Kinetics of nucleation in open and closed systems

Transcription

1 Knetcs of nucleaton n open and closed systems Zdeněk Kožíšek Insttute of Physcs, Academy of Scences of the Czech Republc, Praha, Czech Republc kozsek@fzu.cz MFF UK, November 14, 2006 deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systemsmff Czech Republckozsek@fzu.cz) UK, November 14, / 29

2 Contents 1 Introducton 2 Thermodynamcs 3 Model Assumptons Knetc equatons Transent frequences 4 Results Open system (constant supersaturaton) Closed system (S=3) Closed system (S=5) 5 Conclusons deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systemsmff Czech Republckozsek@fzu.cz) UK, November 14, / 29

3 Introducton Nucleaton process leadng to the formaton of a new phase (sold, lqud) wthn metastable orgnal phase (undercooled melt, supersaturated vapor or soluton) nucle of a new phase (droplet, crystal) Mother phase (supersaturated vapor, soluton or supercooled lqud) homogeneous nucleaton heterogeneous nucleaton 2D, 3D nucleaton nucleaton on actve centers (specal case) deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systemsmff Czech Republckozsek@fzu.cz) UK, November 14, / 29

4 Introducton Motvaton Motvaton Nucleaton n mcroemulsons Mcroemulson thermodynamcally stable dsperson of one lqud phase nto another (ol-n-water, water-n-ol) Droplet dameters nanometers (Measurements at Hroshma Unversty) Model approaches: only thermodynamcal aspects of nucleaton n closed systems or statonary knetcs D. Turnbull: J. Chem. Phys. 20 (1952) 411; D. Kashchev et al.: J. Collod Interface Sc. 208 (1998) 167; B. Mutafschev: The Atomstc Nature of Crystal Growth (Sprnger-Verlag, Berln, 2001). = Nucleaton knetcs n closed systems deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systemsmff Czech Republckozsek@fzu.cz) UK, November 14, / 29

5 Introducton Structure Jean-Patrck Commerade: The scence of clusters: An emergng feld, Europhyscs news 33/6 (2002) 200. deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systemsmff Czech Republckozsek@fzu.cz) UK, November 14, / 29

6 Thermodynamcal aspects W (n kt unts) ( J S = A exp F 0 = B exp Work of formaton of clusters µ 1 µ W k B T ( W k B T ) ) Cluster sze µ 1 > µ 2 statonary nucleaton rate equlbrum dstrbuton functon deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systemsmff Czech Republckozsek@fzu.cz) UK, November 14, / 29

7 Termodynamcal aspects Polymer systems M. Nsh et al.: Polymer Journal 31 (1999) 749. deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systemsmff Czech Republckozsek@fzu.cz) UK, November 14, / 29

8 Model Assumptons k + 1 k + 2 k + 3 k + 4 k 2 k 3 k 4 k 5 k + (k ) attachment (detachment) frequences of molecules Coalescence s neglected attachment (resp. detachment) of sngle molecules plays domnant role n nucleaton and growth process Nucleaton starts at any monomer (nucleaton center) n the bulk of the supersaturated mother phase Zdeněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systemsmff Czech Republckozsek@fzu.cz) UK, November 14, / 29

9 Model Knetc equatons df = J 1 (t) J (t) dt F number densty of nucle of sze J (t) = k + F (t) k +1 F +1(t) Cluster flux densty (nucleaton rate for ) Intal and boundary condtons: N T = 0 =1 F 0 total number of molecules F (t = 0) = F 0 pro 0 F (t = 0) = 0 pro > 0 F 1 >1 F (t) = F 1 = const. F 1 (t) = N T >1 F (t) open system closed system deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systemsmff Czech Republckozsek@fzu.cz) UK, November 14, / 29

10 Model Equlbrum J (t) = 0 k + F 0 = k +1 F +1 0 local equlbrum deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

11 Model Equlbrum J (t) = 0 k + F 0 = k +1 F +1 0 local equlbrum F 0 F 0 3 = k + 2 k 3 F 0 2 = k + 1 k 2 F 0 1 F2 0 = k + 1 k + 2 k 2 k 3... = k k + 1 k 2... k F1 0 = F 1 0 F j=1 k + j k j 1 deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

12 Model Equlbrum J (t) = 0 k + F 0 = k +1 F +1 0 local equlbrum F 0 F 0 3 = k + 2 k 3 F 0 2 = k + 1 k 2 F 0 1 F2 0 = k + 1 k + 2 k 2 k 3... = k k + 1 k 2... k F1 0 = F 1 0 F j=1 ( F 0 = F1 0 exp W ) W = k B T k B T k + j k j 1 ( ) k j ln k + j 1 j=2 deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

13 Model Statonary nucleaton (constant supersaturaton) J (t) = J 1 (t) = const. = J S deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

14 Model Statonary nucleaton (constant supersaturaton) J (t) = J 1 (t) = const. = J S ξ 1 = k + 1 F 1 S ; ξ = k + J S = k + F S k +1 F +1 S ξ ξ k ξ 1 = k + F S k + ξ ξ +1 F +1 S ξ +1 deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

15 Model Statonary nucleaton (constant supersaturaton) J (t) = J 1 (t) = const. = J S ξ 1 = k + 1 F 1 S ; ξ = k + J S = k + F S k +1 F +1 S ξ ξ k ξ 1 = k + F S k + ξ ξ +1 F +1 S ξ +1 J S ξ JS M 1 = J S ξ M 1 j=1 1 ξ j = k + 1 F S 1 ξ 1 k + M F S M ξ M 1 deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

16 Model Statonary nucleaton (constant supersaturaton) J (t) = J 1 (t) = const. = J S ξ 1 = k + 1 F 1 S ; ξ = k + J S = k + F S k +1 F +1 S ξ ξ k ξ 1 = k + F S k + ξ ξ +1 F +1 S ξ +1 J S ξ JS M 1 = J S ξ M 1 j=1 1 ξ j = k + 1 F S 1 ξ 1 k + M F S M ξ M 1 J S = k + 1 F M 1 =2 k 2 k 3...k k + 2 k k + R. Becker, W. Dörng, Ann. Phys. 24 (1935) 719. deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

17 Model Contnuum sze 1 where F (, t) t + J(, t) J(, t) = k + (, t)f 0 () = 0 ( ) F (, t) F 0 () deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

18 Model Contnuum sze 1 where F (, t) t + J(, t) J(, t) = k + (, t)f 0 () = 0 ( ) F (, t) F 0 () Statonary nucleaton rate z = J S = k + zf 0 1 2πk B T ( d 2 ) G d 2 = deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

19 Model Attachment frequency Vapor Lqud Vapor Sold k + = k + = P 2πmkT S ( P S exp E ) 2πmkT kt Lqud Sold k + = ϱ S ( kt h ) ( S exp E ) ( exp q g ) kt kt g = G +1 G ; q = 1 2 [1 + sgn( g )] ϱ S - surface densty of monomers deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

20 Results Expermental data Nucleaton from soluton Z. Kozsek et al.: J. Chem. Phys. 114 (2001) deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

21 Results Expermental data Nucleaton on actve centers H. Kumom and F. G. Sh: Phys. Rev. Lett. 82 (1999) deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

22 Results Transent probabltes Model system: ethanol, V L transton (T = 260 K) 3 k +, k - x 10-9 (s -1 ) k +1 S=4 S=2 S= deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29 k +

23 Results Open system (constant supersaturaton) Dstrbuton functon S=3, * = 75 Log 10 F (m -3 ) υ = F deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

24 Results Open system (constant supersaturaton) Normalzed dstrbuton functon S=3, * = f=f/f υ = Zdeněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

25 Results Open system (constant supersaturaton) Nucleaton rate (S=3) J/J S υ deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

26 Results Closed system (S n =3) Dstrbuton functon F (m -3 ) 1e+09 1e+08 1e+07 1e S n = closed system open system Dmensonless tme deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

27 Results Closed system (S n =5) Dstrbuton functon F x (m -3 ) S n = Dmensonless tme deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

28 Results Closed system (S n =5) Nucleaton rate S n = closed system open system J/J S (m -3 s -1 ) Dmensonless tme deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

29 Results Closed system (S n =5) Normalzed dstrbuton functon f=f/f υ = Zdeněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

30 Results Closed system (S n =5) Dstrbuton functon 3 F x (m -3 ) deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

31 Results Closed system (S n =5) Dstrbuton functon F x (m -3 ) deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

32 Results Closed system (S n =5) Dstrbuton functon F x (m -3 ) υ= deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

33 Results Closed system (S n =5) Dstrbuton functon F x (m -3 ) υ = deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

34 Results Closed system (S n =5) Supersaturaton, crtcal sze 5.0 Supersaturaton Supersaturaton Dmensonless tme Dmensonless tme Crtcal sze Dmensonless tme deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

35 Results Closed system (S n =5) Nucleaton rate 0.01 S J /J n deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

36 Conclusons At low supersaturatons: standard model wth constant supersaturaton works well also n closed systems. At hgh supersaturatons: dfferent behavour of nucleatng system; no statonary regme. Perspectve: drect comparson of the sze dstrbuton functon of nucle wth expermental data. Ths work was supported by Project No. A of the Grant Agency AS CR. kozsek/lectures/mff2006.pdf deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

37 Conclusons At low supersaturatons: standard model wth constant supersaturaton works well also n closed systems. At hgh supersaturatons: dfferent behavour of nucleatng system; no statonary regme. Perspectve: drect comparson of the sze dstrbuton functon of nucle wth expermental data. Ths work was supported by Project No. A of the Grant Agency AS CR. kozsek/lectures/mff2006.pdf deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29

38 Conclusons At low supersaturatons: standard model wth constant supersaturaton works well also n closed systems. At hgh supersaturatons: dfferent behavour of nucleatng system; no statonary regme. Perspectve: drect comparson of the sze dstrbuton functon of nucle wth expermental data. Ths work was supported by Project No. A of the Grant Agency AS CR. kozsek/lectures/mff2006.pdf deněk Kožíšek (Insttute of Physcs, Academy Knetcs of Scences of nucleaton of the Czech n open Republc, and closed Praha, systems Czech MFFRepublckozsek@fzu.cz) UK, November 14, / 29