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

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.


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


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


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)


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


),( 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


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


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


(dispersive kinetics !)

Especially, for readily small it indicates a superdiffusive motion !

RESULTS I


xconst

xxVgr

.

)(

RESULTS II

1

12 2

2)(:0

R

RRAfor

221

2

221

2

2 2

2)(:0

RR

RRRAfor


RESULTS III


SUMMARY – RESULTS (I)


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

http://hyperphysics.phy-astr.gsu.edu/hbase/therm/entrop2.html

SUMMARY – RESULTS (II)

II. On a superdiffusive (Levy flight in the double layer?) motion of nearby particles, feeding the object: 0<<1/2 formally holds


http://classes.yale.edu/fractals/RandFrac/Levy/Levy.html

CONCLUSION


We have designed a purely MASS CONVECTIVE growth model,

the signatures thereof are as follows:

(i) The most (technologically) desired growth speed is a constant speed;

(ii) The flux j involved in MCL is particle concentration x particle velocity, i.e. assumed to be purely convective;

(iii) The most efficient stochastic characteristic of the moving nearby particles appears to be superdiffusive

It is hoped to have the model applicable to PROTEIN CRYSTALS?!

FINALE

REFERENCES


D.Reguera, J.M.Rubì, J. Chem.Phys. 115, 7100 (2001)J.Łuczka, M.Niemiec, R.Rudnicki, Phys. Rev. E 65, 051401 (2002)A.G., J.Siódmiak, Cryst. Res. Technol. 37, 281 (2002)

Thanks go to:

> J.M.Rubì (University of Barcelona)

> I.Santamarìa-Holek (UNAM Mexico)

> J.Siódmiak (UTA Bydgoszcz)

for cooperation on the presented subject matter.

KBN grant no. 2 P03B 032 25 (2003-2006) is acknowledged.

Last but not least: to Prof. Andrzej Fuliński

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

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