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Orientational correlations in stripe patterns
Physica B 322 (2002) 110–115
Orientational correlations in stripe patterns S.R. Ren, I.W. Hamley* School of Chemistry, University of Leeds, LS2 9JT Leeds, UK Received 3 April 2001; received in revised form 15 June 2001
Abstract Stripe phases are observed in many condensed matter systems where there is a competition between long- and shortrange ordering. Cell dynamics simulations are used to generate patterns in which the stripe orientation is enhanced by increasing thermal noise, decreasing temperature or applying shear. Although a steady state degree of order is reached, the non-equilibrium nature of the simulated patterns is demonstrated through the dependence of orientational correlation functions on thermal history. Nevertheless, the results show that the computed orientational correlation function provides a quantitative ‘‘fingerprint’’ that should enable the characterization of equilibrium stripe patterns as well as trapped non-equilibrium states. r 2002 Elsevier Science B.V. All rights reserved. PACS: 47.20.Hw; 83.50.v; 81.05.Lg; 66.30.Lw Keywords: Stripe patterns; Orientational ordering; Computer simulations; Cellular automata
Stripe patterns are observed for a range of condensed matter systems [1], including type I superconductor films [2], ferromagnetic garnet films [1,3–5], Langmuir monolayers of lipid molecules [1,6–8], stationary (Turing) patterns in reaction–diffusion systems [9], and lamellar diblock copolymer films [10–12]. Stripe patterns result from a competition between long-range ordering that favors separation into two distinct phases, and a local term that tends to maintain connectivity of alternating domains. Seul and co-workers have analyzed in detail stripe phases formed in magnetic garnet films in terms of topological defects, the density of which is much greater in the labyrinthine phase [4], in *Corresponding author. Tel.: +113-233-6430; fax: +113233-6565. E-mail address: [email protected] (I.W. Hamley).
which the stripes are orientationally disordered, than in more highly aligned stripe phases [3]. Here we introduce a method to characterize the orientational ordering of the interface at different lengthscales in the system. The approach was inspired by the analysis of the statistics of a random interface in a microemulsion [13]. Rather than analyzing experimental stripe patterns, we have used the cell dynamics simulation (CDS) method to generate them. This offers the advantage that the orientational ordering of the stripe interfaces can be controlled. An increased stripe orientation can be achieved by adding a noise-term into the kinetic equations. The CDS method is a cellular automaton, in which a timedependent spatially varying order parameter cðr; tÞ; is discretized on a lattice, taking values cðt; iÞ in cell i at time t: For the case of a nonconserved order parameter, the time evolution of the order parameter is represented in the form
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 6 0 5 - 1
S.R. Ren, I.W. Hamley / Physica B 322 (2002) 110–115
111
Fig. 1. Stripe patterns with different levels of thermal noise, generated by cell dynamics simulations for a quench to t=0.32. (a) Z=0, (b) Z=0.06, (c) Z=0.1. Inset: Fourier transform patterns.
developed by Bahiana and Oono [14]. cðt þ 1; iÞ ¼ f ðcðt; iÞÞ þ D½//cðt; iÞSS cðt; iÞ cðt; iÞ ¼ F ½cðt; iÞ: ð1Þ Here f is a function that represents the time evolution of the order parameter in a single cell. It has one hyperbolic source at the origin (corresponding to the high temperature c ¼ 0 state) and two hyperbolic sinks corresponding to the ordered state, ca0: The second term on the right-hand side of Eq. (1) allows for spatially cooperative dynamics via a diffusive mechanism due to the connectivity of cells. In Eq. (1), D is a positive constant that plays the role of a diffusion coefficient, and hhX ii X is the isotropized discrete Laplacian [15]. For a conserved order parameter, there should be no net change of order parameter in the region surrounding a given cell. Since the net gain of order parameter in a particular cell is given by F ½cðt; nÞ cðt; nÞ; the CDS model for a conserved order parameter becomes [14] cðt þ 1; iÞ ¼ cðt; iÞ þ F ½cðt; iÞ hhF ½cðt; iÞii Bcðt; iÞ þ Zzðt; iÞ; ð2Þ where B is a term that accounts for long-range ordering [14]. Eq. (2) includes a contribution from thermal noise, zðt; iÞ; of amplitude Z [16,17]. Although the CDS equations, Eqs. (1) and (2), are independent of any model for the free energy functional, it has been shown [18,19] that in the form Eq. (1), the CDS equations correspond to a coarse-grained discretization of the time-depen-
Fig. 2. Interface orientation correlation functions for the patterns in Fig. 1. The distance r is in lattice units. The solid lines indicate fits to Eq. (5). Inset: showing the power law fits on a double logarithmic scale.
dent Ginzburg–Landau equation (TDGL) equation. For a conserved order parameter, Eq. (2) represents the coarse-grained dynamics of the Cahn–Hilliard–Cook (CHC) equation [18–20]. In the theory of dynamic critical phenomena [21] the TDGL equation is also known as model A and the CHC equation is known as Model B. We generated stripe patterns by Cell dynamic simulations of the CHC equation on a 128 128 lattice, using the cell dynamics for the case of a
112
S.R. Ren, I.W. Hamley / Physica B 322 (2002) 110–115
Fig. 3. Stripe patterns corresponding to different effective temperatures t with Z=0.
conserved order parameter, Eq. (2). Calculations were performed for a two-dimensional phase of alternating A and B stripes, in which the volume fraction, fA ¼ fB ¼ 0:5: The map in Eq. (1) was taken to have the form f ðcÞ ¼ ð1 þ tÞc uc3 ; with u ¼ 0:5; and variable temperature-like parameter t: This form of map is described in detail elsewhere [22,23]. Following Bahiana and Oono [14], the parameters D ¼ 0:5 and B ¼ 0:02 were used to simulate a stripe pattern. To ensure approximate conservation of the order parameter when noise is added we have adapted the procedure described elsewhere [24]. Specifically, two independent Gaussian distributed random numbers xl ði; jÞ and xm ði; jÞ are generated at each lattice point ði; jÞ; representing fluctuations in
different directions. The noise term is then taken as Zxðr; tÞ ¼ Z½xl ði; jÞ xl ði71; jÞ þ xm ði; jÞ xm ði; j71Þ at each time step. A third set of uniformly distributed random numbers is used to determine whether ‘7’ takes ‘+’ or ‘’ in the calculation. The amplitude of the noise level is denoted Z (o1). Increasing its amplitude leads to a coarsening of the domain structure [16] and enhanced lamellar orientation due to local thermal annealing, up to some critical value of Z above which the system becomes disordered. To characterize the correlation of stripe orientation over different lengthscales, we compute the correlation function for the angle of the layer normal with respect to an arbitrary direction, fðrÞ; at the interface between A and B stripes. The
S.R. Ren, I.W. Hamley / Physica B 322 (2002) 110–115
Fig. 4. Orientation correlation functions corresponding to the patterns in Fig. 3.
appropriate correlation function for a two-dimensional smectic (stripe) phase is [25] Gðr r Þ ¼ /e 0
2ifðrÞ 2ifðr0 Þ
e
S ¼ /2 cos y 1S; ð3Þ 2
where y ¼ fðrÞ fðr0 Þ: The interface between stripes was located by thresholding the stripe pattern, to produce a step discontinuity in order parameter from A to B stripe. This discontinuity defined the cells at the interface, and the (discrete) normal (orientation angle y ¼ n p=4; n ¼ 0; 1; y; 7) was computed as the vector between adjacent cells at the interface. The correlation function, Eq. (3), has been used to analyze orientational correlations between stripes formed in block copolymer films [26]. The cell dynamic simulations were run until the average order parameter in the system was constant for at least 5000 timesteps. All data shown are averages from seven independent runs. Patterns generated from the CDS simulations with different levels of thermal noise are shown in Fig. 1. From inspection, it is apparent that the mutual orientation of stripes increases as the noise amplitude is increased. However, this is not reflected in the Fourier transforms shown in the insets. This is because the Fourier transform depends on the global alignment of the pattern, and is not sensitive to local orientation. For many
113
stripe patterns where there is no long-range orientational order, Fourier transformation is thus an inappropriate measure of alignment. Furthermore, the labyrinthine nature of the structure prevents unambiguous identification of defects, which thus do not provide a suitable description of the pattern. However, the interface normal correlation function, Eq. (3), is strongly dependent on local stripe orientation, as shown in Fig. 2. This shows that Gðr r0 Þ increases monotonically with Z for small r ¼ jrFr0 j: For large r; Gðr r0 Þ approaches zero, its value for random stripe orientation. Orientational correlations in the stripe phase have been investigated theoretically using a Landau-de Gennes free energy functional [25]. At sufficiently short lengthscales, dislocations do not perturb the stripe orientation and it is predicted that there is long range orientational order. The correlation function for layer normal orientation, nðrÞ; tends to a cutoff-dependent constant at large r [25], lim /nðrÞ nðr0 ÞS ¼ expðkB TL=2pBlÞ;
r-N
ð4Þ
where L is the cutoff, B is the elastic constant associated with stripe compression and l ¼ ðK=BÞ1=2 ; where K is a Frank elastic constant, corresponding to layer bending in two dimensions. However, at large lengthscales, the finite density of dislocations will lead to a two-dimensional nematic phase. It is predicted that for a length greater than xD BexpðED =2kB TÞ; where ED is the energy of an isolated dislocation, the correlation function for the layer normal angle (Eq. (3)) scales as [25] Gðr r0 ÞBrgðTÞ ;
ð5Þ
where gðTÞ ¼ 2kB T=pKðTÞ: The inset to Fig. 2 shows fits of Eq. (5) to Gðr r0 Þ computed from the simulations. The power law fit describes the decay of orientational correlations quite well, although an exponential function was found to give better fits. To determine K from the exponent g we could make use of the relationship between equilibrium noise and temperature, i.e. kB TBZ: However, the cell dynamics simulations do not produce equilibrium stripe patterns, rather steady-state trapped nonequilibrium ones as shown by the path dependence
114
S.R. Ren, I.W. Hamley / Physica B 322 (2002) 110–115
Fig. 5. Effect of oscillatory shear on a stripe pattern at t ¼ 0:35: (a) Initial labyrinthine stripe morphology (Z¼ 0), (b) after oscillatory shear, amplitude G ¼ 0:5; frequency o= 0.006 rad s-1, (c) interface orientation correlation function of pattern in (b). The distance r is in lattice units.
of the stripe orientation following different thermal quenches. As noted by Toner and Nelson [25], the correlation function, Eq. (5), can be used to analyze out of equilibrium systems such as Rayleigh–Benard convective rolls (described by the TDGL equation), however, it is necessary that the system explore all possible configurations and this not the case in the cell dynamics simulations, where in steady state the patterns are trapped. The effect of different thermal quenches on the stripe order is shown in Fig. 3. This indicates that quenching from the disordered state to higher temperatures in the ordered state (smaller t) leads to enhanced orientational correlations between stripes, which we ascribe to the local annealing out
of defects. However, thermal history has a pronounced effect, as apparent when comparing the patterns for a direct quench to t ¼ 0:3 and an indirect quench to the same temperature, following a lower temperature quench (to t ¼ 0:42). The corresponding normal angle correlation functions are shown in Fig. 4. These provide a quantitative measure of the changes in the orientational correlation between stripes. In particular, the difference in the correlation functions for the two paths to t ¼ 0:3 is apparent (the indirect quench produces a correlation function similar to that for a direct quench to t ¼ 0:35; intermediate between the two quench temperatures).
S.R. Ren, I.W. Hamley / Physica B 322 (2002) 110–115
The interface normal correlation function is also sensitive to the orientation of structures induced by shear. This is illustrated by the results from cell dynamics simulations shown in Fig. 5. Here, we used cell dynamics equations for two-dimensional systems under shear developed by Doi and coworkers [27,28]. An initial labyrinthine stripe pattern, with no global orientation (as confirmed by Fourier transformation) was subjected to oscillatory strain gðtÞ ¼ GsinðotÞ with G ¼ 0:5 and o ¼ 0:004 Hz: A steady state was reached after approximately 300,000 timesteps of the simulation and shearing was then stopped. The steady state structure after shearing shown in Fig. 5 shows a high degree of lamellar alignment. This is quantified by the corresponding interface orientation correlation function. In summary, cell dynamics simulations provide a powerful tool to probe the effect of thermal noise, temperature, and shear, all of which change the orientation of stripe patterns. An interface orientation correlation function quantifies the extent of local alignment, information that is lost in a Fourier transformation and is not properly quantified by the defect distribution in the case of a labyrinthine structure. However, due to the nonequilibrium nature of the simulations, it is not possible to determine an elastic constant for stripe bending. We have benefited from stimulating discussions with Peter Olmsted and Paulo Teixeira (Physics, Leeds). This work was supported by grant GR/ M08523 (ESPRC, UK).
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References [1] M. Seul, D. Andelman, Science 267 (1995) 476. [2] R.P. Huebener, Magnetic Flux Structures in Superconductors, Springer, Berlin, 1979. [3] M. Seul, R. Wolfe, Phys. Rev. A 46 (1992) 7519. [4] M. Seul, R. Wolfe, Phys. Rev. A 46 (1992) 7534. [5] M. Seul, L.R. Monar, L. O’Gorman, R. Wolfe, Science 254 (1991) 1616. [6] J. Zasadzinski, M.B. Schneider, J. Phys. (France) 48 (1987) 2001. [7] R.M. Weis, H.M. McConnell, J. Phys. Chem. 89 (1985) 4453. [8] M. Seul, V.S. Chen, Phys. Rev. Lett. 70 (1993) 1658. [9] Q. Ouyang, H.L. Swinney, Nature 352 (1991) 610. [10] M. Park, et al., Science 276 (1997) 1401. [11] R.W. Zehner, L.R. Sita, Langmuir 15 (1999) 6139. [12] I.I. Potemkin, et al., Langmuir 15 (1999) 7290. [13] P.G. de Gennes, C. Taupin, J. Phys. Chem. 86 (1982) 2294. [14] M. Bahiana, Y. Oono, Phys. Rev. A 41 (1990) 6763. [15] S. Puri, Y. Oono, Phys. Rev. A 38 (1988) 1542. [16] T. Taneika, Y. Shiwa, J. Phys.: Condens. Matter 9 (1997) L147. [17] S. Komura, J.-I. Fukuda, G.C. Paquette, Phys. Rev. E 53 (1996) 5588. [18] Y. Oono, S. Puri, Phys. Rev. A 38 (1988) 434. [19] Y. Oono, S. Puri, Phys. Rev. Lett. 58 (1987) 836. [20] A. Shinozaki, Y. Oono, Phys. Rev. E 48 (1993) 2622. [21] P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435. [22] P.I.C. Teixeira, B.M. Mulder, Phys. Rev. E 55 (1997) 3789. [23] I.W. Hamley, Macromol. Theory Simulation 9 (2000) 363. [24] R.C. Ball, R.L.H. Essery, J. Phys.: Condens. Matter 2 (1990) 10303. [25] D. Toner, D.R. Nelson, Phys. Rev. B 23 (1981) 316. [26] C. Harrison, et al., Macromolecules 33 (2000) 857. [27] T. Ohta, Y. Enomoto, J.L. Harden, M. Doi, Macromolecules 26 (1993) 4928. [28] H. Kodama, M. Doi, Macromolecules 29 (1996) 2652.
Orientational correlations in stripe patterns S.R. Ren, I.W. Hamley* School of Chemistry, University of Leeds, LS2 9JT Leeds, UK Received 3 April 2001; received in revised form 15 June 2001
Abstract Stripe phases are observed in many condensed matter systems where there is a competition between long- and shortrange ordering. Cell dynamics simulations are used to generate patterns in which the stripe orientation is enhanced by increasing thermal noise, decreasing temperature or applying shear. Although a steady state degree of order is reached, the non-equilibrium nature of the simulated patterns is demonstrated through the dependence of orientational correlation functions on thermal history. Nevertheless, the results show that the computed orientational correlation function provides a quantitative ‘‘fingerprint’’ that should enable the characterization of equilibrium stripe patterns as well as trapped non-equilibrium states. r 2002 Elsevier Science B.V. All rights reserved. PACS: 47.20.Hw; 83.50.v; 81.05.Lg; 66.30.Lw Keywords: Stripe patterns; Orientational ordering; Computer simulations; Cellular automata
Stripe patterns are observed for a range of condensed matter systems [1], including type I superconductor films [2], ferromagnetic garnet films [1,3–5], Langmuir monolayers of lipid molecules [1,6–8], stationary (Turing) patterns in reaction–diffusion systems [9], and lamellar diblock copolymer films [10–12]. Stripe patterns result from a competition between long-range ordering that favors separation into two distinct phases, and a local term that tends to maintain connectivity of alternating domains. Seul and co-workers have analyzed in detail stripe phases formed in magnetic garnet films in terms of topological defects, the density of which is much greater in the labyrinthine phase [4], in *Corresponding author. Tel.: +113-233-6430; fax: +113233-6565. E-mail address: [email protected] (I.W. Hamley).
which the stripes are orientationally disordered, than in more highly aligned stripe phases [3]. Here we introduce a method to characterize the orientational ordering of the interface at different lengthscales in the system. The approach was inspired by the analysis of the statistics of a random interface in a microemulsion [13]. Rather than analyzing experimental stripe patterns, we have used the cell dynamics simulation (CDS) method to generate them. This offers the advantage that the orientational ordering of the stripe interfaces can be controlled. An increased stripe orientation can be achieved by adding a noise-term into the kinetic equations. The CDS method is a cellular automaton, in which a timedependent spatially varying order parameter cðr; tÞ; is discretized on a lattice, taking values cðt; iÞ in cell i at time t: For the case of a nonconserved order parameter, the time evolution of the order parameter is represented in the form
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 6 0 5 - 1
S.R. Ren, I.W. Hamley / Physica B 322 (2002) 110–115
111
Fig. 1. Stripe patterns with different levels of thermal noise, generated by cell dynamics simulations for a quench to t=0.32. (a) Z=0, (b) Z=0.06, (c) Z=0.1. Inset: Fourier transform patterns.
developed by Bahiana and Oono [14]. cðt þ 1; iÞ ¼ f ðcðt; iÞÞ þ D½//cðt; iÞSS cðt; iÞ cðt; iÞ ¼ F ½cðt; iÞ: ð1Þ Here f is a function that represents the time evolution of the order parameter in a single cell. It has one hyperbolic source at the origin (corresponding to the high temperature c ¼ 0 state) and two hyperbolic sinks corresponding to the ordered state, ca0: The second term on the right-hand side of Eq. (1) allows for spatially cooperative dynamics via a diffusive mechanism due to the connectivity of cells. In Eq. (1), D is a positive constant that plays the role of a diffusion coefficient, and hhX ii X is the isotropized discrete Laplacian [15]. For a conserved order parameter, there should be no net change of order parameter in the region surrounding a given cell. Since the net gain of order parameter in a particular cell is given by F ½cðt; nÞ cðt; nÞ; the CDS model for a conserved order parameter becomes [14] cðt þ 1; iÞ ¼ cðt; iÞ þ F ½cðt; iÞ hhF ½cðt; iÞii Bcðt; iÞ þ Zzðt; iÞ; ð2Þ where B is a term that accounts for long-range ordering [14]. Eq. (2) includes a contribution from thermal noise, zðt; iÞ; of amplitude Z [16,17]. Although the CDS equations, Eqs. (1) and (2), are independent of any model for the free energy functional, it has been shown [18,19] that in the form Eq. (1), the CDS equations correspond to a coarse-grained discretization of the time-depen-
Fig. 2. Interface orientation correlation functions for the patterns in Fig. 1. The distance r is in lattice units. The solid lines indicate fits to Eq. (5). Inset: showing the power law fits on a double logarithmic scale.
dent Ginzburg–Landau equation (TDGL) equation. For a conserved order parameter, Eq. (2) represents the coarse-grained dynamics of the Cahn–Hilliard–Cook (CHC) equation [18–20]. In the theory of dynamic critical phenomena [21] the TDGL equation is also known as model A and the CHC equation is known as Model B. We generated stripe patterns by Cell dynamic simulations of the CHC equation on a 128 128 lattice, using the cell dynamics for the case of a
112
S.R. Ren, I.W. Hamley / Physica B 322 (2002) 110–115
Fig. 3. Stripe patterns corresponding to different effective temperatures t with Z=0.
conserved order parameter, Eq. (2). Calculations were performed for a two-dimensional phase of alternating A and B stripes, in which the volume fraction, fA ¼ fB ¼ 0:5: The map in Eq. (1) was taken to have the form f ðcÞ ¼ ð1 þ tÞc uc3 ; with u ¼ 0:5; and variable temperature-like parameter t: This form of map is described in detail elsewhere [22,23]. Following Bahiana and Oono [14], the parameters D ¼ 0:5 and B ¼ 0:02 were used to simulate a stripe pattern. To ensure approximate conservation of the order parameter when noise is added we have adapted the procedure described elsewhere [24]. Specifically, two independent Gaussian distributed random numbers xl ði; jÞ and xm ði; jÞ are generated at each lattice point ði; jÞ; representing fluctuations in
different directions. The noise term is then taken as Zxðr; tÞ ¼ Z½xl ði; jÞ xl ði71; jÞ þ xm ði; jÞ xm ði; j71Þ at each time step. A third set of uniformly distributed random numbers is used to determine whether ‘7’ takes ‘+’ or ‘’ in the calculation. The amplitude of the noise level is denoted Z (o1). Increasing its amplitude leads to a coarsening of the domain structure [16] and enhanced lamellar orientation due to local thermal annealing, up to some critical value of Z above which the system becomes disordered. To characterize the correlation of stripe orientation over different lengthscales, we compute the correlation function for the angle of the layer normal with respect to an arbitrary direction, fðrÞ; at the interface between A and B stripes. The
S.R. Ren, I.W. Hamley / Physica B 322 (2002) 110–115
Fig. 4. Orientation correlation functions corresponding to the patterns in Fig. 3.
appropriate correlation function for a two-dimensional smectic (stripe) phase is [25] Gðr r Þ ¼ /e 0
2ifðrÞ 2ifðr0 Þ
e
S ¼ /2 cos y 1S; ð3Þ 2
where y ¼ fðrÞ fðr0 Þ: The interface between stripes was located by thresholding the stripe pattern, to produce a step discontinuity in order parameter from A to B stripe. This discontinuity defined the cells at the interface, and the (discrete) normal (orientation angle y ¼ n p=4; n ¼ 0; 1; y; 7) was computed as the vector between adjacent cells at the interface. The correlation function, Eq. (3), has been used to analyze orientational correlations between stripes formed in block copolymer films [26]. The cell dynamic simulations were run until the average order parameter in the system was constant for at least 5000 timesteps. All data shown are averages from seven independent runs. Patterns generated from the CDS simulations with different levels of thermal noise are shown in Fig. 1. From inspection, it is apparent that the mutual orientation of stripes increases as the noise amplitude is increased. However, this is not reflected in the Fourier transforms shown in the insets. This is because the Fourier transform depends on the global alignment of the pattern, and is not sensitive to local orientation. For many
113
stripe patterns where there is no long-range orientational order, Fourier transformation is thus an inappropriate measure of alignment. Furthermore, the labyrinthine nature of the structure prevents unambiguous identification of defects, which thus do not provide a suitable description of the pattern. However, the interface normal correlation function, Eq. (3), is strongly dependent on local stripe orientation, as shown in Fig. 2. This shows that Gðr r0 Þ increases monotonically with Z for small r ¼ jrFr0 j: For large r; Gðr r0 Þ approaches zero, its value for random stripe orientation. Orientational correlations in the stripe phase have been investigated theoretically using a Landau-de Gennes free energy functional [25]. At sufficiently short lengthscales, dislocations do not perturb the stripe orientation and it is predicted that there is long range orientational order. The correlation function for layer normal orientation, nðrÞ; tends to a cutoff-dependent constant at large r [25], lim /nðrÞ nðr0 ÞS ¼ expðkB TL=2pBlÞ;
r-N
ð4Þ
where L is the cutoff, B is the elastic constant associated with stripe compression and l ¼ ðK=BÞ1=2 ; where K is a Frank elastic constant, corresponding to layer bending in two dimensions. However, at large lengthscales, the finite density of dislocations will lead to a two-dimensional nematic phase. It is predicted that for a length greater than xD BexpðED =2kB TÞ; where ED is the energy of an isolated dislocation, the correlation function for the layer normal angle (Eq. (3)) scales as [25] Gðr r0 ÞBrgðTÞ ;
ð5Þ
where gðTÞ ¼ 2kB T=pKðTÞ: The inset to Fig. 2 shows fits of Eq. (5) to Gðr r0 Þ computed from the simulations. The power law fit describes the decay of orientational correlations quite well, although an exponential function was found to give better fits. To determine K from the exponent g we could make use of the relationship between equilibrium noise and temperature, i.e. kB TBZ: However, the cell dynamics simulations do not produce equilibrium stripe patterns, rather steady-state trapped nonequilibrium ones as shown by the path dependence
114
S.R. Ren, I.W. Hamley / Physica B 322 (2002) 110–115
Fig. 5. Effect of oscillatory shear on a stripe pattern at t ¼ 0:35: (a) Initial labyrinthine stripe morphology (Z¼ 0), (b) after oscillatory shear, amplitude G ¼ 0:5; frequency o= 0.006 rad s-1, (c) interface orientation correlation function of pattern in (b). The distance r is in lattice units.
of the stripe orientation following different thermal quenches. As noted by Toner and Nelson [25], the correlation function, Eq. (5), can be used to analyze out of equilibrium systems such as Rayleigh–Benard convective rolls (described by the TDGL equation), however, it is necessary that the system explore all possible configurations and this not the case in the cell dynamics simulations, where in steady state the patterns are trapped. The effect of different thermal quenches on the stripe order is shown in Fig. 3. This indicates that quenching from the disordered state to higher temperatures in the ordered state (smaller t) leads to enhanced orientational correlations between stripes, which we ascribe to the local annealing out
of defects. However, thermal history has a pronounced effect, as apparent when comparing the patterns for a direct quench to t ¼ 0:3 and an indirect quench to the same temperature, following a lower temperature quench (to t ¼ 0:42). The corresponding normal angle correlation functions are shown in Fig. 4. These provide a quantitative measure of the changes in the orientational correlation between stripes. In particular, the difference in the correlation functions for the two paths to t ¼ 0:3 is apparent (the indirect quench produces a correlation function similar to that for a direct quench to t ¼ 0:35; intermediate between the two quench temperatures).
S.R. Ren, I.W. Hamley / Physica B 322 (2002) 110–115
The interface normal correlation function is also sensitive to the orientation of structures induced by shear. This is illustrated by the results from cell dynamics simulations shown in Fig. 5. Here, we used cell dynamics equations for two-dimensional systems under shear developed by Doi and coworkers [27,28]. An initial labyrinthine stripe pattern, with no global orientation (as confirmed by Fourier transformation) was subjected to oscillatory strain gðtÞ ¼ GsinðotÞ with G ¼ 0:5 and o ¼ 0:004 Hz: A steady state was reached after approximately 300,000 timesteps of the simulation and shearing was then stopped. The steady state structure after shearing shown in Fig. 5 shows a high degree of lamellar alignment. This is quantified by the corresponding interface orientation correlation function. In summary, cell dynamics simulations provide a powerful tool to probe the effect of thermal noise, temperature, and shear, all of which change the orientation of stripe patterns. An interface orientation correlation function quantifies the extent of local alignment, information that is lost in a Fourier transformation and is not properly quantified by the defect distribution in the case of a labyrinthine structure. However, due to the nonequilibrium nature of the simulations, it is not possible to determine an elastic constant for stripe bending. We have benefited from stimulating discussions with Peter Olmsted and Paulo Teixeira (Physics, Leeds). This work was supported by grant GR/ M08523 (ESPRC, UK).
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