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Crystallisation of a bent-core liquid crystal mesogen
Physica B 304 (2001) 51–59
Crystallisation of a bent-core liquid crystal mesogen I. Dierking* Institut fu.r Physikalische Chemie, Technische Universita.t Darmstadt, Petersenstr. 20, D-64287 Darmstadt, Germany Received 12 February 2001; received in revised form 21 March 2001
Abstract In a general view, the crystallisation process of a conventional (rod-like) thermotropic liquid crystal can be placed in either of two categories: (i) spherulite growth, which is often observed for crystallisation from nematic or fluid smectic phases and easily observable by polarising microscopy and (ii) gradual ceasing of molecular disorder, a process, which is often observed during the crystallisation from hexatic or higher ordered smectic phases, but may hardly be detected by texture observation. In either case, the crystallisation of rod-like mesogens does not exhibit fractal features. In contrast, here the growth of fractal crystalline aggregates from the liquid crystalline phase is demonstrated for a bent-core, or so called banana-mesogen. Fractal growth was investigated with respect to quench depth at isothermal conditions, sample dimension and constant cooling rate across the liquid crystal to crystalline phase transition. r 2001 Elsevier Science B.V. All rights reserved. PACS: 61.30.@v; 81.10.Aj; 67.70.Md; 61.43.Hv Keywords: Crystallisation; Liquid crystal; Bent-core mesogen; Fractal growth; Phase transition
1. Introduction Liquid crystals (LC) [1,2] are anisotropic fluids, with partially ordered phases thermodynamically located between the isotropic liquid and the threedimensionally ordered crystal. Thus, they combine the properties of a common liquid, such as flow behaviour, with an anisotropy of physical properties, i.e. birefringence. In addition to the classical thermotropic liquid crystal phases of rod-like (calamitic) mesogens, distinct novel phases have recently been reported [3,4] for mesogens of bentcore molecular structure, so called ‘banana-phases’. *Tel.: +49-6151-16-3497; fax: +49-6151-16-4924. E-mail address: [email protected] (I. Dierking).
These have lately attracted great interest in the light of fundamental liquid crystal physics, as they exhibit polar properties, i.e. the occurrence of ferro- and antiferroelectricity, in the absence of molecular chirality. In comparison to the phase ordering process of calamitic phases from the isotropic melt, which is generally accomplished via the growth of spherical (nematic, cholesteric phases) or b#atonnet (smectic phases) shaped nuclei, the formation of LC phases of bent-core mesogens is qualitatively different. While in the former case the liquid crystal aggregates exhibit an euclidean dimension of D ¼ 2, the phase ordering process of bent-core mesogens has been shown [5,6] to be accomplished by the growth of irregular aggregates of a fractal dimension in the order of DE1:8521:89,
0921-4526/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 5 4 9 - X
52
I. Dierking / Physica B 304 (2001) 51–59
typical values observed for percolation growth in a two-dimensional medium at the percolation threshold [7]. Up to now, not much attention has been paid to the liquid crystal–crystal phase transition. The shape and growth of crystallites during the crystallisation process of a (calamitic) liquid crystal generally depends on the nature of the high temperature LC phase. For the crystallisation
isothermal conditions, sample dimension and constant cooling rate.
from nematic/cholesteric phases, which exhibit only orientational order of the long molecular axis of rod-like molecules, most of the times spherulite crystal growth is observed. This is generally also the case for fluid smectic phases (SmA, SmC), which exhibit a one-dimensional positional order of the molecular centres of mass, in addition to the orientational order of the long molecular axis, while here sometimes ‘needlegrowth’ may also be observed. For higher ordered liquid crystal phases, hexatic smectic phases, such as SmBhex, SmI, SmF with bond-orientational order within smectic layers, and especially crystalline smectics, such as SmBcryst, SmG, SmH with long-range order across smectic layers, the crystallisation is generally accomplished by only a freezing-in of motional freedom. These transitions can often easily be followed by differential scanning calorimetry, but are generally much harder to detect by texture observation, as only very subtle changes in the textural appearance may be observed. Generally, the crystallisation of conventional rod-like mesogens is a process, which bears no similarities to fractal growth phenomena. In this study, crystallisation of bent-core mesogens via fractal aggregates is demonstrated and experimentally investigated as a function of quench depth at
Its phase sequence, as determined by polarising microscopy in this study is given by: Iso. 169 B2 150 B3 145 B4. Transition temperatures are slightly different from those of the original report [3], which is however of no direct relevance to the present investigations. The B2 phase is a fluid smectic phase, similar to an antiferroelectric SmCA* phase of calamitic molecules, while the B3 phase seems to be a higher ordered smectic. The B4 phase, whose phase formation is the topic of the reported study, is a crystalline phase, which shows some remarkable physical properties: even though the constituent molecules are achiral, one may observe the spontaneous formation of weakly birefringent chiral domains on very slow cooling, which possess an opposite optical activity [8,9] and reflect circular polarised light of the opposite handedness. This suggests a helical superstructure, which has been demonstrated by atomic force microscopy [9]. Without external electric fields applied, the structure of the B4 phase is noncentrosymmetric, which has been demonstrated by the observation of a pronounced SHG signal [10–12]. The crystallisation of the banana-mesogen from the liquid crystalline state B3-B4 was followed directly by polarising microscopy (Leitz Orthoplan), equipped with a Mettler FP82 hot stage for
2. Experimental details The compound under investigation is the socalled banana-mesogen of the following structural formula:
I. Dierking / Physica B 304 (2001) 51–59
53
Fig. 1. Series of exemplary images illustrating the growth of crystalline B4 clusters (black) from the B3 liquid crystal phase (white) at isothermal conditions at a quench depth of DT=0.5 K for a 2 mm thick cell. The image size is 640 mm 490 mm.
relative temperature control better than 0.1 K. Digital images at a resolution of 720 540 pixels, corresponding to an image of 640 mm 490 mm were recorded with a Sony Hyper HAD model SSC-DC38P video camera in combination with imaging software (MiroVideoCapture). For the investigations the following general experimental conditions were used: commercially available sandwich cells (E.H.C., Japan) of gap d ¼ 2 mm, quench depth DT=0.5 K and quench rate R ¼ 3 K min@1. For the different measurement series only respective parameters were changed. Up to a sample thickness of d ¼ 6 mm, the growth of the crystalline phase could easily be observed as dark areas between crossed polarisers (optic axis along the direction of light propagation), while bright areas, due to birefringence, correspond to the liquid crystalline phase. An exemplary time series of crystal growth at isothermal conditions is depicted in Fig. 1. Time dependent fractal dimensional analysis with BENOIT1.3 (Trusoft International) was carried out by several different methods relating to the covered area of the whole texture, single aggre-
gates, as well as the perimeter of single aggregates. In the following, we restrict our discussion to a quasi-two-dimensional medium, which is reasonably well realised with our sample dimensions (640 mm 490 mm 2 mm). The methods used in relation to covered cluster area were: 1. The box dimension method, which defines the fractal dimension Db through the following relationship: NðdÞB
1 d Db
ð1Þ
with NðdÞ the number of boxes of size d being occupied by the data set. This is a popular method used for fractal dimensional analysis, but it should be noted, that it often yields values which are slightly too small. For objects (aggregates) as obtained for example in domain growth of a calamitic liquid crystal, an euclidian dimension of Db ¼ 2 is obtained. 2. The information dimension method is similar to the box method, but weights individual boxes according to the number of points contained. It
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I. Dierking / Physica B 304 (2001) 51–59
is defined from the relation IðdÞB@Di logðdÞ
ð2Þ
with IðdÞ the information entropy of NðdÞ boxes of size d, given by IðdÞ ¼ @
NðdÞ X
mi logðmi Þ
ð3Þ
i¼1
with mi ¼ Mi =M, where Mi is the number of points in the ith box and M the number of total points in the data set. Values for the fractal dimension Di are generally larger than those from the box dimension method, but often more reliable. 3. The mass dimension method is generally most applicable for patterns of radial symmetry, but can also be employed in the present case with the appropriate choice of circle centre coordinates and radius interval. Dm is defined from the relation mðrÞBrDm
ð4Þ
with mðrÞ ¼ MðrÞ=M the ‘mass’ within a circle of radius r, where MðrÞ is the data set points contained within a circle and M the total number of points in the set. In principle, the mass dimension Dm should be equal to the box dimension Db . In practice, Dm is often slightly larger than Db , more comparable to the information dimension Di .Determination methods for the fractal dimension related to only single aggregates were as follows: 4. The area-perimeter dimension method, which relates the area A of an object (closed loop) to its perimeter P by the dimension Dp , which is defined through ABP2=Dp :
ð5Þ
For euclidian objects like the circular domains of calamitic liquid crystals, a dimension Dp=1 is found, as expected.
5. The ruler dimension method is related solely to the perimeter of an object. It is obtained from the relationship MðlÞBl @Dr
ð6Þ
with MðlÞ the number of steps taken a ruler of length l on the perimeter of the object. If the line is euclidean, the ruler dimension is Dr ¼ 1.
3. Experimental results and discussion Fig. 2 shows a demonstration of the different fractal dimension methods discussed above for a single crystalline aggregate in a 2 mm cell at quench depth DT=0.5 K after t ¼ 20 s of growth. Fractal analysis was carried out by varying the box size, circle radius and ruler length over two orders of magnitude with a variation of the respective dependent variable over approximately four orders of magnitude. In all cases a linear relationship according to Eqs. (1)–(6) is obtained. Fractal dimensions were then evaluated by minimisation of the standard deviation of a linear fit to SDo0.001 (data used in the evaluation is indicated in Fig. 2 by filled symbols). Values of the box dimension Db ¼ 1:81, the information dimension Di ¼ 1:89 and the mass dimension Dm ¼ 1:90 clearly indicate the fractal nature of the crystalline aggregate, as does the area-perimeter dimension of Dp ¼ 1:29 and the ruler dimension Dr ¼ 1:18. Ideal fractal (mathematical) objects display selfsimilarity, i.e. the same fractal dimension, at all length scales. This can obviously not be true for real objects, as they are studied here, with one limitation for example being the restrictions through digital image resolution. There are no ‘‘real’’ fractals in nature. Nevertheless, self-similarity of objects in nature can be at least exhibited over a certain range of length scales. In our case, fractal dimensions obtained from an analysis of differently sized aggregates (at constant
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––Fig. 2. Demonstration of the different fractal estimation methods for a single crystalline cluster in a 2 mm cell at a quench depth of DT=0.5 K after 20 s of growth at isothermal conditions: (a) box dimension method, (b) information dimension method, (c) mass dimension method, (d) area-perimeter dimension method and (e) ruler dimension method. Fractal analysis was carried out by variation of the box size, circle radius and ruler length over two orders of magnitude and subsequent minimisation of the standard deviation of a linear fit to the data in a log–log plot.
I. Dierking / Physica B 304 (2001) 51–59
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I. Dierking / Physica B 304 (2001) 51–59
Fig. 3. Time dependence of the information dimension Di , determined for the whole microscopic texture of a 2 mm cell for various quench depths DT.
Fig. 4. Time dependence of the different fractal dimensions of a single aggregate growing at isothermal conditions at a quench depth DT=0.5 K in a 2 mm cell.
experimental conditions, from one image) on the scale between 50 mm and the whole texture of 640 mm do in fact yield the same fractal dimension, thus exhibit self-similarity at least within this range of scales. Fig. 3 shows the time development of the information dimension Di (of the whole microscopic texture image) at isothermal conditions for a variety of different quench depths DT below the liquid crystal to crystal transition. For short times (t ¼ 0210 s) a strong increase of the fractal dimension is observed, starting from values of Di E1:5, before a plateau is reached (t ¼ 10260 s), where Di E1:8521:88. This behaviour may indicate a growth process as described by percolation clusters, below the percolation threshold for to10 s and at the percolation threshold when Di reaches the plateau at 10 soto60 s. At later times t > 80 s the information dimension slowly approaches values of Di-2, the euclidean dimension of a space filling object in two dimensions. This is obviously due to ‘‘experimental limitations’’ as naturally the microscopic field of view is restricted to a certain observable area and the cell is of finite dimension. It should be mentioned, that a comparable time development is also obtained for the box and the mass dimensions. A basically constant fractal dimension is obtained with time (Fig. 4), when considering a single cluster of the crystalline phase growing at
isothermal conditions at a quench depth of DT=0.5 K from the liquid crystalline phase until aggregation with other clusters takes place. We note, that the observed values correspond to those obtained from a fractal analysis of the whole microscopic texture within the range of the above mentioned plateau, demonstrating self-similarity. Within the limits of error, which is estimated to 70.02, the information method and the mass method give the same fractal dimensions, Di ¼ Dm E1:88. Values obtained for the box dimension Db are somewhat lower, but as discussed above, probably less reliable. The perimeter-area dimension is in the order of Dp E1:35 and the ruler dimension Dr E1:2, clearly demonstrating not only fractal properties of the cluster, but also of its perimeter. Fig. 5 summarises the quench depth dependence of the fractal dimension D(DT) for all five measures under discussion. Within the range of quench depths accessible, where nucleation and growth proceed at isothermal conditions (up to DT=1 K), the fractal dimensions are basically independent on DT with Db ¼ 1:81, Di ¼ 1:90, Dm ¼ 1:89, Dp E1:35 and Dr E1:20. The data obtained suggests, that the growth of the crystalline phase can be described by two-dimensional percolation clusters at the percolation threshold. The time development of the information dimension Di during isothermal crystallisation at
I. Dierking / Physica B 304 (2001) 51–59
57
Fig. 5. Dependence of the different fractal dimensions on quench depth DT for a single aggregate in a 2 mm cell after 20 s of growth at isothermal conditions.
Fig. 6. Time dependence of the information dimension Di determined for the whole microscopic texture at a quench depth of DT=0.5 K for various cell gaps. Growth of crystalline clusters occurs at isothermal conditions.
DT=0.5 K is depicted in Fig. 6 for selected cell gaps between 2 and 6 mm from analysis of the whole microscopic texture. The crystallisation proceeds more quickly for larger cell gaps, as generally observed. Values for the information dimension are rather small at the very early times of the growth process (Di E1:521:65) and then saturate at a plateau with Di E1:8521:95, slightly depending on cell gap. Again, this behaviour points towards a two-dimensional percolation system with growth approaching the percolation threshold at t ¼ 30@10 s for cell gaps between 2 and 6 mm, respectively. For cell gaps greater than 6 mm, the crystallisation does not proceed within two dimensions and the crystal clusters are not uniformly black anymore. In this case the optic axis is not parallel to the direction of light propagation everywhere within a cluster and images are not suitable for fractal dimensional analysis. Fig. 7 depicts the results of an analysis of a single crystal aggregate as a function of cell gap for all five different fractal estimates. Fractal dimensions are basically constant within the accessible range of sample dimensions, with Di and Dm in the order of 1.85–1.90, typical values obtained for percolation clusters in a two-dimensional medium at the percolation threshold. Also the perimeter related dimensions are basically independent on cell gap, with Dp E1:4 and Dr E1:23.
Fig. 7. Cell gap dependence of the different fractal dimensions determined at t=5 s for a single crystalline aggregate growing at isothermal conditions at a quench depth of DT=0.5 K.
Up to now, we have considered a non-equilibrium situation with the energetically favoured, stable low temperature crystalline phase growing the ‘‘sea’’ of the metastable high temperature liquid crystalline phase at isothermal conditions. We will now leave the restrictions of growth under isothermal conditions and investigate the time development, respectively the temperature dependence of growth under cooling at a constant rate R across the LC-crystal transition. For cooling rates R > 5 K min@1 a large number of crystalline germs nucleate, so that nuclei already aggregate
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I. Dierking / Physica B 304 (2001) 51–59
Fig. 8. Dependence of the information dimension Di of the whole microscopic texture on reduced temperature TC@T (respectively time t, see inset) for a 2 mm cell at various cooling rates R.
Fig. 9. Dependence of the information dimension Di on cooling rate R for selected reduced temperatures TC@T.
when they are quite small. For this reason and the limited pixel resolution, we can not follow the growth of a single cluster by fractal dimensional analysis in this section. Fig. 8 shows the dependence of the information dimension on reduced temperature for several cooling rates, with TC
being the LC-crystal transition temperature. The inset of Fig. 8 depicts the respective time development. Also in this series we observe a plateau, although not as pronounced as before. For lower temperatures, respectively longer times, the fractal dimension increases to Di ¼ 2 when the crystalline phase is space filling. This point is reached for all cooling rates at about t ¼ 30 s, thus at increasing reduced temperatures TC @T for increasing cooling rate R. The behaviour is shown more clearly in Fig. 9, depicting the information dimension as a function of cooling rate at several reduced temperatures. Di clearly decreases with increasing R. The same behaviour is obtained for the mass dimension and the box dimension.
4. Conclusions In contrast to the crystallisation of conventional rod-like liquid crystals, bent-core mesogens exhibit crystal clusters or aggregates of a non-euclidean, fractal dimension, where also the perimeter is
I. Dierking / Physica B 304 (2001) 51–59
fractal. As a general trend for five different fractal estimates, it was found that the fractal dimension of crystal clusters is basically independent on quench depth during growth at isothermal conditions. Values of the information dimension and the mass dimension in the order of Di ¼ Dm E1:89 suggest a growth process of aggregates, which can be described by a percolation system in twodimensions at the percolation threshold. A similar behaviour with the same fractal dimensions was obtained as the cell gap was varied. For cooling across the LC to crystalline transition at a constant rate, thus leaving isothermal growth conditions, a decreasing fractal dimension is observed with increasing cooling rate.
Acknowledgements The compound under investigation was kindly supplied by G. Pelzl, Halle. I would also like to acknowledge the financial support of the Fonds der Chemischen Industrie.
59
References [1] P.J. Collings, M. Hird, Introduction to Liquid Crystals, TayloraFrancis, London, 1997. [2] D. Demus, J. Goodby, G.W. Gray, H.-W. Spiess, V. Vill (Eds.), Physical Properties of Liquid Crystals, Wiley-VCH, Weinheim, 1999. [3] T. Sekine, T. Niori, M. Sone, J. Watanabe, S.W. Choi, Y. Takanishi, H. Takazoe, Jpn. J. Appl. Phys. 36 (1997) 6455. [4] G. Pelzl, S. Diele, W. Weissflog, Adv. Mater. 11 (1999) 707. [5] I. Dierking, Chem. Phys. Chem. 1 (2001) 59. [6] I. Dierking, J. Phys.: Condens. Matter 13 (2001) 1353. [7] T. Vicsek, Fractal Growth Phenomena, World Scientific, Singapore, 1989 (Chapter 5). [8] T. Sekine, T. Niori, J. Watanabe, T. Furukawa, S.W. Choi, H. Takezoe, J. Mater. Chem. 7 (1997) 1307. [9] G. Heppke, D. Moro, Science 279 (1998) 1872. [10] R. Macdonald, F. Kentischer, P. Warnick, G. Heppke, Phys. Rev. Lett. 81 (1998) 4408. [11] S.W. Choi, Y. Kinoshita, B. Park, H. Takezoe, T. Niori, J. Watanabe, Jpn. J. Appl. Phys. 37 (1998) 3408. [12] F. Kentischer, R. Macdonald, P. Warnick, G. Heppke, Liq. Cryst. 25 (1998) 341.
Crystallisation of a bent-core liquid crystal mesogen I. Dierking* Institut fu.r Physikalische Chemie, Technische Universita.t Darmstadt, Petersenstr. 20, D-64287 Darmstadt, Germany Received 12 February 2001; received in revised form 21 March 2001
Abstract In a general view, the crystallisation process of a conventional (rod-like) thermotropic liquid crystal can be placed in either of two categories: (i) spherulite growth, which is often observed for crystallisation from nematic or fluid smectic phases and easily observable by polarising microscopy and (ii) gradual ceasing of molecular disorder, a process, which is often observed during the crystallisation from hexatic or higher ordered smectic phases, but may hardly be detected by texture observation. In either case, the crystallisation of rod-like mesogens does not exhibit fractal features. In contrast, here the growth of fractal crystalline aggregates from the liquid crystalline phase is demonstrated for a bent-core, or so called banana-mesogen. Fractal growth was investigated with respect to quench depth at isothermal conditions, sample dimension and constant cooling rate across the liquid crystal to crystalline phase transition. r 2001 Elsevier Science B.V. All rights reserved. PACS: 61.30.@v; 81.10.Aj; 67.70.Md; 61.43.Hv Keywords: Crystallisation; Liquid crystal; Bent-core mesogen; Fractal growth; Phase transition
1. Introduction Liquid crystals (LC) [1,2] are anisotropic fluids, with partially ordered phases thermodynamically located between the isotropic liquid and the threedimensionally ordered crystal. Thus, they combine the properties of a common liquid, such as flow behaviour, with an anisotropy of physical properties, i.e. birefringence. In addition to the classical thermotropic liquid crystal phases of rod-like (calamitic) mesogens, distinct novel phases have recently been reported [3,4] for mesogens of bentcore molecular structure, so called ‘banana-phases’. *Tel.: +49-6151-16-3497; fax: +49-6151-16-4924. E-mail address: [email protected] (I. Dierking).
These have lately attracted great interest in the light of fundamental liquid crystal physics, as they exhibit polar properties, i.e. the occurrence of ferro- and antiferroelectricity, in the absence of molecular chirality. In comparison to the phase ordering process of calamitic phases from the isotropic melt, which is generally accomplished via the growth of spherical (nematic, cholesteric phases) or b#atonnet (smectic phases) shaped nuclei, the formation of LC phases of bent-core mesogens is qualitatively different. While in the former case the liquid crystal aggregates exhibit an euclidean dimension of D ¼ 2, the phase ordering process of bent-core mesogens has been shown [5,6] to be accomplished by the growth of irregular aggregates of a fractal dimension in the order of DE1:8521:89,
0921-4526/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 5 4 9 - X
52
I. Dierking / Physica B 304 (2001) 51–59
typical values observed for percolation growth in a two-dimensional medium at the percolation threshold [7]. Up to now, not much attention has been paid to the liquid crystal–crystal phase transition. The shape and growth of crystallites during the crystallisation process of a (calamitic) liquid crystal generally depends on the nature of the high temperature LC phase. For the crystallisation
isothermal conditions, sample dimension and constant cooling rate.
from nematic/cholesteric phases, which exhibit only orientational order of the long molecular axis of rod-like molecules, most of the times spherulite crystal growth is observed. This is generally also the case for fluid smectic phases (SmA, SmC), which exhibit a one-dimensional positional order of the molecular centres of mass, in addition to the orientational order of the long molecular axis, while here sometimes ‘needlegrowth’ may also be observed. For higher ordered liquid crystal phases, hexatic smectic phases, such as SmBhex, SmI, SmF with bond-orientational order within smectic layers, and especially crystalline smectics, such as SmBcryst, SmG, SmH with long-range order across smectic layers, the crystallisation is generally accomplished by only a freezing-in of motional freedom. These transitions can often easily be followed by differential scanning calorimetry, but are generally much harder to detect by texture observation, as only very subtle changes in the textural appearance may be observed. Generally, the crystallisation of conventional rod-like mesogens is a process, which bears no similarities to fractal growth phenomena. In this study, crystallisation of bent-core mesogens via fractal aggregates is demonstrated and experimentally investigated as a function of quench depth at
Its phase sequence, as determined by polarising microscopy in this study is given by: Iso. 169 B2 150 B3 145 B4. Transition temperatures are slightly different from those of the original report [3], which is however of no direct relevance to the present investigations. The B2 phase is a fluid smectic phase, similar to an antiferroelectric SmCA* phase of calamitic molecules, while the B3 phase seems to be a higher ordered smectic. The B4 phase, whose phase formation is the topic of the reported study, is a crystalline phase, which shows some remarkable physical properties: even though the constituent molecules are achiral, one may observe the spontaneous formation of weakly birefringent chiral domains on very slow cooling, which possess an opposite optical activity [8,9] and reflect circular polarised light of the opposite handedness. This suggests a helical superstructure, which has been demonstrated by atomic force microscopy [9]. Without external electric fields applied, the structure of the B4 phase is noncentrosymmetric, which has been demonstrated by the observation of a pronounced SHG signal [10–12]. The crystallisation of the banana-mesogen from the liquid crystalline state B3-B4 was followed directly by polarising microscopy (Leitz Orthoplan), equipped with a Mettler FP82 hot stage for
2. Experimental details The compound under investigation is the socalled banana-mesogen of the following structural formula:
I. Dierking / Physica B 304 (2001) 51–59
53
Fig. 1. Series of exemplary images illustrating the growth of crystalline B4 clusters (black) from the B3 liquid crystal phase (white) at isothermal conditions at a quench depth of DT=0.5 K for a 2 mm thick cell. The image size is 640 mm 490 mm.
relative temperature control better than 0.1 K. Digital images at a resolution of 720 540 pixels, corresponding to an image of 640 mm 490 mm were recorded with a Sony Hyper HAD model SSC-DC38P video camera in combination with imaging software (MiroVideoCapture). For the investigations the following general experimental conditions were used: commercially available sandwich cells (E.H.C., Japan) of gap d ¼ 2 mm, quench depth DT=0.5 K and quench rate R ¼ 3 K min@1. For the different measurement series only respective parameters were changed. Up to a sample thickness of d ¼ 6 mm, the growth of the crystalline phase could easily be observed as dark areas between crossed polarisers (optic axis along the direction of light propagation), while bright areas, due to birefringence, correspond to the liquid crystalline phase. An exemplary time series of crystal growth at isothermal conditions is depicted in Fig. 1. Time dependent fractal dimensional analysis with BENOIT1.3 (Trusoft International) was carried out by several different methods relating to the covered area of the whole texture, single aggre-
gates, as well as the perimeter of single aggregates. In the following, we restrict our discussion to a quasi-two-dimensional medium, which is reasonably well realised with our sample dimensions (640 mm 490 mm 2 mm). The methods used in relation to covered cluster area were: 1. The box dimension method, which defines the fractal dimension Db through the following relationship: NðdÞB
1 d Db
ð1Þ
with NðdÞ the number of boxes of size d being occupied by the data set. This is a popular method used for fractal dimensional analysis, but it should be noted, that it often yields values which are slightly too small. For objects (aggregates) as obtained for example in domain growth of a calamitic liquid crystal, an euclidian dimension of Db ¼ 2 is obtained. 2. The information dimension method is similar to the box method, but weights individual boxes according to the number of points contained. It
54
I. Dierking / Physica B 304 (2001) 51–59
is defined from the relation IðdÞB@Di logðdÞ
ð2Þ
with IðdÞ the information entropy of NðdÞ boxes of size d, given by IðdÞ ¼ @
NðdÞ X
mi logðmi Þ
ð3Þ
i¼1
with mi ¼ Mi =M, where Mi is the number of points in the ith box and M the number of total points in the data set. Values for the fractal dimension Di are generally larger than those from the box dimension method, but often more reliable. 3. The mass dimension method is generally most applicable for patterns of radial symmetry, but can also be employed in the present case with the appropriate choice of circle centre coordinates and radius interval. Dm is defined from the relation mðrÞBrDm
ð4Þ
with mðrÞ ¼ MðrÞ=M the ‘mass’ within a circle of radius r, where MðrÞ is the data set points contained within a circle and M the total number of points in the set. In principle, the mass dimension Dm should be equal to the box dimension Db . In practice, Dm is often slightly larger than Db , more comparable to the information dimension Di .Determination methods for the fractal dimension related to only single aggregates were as follows: 4. The area-perimeter dimension method, which relates the area A of an object (closed loop) to its perimeter P by the dimension Dp , which is defined through ABP2=Dp :
ð5Þ
For euclidian objects like the circular domains of calamitic liquid crystals, a dimension Dp=1 is found, as expected.
5. The ruler dimension method is related solely to the perimeter of an object. It is obtained from the relationship MðlÞBl @Dr
ð6Þ
with MðlÞ the number of steps taken a ruler of length l on the perimeter of the object. If the line is euclidean, the ruler dimension is Dr ¼ 1.
3. Experimental results and discussion Fig. 2 shows a demonstration of the different fractal dimension methods discussed above for a single crystalline aggregate in a 2 mm cell at quench depth DT=0.5 K after t ¼ 20 s of growth. Fractal analysis was carried out by varying the box size, circle radius and ruler length over two orders of magnitude with a variation of the respective dependent variable over approximately four orders of magnitude. In all cases a linear relationship according to Eqs. (1)–(6) is obtained. Fractal dimensions were then evaluated by minimisation of the standard deviation of a linear fit to SDo0.001 (data used in the evaluation is indicated in Fig. 2 by filled symbols). Values of the box dimension Db ¼ 1:81, the information dimension Di ¼ 1:89 and the mass dimension Dm ¼ 1:90 clearly indicate the fractal nature of the crystalline aggregate, as does the area-perimeter dimension of Dp ¼ 1:29 and the ruler dimension Dr ¼ 1:18. Ideal fractal (mathematical) objects display selfsimilarity, i.e. the same fractal dimension, at all length scales. This can obviously not be true for real objects, as they are studied here, with one limitation for example being the restrictions through digital image resolution. There are no ‘‘real’’ fractals in nature. Nevertheless, self-similarity of objects in nature can be at least exhibited over a certain range of length scales. In our case, fractal dimensions obtained from an analysis of differently sized aggregates (at constant
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––Fig. 2. Demonstration of the different fractal estimation methods for a single crystalline cluster in a 2 mm cell at a quench depth of DT=0.5 K after 20 s of growth at isothermal conditions: (a) box dimension method, (b) information dimension method, (c) mass dimension method, (d) area-perimeter dimension method and (e) ruler dimension method. Fractal analysis was carried out by variation of the box size, circle radius and ruler length over two orders of magnitude and subsequent minimisation of the standard deviation of a linear fit to the data in a log–log plot.
I. Dierking / Physica B 304 (2001) 51–59
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I. Dierking / Physica B 304 (2001) 51–59
Fig. 3. Time dependence of the information dimension Di , determined for the whole microscopic texture of a 2 mm cell for various quench depths DT.
Fig. 4. Time dependence of the different fractal dimensions of a single aggregate growing at isothermal conditions at a quench depth DT=0.5 K in a 2 mm cell.
experimental conditions, from one image) on the scale between 50 mm and the whole texture of 640 mm do in fact yield the same fractal dimension, thus exhibit self-similarity at least within this range of scales. Fig. 3 shows the time development of the information dimension Di (of the whole microscopic texture image) at isothermal conditions for a variety of different quench depths DT below the liquid crystal to crystal transition. For short times (t ¼ 0210 s) a strong increase of the fractal dimension is observed, starting from values of Di E1:5, before a plateau is reached (t ¼ 10260 s), where Di E1:8521:88. This behaviour may indicate a growth process as described by percolation clusters, below the percolation threshold for to10 s and at the percolation threshold when Di reaches the plateau at 10 soto60 s. At later times t > 80 s the information dimension slowly approaches values of Di-2, the euclidean dimension of a space filling object in two dimensions. This is obviously due to ‘‘experimental limitations’’ as naturally the microscopic field of view is restricted to a certain observable area and the cell is of finite dimension. It should be mentioned, that a comparable time development is also obtained for the box and the mass dimensions. A basically constant fractal dimension is obtained with time (Fig. 4), when considering a single cluster of the crystalline phase growing at
isothermal conditions at a quench depth of DT=0.5 K from the liquid crystalline phase until aggregation with other clusters takes place. We note, that the observed values correspond to those obtained from a fractal analysis of the whole microscopic texture within the range of the above mentioned plateau, demonstrating self-similarity. Within the limits of error, which is estimated to 70.02, the information method and the mass method give the same fractal dimensions, Di ¼ Dm E1:88. Values obtained for the box dimension Db are somewhat lower, but as discussed above, probably less reliable. The perimeter-area dimension is in the order of Dp E1:35 and the ruler dimension Dr E1:2, clearly demonstrating not only fractal properties of the cluster, but also of its perimeter. Fig. 5 summarises the quench depth dependence of the fractal dimension D(DT) for all five measures under discussion. Within the range of quench depths accessible, where nucleation and growth proceed at isothermal conditions (up to DT=1 K), the fractal dimensions are basically independent on DT with Db ¼ 1:81, Di ¼ 1:90, Dm ¼ 1:89, Dp E1:35 and Dr E1:20. The data obtained suggests, that the growth of the crystalline phase can be described by two-dimensional percolation clusters at the percolation threshold. The time development of the information dimension Di during isothermal crystallisation at
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Fig. 5. Dependence of the different fractal dimensions on quench depth DT for a single aggregate in a 2 mm cell after 20 s of growth at isothermal conditions.
Fig. 6. Time dependence of the information dimension Di determined for the whole microscopic texture at a quench depth of DT=0.5 K for various cell gaps. Growth of crystalline clusters occurs at isothermal conditions.
DT=0.5 K is depicted in Fig. 6 for selected cell gaps between 2 and 6 mm from analysis of the whole microscopic texture. The crystallisation proceeds more quickly for larger cell gaps, as generally observed. Values for the information dimension are rather small at the very early times of the growth process (Di E1:521:65) and then saturate at a plateau with Di E1:8521:95, slightly depending on cell gap. Again, this behaviour points towards a two-dimensional percolation system with growth approaching the percolation threshold at t ¼ 30@10 s for cell gaps between 2 and 6 mm, respectively. For cell gaps greater than 6 mm, the crystallisation does not proceed within two dimensions and the crystal clusters are not uniformly black anymore. In this case the optic axis is not parallel to the direction of light propagation everywhere within a cluster and images are not suitable for fractal dimensional analysis. Fig. 7 depicts the results of an analysis of a single crystal aggregate as a function of cell gap for all five different fractal estimates. Fractal dimensions are basically constant within the accessible range of sample dimensions, with Di and Dm in the order of 1.85–1.90, typical values obtained for percolation clusters in a two-dimensional medium at the percolation threshold. Also the perimeter related dimensions are basically independent on cell gap, with Dp E1:4 and Dr E1:23.
Fig. 7. Cell gap dependence of the different fractal dimensions determined at t=5 s for a single crystalline aggregate growing at isothermal conditions at a quench depth of DT=0.5 K.
Up to now, we have considered a non-equilibrium situation with the energetically favoured, stable low temperature crystalline phase growing the ‘‘sea’’ of the metastable high temperature liquid crystalline phase at isothermal conditions. We will now leave the restrictions of growth under isothermal conditions and investigate the time development, respectively the temperature dependence of growth under cooling at a constant rate R across the LC-crystal transition. For cooling rates R > 5 K min@1 a large number of crystalline germs nucleate, so that nuclei already aggregate
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Fig. 8. Dependence of the information dimension Di of the whole microscopic texture on reduced temperature TC@T (respectively time t, see inset) for a 2 mm cell at various cooling rates R.
Fig. 9. Dependence of the information dimension Di on cooling rate R for selected reduced temperatures TC@T.
when they are quite small. For this reason and the limited pixel resolution, we can not follow the growth of a single cluster by fractal dimensional analysis in this section. Fig. 8 shows the dependence of the information dimension on reduced temperature for several cooling rates, with TC
being the LC-crystal transition temperature. The inset of Fig. 8 depicts the respective time development. Also in this series we observe a plateau, although not as pronounced as before. For lower temperatures, respectively longer times, the fractal dimension increases to Di ¼ 2 when the crystalline phase is space filling. This point is reached for all cooling rates at about t ¼ 30 s, thus at increasing reduced temperatures TC @T for increasing cooling rate R. The behaviour is shown more clearly in Fig. 9, depicting the information dimension as a function of cooling rate at several reduced temperatures. Di clearly decreases with increasing R. The same behaviour is obtained for the mass dimension and the box dimension.
4. Conclusions In contrast to the crystallisation of conventional rod-like liquid crystals, bent-core mesogens exhibit crystal clusters or aggregates of a non-euclidean, fractal dimension, where also the perimeter is
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fractal. As a general trend for five different fractal estimates, it was found that the fractal dimension of crystal clusters is basically independent on quench depth during growth at isothermal conditions. Values of the information dimension and the mass dimension in the order of Di ¼ Dm E1:89 suggest a growth process of aggregates, which can be described by a percolation system in twodimensions at the percolation threshold. A similar behaviour with the same fractal dimensions was obtained as the cell gap was varied. For cooling across the LC to crystalline transition at a constant rate, thus leaving isothermal growth conditions, a decreasing fractal dimension is observed with increasing cooling rate.
Acknowledgements The compound under investigation was kindly supplied by G. Pelzl, Halle. I would also like to acknowledge the financial support of the Fonds der Chemischen Industrie.
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