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On the roughening transition of anisotropic and (pseudo) hexagonal lattices
Journal of Crystal Growth 191 (1998) 563—572
On the roughening transition of anisotropic and (pseudo) hexagonal lattices L.J.P. Vogels1, P.J.C.M. van Hoof *, R.F.P. Grimbergen RIM Laboratory of Solid State Chemistry, Faculty of Science, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands Received 21 March 1997; accepted 26 January 1998
Abstract In this paper the results of the calculations of the roughening transition temperatures of anisotropic and (pseudo) hexagonal lattices are presented. The calculations are based upon a so-called Ising lattice combined with a simple model for the representation of the lattice. The lattice has three different bonds of which for two the relative strengths are varied. The Ising transition temperatures have been calculated for different lattices and for the case of orthorhombic n-paraffin crystals they have been compared with experimental values obtained from vapour growth experiments. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 05.50; 61.50.A; 61.66.H; 81.10.A Keywords: Roughening temperature; Ising temperature; Hexagonal lattice; Bond energy; Periodic bond chain; Vapour crystal growth
1. Introduction Thermodynamically the equilibrium form of a faceted crystal is determined by the minimisation of the Gibbs free energy [1] G"Nk# + A p , hkl hkl hkl
(1)
* Corresponding author. Fax: #31 24 3553450; e-mail: [email protected]. 1 Present address: MOS4YOU, Philips Semiconductors, Gerstweg 2, 6534 AE Nijmegen, The Netherlands.
where N is the total number of particles in the crystal, k is the chemical potential per particle, A the surface area of a face Mh k lN and p its hkl hkl surface free energy; p is related to the attachhkl ment energy, E!55 [2]. Minimisation of the Gibbs hkl free energy leads to the anisotropic equilibrium form of the crystal [3]. At finite temperature the growth form of a crystal is also determined by a transition temperature ¹R , above which the hkl facets are roughened and below which they grow as flat faces via two-dimensional nucleation or a spiral growth mechanism. This transition temperature is, therefore, called the roughening temperature [4—7]. The roughening temperature of a facet
0022-0248/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 0 1 3 8 - 9
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is determined by the structure, anisotropy and energy of the bonds within a slice d [8—16]. In this hkl paper the calculations of the roughening temperatures of many different (pseudo) hexagonal and other anisotropic lattices are elaborated and presented in one graph. Similar results have been presented earlier by Houtappel [9]. However, here we will use them to derive the roughening temperatures of different faces of orthorhombic n-paraffin crystals (n-C H crystals), of which many have 23 48 a pseudo-hexagonal structure, and compare them with roughening temperatures obtained from vapour growth experiments. Furthermore, (pseudo) hexagonal lattices can be found in compounds crystallising in an FCC structure as well as in many organic crystals [17—20]. 1.1. Ising transition temperature
solid, as well as for the interaction of the solid with the fluid. Furthermore, they are temperature dependent. 1.2. Theoretical stability of facets In this section the method and model to determine the stability of a (pseudo) hexagonal or related facet Mh k lN is described. First of all, a two-dimensional (pseudo) hexagonal lattice is reduced to a so-called two-dimensional crystal graph. In this graph the (chemical) bonds are presented by lines, whereas the molecules or atoms are situated at the intersection points of these lines. The graph of the (pseudo) hexagonal lattice is presented in Fig. 1a and that of the M0 1 0N and M1 0 0N faces of orthorhombic n-paraffin crystals in Fig. 1b. The two-dimensional graph of these lattices is similar to the hexagonal case but differs in the sequence of the
The model we use in the statistical mechanical approach is a two-dimensional Ising model, which generally describes well the surface structures of three-dimensional crystals [8,9,14]. The Ising temperature ¹# (or dimensionless Ising temperature h#), i.e. the temperature that marks the order—disorder transition of the two-dimensional Ising layer, can be determined [8,12—14] and is typically about 10% less than the roughening temperature ¹R of the three-dimensional crystal surface [12,13,21]. In the Ising model representation the crystal structure is partitioned into equally shaped blocks, which are occupied by the building units, and the fluid phase is partitioned into the same blocks. In case of three-dimensional crystals the crystal—fluid interface consists of a few layers of blocks of solid (s) or fluid (f ) type; for the two-dimensional Ising models elaborated in this paper the solid—fluid layer consists of one layer of blocks. Three type of interactions occur between these blocks, a solid—solid, a fluid—fluid and a solid—fluid one, denoted as /44, i /&& and /4& with i i
G
H
/ 1 i " /4&! M/44#/&&N /k¹ i i k¹ 2 i
(2)
being half the energy necessary to break or to form a bond. These interactions account for the behaviour of a growth unit in the mother phase and in the
Fig. 1. (a) A representation of a (pseudo) hexagonal lattice and (b) a representation of a modified hexagonal lattice, e.g. the (0 1 0) and (1 0 0) lattices of n-C H crystals. The bonds 1, 3 23 48 2 and 3 are represented by lines, the atoms or molecules by the intersection of the lines.
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bonds 1, 2 and 3. Each atom or molecule in both graphs has a six-fold coordination. Each unit cell has one bond of types 1, 2 and 3 which may vary in (relative) energy. Due to symmetry reasons the bonds are interchangeable. The bond of type 1 is always chosen to be the strongest with relative energy 1. In case of an exact hexagonal lattice the Ising temperature can be calculated following Houtappel [9], who showed that the following relation has to be solved: sinh(2/ /kh#) sinh(2/ /kh#) 1 2 #sinh(2/ /kh#) sinh(2/ /kh#) 2 3 #sinh(2/ /kh#) sinh(2/ /kh#)"1, (3) 3 1 where h# is the dimensionless Ising temperature which equals 2k¹#//. In order to calculate the Ising temperature of a lattice which is not exactly hexagonal or rectangular the method of Rijpkema and Knops [14] can be used. In this method the lattice has to be transformed into a so-called rectangular Ising lattice. For this purpose bonds of infinite strength (“i” in Fig. 3) are introduced, which so to speak, split one atom or molecule into two (or more) units. The calculated Ising temperatures (h#) depend on the structure of the lattice, the anisotropy and absolute bond strengths of the bonds in the lattice. To be able to compare the effects of the (an)isotropy of one (pseudo) hexagonal lattice with another, the total energy of the lattice should remain constant. Since the total energy of the lattice is given by (4) E4-*#%"+ / "2M/ #/ #/ N, hkl i 1 2 3 i and the Ising temperature can be normalised by using the maximum energy of the lattice (E4-*#%/k¹"6): .!9 E4-*#% h#{"h# .!9 , (5) E4-*#% hkl where h#{ will be denoted as the “normalised Ising temperature” which is scaled to a constant slice energy. To be able to compare also lattices with negative (repulsive) bonds we assume that the negative bonds in a lattice lower the slice energy with their according value. Because the bond energies
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are scaled according to the strongest bond in the lattice (crystal), the calculated Ising temperature is also normalised to this bond. Therefore, the roughening temperature can be calculated using the strongest bond energy: h#/453 ¹R+ . 2k
(6)
2. Anisotropy and Ising temperatures Fig. 2 shows the normalised Ising temperatures of a hexagonal lattice, calculated by using Eq. (3). In this figure / has been kept constant ("1) and 1 bonds / and / have been varied relative to this 2 3 from !1 to 1. Due to the possibility of interchanging bonds 2 and 3, without changing the hexagonal lattice, this figure is symmetrical, with a mirror plane on the x"y diagonal. The figure can roughly be divided into three regions. Region one is the part where both bonds 2 and 3 are both positive. This region is bounded by thick lines in Fig. 2. If both bonds 2 and 3 are large (/'0.3) the hexagonal lattice clearly stays stable; the roughening transition temperature is higher than that of a rectangular lattice (h#'2.269) [4]. If the energy of the bonds 2 and 3 are taken to be 1, the hexagonal lattice is isotropic, that is an FCC(1 1 1) [4]. In this case the maximal value of the relative roughening transition temperature is h#{"3.636. If one bond is taken to be zero the roughening transition temperature of an anisotropic rectangular lattice can be found. These temperatures are indicated by the thick lines in Fig. 2 and are the same as the ones that have been calculated before [8]. If one bond is taken zero and the other is taken equal to one, the Ising temperature can be found which is equal to that of a rectangular lattice, h#{"3.404 or h#"2.269. This value is still somewhat smaller than the value found for an isotropic hexagonal lattice (h#+3.636). The difference indicates that a hexagonal lattice, with an equal slice energy per unit area as a rectangular lattice, is still somewhat more stable, due to the structure of the lattice. These two limiting values agree with the theoretically predicted values for a BCCM1 1 0N or
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Fig. 2. (a) The normalised roughening transition temperature as a function of the strengths of bonds 2 and 3 for a (pseudo) hexagonal lattice (Fig. 1a). Bond 1 is constant (/ "1). Between the thick lines is the area where / and / are positive. (b) Contour plot of (a). 1 2 3
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an FCCM1 0 0N lattice for which a value of h#"2.269 is found and for an FCCM1 1 1N lattice for which a value of h#"3.641 is found [4]. Further it can be seen that rather anisotropic lattices still have relatively high Ising temperatures. A rectangular lattice with an anisotropy of 0.1, that is one bond which is ten times stronger than the other bond, has a normalised Ising temperature (h#{) of 2.5, which is still 68% of that of an isotropic rectangular lattice. Also a hexagonal lattice where both / and / are ten times as small as / has 2 3 1 a normalised Ising temperature of about 2.8, which is still 77% of that of an isotropic hexagonal lattice. If both bonds 2 and 3 are smaller than 0.1 the (pseudo) hexagonal net roughens at very low temperatures. This is due to the fact that only one strong direction, parallel to bond 1, remains. The value of the roughening temperature of such a net is just larger than that of a step, which is a onedimensional chain, with a roughening transition temperature of 0 K. ¹wo equivalent regions appear, where either bond 2 or 3 is smaller than 0, and the other bonds remain positive. It is of importance to notice that if
the positive bond, a very sharp decrease in the roughening transition temperature is observed. A slow decrease in the value of the roughening transition temperature is observed if one of the bonds 2 or 3 is still close to unity (/+1) and the other is approaching !1. Region three is the part where both bonds 2 and 3 are negative (/ (0), Eq. (3) cannot be solved 2,3 and therefore the corresponding lattice is given a roughening transition temperature of zero. Table 1 shows the Ising temperatures of some pseudo-hexagonal faces of organic crystals found in literature. It shows that these Ising temperatures can be calculated with the help of Eq. (3) and Fig. 1. Many organic compounds crystallising in the space group P show pseudo-hexagonal sym21 metry and pseudo-hexagonal faces [17] of which the corresponding roughening temperatures can be found using Fig. 1.
D/ D(D/ D (7) 104 /%' Eq. (3) cannot be solved [9] and in case of the method of Rijpkema and Knops no convergence takes place. In these cases the roughening transition temperatures are set to zero. If the absolute size of the negative bond increases to that of
The roughening temperatures of the possible faces of n-C H have been calculated and mea23 48 sured experimentally. In this section we assume that the roughening temperature is approximately the same as the calculated Ising temperatures and therefore we only talk about roughening temperatures (¹R) and not about Ising temperatures (h#).
3. Roughening temperatures of the faces of n-C H 23 48
Table 1 Comparison of calculated Ising temperatures with values found in literature, / "1 1 Compound
Ref.
(h k l )
/ 2
/ 3
h#
h# (ref.)
n-C H 10 22
[27] [27] [28] [28] [28] [28] [20] [20] [20] [20] [20]
(0 1 0) (1 1 1) (1 1 0) (1 1 11 ) (0 1 0) (2 0 11 ) (0 0 1) (1 0 0) (2 0 11 ) (1 1 0) (11 1 1)
0.1383 0.2308 0.08283 0.08283 0.15063 1 0.3741 0.7257 0.3815 0.5155 0.9804
0.07267 0.3130 0.08346 0.04558 0.08227 0.8805 0.3741 0.3017 0.3815 0.5258 0.3954
1.1733 0.7699! 1.0753 0.9819 0.6750! 0.2890! 2.0350 2.3973 2.0557 1.7663! 1.0818!
1.1696 0.7698 1.0736 0.9828 0.6726 0.2891 2.0408 2.3981 2.0619 1.7699 1.0707
n-C H 10 22
p-xylene
!h# had to be rescaled because the strongest bond of the crystal was not in the face (h k l ).
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The morphology of these crystals and the roughening temperature of the crystal faces have been studied in great detail [6,18,22,23]; in this paper we will use some of the results published before, extend the concept of anisotropy [6] and show new experimental results obtained from near-equilibrium vapour “growth” experiments. 3.1. Calculations To determine the most important faces (F faces) that determine a crystal morphology the crystal graph and subsequently the connected nets have to be derived from the crystal structure [4]. Bennema et al. [18] published the most important connected nets and later Grimbergen et al. [23] showed that some additional nets can be found. We will use the same notation as Bennema et al. [18] for these connected nets as well as their bonds, further force field calculations (with the help of the Cerius2 modelling package using the Dreiding 2.21 force field and MNDO-ESP charges [24]) are applied to get more accurate values for the bond energies, which have been listed in Table 2. It has been shown [23] that the two (1 1 0) and (1 1 0) connected nets can 1 2 be combined to form a substitute connected net (1 1 0) with bonds / !/ in one direction and 4 e f / and / in the other direction. For more details d d{ we refer to the corresponding publication. The anisotropy and the slice energy of the different connected nets have been determined and listed in Table 3. The anisotropy is defined by / / d " 2 and d " 3, 1 / 2 / 1 1
(8)
where d is the anisotropy in the direction of bonds 1 / and d the anisotropy in the direction of bonds 2 2 / ; / is defined as the strongest bond. Although 3 1 the different connected nets listed in Table 3 have similar structures as the hexagonal lattice the sequence of the bonds is different and for each lattice the corresponding roughening temperature has to be calculated separately. The roughening temperatures listed in Table 3 have been calculated using the method of Rijpkema and Knops [14]. Finally, the effect of the anisotropy on the roughening temperatures of the face (0 1 0) (Fig. 1b), which is differ-
Table 2 Bond energies calculated using the Cerius2 modelling package and the Dreiding 2.21 force field and MNDO-ESP charges [24] for an n-C H crystal 23 48 Bond!
Energy (kcal/mol)
/ d / d{ / a / b / g{ / g / e / f
14.885 14.642 8.910 1.587 0.644 0.568 0.556 0.286
!For the bonds the same notation as in Ref. [18] is used.
Table 3 Calculated slice energies (E ), anisotropies (d) and Ising temper4atures (¹ ) for different connected nets of n-C H [18] # 23 48 Connected net (h k l )
E 4(kcal/mol)
d
(0 0 1) (1 1 0) 1 (1 1 1) (1 1 0) 2 (1 1 0) 4 (0 1 0) (0 1 1) (1 0 0) 3 (1 0 1)
40.02 15.93 15.79 15.65 15.03 10.12 9.33 2.43 2.19
1.000 0.040 0.042 0.031 0 0.068 0 0.27 0
1
d 2
¹ # (K)
0.61 0.040 0.029 0.031 0.019 0.068 0.047 0.27 0.38
5834 1563 1504 1448 1073 1123 805 336 292
ent from that of a hexagonal lattice (Fig. 1a), has been calculated and its graph is given by Fig. 3. 3.2. Discussion Fig. 3 shows that the roughening temperatures of the connected net (0 1 0) show the same behaviour on varying bonds / and / from 0 to 1 as an 2 3 hexagonal lattice (Fig. 2), which has been discussed in Section 2. This shows that the anisotropy of the lattice is of more importance for the roughening temperature than a small change in the structure of the lattice. Therefore, we will compare the roughening temperatures of the different pseudo-hexagonal
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Fig. 3. (a) Representation of the (0 1 0) and (1 0 0) Ising lattices of orthorhombic n-paraffin crystals, deduced from the hexagonal 3 lattices in Fig. 1b. The bonds 1, 2 and 3 are represented by lines, the atoms or molecules by the intersection of the lines. Also extra bonds of infinite strength are added (see text). (b) The normalised roughening transition temperature as a function of the strengths of bonds 2 and 3 for the modified hexagonal lattice (a). Bond 1 is constant (/ "1). 1
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connected nets of n-paraffin with slightly different structures, that are (0 0 1), (1 1 0) , (1 1 0) , (1 1 0) , 1 2 4 (1 1 1), (0 1 0) and (1 0 0) , with each other and with 3 the isotropic hexagonal lattice. To be able to get a rectangular Ising lattice for the (0 0 1) lattice the bond “b” is left out, because it is a so-called crossing bond with relatively low energy. Fig. 4 shows the roughening temperatures of all connected nets that are listed in Table 3 as a function of their slice energy. The solid lines in this figure indicate the dependence of the ¹R of an isotropic hexagonal lattice (d "d "1) and of 1 2 some anisotropic hexagonal lattices (d "0, 1 d "0.5 and d "d "0.05) on E , which show 2 1 2 4-*#% a linear dependence. These roughening temperatures have been scaled to a maximum E which 4-*#% is equivalent to that of the (0 0 1) face. The figure shows that ¹R is determined to a great extent by E and not so much by the anisotropy of the 4-*#% connected net. All deviations from the solid line (d "d "1) are because of anisotropy in the con1 2 nected net or because the structure of the connected net differs somewhat from that of an hexagonal net. It shows that although (0 0 1) is not a hexagonal net, like shown in Fig. 1a the roughening temperature is almost the same. The small deviation is because the anisotropy in one direction of this net is 0.61.
More interesting are the faces (1 1 0) , (1 1 1), 1 (1 1 0) and (1 1 0) , which are all highly anisotropic 2 4 (d’s ranging from 0.019 to 0.040) and have almost the same slice energy (from 15.0 to 15.9 kcal/mol). As expected the roughening temperatures of these connected nets are below the line indicating the roughening temperatures of structures with an anisotropy of d "d "0.05, because for those faces 1 2 both anisotropies d and d are less than 0.05. 1 2 Table 3 shows that the roughening temperatures of these connected nets are directly correlated with their anisotropy: the more anisotropic the nets, the lower are the roughening temperatures. When (1 1 0) and (1 1 0) are replaced by the substitute 1 2 connected net (1 1 0) [23], an anisotropic rectan4 gular net is obtained with a roughening temperature which is much lower than that of an isotropic rectangular net. Although this connected net has a E comparable to (1 1 1) it has a lower 4-*#% ¹R because of its higher anisotropy. For the same reason ¹R of (1 1 0) is even lower than ¹R of (0 1 0), 4 which has a considerable lower slice energy. The difference in the roughening temperature of the faces (0 1 0) and (0 1 1), which have almost the same slice energy (9.33 and 10.12 kcal/mol), can be explained by the fact that (0 1 1) is a rectangular net with d "0 and d "0.047, whereas (0 1 0) has 1 2 a lower anisotropy of d "d "0.068. It can be 1 2 seen that the roughening temperature of the latter one is slightly higher than the line indicating d "d "0.05. The roughening temperatures of 1 2 the connected nets (1 0 0) and (1 0 1), with aniso3 tropies of, respectively, d "d "0.27 and d "0; 1 2 1 d "0.38, are very close to the roughening temper2 atures of the isotropic hexagonal lattice (d "d "1). 1 2 3.3. Near-equilibrium vapour experiments
Fig. 4. The dependence of the calculated Ising temperatures on the slice energy of the lattices listed in Table 3. The straight line represents the Ising temperatures of an isotropic hexagonal lattice with d "d "1 and of two anisotropic lattices with 1 2 d "0, d "0.5 and d "d "0.05. 1 2 1 2
We examined the morphology of n-C H crys23 48 tals and the roughening temperatures of its different faces at near equilibrium conditions and high temperatures using a vapour growth cell, which is shown in Fig. 5. In this cell bulk material of nC H is heated and sublimated under vacuum 23 48 conditions and then condensed on a cold finger. A window above the cold finger enables us to follow the growth of crystals in situ, by making use
L.J.P. Vogels et al. / Journal of Crystal Growth 191 (1998) 563–572
Fig. 5. A schematic drawing of a vapour growth cell used in our experiments. (1) Glass window, (2) vacuum flange, (3) heating tube, (4) cold finger with a crystal, (5) thermostated wall with bulk material, (6) thermocouple and (7) cold water inlet.
of a Nikon optical microscope with long working distance objectives. Further some scanning electron micrographs have been made to examine the crystal faces more closely. The temperature of the cold finger was kept constant at 287, 303 and 312 K which is 1.7 K below the transition temperature from the orthorhombic phase to the rotator phase [25]. Note that this transition temperature is the highest temperature possible to observe the faces that are listed in Table 3 for n-C H crys23 48 tals. The temperature of the bulk material has been kept constant at 2 K above the temperature of the cold finger, so that *¹ was 2 K. These near-equilibrium conditions (instead of “real” equilibrium conditions) were applied to shorten the experimental time to a few weeks. At the temperature of 287 K kinetic roughening experiments have been performed at supersaturations of *¹"2, 4, 6 and 8 K.
571
faceted. At a temperature of 303 K and a supersaturation of 2 K faceted M0 0 2N, M1 1 1N, M1 1 0N and M0 2 0N faces have been found, and at a temperature of 313 K at *¹ of 2 K only faceted M0 0 2N faces were present. At this temperature the faces M1 1 0N, M1 1 1N and M0 2 0N were rough. It is clear that, as predicted, the M0 0 2N faces have the highest roughening temperature, which is supposed to be at least higher than 312 K. The faces M1 1 0N, M1 1 1N and M0 2 0N have roughening temperatures which are higher than 303 K. That those faces are rough at 312 K is probably due to the transition from the orthorhombic phase to the pseudo-hexagonal phase [25]. Although 312 K is still 1.7 K below this transition temperature it is known that homologous impurities lower this temperature [26]. The bond energies in the pseudohexagonal phase are smaller than of those in the orthorhombic phase which results in lower roughening temperatures. So it is still possible that the roughening temperatures of the M1 1 0N, M1 1 1N and M0 2 0N faces in the orthorhombic phase are much higher than 313.7 K. Although it is not possible to determine the exact values of the roughening temperatures, because they are higher than the transition temperature from the orthorhombic phase to the pseudo-hexagonal phase, the kinetic roughening experiments can be used to determine the sequence of roughening temperatures of the different faces. Here we will assume that a high roughening temperature leads to a high edge free energy and thus to a high critical supersaturation. We observed that the order in kinetic roughening of the faces is first the faces M1 0 0N and M0 2 0N then the faces M1 1 0N and M1 1 1N, whereas the M0 0 2N faces will not roughen. This sequence is the same as that of the roughening temperatures of those faces.
3.4. Results 4. Conclusions At the lowest temperature and a supersaturation of 2 K faceted M0 0 2N, M1 1 1N, M1 1 0N, M0 2 0N and M1 0 0N faces were present. The faces M1 0 0N have only been observed for 10% of the crystals. Upon increasing the supersaturation first the faces M1 0 0N and M0 2 0N roughen, then is followed by M1 1 0N and M1 1 1N, whereas the faces M0 0 2N always stay
We calculated the normalised Ising temperatures of (pseudo) hexagonal lattices for both positive and negative bond energies. To be able to compare the roughening temperatures of different lattices with different total slice energies they have been normalised according to a maximum slice energy. If in
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a (pseudo) hexagonal Ising lattice the relative strength of one of the bonds is less than 0.1, the lattice reduces effectively to a rectangular Ising lattice. Considering the roughening transition temperature, this bond may therefore be neglected. If one of the bonds becomes negative in a (pseudo) hexagonal lattice, the lattice remains stable, that is it keeps a relatively high roughening transition temperature if the absolute value of its bond energy is much smaller than the smallest of the other two bonds. In fact, Fig. 2 shows normalised Ising temperatures h#{ of simple ionic (pseudo) hexagonal crystal faces. If the absolute strength of the negative bond increases to that of the smallest of the other two, the stability of the lattice decreases dramatically. Effectively, the lattice reduces to a step, with a roughening transition temperature of 0 K. If the absolute strength of the negative bond becomes larger than one of the positive bonds, no roughening transition temperature can be calculated and is therefore set to be 0 K. If the bond energies of a crystal face are known, Fig. 2 can be used to calculate its roughening temperature. From the n-paraffin example it may be concluded that the roughening temperature is mainly determined by the slice energy of the connected net and not so much by the anisotropy of that net. It has been shown that the difference in roughening temperatures of faces with almost the same slice energy like (1 1 0) , (1 1 1), (1 1 0) and (1 1 0) will 1 2 4 be determined by the anisotropy of these faces. That is, the face with the highest anisotropy will have the lowest roughening temperature. In case of the face (1 1 0) it turns out that although this face 4 has a higher slice energy than the face (0 1 0) it has a lower roughening temperature. The roughening temperatures could not be determined experimentally because they were all higher than the highest experimentally accessible temperature, that is the temperature of the transition from the orthorhombic phase to the pseudohexagonal phase. The sequence of roughening temperatures is likely to be equal to the sequence of the kinetic roughening of the faces, which is M1 0 0N, M0 2 0N, M1 1 0N, M1 1 1N and M0 0 2N. The calculated roughening temperatures show the same sequence.
Acknowledgements P. van Hoof acknowledges the financial support of Shell International Oil Products B.V. Furthermore we would like to thank Dr. W. van Enckevort and Dr. H. Meekes for carefully reading the manuscript. References [1] M. Volmer, Kinetik der Phasenbildung, Steinkopff, Leipzig, 1939. [2] P. Hartman, P. Bennema, J. Crystal Growth 49 (1980) 145. [3] G. Wulff, Z. Kristallogr. Miner. 34 (1901) 449. [4] P. Bennema, J.P. van der Eerden, in: Morphology of Crystals, Part A: Fundamentals, ch. 1, Terra Scientific Publishing Company, 1987, pp. 1—75. [5] P. Bennema, in: Morphology and Growth Units of Crystals in Proc., Terra Publisher, Tokyo, 1989, p. 219. [6] P. Bennema, in: Handbook of Crystal Growth, vol. 1, ch. 7, Elsevier, Amsterdam, 1993, p. 481. [7] K.A. Jackson, in: Crystal Growth, Pergamon, Oxford, 1967, p. 17. [8] J.P. van der Eerden, C. van Leeuwen, P. Bennema, W.L. van der Kruk, B.P.Th. Veltman, J. Appl. Phys. 48 (1977) 2124. [9] R.M.F. Houtappel, Physica 16 (1950) 425. [10] P. Hartman, W.G. Perdok, Acta Crystallogr. 8 (1955) 49. [11] P. Hartman, W.G. Perdok, Acta Crystallogr. 8 (1955) 521. [12] J.D. Weeks, G.H. Gilmer, in: Advances in Chemical Physics, Wiley, New York, 1979, p. 157. [13] H. van Beijeren, Commun. Math. Phys. 40 (1975) 1. [14] J.M. Rijpkema, H.J.F. Knops, P. Bennema, J.P. van der Eerden, J. Crystal Growth 61 (1982) 295. [15] P. Hartman, in: Crystal Growth: An Introduction, NorthHolland, Amsterdam, 1973, p. 367. [16] L.J.P. Vogels, K. Balzuweit, H. Meekes, P. Bennema, J. Crystal Growth 116 (1992) 397. [17] P. Hartman, J. Crystal Growth 110 (1991) 559. [18] P. Bennema, X.Y. Liu, K. Lewtas, R.D. Tack, J.J.M. Rijpkema, K.J. Roberts, J. Crystal Growth 121 (1992) 679. [19] P. Bennema, L.J.P. Vogels, S. de Jong, J. Crystal Growth 123 (1992) 141. [20] L.J.P. Vogels, R.F.P. Grimbergen, R.F. Blanks, Chem. Phys. 203 (1996) 69. [21] J.P. van der Eerden, P. Bennema, T.A. Cherepanova, Prog. Crystal Growth Characterization 1 (1978) 219. [22] C.S. Strom, P. Bennema, to be published. [23] R.F.P. Grimbergen, P.J.C.M. van Hoof, H. Meekes, P. Bennema, J. Crystal Growth 191 (1998), in press. [24] Cerius2 modeling package V 2.0, Molecular Simulations Incorporated, San Diego, 1997. [25] M.G. Broadhurst, J. Res. A 66a (1962) 241. [26] W. Pechhold, W. Dollhopf, A. Engel, Acustica 17 (1996) 61. [27] X.Y. Liu, P. Bennema, J. Crystal Growth 135 (1994) 209. [28] X.Y. Liu, P. Bennema, J. Appl. Crystallogr. 26 (1993) 229.
On the roughening transition of anisotropic and (pseudo) hexagonal lattices L.J.P. Vogels1, P.J.C.M. van Hoof *, R.F.P. Grimbergen RIM Laboratory of Solid State Chemistry, Faculty of Science, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands Received 21 March 1997; accepted 26 January 1998
Abstract In this paper the results of the calculations of the roughening transition temperatures of anisotropic and (pseudo) hexagonal lattices are presented. The calculations are based upon a so-called Ising lattice combined with a simple model for the representation of the lattice. The lattice has three different bonds of which for two the relative strengths are varied. The Ising transition temperatures have been calculated for different lattices and for the case of orthorhombic n-paraffin crystals they have been compared with experimental values obtained from vapour growth experiments. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 05.50; 61.50.A; 61.66.H; 81.10.A Keywords: Roughening temperature; Ising temperature; Hexagonal lattice; Bond energy; Periodic bond chain; Vapour crystal growth
1. Introduction Thermodynamically the equilibrium form of a faceted crystal is determined by the minimisation of the Gibbs free energy [1] G"Nk# + A p , hkl hkl hkl
(1)
* Corresponding author. Fax: #31 24 3553450; e-mail: [email protected]. 1 Present address: MOS4YOU, Philips Semiconductors, Gerstweg 2, 6534 AE Nijmegen, The Netherlands.
where N is the total number of particles in the crystal, k is the chemical potential per particle, A the surface area of a face Mh k lN and p its hkl hkl surface free energy; p is related to the attachhkl ment energy, E!55 [2]. Minimisation of the Gibbs hkl free energy leads to the anisotropic equilibrium form of the crystal [3]. At finite temperature the growth form of a crystal is also determined by a transition temperature ¹R , above which the hkl facets are roughened and below which they grow as flat faces via two-dimensional nucleation or a spiral growth mechanism. This transition temperature is, therefore, called the roughening temperature [4—7]. The roughening temperature of a facet
0022-0248/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 0 1 3 8 - 9
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is determined by the structure, anisotropy and energy of the bonds within a slice d [8—16]. In this hkl paper the calculations of the roughening temperatures of many different (pseudo) hexagonal and other anisotropic lattices are elaborated and presented in one graph. Similar results have been presented earlier by Houtappel [9]. However, here we will use them to derive the roughening temperatures of different faces of orthorhombic n-paraffin crystals (n-C H crystals), of which many have 23 48 a pseudo-hexagonal structure, and compare them with roughening temperatures obtained from vapour growth experiments. Furthermore, (pseudo) hexagonal lattices can be found in compounds crystallising in an FCC structure as well as in many organic crystals [17—20]. 1.1. Ising transition temperature
solid, as well as for the interaction of the solid with the fluid. Furthermore, they are temperature dependent. 1.2. Theoretical stability of facets In this section the method and model to determine the stability of a (pseudo) hexagonal or related facet Mh k lN is described. First of all, a two-dimensional (pseudo) hexagonal lattice is reduced to a so-called two-dimensional crystal graph. In this graph the (chemical) bonds are presented by lines, whereas the molecules or atoms are situated at the intersection points of these lines. The graph of the (pseudo) hexagonal lattice is presented in Fig. 1a and that of the M0 1 0N and M1 0 0N faces of orthorhombic n-paraffin crystals in Fig. 1b. The two-dimensional graph of these lattices is similar to the hexagonal case but differs in the sequence of the
The model we use in the statistical mechanical approach is a two-dimensional Ising model, which generally describes well the surface structures of three-dimensional crystals [8,9,14]. The Ising temperature ¹# (or dimensionless Ising temperature h#), i.e. the temperature that marks the order—disorder transition of the two-dimensional Ising layer, can be determined [8,12—14] and is typically about 10% less than the roughening temperature ¹R of the three-dimensional crystal surface [12,13,21]. In the Ising model representation the crystal structure is partitioned into equally shaped blocks, which are occupied by the building units, and the fluid phase is partitioned into the same blocks. In case of three-dimensional crystals the crystal—fluid interface consists of a few layers of blocks of solid (s) or fluid (f ) type; for the two-dimensional Ising models elaborated in this paper the solid—fluid layer consists of one layer of blocks. Three type of interactions occur between these blocks, a solid—solid, a fluid—fluid and a solid—fluid one, denoted as /44, i /&& and /4& with i i
G
H
/ 1 i " /4&! M/44#/&&N /k¹ i i k¹ 2 i
(2)
being half the energy necessary to break or to form a bond. These interactions account for the behaviour of a growth unit in the mother phase and in the
Fig. 1. (a) A representation of a (pseudo) hexagonal lattice and (b) a representation of a modified hexagonal lattice, e.g. the (0 1 0) and (1 0 0) lattices of n-C H crystals. The bonds 1, 3 23 48 2 and 3 are represented by lines, the atoms or molecules by the intersection of the lines.
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bonds 1, 2 and 3. Each atom or molecule in both graphs has a six-fold coordination. Each unit cell has one bond of types 1, 2 and 3 which may vary in (relative) energy. Due to symmetry reasons the bonds are interchangeable. The bond of type 1 is always chosen to be the strongest with relative energy 1. In case of an exact hexagonal lattice the Ising temperature can be calculated following Houtappel [9], who showed that the following relation has to be solved: sinh(2/ /kh#) sinh(2/ /kh#) 1 2 #sinh(2/ /kh#) sinh(2/ /kh#) 2 3 #sinh(2/ /kh#) sinh(2/ /kh#)"1, (3) 3 1 where h# is the dimensionless Ising temperature which equals 2k¹#//. In order to calculate the Ising temperature of a lattice which is not exactly hexagonal or rectangular the method of Rijpkema and Knops [14] can be used. In this method the lattice has to be transformed into a so-called rectangular Ising lattice. For this purpose bonds of infinite strength (“i” in Fig. 3) are introduced, which so to speak, split one atom or molecule into two (or more) units. The calculated Ising temperatures (h#) depend on the structure of the lattice, the anisotropy and absolute bond strengths of the bonds in the lattice. To be able to compare the effects of the (an)isotropy of one (pseudo) hexagonal lattice with another, the total energy of the lattice should remain constant. Since the total energy of the lattice is given by (4) E4-*#%"+ / "2M/ #/ #/ N, hkl i 1 2 3 i and the Ising temperature can be normalised by using the maximum energy of the lattice (E4-*#%/k¹"6): .!9 E4-*#% h#{"h# .!9 , (5) E4-*#% hkl where h#{ will be denoted as the “normalised Ising temperature” which is scaled to a constant slice energy. To be able to compare also lattices with negative (repulsive) bonds we assume that the negative bonds in a lattice lower the slice energy with their according value. Because the bond energies
565
are scaled according to the strongest bond in the lattice (crystal), the calculated Ising temperature is also normalised to this bond. Therefore, the roughening temperature can be calculated using the strongest bond energy: h#/453 ¹R+ . 2k
(6)
2. Anisotropy and Ising temperatures Fig. 2 shows the normalised Ising temperatures of a hexagonal lattice, calculated by using Eq. (3). In this figure / has been kept constant ("1) and 1 bonds / and / have been varied relative to this 2 3 from !1 to 1. Due to the possibility of interchanging bonds 2 and 3, without changing the hexagonal lattice, this figure is symmetrical, with a mirror plane on the x"y diagonal. The figure can roughly be divided into three regions. Region one is the part where both bonds 2 and 3 are both positive. This region is bounded by thick lines in Fig. 2. If both bonds 2 and 3 are large (/'0.3) the hexagonal lattice clearly stays stable; the roughening transition temperature is higher than that of a rectangular lattice (h#'2.269) [4]. If the energy of the bonds 2 and 3 are taken to be 1, the hexagonal lattice is isotropic, that is an FCC(1 1 1) [4]. In this case the maximal value of the relative roughening transition temperature is h#{"3.636. If one bond is taken to be zero the roughening transition temperature of an anisotropic rectangular lattice can be found. These temperatures are indicated by the thick lines in Fig. 2 and are the same as the ones that have been calculated before [8]. If one bond is taken zero and the other is taken equal to one, the Ising temperature can be found which is equal to that of a rectangular lattice, h#{"3.404 or h#"2.269. This value is still somewhat smaller than the value found for an isotropic hexagonal lattice (h#+3.636). The difference indicates that a hexagonal lattice, with an equal slice energy per unit area as a rectangular lattice, is still somewhat more stable, due to the structure of the lattice. These two limiting values agree with the theoretically predicted values for a BCCM1 1 0N or
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Fig. 2. (a) The normalised roughening transition temperature as a function of the strengths of bonds 2 and 3 for a (pseudo) hexagonal lattice (Fig. 1a). Bond 1 is constant (/ "1). Between the thick lines is the area where / and / are positive. (b) Contour plot of (a). 1 2 3
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an FCCM1 0 0N lattice for which a value of h#"2.269 is found and for an FCCM1 1 1N lattice for which a value of h#"3.641 is found [4]. Further it can be seen that rather anisotropic lattices still have relatively high Ising temperatures. A rectangular lattice with an anisotropy of 0.1, that is one bond which is ten times stronger than the other bond, has a normalised Ising temperature (h#{) of 2.5, which is still 68% of that of an isotropic rectangular lattice. Also a hexagonal lattice where both / and / are ten times as small as / has 2 3 1 a normalised Ising temperature of about 2.8, which is still 77% of that of an isotropic hexagonal lattice. If both bonds 2 and 3 are smaller than 0.1 the (pseudo) hexagonal net roughens at very low temperatures. This is due to the fact that only one strong direction, parallel to bond 1, remains. The value of the roughening temperature of such a net is just larger than that of a step, which is a onedimensional chain, with a roughening transition temperature of 0 K. ¹wo equivalent regions appear, where either bond 2 or 3 is smaller than 0, and the other bonds remain positive. It is of importance to notice that if
the positive bond, a very sharp decrease in the roughening transition temperature is observed. A slow decrease in the value of the roughening transition temperature is observed if one of the bonds 2 or 3 is still close to unity (/+1) and the other is approaching !1. Region three is the part where both bonds 2 and 3 are negative (/ (0), Eq. (3) cannot be solved 2,3 and therefore the corresponding lattice is given a roughening transition temperature of zero. Table 1 shows the Ising temperatures of some pseudo-hexagonal faces of organic crystals found in literature. It shows that these Ising temperatures can be calculated with the help of Eq. (3) and Fig. 1. Many organic compounds crystallising in the space group P show pseudo-hexagonal sym21 metry and pseudo-hexagonal faces [17] of which the corresponding roughening temperatures can be found using Fig. 1.
D/ D(D/ D (7) 104 /%' Eq. (3) cannot be solved [9] and in case of the method of Rijpkema and Knops no convergence takes place. In these cases the roughening transition temperatures are set to zero. If the absolute size of the negative bond increases to that of
The roughening temperatures of the possible faces of n-C H have been calculated and mea23 48 sured experimentally. In this section we assume that the roughening temperature is approximately the same as the calculated Ising temperatures and therefore we only talk about roughening temperatures (¹R) and not about Ising temperatures (h#).
3. Roughening temperatures of the faces of n-C H 23 48
Table 1 Comparison of calculated Ising temperatures with values found in literature, / "1 1 Compound
Ref.
(h k l )
/ 2
/ 3
h#
h# (ref.)
n-C H 10 22
[27] [27] [28] [28] [28] [28] [20] [20] [20] [20] [20]
(0 1 0) (1 1 1) (1 1 0) (1 1 11 ) (0 1 0) (2 0 11 ) (0 0 1) (1 0 0) (2 0 11 ) (1 1 0) (11 1 1)
0.1383 0.2308 0.08283 0.08283 0.15063 1 0.3741 0.7257 0.3815 0.5155 0.9804
0.07267 0.3130 0.08346 0.04558 0.08227 0.8805 0.3741 0.3017 0.3815 0.5258 0.3954
1.1733 0.7699! 1.0753 0.9819 0.6750! 0.2890! 2.0350 2.3973 2.0557 1.7663! 1.0818!
1.1696 0.7698 1.0736 0.9828 0.6726 0.2891 2.0408 2.3981 2.0619 1.7699 1.0707
n-C H 10 22
p-xylene
!h# had to be rescaled because the strongest bond of the crystal was not in the face (h k l ).
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The morphology of these crystals and the roughening temperature of the crystal faces have been studied in great detail [6,18,22,23]; in this paper we will use some of the results published before, extend the concept of anisotropy [6] and show new experimental results obtained from near-equilibrium vapour “growth” experiments. 3.1. Calculations To determine the most important faces (F faces) that determine a crystal morphology the crystal graph and subsequently the connected nets have to be derived from the crystal structure [4]. Bennema et al. [18] published the most important connected nets and later Grimbergen et al. [23] showed that some additional nets can be found. We will use the same notation as Bennema et al. [18] for these connected nets as well as their bonds, further force field calculations (with the help of the Cerius2 modelling package using the Dreiding 2.21 force field and MNDO-ESP charges [24]) are applied to get more accurate values for the bond energies, which have been listed in Table 2. It has been shown [23] that the two (1 1 0) and (1 1 0) connected nets can 1 2 be combined to form a substitute connected net (1 1 0) with bonds / !/ in one direction and 4 e f / and / in the other direction. For more details d d{ we refer to the corresponding publication. The anisotropy and the slice energy of the different connected nets have been determined and listed in Table 3. The anisotropy is defined by / / d " 2 and d " 3, 1 / 2 / 1 1
(8)
where d is the anisotropy in the direction of bonds 1 / and d the anisotropy in the direction of bonds 2 2 / ; / is defined as the strongest bond. Although 3 1 the different connected nets listed in Table 3 have similar structures as the hexagonal lattice the sequence of the bonds is different and for each lattice the corresponding roughening temperature has to be calculated separately. The roughening temperatures listed in Table 3 have been calculated using the method of Rijpkema and Knops [14]. Finally, the effect of the anisotropy on the roughening temperatures of the face (0 1 0) (Fig. 1b), which is differ-
Table 2 Bond energies calculated using the Cerius2 modelling package and the Dreiding 2.21 force field and MNDO-ESP charges [24] for an n-C H crystal 23 48 Bond!
Energy (kcal/mol)
/ d / d{ / a / b / g{ / g / e / f
14.885 14.642 8.910 1.587 0.644 0.568 0.556 0.286
!For the bonds the same notation as in Ref. [18] is used.
Table 3 Calculated slice energies (E ), anisotropies (d) and Ising temper4atures (¹ ) for different connected nets of n-C H [18] # 23 48 Connected net (h k l )
E 4(kcal/mol)
d
(0 0 1) (1 1 0) 1 (1 1 1) (1 1 0) 2 (1 1 0) 4 (0 1 0) (0 1 1) (1 0 0) 3 (1 0 1)
40.02 15.93 15.79 15.65 15.03 10.12 9.33 2.43 2.19
1.000 0.040 0.042 0.031 0 0.068 0 0.27 0
1
d 2
¹ # (K)
0.61 0.040 0.029 0.031 0.019 0.068 0.047 0.27 0.38
5834 1563 1504 1448 1073 1123 805 336 292
ent from that of a hexagonal lattice (Fig. 1a), has been calculated and its graph is given by Fig. 3. 3.2. Discussion Fig. 3 shows that the roughening temperatures of the connected net (0 1 0) show the same behaviour on varying bonds / and / from 0 to 1 as an 2 3 hexagonal lattice (Fig. 2), which has been discussed in Section 2. This shows that the anisotropy of the lattice is of more importance for the roughening temperature than a small change in the structure of the lattice. Therefore, we will compare the roughening temperatures of the different pseudo-hexagonal
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569
Fig. 3. (a) Representation of the (0 1 0) and (1 0 0) Ising lattices of orthorhombic n-paraffin crystals, deduced from the hexagonal 3 lattices in Fig. 1b. The bonds 1, 2 and 3 are represented by lines, the atoms or molecules by the intersection of the lines. Also extra bonds of infinite strength are added (see text). (b) The normalised roughening transition temperature as a function of the strengths of bonds 2 and 3 for the modified hexagonal lattice (a). Bond 1 is constant (/ "1). 1
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connected nets of n-paraffin with slightly different structures, that are (0 0 1), (1 1 0) , (1 1 0) , (1 1 0) , 1 2 4 (1 1 1), (0 1 0) and (1 0 0) , with each other and with 3 the isotropic hexagonal lattice. To be able to get a rectangular Ising lattice for the (0 0 1) lattice the bond “b” is left out, because it is a so-called crossing bond with relatively low energy. Fig. 4 shows the roughening temperatures of all connected nets that are listed in Table 3 as a function of their slice energy. The solid lines in this figure indicate the dependence of the ¹R of an isotropic hexagonal lattice (d "d "1) and of 1 2 some anisotropic hexagonal lattices (d "0, 1 d "0.5 and d "d "0.05) on E , which show 2 1 2 4-*#% a linear dependence. These roughening temperatures have been scaled to a maximum E which 4-*#% is equivalent to that of the (0 0 1) face. The figure shows that ¹R is determined to a great extent by E and not so much by the anisotropy of the 4-*#% connected net. All deviations from the solid line (d "d "1) are because of anisotropy in the con1 2 nected net or because the structure of the connected net differs somewhat from that of an hexagonal net. It shows that although (0 0 1) is not a hexagonal net, like shown in Fig. 1a the roughening temperature is almost the same. The small deviation is because the anisotropy in one direction of this net is 0.61.
More interesting are the faces (1 1 0) , (1 1 1), 1 (1 1 0) and (1 1 0) , which are all highly anisotropic 2 4 (d’s ranging from 0.019 to 0.040) and have almost the same slice energy (from 15.0 to 15.9 kcal/mol). As expected the roughening temperatures of these connected nets are below the line indicating the roughening temperatures of structures with an anisotropy of d "d "0.05, because for those faces 1 2 both anisotropies d and d are less than 0.05. 1 2 Table 3 shows that the roughening temperatures of these connected nets are directly correlated with their anisotropy: the more anisotropic the nets, the lower are the roughening temperatures. When (1 1 0) and (1 1 0) are replaced by the substitute 1 2 connected net (1 1 0) [23], an anisotropic rectan4 gular net is obtained with a roughening temperature which is much lower than that of an isotropic rectangular net. Although this connected net has a E comparable to (1 1 1) it has a lower 4-*#% ¹R because of its higher anisotropy. For the same reason ¹R of (1 1 0) is even lower than ¹R of (0 1 0), 4 which has a considerable lower slice energy. The difference in the roughening temperature of the faces (0 1 0) and (0 1 1), which have almost the same slice energy (9.33 and 10.12 kcal/mol), can be explained by the fact that (0 1 1) is a rectangular net with d "0 and d "0.047, whereas (0 1 0) has 1 2 a lower anisotropy of d "d "0.068. It can be 1 2 seen that the roughening temperature of the latter one is slightly higher than the line indicating d "d "0.05. The roughening temperatures of 1 2 the connected nets (1 0 0) and (1 0 1), with aniso3 tropies of, respectively, d "d "0.27 and d "0; 1 2 1 d "0.38, are very close to the roughening temper2 atures of the isotropic hexagonal lattice (d "d "1). 1 2 3.3. Near-equilibrium vapour experiments
Fig. 4. The dependence of the calculated Ising temperatures on the slice energy of the lattices listed in Table 3. The straight line represents the Ising temperatures of an isotropic hexagonal lattice with d "d "1 and of two anisotropic lattices with 1 2 d "0, d "0.5 and d "d "0.05. 1 2 1 2
We examined the morphology of n-C H crys23 48 tals and the roughening temperatures of its different faces at near equilibrium conditions and high temperatures using a vapour growth cell, which is shown in Fig. 5. In this cell bulk material of nC H is heated and sublimated under vacuum 23 48 conditions and then condensed on a cold finger. A window above the cold finger enables us to follow the growth of crystals in situ, by making use
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Fig. 5. A schematic drawing of a vapour growth cell used in our experiments. (1) Glass window, (2) vacuum flange, (3) heating tube, (4) cold finger with a crystal, (5) thermostated wall with bulk material, (6) thermocouple and (7) cold water inlet.
of a Nikon optical microscope with long working distance objectives. Further some scanning electron micrographs have been made to examine the crystal faces more closely. The temperature of the cold finger was kept constant at 287, 303 and 312 K which is 1.7 K below the transition temperature from the orthorhombic phase to the rotator phase [25]. Note that this transition temperature is the highest temperature possible to observe the faces that are listed in Table 3 for n-C H crys23 48 tals. The temperature of the bulk material has been kept constant at 2 K above the temperature of the cold finger, so that *¹ was 2 K. These near-equilibrium conditions (instead of “real” equilibrium conditions) were applied to shorten the experimental time to a few weeks. At the temperature of 287 K kinetic roughening experiments have been performed at supersaturations of *¹"2, 4, 6 and 8 K.
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faceted. At a temperature of 303 K and a supersaturation of 2 K faceted M0 0 2N, M1 1 1N, M1 1 0N and M0 2 0N faces have been found, and at a temperature of 313 K at *¹ of 2 K only faceted M0 0 2N faces were present. At this temperature the faces M1 1 0N, M1 1 1N and M0 2 0N were rough. It is clear that, as predicted, the M0 0 2N faces have the highest roughening temperature, which is supposed to be at least higher than 312 K. The faces M1 1 0N, M1 1 1N and M0 2 0N have roughening temperatures which are higher than 303 K. That those faces are rough at 312 K is probably due to the transition from the orthorhombic phase to the pseudo-hexagonal phase [25]. Although 312 K is still 1.7 K below this transition temperature it is known that homologous impurities lower this temperature [26]. The bond energies in the pseudohexagonal phase are smaller than of those in the orthorhombic phase which results in lower roughening temperatures. So it is still possible that the roughening temperatures of the M1 1 0N, M1 1 1N and M0 2 0N faces in the orthorhombic phase are much higher than 313.7 K. Although it is not possible to determine the exact values of the roughening temperatures, because they are higher than the transition temperature from the orthorhombic phase to the pseudo-hexagonal phase, the kinetic roughening experiments can be used to determine the sequence of roughening temperatures of the different faces. Here we will assume that a high roughening temperature leads to a high edge free energy and thus to a high critical supersaturation. We observed that the order in kinetic roughening of the faces is first the faces M1 0 0N and M0 2 0N then the faces M1 1 0N and M1 1 1N, whereas the M0 0 2N faces will not roughen. This sequence is the same as that of the roughening temperatures of those faces.
3.4. Results 4. Conclusions At the lowest temperature and a supersaturation of 2 K faceted M0 0 2N, M1 1 1N, M1 1 0N, M0 2 0N and M1 0 0N faces were present. The faces M1 0 0N have only been observed for 10% of the crystals. Upon increasing the supersaturation first the faces M1 0 0N and M0 2 0N roughen, then is followed by M1 1 0N and M1 1 1N, whereas the faces M0 0 2N always stay
We calculated the normalised Ising temperatures of (pseudo) hexagonal lattices for both positive and negative bond energies. To be able to compare the roughening temperatures of different lattices with different total slice energies they have been normalised according to a maximum slice energy. If in
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a (pseudo) hexagonal Ising lattice the relative strength of one of the bonds is less than 0.1, the lattice reduces effectively to a rectangular Ising lattice. Considering the roughening transition temperature, this bond may therefore be neglected. If one of the bonds becomes negative in a (pseudo) hexagonal lattice, the lattice remains stable, that is it keeps a relatively high roughening transition temperature if the absolute value of its bond energy is much smaller than the smallest of the other two bonds. In fact, Fig. 2 shows normalised Ising temperatures h#{ of simple ionic (pseudo) hexagonal crystal faces. If the absolute strength of the negative bond increases to that of the smallest of the other two, the stability of the lattice decreases dramatically. Effectively, the lattice reduces to a step, with a roughening transition temperature of 0 K. If the absolute strength of the negative bond becomes larger than one of the positive bonds, no roughening transition temperature can be calculated and is therefore set to be 0 K. If the bond energies of a crystal face are known, Fig. 2 can be used to calculate its roughening temperature. From the n-paraffin example it may be concluded that the roughening temperature is mainly determined by the slice energy of the connected net and not so much by the anisotropy of that net. It has been shown that the difference in roughening temperatures of faces with almost the same slice energy like (1 1 0) , (1 1 1), (1 1 0) and (1 1 0) will 1 2 4 be determined by the anisotropy of these faces. That is, the face with the highest anisotropy will have the lowest roughening temperature. In case of the face (1 1 0) it turns out that although this face 4 has a higher slice energy than the face (0 1 0) it has a lower roughening temperature. The roughening temperatures could not be determined experimentally because they were all higher than the highest experimentally accessible temperature, that is the temperature of the transition from the orthorhombic phase to the pseudohexagonal phase. The sequence of roughening temperatures is likely to be equal to the sequence of the kinetic roughening of the faces, which is M1 0 0N, M0 2 0N, M1 1 0N, M1 1 1N and M0 0 2N. The calculated roughening temperatures show the same sequence.
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