# adam gadomski institute of mathematics and physics university of technology and agriculture

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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. - PowerPoint PPT PresentationTRANSCRIPT

Adam Gadomski

Institute of Mathematics and PhysicsUniversity of Technology and AgricultureBydgoszcz, PolandKinetics of growth process controlled by convective fluctuations as seen by mesoscopic non-equilibrium thermodynamics17th 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 2004Cl- ionDOUBLE LAYERsurface of the growing crystalNa+ ionwater dipoleLyzosyme proteinrandom walk

GROWTH OF A SPHERE: two stages17th Marian Smoluchowski Symposium on Statistical Physics Zakopane, Poland, September 4-9 2004

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

MODEL OF GROWTH: a deterministic viewUnder assumptions [A.G., J.Sidmiak, 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 length17th Marian Smoluchowski Symposium on Statistical Physics Zakopane, Poland, September 4-9 2004

MODEL OF GROWTH (continued): specification of andFor A(R) from r.h.s. of GR reduces to17th Marian Smoluchowski Symposium on Statistical Physics Zakopane, Poland, September 4-9 2004For nonzero -s: R~t is an asymptotic solution to GR constant tempo !

MODEL OF GROWTH: stochastic partwhereAssumption about time correlations within the particles velocity field [see J.uczka et al., Phys. Rev. E 65, 051401 (2002)]K a correlation function to be proposed17th Marian Smoluchowski Symposium on Statistical Physics Zakopane, Poland, September 4-9 2004Question: 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:Fokker-Planck representation:withand(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 clustersDescribed 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.Rub, J. Chem.Phys. 115, 7100 (2001)] :withand where - Onsager coefficient17th Marian Smoluchowski Symposium on Statistical Physics Zakopane, Poland, September 4-9 2004

THE GROWTH OF THE SPHERE IN TERMS OF MNETwhere the energy (called: entropic potential)and the diffusion functionThe matter flux:Most interesting: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 I17th Marian Smoluchowski Symposium on Statistical Physics Zakopane, Poland, September 4-9 2004

RESULTS II17th Marian Smoluchowski Symposium on Statistical Physics Zakopane, Poland, September 4-9 2004

RESULTS III17th 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 2004Multiplicity = W Entropy = kB lnWIn 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
CONCLUSION17th Marian Smoluchowski Symposium on Statistical Physics Zakopane, Poland, September 4-9 2004We 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 superdiffusiveIt is hoped to have the model applicable to PROTEIN CRYSTALS?!

FINALEREFERENCES17th Marian Smoluchowski Symposium on Statistical Physics Zakopane, Poland, September 4-9 2004D.Reguera, J.M.Rub, J. Chem.Phys. 115, 7100 (2001)J.uczka, M.Niemiec, R.Rudnicki, Phys. Rev. E 65, 051401 (2002)A.G., J.Sidmiak, Cryst. Res. Technol. 37, 281 (2002)Thanks go to:> J.M.Rub (University of Barcelona)> I.Santamara-Holek (UNAM Mexico)> J.Sidmiak (UTA Bydgoszcz)for cooperation on the presented subject matter.KBN grant no. 2 P03B 032 25 (2003-2006) is acknowledged.

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