aggregation with increasing reactivity -...

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1Challenge the future

Monte Carlo calculations of cluster-cluster aggregation with increasing reactivity

L.F. van Heijkamp, J.R. Heringa, I.M. de Schepper, W.G.Bouwman


Outline

IntroductionModelInitial phaseCross-overPercolationConclusion


SamplesSamples• What’s the matter?

� Colloids� Gels

• What’s going on?� Aggregation� Gelation

• Who cares?� Biology� Forensics� Health care� Cosmetic industry� Food products


Single casein micelle

50-200nm

From milk to yogurt and curd


AggregationThe aggregate is expected to grow initially as a fractalSelf organized criticality

1.90percolation

1.56RLCA

1.45DLCA2

2.52percolation

2.1RLCA

1.8DLCA3

Fractal dimension

growthdimensionality


Model

Particles on a latticeParticles move and neighbors stick with reaction constant kTime normalization: Monomer diffusion constant = 1

Random initialization with given density ϕRepeat until 1 aggregate left:

Select randomly one aggregateMove with probability 1/mα (α is chosen ½ for simplicity) Increment time with 1/(number of aggregates)Neighboring particles stick with stick probability in increment


Time dependent reaction constant

For yogurt the reactivity is expected to increase in timeWe chose k(t)=kmax(1-e-t/θ).

For small sticking probability randomly choose which bond to add (Luijten and Blöte)

For small probability of movement randomly determine time of movement


Cluster sizes


Relaxation time 1/kmax if θ smallGaussian relaxation for large θ with

From Nu we calculate the number of clusters assuming every bond joins 2 clusters

Approximation no Diffusion

d ( ) ( ) ( )du

uN t k t N tt

= −

0 max( ) 3 exp( ( exp( / )))uN t N k t tφ θ θ θ= − − −

N0:number of particlesφ: density of particlesInitially we expect 3N0φ pairs of neighboring occupied sites. Nu is the number of unbound pairs.

maxkθ


Relaxation time 1/φkmax if θ smallGaussian relaxation for large θ with

Approximation RLCA

d ( ) ( ) ( )du

uN t k t N tt

φ= −

0 max( ) 3 exp( ( exp( / )))uN t N k t tφ θ θ θ= − − −

maxkθ

φ


Structure

Aspects contributing to configuration

• initial configuration• growth reaction limited or diffusion limited

Diffusion time changes in time:1. Larger cluster moves slower2. Average distance between clusters changes


θ dependence


Density dependence


Aggregate structure

• DLCA


Radial distribution function


Percolation behavior

Distance between cluster centers

Size of cluster ~

Percolation behavior emerges, when twice the cluster size nears the distance between clusters. After the transition we have a percolating structure of blobs.

13( / )cm φ

1

DLCADcm


Cluster size distribution

22 2

0

1 ( )ii

c mN

= ∑

44 4

0

1 ( )ii

c mN

= ∑2

222 43 2cQ

c c< >

=< > − < >

Use cumulant to locate transition:


Locate transition time


Conclusions

Final percolating structure consists of fractal blobs.The effect of the initial reaction limited growth may be

restricted to short distance scales.

aggregation with increasing reactivity -...

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