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’ 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

A PHENOMENOLOGY BASED UPON A HALL-PETCH LIKE CONJECTURE FOR CLUSTER-CLUSTER LATE-TIME AGGREGATION ACCOMPANIED BY STRAIN-STRESS FIELDS Verhulst’200 - 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

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:

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 STRUCTUREFINAL 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, (2001)): a flux of matter specified in the space of cluster sizes Verhulst’200 - hypervolume of a single cluster (state variable) -independent parameters (temperature and diffusion constant) <-Note: cluster surface is crucial! drift term diffusion term surface - to - volume characteristic exponent scaling: holds !

GIBBS EQUATION OF ENTROPY VARIATION AND THE FORM OF DERIVED POTENTIALS AS ‘STARTING FUNDAMENTALS’ OF CLUSTER-CLUSTER LATE-TIME AGGREGATION Verhulst’200 -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:

Local conservation law: IBCs (Remark4: ICs OF ANOMALOUS TYPE MAY CAUSE PROBLEMS!?) : 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 IS OBTAINED USEFUL PHYSICAL QUANTITIES: TAKEN MOST FREQUENTLY (see, discussion in: A. Gadomski et al. Physica A 325, 284 (2003)) FOR THE MODELING where Verhulst’200

Dense merging of clusters: Undense merging of clusters: the 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 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. Remark6: Bethe-lattice (odd-number based generator): a signature of mean-field approximation for the relaxation and a mark of deterministic chaos? It builds Bethe latt. in 3-2 mode Verhulst’200 - 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 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 * 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; AN OPTIMAL PHASE- SEPARATING BEHAVIOR  ALL THE SCENARIO DESCRIBED LEADS TO AN OPTIMAL PHASE- SEPARATING 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.