Crystal nucleation in sub-microemulsions

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1 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: kozsek/lectures/mcro211.pdf Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 1

2 Contents 1 Standard nucleaton model Work of formaton of clusters Knetc equatons Transent probabltes 2 Nucleaton n sub-mcroemulsons Work of formaton of nucle Intal and boundary condtons 3 Numercal soluton of knetc equatons Standard model: homogeneous nucleaton Nucleaton n sub-mcroemulsons 4 Conclusons Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 2

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 (HON) heterogeneous nucleaton (HET) 2D, 3D nucleaton nucleaton on actve centers (specal case) Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 3

4 Example 1: Formaton of n-alcohol crystalltes from soluton Z. Kožíšek, T. Koga, K. Sato, P. Demo, J. Chem. Phys. 114 (21) Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 4

Example 2: Formaton of S crystalltes on substrate H. Kumom, F.G. Sh: Phys. Rev. Lett. 82 (1999) 2717.

5 Example 2: Formaton of S crystalltes on substrate H. Kumom, F.G. Sh: Phys. Rev. Lett. 82 (1999) Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 5

6 Introducton Motvaton Nucleaton n sub-mcroemulsons Mcroemulson thermodynamcally stable dsperson of one lqud phase nto another (ol-n-water, water-n-ol) Droplet dameters 1 1 nanometers (Measurements at Hroshma Unversty) Generally: Nucleaton n small volumes occurs at hgher supersaturatons depleton (decrease of supersaturaton) plays mportant role Analytcal soluton of the statonary nucleaton rate (no depleton) gves upper lmt of real nucleaton rate Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 6

7 Standard nucleaton model Work of formaton of clusters Formaton of phase nterface s energetcally dsadvantageous = One needs to determne work of formaton of clusters Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 7

8 Standard nucleaton model Work of formaton of clusters Formaton of phase nterface s energetcally dsadvantageous = One needs to determne work of formaton of clusters Homogeneous nucleaton: Capllarty approxmaton W = µ + γ 2/3 4 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 Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 7

9 Work of formaton of clusters 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; Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 8

10 Work of formaton of clusters Heterogeneous nucleaton on planar surface From: K.F. Kelton, A.L. Greer, Nucleaton n Condesed Matter, 21 Contact angle φ s determned by Young equaton: σ αβ cos φ = σ αn σ βn Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 9

11 Work of formaton of clusters Heterogeneous nucleaton on planar surface where W HET = W HOM Ψ(φ) Ψ(φ) = 1 4 (1 cos φ)2 (2 + cos φ) W HET = W HOM only for φ = 18 o (Ψ(18) = 1) 18 > φ > W HET < W HOM Hgher probablty of formaton of heterogeneous nucle It s necessary to overcome nucleaton barrer W to form crtcal nucleus. Nucleaton rate J = B 1 exp( W kt ) Knetc factor B 1 depends on the transport near the phase nterface. Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 1

12 Nucleaton knetcs 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, ) Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

13 Knetc equatons k + 1 k 2 k + 2 k 3 k + 3 k 4 k + 4 k 5 Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

14 Knetc equatons k + 1 df dt k 2 = k + 1 F 1 [k + k + 2 k 3 k + 3 k 4 + k ]F + k +1 F +1 = J 1 (t) J (t) J (t) = k + F (t) k +1 F +1(t) F number densty of nucle of sze J cluster flux densty (nucleaton rate for ) ) attachment (detachment) frequences k + (k Total number of nucle greater than m Z m (t) = >m F (t) = t J m (t )dt k + 4 k 5 Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

15 Knetc equatons Equlbrum F J = k + F = k +1 F +1 (1) F = F1 k + 1 k + 2 k k + 1 k 2 k 3 k 4... k 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. (1). Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

16 Knetc equatons 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 Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

17 Knetc equatons 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 Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

18 Knetc equatons 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 Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

19 Knetc equatons 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 F3 S + F 4 S +... = 1+ F M S 1 FM for M Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

20 Knetc equatons 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 F3 S + F 4 S +... = 1+ F M S 1 FM for M J S = ( =1 ) 1 1 k + F exact analytcal formula Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

21 Knetc equatons Analytcal soluton of nucleaton rate Statonary nucleaton J S = k + zf, where Zeldovch factor: z = 1 2πkT ( d ) 2 W d 2 = Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

22 Knetc equatons Analytcal soluton of nucleaton rate Statonary nucleaton J S = k + zf, where Zeldovch factor: z = 1 2πkT Non-statonary nucleaton (non-steady state) at = D. Kashchev: Surf. Sc. 18 (1969) 293. J(t) = J S [1 + 2 ( 1) k exp k=1 ( k 2 ) ] t ; τ K = τ K ( d ) 2 W d 2 = 4 π 3 z 2 k + Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

23 Knetc equatons Analytcal soluton of nucleaton rate Statonary nucleaton J S = k + zf, where Zeldovch factor: z = 1 2πkT Non-statonary nucleaton (non-steady state) at = D. Kashchev: Surf. Sc. 18 (1969) 293. J(t) = J S [1 + 2 ( 1) k exp k=1 Non-statonary nucleaton at > V.A. Shnedman, Sov. [ Phys. Tech. Phys. 33 (1988) J (t) = J S exp exp U TF = ( (t t ) τ )] ; t = r U TF + τ ( k 2 ) ] t ; τ K = τ K [ ln ( 6W kt ( d ) 2 W d 2 = 2r µ S τ snh(.5 µs ); µ S = µ kt ; τ = 1 2πk + z 2 4 π 3 z 2 k + ) ( r ) ] + ln r 1 2 Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

Knetc equatons Transent probabltes k follows from the prncple of local equlbrum (1) Crystal nucleaton Crystal phase corresponds a stable phase, lqud a metastable phase, and n between s the dffuson

24 Knetc equatons Transent probabltes k follows from the prncple of local equlbrum (1) 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 (deposton rate) Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

25 Knetc equatons Transent probabltes Vapor crystal R D = P 2πm1 kt (deposton rate); P vapor pressure; m 1 molecular mass Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

26 Knetc equatons Transent probabltes 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 Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

27 Knetc equatons Transent probabltes 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 R D = CN S ( kt h ) ; C - concentraton; nucleaton knetcs s restrctve R D ncomng dffuson flux of monomers; HON s controlled by volume dffuson Detals: D. Kashev, Cryst. Res. Technol. 38 (23) 555. Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

28 Nucleaton n sub-mcroemulsons homogeneous nucleaton heterogeneous nucleaton Thermodynamc aspects of heterogeneous nucleaton on sphercal substrate: Convex: D. Xu, W.L. Johnson, Phys. Rev. B 72 (25) Concave: S.J. Cooper, C.E. Ncholson, J. Lu: J. Chem. Phys. 129 (28) Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

29 Nucleaton n sub-mcroemulsons Work of formaton of nucle on concave substrate R ϑ r ϕ R droplet radus (substrate) r nucleus radus ϑ contact angle between the nucleus and the sphercal substrate ϕ angle between the sphercal substrate and the plane connectng the nucleus edge Fxed: R and ϑ (Young s equaton) Comment: R and ϕ are negatve for concave substrate, postve for convex substrate the same equatons for W Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

30 Nucleaton n sub-mcroemulsons Work of formaton of nucle on concave substrate W het = 4π 3v 1 r 3 µ[ψ(ϑ + ϕ) (R/r) 3 ψ(ϕ)] +2[1 cos(ϑ + ϕ)]πr 2 σ 2 cos ϑ(1 cos ϕ)πr 2 σ (2) where v 1 s molecular volume n crystallne phase and ψ(α) = 1 4 (2 3 cos α + cos3 α) = 1 4 (1 cos α)2 (2 + cosα) σ the nterfacal energy between the nucleus and the surroundng melt Eq. (2) s vald for convex and concave sphercal substrate. Knetc equatons one needs r() dependence, but ϕ and r depends on. r sn ϕ = R sn(ϕ + ϑ) ϕ = arctg sn ϑ R/r cos ϑ ϱv N = m 1, V N = 4 3 π [ r 3 ψ(ϑ + ϱ) R 3 ψ(ϕ) ] (V N -nucleus volume) r ϕ.e. one has (r) dependece. Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 2

31 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) standard model (HON+HEN) N n >1 F Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

32 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) Sub-mcroemulsons F 1 (t) = N 1 (t = ) >1 F (t) standard model (HON+HEN) N n >1 F closed system (HON) Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

33 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) Sub-mcroemulsons F 1 (t) = N 1 (t = ) >1 F (t) standard model (HON+HEN) N n >1 F closed system (HON) F 1 (t) = N n(t = ) N s (t) closed system (HEN) >1 free substrate surface }{{} number of nucleaton stes occuped by nucle Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

34 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) Sub-mcroemulsons F 1 (t) = N 1 (t = ) >1 F (t) standard model (HON+HEN) N n >1 F closed system (HON) F 1 (t) = N n(t = ) N s (t) closed 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) closed system (HEN) volume of parent phase Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

35 Numercal soluton of knetc equatons Standard model: homogeneous nucleaton N 1 s constant (N 1 >1 F ) Transent probabltes Normalzed transent frequences k + k Cluster sze Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

36 Standard model (N 1 = const.) Sze dstrbuton functon Log 1 F (F/V n m -3 ) F Cluster sze Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

37 Standard model Sze dstrbuton functon 1.2e+17 F for = 5, 1, 2, 4, 6, 8, and 1 1e+17 F (m -3 ) 8e+16 6e+16 4e+16 2e Dmensonless tme Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

38 Standard model Nucleaton rate J /J S Dmensonless tme Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

39 Standard model Nucleaton rate J /J S υ 6 Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

40 Standard model Total number of nucle Z m = >m F Z k 1 + /J S Dmensonless tme Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

41 Standard model Total number of nucle: expermental data exp Z x 1-13 (m -3 ) t (s) Nucleaton of polyethylene on actve centers [[Z. Kozsek, M. Hkosaka, K. Okada, P. Demo, J. Chem. Phys. 143 (211) ] Nucleaton of glass [V.M. Fokn et al., J. Cryst. Growth 52 (1981) 115.] Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

42 Nucleaton n sub-mcroemulsons N 1 decreases wth tme,.e. decrease of supersaturaton. Sze dstrbuton of nucle Log 1 F Cluster sze Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

43 Nucleaton n sub-mcroemulsons Sze dstrbuton of nucle Log 1 F Cluster sze Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13, 211 3

44 Nucleaton n sub-mcroemulsons Sze dstrbuton of nucle F F(r) r (nm) Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

45 Nucleaton n sub-mcroemulsons Sze dstrbuton of nucle F F(r) r (nm) Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

46 Nucleaton n sub-mcroemulsons Nucleaton rate 1.8 J/J S n υ Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

47 Nucleaton n sub-mcroemulsons Nucleaton rate (hgher supersaturaton) J/J S n Nucleus sze Dmensonless tme Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

48 Nucleaton n sub-mcroemulsons Total number of nucle Z m = >m F Z k + */JS Dmensonless tme Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

49 Conclusons In standard model, sze dstrbuton of nucle reaches statonary sze dstrbuton at suffcently long tme. In sub-mcroemulsons the sze dstrbuton reaches some extremal value. It s consequence of depleton of parent phase. Nucleaton rate reaches some extremum wth tme. Statonary nucleaton rate, J S, gves upper lmt of nucleaton rate n mcroemulsons. Nucleaton rate goes to neglgble value at suffcntly long tme,.e., the system goes to equlbrum. Ths work was supported by Project No. IAA1186 of the Grant Agency AS CR. Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

50 Conclusons In standard model, sze dstrbuton of nucle reaches statonary sze dstrbuton at suffcently long tme. In sub-mcroemulsons the sze dstrbuton reaches some extremal value. It s consequence of depleton of parent phase. Nucleaton rate reaches some extremum wth tme. Statonary nucleaton rate, J S, gves upper lmt of nucleaton rate n mcroemulsons. Nucleaton rate goes to neglgble value at suffcntly long tme,.e., the system goes to equlbrum. Ths work was supported by Project No. IAA1186 of the Grant Agency AS CR. Kozsek (Insttute of Physcs, Praha) Crystal nucleaton n sub-mcroemulsons Hroshma Unversty, October 13,

Nucleation kinetics in closed systems

Nucleation kinetics in closed systems 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