ChEn 610:  Applied Statistical Thermodynamics

MonteCarlo Simulated Agglomeration:Based on the Smoluchowski Equation

Patrick  J. RensingApril 26th, 2006

Table of Contents

Summary....................................................................................................................................................1

Introduction................................................................................................................................................2

Discussion..................................................................................................................................................4

Results........................................................................................................................................................5

Conclusions and Recommendations..........................................................................................................9

Table of Nomenclature.............................................................................................................................11

References................................................................................................................................................12

Appendix A: Simulation Source Code....................................................................................................13

List of Figures

Figure 1: Sierpinski Triangle, Df=1.585....................................................................................................2

Figure 2: Average Fractal Dimension Over Time.....................................................................................5

Figure 3: Mass Weighted Average Fractal Dimension Over Time............................................................6

Figure 4: Normal Q-Q Plot of Simulation Time.......................................................................................7

Figure 5: Final Aggregate Structures........................................................................................................8

Summary

Aggregation is important to many fields in the sciences from polymer processing, to blood flow, to

hydrate slurry rheology. This process has been studied in a simplified case here by conducting a

dynamic Monte-Carlo simulation, with the successes based on the agglomeration rate kernel. The

simulations show surprising similarities in the evolution of fractal dimension over time between

different simulations. All simulations had similar trends and all settled at a final fractal dimension

between 1.6 and 1.7. In addition the time for 1000 hard disks to completely agglomerate is shown to be

Gaussian in nature, with a large variance. This simulation shows promising results for the study of

aggregation via dynamic Monte-Carlo simulation. There is still more work that needs to be done to

include more physical reality, i.e. three dimensions and breakup, in the simulations and to relate Monte-

Carlo steps to time.

Introduction

In many systems particles will form aggregates through time dependent agglomeration. This process is

important in many areas of science and research including polymerization slurry processing, metallic

alloy solidification, food processing, blood circulation, and coal cleaning [1,2]. In addition it is also

important in ice and hydrate slurries, where aggregates can alter the flow behavior of these slurries,

causing difficulties transporting the slurry. It is hoped that a transient agglomeration model can be

implemented to help understand and model the transient slurry rheology.

When systems like these form agglomerates it is very rare that they will form a compact aggregate,

rather they tend to form voluminous fractal aggregates. While these aggregates lack exact fractal

structure the aggregates formed are still fairly fractal in nature, so the term fractal dimension is often

used to characterize the equations. The fractal dimension, Df, relates the scaling of the characteristic

length, l, to the mass of the object, M, as suggested by Mandelbrot [3]. (Equation 1)

(1)

Thus a perfect sphere will yield a fractal dimension of three, a circular disk will yield a fractal

dimension of two, and a straight line will yield a fractal dimension of one. Other fractal objects may

yield non-integer fractal dimensions, i.e. the Sierpinski Triangle with a fractal dimension of 1.585 [4]

(Figure 1).

M ∝ l D f

Figure 1: Sierpinski Triangle, Df=1.585

In this paper the proportionality constant is assumed to be the mass of one particle which gives

equation 2, which relates the number of particles, N, to the diameter of the particles, DP, and the diameter of the agglomerate, DA [5].

(2)

In addition to being fractal solids these agglomerates sizes will vary over time. The most common

method to model this agglomeration is the Smoluchowski population balance equation [6] (Equation 3).

(3)

In which Ni(t) is the number of agglomerates of size i as a function of time (t), Bi,j is the aggregation

kernel of particles of size i and size j. Where the aggregation kernel is a function that accounts for

mobility of particles, fractal dimension, and other properties. The result of this is that this equation can

be used to track the size of the agglomerates over time, which will be critical in transient modeling of

agglomerates and the effect they have on slurry flow rheology.

N= DADP D f

dN i t dt

=12∑j=1

i− 1

Bi − j , j N i − j t N j t − ∑j=1

i max

B i , j N i t N j t

Discussion

A dynamic Monte-Carlo simulation of irreversible agglomeration of hard disks over time was

conducted on a system consisting of 1000 initially dispersed hard disks of uniform size. Each Monte-

Carlo step two random agglomerates (or single disks) from the list of particles were selected to be

combined. A probability was calculated, based on the Smoluchowski aggregation kernel of Flesh,

Spicer, and Pratsinis [5], for the successful combination of particles in turbulent flow (Equation 4).

(4)

In which the aggregation kernel is a function of the volume average shear rate, G, the particle volume,

νp, and the size of the aggregates, xi. The probability was taken to be the ratio of Bi, j against the maximum possible Bi, j. In addition the exponent was reduced to a power of two since the simulated

system was done in two dimensions instead of three. Thus the probability of accepting follows in

equation 5 as:

(5)

Each Monte-Carlo step a random probability was compared to the Paccept. If the random probability was

greater than Paccept then no action was taken, but if this random probability was less than or equal to Paccept then the two agglomerates would be combined into one new agglomerate. To combine the two

agglomerates into a new agglomerate each agglomerate was rotated by a random angle, between 0° and

360°, to simulate a random collision vector. Next the particles were brought together along the x axis

until two particles were just touching, with their center of masses kept along the x axis. In reality

agglomerates would not always collide along their center of mass, however this was done to simplify

the simulation. Once the two agglomerates had been combined the new fractal dimension of this

agglomerate would be calculated (via equation 6) with the diameter of the aggregate taken to be the

largest distance between two particles. Once the new agglomerate was calculated it would be returned

to the list of agglomerates.

(6)

Finally, after each Monte-Carlo step this process was then repeated ad infinitum until all the hard disks

had aggregated into one large agglomerate.

Bi , j=0.31G v p x i1/ D f x j1/D f 3

Paccept=x i1 /D f x j1 /D f

2

N 2

D f =ln N

ln DADP

Results

Ten simulations were run with 1000 hard disks, with the initial condition of all particles being

dispersed. For these simulations fractal dimension was analyzed over time and the final agglomerate

structure was analyzed. It was found that the average fractal dimension and mass weighted average

fractal dimension decreased over time in a linear fashion that was consistent between simulations

(Figures 2-3).

Figure 2: Average Fractal Dimension Over Time

The mass weighted fractal dimension decreases faster than the unweighted fractal dimension. This is

expected since all individual hard disks have a fractal dimension of two. This means that the shear

number of single particles will push the average near two at the beginning of the simulation. Near the

end of these simulations, when the number of aggregates are very low, there are wide fluctuations in

the fractal dimension. Again this is to be expected since a small number of observations will yield a

large variance. The mass weighted fractal dimension shows less variance near the end of the

simulations since there is typically one or two large aggregates which dominate all the statistics in these

cases. Another noteworthy point is that all final agglomerates had a final fractal dimension between

1.6 and 1.7. This shows, as do the similar trends in fractal dimension over time, that a stochastic

process with a 'large' number of particles still yield similar behavior. While running simulations the

qualitative trend that large agglomerates were much more likely to agglomerate than smaller

agglomerates was observed, which is consistent with the agglomeration kernel. This will frequently

lead to the development of one or two large agglomerates which will then 'consume' all the smaller

agglomerates and dispersed hard disks. This also makes qualitative sense since a large agglomerate

will have a large collision radius and be very likely to collide with most other particles.

These simulations also show scatter in the amount of time to reach the final agglomerate size of 1000

hard disks. The final agglomeration time varies by about 2x106 Monte-Carlo steps, which is around

50% of the average time to reach the final agglomerate size. When preforming statistical analysis of

Figure 3: Mass Weighted Average Fractal Dimension Over Time

the time for complete agglomeration it was found to be very Gaussian (a Shaprio-Francia statistic of

0.98, which strongly suggests Gaussianity for 10 points) [7]. This Gaussianity can clearly be seen by

linearity of the normal Q-Q (quantile-quantile) plot (Figure 4) which is a plot of a CDF (cumulative

distribution functions) of the final time of the simulations vs. the CDF of a uniform Gaussian function

centered at 0.

In figure 5a-j the ten final agglomerate structures are shown. Here the agglomerates show a radial

branch like agglomerate structure. The fact that many of these agglomerates are very radial in shape is

most likely an artifact due to the forced collisions along the centers of mass, but it is also qualitatively

consistent with a system that has undergone shear deformation. In the images occasionally the

agglomerate appears as two agglomerates stuck together, this is not consistent with what would be

expected in a real system. Shear breakages and deformation will remove these second agglomerates by

breaking off a new agglomerates or compacting them into a more complete part of the agglomerate. It

can also be noticed by the observant reader that not all the particles in the agglomerates are attached.

This is an artifact from the code used to draw the images. All particle positions were recorded as

double floats, while the drawing functions required integer positions. All positions were rounded down

which resulted in the occasional appearance of separation of hard disks in the agglomerate.

Figure 4: Normal Q-Q Plot of Simulation Time

a. b. c. d. e.

f. g. h. i. j.

Figures 5a-j: Final Aggregate Structures

Conclusions and Recommendations

This simulation shows how a dynamic Monte-Carlo simulation can be used to study the agglomeration

of hard disks in a system. The simulations here illustrate a proof of concept that such a simulation

should be useful and feasible to study this stochastic process in the framework of statistical mechanics.

Improved versions of this simulation may prove to be powerful modeling tools for future studies of

agglomeration of hydrates in slurry flow.

This code has been a first pass attempt to model agglomeration of particles. The model seems to

preform adequately and yields results consistent with ones qualitative expectations. Since this is a first

pass model there is still much room for improvement. The first improvement to be considered is

extending the simulation to include breakup as well as agglomeration. This will give more realistic

results and may provide insight on how repeated agglomeration and breakup may effect agglomerates.

Another change related to this would be to include shear compaction of the aggregates. Allowing the

particles in the aggregate to slip and compact could cause changes that one would expect in a flowing

system. In addition to these two improvements another vital improvement would be to extend the

simulation into three dimensional space. The code has been written with this future path in mind and

should be a reasonably easy improvement to incorporate. If three dimensions are implemented another

change that will need to be included is an improved visualization kit. Currently this code only

produces an image of the final two dimensional structure, and has some problems with round off errors

while drawing the images. An improved drawing package, that supports three dimensions may

improve information gained from the program. In addition allowing the visualization software to tract

aggregates over time may give some insight into the agglomeration/breakup process and shape

evolution over time. Another important improvement to increase the physicality of the simulation

would be to include non-centerline collisions between aggregates, which could be implemented by

simply picking a random collision offset. Another improvement that could be of interest would be

allowing the particles to randomly come from a distribution. Since it is rarely the case that particles are

of a single size in a hydrate systems this could give some insight on how different distributions effect

agglomeration and the agglomerates properties. Probably the most important thing to be include in

future work is to compare these simulations with experimental data on aggregation to test the theory

behind the agglomeration rates. Another improvement that should be made is to clean and optimize the

code. As with any typical programming project this simulation has been written in the “make it work,

then make it work fast” mindset. As this is the case the program runs well but has some severe

inefficiencies with the way it deals with linked lists. It should be possible to increase the efficiency of

the code. As simulations become large (>1000 hard spheres) this consideration will become a more

pressing concern. The last improvement that needs to be made is translating the Monte-Carlo steps into

real time. There is a problem with the current Monte-Carlo steps as they ignore the total number of

aggregates in the system. As the number of aggregates in the system decreases the likelihood of a

collision will decrease, however this relationship is not taken into account in the simulation, but should

be correctable with a simple scaling relationship or by incorporating this consideration into the

probability of acceptance function.

Table of Nomenclature

Bi,j.........................................................................................................Collision Kernel for Particle i and j

CDF........................................................................................................Cumulative Distribution Function

DA.............................................................................................................................Diameter of Aggregate

Df.....................................................................................................................................Fractal Dimension

DP.........................................................................................................................Diameter of Hard Sphere

G......................................................................................................................Volume Average Shear Rate

i.............................................................................................................................................Index Variable

j.............................................................................................................................................Index Variable

l..................................................................................................................................Characteristic Length

M....................................................................................................................................Mass of Aggregate

N.........................................................................................................Number of Hard Disks in Aggregate

Paccept......................................................................................Probability of Accepting Monte-Carlo Move

Q-Q..................................................................................................................................Quantile-Quantile

t............................................................................................................................................................Time

xi.......................................................................................................Number of Hard Disks in Aggregate i

νP..................................................................................................................................Volume of a Particle

References

1. Barthelmes G., Pratsinis S. E., Buggisch H., 2003. Particle Size Distributions and Viscosity of Suspensions Undergoing Shear-Induced Coagulation and Fragmentation. Chemical Engineering Science 58, 2893-2902.

2. Capes C. E., Darcovich K., 1984. A Survey of Oil Agglomeration in Wet Fine Coal Processing. Powder Technology 40, 43-52.

3. Mandelbrot, B., 1982. The Fractal Geometry of Nature. W.H. Freeman, San Fransisco.4. Sierpinski, W. 1915. "Sur une courbe dont tout point est un point de ramification." C. R. A. S.

160, 302-305.5. Flesch J. C., Spicer P. T., Pratsinis S. E., 1999. Laminar and Turbulent Shear-Induced

Flocculation of Fractal Aggregates. Materials, Interfaces and Electrochemical Phenomena 45, 1114-1124.

6. Lattuada M., Wu H., Sandkühler P., Sefcik J., Morbidelli M., 2004. Modelling of Aggregation Kinetics of Colloidal Systems and its Validation by Light Scattering Measurements. Chemical Engineering Science 59, 1783-1798.

7. Shaprio S.S., Francia R.S., 1972. Approximate Analysis of Variance Test for Normality. Journal of The American Statistical Association 67, 215-225.

Appendix A:

Simulation Source Code*

*Source Code has not been included in this pdf. Please contact Patrick Rensing for a copy.