Number-Theoretic Aspects
of Matter Agglomeration/Aggregation
Modelling in Dimension d
Adam Gadomski
Institute of Mathematics and Physics
U.T.A. Bydgoszcz, Poland
in cooperation with:
Marcel Ausloos
SUPRATECS
University of Liège, Liège, Belgium
Verhulst’200
16-18 September 2004, Royal Military Academy, Brussels, Belgium
Verhulst’200
OBJECTIVE: TO REVEAL NUMBER-THEORETIC
ASPECTS OF ADVANCED STAGES OF A MODEL
CLUSTER-CLUSTER AGGREGATION WITH STRESSSTRAIN FIELDS INVOLVED, EXAMINED IN A
MESOSCOPIC SCALE, AND LEADING TO A PHASE
SEPARATION HIGH TEMPERATURE EFFECT
Verhulst’200
A PHENOMENOLOGY BASED UPON A HALL-PETCH LIKE
CONJECTURE FOR CLUSTER-CLUSTER LATE-TIME
AGGREGATION ACCOMPANIED BY STRAIN-STRESS FIELDS
 m  1/ R
m
R
- internal stress accumulated in the inter-cluster
spaces
-average cluster radius, to be inferred from the
growth model; a possible extension, with a q,
like
 m  1/ R ; q  1 2
q
 m   m t ; R  Rt ; t  1
Verhulst’200
Remark1: WE MAY HAVE AT LEAST
MOLECULAR CHAOS ...
TWO-PHASE
ENTROPIC SYSTEM
Model clustercluster aggregation
of one-phase
molecules, forming
a cluster, in a
second phase
(solution): (A) An
early growing stage
– some single
cluster (with a
double layer) is
formed; (B) A later
growing stage –
many more clusters
are formed
Verhulst’200
TYPICAL CLUSTER-MERGING (3 GRAINS) MECHANISMS:
Dense Merging (left) vs Undense Merging (right)
(see, Meakin & Skjeltorp, Adv. Phys. 42, 1 (1993), for colloids)
3
3
1
1
2
2
t1
t1
3
3
2
2
t2
t2
A : Vtotal  Const.
B : Vtotal  Const.
Verhulst’200
RESULTING 2D-MICROSTRUCTURE IN TERMS OF
DIRICHLET-VORONOI MOSAIC REPRESENTATION (for model
colloids – Earnshow & Robinson, PRL 72, 3682 (1994)):
Remark2: Depletion zones in case B can be expected
INITIAL STRUCTURE
FINAL STRUCTURE
Verhulst’200
„Two-grain” model: a link
between growth&relaxation
„Two-grain”
spring-and-dashpot
Maxwell-like model
with (un)tight piston:
a quasi-fractional
viscoelastic element,
see A.G., J.M. Rubi, J.
Luczka, M.A.,
submitted to Chem,.
Phys.
Remark3: Untight =
competiotion and loss
Verhulst’200
THE GROWTH MODEL COMES FROM MNET (Mesoscopic
Nonequilibrium Thermodynamics, Vilar & Rubi, PNAS 98,
11091 (2001)): a flux of matter specified in the space of
cluster sizes

f x,t 
j x,t   bx 
f ( x, t )  D  x 
x
x
(!)
diffusion term
drift term
x - hypervolume of a single cluster (state variable)
T, D 0
-independent parameters (temperature
and diffusion constant)
D x   D0 x α ,
<-Note: cluster surface is crucial!
d  1 surface - to - volume
b x   D0 k BT x α

d characteristic exponent
scaling: x  R d holds ! f ;   kinetic & thermodyna mic
Verhulst’200
GIBBS EQUATION OF ENTROPY VARIATION AND THE FORM
OF DERIVED POTENTIALS AS ‘STARTING
FUNDAMENTALS’

OF CLUSTER-CLUSTER LATE-TIME AGGREGATION
   ( x, t )

S  1 T   ( x, t )fdx
-state variable and time dependent chemical
potential
-denotes variations of entropy S and
f  f ( x, t )
(i) Potential for dense micro-aggregation
(curvature-driven growth in a competing
manner: „the smaller the worse”):
   ( x)  ln( x)
(ii) Potential for undense micro-aggregation:
1d
   ( x)  x
Verhulst’200
Local conservation law and IBCs
Local
conservation
law:

t
divergence
operator
f  div ( j )  0,
f  f x, t ; j  j  x, t 
no additional sources
IBCs
(Remark4: ICs
OF
ANOMALOUS f ( x  0, t )  f ( x  , t )  0normality 

TYPE MAY
s tan dard!?
CAUSE
Remark5: typical BCs
PROBLEMS!?)
prescribed but abnormalies may
:
occur...
Verhulst’200
AFTER SOLVING THE STATISTICAL PROBLEM
f x, t IS OBTAINED
 
USEFUL PHYSICAL QUANTITIES:
x t  :
n
V fin
 x f x, t dx
n
0
where
V fin  
TAKEN MOST FREQUENTLY (see, discussion in: A.
Gadomski et al. Physica A 325, 284 (2003)) FOR THE
MODELING
Verhulst’200
REDUCED VARIANCES AS MEASURES OF
HYPERVOLUME FLUCTUATIONS
specific volume
fluctuations
Dense merging
of clusters:
 (t )  t
2
Undense
merging of
clusters:
d d 1
, t  1
the exponent reads: space
dimension over space
superdimension
 (t )  t
2
1 d 1
, t  1
the exponent reads: one
over superdimension
(cluster-radius
fluctuations)
Verhulst’200
An important
fluctuational regime of
d-DIMENSIONAL MATTER AGGREGATION COUPLED TO
STRESS RELAXATION FIELD – a metastable regime
 m  R 1 2   1
Hall-Petch
fluctuational
stress-involved
growth mode
contribution
Verhulst’200
AT WHICH BASIC GROWTH RULE DO WE ARRIVE ?
HOW DO THE INTERNAL STRESS RELAX ?
Answer: We anticipate appearence of power laws.
1 d 1
R  R(t )  t ,
t  1
1  1    ( d );  ( d )  2d  3
 m (t )  t
,

1
It builds Bethe latt. in 3-2 mode
- d-dependent quantity
 1
- a relaxation exponent
based on the above
Remark6: Bethe-lattice
(odd-number based generator):
a signature of mean-field
approximation for the relaxation
and a mark of deterministic
chaos?
Verhulst’200
Bethe
lattice, a
signature of
structural
irregularity
Verhulst’200
ABOUT A ROLE OF MEAN HARMONICITY: TOWARD A
‘PRIMITIVE’ BETHE LATTICE GENERATION (model colloids)?
Remark7: Mean harmonicity means order coming from
disorder
2 sp   sp , d  1,2,3M .H .
(d )
 sp :  ln  m (t ) / ln t ,
(d )
 sp
(d )
(d )


: ln  t  / ln t .
2
They both obey MEAN HARMONICITY [M.H.] rule, indicating, that the
case d=2 is the most effective !!!
CONCLUSION: Matter aggregation (in its late stage) and mechanical
relaxation are also coupled linearly by their characteristic
exponents ...
Verhulst’200
CONCEPT of Random Space – Filling Systems*
d=1
d=2
Problem looks dimensionality
dependent (superdimension!):
d=3
Any reasonable characteristics
is going to have (d+1) – account
in its exponent’s value, see A.G.,
J.M. Rubi, Chem. Phys. 293, 169
(2003). Remark8: Is this a
signature of existence of RCP
(randomly close-packed) phases
(see, Remark 7)?
*
R.Zallen, The Physics of Amorphous Solids, Wiley, NY,1983
Verhulst’200
CONCLUSIONS

THE MODEL IS GOVERNED BY SPACE DIMENSION d AND TEMPERATURE T;

THE MOST INTRIGUING THINGS HAPPEN IN SUFFICIENTLY HIGH T LIMIT;

THOUGH THE GROWTH EXPONENT REMAINS AS FOR LOW T CASE, THE
GROWTH TEMPO IS BETTER OPTIMISED – IT LEADS TO MEAN
HARMONICITY RULE! THE CASE OF d=2 IS THE MOST EFFICIENT;

THE STRESS RELAXATION SPEED IS ALSO WELL OPTIMISED IN HIGH T
LIMIT, AND BECAUSE OF HALL-PETCH CONJECTURE, MEAN HARMONICITY
RULE APPEARS AGAIN, AND A BETHE LATTICE GENERATOR ARISES;

THE RELAXATION EXPONENT IS A HALF OF THE GROWTH EXPONENT
WHEN EXACTLY THE HALL-PETCH CONJECTURE IS APPLIED;

BOTH EXPONENTS BEAR A „NUMERIC” SIGNATURE OF CLOSE-PACKING,
NAMELY A (d+1)-ACCOUNT, SEEN ALSO IN GROWTH & RELAXATION
EXPONENTS;

ALL THE SCENARIO DESCRIBED LEADS TO AN OPTIMAL PHASESEPARATING BEHAVIOR, WITH AN ‘EARLY SIGNATURE’ OF FIBONACCI
NUMBERING COMING FROM SCALING A SPACE DIMENSION DEPENDENT
PREFACTOR OF THE STATE VARIABLE DEPENDENT DIFFUSION
COEFFICIENT
FINALE (especially, for Verhulst’200 ?):
A HIGH T AND d DEPENDENT PHASE SEPARATION
EFFECT WOULD BE SEEN AS A MANIFESTATION
OF A METASTABLE CHAOTIC BEHAVIOR IN SPACE
(THE CLUSTERS GET SLIGHTLY APART!) BUT
THE PROCESS GOES MORE SMOOTHLY AND IN A
MORE ORDERED MANNER IN TIME THAN ITS LOW
T, CLUSTER CURVATURE DRIVEN & READILY
SPACE-FILLING (STABLE) COUNTERPART !!!
A.G. thanks COST P10 (Prof. P.
Richmond) for financial support.