Particle Dynamics: Brownian Diffusion Prof. Sotiris E. Pratsinis Particle Technology Laboratory Department of Mechanical and Process Engineering, ETH Zürich, Switzerland www.ptl.ethz.ch 1 or or or or nucleation inception condensation surface growth v or v evaporation flocculation coalescence aggregation sintering chemical bonding Aerosol-based Technologies in Nanoscale Manufacturing: from Functional Materials to Devices through Core Chemical Engineering, AIChE J., 56, 3028-3035 (2010) agglomeration attachement physical adhesion
Particle Dynamics Coagulation Fragmentation Convection in Shrinking by evaporation or dissolution Growth by condensation or chemical reaction Convection out Diffusion Settling 3 Theory: Population Balance Equation n t n u Dn dv n v dt c n convection diffusion growth external force 1 v 2 0 v~,v v~ n v~ n v v~ S v dv ~ coagulation v 0 fragmentation v,v ~ nv v,v ~ Snv~ nv nv~ dv ~ dv ~ u D c S u,u, u = gas velocity vector x y z = particle diffusivity = velocity of particles of size v (e.g. settling) = coagulation rate = fragmentation rate = fragment size distribution n u un n u 0 continuity 4
Mean Free Path: Continuum vs. Free-molecule regime The mean free path of a gas,, is the average distance traveled by gas molecules between their collisions. When particles are much larger than (e.g. d p > 10), they do not sense individual collisions with molecules feeling a continuum so particle motion takes place in the so-called continuum regime and described by the standard or classic Navier-Stokes equations, the gospel of engineering. When particles are much smaller than (e.g. d p < ), they are in the so-called free-molecule regime and their motion is described by the kinetic theory of gases. Inbetween, interpolations are devised specific to the process (e.g. for diffusion, coagulation or condensation) 5 1. DIFFUSION Particles suspended in a fluid medium exhibit a haphazard dancing motion Botanist Robert Brown discovered that this motion was a general property of matter regardless of its origin (dust vs. pollen) Hands-on experiment resembling the motion of oil droplets in water with ball bearings and metal rings on a vibrating table at the Museum of Fine Arts at the San Francisco Exploratorium organized by Dr. Frank Oppenheimer (brother of Robert the father of the Atomic Bomb) 6
Diffusion is the net migration of particles from regions of HIGH to LOW concentration Net rate of transport into that element: Friedlander, S.K., Smoke, Dust and Haze, Chapter 2, Oxford Press, 2nd Edition, New York, 2000 7 The rate of change of the number of particles per unit volume (& size), n, in the elemental volume δxδyδz is: From experimental observations: Fick s first law Substituting in the above gives second Fick s law: Coefficient of Diffusion or Diffusivity, D 8
D = f (particle size and gas properties) Consider particle transport in one dimension, x Release equally sized particles, N 0, at t=0 and observe the n distribution in space and time For the boundary conditions at x = 0, x = & t =0, the particle concentration distribution in x,t is 9 The mean square displacement of the particles from x=0 at time t is: We can measure chequered glass. by putting spheres in a liquid and follow their motion through a 10
The goal is to relate the mean square displacement of a particle with the energy required for this job. Force balance on a particle in Brownian motion: Now multiply both sides of eq. (5) by the displacement x and divide by m. For a single particle: x du f ux F() t x dt m m (6) 11 define as β = f/m and A = F(t)/m and remember that: Using these expressions eq. (6) becomes Integrate from t=0 to t and obtain: where t is a variable of integration representing time. 12
Average over all particles: Since there is no correlation between displacement x and kick, A, the second term of eq. (7) vanishes: You can also write: 13 Because the derivative of the mean over particles with respect to time is equal to the mean of the derivative: From eq. (8) & (9): Integrate over time from t = 0 to t for t >> 1/β (or β t >> 1): 14
Invoke the equipartition of energy, meaning that the kinetic energy of particles is equal to the kinetic energy of the surrounding gas molecules: This is the Stokes-Einstein expression for D. It relates D to the properties of the fluid and the particle through the friction coefficient. 15 Perrin s (1910) study allowed calculation of the Avogadro number N AV. By observing the motion of an emulsion he calculated the number of (attacking) molecules: where R is the gas constant He gave an experimental proof of the kinetic theory by measuring the net displacement Modern methods show that N AV =6.023 10 23 molecules/mol 16
Friction coefficient mean free path of gas medium with ρ : density of the medium (e.g. air) m 1 : molecular mass of the medium In the continuum regime (d P >> λ ): f = 3d P In the free molecular regime (d P << λ ): with a: accomodation coefficient 0.9 In the entire range: with C: Cunningham correction factor 17 Coefficients A1, A2, and A3 are empirical constants that have been obtained by measuring the settling velocities of particles in various gases. Table 1 gives these constants for various gases (Rader, D. J., Momentum slip correction factor for small particles in nine common gases, J. Aerosol Sci., 21 (1990), 161-168) The ratio of the mean free path of the gas and the particle radius is the Knudsen number Kn = 2λ/dp. The Cunningham correction factor does not change very much with different gases for the same Kn (Table 2). 18
Diffusion and sedimentation dominate the particles motion at opposite size regimes (Table 3). 19 DIFFUSION during LAMINAR PIPE FLOW Entrance length: L = 0.04dRe (laminar flow) d : pipe diameter Re : Reynolds-number The momentum boundary layer develops rapidly while the concentration boundary layer follows: 20
Separation of variables: Result: So the average particle concentration is defined as: And in general it is given as: Where G k and λ k2 are given in Table 3.1 in the book by Friedlander (1977). The above ratio is called also the particle penetration, P, and it is defined as: 21 Penetration curves for particle diffusion to pipe walls Penetration versus deposition parameter for circular tubes and rectangular cross section channels: D: particle Diffusivity, L: tube or channel length, Q: gas flowrate, h: interplate distance, W: channel width (Hinds, 1982) 22
Rowell, J. M., Scientific American, October 1986, 147. 23
MacChesney, J. B., O Connor, P. B. and Presby, H. M., 1974, A new technique for preparation of low-loss and graded index optical fibers. Proc. IEEE 62, 1280-1281. J. Aerosol Sci., 20, 101-111 (1989)