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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 Presentation

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  • 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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