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

Verhulst’200

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.