Critical Size and Particle Growth

Transcription

1 Critical Size an article Growth rof. Sotiris E. ratsinis article Technology Laboratory Department of Mechanical an rocess Engineering, ETH Zürich, Switzerlan 1 Nucleation-Conensation A phase transition is encountere in many inustrial (e.g. crystallization, carbon black prouction) an environmental (e.g. smog formation) processes The funamental equation that escribes these processes is: n t vn i 0 With bounary conitions: at nv I nucleation t 0 n n 0 initial istribution i i

2 The goal is to etermine: 1. the critical iameter for particle formation which is ictate by thermoynamics. the growth rate that is etermine by thermoynamics an transport 3. the nucleation rate which is etermine by thermoynamics an kinetic theory by physical (e.g.cooling) or chemical (e.g. reactions) riving forces 3 1 Critical article Size Key feature: The curve interface The goal is to erive an expression relating the concentration (vapor pressure) of species A with a particle (roplet) of raius at equilibrium (Seinfel, 1986) If the interface was flat which is, for example, the tabulate equilibrium concentration or vapor pressure at a given temperature an pressure. Consier the change in Gibbs free energy accompanying the formation of a single rop (embryo) of pure material A of iameter containing g molecules of A: G G G embryo system pure vapor (1) 4

3 Now let s say that the number of molecules in the starting conition of pure vapor is n T. After the embryo forms, the number of vapor molecules remaining is nnt g. Then the above equation is written as: G ng v gg l n T G v () where G V an G l are the free energies of a molecule in a liqui an vapor phases an is the surface energy G ggl Gv v G G 6 Noting that gvl 3 6 Where v l is the volume occupie by a molecule in the liqui phase (equivalent sphere in liqui phase). 3 l l v (3) 5 Before we go further let s evaluate the ifference in Gibbs free energy: G = V then G = (v l -v v ) But v l << v v then G = - v v Accoring to ieal gas law v v = k B T/ Then A G G k T k T v l A B B ln kbtlns A 0 A0 Where S is the saturation ratio. 6

4 Now equation 3 becomes: G Now plot G as a function of 3 v k B T S ln 6 l volume free energy of an embryo G surface free energy roplet at equilibrium with surrouning vapor S 1 S 1 S <1 monotonic increase in G S > 1 positive an negative contributions at small the surface tension ominates an the behavior of G as a function of is close to that for S <1. For larger the first term becomes 7 important. At G 0 4 vl k TlnS B This is the minimum possible particle size. This equation relates the equilibrium raius of a roplet of a pure substance to the physical properties of the substance an the saturation ratio of its environment. It is calle also the Kelvin equation an the critical iameter is calle the Kelvin iameter. 8

5 This equation relates the equilibrium raius of a roplet of a pure substance to the physical properties of the substance an the saturation ratio of its environment. It is calle also the Kelvin equation an the critical iameter is calle the Kelvin iameter. The Kelvin equation states that the vapor pressure over a curve interface always excees that of the same substance over a flat surface: See the anchoring of the surface molecules on a flat an a curve surface. Surface molecules are anchore on two molecules on the layer below flat surfaces while on curve interfaces some are anchore on just one! These can easily escape (evaporate) from the conense (liqui or soli) phase. 9 article Growth The mechanism for particle growth refers to roplet or particle growth from gas (conensation), to crystal growth from solution etc.. In all cases mass shoul be transporte to the particle surface. In principle, two steps are require, a iffusional step followe by a surface reaction or rearrangement step. In conensation the former is ominant while in crystallization is the latter. In many processes both can be ominant. 10

6 .1 Mass transfer to a particle surface (continuum) Consier a single roplet growing by conensation without convection at rather ilute conitions. The goal is to etermine the flux of mass to its surface. For this the vapor concentration profile aroun the roplet is neee at steay state: roplet t D r r r C C 0 r (1) D = vapor iffusivity C = vapor concentration (moles/cm 3 ) vapor molecules 11 With bounary conitions: at r = / C = C the equilibrium concentration at the roplet surface at r = C = C bulk vapor concentration Solving the above equation for C as a function of r gives: C C 1 () C C r Then the rate of conensation F is: F C D r r D C C 0 C D C (3) 1

7 An the rate of particle volume growth is: v 6 FMW DC C 3 MW (4) where MW an r are the molecular weight an ensity of the conensing material So the iameter growth rate is (cm/s): 4D C C MW C C e exp 4 v l k B T (5) 133. Mass transfer to a particle surface (free molecule) The collision rate per unit area is: z N Cc 4 AV (6) where c an m 1 are the molecular velocity an mass an N AV the Avogaro number 1 N so z becomes AV C C 8k BT z (7) 4 m1 14

8 Then the rate of conensation F to particle surface is: F z area / N AV kbt m 1 1 C C (8) An the rate of particle volume growth is: v FMW k BT m 1 1 C C MW (9) So the iameter growth rate is: MW kbt m 1 1 C C (10) 15.3 Mass transfer to a particle surface (entire spectrum) For particle growth from the free molecule to continuum regime, the expression for the continuum regime is extene by an interpolation factor: 4D C C MW 1 Kn 11.71Kn 1.33Kn (11) where the Knusen number is Kn= / This is calle the Fuchs effect. 16

9 The effect of of temperature epression is to reuce the partial pressure of vapor at the roplet surface an slow the rate of evaporation. Similarly a temperature enhancement slows the rate of conensation. (aapte from Hins (198)) 17 (aapte from Hins (198)) 18

10 (aapte from Hins (198)) 19.4 Growth ominate by article hase Reactions The growth of the conense phase is limite by particle or film volume or surface reaction or the so-calle rearrangement step. Transport to the conense phase is much faster than this step. This is the case, for example, in formation of aci rain where gas transport to the roplet surface rapily takes place while its conversion to aci may take ays epening on meteorology. Similarly, in soot formation an growth uring hyrocarbon combustion, the rate-limiting step is the growth of soot layers on each particle surface by reaction of hyrocarbon raicals onto the soot particle surface. Also, in epitaxial eposition of thin films for microelectronics by chemical vapor eposition, the rate limiting step is reaction on the film surface. Crystal forming (rearrangement) reactions ominate 0 crystal growth in crystallization processes.

11 In general, consier a roplet/particle that grows by reaction at/in the particle phase. Then the volume of the particle, v, changes with time as: v vi (1) Where v i is the volume of species i ab()sorbe by the particle or roplet. If the rate of uptake of species is equal to the rate of conversion by chemical reaction, then v i MWi C pi i MWi vr pi (13) 1 Where C i is the number of moles of species i an i is the stoichiometric coefficient for species in the reaction of rate r (moles/cm 3 s): C r 1 v i (14) By combining equations 1-14, the particle growth rate is: i v MW v i i r pi (15) The above treatment can be generalize to multiple reactions.

12 To better unerstan this, consier, for example, growth of TiO by surface reaction of TiCl 4 on the surface of TiO film: TiCl4 O TiO Cl (16) Accoring to Ghostagore (J. Electrochem. Soc. Soli State Sci. 117, 59-34, 1970) an ratsinis & Spicer (Chem. Eng. Sci. 53, , 1998) this is a first orer reaction with respect to TiCl 4 so: C C TiCl4 TiO k AC s TiCl4 (17) 3 Here k s is the surface reaction rate constant, A the film surface area for a suspension of N particles of iameter p then A=N p, i = 1, only one species (=1, TiO ) ens in the conense phase so eq. 17 becomes: v kmw s TiO ptio, p C TiCl4 (18) The equation for the particle iameter growth rate can be reaily obtaine as: kmw p s TiO ptio, C TiCl 4 4

13 .5 Transport an article Limite article Growth Uner certain conitions both transport to the particle surface an reactions in the particle phase can control the growth rate of the particles. Consier a magnification of the particle transport/reaction system such that the reaction is first orer with respect to the gas species: C C s C Such conitions can be only etermine when the respective rates are known. 5 For transport-limite growth let us simplify eq. 4 or 9 to: v K ( C C ) g (19) where K g lumps all terms but C in eq. 4 or 9. For reaction-limite growth let us simplify eq.15 for a first orer reaction such as eq. 18 to: C v K C C K C s s s TiCl 4 (0) where K s lumps all terms but C TiCl4 in eq

14 By matching the flux at C (Ranolph an Larson, Theory of articulate rocesses, Acaemic, 1988) which is equivalent also to the funamental treatment of transport between phases, or to couple mass transfer an heterogeneous chemical reactions (Cussler, Diffusion, Cambrige, 1988), it can be shown that: v KC TiCl 4 (1) where the combine constant K is the harmonic average of K g an K s an : KgKs K () Kg Ks From this, the iameter growth rate can be easily obtaine Nucleation Rate. article formation by nucleation from one phase to the other remains one of the last challenges in particle ynamics. Homogeneous or heterogeneous nucleation refers to particle formation in the absence or presence, respectively of external surfaces while homo- an hetero-molecular nucleation refers to formation of pure or mixe composition particles. There are a number of literature expressions for nucleation that have been harly verifie with ata of the correct (critical) particle size DURING particle formation. Typically theory preicts nm size particles but ata employe are those of msize particles when conensation, iffusion an coagulation coul have ha contribute alreay to the evolution of particle size. So procee with caution & check most recent literature. 8