Adam Gadomski Institute of Mathematics and Physics University of Technology and Agriculture Bydgoszcz, Poland Kinetics of growth process controlled by convective fluctuations as seen by mesoscopic non-equilibrium thermodynamics 17 th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
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Adam Gadomski Institute of Mathematics and Physics University of Technology and Agriculture Bydgoszcz, Poland Kinetics of growth process controlled by.
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Adam Gadomski
Institute of Mathematics and PhysicsUniversity of Technology and Agriculture
Bydgoszcz, Poland
Kinetics of growth process controlled by
convective fluctuations as seen by
mesoscopic non-equilibrium thermodynamics
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
OBJECTIVE:
To offer a refreshed view of a growth process controlled by time-dependent fluctuations of a
velocity field nearby the growing object.
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
Cl- ion
DOUBLE LAYER
surface of the growing crystal
Na+ ion
water dipole Lyzosyme protein
random walk
Vt
Cc
- volume
- surface
- time
- internal concentration (density)
- external concentration
r
- position vector
tV tV1tV1tV
t t
1t 1t
rc
rc
rc
rC
rCrC
GROWTH OF A SPHERE: two stages
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
GROWTH OF A SPHERE: mass conservation law (MCL)
tVtV
dVrcrCttt
m
11
1
t
drct
mSj )]([
ttV
drcdVrcrCdt
dSj
dVrCtmtV
1
1
tVtVtV
dVrcdVrCtm1
tV tV1tV1tV
t t
1t 1t
rc
rc
rc
rC
rCrC
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
MODEL OF GROWTH: a deterministic view
Under assumptions [A.G., J.Siódmiak, Cryst. Res. Technol. 37, 281 (2002)]:
(i) The growing object is a sphere of radius: ;
(ii) The feeding field is convective: ;
(iii) The generalized Gibbs-Thomson relation:
where: ; (curvatures !)
and when (on a flat surface)
: thermodynamic parameters
i=1 capillary (Gibbs-Thomson) length
i=2 Tolman length
0)( tRR
rtRvRcrc ej ),()()],,~([
)1()(),,~( 222110 KKcRcrc
RK
21 22
1
RK
)(0 Rcc R
i
),()( tRvRAdt
dR
Growth Rule (GR)
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
MODEL OF GROWTH (continued): specification of
and
)(RA
),( tRv
11 2where,
2)(
ccc
RRRRR
RRA
221
2
221
2
2
2)(
RR
RRRA
For A(R) from r.h.s. of GR reduces to02
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
),( tRv velocity of the particles nearby the object
Could v(R,t) express a truly convective nature? What for?
- supersaturation dimensionless parameter
For nonzero -s: R~t is an asymptotic solution to GR – constant tempo !
MODEL OF GROWTH: stochastic part
)(),( tVtRv
where
)()()(,0)( stKsVtVtV
Assumption about time correlations within the particles’ velocity field [see J.Łuczka et al., Phys. Rev. E 65, 051401 (2002)]
K – a correlation function to be proposed
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
Question: Which is a mathematical form of K that suits optimally to a growth with constant tempo?
MODEL OF GROWTH: stochastic part (continued)
Langevin-type equation with multiplicative noise:
)()( tVRAdt
dR
Fokker-Planck representation:
),(),( tRJR
tRPt
with ),()]()[(),()()()(),( 2 tRPR
RAtDtRPRAR
RAtDtRJ
and dssKtDt
0
)()( (Green-Kubo formula),
with corresponding IBC-s17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
MESOSCOPIC NONEQUILIBRIUM THERMODYNAMICS (MNET): a simple crystallization of spherical clusters
Described in terms of the Kramers picture: As a diffusion over an energetic barrier !
An overview: Basic equation for the objects’ distribution function of „size” reads [see D.Reguera, J.M.Rubì, J. Chem.Phys. 115,
7100 (2001)] :),( tff
f
tDftbt
f),(),(
with
and where - Onsager coefficient
),(
)(),(
tTf
Ltb
),(),( tTbktD B )(L
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
THE GROWTH OF THE SPHERE IN TERMS OF MNET
R
tRPtRDtRP
RTk
tRD
Rt
tRP
B
),(),(),(
),(),(
where the energy (called: entropic potential) )(ln RATkB
and the diffusion function 2)()(),( RAtDtRD
R
tRPtRDtRP
RTk
tRDtRJ
B
),(
),(),(),(
),(
The matter flux:
Most interesting: 01 for)( ttttD
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
(dispersive kinetics !)
Especially, for readily small it indicates a superdiffusive motion !
RESULTS I
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
xconst
xxVgr
.
)(
RESULTS II
1
12 2
2)(:0
R
RRAfor
221
2
221
2
2 2
2)(:0
RR
RRRAfor
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
RESULTS III
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
SUMMARY – RESULTS (I)
17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004
Multiplicity =
Entropy = kB ln
In order to achieve a ‘technologically favorable’ constant tempo of growth, „an experimenter” would try to keep:
I. Entropic (Boltzmann) character of the free energy