COUPLING MATTER AGGLOMERATION WITH MECHANICAL STRESS RELAXATION AS A WAY OF

MODELING THE FORMATION OF JAMMED MATERIALS

Adam GadomskiInstitute 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

OBJECTIVE: TO COUPLE, ON A CLUSTER MESOSCOPIC LEVEL & IN A PHENOMENOLOGICAL WAY, ADVANCED STAGES OF CLUSTER-CLUSTER AGGREGATION WITH STRESS-STRAIN FIELDS

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

Rm /1

THE PHENOMENOLOGY BASED UPON A HALL-PETCH LIKE RELATIONSHIP CONJECTURE FOR CLUSTER-CLUSTER

LATE-TIME AGGREGATION

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

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 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

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

Dense Merging (left) vs Undense Merging (right)

(see, Meakin & Skjeltorp, Adv. Phys. 42, 1 (1993), for colloids)

TYPICAL CLUSTER-MERGING (3 GRAINS) MECHANISMS:

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

.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))

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

INITIAL STRUCTURE FINAL STRUCTURE

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE DEFORMATION „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

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

),(

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

0D T,

x - hypervolume of a single cluster (internal variable)

-independent parameters

<-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

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

),( tx

-internal variable and time dependent chemical

potential -denotes variations of entropy S and

(i) Potential for dense micro-aggregation (another one for nano-aggregation is picked up too):

(ii) Potential for undense micro-aggregation: dxx 1)(

)ln()( xx

),( txff

Local conservation law: txjjtxffjdivf

t,;,,0)(

IBCs (IC usually of minor importanmce):

!tan

0),(),0(dards

normalitytftf

a typical BCs prescribed

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

additional sources = zero

divergence operator

Local conservation law and IBCs

AFTER SOLVING THE STATISTICAL PROBLEM txf , IS OBTAINED

USEFULL 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

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

Dense merging of clusters:

1,)( 12 ttt dd

Undense merging of clusters:

1,)( 112 ttt dthe exponent reads: one

over superdimension (cluster-radius fluctuations)

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

the exponent reads: space dimension over space superdimension

specific volume fluctuations

REDUCED VARIANCES AS MEASURES OF HYPERVOLUME FLUCTUATIONS

An important fluctuational regime of

d-DIMENSIONAL MATTER AGGREGATION COUPLED TO STRESS RELAXATION FIELD

121 Rm

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

fluctuational modeHall-Petch 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

Bethe-lattice generator: a

signature of mean-field approximation

for the relaxation ?

It builds Bethe latt. in 3-2 mode

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

,)( 11 ttm

11

- d-dependent quantity

- a relaxation exponent based on the above

ABOUT A ROLE OF MEAN HARMONICITY: TOWARD A ‘PRIMITIVE’ FIBONACCI SEQUENCING (model colloids)?

Remark: No formal proof is presented so far but ...

..3,2,1,2 )()( HMddsp

dsp

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 ...

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

,ln/)(ln:)( ttmd

sp

.ln/ln: 2)( ttdsp

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATION

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. Is this a

signature of existence of RCP

(randomly close-packed) phases ?

* R.Zallen, The Physics of Amorphous Solids, Wiley, NY,1983

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)

XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE

DEFORMATIONCONCLUSIONS

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 !!!