lecture on polarisation

8/18/2019 Lecture on Polarisation

1/41

R. Botet 04/2016

Module B:aggregates of grains

-formation and simulation -

LECTURE 6 :

physics of aggregates


2/41

R. Botet 04/2016

VI. Details about the numerical algorithms

VI.1 the hierarchical model VI.2 hierarchical BaCCA

VI.3 hierarchical DLCCA

VI.4 hierarchical RCCA

VII. Fractal structure

VII.1 fundamentals

VI I.2 the mass-radius relation

VII.3 the pair-correlation function

VII.4 values of the fractal dimension

VIII. Fractal aggregates

VIII.1 fundamentals

VIII.2 definitions of the aggregate radius

VIII.3 physical properties depending only on the fractal dimension

VIII.4 definitions of the porosity

OUTLINE - LECTURE 6 -


3/41

R. Botet 04/2016

VI.1 DETAILS ABOUT THE NUMERICALALGORITHMS

- the hierarchical model -

aggregation models are based on the double process : diffusion + sticking

full-system simulations are possible but the codes are a bit difficult to write because

of the (generally : periodic) boundary conditions, and the simulations are slow in

the usual dilute regime

in the following pages, some remarks about :

• how to model diffusion (Brownian, ballistic or reaction-limited)

• how to manage the sticking event


4/41

R. Botet 04/2016

VI.2 DETAILS ABOUT THE NUMERICAL ALGORITHMS- Hierarchical Ballistic Cluster-Cluster Aggregation model -

along the simulation, the most demanding part is the diffusion, that is :

consider two clusters moving along random linear trajectories, when do they

collide?

• the full-system problem is tricky because of the periodic boundary conditions(and kind of billiard if hard boundaries)


5/41

R. Botet 04/2016


in principle, one should take the initial distance, Do, between the two clusters to be

1/n1/3, with n the number of clusters per unit of volume

generally, one does not know the value of Do,

but it was proved that the resulting clusters

do not depend on the choice of Do if

Do > 1.2( Rmax,1+ Rmax,2)

Do

Rmax,2

Rmax,1

computing time is Do

2 take Do as small as possible

the hierarchical algorithm is much simpler (because only 2 clusters in free space)


6/41

R. Botet 04/2016


consider two clusters moving along random linear trajectories, when do theycollide?

l u

for each couple of grains (i, j) with i cluster 1 and j cluster 2,

is there a positive solution in l to the quadratic equation :

?22 )2()( a ji rur l

i

j


7/41R. Botet 04/2016


l u

yes if :0)( i j rru

222 4)()( ai ji j rrrru

ri

r jr j-r

i

(and starting from the circumscribed spheres)

positive solution

real solution

much faster

than N 2 process

consider two clusters moving along random linear trajectories, when do theycollide?


is there a positive solution in l to the quadratic equation :

?22 )2()( a ji rur l


8/41R. Botet 04/2016

VI.3 DETAILS ABOUT THE NUMERICAL ALGORITHMS- Hierarchical Brownian Cluster-Cluster Aggregation model -

same idea as for BaCCA, but diffusion = random walk


is there a positive solution in l < dl to the quadratic equation :

dl u

yes if :

rir j r j-ri

(and starting from the circumscribed spheres)

positive solution

real solution

at each step :

22 )2()( a ji rur l

0)( i j rru

222 4)()( ai ji j rrrru


9/41R. Botet 04/2016

Do

VI.3 DETAILS ABOUT THE NUMERICAL ALGORITHMS- Hierarchical Brownian Cluster-Cluster Aggregation model -

one can take the initial distance, Do, between the two clusters as small as possible

(for example : Rmax,1 + Rmax,2 + dl )

select a large sphere (radius Dmax) : if the diffusing cluster crosses the limit of

this large sphere : it is removed and replaced on the small sphere of radius Do

it was proved that the resulting clusters do not depend on the choice of Dmax if Dmax > 5 Do

Dmax

computing time is Dmax2 take Dmax as small as possible


10/41R. Botet 04/2016

VI.4 DETAILS ABOUT THE NUMERICAL ALGORITHMS- Hierarchical Reaction-Limited Cluster-Cluster Aggregation -

Meakin’s algorithm = no diffusion

the full-system simulations are impracticable ( sticking probability very small )

find a random couple of grains (i, j) with i cluster 1 and j cluster 2,

and a random tangent position between them, such that :

for all the other couples (i’, j’) of grains

(we do not know tricks to make it faster)

22

'' )2()( a ji rr

at each step :


11/41R. Botet 04/2016


VI.1 the hierarchical model

VI.2 hierarchical BaCCA




VII.1 fundamentals





VIII.1 fundamentals






12/41R. Botet 04/2016

VII.1 FRACTAL STRUCTURE- FUNDAMENTALS -

Witten and Sander proved in 1981 that natural aerosol aggregates have fractal structure…

what is a fractal ?


13/41R. Botet 04/2016

VII.1 FRACTAL STRUCTURE- FUNDAMENTALS -

FRACTAL IS CHARACTERIZED BY SELF-SIMILARITY

that is : part is similar to the whole, just after changing the length-scale

example : Vicsek fractal

grain 5 grains

length-scale divided by 3

25 grains

length-scale divided by 9 5n

grainslength-scale divided by 3n

…


14/41R. Botet 04/2016

VII.2 FRACTAL STRUCTURE- THE MASS-RADIUS RELATION -

FRACTAL IS CHARACTERIZED BY SCALING-LAW

in particular : between mass and diameter (or : number of grains and radius)

example : finite-size Vicsek fractals

N = 5 grains

diameter = 3

N = 25 grains

diameter = 9

N = 5n grains

diameter = 3n

f d R N

in this example d f = log(5)/log(3) = 1.46…

(5n = (3n)d f )


15/41R. Botet 04/2016




check and measurement of the fractal dimension

is straightforward if we can measure both mass and radius

f d R N

Au colloids, Weitz et al 1984log R

log N


16/41R. Botet 04/2016




a real experiment :

in this example

d f

1.7

f d R N

carbon black smoke, Xiong and Frielander 2001


17/41R. Botet 04/2016

VII.3 FRACTAL STRUCTURE- THE PAIR-CORRELATION FUNCTION -


g(r) = probability to find grain at the distance r from a given grain

g d Rr hr r g f /1)( 3with h a cutoff function

h(0) = 1 ; h() = 0

log r

log g (r )

Ag grains in ait-waterinterface, Yongdong et al 2002

rr dr


18/41R. Botet 04/2016

VII.3 FRACTAL STRUCTURE- THE PAIR-CORRELATION FUNCTION -


g(r) = probability to find grain at the distance r from a given grain

f d R N

g d Rr hr r g f /1)( 3with h a cutoff function

h(0) = 1 ; h() = 0

number of grains inside a sphere of radius r centered on the

center of mass r d f r dr r r g 0

2 ''4)'(


19/41R. Botet 04/2016

for a fractal particle (definition) :

then :

such that if qRg >> 1 :

RAYLEIGH-GANS-DEBYE THEORY

- THE POWER-LAW REGIME-

power-law regime

g d Rr hr r g f /1)( 3

022

4)(

sin

),( dr r r g qr

qr

F

02

2

)/()sin(1),( duqRuhuuq

F g d d f

f

with h a cutoff functionh(0) = 1 ; h() = 0

f d q F

1

),(

2

Menger sponge (d f = 2.57)

direct measurement of the

fractal dimension of the particles


20/41R. Botet 04/2016

RAYLEIGH-GANS-DEBYE THEORY

- THE POWER-LAW REGIME-

f d q F

1),(

2

I sca(q)

q

Guinier regime ( R g )

power-law regime ( d f )


21/41R. Botet 04/2016

VII.4 FRACTAL STRUCTURE- VALUES OF THE FRACTAL DIMENSION -

FOR AGGREGATES 1 d f 3

linear aggregate homogeneous aggregate

R N

3 R N


22/41R. Botet 04/2016

VII.4 FRACTAL STRUCTURE- VALUES OF THE FRACTAL DIMENSION -

FOR AGGREGATES 1 d f 3

linear aggregate homogeneous aggregate

R N

3 R N but : fractal dimension is related to correlationsnot to porosity

porous d f 3 polymeric sphere (Lapierre, 2014)

3

)1( R p N volume fraction


23/41

R. Botet 04/2016


VI.1 the hierarchical model

VI.2 hierarchical BaCCA




VII.1 fundamentals





VIII.1 fundamentals






24/41

R. Botet 04/2016

VIII.1 FRACTAL AGGREGATES- FUNDAMENTALS -

Witten and Sander proved in 1981 that natural aerosol aggregates have fractal structureof fractal dimension d f = 1.8

…this result agrees with numerical results on the aggregate models that we know:

Brownian Ballistic Reaction-Limited

Cluster-Cluster 1.8 1.9 2.0

Particle-Cluster 2.5 3 3

fractal dimensions


25/41

R. Botet 04/2016

P A R T I C L E - C L U S T

E R

C L U S T E R - C L U S T E R

BROWNIAN BALLISTIC REACTION-LIMITED

d f = 1.8 d f = 1.9 d f = 2.0

d f = 2.5 d f = 3 d f = 3


26/41

R. Botet 04/2016

VIII.2 FRACTAL AGGREGATES- DEFINITIONS OF THE AGGREGATE RADIUS -

1) the relevant definition of the radius for dust particle is the radius of gyration :

where rcm is the location of the center of mass :

N

i

cmi g N

R1

22 )(1

rr

N

i

icm

N 1

1rr

f d R N


27/41

R. Botet 04/2016




2) maximum radius (= half diameter of the particle):

N

i

cmi g N

R1

22 )(1

rr

N

i

icm

N 1

1rr

2,2 )(max4

1max ji ji

R rr

f d R N

f d R N

2


28/41

R. Botet 04/2016




2) maximum radius (= half diameter of the particle):

3) effective radius ( = radius of the sphere of same volume) :

N

i

cmi g N

R1

22 )(1

rr

N

i

icm

N 1

1rr

2,2 )(max4

1max ji ji

R rr

4

33 V Reff f d

eff R N

f d R N

f d R N

VIII 2


29/41

R. Botet 04/2016


for optical properties of fractal dust, it is important to know if the fractal dimension of thescattering particles is < 2 or > 2

d f 1.9

no shadowing

all the grains feel directly the incident light

d f 2.5

shadowing

most of the grains are hidden behind others

VIII 2


30/41

R. Botet 04/2016


for optical properties of fractal dust, it is important to know if the fractal dimension of thescattering particles is < 2 or > 2

d f 1.9 d f 2.5

S R

pro

2

4) the projected radius (= radius of the circle of same area as projection)

f d

pro R N

VIII 2


31/41

R. Botet 04/2016


Particle-Cluster aggregates are isotropic in average

aggregate radius is well-defined

d f = 2.5 d f = 3 d f = 3

VIII 2


32/41

R. Botet 04/2016


Cluster-Cluster aggregates are essentially anisotropic one can define three aggregate radius

they are the eigenvalues of the inertia matrix:

d f = 2.0

i iii iii ii

i

ii

i

ii

i

ii

i

ii

i

ii

i

ii

y x y z x z

z y z x x y

z x y x z y

)(

)(

)(

22

22

22

for the RCCA model:

R g 1/ R g 3 = 1.75

R g 2/ R g 3 = 1.63

VIII 3


33/41

R. Botet 04/2016

VIII.3 FRACTAL AGGREGATES- PHYSICAL PROPERTIES DEPENDING ONLY ON d f -

• in the domain : 1/ R < q 4sin( /2)/l < 1/a , we have seen that there exists the power-law

behavior of the X-ray scattering (when Rayleigh-Gans-Debye theory is valid) :

because I (q) is essentially the Fourier transform of the pair-correlation function g (r )

it is robust power-law as it remains true even if the conditions are beyond the RGD theory

1. electromagnetic wave scattering

f d qq I /1)(

VIII 3


34/41

R. Botet 04/2016

VIII.3 FRACTAL AGGREGATES- PHYSICAL PROPERTIES DEPENDING ONLY ON d f -

• in the solar system, orbits of dust particles around the Sun are essentially ruled by the ratio,

b , between the force due to radiative pressure and the gravitational force :

if the particle is compact : b 1/ L ( L = typical size of the particle), hence big difference

between small and large particles

it the particle is fractal of fractal dimension d f < 2, the system is so fluffy that almost all

the grains are in surface, then the surface facing the sun is just N , similarly to the mass.

Then:

and no big difference between small and large particles is expected in the fractal d f < 2 case

2. orbits of dust particles

M

S Q pr b

surface of the particle

mass of the particle

pr Q b

VIII 4


35/41

R. Botet 04/2016

VIII.4 FRACTAL AGGREGATES- DEFINITIONS OF THE POROSITY -

• the porosity, p, of a material is generally defined as the void fraction, that is :

where f is the volume fraction (proportion of matter = N (4a3/3)/(4 R3/3))

For a fractal aggregate of size N and radius R for which : N k o ( R/a)d f , porosity is :

which depends on the size , R, of the aggregate if d f < 3

f 1 p

f d

o R

ak p

3

11

p 1 for the large aggregates d f < 3

p constant < 1 for the large aggregates d f 3

VIII 4


36/41

R. Botet 04/2016


precise definition of the porosity depends on the context(it may depend on the physical features)

a few contexts :

• light scattering using the Effective-Medium theory (aggregate = sphere + inclusions)

• stocking chemical compounds (aggregate = high specific surface)

• chemical reactor (aggregate = catalyst)

• transport of fluid throughout the aggregate (aggregate = permeable material)

• …

is only related to the inertia of the aggregate

(how fluffy it is)

f d

o R

ak p

3

11

VIII 4


37/41

R. Botet 04/2016


example : the Effective-Medium Theory

good choice for the effective refractive index:

Maxwell -Garnett formula22

22

22

22

22 m s

m s

m

m

mm

mm p

mm

mm

volume fraction of the inclusions

here : refractive index m s for the inclusions

and m m for the matrix

m smm

Ag inclusions in C grains, Tang et al, 2010

VIII 4


38/41

R. Botet 04/2016



inclusions = grains (d f < 2) :

Maxwell -Garnett formula

2

1)1(

2

12

2

2

2

g

g

m

m p

m

m

volume fraction of the grains

m g

aggregate d f < 2

1

refractive index m g for the grains

and 1 for the voids

VIII 4


39/41

R. Botet 04/2016



inclusions = voids (d f > 2) :

Maxwell -Garnett formula2

2

22

22

21

1

2 g

g

g

g

m

m p

mm

mm

volume fraction of the voids

refractive index m g for the grains

and 1 for the voids

m g1

aggregate d f > 2

VIII 4


40/41

R. Botet 04/2016



better formula

another possible definition of the “porosity” in

this context :

• determine the best value of the refractive index m forwhich the Mie sphere theory reproduces the results from

T-Matrix or DDA

• warning 1 : the resulting porosity depends on the size R

of the aggregate• warning 2 : the resulting porosity should be close to

(but a bit different from) the Maxwell-Garnett formula

• the corresponding “porous” sphere scatters the same asthe aggregate


41/41

Summary of the Lecture 6

Cluster-Cluster aggregates are fractal fractal dimension depends on the diffusion process

d f for Particle-Cluster > d f for Cluster-Cluster

physical properties of fractal clusters depend on d f d f can be measured through scattering experiments

some geometric features, such as porosity, are not well defined

lecture on polarisation

Documents