MICROPARACRYSTALS - THE BUILDING MATERIAL IN NATURE

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MICROPARACRYSTALS - THE BUILDING

MATERIAL IN NATURE

R. Hosemann

To cite this version:

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MICROPARACRYSTALS

-

T H E BUILDING M A T E R I A L IN NATURE

R . Hosemann

Grztppe ParakristaZZforschwzg, c / o BAM, Unter den Eichen 8 7 ,

1000 B e r l i n 45, F.R.G.

4bstract The positions of atoms in materials can b e explored most directly by diffraction of X-rays. This was demonstrated 1912 for crystals by Laue, Friedrich and Knipping /I/. The positions of atoms in polymers could be analysed in a similar way by transforming the intensity of scattered X-rays or neutrons into detailed structural information. Words such as short range order, medium range order, clusters, micelles, mesophases or amorphous phases, semi-crystalline or liquid-crystalline materials may be used. New definitions such as alignments, nodules, inclinations and meander-cubes appear. Another kind of research applies mathematical language: beginning with the results by Laue /2/, the reciprocal space notation by Ewald / 3 / and the idea of the same a-priori probability function for allflatoms in the noncrystalline state by Debye

/4/,

the theory of the ideal paracrystal" was published in 1950 /5/. Laue's idea of lattices is blended with Debye's a-priori probability function: The three lattice cell vectors a each have a priori three rela- tive variances gki in the threzk directions

ai

,

e.g.

The lattice factor Z then is no longer Ewald's lattice peak function, but the three-dimensional Fourier transform of Landau's con- volution polynomial / 6 / . The Fourier transform F (b) of three a-priori distance statistics H (x) of the vectkr~ a define the lattice factor. In the followlfng table all essentid formula are collected:

FUNDAMENTALS OF THE GENERALIZED CRYSTALLOGRAPHY

E:

. v.,n Laue (19121: sk.b = hk ; b = (s-s~)/A

1

P. Debye (1930): The "a priori probability functionn

I

F. Rinne (1933 ): "Paracrystals and Bioparacrgstalsn 3 l+Fk -22!i:.% R. H o s e m m (19501: Z(&)

-

Re ; Fk - 1 ~ ~ J e

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C8-380 JOURNAL DE PHYSIQUE

The name "paracrystal" was introduced by Rinne

/7/

for liquid crystals. Laue recognized the importance of the theory of paracrystals for all materials internediate between crystalline and liquid stzte. Never- theless, the new theory is still a plied merely in USA and the U.K.

preferably to promoted catalysts / 8 , 9 / . The caution of most scientists may be due to the new mathematical language and the fact that ideal paracrystals consist of lattice cells shaped like paralleloepipeds. On this account their lattices contain particular irrational statistical correlations. Recently it was proved that in all paracrystals only certain partial correlations exist which, on account of the so called &-relation, affect the intensity function of the ideal paracrystal by less than 1,5 $ and hence can be neglected /lo/. The theory of paracrystals offers fundamental equations by which most substances can be deciphered as microparacrystals /11/ by measuring their int ral widths

sg

&b of reflections. These widths increase linearly with h (h = order of reflection). From the slope of this straight line and from its intersection with the ordinate, gkk and the number, Nk,of netplanes can be calculated. All materials Investigated by us until now are submitted to the &-relation which connects Nk with the

gkdvalue of the bearing netplanes k (see below).

$L

*

g k k f i k = OL ; C( = 0,15t 0,05. (1)

Figure 1 gives some examples of organic and inorganic materials in a plot against 1/g. There is no doubt that this e*-relation con- ceals a law in nature which is of fundamental importance for most noncrystalline materials. The model of Figure 2 shows all essentials of a two-dimensional paracrystal in equilibrium state. Some larger coins are statistically mixed with smaller ones and represent atoms with four orthogonal valences.

Conventional crystals

7

1

a*=0,2 ,'

50

100

150

4

reciprocal g -value 4r

Fig

.

1

-

The C4 -relation (Eq. 1 ).

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other hand become active, because the coins lie in statistically curved lines and SOWS. These deviations resulting from straight lines Teach maximal 360 cA*/2% and influence the diffraction by less than

-

1/20(*2. Those rows where atoms directly touch each other are called "bearing netplanes" because tangential strains are transfered by them. The consequently arising distortions of the valence angles require a new kind of energy which in crystalline lattices is unknown. This leads tg a new term of free enthalpy

AG

which for cubic paracrystals with N -lattice cells is given by /12/:~

bP

= 3/2

1

N3 Nk A. gEk

k (2)

A

depends on the form of the tangential potentials and determines the v h u e of a(*. Figure

3

shows a characteristic example with a particular value of the surface free energy

C

Y

and volume enthalpy

A G

,

g = 0

%

for crystals, g = 4,2

%

for polymer blends and g =

7,5

$ fo? melts. After attaining the crltical sizs, the paracrystals grow to a minimum of enthalpy where Nk =

(d/gkk)

.

Fig.

3 -

Free enthalpy of crystals (g = 0 %), polymers (g = 4,2 %), melts and glasses (g

7,5

%)

.

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C8-382 JOURNAL DE PHYSIQUE

The thus constructed crystal has a much too large density. It becomes somewhat toosmallbyusing the lattice cell of Figure 5a as tQe b ~ i c k of a kwo tines larger fcc-lattice (Fig. 9). An alternating 2 11 -twist

of the SiO -tetrahedra (~ig. 5c) leads to the correct density and in- troduces pzracrystalline distortions for simplicity not drawn in Fig.

5.

Nevertheless, one obtains herefrom all the positions of the pair distribution function and their weights. The residual electron density distribution at the Si-Si- and 0-0-peaks are negligible. Now the RRDF depends only on the pair-distribution of atom positions. One has to adapt solely the widths of the single pair-peaks to the observed RRDF and one finds automatically the paracrystalline distortions of g - 1 0 $. It is inte- resting to note that the 11 -twists of the SiO -tetrahedra are obser- vable in the RRDF, because some of the 0-0-max4k cleave into two parts

(Fig.

4).

After introducing paracrystalline distortions, four inter- penetrating families of rings with six SiO -tetrahedra of Figure

5

still build up a three-dimensional network!/2~he paracrystalline lattice factor Z ( b ) ofthe table describes the structure of the microparacrystal- line powder diagram similar as the conventional lattice factor averaged over all directions in a Debye-diagram,including the atoms in the grain boundaries. In melts and glasses this is performed by bimodal statistics which in the case of Si02 build up the "amorphous network".

Fig, 4

-

Radial density function of Si02 reduced to SiO (RRDF).

Fig,

5 -

Undistorted lathice cell of the SiO The arrows -axis. The dotted lines

indicate the 11 -twists around the z - ~ ~ ~ ~ ~ ~ ~ ~ ~ *

represent the "bearing netplanes".

Figure

6

shows schematically the Si-atoms of one SiO -microparacrystal of w 12

a

diameter. The "bearing netplanes" of tge fcc- crystal are the (111)-netplanes which define the surfaces o

crystalline octahedra and produce the above mentioned 0,24

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et al. /14/, Zachariasen /15/, and the recently mentioned 'clusters by Phillips /16/.

Fig.

6 -

Schematic drawing of an undistorted octahedral microparacrystal of Si02-glass.

REFERENCES

/1/ Laue, M., Friedrich, F. and Knipping, P., Sitzungs-Ber. Bayer. Akad. Wiss. Math. Phys., K1.

303

(1912),

363.

/2/ Laue, M.v., RSntgenstrahl-Interferenzen,

3.

Aufl., (1960), Akad. Verl. Ges., ~rankfurt/~ain.

/3/

Ewald, P.P., Proc. Phys. Soc.

o on don),

2

(1940)~ 167.

/4/

Debye, P., Z. PhyS., 28 (1927),

135. /5/

Hosemann, R.,

Z.

~hys,

128

(1950), 1 and 465.

/ 6 / Landau,

L.D.

and Lifshitz, E.M.,Statistical Physics, Oxford (1938).

/7/

Rinne, F., , Trans. Act. Faraday Soc.,

292

(19331, 1016.

/8/~orghard, W.S. and Boudart, M., J. Catal.,

80 (1983)~

194.

/9/

Puxley D.C., Wright, C. J. and Windsor, C.G., J. Catal.,

78

(1962)~ 257.

/lo/ Esemann, R., Schmidt, W., Fischer, A. and Balt6-Calleja, F.J., Anales de Fisica,

A79

(1983)~

145.

/11/ Hosemann, R., Ber. Bunsen-Ges. phys. Chem.,

74

(19701, 755. /12/ Hosemann, R., Colloid & Polymer Scl,,

260

(l92),

86 .

/l3/

Hosemann, R., Hentschel, M., Lange, A., Uther, B. and Bruckner, R., Z. Kristallogr.,

169 (1984)~

13.

/14/ Valenkov, N. and Porai-Koshitz,

Z.

Kristallogr., 91 (1936), 195. /15/ Zachariasen, W.H., J. Am. Chem. Soc.,

3

(1932), 3 4 1 .