z’). If a cut with ψ ϕ intersects the grain, then there are no bonds centered in this region. In this case, the total area sampled by the cut is still given by 2πR(z’ – z”), but the total area over which bonds can occur is 4πR2 – 2πR(R – z’). Thus, for ψ ‹ ϕ, a cut at z samples F2 of the total bonds, where
Geometry for inscribed grain bonds, with plane of section near center of grain.
By direct geometry, we can show that when ϕ = ψ, z = R√1–α2. This allows us to write the fraction of bonds, F, intersected by a cut at z, as
Because of the spherical symmetry of the problem, the average fraction of bonds sampled by a cut through a grain, Γins, is simply the average of F over all z along any diameter of the spherical grain. Thus
The first integral in Equation (12) can be evaluated directly, and Equation (12) becomes
The remaining integral can be evaluated numerically without great difficulty and is a function of a only.
We should note here that the second integral in Equation (12) is a small correction term except at large α. Had we ignored this term and evaluated the first integral from –R to R, the result would have been
Equations (13) and (14) differ by only 8% for α = 1.0, by only 0.2% for α = 0.7, and are identical to four significant figures for α = 0.5; thus, Equation (14) could be used in place of Equation (13) in most cases.
Next, consider the tangential case shown in Figure 4. The tangential bond to a sphere of radius R is equivalent to an inscribed bond in a sphere of radius R√1+α2. The tangential case with relative bond size α thus behaves like the inscribed case with bond size α ’, where
Geometry for inscribed grain bonds, with plane of section near center of grain.
and
The true r for a given grain lies between rins and rtan’ so we choose as the best estimate
Values of Γ(α) are listed in Table II, together with the relative difference between Γ and Γins or Γ tan. Based on experience, we do not expect natural values of α to exceed 0.7 commonly, so Table II shows that the maximum uncertainty introduced by choosing Γ rather than Γins or Γtan is less than 10%. In most cases, the error introduced by using Γ should be very small.
Γ Versus a from Equation (17), Andmaximum Relative Error in Γ from Assumed Shape
Equation (17) and Table II represent our best estimate of Γ. If we use the approximation in Equation (14), then we can estimate Γ as
This is an excellent approximation of Equation (17) except at large α.
Calculations
This method requires that r and R, the average radii of bonds and grains, be known so that α can be calculated. For circular bonds that may exhibit a bond-size distribution, Reference FullmanFullman (1953) showed that
where E is the harmonic mean of the lengths of inter- sections of bonds with the plane of section. For mono-sized spherical grains, it is not difficult to show that
where Rm is the average of individual radii of grains seen on the plane of section. This may be a good estimate for non-spherical grains of different sizes (Reference MendelsonMendelson, 1969), although this is difficult to demonstrate rigorously. Methods for determining R from measured intercept lengths were also discussed by Reference MendelsonMendelson (1969). (We recognize the imprecision introduced by not treating explicitly the effect on R and α of a distribution of grain shapes and sizes, but grain-size itself is not a well-defined quantity unless all grains have a specified shape (Reference UnderwoodUnderwood, 1970); we are continuing to investigate this problem.) Once r and R are known, then α is calculated from
In many cases, it is easier to calculate α from r’ and R’, which are calculated from average areas. For circular bonds of different sizes, Reference FullmanFullman (1953) showed that the average bond area, a, is given by
The true average cross-sectional area of grains, A, is related in some fashion to the measured cross-sectional area on the plane of section, Am. For mono-sized spherical grains, A is given exactly by
This is a slight overestimate of A for a sample consisting of spherical grains of different sizes and a slight underestimate of A for mono-sized, non-spherical grains, and so should be a good estimate for non-spherical grains of different sizes (paper in preparation by R.B. Alley). Then
Empirically, we find that Equations (24) and (21) differ by less than 5% in most cases, so the investigator should choose the more convenient.
Discussion
Because of the near-impossibility of learning n 3 exactly in a real sintered material, we cannot provide a rigorous test of our new method for calculating n3. As discussed below, however, several factors recommend our method, including its insensitivity to deviations from assumptions, its independence from shape factors or tunable parameters, its accuracy, and its ease of computation.
No real system will match exactly the geometry assumed in our derivation. A grain can deviate from a spherical form toward some other convex form or toward a non-convex form. In either case, however, the deviation will affect in a similar manner both the total surface area of the grain and the average surface sampled by a plane of section; thus, Γ will vary more slowly than any single measure of grain shape. As an extreme example, the value of r for a thin disc of radius R is only about (4/π) times Γ for a sphere of radius R, if r is held fixed.
As we discussed above, some other published methods of calculating n 3 rely on shape factors or tunable parameters that cannot be estimated well. Our new model does not require any of these and so is preferable. (Deviations from sphericity could be corrected by a shape factor but the slow variation of r with grain shape allows us to set this shape factor to 1.)
A major objection to other methods of calculating n3 is their reliance on quantities, particularly Nv, that cannot be determined accurately. Our method depends on n2 and on ∝, the latter does depend on a number of measured and calculated quantities. However, both experience and theory (Reference AlleyAlley and others, 1982; paper in preparation by R.B. Alley) indicate that α (and thus Γ) varies by a factor of 2 or less after a bonded structure is developed. Most variations in n3 thus result from variations in n 2. The value of n2 can be determined with considerable accuracy. In typical firn of density 0.55 Mg m-3 from “Upstream B” on the Siple Coast of West Antarctica, a t-test on a count of 100 grains typically yields n 2 = 2.5 ± 0.17 with 90% confidence. Counting more grains would narrow the confidence interval further. We believe that the total accuracy of our method is better than 20%, although we cannot demonstrate this rigorously. (For values of α less than 0.1, difficulty in recognizing a contact and a large standard deviation on n 2 will decrease the accuracy.)
Finally, our new method allows easy computation. Although r and R can be determined only after substantial effort, they are frequently of interest in their own right. Once r and R (or r’ and R’) are known, our model requires only that n2 be measured and Γ determined from Table II or Equation (18), and these substituted into Equation (5). The total time required for measurement and computation after r and R are known is typically 10–15 min.
As an exercise, we compared n 3calculated using our new method with n 3 calculated after Reference AlleyAlley and others (1982) for 20 samples from “Upstream B” on the Siple Coast of West Antarctica. Values of n3, are plotted in Figure 5. If we take C’ = 1 following Reference AlleyAlley and others (1982), then n 3 ’ n 3 in every case considered. If we assume that n 3 from our new method is exact and calculate C’ for the samples, we obtain a mean value of C’ = 1.29 with a standard deviation of 0.15. Actual grains in shallow firn probably range from spheres to prolate spheroids and become more like truncated octahedra (tetrakaidecahedra) or dodecahedra with increasing depth; also, actual grains probably have some surface irregularities which would tend to increase C’. In the light of these considerations, Table I shows that C’ = 1.3 is a reasonable value, which tends to lend credence to our model. We emphasize, however, that C’ need not be the same in different samples and cannot be known a priori.
Values of n3 versus density for ‘ UpstreamB ” on the Siple Coast of West Antarctica. Values of n 3 for anisotropic firn were obtained by averaging values for horizontal and vertical sections from the same sample.
We also tested our new model against n3” from Reference Dehoff and RhinesFullman (1953) and Reference Kry Kry (1975), assuming spherical grains. Results show some variability but in general ns” is 10–20% less than n 3. Grains are not spherical in real firn but resemble prolate ellipsoids. Agreement betweenn 3 ” and n3 would be improved significantly if we assumed the grains to be prolate ellipsoids of axial ratio 0.9 (Reference Dehoff and RhinesDeHoff and Rhines, 1961). This is a reasonable value based on observation but cannot be derived readily from measurements.
Conclusions
We have presented a new method for calculating n 3 , the average three-dimensional coordination number in a granular material, from stereological measurements on a section plane. The model is computationally simple, accurate, insensitive to deviations from assumptions used in its derivation, and requires no shape factors or tunable parameters. Differences between results from our new model and previous models are explicable based on known weaknesses in the previous models. Thus, we believe that our model provides a useful way to estimate ns. We now are using data on n 3 to study densification processes in firn (Reference Alley and BentleyAlley and Bentley, in press).
Acknowledgements
Financial support for this work was provided by the U.S. National Science Foundation under grant DPP-8315777. We thank C.R. Bentley, J.F. Bolzan, J.H. Perepezko, H.F. Wang, and I.M. Whillans for helpful suggestions and A.N. Mares and S.H. Smith for manuscript preparation. This is contribution No. 450 of the Geophysical and Polar Research Center, University of Wisconsin–Madison.
