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
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
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
THE PHENOMENOLOGY BASED UPON A HALL-PETCH LIKE
RELATIONSHIP CONJECTURE FOR CLUSTER-CLUSTER
LATE-TIME AGGREGATION
 m  1/ R
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
 m  1/ R ; q  1 2
q
 m   m t ; R  Rt ; t  1
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
TWO-PHASE
SYSTEM
Model clustercluster 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
TYPICAL CLUSTER-MERGING (3 GRAINS) MECHANISMS:
Dense Merging (left) vs Undense Merging (right)
(see, Meakin & Skjeltorp, Adv. Phys.3 42, 1 (1993), for colloids)
3
1
1
2
2
t1
t1
3
3
2
2
t2
t2
A : Vtotal  Const.
B : Vtotal  Const.
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
RESULTING 2D-MICROSTRUCTURE IN TERMS OF
DIRICHLET-VORONOI MOSAIC REPRESENTATION (for model
colloids – Earnshow & Robinson, PRL 72, 3682 (1994))
INITIAL STRUCTURE
FINAL STRUCTURE
XIX SITGES CONFERENCE JAMMING,YIELDING, AND IRREVERSIBLE DEFORMATION
„Two-grain” model: a merger
between growth&relaxation
• „Two-grain”
spring-anddashpot Maxwelllike model with
(un)tight piston: a
quasi-fractional
viscoelastic element
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
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

f x,t 
j x,t   bx 
f ( x, t )  D  x 
x
x
drift term
(!)
diffusion term
x - hypervolume of a single cluster (internal variable)
-independent parameters
T, D 0
Dx   D0 x α ,<-Note: cluster surface is crucial!
d  1 surface - to - volume
bx   D0 k BT x α

d characteristic exponent
scaling: x  R d holds ! f ;   kinetic & thermodyna mic
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
GIBBS EQUATION OF ENTROPY VARIATION AND THE FORM
OF DERIVED POTENTIALS AS ‘STARTING
FUNDAMENTALS’

OF CLUSTER-CLUSTER LATE-TIME AGGREGATION
   ( x, t )

S  1 T   ( x, t )fdx
-internal variable and time dependent chemical
potential
-denotes variations of entropy S and
f  f ( x, t )
(i) Potential for dense micro-aggregation
(another one for nano-aggregation is picked up
too):
   ( x)  ln( x)
(ii) Potential for undense micro-aggregation:
   ( x)  x1 d
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
Local conservation law and IBCs
Local
conservation
law:

t
divergence
operator
f  div ( j )  0,
f  f x, t ; j  j  x, t 
additional sources = zero
IBCs (IC
usually of
minor
importanmce):
f (0, t )  f (, t )  0normality 

s tan dard!
a typical BCs prescribed
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
AFTER SOLVING THE STATISTICAL PROBLEM
f x, t IS OBTAINED
 
USEFULL PHYSICAL QUANTITIES:
x t  :
n
V fin
 x f x, t dx
n
0
where
V fin  
TAKEN MOST FREQUENTLY (see, discussion in: A.
Gadomski et al. Physica A 325, 284 (2003)) FOR THE
MODELING
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
REDUCED VARIANCES AS MEASURES OF HYPERVOLUME
FLUCTUATIONS
specific volume
fluctuations
Dense merging
of clusters:
 (t )  t
2
Undense
merging of
clusters:
d d 1
, t  1
the exponent reads: space
dimension over space
superdimension
 (t )  t
2
1 d 1
, t  1
the exponent reads: one
over superdimension
(cluster-radius
fluctuations)
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
An important
fluctuational regime of
d-DIMENSIONAL MATTER AGGREGATION COUPLED TO
STRESS RELAXATION FIELD
 m  R 1 2   1
Hall-Petch
contribution
fluctuational mode
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
AT WHICH BASIC GROWTH RULE DO WE ARRIVE ?
HOW DO THE INTERNAL STRESS RELAX ?
Answer: We anticipate appearence of power laws.
1 d 1
R  R(t )  t ,
t  1
1  1    ( d );  ( d )  2d  3
 m (t )  t
,

1
It builds Bethe latt. in 3-2 mode
Bethe-lattice
- d-dependent quantity
generator: a
signature
of mean-field
- a relaxation exponent
approximation
based on the above
for the relaxation ?
 1
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
ABOUT A ROLE OF MEAN HARMONICITY: TOWARD A
‘PRIMITIVE’ FIBONACCI SEQUENCING (model colloids)?
Remark: No formal proof is presented so far but ...
2 sp   sp , d  1,2,3M .H .
(d )
 sp :  ln  m (t ) / ln t ,
(d )
 sp
(d )
(d )


: ln  t  / ln t .
2
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
CONCEPT of Random Space – Filling Systems*
d=1
d=2
Problem looks dimensionality
dependent (superdimension!):
d=3
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
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE
DEFORMATION
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 TWOAS 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)
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 !!!