Nucleation kinetics in closed systems

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1 Nucleaton knetcs n closed systems COST Acton CM142: From molecules to crystals how do organc molecules form crystals? Zdeněk Kožíšek Insttute of Physcs of the Czech Academy of Scences, Prague, Czech Republc kozsek@fzu.cz kozsek/ 2-6 November 215, Lyon, France Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 1 / 37

2 Prague Charles Brdge, Prague Castle Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 2 / 37

3 Insttute of Physcs CAS, Prague Phase transton theory group (Dept. of Optcal Materals): Zdeněk Kožíšek, Pavel Demo, Alexe Sveshnkov, Jan Kulvet (PhD Student) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 3 / 37

4 Contents 1 Introducton 2 Standard nucleaton model Knetc equatons Work of formaton of clusters 3 Selected Applcatons Homogeneous nucleaton Heterogeneous nucleaton on actve centers 4 Summary Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 4 / 37

5 Introducton Nucleaton process leadng to the formaton of a new phase (sold, lqud) wthn metastable orgnal phase (undercooled melt, supersaturated vapor or soluton) frst step n crystallzaton process; plays a decsve role n determnng the crystal structure and the sze dstrbuton of nucle nucle of a new phase (droplet, crystal) parent phase (vapor, soluton or lqud) homogeneous nucleaton (HON) (at random stes n the bulk of a parent phase) heterogeneous nucleaton (HEN) (on foregn substrate, mpurtes, defects, actve centres) nucleus:? the smallest observable partcle? (often 1µm) Clusters of a new phase are formed on nucleaton stes due to fluctuatons and after overcomng some crtcal sze (< 1nm) become nucle (overcrtcal clusters). Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 5 / 37

6 Introducton Crtcal supersaturaton Expermental data: H.R. Pruppacher: A new look at homogeneous ce nucleaton n supercooled water drops, J. Atmospherc Scences 52(11) (1995) Supersaturaton (supercoolng) ncreases wth volume decrease. nucleaton knetcs n confned volumes Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 6 / 37

7 Standard nucleaton model UNARY NUCLEATION (sngle component system of a new phase) k 1 + k 2 + k 3 + k 4 + k2 k3 k4 k5 BINARY NUCLEATION (A, B components) k + B(, 2) k B(, 3) k + A(, 2) k A(1, 2) k + B(, 1) k B(, 2) k + B(1, 1) k B(1, 2) k + A(, 1) k A(1, 1) k + A(1, 1) k A(2, 1) k + B(1, ) k B(1, 1) k + B(2, ) k B(2, 1) k + A(1, ) k A(2, ) k + A(2, ) k A(3, ) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 7 / 37

8 Standard nucleaton model Knetc equatons k + (k ) attachment (detachment) frequences of molecules df dt = k + 1 F 1 [k + F number densty of clusters of sze J - cluster flux densty (nucleaton rate for ) ) attachment (detachment) frequences k + (k + k ]F + k +1 F +1 = J 1 (t) J (t) (1) J (t) = k + F (t) k +1 F +1(t) (2) Total number of nucle greater than m : Z m (t) = >m F (t) = t J m(t )dt Local equlbrum: J = k + F = k +1 F +1 k +1 = k + F F +1 = k + exp ( ) W +1 W k B T (3) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 8 / 37

9 Knetc equatons Equlbrum J = k + F = k +1 F +1 (4) F = F1 k + 1 k + 2 k k + 1 k 2 k 3 k 4... k F equlbrum number of cluster formed by molecules It can be shown that ( F = B 2 exp W ) kt ( ) Homogeneous nucleaton, self-consstent model: B 2 = N 1 exp W1 kt N 1 number of molecules wthn parent phase Knowng F and k + k from Eq. (4). Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 9 / 37

10 Standard nucleaton model Classcal nucleaton theory (CNT) Intal and boundary condtons N 1 - number of molecules (number of nucleaton stes) F 1 (t = ) = N 1, F >1 (t = ) = F 1 (t) = N 1 CNT t nducton tme Statonary nucleaton (steady-state): J (t) = J S = const. F J S = ( =1 ) 1 1 k + F exact analytcal formula J S = k + zf, where Zeldovch factor: z = ( = B exp W ) ; B = N 1 exp kt ( W1 kt ( 1 d ) 2 W 2πkT d 2 = ) k +, W =?; small clusters? Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 1 / 37

11 Standard nucleaton model Statonary nucleaton (steady-state) J = J S = const. Boundary condtons: for any cluster sze (N 1 = const.!) F S F for 1; F S for J S = k + F S k +1 F +1 S Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 11 / 37

12 Standard nucleaton model Statonary nucleaton (steady-state) J = J S = const. Boundary condtons: M 1 =1 F S F for 1; F S J S = k + F S k +1 F +1 S = k + J S k + F for any cluster sze (N 1 = const.!) F F S F k +1 F +1 }{{} k + F for ( F+1 S F+1 = k + F F S F F ) +1 S F+1 Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 11 / 37

13 Standard nucleaton model Statonary nucleaton (steady-state) J = J S = const. Boundary condtons: M 1 =1 F S F for 1; F S J S = k + F S k +1 F +1 S = k + J S ( F S k + F = 1 F1 for any cluster sze (N 1 = const.!) F F 2 S ) ( F2 S F2 + F2 F S F k +1 F +1 }{{} k + F F S 3 F 3 ) + for ( F+1 S F+1 = k + F F S F ( F S 3 F 3 F S 4 F 4 ) +... F ) +1 S F+1 Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 11 / 37

14 Standard nucleaton model Statonary nucleaton (steady-state) J = J S = const. Boundary condtons: M 1 =1 F S F for 1; F S J S = k + F S k +1 F +1 S = k + J S ( F S k + F = 1 F1 F 2 for any cluster sze (N 1 = const.!) F F 2 F S F k +1 F +1 }{{} k + F F 3 for ( F+1 S F+1 = k + F F S F F 3 F 4 F ) +1 S F+1 F 2 S ) ( F2 S + F ) ( ) 3 S F S + 3 F 4 S +... = 1 + F M S 1 FM for M Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 11 / 37

15 Standard nucleaton model Statonary nucleaton (steady-state) J = J S = const. Boundary condtons: M 1 =1 F S F for 1; F S J S = k + F S k +1 F +1 S = k + J S ( F S k + F = 1 F1 F 2 for any cluster sze (N 1 = const.!) F F 2 F S F k +1 F +1 }{{} k + F F 3 for ( F+1 S F+1 = k + F F S F F 3 F 4 F ) +1 S F+1 F 2 S ) ( F2 S + F ) ( ) 3 S F S + 3 F 4 S +... = 1 + F M S 1 FM for M J S = ( =1 ) 1 1 k + F exact analytcal formula Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 11 / 37

Standard nucleaton model k + - attachment frequency k follows from the prncple of local equlbrum (3) Crystal nucleaton Crystal phase corresponds a stable phase, lqud a metastable phase, and n between

16 Standard nucleaton model k + - attachment frequency k follows from the prncple of local equlbrum (3) Crystal nucleaton Crystal phase corresponds a stable phase, lqud a metastable phase, and n between s the dffuson actvaton energy. k + From: Yuko Sato,Statstcal Physcs of Crystal Growth, Word Scentfc (1996). ( = R D A exp E ) ( D exp q(w ) +1 W ) ; q =.5[1 + sgn(w +1 W )] kt kt A = γ 2/3 = 4πr 2 surface area R D mean number of molecules strkng on unt nucleus surface Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 12 / 37

17 Standard nucleaton model k + - attachment frequency Vapor crystal R D = P 2πm1 kt (deposton rate); P vapor pressure; m 1 molecular mass Melt crystal (unary parent phase) D. Turnbull, J. Fsher, Rate of nucleaton n condensed systems, J. Chem. Phys. 17 (1949) 145. ( ) kt R D = N S ; N S = ϱ S A h N S number of nucleaton stes on the nucelus surface; ϱ S surface densty of molecules Soluton crystal HON: (A) Drect-mpngement control / (B) Volume-dffuson control ( ) kt R D = CN S ; C - concentraton; (A) nucleaton knetcs s restrctve h R D ncomng dffuson flux of monomers; (B) HON s controlled by volume dffuson Detals: Z. Kozsek D. (Prague, Kashev, CzechCryst. Republc) Res. Technol. Nucleaton 38 (23) knetcs Nov 215 Lyon 13 / 37

18 Standard nucleaton model Work of formaton of clusters Formaton of phase nterface s energetcally dsadvantageous Homogeneous nucleaton: Capllarty approxmaton 4 W = µ + γ 2/3 σ }{{} = 3 πr 3 µ + 4πr 2 σ v 1 W S = k A kσ k surface energy cluster sze (number of molecules wthn cluster) r cluster radus; σ nterfacal energy; v 1 molecular volume µ dfference of chemcal potentals A k surface areas; σ k correspondng nterfacal energes V n ϱ = m 1 r() V n - nucleus volume; ϱ densty of crystal phase; m 1 molecular mass Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 14 / 37

19 Standard nucleaton model Work of formaton of clusters Homogeneous nucleaton 3 W (n k B T unts) W = = ( ) 3 2γσ ; 3 µ crtcal sze; W = W nucleaton barrer melt crystal: µ = h E N A T E (T E T ) soluton crystal: µ = k B T ln S h E heat of fuson; N A Avogadro constant; T E equlbrum temperature; T temperature k B Boltzmann constant; S supersaturaton; Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 15 / 37

20 Standard nucleaton model Work of formaton of clusters Nucleaton n polymer systems: thermodynamc aspects M. Nsh et al., Polymer Journal 31 (1999) 749. Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 16 / 37

21 Standard nucleaton model Classcal nucleaton theory Classcal nucleaton theory (CNT): fals to explan exp. data (W =??) Argon Lennard-Jones nucleaton MC smulatons δ W n = W n W n 1 Down to very small cluster szes, classcal nucleaton theory bult on the lqud drop model can be used very accurately to descrbe the work requred to add a monomer to the cluster! However, erroneous absolute value for the cluster work of formaton, W. B. Hale, G. Wlemsk, 18th ICNAA conference (29) 593. J. Merkanto et al., PRL 98 (27) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 17 / 37

22 Standard nucleaton model Confned systems Intal and boundary condtons F (t = ) = F pro (usually /2) F - equlbrum dstrbuton of nucle F (t = ) = pro > F 1 (t) = F1 = Nn = const. N n number of nucleaton stes (N n = N 1 for HON) CNT model (HON+HEN) N n >1 F Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 18 / 37

23 Standard nucleaton model Confned systems Intal and boundary condtons F (t = ) = F pro (usually /2) F - equlbrum dstrbuton of nucle F (t = ) = pro > F 1 (t) = F1 = Nn = const. N n number of nucleaton stes (N n = N 1 for HON) F 1 (t) = N 1 (t = ) >1 F (t) CNT model (HON+HEN) N n >1 F confned system (HON) F 1 (t) = N n(t = ) N s (t) confned system (HEN) >1 free substrate surface }{{} number of nucleaton stes occuped by nucle N 1 (t) = N T >1 F (t) N 1 - number of molecules wthn parent phase N T - total number of molecules wthn system (lqud + sold phase) confned system (HEN) volume of parent phase Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 18 / 37

24 Selected Applcatons Homogeneous nucleaton HON: ethanol, V L transton (T = 26 K) [Z. Kožíšek et al., J. Chem. Phys. 125 (26) 11454] k +, k - x 1-9 (s -1 ) k +1 S=4 S=2 S= k + Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 19 / 37

25 Selected Applcatons Homogeneous nucleaton Sze dstrbuton of nucle S=3, * = 75 Log 1 F (m -3 ) υ = F Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 2 / 37

26 Selected Applcatons Homogeneous nucleaton Dstrbuton functon - tme dependence S n = 3 S n = 5 1e+9 1e F (m -3 ) 1e+7 1e closed system open system F x 1-17 (m -3 ) Dmensonless tme Dmensonless tme Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 21 / 37

27 Selected Applcatons Homogeneous nucleaton Dstrbuton functon - sze dependence (S n = 5) F x 1-15 (m -3 ) 2 1 F x 1-14 (m -3 ) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 22 / 37

28 Selected Applcatons Homogeneous nucleaton Nucleaton rate - tme dependence S n =3 S n = 5 J/J S (m -3 s -1 ) closed system open system J/J S (m -3 s -1 ) closed system open system Dmensonless tme Dmensonless tme Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 23 / 37

29 Selected Applcatons Homogeneous nucleaton Nucleaton rate (S n = 5, J S = J S at S n ) Open system (S n = const.) Closed system J/J S 2 J/J S υ υ Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 24 / 37

30 Selected Applcatons Homogeneous nucleaton Nucleaton rate (S n = 5) J/J J/J S Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 25 / 37

31 Selected Applcatons Homogeneous nucleaton Supersaturaton 5. Supersaturaton Supersaturaton Dmensonless tme Dmensonless tme 5 4 Crtcal sze Z. Kozsek (Prague, Czech Republc) Dmensonless Nucleaton knetcs tme 2-6 Nov 215 Lyon 26 / 37

32 Selected Applcatons Homogeneous nucleaton HON: Lqud/Soluton Sold transton Z. Kožíšek, CrystEngComm 15 (213) 2269 Sze dstrbuton of nucle CNT Encapsulated system F x 1-19 (m -3 ) F x 1-19 (m -3 ) r (nm) N 1 (t) = const r (nm) N 1 decreases wth tme Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 27 / 37

33 Selected Applcatons Homogeneous nucleaton Model system: nucleaton of N melt DSC experments and MC smulatons [J. Bokeloh et al., PRL 17 (211) J. Bokeloh et al., Eur. Phys. J. Specal Topcs 223 (214) 511] J was obtaned from a statstcal evaluaton of crystallzaton behavor durng contnuous coolng. A sngle N sample was repeatedly heated up to 1773 K an subsequently cooled down to 1373 K. sample masses: 23 µg 63 mg survvorshp functon: F sur ( T ) = 1 exp( J( T )dt) T J = Γ exp( W k B T ) MC smulatons show a devaton of the energy of formaton W from CNT. However, the actual heght of the energy barrer s n good agreement wth CNT. All system parameters are known we can determne the sze dstrbuton of nucle F Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 28 / 37

34 Selected Applcatons Homogeneous nucleaton N melt: h E = kj mol 1, T E = 1748 K, ϱ S = 8357 kg m 3, σ = σ mt /T E, where σ m =.275 J m 2, E = kj mol 1 F x 1-4 (m -3 ) Tme (s) T = 1449 K = F V = 1 one -szed nucleus s formed n V F456 S = 25 m 3 V 1 = 1/F S = 4 cm 3 sample masses: 23 µg 63 mg V = m m 3 In dfference of expermental data, no crtcal nucle are formed n N melt. Soluton: reduce the nterfacal energy or take nto account σ() dependency lower nucleaton barrer Maybe heterogeneous nucleaton occurs? Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 29 / 37

35 Selected Applcatons Nucleaton on actve centers CNT model of nucleaton on actve centers one addtonal equaton s needed [S. Toshev, I. Gutzow, Krst. Tech. 7 (1972) 43; P. Ascarell, S. Fontana: Damond Rel. Mater. 2 (1993) ; D. Kashchev: Nucleaton, Butterworth-Henemann Boston, 2] Z/N Dmensonless tme Our approach Avram model Open system Addtonal equaton: dz (t) dt = [N Z (t)]j S (t) Z (t) - the total number of nucle at tme t N - the number of actve stes; J S - tme dependent nucleaton frequency (usually taken as ft<f8> parameter) Our new model does not need addtonal equaton!! only modfcaton of boundary condtons Z. Kozsek et al., Transent nucleaton on nhomogeneous foregn substrate, J. Chem. Phys. 18 (1998) 9835; Nucleaton on actve stes: evoluton of sze dstrbuton, J. Cryst. Growth 29 (2) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 3 / 37

36 Selected Applcatons Nucleaton on actve centers HEN: Vapor Sold transton F/N A r (nm) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 31 / 37

37 Selected Applcatons Nucleaton on actve centers HEN: Vapor Sold transton H. Kumom, F.G. Sh: Phys. Rev. Lett. 82 (1999) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 32 / 37

38 Selected Applcatons Nucleaton on actve centers HEN: Vapor Sold transton F/N A x r (nm) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 33 / 37

39 Selected Applcatons Nucleaton on actve centers HEN: polymer nucleaton from melt (polyethylene) T = 1.4 K data 6 Z x 1-13 (m -3 ) Tme (s) Z. Kožíšek et al. Nucleaton knetcs of folded chan crystals of polyethylene on actve centers, J. Chem. Phys. 121 (24) Z. Kožíšek et al. Nucleaton on actve centers n confned volumes, J. Chem. Phys. 134 (211) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 34 / 37

40 Selected Applcatons Nucleaton on actve centers HEN: polymer nucleaton from melt (polyethylene) 27: K. Okada et al.: Polymer 48 (27) Log 1 F(, t) Expermental data t (mn) = σ nano (nstead of σ) ntroduced to ft F S Log 1 F data: Okada et al.: Polymer 48 (27) mn 14 mn 21 mn 35 mn 64 mn 98 mn Log 1 k = 9, l = 24.9, m = 4.5 3D model Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 35 / 37

41 Selected Applcatons Polymer nucleaton on actve centers (polyethylene) Log 1 F 3 2 2Dmodel: macro parameters, * =666, c N = *c 2 N -5 94*c -5 N 94 69*c N t (mn) t (mn) Log 1 F (F*cn n m -3 ) 2 1 3D: nano parameters, * =487, exp =5,63 Z. Kožíšek, M. Hkosaka, K. Okada, P. Demo: J. Chem. Phys. 134 (211) & 136 (212) Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 36 / 37

42 Summary Advantages Relatvely smple model enables to determne basc characterstcs of nucleaton n real tme. Model takes nto account depleton of the parent phase durng phase transformaton. Model ncludes exhauston of actve centres (new approach to heterogeneous nucleaton). Ths work was supported by the Project No. LD154 (VES15 COST CZ) of the Mnstry of Educaton of the Czech Republc. Thank you for your attenton. Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 37 / 37

43 Summary Advantages Relatvely smple model enables to determne basc characterstcs of nucleaton n real tme. Model takes nto account depleton of the parent phase durng phase transformaton. Model ncludes exhauston of actve centres (new approach to heterogeneous nucleaton). Ths work was supported by the Project No. LD154 (VES15 COST CZ) of the Mnstry of Educaton of the Czech Republc. Thank you for your attenton. Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 37 / 37

Summary Advantages Relatvely smple model enables to determne basc characterstcs of nucleaton n real tme. Model takes nto account depleton of the parent phase durng phase transformaton.

44 Summary Advantages Relatvely smple model enables to determne basc characterstcs of nucleaton n real tme. Model takes nto account depleton of the parent phase durng phase transformaton. Model ncludes exhauston of actve centres (new approach to heterogeneous nucleaton). Ths work was supported by the Project No. LD154 (VES15 COST CZ) of the Mnstry of Educaton of the Czech Republc. Thank you for your attenton. Z. Kozsek (Prague, Czech Republc) Nucleaton knetcs 2-6 Nov 215 Lyon 37 / 37

Crystal nucleation in sub-microemulsions

Crystal nucleation in sub-microemulsions Crystal nucleaton n sub-mcroemulsons Zdeněk Kožíšek Insttute of Physcs AS CR, Praha, Czech Republc kozsek@fzu.cz Hroshma Unversty, Japan, October 13, 211 Web page: http://www.fzu.cz/ kozsek/lectures/mcro211.pdf