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, 2006 1 / 29
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, 2006 2 / 29
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, 2006 3 / 29
Introducton Motvaton Motvaton Nucleaton n mcroemulsons Mcroemulson thermodynamcally stable dsperson of one lqud phase nto another (ol-n-water, water-n-ol) Droplet dameters 10 1000 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, 2006 4 / 29
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, 2006 5 / 29
Thermodynamcal aspects W (n kt unts) 9 8 7 6 5 4 3 2 1 0 ( J S = A exp F 0 = B exp Work of formaton of clusters µ 1 µ 2 0 10 20 30 40 50 60 70 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, 2006 6 / 29
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, 2006 7 / 29
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, 2006 8 / 29
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, 2006 9 / 29
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, 2006 10 / 29
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 + 1... k + 1 k 2... k F1 0 = F 1 0 F 0 1 1 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, 2006 10 / 29
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 + 1... k + 1 k 2... k F1 0 = F 1 0 F 0 1 1 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, 2006 10 / 29
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, 2006 11 / 29
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, 2006 11 / 29
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 1 +... + 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, 2006 11 / 29
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 1 +... + 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 0 1 1 + M 1 =2 k 2 k 3...k k + 2 k + 3...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, 2006 11 / 29
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, 2006 12 / 29
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, 2006 12 / 29
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, 2006 13 / 29
Results Expermental data Nucleaton from soluton Z. Kozsek et al.: J. Chem. Phys. 114 (2001) 7622. 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, 2006 14 / 29
Results Expermental data Nucleaton on actve centers H. Kumom and F. G. Sh: Phys. Rev. Lett. 82 (1999) 2717. 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, 2006 14 / 29
Results Transent probabltes Model system: ethanol, V L transton (T = 260 K) 3 k +, k - x 10-9 (s -1 ) 2.5 2 1.5 1 0.5 0 - k +1 S=4 S=2 S=1 50 100 150 200 250 300 350 400 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, 2006 15 / 29 k +
Results Open system (constant supersaturaton) Dstrbuton functon S=3, * = 75 Log 10 F (m -3 ) 20 15 10 5 υ = 1 11 21 31 41 51 F 0 0 0 50 100 150 200 250 300 350 400 450 500 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, 2006 16 / 29
Results Open system (constant supersaturaton) Normalzed dstrbuton functon S=3, * = 75 1 0.8 f=f/f 0 0.6 0.4 υ = 1 6 11 21 41 101 0.2 0 0 20 40 60 80 100 120 140 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, 2006 17 / 29
Results Open system (constant supersaturaton) Nucleaton rate (S=3) J/J S 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1.4 1.2 1 0.8 0.6 0.4 0.2 0 180 200 0 120 140160 100 200 300 400 60 80100 υ 500 600 700 800 900 0 2040 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, 2006 18 / 29
Results Closed system (S n =3) Dstrbuton functon F (m -3 ) 1e+09 1e+08 1e+07 1e+06 100000 10000 1000 100 10 1 S n = 3 75 150 closed system open system 0 20 40 60 80 100 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, 2006 19 / 29
Results Closed system (S n =5) Dstrbuton functon F x 10-17 (m -3 ) 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 S n = 5 24 40 100 0 20 40 60 80 100 120 140 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, 2006 20 / 29
Results Closed system (S n =5) Nucleaton rate S n = 5 1 0.8 24 closed system open system J/J S (m -3 s -1 ) 0.6 0.4 0.2 100 300 0 0 20 40 60 80 100 120 140 160 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, 2006 21 / 29
Results Closed system (S n =5) Normalzed dstrbuton functon f=f/f 0 4 3.5 3 2.5 2 1.5 1 0.5 0 υ =2 5 70 100 110 120 130 0 10 20 30 40 50 60 70 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, 2006 22 / 29
Results Closed system (S n =5) Dstrbuton functon 3 F x 10-15 (m -3 ) 2 1 0 160 40 80 120 140 0 10000 20000 30000 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, 2006 23 / 29
Results Closed system (S n =5) Dstrbuton functon F x 10-14 (m -3 ) 14 12 10 8 6 4 2 0 160 200 240 320 0 20000 40000 60000 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, 2006 24 / 29
Results Closed system (S n =5) Dstrbuton functon F x 10-14 (m -3 ) 6 5 4 3 2 1 υ=320 500 700 1000 0 0 20000 40000 60000 80000 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, 2006 25 / 29
Results Closed system (S n =5) Dstrbuton functon F x 10-14 (m -3 ) 6 5 4 3 2 1 υ = 1000 1400 2000 2500 3000 0 0 40000 80000 120000 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, 2006 26 / 29
Results Closed system (S n =5) Supersaturaton, crtcal sze 5.0 Supersaturaton 4.0 3.0 2.0 Supersaturaton 1.2 1.0 0 200 400 600 800 Dmensonless tme 50000 1.1 0 1000 2000 3000 Dmensonless tme 40000 Crtcal sze 30000 20000 10000 0 0 500 1000 1500 2000 2500 3000 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, 2006 27 / 29
Results Closed system (S n =5) Nucleaton rate 0.01 S J /J n 0-0.01 400 600 1000 1400 1800 2200 2600 3000 0 50000 100000 150000 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, 2006 28 / 29
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. A1010311 of the Grant Agency AS CR. http://www.fzu.cz/ 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, 2006 29 / 29
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. A1010311 of the Grant Agency AS CR. http://www.fzu.cz/ 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, 2006 29 / 29
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. A1010311 of the Grant Agency AS CR. http://www.fzu.cz/ 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, 2006 29 / 29