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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
- internal stress accumulated in the inter-cluster spaces
R
-average cluster radius, to be inferred from the growth model; a possible extension, with a
q
, like
m
m
1 /
q
m t
;
R R
;
q
1 2
R t
;
t
1
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE DEFORMATION
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
TYPICAL CLUSTER-MERGING (3 GRAINS) MECHANISMS: Dense Merging (left) vs Undense Merging (right) (see,
Meakin & Skjeltorp, Adv. Phys. 42, 1 (1993),
3 3 1 2 1 2 for colloids)
t
1
t
1 3 2 3 2
t
2
t
2
A : V total
Const
.
B : V total
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 J
AMMING,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
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
j
b
x f
(
x
,
t
)
D
x (!)
diffusion term drift term x - hypervolume of a single cluster (internal variable)
D b
T, D
x x
0 -independent parameters
D
0
D
0
x α k
, < Note:
cluster surface is crucial!
B T
x α
d d
1 surface - to - volume characteristic exponent
x
R 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
S
(
x
,
t
)
fdx
(
x
,
t
) -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
)
x
1
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
;
j
j
additional sources = zero IBCs (IC usually of minor importanmce):
f
( 0 ,
t
)
f
( ,
t
) 0
s
tan
dard
!
a typical BCs prescribed
XIX SITGES CONFERENCE
JAMMING,YIELDING, AND IRREVERSIBLE DEFORMATION
f
IS OBTAINED USEFULL PHYSICAL QUANTITIES:
x n
where :
V
0
fin x n f
dx 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
Dense merging of clusters:
2 (
t
)
t d specific volume fluctuations d
1 ,
t
1 Undense merging of clusters: the exponent reads: space dimension over space 2 (
t
)
t
1
d
superdimension 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:
m
(
t
)
We anticipate appearence of power laws
.
R t
1
R
(
t
1 , )
t
1
d
1 , (
d
);
t
(
d
) 1 2
d
3 1 - d-dependent quantity 1 It builds Bethe latt. in 3-2 mode Bethe-lattice generator: a signature - a relaxation exponent based on the above of mean-field approximation for the relaxation ?
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 (
d sp
sp
( )
d
)
:
sp
(
d
) ln ,
d
sp
(
d
) : ln
m
1 , (
t
) 2 , 3 / ln
M
.
H
, / ln .
.
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!):
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 ?
d=3 * 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 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)
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)