Adam Gadomski Institute of Mathematics and Physics University of Technology and Agriculture

1. COUPLING MATTER AGGLOMERATION WITH MECHANICAL STRESS RELAXATION AS A WAY OF MODELING THE FORMATION OF JAMMED MATERIALS Adam Gadomski Institute of Mathematics and Physics University of Technology and Agriculture Bydgoszcz, Poland XIX SITGES CONFERENCE JAMMING, YIELDING, AND IRREVERSIBLE DEFORMATION 14-18 June, 2004, Universitat de Barcelona, Sitges, Catalunya

2. OBJECTIVE: TO COUPLE, ON A CLUSTER MESOSCOPIC LEVEL & IN A PHENOMENOLOGICAL WAY, ADVANCED STAGES OF CLUSTER-CLUSTER AGGREGATION WITH STRESS-STRAIN FIELDS XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION

3. THE PHENOMENOLOGY BASED UPON A HALL-PETCH LIKE RELATIONSHIP CONJECTURE FOR CLUSTER-CLUSTER LATE-TIME AGGREGATION XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION - 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

4. XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION TWO-PHASE 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

5. Dense Merging (left) vs Undense Merging (right) (see, Meakin & Skjeltorp, Adv. Phys. 42, 1 (1993), for colloids) XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION TYPICAL CLUSTER-MERGING (3 GRAINS) MECHANISMS:

6. RESULTING 2D-MICROSTRUCTURE IN TERMS OF DIRICHLET-VORONOI MOSAIC REPRESENTATION (for model colloids – Earnshow & Robinson, PRL 72, 3682 (1994)) XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION INITIAL STRUCTURE FINAL STRUCTURE

7. XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION„Two-grain” model: a merger between growth&relaxation • „Two-grain” spring-and-dashpot Maxwell-like model with (un)tight piston: a quasi-fractional viscoelastic element

8. 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 XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLE DEFORMATION surface - to - volume characteristic exponent diffusion term drift term - hypervolume of a single cluster (internal variable) • independent parameters • <-Note: cluster surface is crucial! scaling: holds !

9. GIBBS EQUATION OF ENTROPY VARIATION AND THE FORM OF DERIVED POTENTIALS AS ‘STARTING FUNDAMENTALS’ OF CLUSTER-CLUSTER LATE-TIME AGGREGATION XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION • internal variable and time dependent chemical • potential • denotes variations of entropyS and • (i) Potential for dense micro-aggregation (another one for nano-aggregation is picked up too): • (ii) Potential for undense micro-aggregation:

10. XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION Local conservation law and IBCs divergence operator additional sources = zero a typical BCs prescribed Local conservation law: IBCs (IC usually of minor importanmce):

11. XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION where TAKEN MOST FREQUENTLY(see, discussion in: A. Gadomski et al. Physica A 325, 284 (2003)) FOR THE MODELING AFTER SOLVING THE STATISTICAL PROBLEM IS OBTAINED USEFULL PHYSICAL QUANTITIES:

12. XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION REDUCED VARIANCES AS MEASURES OF HYPERVOLUME FLUCTUATIONS specific volume fluctuations the exponent reads: space dimension over space superdimension the exponent reads: one over superdimension (cluster-radius fluctuations) Dense merging of clusters: Undense merging of clusters:

13. An important fluctuational regime of d-DIMENSIONAL MATTER AGGREGATION COUPLED TO STRESS RELAXATION FIELD XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION Hall-Petchcontribution fluctuational mode

14. XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION AT WHICH BASIC GROWTH RULE DO WE ARRIVE ? HOW DO THE INTERNAL STRESS RELAX ? Answer:We anticipate appearence of power laws. It builds Bethe latt. in 3-2 mode - d-dependent quantity Bethe-lattice generator: a signature of mean-field approximation for the relaxation ? - a relaxation exponent based on the above

15. XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION ABOUT A ROLE OF MEAN HARMONICITY: TOWARD A ‘PRIMITIVE’ FIBONACCI SEQUENCING (model colloids)? Remark: No formal proof is presented so far but ... They both obey mean harmonicity rule, indicating, see [M.H.] 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 ...

16. XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION Problem looks dimensionality dependent(superdimension!): Any reasonable characteristics is going to have (d+1) – account in its exponent’s value. Is this a signature of existence of RCP (randomly close-packed) phases ? CONCEPT of Random Space – Filling Systems* * R.Zallen, The Physics of Amorphous Solids, Wiley, NY,1983

17. XIX SITGES CONFERENCEJAMMING,YIELDING, AND IRREVERSIBLEDEFORMATION CONCLUSIONS • UTILISING A HALL-PETCH (GRIFFITH) LIKE CONJECTURE ENABLES TO COUPLE LATE-STAGE MATTER AGGREGATION AND MECHANICAL RELAXATION EFFECTIVELY • SUCH A COUPLING ENABLES SOMEONE TO STRIVE FOR LINKING TOGETHER BOTH REGIMES, USUALLY CONSIDERED AS DECOUPLED, WHICH IS INCONSISTENT WITH EXPERIMENTAL OBSERVATIONS FOR TWO- AS WELL AS MANY-PHASE (SEPARATING) VISCOELASTIC SYSTEMS • THE ON-MANY-NUCLEI BASED GROWTH MODEL, CONCEIVABLE FROM THE BASIC PRINCIPLES OF MNET, AND WITH SOME EMPHASIS PLACED ON THE CLUSTER SURFACE, CAPTURES ALMOST ALL THE ESSENTIALS IN ORDER TO BE APPLIED TO SPACE DIMENSION AS WELL AS TEMPERATURE SENSITIVE INTERACTING SYSTEMS, SUCH AS COLLOIDS AND/OR BIOPOLYMERS (BIOMEMBRANES; see P.A. Kralchevsky et al., J. Colloid Interface Sci. 180, 619 (1996)) • IT OFFERS ANOTHER PROPOSAL OF MESOSCOPIC TYPE FOR RECENTLY PERFORMED 2D EXPERIMENTS CONSIDERED BASED ON MICROSCOPIC GROUNDS, e.g. F. Ghezzi et al. J. Colloid Interface Sci. 251, 288 (2002)

18. LITERATURE:- A.G. (mini-review) Nonlinear Phenomena in Complex Systems 3, 321-352 (2000) http://www.j-npcs.org/online/vol2000/v3no4/v3no4p321.pdf - J.M. Rubi, A.G. Physica A 326, 333-343 (2003) - A.G., J.M. Rubi Chemical Physics 293, 169-177 (2003) -A.G. Modern Physics Letters B 11, 645-657 (1997) ACKNOWLEDGEMENT !!!