입자침적 해석용 모델의 비교

김호연, 김 헌, 편무실

명지대학교 화학공학과

Comparison between Models for the Analysis of Particle Deposition

Ho-Yeon Kim, Hern Kim, Moo-Sil Pyun

Department of Chemical Engineering, Myong Ji University

Introduction

The influence of deposition and packing of fine powders has received considerable attention in recent years because of the importance of the resulting microstructures on the properties of the materials formed by particle assemblies[1-3].  In chemical engineering literature, packing problems have been a focus of examination usually in the context of structure of packed beds and its effects on flow distribution and attendant transport phenomena[4,5].  One should note that a number of factors affect packing processes[6-8] and can cause flaws (or defects) in the resulting microstructures.

Computer simulation methods based on a Monte Carlo approach can be effectively used for studying the packing problems relevant to the formation of green compacts in ceramic powder processing.  Individual and/or simultaneous effects of adhesion (or sticking) and/or polydispersity on the formation of structures and geometrical properties have been examined[9-11].  A number of cooperative phenomena, such as Brownian motion, shaking, vibration, etc., have been investigated[12,13].  The structures of the packings, contact networks, diffraction patterns, radial distribution functions, and angular distribution functions can be used for the analysis of the structures.  Bulk and local geometrical properties such as packing fraction, particle number density, the number of contacts, etc., can be examined.

The objective of the present work is to compare, for both noncooperative and cooperative packing processes, computer simulation models which can provide a better understanding of a variety of packing phenomena where inhomogeneities of the packing structures are important.

Packing Algorithms

In general, the packing (or deposition) process can be classified into two types, according to the method of packing the particles.  When the particles are added collectively, the deposition is affected by the collective movement of all the particles; we call this process 'cooperative packing process'.  When the particles are added one by one, the deposition of one particle is completed before a second particle arrives in its vicinity.  We call this process 'noncooperative packing process'.  In practice, this process corresponds to the case when the rate of addition of particles is slow or the concentration of particulate system is small.

(1) Noncooperative Packing Process

We begin with a vertical strip (of width L) into which the particles are deposited.  The particles, whose radii are already chosen from a given size distribution, enter this 'chamber' one by one from the top at random positions, distributed uniformly along the width of the chamber.  Each particle falls vertically, i.e., along the direction of the external field (in this case, gravity) and comes into contact with either a previously deposited particle or the substrate.  If the particle reaches and makes contact with the substrate, it is assumed to rest there permanently.  If, on the other hand, it comes into contact with a previously deposited particle, it is assumed to either stick to the deposited particle with a probability s, or roll down until it finds another particle or the substrate.  If it rolls down and makes contact with another particle, one first determines whether the rolling particle sticks to the other particle or not (using, again, s as the probability of sticking).  If it does not stick, then one determines if the current position of the rolling particle is stable.  For this purpose, the position is assumed to be stable if the center of the rolling particle lies between the centers of two contacting particles.  Otherwise, the rolling particle is allowed to continue to move (roll or fall, as the case may be), until a suitable host is found.  (If a particle makes contact with more than two particles at a time, the above procedure will automatically detect the situation since the third particle will prevent further motion of the rolling particle.)  This entire procedure is repeated until a predetermined number of particles is deposited.

(2) Cooperative Packing Process

In this simulation, the volume containing the particles is defined by a vertical strip of width L and an initial height of Ho.  Initially, the particles are randomly placed in the given volume, without overlapping.  The particles are then moved one at a time with a small displacement {□x,□y} in a random deposition.  For this, we chose the trial position of particle i to be

xi,trial = xi + □x	(1)

yi,trial = yi + □y

The random displacements in vertical and horizontal directions, □x and □y, are determined by values chosen within the range of □□x□≤□ and □□y□≤□, respectively, where the maximum value of the displacement, □, is arbitrarily set (here, at 1.0 Rb), where Rb is a reference length, chosen as the radius of one of the particles; the subscript, 'b' stands for 'basis', as mentioned below), depending on the system size, particle size, and so on.  If the trial position results in an overlap between particles, it is rejected.  Otherwise, the trial position is accepted or rejected according to the transition probability calculated from the change in the potential energy of the system.  The transition probability is taken to be

P = exp[-□E/kBT] = exp[-Pe□*V*□y*],	(2)

where the dimensionless terms are defined as follows: Pe=□□VbFggRb/kBT, □*= □□/□□b, □□=□p-□m  □□b=□b-□m, V*=V/Vb, and y*=y/Rb, where g is the acceleration due to gravity, Fg is the field strength (=1 for normal gravity), T is the absolute temperature of the system, and kB is the Boltzmann constant, respectively.  The change of particle position is accepted or rejected according to the resulting transition probability.  The above procedure is repeated for each particle, and each such repetition (i.e., for all the N particles) is called a Monte Carlo Step (MCS).

We have used a much smaller number of particles in these simulations (namely, 150 particles in a strip having a length of 10).  The results are based on averages over 10 configurations; this was found to be sufficient for the structural features examined here.  Periodic boundary conditions are used in the horizontal direction.  Finally, in the results presented in this work we use a normal field strength of unity, i.e., Fg=1.

Characterization of Structures

The structure of the deposits generated in the simulations is analyzed using diffraction patterns and radial distribution functions of particle positions.  For calculating these, only those particles within □30% from the mid point are taken so that the influence of the substrate and of the free 'surface' at the top of the packing is avoided.  The diffraction patterns are prepared using the structure factor S(q), which is defined by

S(q) = (1/M) < □□exp(iq□rj)□2 >; j=1,...,M	(3)

where rj is the position vector of the j-th particle, the summation is over all M particles in the chosen region, and the bracket <...> denotes the statistical average over the configurations.

The radial distribution function, g(r), is a measure of the probability of finding two particles at any center-to-center separation r and is given by

g(r) = [□n(r)/2□r□r]/□avg	(4)

where [□n(r)/2□r□r] is equal to □(r), the local particle density, with □n(r) equal to the number of particles in the interval [r,r+□r] and □avg is the overall average density of the packing.

Results and Discussions

It has been observed in many experiments[6,14], expecially for the packing of non-Brownian particles, that packings that result from collective deposition (i.e., in cases where particles move as packets) can have structures with small or large bridge-like formations or 'arches' whose sizes depend on factors such as the stresses imposed on the packing, the size of the container, and the frictional effects among the particles and between the particles and the container wall.  The presence of such arches leads to packing fractions and mean coordination numbers that are lower than the characteristic values of packings formed by noncooperative methods.  In order to obtain some insights concerning there, one can compare the structures generated by cooperative and noncooperative packing processes.  To guarantee reasonably similar packing conditions for such a comparison, one can assume that (i) the particles are non-Brownian, (ii) the particles are packed gravitationally, (iii) the Peclet numbers are very large, and (iv) the sticking probability is zero.  Within these restrictions, the noncooperative packing corresponds to packing the particles one by one, with no initiation of cooperative effects as a particle finds it final resting place, where as the cooperative packing scenario is similar to the one described in the previous paper, except that Brownian motion is not considered.  Such a comparison, using packing structures, radial distribution functions, diffraction patterns, and bulk properties for noncooperative and cooperative processes, is shown in Figs. 1a and 1b.  As mentioned above, these results have been obtained for a very large value of Pe (1016), i.e., when particles of large size are used and the particle density is large.  From direct observations of packing structures in the above figures one notes that, while noncooperative packing methods produce structures comprising of rhombic structural units, cooperative packing methods produce many bridging structures as arches or polygons, as often observed in practical instances of packing of powders.  Correspondingly, the radial distribution functions obtained from noncooperative packing methods show distinct peaks over long ranges, but those from cooperative packing techniques have small peaks over short ranges of r.  Cooperative packing can lead to ordered domains, but the above result shows that ordered patterns cannot be obtained from cooperative packing mode without the benefit of other mechanisms such as Brownian motion, shaking, vibration, etc.  In general, cooperative packing tends to favor six-fold-coordinated configurations, while four-fold-coordinated ones dominate noncooperative packing structures; this can be also seen from the diffraction patterns shown in Fig. 1.  Thus, one sees that the differences in physical interactions between particles during packing determine structural changes from four-fold to six-fold configurations.  Physical experiments[15] lead to packing structures consisting mainly of six-fold hexagonal subunits and relatively fewer four-fold rhombic subunits.  This suggests that both cooperative and noncooperative mechanisms may be important in simulating real packings.  Nevertheless, the present results based on cooperative packing methods indicate that the latter lead to realistic picture of the packing structures and properties.  On the other hand, it would be advantageous computationally if noncooperative techniques could be used instead.  It has been noted[9] that the packing structures obtained from noncooperative packing methods for certain values of the sticking probability, s, do lead to small or large bridges, similar to those observed in physical experiments.

In order to see if noncooperative methods with nonzero sticking probabilities can provide packings with properties similar to those in cooperative packings, one examine some bulk properties of the structures obtained from two different packing methods.  Fig. 2 shows that the packing fraction and average height of the structures obtained by cooperative packing correspond to those obtained by noncooperative packing in the range of sticking probability roughly between 0.05 and 0.15.  This suggests that a noncooperative packing technique, with a sticking probability around 0.1, may be used to produce structures that mimic the packing fractions and average heights of those that are typically seen in cooperative packings,  Since, as mentioned above, simulating noncooperative packing methods is relatively simpler, the above possibility reduces the computational efforts significantly.

Formation of bridges due to the cooperative packing of non-Brownian particles or due to sticky particles in noncooperative packing are detrimental to many practical applications where compact packing structures are needed.  Therefore, in practice, additional packing mechanisms such as vibration, shaking, and tapping, etc., are often used to promote the formation of denser compacts.  The effects of such additional methods on packing structures have been studied[13].

References

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5. Chan, S. K. and Ng, K. M., Chem. Eng. Commun., 48, 215 (1986).

6. Gray, W. A., The Packing of Solid Particles, Chapman and Hall, London, 1968.   7. Hausner, H. H. and Kumar Mal, M., Handbook of Powder Metallurgy, 2nd ed.,

Chemical Publishing Co., New York, 1982.

8. Nelson, R. D., Jr., Dispersing Powders in Liquids, Elsevier, New York, 1988.

9. Kim, H. and Rajagopalan, R., Chem. Eng. Commun., 108, 147 (1991).

10. Kim, H. and Rajagopalan, R., KJChE, 11, 239 (1994).

11. Kim, H. and Rajagopalan, R., KJChE, 11, 290 (1994).

12. Kim, H. and Rajagopalan, R., HWAHAK KONGHAK, 32, 659 (1994).

13. Kim, H. and Pyun, M. S., KJChE, 12, 488 (1995).

14. Cumberland, D. J. and Crawford, R. J., The Packing of Particles, Elsevier,

New York, 1987.

15. Edelson, L. H. and Glaeser, A. M., J. Am. Ceram. Soc., 71, 225 (1988).

Acknowledgement

This paper was supported in part by Special Fund for University Research Institute, Korea Research Foundation.

Fig. 1 Comparison of the packing structures obtained by (a) noncooperative and (b) cooperative packing processes, and the diffraction patterns and radial distribution functions corresponding to the structures.	Fig. 2 Comparison of the packing fraction and the average height of the deposit obtained by noncooperative and cooperative packing processes.