1. 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

2. OBJECTIVE: To offer a refreshed view of a growth process controlled by time-dependent fluctuations of a velocity field nearby the growing object. 17 th 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

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

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

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

6. MODEL OF GROWTH (continued): specification of and For A(R) from r.h.s. of GR reduces to 17 th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004 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 !

7. MODEL OF GROWTH: stochastic part where 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 17 th 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?

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 17 th 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 17 th 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 The matter flux: Most interesting: 17 th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004 (dispersive kinetics !) Especially, for readily small  it indicates a superdiffusive motion !

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

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

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

14. SUMMARY – RESULTS (I) 17 th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004 Multiplicity =  Entropy = k B 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

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 17 th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004 http://classes.yale.edu/fractals/RandFrac/Levy/Levy.html

16. CONCLUSION 17 th Marian Smoluchowski Symposium on Statistical Physics – Zakopane, Poland, September 4-9 2004 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?!

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