Estimation of Mixing Index and Contact Number by Coordination Number Sampling in an Incompletely Mixed State t

A statistical model of the distribution of the contact number for a binary mixture in an incompletely mixed state is introduced. The model is derived from the beta binomial model proposed by Yoshizawa and Shindo 3 J. The mixing index based on the contact number is defined and described by only one parameter in this model. The precision of the estimation of the mixing index, based on the contact number by coordination number sampling in an incompletely mixed state, has been derived and simulated on a computer.


Introduction
Many of conventional solid mixes indicates their mixing indexes based on the variance of target components among spot samples. However, the variance is macroscopic and cannot provide any definite physical meaning, though it is very important statistically. Thus, it is difficult to get a definite meaning of variance when a mixing index is applied to the microscopic analysis of a reaction rate in a solid reaction.
From the above viewpoint, Akao, Fan and others have proposed the representation of a mixing index with a contact number. The purpose of this is to estimate the number of contacts in a mixed substance to represent the mixing index by a sampling method, called coordination number sampling, which measures the number of target component particles in contact with specific component particles extracted randomly. The mixing index based on the contact number, which is both microscopic and geometric, can provide a definite physical meaning. t This report was originally printed in J. Soc. Powder Tech· nology, Japan, 24, 521-526 (1987) in Japanese, before being translated into English with the permission of the editorial committee of the Soc. Powder Technology, 1 apan.
KONA No. 7 (1989) Yoshizawa and Shindo 3 ) have proposed a model based on the ,(3-binomial model to describe any mixed state of a binary mixture and investigated the precision of the estimation of the Lacey's mixing index 4 ) in an incompletely mixed state as well as the relationship of the model with the Polya urn scheme 5 ).
We found the distribution of the contact number in an incompletely mixed state, upuntil-now still unknown, with the same model configuration as that of Yoshizawa and Shindo, discussed about an estimation method for the population contact number and investigated the validity of the theory through computer simulation. The following is a description of our findings.
2. Distribution of contact number and its mixing index 2. 1 Completely segregated state and completely random state Now, we consider a mixture of a two-dimensional regular array consisting of two kinds of equal diameter particles, A 0 and A 1 • Their population concentration is assumed to be X 0 and X 1 (=1 -X 0 ), respectively. A particle extracted randomly from the mixture is called a sample particle and the total number of particles in contact with the sample particle is called the total coordination number n*. Total coordination number n*, which is determined by the array structure of the mixture, is 4 for the square array and 6 for the hexagonal array. When the sample particles are restricted to A . 0 particles, and Ao particle is called the specified particle. The number of A 1 particles in contact with specified particle A 0 is defined as the contact number (see Fig. 1 ). The sampling on the basis of the definition, is called the c~ordina tion number sampling of size n *.
The random variable indicating the contact number and the realized value are denoted by C, (o) and c, (o), respectively. In the completely segregated state of the particles of two kinds it is clear that the contact number C 1 (o) can 'be expressed by the one-point distribution shown below, with the boundary ignored: In the completely random state, the contact number C, (o) follows the binomial distribution as thus, as is well known, the expected value, l1r and the variance, a/, are expressed as follows: Now focusing on the changes in the expected values in Eqs.
(2) and (5), we define mixing index M of the two kinds of particles with the contact number as shown in the equation below: where 11 is the expected value of the contact number in any mixed state. Therefore, contact number mixing index M is 0 in the completely segregated state and I in the completely random state. In addition, the mixing index can take an exclusive value between 0 and I because 0:::;: 11 :::;: n* X,.
While conventional mixing indexes such as Lacey's mixing index 4 ) is based on the variance or secondary moment, contact number mixing index M is based on the expected value or primary moment, as known from Eq. (7).

3 Incompletely mixed state
In an incompletely mixed state, the local concentration of the target particles is largely non-uniform and the concentration of the entire mixture is not capable of being uniform. Thus, it should be considered that the local concentration itself is a random variable following a certain distribution. Therefore, we assume the random variable z of the local concentration and the probability density function f(z), which has the {3 density with parameters T and X 1 as expressed by : where Tis the positive parameter and B( • , ·)is the {3 function.
Now, under the condition that local concentration Z is known, the probability that the sample particle is A 0 and the y particles in the n* particles around the sample particle are A 1 particles is expressed by the equation, where Y is the random variable against y. Therefore, Pr [ Y = y, A 0 ] can be expressed by the equation below as the marginal probability of the simultaneous probabilities in Eqs. (8) and (9): This gives us the probability that they A 1 particles are in contact with the sample particle which is assumed to be an A 0 particle to be expressed by the following equation: the distribution presented by Eq. (11) indicates the contact number distribution in an incompletely mixed state.
In contact number distribution Eq. (11 ), i-th where n*U) = n*(n*-l)···(n*-i+ 1). This enables expected value 11 and variance a 2 of the contact number C 1 (o) in an incompletely mixed state to be determined by the following equations, respectively: The limits of the expected value and variance when parameter T is made close to zero or infinity are, lim /1 = n * X 1 = /1 r lim a 2 = n * X 1 (1 -X 1 ) = a'/ The changes when parameter T is altered from zero to infinity corresponds to the status that the distribution of contact number C 1 (o) moves gradually from the one-point distribution of the completely segregated state in Eq.
(1) to the binomial distribution of the completely random state in Eq. (4), as shown in Fig. 2. Substituting the expected value (13) of contact number C 1 (o) in an incompletely mixed state for the mixture defining Eq. (7), we come up with the following equation associating mixture M with parameter: This relationship coincides with the relationship between Lacey's mixing index and the parameter r in the {3-binomial model proposed by Yoshizawa and Shindo.

Estimation of mixing index and its accuracy
We now consider the problem that the mixture is estimated with k coordination number samples of size n* from the mixture including target particles of population concentration X 1 • In the mixing problem, X 1 is assumed to be already known because it is often given when the mixture is supplied. This causes the denominator of the mixture defining Eq. (7) to be constant. At last, we may estimate the contact number through the equation,  These equations can be obtained by estimate Eq. (5) and variance Eq. (6) of the binomial distribution Eq. (4), which is the distribution of the contact number in the completely random state.

Results and discussions of computer simulation
To determine the validity of estimate accuracy Eqs. (22) and (25), we carried out computer simulation. As shown in Fig. 2, the distribution of the contact number is q discrete distribution. Using the following recurrent equation which satisfied between adjacent probabilities, we found the accumulated probability and generated random numbers obeying Eq. (11) with the reverse conversion method 6 ) . We made a simulation using the following procedure: (i) First mixture M, population concentration X 1 and sample size n* were specified.
(ii) Next, the number of samples k to a certain value were set. (iii) Then, assuming the j-th sample in the i-th COUnt to be Cij' mean contact number Ci (i = 1, · · ·, N) with the following equation was estimated:  always drawn below those with X 1 = 0.1, the relative standard error is smaller to have higher estimate precision as the population concentration X 1 is closer to 0.5.
As a sample result in an incompletely mixed state, Fig. 3 (b) shows the same relationship as in Fig. 3 (a) with mixture M = 0.5. The solid lines in the figure are the theoretical values found from Eq. (22). The relative standard error is larger to be less estimate precision in comparison with Fig. 3 (a). number by the coordination number sampling for an appropriate number of samples prior to mixing. Figure 4 shows the relationship of the relative standard error with mixture for population concentration X 1 = 0.1 and 0.5 with the number of samples k fixed to 20 and total coordination number n* = 4. This indicates that the relative standard error is decreased gradually to make the estimate precision higher prior to mixing. In designing sampling operation, it is easy to use Figs. 3(a) and 3(b), while it can be designed by using Eqs. (23) and (26). Table 1 shows the results of calculating the number of samples using Eqs. (23) and (26) to estimate the contact number with the relative standard error of 10% in two cases of the population concentration X 1 of 0.1 and 0.5 with the total coordination number n* = 4. This also indicates that the estimate precision is higher and the number of samples may be less as population concentration X 1 becomes closer to 0.5 and the mixture becomes closer to the completely random state.

Conclusion
Akao and others have already proposed a method to indicate mixture using the contact number. However, with this method, we could not discuss the estimate precision because it did not clarify the distribution of the contact number in an incompletely mixed state.
This report introduced the theoretical distri- In estimation of the contact number, spot sampling can be used instead of coordination number sampling. The spot sampling method has already been reported.  15, 267 (1976).