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 AusloosSUPRATECS
University of Liège, Liège, Belgium
Verhulst’200
16-18 September 2004, Royal Military Academy, Brussels, Belgium
OBJECTIVE: TO REVEAL NUMBER-THEORETIC ASPECTS OF ADVANCED STAGES OF A MODEL CLUSTER-CLUSTER AGGREGATION WITH STRESS-STRAIN FIELDS INVOLVED, EXAMINED IN A MESOSCOPIC SCALE, AND LEADING TO A PHASE SEPARATION HIGH TEMPERATURE EFFECT
Verhulst’200
Rm /1
A PHENOMENOLOGY BASED UPON A HALL-PETCH LIKE CONJECTURE FOR CLUSTER-CLUSTER LATE-TIME
AGGREGATION ACCOMPANIED BY STRAIN-STRESS FIELDS
Verhulst’200
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
1;; ttRRtmm
21;/1 qRqm
TWO-PHASE ENTROPIC SYSTEM
Model cluster-cluster 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
Remark1: WE MAY HAVE AT LEAST
MOLECULAR CHAOS ...
Verhulst’200
Dense Merging (left) vs Undense Merging (right)
(see, Meakin & Skjeltorp, Adv. Phys. 42, 1 (1993), for colloids)
TYPICAL CLUSTER-MERGING (3 GRAINS) MECHANISMS:
.V:A total Const .V:B total Const
1
1 1
22
12
3
3 3
3
2 2
2
t t
tt
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
Verhulst’200
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
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
(!)x
x,tfxDtxf
xxbx,tj
),(
Verhulst’200
0D T,
x - hypervolume of a single cluster (state variable)-independent parameters (temperature and diffusion constant)
<-Note: cluster surface is crucial!
drift term diffusion term
α
B
α
xTkDxb
xDxD
0
0 ,
surface - to - volume characteristic exponentd
d 1
scaling: holds ! dRx micthermodyna&kinetic; f
fdxtxTS ),(1
GIBBS EQUATION OF ENTROPY VARIATION AND THE FORM OF DERIVED POTENTIALS AS ‘STARTING
FUNDAMENTALS’ OF CLUSTER-CLUSTER LATE-TIME AGGREGATION
Verhulst’200
),( tx
-state variable and time dependent chemical
potential -denotes variations of entropy S and
(i) Potential for dense micro-aggregation (curvature-driven growth in a competing manner: „the smaller the worse”):
(ii) Potential for undense micro-aggregation: dxx 1)(
)ln()( xx
),( txff
Local conservation law: txjjtxffjdivf
t,;,,0)(
IBCs (Remark4: ICs OF ANOMALOUS TYPE MAY CAUSE PROBLEMS!?):
?!tan
0),(),0(dards
normalitytxftxf
Remark5: typical BCs prescribed but abnormalies may
occur...
Verhulst’200
no additional sources
divergence operator
Local conservation law and IBCs
AFTER SOLVING THE STATISTICAL PROBLEM txf , IS OBTAINED
USEFUL PHYSICAL QUANTITIES:
TAKEN MOST FREQUENTLY (see, discussion in: A. Gadomski et al. Physica A 325, 284 (2003)) FOR THE
MODELING
fin
V
nn
V
dxtxfxtxfin
0
,:
where
Verhulst’200
Dense merging of clusters:
1,)( 12 ttt dd
Undense merging of clusters:
1,)( 112 ttt dthe exponent reads: one
over superdimension (cluster-radius fluctuations)
the exponent reads: space dimension over space superdimension
specific volume fluctuations
Verhulst’200
REDUCED VARIANCES AS MEASURES OF HYPERVOLUME FLUCTUATIONS
An important fluctuational regime of
d-DIMENSIONAL MATTER AGGREGATION COUPLED TO STRESS RELAXATION FIELD – a metastable regime
121 Rm
Verhulst’200
fluctuational
growth mode
Hall-Petch
stress-involved
contribution
AT WHICH BASIC GROWTH RULE DO WE ARRIVE ?
HOW DO THE INTERNAL STRESS RELAX ?
Answer: We anticipate appearence of power laws.
1,)( 11 tttRR d
32)();( ddd
Remark6: Bethe-lattice(odd-number based generator):
a signature of mean-field approximation for the relaxationand a mark of deterministic chaos?
It builds Bethe latt. in 3-2 mode
Verhulst’200
,)( 11 ttm
11
- d-dependent quantity
- a relaxation exponent based on the above
Verhulst’200
Bethe
lattice, a signature of
structural irregularity
ABOUT A ROLE OF MEAN HARMONICITY: TOWARD A ‘PRIMITIVE’ BETHE LATTICE GENERATION (model colloids)?
Remark7: Mean harmonicity means order coming from disorder
..3,2,1,2 )()( HMddsp
dsp
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
,ln/)(ln:)( ttmd
sp
.ln/ln: 2)( ttdsp
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CONCEPT of Random Space – Filling Systems* d=1 d=2
d=3
Problem looks dimensionality
dependent (superdimension!):
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
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 PHASE-AN OPTIMAL PHASE-SEPARATING BEHAVIORSEPARATING BEHAVIOR, WITH AN ‘EARLY SIGNATURE’ OF FIBONACCI NUMBERING COMING FROM SCALING A SPACE DIMENSION DEPENDENT PREFACTOR OF THE STATE VARIABLE DEPENDENT DIFFUSION COEFFICIENT
Verhulst’200 CONCLUSIONS
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.