Lecture on Polarisation
Transcript of Lecture on Polarisation
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Module B:aggregates of grains
-formation and simulation -
LECTURE 6 :
physics of aggregates
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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 -
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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
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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)
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VI.2 DETAILS ABOUT THE NUMERICAL ALGORITHMS- Hierarchical Ballistic Cluster-Cluster Aggregation model -
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)
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VI.2 DETAILS ABOUT THE NUMERICAL ALGORITHMS- Hierarchical Ballistic Cluster-Cluster Aggregation model -
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
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VI.2 DETAILS ABOUT THE NUMERICAL ALGORITHMS- Hierarchical Ballistic Cluster-Cluster Aggregation model -
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?
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
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VI.3 DETAILS ABOUT THE NUMERICAL ALGORITHMS- Hierarchical Brownian Cluster-Cluster Aggregation model -
same idea as for BaCCA, but diffusion = random walk
for each couple of grains (i, j) with i cluster 1 and j cluster 2,
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
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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
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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 :
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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 -
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VII.1 FRACTAL STRUCTURE- FUNDAMENTALS -
Witten and Sander proved in 1981 that natural aerosol aggregates have fractal structure…
what is a fractal ?
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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
…
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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 )
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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)
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
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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)
a real experiment :
in this example
d f
1.7
f d R N
carbon black smoke, Xiong and Frielander 2001
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VII.3 FRACTAL STRUCTURE- THE PAIR-CORRELATION FUNCTION -
FRACTAL IS CHARACTERIZED BY SCALING-LAW
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
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VII.3 FRACTAL STRUCTURE- THE PAIR-CORRELATION FUNCTION -
FRACTAL IS CHARACTERIZED BY SCALING-LAW
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)'(
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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
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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 )
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VII.4 FRACTAL STRUCTURE- VALUES OF THE FRACTAL DIMENSION -
FOR AGGREGATES 1 d f 3
linear aggregate homogeneous aggregate
R N
3 R N
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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
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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 -
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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
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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
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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
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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 :
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
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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 :
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
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VIII.2 FRACTAL AGGREGATES- DEFINITIONS OF THE AGGREGATE RADIUS -
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
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VIII.2 FRACTAL AGGREGATES- DEFINITIONS OF THE AGGREGATE RADIUS -
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
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VIII.2 FRACTAL AGGREGATES- DEFINITIONS OF THE AGGREGATE RADIUS -
Particle-Cluster aggregates are isotropic in average
aggregate radius is well-defined
d f = 2.5 d f = 3 d f = 3
VIII 2
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VIII.2 FRACTAL AGGREGATES- DEFINITIONS OF THE AGGREGATE RADIUS -
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
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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
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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
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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
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VIII.4 FRACTAL AGGREGATES- DEFINITIONS OF THE POROSITY -
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
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VIII.4 FRACTAL AGGREGATES- DEFINITIONS OF THE POROSITY -
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
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VIII.4 FRACTAL AGGREGATES- DEFINITIONS OF THE POROSITY -
example : the Effective-Medium Theory
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
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VIII.4 FRACTAL AGGREGATES- DEFINITIONS OF THE POROSITY -
example : the Effective-Medium Theory
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
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VIII.4 FRACTAL AGGREGATES- DEFINITIONS OF THE POROSITY -
example : the Effective-Medium Theory
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
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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