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

1. Kinetics of growth process controlled by convective fluctuations as seen by mesoscopic non-equilibrium thermodynamics Adam Gadomski Institute of Mathematics and Physics University of Technology and Agriculture Bydgoszcz, Poland 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

2. OBJECTIVE: To offer a refreshed view of a growth process controlled by time-dependent fluctuations of a velocity field nearby the growing object. Na+ion Lyzosyme protein water dipole random walk DOUBLE LAYER surface of the growing crystal Cl-ion 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

3. - volume - internal concentration (density) - surface - external concentration - time - position vector GROWTH OF A SPHERE: two stages 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

4. GROWTH OF A SPHERE: mass conservation law (MCL) 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

5. Growth Rule (GR) 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 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

6.  - supersaturation dimensionless parameter velocity of the particles nearby the object Could v(R,t) express a truly convective nature? What for? MODEL OF GROWTH (continued): specification of and For A(R) from r.h.s. of GR reduces to For nonzero -s: R~t is an asymptotic solution to GR – constant tempo ! 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

7. MODEL OF GROWTH: stochastic part Assumption about time correlations within the particles’ velocity field [see J.Łuczka et al., Phys. Rev. E 65, 051401 (2002)] where K – a correlation function to be proposed Question: Which is a mathematical form of K that suits optimally to a growth with constant tempo? 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

8. MODEL OF GROWTH: stochastic part (continued) Langevin-type equation with multiplicative noise: Fokker-Planck representation: with and (Green-Kubo formula), with corresponding IBC-s 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

9. 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)]: with and where - Onsager coefficient 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

10. THE GROWTH OF THE SPHERE IN TERMS OF MNET where the energy (called: entropic potential) and the diffusion function Most interesting: (dispersive kinetics !) Especially, for readily small  it indicates a superdiffusive motion ! The matter flux: 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

11. RESULTS I 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

12. RESULTS II 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

13. RESULTS III 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

14. SUMMARY – RESULTS (I) 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 Multiplicity = W Entropy = kB lnW http://hyperphysics.phy-astr.gsu.edu/hbase/therm/entrop2.html 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

15. 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 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

16. CONCLUSION • We have designed a purely MASS CONVECTIVE growth model, • the signatures thereof are as follows: • The most (technologically) desired growth speed is a constant speed; • The flux j involved in MCL is particle concentration x particle velocity, i.e. assumed to be purely convective; • 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?! 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004

17. Last but not least: to Prof. Andrzej Fuliński FINALE 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. 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) 17th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004