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Complex growth polytypism and periodic polysynthetic twins on octacosane crystals (n-C28H58)
Journal of Crystal Growth 37 (1977) 215—228 © North-Holland Publishing Company
COMPLEX GROWTH POLYTYPISM AND PERIODIC POLYSYNTHETIC TWINS ON OCTACOSANE CRYSTALS (n-C
28H58)
D. AQUILANO * Laboratoire des Mécanismes de Ia Croissance Cristalline, Unii’ersité d ‘Aix/Marseille HI, Centre Scientijique de Saint-férônie, 13397 Marseille, Cédex 4, France Received 2 November 1976
Complex growth polytypic structures arising from evaporated solutions of n-C28H58 (octacosane) in petroleum ether are described. Dihedral concave angle effect on the step velocity is discussed in detail as the key for the determination of the mutual orientation of two layers in twin position. Several periodic polysynthetic twins are described as growth phenomena generated by screw dislocations. The (110) twin laws are discussed in detail. An attempt is made to correlate the observed occurrence of both complex polytypes and periodic polysynthetic twins with the conditions of crystallization.
1.Introduction
C36H74. Recent observations in our laboratories [131 seem to confirm this prediction, since the triclinic phase was observed for n-C23H58 in the range of ternperature 0—10°C. It must be emphasized that the work done in this field concerns principally the polymorphism of these compounds; consequently we have little information at our disposal about their polytypic structures. The first contribution is due to Amelinckx [15—17}who, applying the phase contrast microscopy and the multiple beam interferometry to the examination of the growth features of n-C34H70, found the basic polytypes 1 M[0], 20 [180] and the complex polytypes: 3M[180(0)1 180], 4O[(O)~ l8O(O)~180], 5M[(0)2 1 80(0)~180] *• More recently, Boistelle, Simon and Pèpe [12] have determined the first structure of a basic polytype 20[1801 for both n-C28H58 and n-C36H74. The purpose of our work is firstly to extend the investigation on the complex polytypism of the longchain n-paraffins, using their spiral growth patterns as
The structural aspect of the normal paraffins C~H2~+2was extensively studied by numerous authors [1-12]. Among them, Muller [1,2] and Mazee [4] determined the polymorphic modifications and their respective stability domains for several longchain members of the series. MUller and Lonsdale [3], Smith [5], Schearer and Vand [6], Teare [7], Nyburg and LUth [9], and Broadhurst [10] give a determinative contribution to the structure resolution of the fundamental polymorphs for both the even and odd members in the range 11 ~ n ( 39. From these studies it appears that, according to the conditions of crystallization, to the purity of the materials and to the number of carbon atoms in the chain, the paraffins usually crystallize in one of three systems, triclinic [n(even) ~ 26], monoclinic or, if not quite pure, orthorhombic [26 ~ n(even) ~ 36; 11 ~ n(odd)~<39]. In their paper on the prediction of unit cells and atomic coordinates of n-alkanes, Nyburg and Potworowski [11] predicted triclinic structures up to *
*
NATO Science Fellow; on leave from the Istituto di Mineralogia e Cristallografia dell’Università di Torino, Via S. Massimo 24, Torino 10123, Italy. 215
The Ramsdell polytype designation 114] is adopted throughout this paper. The number of layers in the unit cell is followed by the symmetry symbol of the same cell T~= triclinic, M = monoclinic, 0 = orthorhombic, etc. We will explain in section 3.2.1 the meaning of the symbols contained in the brackets.
216
D. Aquilano
/ Complex growth
a tool to reveal the anomalies in the normal layer stacking. Secondly, considering that n-paraffins present a layered structure which is favourable to the twinning [18,19], we are interested in the periodic polysynthetic twins which may be generated by the spiral growth from a normally twinned crystal or from two crystals, belonging to the same polytype, which collide in twin position. Up till today, this peculiar kind of twins, the structures of which are nearly identical to the polytypic one, were not extensively studied. Amelinckx [16] and Baronnet [101 have treated this argument respectively for the B-form of the behenic-acid and for the mica muscovite, in relation to their growth mechanism. Among the different paraffins in which we are interested, the octacosane (n-C23H58) was chosen. This paper deals with the results obtained by crystallization of the octacosane in a well defined supersaturation range (a ( 0.05). At the temperatures of crystallization we have chosen, the monoclinic polymorph is the stable one. The complex polytypism, and the periodic polysynthetic twinning connected with this polymorph will be described here in their crystallographical and growth aspects,
polytypism on octacosane crystals
small crystals (<500 pm) in order to observe the surface phenomena as they appear during the first growth stages. As a consequence is was not possible, even if not necessary, to study the crystals by X-ray investigations. The observations of the crystal surfaces by electron microscopy with the platinum—carbon replica technique was quite difficult to perform, because the surfaces of the octacosane crystals (melting point 6 1°C)are more or less destroyed by the carbon flux. For this reason the crystals, after silvering, were observed at the metallurgical microscope with or without a phase contrast device. In some cases a different method has been used: a thick layer of silver (>1000 A) was first evaporated on the bottom of the crystallizer; then the crystals were obtained in the usual way and later on covered with a half-reflecting silver layer (200—300 A). By these means the double silver layer including the crystals, observed in reflected light, caused the interference colours to occur. The thickness of the growth steps on the (001) faces of the crystals were estimated with this technique; the measurement are in good agreement with those performed with the aid of a double-beam interferometer.
2. Experimental Crystals of octacosane n-C28H58 were obtained by very slow evaporation from weakly concentrated solutions of n-C28H58 in light petroleum ether. Ligth petroleum ether, frequently used as a solvent for paraffmns is a complex mixture. The commercial available product (Prolabo) has a mean molar mass of 78.5 g and contains essentially pentane, 2-methylpentane, cyclopentane and 3-methylpentane in the amount of 39.4, 27.4, 12.2 and 8.1 mol per 100 mol of solvent. As for the employed paraffin, its purity is better than 95% (Fluka), the “impurities” probably being paraffins with slightly different numbers of carbon atoms in the chain. The crystallization was carried out in the temperature range 15—27°C, because, at temperatures lower than 15°C appears the triclinic polymorph, and at temperatures higher than 27°C,petroleum ether evaporates too quickly (boiling point 3 1°C).Highly supersaturated solutions have not been used in order to minimize the dendritic growth. Our aim was to obtain
3. Growth features on the (001) faces 3.1. Simple growth spirals Basic polytype JM[OJ. —
The crystals of the monoclinic polymorph are generally lozenge-shaped thin plates having only the [001} form well developed and limited by (110> edges. The lateral faces (11—) are too stepped to be directly observed. The average value of the acute angle x of the lozenge is 74 ±2°;this value agrees very well with the calculated one between the directions (110> in the Shearer and Vand [6] cell: a0 = 5,59 A, b0 = 7,42 A, c0 = 38,18 A, i3 = 1 19°56’,S.G.P 21/a. The molecular chains make an angle of about 30°with the normal to the (001) face (c* direction) and run nearly parallel to the c axis. Thus the thickness of a monomolecular layer is d001 = 33,25 A. The (001) faces are generally characterized by growth spirals, simple and polygonized, the growth steps of which are parallel to the (110> directions. Rarely the edges [100] truncate the acute angle (figs. la,b): this happens when the crystal grows at very
D. Aquilano
/ Complex growth polytypism on
octacosane crystals
(Sto)
217
A
1100]
[~io]
(no)
(010)
rn
Hg. Ia. Simple polygonized grov~th on the (01)11 face of an octacosane crystal, slio~~ ing ssell developed steps along the (110) directions, less deseloped steps parallel to the 11001 direction and the growth velocities anisotropy.
Fig. lb. The drawing represents the pattern in fig. Ia; the line m is the symmetry element of the pattern: direction 11001. The symbols (l10),(110), (110), (110), (010) and (010) mdicate the microfacets corresponding to the ledges of the nonmonomolecular steps. I~7r
A
t(oiO)
~(oio)
r;~
/
A
of acute angles are deterFig.the ld.pattern Fateral ib); view the along the and [0101obtuse direction (section AA’ mined by the tilt of the paraffinic chains which are subparallel to the c ax is.
V( 1T0I
rn
V
1
(110)
Fig. Ic. Polar diagram corresponding to the pattern in figs la, lb. The depths of the minima are (qualitatively) inversely proportional to the corresponding normal growth velocitie~ of the steps. The arrow indicates the projection of the c axis onto the (001) plane, i.e. the tilt of the paraffinic chains.
b,,. p.d00,
(ooi)
doe,
Fig. le. The height of the exposed ledge (p X dtyji) is imposed by the normal component b~of the Burgers vector.
l’ig. If. The light reflecting macrosteps are clearly visible; they make an obtuse angle with the (001) plane. On the contrary the steps making an acute angle and the composed ledges (concave angle) are not reflecting the incident light (arrow direction).
218
D. Aquilano
/ Complex growth polvtypism on
low supersaturation and its morphology goes towards the equilibrium form. The corresponding polar diagrams of the step velocities (fig. lc)shows a line m of symmetry which coincides with the direction [100], according to the symmetry of the (001) faces in the Schearer and Vand cell. The step heights are variable, in relation to the strength of the screw dislocation which generates the spiral; step heights up to 3500 A have been observed, corresponding to about one hundred monomolecular layers. In this case, if these macrosteps are undissociated, we consider the spiral ledges as microtacets belonging to the form {1 10}. The anisotropy of the step velocities for a simple spiral (figs. la,b,c) is due to the different angles formed by the microfacets {1 10} with the (001) surface. As it may be deduced from the figs. lc,d, the microfacets (110) and (110) make an obtuse dihedral angle with the (001)_plane; on the contrary the microfacets (110) and (110) make, with the same plane, an acute angle, supplementary to the previous one, In his work on the polytypism of n-C34H70. Arnelinckx [17] suggested that the steps forming an obtuse angle with their substrate, spread over this substrate faster than the ones forming the corresponding acute angle. From our observations, drawing advantage from the reflecting power of the very high steps, and knowing the direction of the incident light (fig. If), we come to the opposite conclusion. Only after this statement it is possible to know the orientation of each spiral we have observed in relation to the crystallographic axes of the chosen lattice, All the simple spirals, as represented in fig. la, belong to the polytype IM[0], irrespectively of the height of their growth steps. According to the Ramsdell notation [141the symbols contained in 1M[0] allow us to describe the basic structure revealed by the spiral growth. Let us consider a structure defined by the stacking of monoclinic layers (M) the thickness of which is I X d001 (1). If the rotation angle between successive stacked layers is 1 = 0 (i.e. they are arranged in the normal way), the resulting structure belong to the polytype IM[0]. Now, if a screw dislocation which has its component normal to the (001) face, b~,different form zero, becomes active on the (001) face of such a crystal (fig. le), the resulting growth pattern will be like that represented in fig. la, and the undissociated growth steps of the spiral will have a thickness b0 = p X d001 (p integer),
octacosane crystals
3.2. 3.2.1. Interlaced growth spirals complex polytypism When the temperature of crystallization exceeds 18°C the growth patterns on the (001) faces of several crystals become more complex. Two spirals of the same sign, generated by the same screw dislocation, show homologous growth steps strictly parallel to the (110> directions, but the growth rates of the steps belonging to the two spirals, are different from one another (fig. 2a,a’). Consequently, depending on the difference of the growth rates, the fronts, belong. ing to one spiral, may reach, after a certain time, the fronts of the other one (fig. 2b,b’). From the interaction between them a common front will result (front BC in fig. 2b) the growth rate of which is greater than the one of the residual front (AB’ in fig. 2c,c’). The overall interaction between the two spirals produces an interlaced growth pattern (figs. 2d,e). By drawing —
up its associated polar diagram and taking into account our conclusions on the orientation of a single spiral in respect to the crystallographic axes (3.1), we deduce that the two spirals are mutually rotated through an angle 1 = 180° around the normal to (001) face. In order to deduce a structural interpretation about the crystal generated by these interacting spirals, we will use this last conclusion together with a detailed analysis of the polar diagram (fig. 2fl in the following way: (i) Let us consider the spiral A composed by p monomolecular layers (3.1) as represented in full line in figs. 2f,e. A second spiral B (dotted line) composed by q monomolecular layers and turned through an angle of 180° around the c’~direction, is superim. posed to the spiral A. (ii) Let now their respective growth steps interact: in the sector I of the corresponding polar diagram (2f) the growth front of the “upper spiral” B forms an acute angle with the substrate; on the contrary the growth front of the “lower spiral” A forms the supplementary obtuse angle with its substrate owing to the mutual rotation of the two spirals. (iii) Let us apply the same reasoning to the other sectors of the polar diagram, considering the relationship among the different growth rates of the steps for each spiral. Labelling VAM, VBM and VAm, VBm
D. Aquilano
/ Complex growth polytypism on
Figs. _ 2a,2a’. The growth steps of the two spirals emerging form the different velocities (VB > VA), due to the acute—obtuse angle effect.
octacosane crystals
same ~VBVAa~ core spread out
219
parallel onto the (001) plane, but with
Figs. 2b,2b’. The faster step (VB) reaches the slower one (VA): a dihedral concave angle is produced at the common front BC.
C
C’
Figs. 2c,2c’. The common Iront velocity (V~i.)is higher than that of the residual step AB’ (VA) and probably lower than that of the fastest step (Vgl.
220
D. Aquilano
/ Complex growth polytypism
on octacosane crystals
Fig. 2e. Decomposition of the interlaced pattern in fig. 2d: I ig. 2d. The interlaced pattern corresponding to the complex 8O(O)q_ 1 180]. polytype (p + q) M[ (O)~_1 l
the upper spiral (dotted line) is turned through an angle 180° in respect to the lower spiral (full line).
‘~
semr~4~s(1
1~2
Fig. 2f. The polar diagram corresponding to the pattern in figs. 2d,e: upper spiral diagram (dotted line), lower spiral diagram (full line). The two arrows indicate the projections of the c axes associated to the orientations of the spirals on the (001) plane,
Fig. 2g. The lateral view along the [010] direction (section AA’ of the pattern in figs. 2d,e). In the sectors 1 and 2 the common fronts are characterized by the same concave angle as in the sectors 3 and 4, but they have different configuralions in relation to their substrates, being p ~ a.
respectively the maxima and the minima of the rates of growth for the spirals A and B, and remembering the growth anisotropy due to the different inclinations of the steps, we obtain for each spiral and each sector the growth velocities shown in table 1, where VAM > VAm, VBM > VBm, VAM> VBm, and VBM>
If in every sector the fast growth step is able to reach the slower one, there are then four common fronts, two by two identical. Namely in the sectors 1 and 2, the common fronts define two dihedral angles, concave and equal one to the other. Their angles are built up by the p layers of the spiral A plus the q layers of the spiral B. On the other hand in the sectors 3 and 4 two other angles will result, both concave and equal one to the other, but different from the previous ones, as regards their configuration in relation to the substrate (fig. 2g). The concavity of these dihedral angles, together with their different configuration in relation to the
VAm.
Table 1 Sector
1
2
3
Upper spiral B Lower spiral A
VBM VAm
VBM VAm
VBm 17AM
VBm 17AM
substrate can be proved to be well founded statements with the aid of the following considerations:
D. Aquilano
/ Complex gro seth polyrypism
(i) The growth rate of the common front becomes higher than that of the residual one, as emphasized in figs. 2c,c’. Its acceleration can be attributed only to the formation of a concave angle. A few interpretations have been proposed to explain this fact. Dawson [19] observed this phenomenon in the formation of concave angle in microscopic twinned laths on n-C100H202. Hartman [21] and Frank [22] suggested that two-dimensional nucleation is promoted in the concave angle of the macroscopic twins near the twin boundary. Finally, Billig [23] showed that the growth anisotropy of the twinned lamellae is due to the concave angle effect, (ii) All the interlaced growth patterns we have analyzed show that the distances between two parallel common fronts in the sectors 1 and 2 are not the same as the in sectors 3 and 4. That means different growth rates in the above-mentioned sectors, together with a symmetry line parallel to the short diagonal of the lozenge shaped interlaced pattern, Consequently, there is no reason that the common growth fronts are not related by the symmetry line must havewhich the same configuration in relation to the substrate. This proves furthermore that the step
221
on octacosane crystals
height is not the same for the spirals A and B, that is to say, p is different from q. We can now define a complex polytypic structure associated to the growth pattern we have described. Following the Ramsdell notation this new polytype may be represented by (p + q) M[(0)~_1 l8O(O)q_i 180] where (p + q) X d 001 is the period of the polytype along the c~direction. The length of this period is equal to b0, the component parallel to the c~direction of the Burgers vector b associated with the screw dislocation which generates the interlaced growth spiral. The symbol M is related to the monoclinic symmetry of the polytype; (p 1) indicates the number of monomolecular layers, all parallel, which are stacked in the same orientation (0 degree) as the first one; (q 1) mdicates the number of the monomolecular layers, all parallel to the first one (0) which is turned through 180°in respect to the p-th underlying layer. Let us now put the case p = q ° 1. According to the previous considerations, this will lead to a cornplex2P0[(O)p_i orthorhômbic polytype, the The notation of which l8O(U)p_i 180]. corresponding is interlaced growth pattern will be characterized by a symmetry mm. We have not observed this kind of —
-—
rn\~
Fig. 3a.The growth patterns is composed by two spirals emerging from the same core and mutally rotated through and angle CV.
Fig. 3b. The decomposition of the growth pattern represented in fig. 3a. Each pattern has a line of symmetry lying in the (001) plane: m (full line spiral), m’ (dotted line spiral), i.e. each spiral is associated to the polytype lMEOl. From the corresponding polar diagram the rotation angle between the two spirals is CV 01~ 29°.
222
D. Aquilano
\
bn;(Ps+Qs)xdooi
:~ç\ fl~-~
I c
on octacosane crystals
_________~5SS~\
~ L
/ Complex growth polytypism
L ,
.
r
‘~
I I ‘k~
~
,,.~‘
is
~
.2
higherof nent than a screw ~B x ddislocation normal to the (001) face, is (p~+ au) X d001 001,may an exposed occur onledge the the freeheight surface of of which the crystal B.
~
Fig. 3c. The two-dimensional common lattice (2D~CL)which characterizes the interface between the two mutually rotated 1M[Ol structures, as represented in figs. 3a,b. The circled intersecting arrows correspond to the points defining the 2D.CL cell (white surface). Twin axis 15101.
V~106
with Fig. 3d. orientation A crystallite A,and is by composed a B (001)bylayers ~ +allPA) turned (001) through layers and angle CV as regards to the orientation A. lf bn, the compo-
A
Fig. 3e. The spiral B (dotted line) is rotated through and angle CV = 106° as regards to the underlying spiral A (full line). From this construction, with the aid of table 3, we come to the idealized interlaced pattern in fig. 3h. Twin axis normal to 11101 direction.
sectn(3)q
Fig. 3g. Lateral view along the [1101 direction (section AA’ of the pattern in fig. 3h). The concave angles in the sector 1 have a different configuration, in relation to the (001) plane, as regards to the sector 3.
--
a
/~“‘~‘~,
2
Fig. 3f. The polar diagram of the step velocities corresponding to the construction in fig. 3e. In the sectors 1, 2, 3 and 4 the relations between the step velocities are chosen according to table 3.
/ [11o7/~’
~~9/2
7
Fig. 3h. Idealized growth pattern as resulting from the conditions described in figs. 3e,f and in table 3. The pattern is more developed along the [110] direction, due to the concave angle effect in the sectors 1 and 3.
D. Aquilano
/ Complex growth polytypism on
octacosane crystals
223
\
8--~ C
A
3 7’t’\ ‘~\\\ ..---///A. ‘\ ~\\\
“ \“ ~‘
—
C
p
\‘\‘\~\-
~
/
/
/
r Fig. 3i. The spiral B (dotted line) is rotated through an angle = 74° with regard to the underlying spiral A (full line). From this construction and according to table 4 we conic to the idealized interlaced pattern in fig. 3t. The twin axis is along the [110] direction.
I Fig. 3h’. An example of the interlaced growth pattern idealized in fig. 3ls. The elongation of this growth polysynthetie twin is in the [110] direction.
sectors (a,a)
a
~
B
~C~Y /
1
/
~
..
-.
secrors(r)
2
-
Fig. 3j. The polar diagram of the step velocities corresponding to the construction in fig. 3i. In the sectors 1,2,3 and 4 the relations between the step velocities are chosen according totable4.
Fig. 35. Idealized growth pattern as resulting from the conditions described in figs. 3ij and in table 4. In this ease the lateral anisotropy of the twin is along the direction AA’, i.e. normally to the [110].
Fig. 3k. Lateral view along the [110] direction (section AN of the pattern in fig. 3t). In this case the concave angle effect is missing and the result of the step inieraction can be the occurring of high microfacets having the profile of the (110) faces.
a
Fig. 32’. The growth pattern shows an example of the idealized one in fig. 32.
224
D. Aquilano
/ Complex growth polytypism
growth pattern. If p = q = 1 we find again the particular case of the basic orthorhombic polytype 20[180] [12].
3.2.2. Periodic poiysynthetic twins When the crystallization temperature is lying in the interval 1 8—27°C,a new kind of growth patterns is observed. The occurrence frequency of these new patterns is considerably reduced with regard to the previous ones which appear also. The morphology of the (001) faces of the crystals is then characterized by the fact that, while one spiral has its steps parallel to the (110) directions of the crystal, the other is rotated in relation to the former through an angle around the normal to the (001) face (fig. 3a). The m symmetry of each spiral indicates that they belong individually to the polytype 1 M [0]. The step height is variable from one spiral to the other and it remains of the same order of magnitude as the previous ones. The angle CF can be measured, for every couple of spirals on the corresponding polar diagram (fig. 3b), according to the same reasoning we have explained in the preceding section. In order to relate the observed CF values to the geometrical properties of the interface defined at the boundary layers between the two spirals, it it sufficient to consider, for the sake of simplicity, only two monoclinic layers in contact on the (001) plane, and to calculate all the rotation angles CF which allow two-dimensional common lattices (2D-CL) to occur at their interface. The 2D-CL cells were chosen according to the following conditions: (i) Let us represent with an arrow the orientation (projected on the (001) plane) of a molecular chain (full arrow lower layer chains; dotted arrow upper layer chains) (fig. 3c). The origin of the cell coincides with the gravity centers of two molecules (projected on the (001) plane), one belonging to the upper layer, the other to the lower one. At CF = 0, the arrows have the same orientation, i.e. the two layers are stacked in a normal way. (ii) The rotation axis of the system is along the normal to the (001) plane and passes through the origin, (iii) The multiplicity of the cell is defined by assuming as equivalent points of the cell only those in which the intersecting arrows have the same mutual orientation. The observed CF values and the calculated ones are both summarized in table 2; the structural —
—
on octacosane crystals
relations between the layers may be described either by the rotation angle ‘1 or by simple twin laws, (001) being the contact plane for all the twins and [uvO] the directions of the twin axes. Among the twin laws, two are very interesting for their occurence and for the peculiarity of their growth morphology. We limit our discussion to both these cases. 3.2.2.1. Twin laws: contact plane (001), twin axis normal to [110]. Following the conditions exposed in the preceding section we find four angles for which the corresponding 2D-CL cells have the same geometry; they are °F~ = 73°58’, CF 2 = 106°02’, CF3 = 73°58’+ 1800, CF.~= 106°58’+ 180°. In all these cases, the center of gravity of the molecules belonging to the two layers in twin position are lying on parallel rows along the directions [110] (CF1, CF3) and [110] (CF2, CF4). But even if the geometrical configuration of the respective 2D-CL cells is the same (except their absolute orientation), the kinetic effect, due to the tilt of the chains in the layers, may cause different patterns in the final growth morphologies. Let us imagine two crystals in contact on their (001) plane and mutually rotated through the considered angle CF; they belong both to the polytype 1M[0]. The f’irst crystal is built up by (n + p) monomolecular layers in positions A; the second one by q layers in position B (fig. 3d). 1.et us now assume that a screw dislocation appears during the growth of this crystal couple. If its component b~is equal to (PA + q~)X d00~,an exposed ledge of the same height occurs on the free surface of the crystal B. Owing to the growth conditions this exposed ledge can generate two spirals which develop mutually rotated through the angle CF. Let us consider the case CF = 106°02’;as may be seen in fig. 3e, we assume that the two spirals develop independently one form the other; then we analyze the interaction of their growth fronts, taking into account the conditions of the preceding section (3.2.1). In this case we predict the polar diagram of the step growth velocities with the aid of table 3, where RAm, VBm, VAM, VBM are obviously subjected to the same relations as indicated in table 1. In the sectors I and 3 the growth steps of the two spirals are parallel, but they have different inclinations with respect of the (001) plane (figs. 3f,g) as in the case of the sectors 3 and 1 respectively of section
D. Aquilano
/ Complex growth polytypism
on octacosane crystals
225
Table 2 CVobs
(a) 0° (b) 180°
~ca1c
Components of the 2D-CL cell vectors: (xa +yb) (x’a+x’b)
0°
x = 1, x’O,
180°
9°49’ 12°l7’
—
lS°30’
17°06’
16°
2D-CL cell multiplicity
= 0
2D-CL reticular density
Twin axis
[uv0]
Number of observed crystals
1
2
—
Very high
1
2
(001) plane
High
68
0.06
[190]
—
46
0.087
[170]
23
0.087
[150]
2
29
0.07
[140]
2
y=l
1, 0, —8,
0 1 4
7,
5
—7,
3
6, —5, 4, 7, —1,
4 2 3 1 4
—
21°
2l°20’
26°,27°50’ 29° (twice)
28°10’
2, —3,
2 1
8
0.50
[130]
42°
42°12
3, —4,
2 0
8
0.125
[120]
48°40’
3, —3,
1 5
19
0.21
[350]
53°30’
53°20’
5, —2,
0 3
15
0.066
[230]
1
59°(twice)
58°56’
—3, 7,
4 3
37
0.054
[340]
2
74°
73°58’
1,
—l
18
0.222
[110]
106°
106°02
1, 5,
—1 4
18
0.222
normal to
—
—
14 [110]
87°
86°33’
—5,
4
53
0.037
[5401
93°
92°56’
5, —7,
4 5
53
0.037
[750]
89°44’
4, 0,
0 3
12
0.083
[430]
10
—
—
The vectors defining the 2D-Cl cells are referred to the (a, b) axes of the lower paraffinic layer, the rotation angle CV of the upper layer in respect to the lower one is anticlockwise. The values of the angle CVcalc > 90°are missing (except CV = 92°02’, 106°02’ which are quoted as examples) because if there exists a 2D-CL for CV = CV0, there will exist three other equivalent cells for CV = 180 CV0, 180 + CV0, 360 —
(a) Basic polytype 1Mb]. 80(O)q—1 180]. (b) Complex polytype (p + q)M[(0)~_1 l
—
226
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/ C’oinplex growth polytypism on
Table 3
octacosane crystals
Table 4
Sector
1
2
3
4
Sector
1
2
3
Upper Lower spiral spiral B A
VBm VAM
VBM VAM
VBM t7Arn
VBm VAm
Upper Lower spiraiB spiral A
VBm LAm
VBm LAM
VBM LAM
(3.2.1). If we apply the same reasoning explained before about the kinetic of the common fronts, we may deduce that the lateral development of this twin will be controled by the velocity VBm in the sector 1 and by VAm in the sector 3. In the sectors 2 and 4, however, where the growth steps of the two spirals make an angle of ~148°, there is no possibility that a common front will be built up. Even if we are not able to predict this complex interaction, we may outline that in the sector 2 the growth fronts make the same acute angle with the (001) plane; consequently they will spread at the maximum velocities VAM and VBM. On the contrary, in the sector 4, where both the growth fronts make the same obtuse angle with the (001) plane, the limiting growth rates will be VAm and VBm. If now we choose the emergency of the screw dislocation as the reference point, we will observe that the lateral devel. opment of the resulting polysynthetic twin will be enhanced in the sector 2 and the growth pattern will take a rectangular shape elongated in the direction [110] (figs. 3h,3h’). The elongation depends obviously from the anisotropy ratios VAM/VAm and VBM/VBm, and it may be furthermore increased by the effect of the concave angle, as described by Dawson [211 for the (110) normal twins in n-hectane. We may find a similar growth pattern for the rotation angle r1 = 254°; in this case the twin axis is normal to the [110] direction and the growth pattern will result in enantiomorphic position in respect of the previous one, (010) being the mirror plane. 3.2.2.2. Twin law: contact plane (001), twin axis: Applying the same reasoning as for the just described twins, we may deduce that for a rotation angle 1 74°, we obtain the growth rates reported in table 4, together with the associated polar diagram (figs. 3i,j). In the sectors 1 and 3 where the growth steps of
/ 110J.
4 17BM 1~Am
the two spirals are parallel, the growth fronts have, two by two, the same inclination, i.e. they make the same obtuse angle with the (001) plane in the sector 1 and the same corresponding acute angle in the seetor 3. For VBm VAm, after a certain period of growth, in both these sectors, the fastest front will reach the slowest one. At this moment a common front will occur, the morphology of which is that of a microfacet (fig. 3k). Its height is (p + q) X d 1~01its velocity will be reduced with respect to that of the slowest front, in both the sectors. For instance, in the sector 1 the velocity of the common front will be slower than the smallest one of (VAII,, VBm). In the sector 3, the common front will spread more rapidly than in the sector 1, for the microfacet makes an acute angle with the (001) plane. In the sectors 2 and 4 the growth steps make, also in this case, an angle of 148w, but the interaction between the growth fronts is more complex than in the previous case. [n the sector 2 (as in the sector 4), the two interacting fronts have opposite inclinations. Amelinckx [16] discussed this interaction for the B-form of the behenic acid and he deduced the final equilibrium form of this polysynthetic twin. The idealized growth pattern we have obtained and the observed ones (figs. 31,1’) are in good agreement with Amelinckx’s conclusions. Furthermore we may establish that in the sectors 2 and 4 this twin would have the same development, owing to the symmetrical interaction of the growth fronts in both the sectors. Summarizing we may conclude that this twin will be recognizable from the lateral anisotropy of growth in the sectors 1 and 2 where the common growth fronts develop. As in the previous case we may find an analogous but enantiomorphous growth pattern, if the rotation angle between the twinned layers is 286°. In this case the twin axis will be along the direction [1101. Remark: Throughout this paper we have used the ~°
terms: complex growth polytypes and periodic poly-
D. Aquilano / Complex growth polytypism on octacosane crystals
synthetic twins. These terms would be correct if the corresponding structures are seen by means of X-ray methods. Indeed, if X-ray investigations can be carned out on such crystals, we should find the reflections corresponding to the long periodicity associated to the spiral growth promoted by high Burgers vectors. By this way the commonly used X-ray methods are not able to distinguish a structure generated by the activity of a screw dislocation from the same structure generated by 3D-nucleation. Strictly speaking both kinds of structures we have described, in their parts generated by spiral growth, are not a penodic stacking of crystalline individuals related by twinning operation; on the contrary they are two individuals only, related by twinning operation, which develop continuously describing the helicoidal surface imposed by the screw dislocation,
4. Discussion Summarizing the results obtained in this work, we may deduce the following considerations: (i) The acute angle effect (section 3.1) and the dthedral concave angle effect (section 3.2.1) have been revealed as a useful tool to define unambiguously the polytype and twin laws we have found starting from the corresponding interlaced spiral growth patterns. Consequently, the application of these effects can be generalized successfully to all the polygonized patterns in which the growth steps mak~no right angles with their substrate. (ii) As may be seen in table 2 the values of the rotation angles ~calc are in good correspondance with those we have measured by the decomposition of the interlaced growth patterns (~obs). But there is no relation between the reticular densities of 2D-CL cells
227
impingement mechanism suggested by Fisher et al.
[24] and observed by Newkirk [25], Amelinckx and Dekeyser [27], and Baronnet [201 on crystals grown from solutions. In our opinion this mechanism may be promoted by the weak convection currents present in the evaporating solutions and by the competition of the crystals on the bottom of the crystallizer in the final period of their growth. (iv) As we have described in sections 3.1 and 3.2, the frequency of the occurrence of polytytpes and twins, independently from their crystallographical laws, depends on the temperature of crystallization T~,i.e., the higher is T~,the higher is the probability of the observation of a polytype or a twin. Obviously it is not only T~which causes this behaviour, but also the rate of evaporation of the solvent and consequently the rate of the supersaturation change in the solution. The supersaturation determines the probability of anomalous nucleation, spiral growth activation, etc. At the same time, it is well known that the higher is T,~,the higher is the probability that an anomalous nucleus can adjust and fix on a normally grown substrate. In order to dissociate the different influencies of the crystallization temperature of the solution, and of the solvents, and to generalize the twin laws for paraffins having different chain lengths, a research is in progress on octacosane and hexatriacontane crystals obtained from evaporated solutions in which the solvents employed are the pure components of light petroleum ether [29].
Acknowledgements The author thanks Drs. A. Baronnet, R. Boistelle, and B. Simon of the laboratory for helpful discus-
sions and valuable criticisms on this work.
and the frequency of the occurrence of the corresponding twin laws. This disagreement is due to the
roughness of the geometrical approach. Calculations of the interaction energies associated to the observed twin laws are in progress. The first results we have obtamed are in good agreement with the observed frequencies of occurrence, for both polytypes and twin -
.
-
.
laws [28]. (iii) The surprisingly elevated values of the Burgers vector b we have encountered in all the spiral pat. terns, may be explained if we consider the edgewise
References [1] A. Muller, Proc. Roy. Soc. (London) A120 (1928) 137. [2] A. Muller, Proc. Roy. Soc. (London) A138 (1932) 514. [31A. Muller and K. Lonsdale, Acta Cryst. 1 (1948) 129. [4] W.M. Mazee, Travaux Chim. Pays Bas, 67 (1948) 197. [5] A.E. Smith, J. Chem. Phys. 21(1953) 2229. [6] H.M.M. Shearer and V. Vand, Acta Cryst. 9 (1956) 379. [7] P.W. Teare, Acta Cryst. 12 (1959) 294.
[8] A.R. Ubbelohde, Trans. Faraday Soc. 34 (1938) 296.
228
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/ Complex growth polytypism on
[9] S.C. Nyburg and H. Luth, Acta Cryst. B28 (1972) 2992. 1101 M.G. Broadhurst, J. Res. Natl. Bur. Stand. 66A (1962) 241. [11] S.C. Nyburg and J.A. Potworowski, Acta Cryst. B28
(1973) 347. [121 R. Boistelle, B. Simon and G. Pêpe, Acta Cryst. B32
(1976) 1240. [13] II.E.L. Madsen and R. Boistelle, Acta Cryst. A32 (1976)
(in press). [14] [15] [16] [17] [18]
L.S. Ramsdell, Am. Mineralogist. 32(1947)64. S. Amelinckx, Acta Cryst. 8 (1955) 530. S. Amelinckx, Acta Cryst. 9 (1956) 16. 5. Amelinckx, Acta Cryst. 9 (1956) 217. F. Khoury, J. App. Phys. 34 (1963) 73.
octacosane crystals
[19] l.M. Dawson, Proc. Roy. Soc. (London) A214 (1952) 72. [201 A. Baronnet, J. Crystal Growth 19(1973)193. [21] P. I-lartman, Z. Krist. 107 (1956) 225. [22] F.C. Frank, Discussions Faraday Soc. 5 (Crystal
Growth) (1949) 48. [23] F. Billig, J. Inst. Metals 83 (1954), 53. [24] J.C. Fisher, R.L. Fullman and G.W. Sears, Acta Met. 2 (1954) 344. [25 J.B. Newkirk, Bull. Am. Phys. Soc. 29 (1954) 28. [26] J.B. Newkirk, Acta Met. 3 (1955) 121. [27] W. Dekeyser and S. Amelinckx, in: Les Dislocations et Ia Croissance des Cristaux (Masson, Paris, 1955) p. 146. [28] R. Boistelle and D. Aquilario, to be published. [29] D. Aquilano and R. Boistelle, to be published.
COMPLEX GROWTH POLYTYPISM AND PERIODIC POLYSYNTHETIC TWINS ON OCTACOSANE CRYSTALS (n-C
28H58)
D. AQUILANO * Laboratoire des Mécanismes de Ia Croissance Cristalline, Unii’ersité d ‘Aix/Marseille HI, Centre Scientijique de Saint-férônie, 13397 Marseille, Cédex 4, France Received 2 November 1976
Complex growth polytypic structures arising from evaporated solutions of n-C28H58 (octacosane) in petroleum ether are described. Dihedral concave angle effect on the step velocity is discussed in detail as the key for the determination of the mutual orientation of two layers in twin position. Several periodic polysynthetic twins are described as growth phenomena generated by screw dislocations. The (110) twin laws are discussed in detail. An attempt is made to correlate the observed occurrence of both complex polytypes and periodic polysynthetic twins with the conditions of crystallization.
1.Introduction
C36H74. Recent observations in our laboratories [131 seem to confirm this prediction, since the triclinic phase was observed for n-C23H58 in the range of ternperature 0—10°C. It must be emphasized that the work done in this field concerns principally the polymorphism of these compounds; consequently we have little information at our disposal about their polytypic structures. The first contribution is due to Amelinckx [15—17}who, applying the phase contrast microscopy and the multiple beam interferometry to the examination of the growth features of n-C34H70, found the basic polytypes 1 M[0], 20 [180] and the complex polytypes: 3M[180(0)1 180], 4O[(O)~ l8O(O)~180], 5M[(0)2 1 80(0)~180] *• More recently, Boistelle, Simon and Pèpe [12] have determined the first structure of a basic polytype 20[1801 for both n-C28H58 and n-C36H74. The purpose of our work is firstly to extend the investigation on the complex polytypism of the longchain n-paraffins, using their spiral growth patterns as
The structural aspect of the normal paraffins C~H2~+2was extensively studied by numerous authors [1-12]. Among them, Muller [1,2] and Mazee [4] determined the polymorphic modifications and their respective stability domains for several longchain members of the series. MUller and Lonsdale [3], Smith [5], Schearer and Vand [6], Teare [7], Nyburg and LUth [9], and Broadhurst [10] give a determinative contribution to the structure resolution of the fundamental polymorphs for both the even and odd members in the range 11 ~ n ( 39. From these studies it appears that, according to the conditions of crystallization, to the purity of the materials and to the number of carbon atoms in the chain, the paraffins usually crystallize in one of three systems, triclinic [n(even) ~ 26], monoclinic or, if not quite pure, orthorhombic [26 ~ n(even) ~ 36; 11 ~ n(odd)~<39]. In their paper on the prediction of unit cells and atomic coordinates of n-alkanes, Nyburg and Potworowski [11] predicted triclinic structures up to *
*
NATO Science Fellow; on leave from the Istituto di Mineralogia e Cristallografia dell’Università di Torino, Via S. Massimo 24, Torino 10123, Italy. 215
The Ramsdell polytype designation 114] is adopted throughout this paper. The number of layers in the unit cell is followed by the symmetry symbol of the same cell T~= triclinic, M = monoclinic, 0 = orthorhombic, etc. We will explain in section 3.2.1 the meaning of the symbols contained in the brackets.
216
D. Aquilano
/ Complex growth
a tool to reveal the anomalies in the normal layer stacking. Secondly, considering that n-paraffins present a layered structure which is favourable to the twinning [18,19], we are interested in the periodic polysynthetic twins which may be generated by the spiral growth from a normally twinned crystal or from two crystals, belonging to the same polytype, which collide in twin position. Up till today, this peculiar kind of twins, the structures of which are nearly identical to the polytypic one, were not extensively studied. Amelinckx [16] and Baronnet [101 have treated this argument respectively for the B-form of the behenic-acid and for the mica muscovite, in relation to their growth mechanism. Among the different paraffins in which we are interested, the octacosane (n-C23H58) was chosen. This paper deals with the results obtained by crystallization of the octacosane in a well defined supersaturation range (a ( 0.05). At the temperatures of crystallization we have chosen, the monoclinic polymorph is the stable one. The complex polytypism, and the periodic polysynthetic twinning connected with this polymorph will be described here in their crystallographical and growth aspects,
polytypism on octacosane crystals
small crystals (<500 pm) in order to observe the surface phenomena as they appear during the first growth stages. As a consequence is was not possible, even if not necessary, to study the crystals by X-ray investigations. The observations of the crystal surfaces by electron microscopy with the platinum—carbon replica technique was quite difficult to perform, because the surfaces of the octacosane crystals (melting point 6 1°C)are more or less destroyed by the carbon flux. For this reason the crystals, after silvering, were observed at the metallurgical microscope with or without a phase contrast device. In some cases a different method has been used: a thick layer of silver (>1000 A) was first evaporated on the bottom of the crystallizer; then the crystals were obtained in the usual way and later on covered with a half-reflecting silver layer (200—300 A). By these means the double silver layer including the crystals, observed in reflected light, caused the interference colours to occur. The thickness of the growth steps on the (001) faces of the crystals were estimated with this technique; the measurement are in good agreement with those performed with the aid of a double-beam interferometer.
2. Experimental Crystals of octacosane n-C28H58 were obtained by very slow evaporation from weakly concentrated solutions of n-C28H58 in light petroleum ether. Ligth petroleum ether, frequently used as a solvent for paraffmns is a complex mixture. The commercial available product (Prolabo) has a mean molar mass of 78.5 g and contains essentially pentane, 2-methylpentane, cyclopentane and 3-methylpentane in the amount of 39.4, 27.4, 12.2 and 8.1 mol per 100 mol of solvent. As for the employed paraffin, its purity is better than 95% (Fluka), the “impurities” probably being paraffins with slightly different numbers of carbon atoms in the chain. The crystallization was carried out in the temperature range 15—27°C, because, at temperatures lower than 15°C appears the triclinic polymorph, and at temperatures higher than 27°C,petroleum ether evaporates too quickly (boiling point 3 1°C).Highly supersaturated solutions have not been used in order to minimize the dendritic growth. Our aim was to obtain
3. Growth features on the (001) faces 3.1. Simple growth spirals Basic polytype JM[OJ. —
The crystals of the monoclinic polymorph are generally lozenge-shaped thin plates having only the [001} form well developed and limited by (110> edges. The lateral faces (11—) are too stepped to be directly observed. The average value of the acute angle x of the lozenge is 74 ±2°;this value agrees very well with the calculated one between the directions (110> in the Shearer and Vand [6] cell: a0 = 5,59 A, b0 = 7,42 A, c0 = 38,18 A, i3 = 1 19°56’,S.G.P 21/a. The molecular chains make an angle of about 30°with the normal to the (001) face (c* direction) and run nearly parallel to the c axis. Thus the thickness of a monomolecular layer is d001 = 33,25 A. The (001) faces are generally characterized by growth spirals, simple and polygonized, the growth steps of which are parallel to the (110> directions. Rarely the edges [100] truncate the acute angle (figs. la,b): this happens when the crystal grows at very
D. Aquilano
/ Complex growth polytypism on
octacosane crystals
(Sto)
217
A
1100]
[~io]
(no)
(010)
rn
Hg. Ia. Simple polygonized grov~th on the (01)11 face of an octacosane crystal, slio~~ ing ssell developed steps along the (110) directions, less deseloped steps parallel to the 11001 direction and the growth velocities anisotropy.
Fig. lb. The drawing represents the pattern in fig. Ia; the line m is the symmetry element of the pattern: direction 11001. The symbols (l10),(110), (110), (110), (010) and (010) mdicate the microfacets corresponding to the ledges of the nonmonomolecular steps. I~7r
A
t(oiO)
~(oio)
r;~
/
A
of acute angles are deterFig.the ld.pattern Fateral ib); view the along the and [0101obtuse direction (section AA’ mined by the tilt of the paraffinic chains which are subparallel to the c ax is.
V( 1T0I
rn
V
1
(110)
Fig. Ic. Polar diagram corresponding to the pattern in figs la, lb. The depths of the minima are (qualitatively) inversely proportional to the corresponding normal growth velocitie~ of the steps. The arrow indicates the projection of the c axis onto the (001) plane, i.e. the tilt of the paraffinic chains.
b,,. p.d00,
(ooi)
doe,
Fig. le. The height of the exposed ledge (p X dtyji) is imposed by the normal component b~of the Burgers vector.
l’ig. If. The light reflecting macrosteps are clearly visible; they make an obtuse angle with the (001) plane. On the contrary the steps making an acute angle and the composed ledges (concave angle) are not reflecting the incident light (arrow direction).
218
D. Aquilano
/ Complex growth polvtypism on
low supersaturation and its morphology goes towards the equilibrium form. The corresponding polar diagrams of the step velocities (fig. lc)shows a line m of symmetry which coincides with the direction [100], according to the symmetry of the (001) faces in the Schearer and Vand cell. The step heights are variable, in relation to the strength of the screw dislocation which generates the spiral; step heights up to 3500 A have been observed, corresponding to about one hundred monomolecular layers. In this case, if these macrosteps are undissociated, we consider the spiral ledges as microtacets belonging to the form {1 10}. The anisotropy of the step velocities for a simple spiral (figs. la,b,c) is due to the different angles formed by the microfacets {1 10} with the (001) surface. As it may be deduced from the figs. lc,d, the microfacets (110) and (110) make an obtuse dihedral angle with the (001)_plane; on the contrary the microfacets (110) and (110) make, with the same plane, an acute angle, supplementary to the previous one, In his work on the polytypism of n-C34H70. Arnelinckx [17] suggested that the steps forming an obtuse angle with their substrate, spread over this substrate faster than the ones forming the corresponding acute angle. From our observations, drawing advantage from the reflecting power of the very high steps, and knowing the direction of the incident light (fig. If), we come to the opposite conclusion. Only after this statement it is possible to know the orientation of each spiral we have observed in relation to the crystallographic axes of the chosen lattice, All the simple spirals, as represented in fig. la, belong to the polytype IM[0], irrespectively of the height of their growth steps. According to the Ramsdell notation [141the symbols contained in 1M[0] allow us to describe the basic structure revealed by the spiral growth. Let us consider a structure defined by the stacking of monoclinic layers (M) the thickness of which is I X d001 (1). If the rotation angle between successive stacked layers is 1 = 0 (i.e. they are arranged in the normal way), the resulting structure belong to the polytype IM[0]. Now, if a screw dislocation which has its component normal to the (001) face, b~,different form zero, becomes active on the (001) face of such a crystal (fig. le), the resulting growth pattern will be like that represented in fig. la, and the undissociated growth steps of the spiral will have a thickness b0 = p X d001 (p integer),
octacosane crystals
3.2. 3.2.1. Interlaced growth spirals complex polytypism When the temperature of crystallization exceeds 18°C the growth patterns on the (001) faces of several crystals become more complex. Two spirals of the same sign, generated by the same screw dislocation, show homologous growth steps strictly parallel to the (110> directions, but the growth rates of the steps belonging to the two spirals, are different from one another (fig. 2a,a’). Consequently, depending on the difference of the growth rates, the fronts, belong. ing to one spiral, may reach, after a certain time, the fronts of the other one (fig. 2b,b’). From the interaction between them a common front will result (front BC in fig. 2b) the growth rate of which is greater than the one of the residual front (AB’ in fig. 2c,c’). The overall interaction between the two spirals produces an interlaced growth pattern (figs. 2d,e). By drawing —
up its associated polar diagram and taking into account our conclusions on the orientation of a single spiral in respect to the crystallographic axes (3.1), we deduce that the two spirals are mutually rotated through an angle 1 = 180° around the normal to (001) face. In order to deduce a structural interpretation about the crystal generated by these interacting spirals, we will use this last conclusion together with a detailed analysis of the polar diagram (fig. 2fl in the following way: (i) Let us consider the spiral A composed by p monomolecular layers (3.1) as represented in full line in figs. 2f,e. A second spiral B (dotted line) composed by q monomolecular layers and turned through an angle of 180° around the c’~direction, is superim. posed to the spiral A. (ii) Let now their respective growth steps interact: in the sector I of the corresponding polar diagram (2f) the growth front of the “upper spiral” B forms an acute angle with the substrate; on the contrary the growth front of the “lower spiral” A forms the supplementary obtuse angle with its substrate owing to the mutual rotation of the two spirals. (iii) Let us apply the same reasoning to the other sectors of the polar diagram, considering the relationship among the different growth rates of the steps for each spiral. Labelling VAM, VBM and VAm, VBm
D. Aquilano
/ Complex growth polytypism on
Figs. _ 2a,2a’. The growth steps of the two spirals emerging form the different velocities (VB > VA), due to the acute—obtuse angle effect.
octacosane crystals
same ~VBVAa~ core spread out
219
parallel onto the (001) plane, but with
Figs. 2b,2b’. The faster step (VB) reaches the slower one (VA): a dihedral concave angle is produced at the common front BC.
C
C’
Figs. 2c,2c’. The common Iront velocity (V~i.)is higher than that of the residual step AB’ (VA) and probably lower than that of the fastest step (Vgl.
220
D. Aquilano
/ Complex growth polytypism
on octacosane crystals
Fig. 2e. Decomposition of the interlaced pattern in fig. 2d: I ig. 2d. The interlaced pattern corresponding to the complex 8O(O)q_ 1 180]. polytype (p + q) M[ (O)~_1 l
the upper spiral (dotted line) is turned through an angle 180° in respect to the lower spiral (full line).
‘~
semr~4~s(1
1~2
Fig. 2f. The polar diagram corresponding to the pattern in figs. 2d,e: upper spiral diagram (dotted line), lower spiral diagram (full line). The two arrows indicate the projections of the c axes associated to the orientations of the spirals on the (001) plane,
Fig. 2g. The lateral view along the [010] direction (section AA’ of the pattern in figs. 2d,e). In the sectors 1 and 2 the common fronts are characterized by the same concave angle as in the sectors 3 and 4, but they have different configuralions in relation to their substrates, being p ~ a.
respectively the maxima and the minima of the rates of growth for the spirals A and B, and remembering the growth anisotropy due to the different inclinations of the steps, we obtain for each spiral and each sector the growth velocities shown in table 1, where VAM > VAm, VBM > VBm, VAM> VBm, and VBM>
If in every sector the fast growth step is able to reach the slower one, there are then four common fronts, two by two identical. Namely in the sectors 1 and 2, the common fronts define two dihedral angles, concave and equal one to the other. Their angles are built up by the p layers of the spiral A plus the q layers of the spiral B. On the other hand in the sectors 3 and 4 two other angles will result, both concave and equal one to the other, but different from the previous ones, as regards their configuration in relation to the substrate (fig. 2g). The concavity of these dihedral angles, together with their different configuration in relation to the
VAm.
Table 1 Sector
1
2
3
Upper spiral B Lower spiral A
VBM VAm
VBM VAm
VBm 17AM
VBm 17AM
substrate can be proved to be well founded statements with the aid of the following considerations:
D. Aquilano
/ Complex gro seth polyrypism
(i) The growth rate of the common front becomes higher than that of the residual one, as emphasized in figs. 2c,c’. Its acceleration can be attributed only to the formation of a concave angle. A few interpretations have been proposed to explain this fact. Dawson [19] observed this phenomenon in the formation of concave angle in microscopic twinned laths on n-C100H202. Hartman [21] and Frank [22] suggested that two-dimensional nucleation is promoted in the concave angle of the macroscopic twins near the twin boundary. Finally, Billig [23] showed that the growth anisotropy of the twinned lamellae is due to the concave angle effect, (ii) All the interlaced growth patterns we have analyzed show that the distances between two parallel common fronts in the sectors 1 and 2 are not the same as the in sectors 3 and 4. That means different growth rates in the above-mentioned sectors, together with a symmetry line parallel to the short diagonal of the lozenge shaped interlaced pattern, Consequently, there is no reason that the common growth fronts are not related by the symmetry line must havewhich the same configuration in relation to the substrate. This proves furthermore that the step
221
on octacosane crystals
height is not the same for the spirals A and B, that is to say, p is different from q. We can now define a complex polytypic structure associated to the growth pattern we have described. Following the Ramsdell notation this new polytype may be represented by (p + q) M[(0)~_1 l8O(O)q_i 180] where (p + q) X d 001 is the period of the polytype along the c~direction. The length of this period is equal to b0, the component parallel to the c~direction of the Burgers vector b associated with the screw dislocation which generates the interlaced growth spiral. The symbol M is related to the monoclinic symmetry of the polytype; (p 1) indicates the number of monomolecular layers, all parallel, which are stacked in the same orientation (0 degree) as the first one; (q 1) mdicates the number of the monomolecular layers, all parallel to the first one (0) which is turned through 180°in respect to the p-th underlying layer. Let us now put the case p = q ° 1. According to the previous considerations, this will lead to a cornplex2P0[(O)p_i orthorhômbic polytype, the The notation of which l8O(U)p_i 180]. corresponding is interlaced growth pattern will be characterized by a symmetry mm. We have not observed this kind of —
-—
rn\~
Fig. 3a.The growth patterns is composed by two spirals emerging from the same core and mutally rotated through and angle CV.
Fig. 3b. The decomposition of the growth pattern represented in fig. 3a. Each pattern has a line of symmetry lying in the (001) plane: m (full line spiral), m’ (dotted line spiral), i.e. each spiral is associated to the polytype lMEOl. From the corresponding polar diagram the rotation angle between the two spirals is CV 01~ 29°.
222
D. Aquilano
\
bn;(Ps+Qs)xdooi
:~ç\ fl~-~
I c
on octacosane crystals
_________~5SS~\
~ L
/ Complex growth polytypism
L ,
.
r
‘~
I I ‘k~
~
,,.~‘
is
~
.2
higherof nent than a screw ~B x ddislocation normal to the (001) face, is (p~+ au) X d001 001,may an exposed occur onledge the the freeheight surface of of which the crystal B.
~
Fig. 3c. The two-dimensional common lattice (2D~CL)which characterizes the interface between the two mutually rotated 1M[Ol structures, as represented in figs. 3a,b. The circled intersecting arrows correspond to the points defining the 2D.CL cell (white surface). Twin axis 15101.
V~106
with Fig. 3d. orientation A crystallite A,and is by composed a B (001)bylayers ~ +allPA) turned (001) through layers and angle CV as regards to the orientation A. lf bn, the compo-
A
Fig. 3e. The spiral B (dotted line) is rotated through and angle CV = 106° as regards to the underlying spiral A (full line). From this construction, with the aid of table 3, we come to the idealized interlaced pattern in fig. 3h. Twin axis normal to 11101 direction.
sectn(3)q
Fig. 3g. Lateral view along the [1101 direction (section AA’ of the pattern in fig. 3h). The concave angles in the sector 1 have a different configuration, in relation to the (001) plane, as regards to the sector 3.
--
a
/~“‘~‘~,
2
Fig. 3f. The polar diagram of the step velocities corresponding to the construction in fig. 3e. In the sectors 1, 2, 3 and 4 the relations between the step velocities are chosen according to table 3.
/ [11o7/~’
~~9/2
7
Fig. 3h. Idealized growth pattern as resulting from the conditions described in figs. 3e,f and in table 3. The pattern is more developed along the [110] direction, due to the concave angle effect in the sectors 1 and 3.
D. Aquilano
/ Complex growth polytypism on
octacosane crystals
223
\
8--~ C
A
3 7’t’\ ‘~\\\ ..---///A. ‘\ ~\\\
“ \“ ~‘
—
C
p
\‘\‘\~\-
~
/
/
/
r Fig. 3i. The spiral B (dotted line) is rotated through an angle = 74° with regard to the underlying spiral A (full line). From this construction and according to table 4 we conic to the idealized interlaced pattern in fig. 3t. The twin axis is along the [110] direction.
I Fig. 3h’. An example of the interlaced growth pattern idealized in fig. 3ls. The elongation of this growth polysynthetie twin is in the [110] direction.
sectors (a,a)
a
~
B
~C~Y /
1
/
~
..
-.
secrors(r)
2
-
Fig. 3j. The polar diagram of the step velocities corresponding to the construction in fig. 3i. In the sectors 1,2,3 and 4 the relations between the step velocities are chosen according totable4.
Fig. 35. Idealized growth pattern as resulting from the conditions described in figs. 3ij and in table 4. In this ease the lateral anisotropy of the twin is along the direction AA’, i.e. normally to the [110].
Fig. 3k. Lateral view along the [110] direction (section AN of the pattern in fig. 3t). In this case the concave angle effect is missing and the result of the step inieraction can be the occurring of high microfacets having the profile of the (110) faces.
a
Fig. 32’. The growth pattern shows an example of the idealized one in fig. 32.
224
D. Aquilano
/ Complex growth polytypism
growth pattern. If p = q = 1 we find again the particular case of the basic orthorhombic polytype 20[180] [12].
3.2.2. Periodic poiysynthetic twins When the crystallization temperature is lying in the interval 1 8—27°C,a new kind of growth patterns is observed. The occurrence frequency of these new patterns is considerably reduced with regard to the previous ones which appear also. The morphology of the (001) faces of the crystals is then characterized by the fact that, while one spiral has its steps parallel to the (110) directions of the crystal, the other is rotated in relation to the former through an angle around the normal to the (001) face (fig. 3a). The m symmetry of each spiral indicates that they belong individually to the polytype 1 M [0]. The step height is variable from one spiral to the other and it remains of the same order of magnitude as the previous ones. The angle CF can be measured, for every couple of spirals on the corresponding polar diagram (fig. 3b), according to the same reasoning we have explained in the preceding section. In order to relate the observed CF values to the geometrical properties of the interface defined at the boundary layers between the two spirals, it it sufficient to consider, for the sake of simplicity, only two monoclinic layers in contact on the (001) plane, and to calculate all the rotation angles CF which allow two-dimensional common lattices (2D-CL) to occur at their interface. The 2D-CL cells were chosen according to the following conditions: (i) Let us represent with an arrow the orientation (projected on the (001) plane) of a molecular chain (full arrow lower layer chains; dotted arrow upper layer chains) (fig. 3c). The origin of the cell coincides with the gravity centers of two molecules (projected on the (001) plane), one belonging to the upper layer, the other to the lower one. At CF = 0, the arrows have the same orientation, i.e. the two layers are stacked in a normal way. (ii) The rotation axis of the system is along the normal to the (001) plane and passes through the origin, (iii) The multiplicity of the cell is defined by assuming as equivalent points of the cell only those in which the intersecting arrows have the same mutual orientation. The observed CF values and the calculated ones are both summarized in table 2; the structural —
—
on octacosane crystals
relations between the layers may be described either by the rotation angle ‘1 or by simple twin laws, (001) being the contact plane for all the twins and [uvO] the directions of the twin axes. Among the twin laws, two are very interesting for their occurence and for the peculiarity of their growth morphology. We limit our discussion to both these cases. 3.2.2.1. Twin laws: contact plane (001), twin axis normal to [110]. Following the conditions exposed in the preceding section we find four angles for which the corresponding 2D-CL cells have the same geometry; they are °F~ = 73°58’, CF 2 = 106°02’, CF3 = 73°58’+ 1800, CF.~= 106°58’+ 180°. In all these cases, the center of gravity of the molecules belonging to the two layers in twin position are lying on parallel rows along the directions [110] (CF1, CF3) and [110] (CF2, CF4). But even if the geometrical configuration of the respective 2D-CL cells is the same (except their absolute orientation), the kinetic effect, due to the tilt of the chains in the layers, may cause different patterns in the final growth morphologies. Let us imagine two crystals in contact on their (001) plane and mutually rotated through the considered angle CF; they belong both to the polytype 1M[0]. The f’irst crystal is built up by (n + p) monomolecular layers in positions A; the second one by q layers in position B (fig. 3d). 1.et us now assume that a screw dislocation appears during the growth of this crystal couple. If its component b~is equal to (PA + q~)X d00~,an exposed ledge of the same height occurs on the free surface of the crystal B. Owing to the growth conditions this exposed ledge can generate two spirals which develop mutually rotated through the angle CF. Let us consider the case CF = 106°02’;as may be seen in fig. 3e, we assume that the two spirals develop independently one form the other; then we analyze the interaction of their growth fronts, taking into account the conditions of the preceding section (3.2.1). In this case we predict the polar diagram of the step growth velocities with the aid of table 3, where RAm, VBm, VAM, VBM are obviously subjected to the same relations as indicated in table 1. In the sectors I and 3 the growth steps of the two spirals are parallel, but they have different inclinations with respect of the (001) plane (figs. 3f,g) as in the case of the sectors 3 and 1 respectively of section
D. Aquilano
/ Complex growth polytypism
on octacosane crystals
225
Table 2 CVobs
(a) 0° (b) 180°
~ca1c
Components of the 2D-CL cell vectors: (xa +yb) (x’a+x’b)
0°
x = 1, x’O,
180°
9°49’ 12°l7’
—
lS°30’
17°06’
16°
2D-CL cell multiplicity
= 0
2D-CL reticular density
Twin axis
[uv0]
Number of observed crystals
1
2
—
Very high
1
2
(001) plane
High
68
0.06
[190]
—
46
0.087
[170]
23
0.087
[150]
2
29
0.07
[140]
2
y=l
1, 0, —8,
0 1 4
7,
5
—7,
3
6, —5, 4, 7, —1,
4 2 3 1 4
—
21°
2l°20’
26°,27°50’ 29° (twice)
28°10’
2, —3,
2 1
8
0.50
[130]
42°
42°12
3, —4,
2 0
8
0.125
[120]
48°40’
3, —3,
1 5
19
0.21
[350]
53°30’
53°20’
5, —2,
0 3
15
0.066
[230]
1
59°(twice)
58°56’
—3, 7,
4 3
37
0.054
[340]
2
74°
73°58’
1,
—l
18
0.222
[110]
106°
106°02
1, 5,
—1 4
18
0.222
normal to
—
—
14 [110]
87°
86°33’
—5,
4
53
0.037
[5401
93°
92°56’
5, —7,
4 5
53
0.037
[750]
89°44’
4, 0,
0 3
12
0.083
[430]
10
—
—
The vectors defining the 2D-Cl cells are referred to the (a, b) axes of the lower paraffinic layer, the rotation angle CV of the upper layer in respect to the lower one is anticlockwise. The values of the angle CVcalc > 90°are missing (except CV = 92°02’, 106°02’ which are quoted as examples) because if there exists a 2D-CL for CV = CV0, there will exist three other equivalent cells for CV = 180 CV0, 180 + CV0, 360 —
(a) Basic polytype 1Mb]. 80(O)q—1 180]. (b) Complex polytype (p + q)M[(0)~_1 l
—
226
D. Aquilano
/ C’oinplex growth polytypism on
Table 3
octacosane crystals
Table 4
Sector
1
2
3
4
Sector
1
2
3
Upper Lower spiral spiral B A
VBm VAM
VBM VAM
VBM t7Arn
VBm VAm
Upper Lower spiraiB spiral A
VBm LAm
VBm LAM
VBM LAM
(3.2.1). If we apply the same reasoning explained before about the kinetic of the common fronts, we may deduce that the lateral development of this twin will be controled by the velocity VBm in the sector 1 and by VAm in the sector 3. In the sectors 2 and 4, however, where the growth steps of the two spirals make an angle of ~148°, there is no possibility that a common front will be built up. Even if we are not able to predict this complex interaction, we may outline that in the sector 2 the growth fronts make the same acute angle with the (001) plane; consequently they will spread at the maximum velocities VAM and VBM. On the contrary, in the sector 4, where both the growth fronts make the same obtuse angle with the (001) plane, the limiting growth rates will be VAm and VBm. If now we choose the emergency of the screw dislocation as the reference point, we will observe that the lateral devel. opment of the resulting polysynthetic twin will be enhanced in the sector 2 and the growth pattern will take a rectangular shape elongated in the direction [110] (figs. 3h,3h’). The elongation depends obviously from the anisotropy ratios VAM/VAm and VBM/VBm, and it may be furthermore increased by the effect of the concave angle, as described by Dawson [211 for the (110) normal twins in n-hectane. We may find a similar growth pattern for the rotation angle r1 = 254°; in this case the twin axis is normal to the [110] direction and the growth pattern will result in enantiomorphic position in respect of the previous one, (010) being the mirror plane. 3.2.2.2. Twin law: contact plane (001), twin axis: Applying the same reasoning as for the just described twins, we may deduce that for a rotation angle 1 74°, we obtain the growth rates reported in table 4, together with the associated polar diagram (figs. 3i,j). In the sectors 1 and 3 where the growth steps of
/ 110J.
4 17BM 1~Am
the two spirals are parallel, the growth fronts have, two by two, the same inclination, i.e. they make the same obtuse angle with the (001) plane in the sector 1 and the same corresponding acute angle in the seetor 3. For VBm VAm, after a certain period of growth, in both these sectors, the fastest front will reach the slowest one. At this moment a common front will occur, the morphology of which is that of a microfacet (fig. 3k). Its height is (p + q) X d 1~01its velocity will be reduced with respect to that of the slowest front, in both the sectors. For instance, in the sector 1 the velocity of the common front will be slower than the smallest one of (VAII,, VBm). In the sector 3, the common front will spread more rapidly than in the sector 1, for the microfacet makes an acute angle with the (001) plane. In the sectors 2 and 4 the growth steps make, also in this case, an angle of 148w, but the interaction between the growth fronts is more complex than in the previous case. [n the sector 2 (as in the sector 4), the two interacting fronts have opposite inclinations. Amelinckx [16] discussed this interaction for the B-form of the behenic acid and he deduced the final equilibrium form of this polysynthetic twin. The idealized growth pattern we have obtained and the observed ones (figs. 31,1’) are in good agreement with Amelinckx’s conclusions. Furthermore we may establish that in the sectors 2 and 4 this twin would have the same development, owing to the symmetrical interaction of the growth fronts in both the sectors. Summarizing we may conclude that this twin will be recognizable from the lateral anisotropy of growth in the sectors 1 and 2 where the common growth fronts develop. As in the previous case we may find an analogous but enantiomorphous growth pattern, if the rotation angle between the twinned layers is 286°. In this case the twin axis will be along the direction [1101. Remark: Throughout this paper we have used the ~°
terms: complex growth polytypes and periodic poly-
D. Aquilano / Complex growth polytypism on octacosane crystals
synthetic twins. These terms would be correct if the corresponding structures are seen by means of X-ray methods. Indeed, if X-ray investigations can be carned out on such crystals, we should find the reflections corresponding to the long periodicity associated to the spiral growth promoted by high Burgers vectors. By this way the commonly used X-ray methods are not able to distinguish a structure generated by the activity of a screw dislocation from the same structure generated by 3D-nucleation. Strictly speaking both kinds of structures we have described, in their parts generated by spiral growth, are not a penodic stacking of crystalline individuals related by twinning operation; on the contrary they are two individuals only, related by twinning operation, which develop continuously describing the helicoidal surface imposed by the screw dislocation,
4. Discussion Summarizing the results obtained in this work, we may deduce the following considerations: (i) The acute angle effect (section 3.1) and the dthedral concave angle effect (section 3.2.1) have been revealed as a useful tool to define unambiguously the polytype and twin laws we have found starting from the corresponding interlaced spiral growth patterns. Consequently, the application of these effects can be generalized successfully to all the polygonized patterns in which the growth steps mak~no right angles with their substrate. (ii) As may be seen in table 2 the values of the rotation angles ~calc are in good correspondance with those we have measured by the decomposition of the interlaced growth patterns (~obs). But there is no relation between the reticular densities of 2D-CL cells
227
impingement mechanism suggested by Fisher et al.
[24] and observed by Newkirk [25], Amelinckx and Dekeyser [27], and Baronnet [201 on crystals grown from solutions. In our opinion this mechanism may be promoted by the weak convection currents present in the evaporating solutions and by the competition of the crystals on the bottom of the crystallizer in the final period of their growth. (iv) As we have described in sections 3.1 and 3.2, the frequency of the occurrence of polytytpes and twins, independently from their crystallographical laws, depends on the temperature of crystallization T~,i.e., the higher is T~,the higher is the probability of the observation of a polytype or a twin. Obviously it is not only T~which causes this behaviour, but also the rate of evaporation of the solvent and consequently the rate of the supersaturation change in the solution. The supersaturation determines the probability of anomalous nucleation, spiral growth activation, etc. At the same time, it is well known that the higher is T,~,the higher is the probability that an anomalous nucleus can adjust and fix on a normally grown substrate. In order to dissociate the different influencies of the crystallization temperature of the solution, and of the solvents, and to generalize the twin laws for paraffins having different chain lengths, a research is in progress on octacosane and hexatriacontane crystals obtained from evaporated solutions in which the solvents employed are the pure components of light petroleum ether [29].
Acknowledgements The author thanks Drs. A. Baronnet, R. Boistelle, and B. Simon of the laboratory for helpful discus-
sions and valuable criticisms on this work.
and the frequency of the occurrence of the corresponding twin laws. This disagreement is due to the
roughness of the geometrical approach. Calculations of the interaction energies associated to the observed twin laws are in progress. The first results we have obtamed are in good agreement with the observed frequencies of occurrence, for both polytypes and twin -
.
-
.
laws [28]. (iii) The surprisingly elevated values of the Burgers vector b we have encountered in all the spiral pat. terns, may be explained if we consider the edgewise
References [1] A. Muller, Proc. Roy. Soc. (London) A120 (1928) 137. [2] A. Muller, Proc. Roy. Soc. (London) A138 (1932) 514. [31A. Muller and K. Lonsdale, Acta Cryst. 1 (1948) 129. [4] W.M. Mazee, Travaux Chim. Pays Bas, 67 (1948) 197. [5] A.E. Smith, J. Chem. Phys. 21(1953) 2229. [6] H.M.M. Shearer and V. Vand, Acta Cryst. 9 (1956) 379. [7] P.W. Teare, Acta Cryst. 12 (1959) 294.
[8] A.R. Ubbelohde, Trans. Faraday Soc. 34 (1938) 296.
228
D. Aquilano
/ Complex growth polytypism on
[9] S.C. Nyburg and H. Luth, Acta Cryst. B28 (1972) 2992. 1101 M.G. Broadhurst, J. Res. Natl. Bur. Stand. 66A (1962) 241. [11] S.C. Nyburg and J.A. Potworowski, Acta Cryst. B28
(1973) 347. [121 R. Boistelle, B. Simon and G. Pêpe, Acta Cryst. B32
(1976) 1240. [13] II.E.L. Madsen and R. Boistelle, Acta Cryst. A32 (1976)
(in press). [14] [15] [16] [17] [18]
L.S. Ramsdell, Am. Mineralogist. 32(1947)64. S. Amelinckx, Acta Cryst. 8 (1955) 530. S. Amelinckx, Acta Cryst. 9 (1956) 16. 5. Amelinckx, Acta Cryst. 9 (1956) 217. F. Khoury, J. App. Phys. 34 (1963) 73.
octacosane crystals
[19] l.M. Dawson, Proc. Roy. Soc. (London) A214 (1952) 72. [201 A. Baronnet, J. Crystal Growth 19(1973)193. [21] P. I-lartman, Z. Krist. 107 (1956) 225. [22] F.C. Frank, Discussions Faraday Soc. 5 (Crystal
Growth) (1949) 48. [23] F. Billig, J. Inst. Metals 83 (1954), 53. [24] J.C. Fisher, R.L. Fullman and G.W. Sears, Acta Met. 2 (1954) 344. [25 J.B. Newkirk, Bull. Am. Phys. Soc. 29 (1954) 28. [26] J.B. Newkirk, Acta Met. 3 (1955) 121. [27] W. Dekeyser and S. Amelinckx, in: Les Dislocations et Ia Croissance des Cristaux (Masson, Paris, 1955) p. 146. [28] R. Boistelle and D. Aquilario, to be published. [29] D. Aquilano and R. Boistelle, to be published.