CARBON PERGAMON Carbon40(2002)2647-2660 Theory and simulation of texture formation in mesophase carbon fib J.Yan,AD.Rey* Received 6 March 2002: accepted 8 May 2002 Abstract Carbonaceous mesophase precursors are spun into high-performance commercial carbon fibers using the standard melt spinning process. The spinning process produces a wide range of cross-sectional fiber textures whose origins are not currently well understood. The planar polar(PP)and planar radial(PR) textures are two frequently observed textures. This paper presents theory and simulations of the elasticity-driven formation process of the Pp texture using the classical Landau-de gennes mesoscopic theory for discotic liquid crystals, including defect nucleation, defect migration, and overall texture geometry. The main characteristic of the real PP texture is the presence of a pair of defects equidistant from the fiber xis. In this research it is analytically and numerically found that, under elastic isotropy, the ratio of the equilibrium defect-defect separation distance to the fiber diameter is always equal to 1/V5. The computed PP and PR textures phase diagram, given in terms of temperature and fiber radius, is used to establish the processing conditions and geometric factors that lead to the selection of these two textures o 2002 Published by Elsevier Science Ltd Keywords: A. Carbon fibers, Mesophase pitch; C Modeling 1. Introduction h as coal tar and petro- leum pitches, are used in the industrial manufacturing of mesophase carbon fiber [1]. This relatively new carbon fiber is more competitive than the conventional fiber mad from acrylic precursors in several application areas [1] he thermodynamic phase that describes carbonaceous Liquid crystals are intermediate (ie. mesophase) phases, typically found for anisodiametric organic molecules hich exist between the higher temperature isotropic liquid Uk state and the lower temperature crystalline state. Car bonaceous mesophases are composed of disk -like mole cules. Fig. I shows the molecular geometry, positional disorder. and uniaxial orientational order of discotic atic liquid crystals. The partial orientational order of nolecular unit normal u is along the average orienta- or director n(nn=1). The name discotic dis- Fig. 1. Definition irector orientation of a uniaxial discotic orresponding author. Tel +1-514-398-4196; fax: +1-514- nematic liquid crystalline material. The director n is the average 398-6678 orientation of the unit normals to the disk -like molecules in a E-mail address. alejandro. rey(@mcgill. ca(A D. Rey) discotic nematic phase. 0008-6223/02/S-see front matter 2002 Published by Elsevier Science Ltd PII:S0008-6223(02)00166-5
Carbon 40 (2002) 2647–2660 T heory and simulation of texture formation in mesophase carbon fibers J. Yan, A.D. Rey* Department of Chemical Engineering, McGill University, 3610 University St., Montreal, Quebec H3A 2B2, Canada Received 6 March 2002; accepted 8 May 2002 Abstract Carbonaceous mesophase precursors are spun into high-performance commercial carbon fibers using the standard melt spinning process. The spinning process produces a wide range of cross-sectional fiber textures whose origins are not currently well understood. The planar polar (PP) and planar radial (PR) textures are two frequently observed textures. This paper presents theory and simulations of the elasticity-driven formation process of the PP texture using the classical Landau–de Gennes mesoscopic theory for discotic liquid crystals, including defect nucleation, defect migration, and overall texture geometry. The main characteristic of the real PP texture is the presence of a pair of defects equidistant from the fiber axis. In this research it is analytically and numerically found that, under elastic isotropy, the ratio of the equilibrium 4 ] defect–defect separation distance to the fiber diameter is always equal to 1/ 5. The computed PP and PR textures phase Œ diagram, given in terms of temperature and fiber radius, is used to establish the processing conditions and geometric factors that lead to the selection of these two textures. 2002 Published by Elsevier Science Ltd. Keywords: A. Carbon fibers, Mesophase pitch; C. Modeling 1. Introduction Carbonaceous mesophases, such as coal tar and petroleum pitches, are used in the industrial manufacturing of mesophase carbon fiber [1]. This relatively new carbon fiber is more competitive than the conventional fiber made from acrylic precursors in several application areas [1]. The thermodynamic phase that describes carbonaceous mesophases is the discotic nematic liquid crystal state [2]. Liquid crystals are intermediate (i.e. mesophase) phases, typically found for anisodiametric organic molecules, which exist between the higher temperature isotropic liquid state and the lower temperature crystalline state. Carbonaceous mesophases are composed of disk-like molecules. Fig. 1 shows the molecular geometry, positional disorder, and uniaxial orientational order of discotic nematic liquid crystals. The partial orientational order of the molecular unit normal u is along the average orientation or director n (n ? n 5 1). The name discotic disFig. 1. Definition of director orientation of a uniaxial discotic *Corresponding author. Tel.: 11-514-398-4196; fax: 11-514- nematic liquid crystalline material. The director n is the average 398-6678. orientation of the unit normals to the disk-like molecules in a E-mail address: alejandro.rey@mcgill.ca (A.D. Rey). discotic nematic phase. 0008-6223/02/$ – see front matter 2002 Published by Elsevier Science Ltd. PII: S0008-6223(02)00166-5
2648 J. Yan, A.D. Rey / Carbon 40 (2002)2647-2660 tinguishes the molecular geometry and the name nematic fiber process-induced structuring and cross-sectional fiber identifies the type of liquid crystalline orientational order. textures'selection. When considering elastic mechanisms The industrial fabrication of mesophase carbon fiber it is necessary to identify the three fundamental elastic using the conventional melt spinning process typically modes of these materials. Fig. 2 shows the three types of produces micrometer-sized cylindrical filaments whose elastic deformation, splay, twist, and bend, and their cross-sectional area displays a variety of transverse tex- corresponding modulus Ku, K22, and K33, known as Frank tures [3], that is, different spatial arrangements of the elasticity constants [6]. The Frank elastic (long range) average orientation n on the plane perpendicular to the energy density f for uniaxial nematic liquid crystals is fiber axis. The selection mechanisms that drive the texture defined by [6] ormation pattern during fiber spinning are at present not ompletely understood, but due to the strong structure- f=vn)2+ n×vm)+-2|nxv×n properties correlations, they are essential for product optimization. Significant progress in the fundamental derstanding of structure formation of mesophase fibers has been presented [4, 5]. The fiber structure development is The type of liquid crystal elasticity is known as orientation the result of the application of a extreme elasticity and is the analogue of Hooke' s elasticity of complex stress and thermal fields on a complex textured isotropic materials In liquid crystals the strains are due to anisotropic viscoelastic material, The scope of the present spatial orientation gradients, and are analogous to positi aper is the characterization of temperature, geometry, and al displacements in isotropic materials, and the the Frank elastic anisotropy effects on texture selection when elasti elastic constants play the role of Hooke's modulus of ffects are dominant, and viscous effects are neglected sotropic materials. The reason that there are three different he idea behind the use of this simplifying assumpti constants is that the material is anisotropic, and different (i.e. neglection of viscous effects)is that an understanding directions exhibit different degrees of distortions under of elastic effects is a pre-requisite to understand the more applied loads, as in composites materials. Assuming elastic complex viscoelastic response. In addition, if this kind of isotropy(one constant approximation),K=Kn=K modeling shows that elastic effects produce microstruc- K33, the Frank energy simplifies to [6 tures compatible with those observed in the real world, one may then be able to conclude that viscous effects reinforce f=I(vn)+lv xnIl the emergence of the textures selected by elasticity Discotic nematic liquid crystals, such as carbonaceous which, due to its simplicity, is widely used to perform mesophases, are anisotropic visco-elastic materials, whose analytical calculations(6]. We note that, in reality the melt spinning of carbonaceous mesophases is to determine how elastic and viscous mechanisms affect the (3a) 92 30 K K Fig. 2. Schematics of the elastic splay (left), twist(center), and bend (night) deformation for uniaxial discotic nematics. Note that the splay ( bend)mode involves bending(splaying)of the disks trajectories, in contrast to the case of uniaxial rod-like nematics. A disk trajectory is a curve locally orthogonal to the director. Adapted from Ref. [5]
2648 J. Yan, A.D. Rey / Carbon 40 (2002) 2647–2660 tinguishes the molecular geometry and the name nematic fiber process-induced structuring and cross-sectional fiber identifies the type of liquid crystalline orientational order. textures’ selection. When considering elastic mechanisms, The industrial fabrication of mesophase carbon fiber it is necessary to identify the three fundamental elastic using the conventional melt spinning process typically modes of these materials. Fig. 2 shows the three types of produces micrometer-sized cylindrical filaments whose elastic deformation, splay, twist, and bend, and their cross-sectional area displays a variety of transverse tex- corresponding modulus K , K , and K , known as Frank 11 22 33 tures [3], that is, different spatial arrangements of the elasticity constants [6]. The Frank elastic (long range) average orientation n on the plane perpendicular to the energy density f for uniaxial nematic liquid crystals is n fiber axis. The selection mechanisms that drive the texture defined by [6] formation pattern during fiber spinning are at present not completely understood, but due to the strong structure– ]] ] KK K 11 22 33 22 2 f 5 (\ ? n) 1 (n 3 \ ? n) 1 un 3 \ 3 nu n 22 2 properties correlations, they are essential for product optimization. Significant progress in the fundamental un- (1) derstanding of structure formation of mesophase fibers has been presented [4,5]. The fiber structure development is The type of liquid crystal elasticity is known as orientation elasticity and is the analogue of Hooke’s elasticity of the result of the application of a series of extremely complex stress and thermal fields on a complex textured isotropic materials. In liquid crystals the strains are due to spatial orientation gradients, and are analogous to position- anisotropic viscoelastic material. The scope of the present paper is the characterization of temperature, geometry, and al displacements in isotropic materials, and the the Frank elastic constants play the role of Hooke’s modulus of elastic anisotropy effects on texture selection when elastic isotropic materials. The reason that there are three different effects are dominant, and viscous effects are neglected. The idea behind the use of this simplifying assumption constants is that the material is anisotropic, and different directions exhibit different degrees of distortions under (i.e. neglection of viscous effects) is that an understanding of elastic effects is a pre-requisite to understand the more applied loads, as in composites materials. Assuming elastic complex viscoelastic response. In addition, if this kind of isotropy (one constant approximation), K 5 K11 22 5 K 5 modeling shows that elastic effects produce microstruc- K , the Frank energy simplifies to [6] 33 tures compatible with those observed in the real world, one may then be able to conclude that viscous effects reinforce K 2 2 f 5 ][(\ ? n) 1 u\ 3 nu ] (2) n 2 the emergence of the textures selected by elasticity. Discotic nematic liquid crystals, such as carbonaceous which, due to its simplicity, is widely used to perform mesophases, are anisotropic visco-elastic materials, whose analytical calculations [6]. We note that, in reality, for properties depend on the average molecular orientation. As disc-like liquid crystals, the twist constant (K ) is greater 22 mentioned above a question of fundamental importance to than the splay (K ) and bend (K ) constants [7]: 11 33 the melt spinning of carbonaceous mesophases is to determine how elastic and viscous mechanisms affect the K . K (3a) 22 11 Fig. 2. Schematics of the elastic splay (left), twist (center), and bend (right) deformation for uniaxial discotic nematics. Note that the splay (bend) mode involves bending (splaying) of the disk’s trajectories, in contrast to the case of uniaxial rod-like nematics. A disk trajectory is a curve locally orthogonal to the director. Adapted from Ref. [5]
J. Yan, A.D. Rey / Carbon 40 (2002)2647-2660 Note that, in contrast to rod-like nematics, for disc-like c. Theory and simulation of liquid crystalline materials K22 ontinues to be performed using macroscopic, mesoscopic nematics the bending disc's trajectories give rise to a splay and molecular models [6]. Macroscopic models based on the Leslie-Ericksen director equations are unsuitable to to a bend deformation; by disc trajectory it means the simulate texture formation because defects are singularitie curve locally orthogonal to the director. The Frank moduli in the orientation field [6]. On the other hand, mesoscopic are functions of temperature and have units of energy per models based on the second moment of the orientation unit length. Heating up a discotic mesophase above the nematic-isotropic transition temperature, it is found that crystalline textures, because defects are non-singular solu- K1=K22=K3=0, that is, the Frank elasticity is due to tions to the governing equations. A very well established orientational liquid crystalline order [4] mesoscopic model in liquid crystalline materials is based It is known [3] that the observed cross-section fiber on the Landau-de gennes free energy [6] and is adopted textures belong to a number of families, such as onion, and used in this work. Related work on fiber structure is radial, mixed, PAN-AM, to name a few. Fig 3 shows the given in [8,9 schematics of two cross-sectional textures most commonly The objectives of this paper are: seen in mesophase carbon fibers. The dashed line indicates the trajectories of the molecular planes,(a) shows the 1. to simulate the transient formation of the planar polar slanar radial(PR)texture, in which only the pure bend texture that is commonly observed during the melt mode exists with one defect in the center of strength +I spinning of carbonaceous mesophase, and(b)shows the planar polar(PP)texture, in which two 2. to characterize the elastic driving forces that promote modes of deformation, splay and bend, couple in the system with two defects of strength +1/2. The defects in the selection of the planar polar texture these textures arise because, in a cylindrical geometry, it is 3. to provide a full geometric characterization of the anar polar textures in terms of defect locations impossible to tangentially align the directors at the surface 4. to present and discuss the planar radial-planar polar without introducing singularities. Defects are singularities in the director field and are characterized by strength ber texture phase diagram, given in terms of tempera- (1/2, 1,.) and sign(+)6. The strength of a discina- ture and fiber radius, and to establish the geometric and tion determines the amount of orientation distortion and operating conditions that lead to the characteristic the sign corresponds to the sense (i.e. clockwise or anti- textures clockwise) of orientation rotation while circling the de- fects. Since the energy of a defect scales with the square of This paper is organized as follows. Section 2 presents defect strength [6], the planar polar texture would seem to the theory and the landau-de gennes governing equa- merge so as to minimize the elastic energy associated tions. Section 3 presents an analytic geometric analysis of with defect distortions. In addition, defects of equal sign the planar polar texture that yields closed form results repel each other, while defects of different sign attract. As Section 4 shows the numerical solutions of our model that shown below. in the pp texture defect-defect interaction erify the analytical predictions made in Section 3, and plays a critical role in the geometry of the texture. also discusses the characteristics of the texture evolution Fig 3. Schematics of transverse textures of actual mesophase carbon fibers. (a) The planar radial(PR)texture, in which the pure bend mode K3)exists with one defect in the center of strength +1.(b)The planar polar(PP)texture, in which two modes of deformation, splay(K,) and bend (K,), couple in the system with two defects of strength +1/2
J. Yan, A.D. Rey / Carbon 40 (2002) 2647–2660 2649 K . K (3b) Theory and simulation of liquid crystalline materials 22 33 continues to be performed using macroscopic, mesoscopic, Note that, in contrast to rod-like nematics, for disc-like and molecular models [6]. Macroscopic models based on nematics the bending disc’s trajectories give rise to a splay the Leslie–Ericksen director equations are unsuitable to deformation, and the splaying disc’s trajectories give rise simulate texture formation because defects are singularities to a bend deformation; by disc trajectory it means the in the orientation field [6]. On the other hand, mesoscopic curve locally orthogonal to the director. The Frank moduli models based on the second moment of the orientation are functions of temperature and have units of energy per distribution function are well suited to capture liquid unit length. Heating up a discotic mesophase above the crystalline textures, because defects are non-singular solu- nematic–isotropic transition temperature, it is found that K 5 K 5 K 5 0, that is, the Frank elasticity is due to tions to the governing equations. A very well established 11 22 33 orientational liquid crystalline order [4]. mesoscopic model in liquid crystalline materials is based It is known [3] that the observed cross-section fiber on the Landau–de Gennes free energy [6] and is adopted textures belong to a number of families, such as onion, and used in this work. Related work on fiber structure is radial, mixed, PAN-AM, to name a few. Fig. 3 shows the given in [8,9] schematics of two cross-sectional textures most commonly The objectives of this paper are: seen in mesophase carbon fibers. The dashed line indicates the trajectories of the molecular planes, (a) shows the 1. to simulate the transient formation of the planar polar planar radial (PR) texture, in which only the pure bend texture that is commonly observed during the melt mode exists with one defect in the center of strength 11, spinning of carbonaceous mesophase; and (b) shows the planar polar (PP) texture, in which two 2. to characterize the elastic driving forces that promote modes of deformation, splay and bend, couple in the the selection of the planar polar texture; system with two defects of strength 11/2. The defects in 3. to provide a full geometric characterization of the these textures arise because, in a cylindrical geometry, it is planar polar textures in terms of defect locations; impossible to tangentially align the directors at the surface 4. to present and discuss the planar radial–planar polar without introducing singularities. Defects are singularities fiber texture phase diagram, given in terms of tempera- in the director field and are characterized by strength ture and fiber radius, and to establish the geometric and (1/2,1, . . . ) and sign (6) [6]. The strength of a disclina- operating conditions that lead to the characteristic tion determines the amount of orientation distortion and textures. the sign corresponds to the sense (i.e. clockwise or anticlockwise) of orientation rotation while circling the defects. Since the energy of a defect scales with the square of This paper is organized as follows. Section 2 presents defect strength [6], the planar polar texture would seem to the theory and the Landau–de Gennes governing equaemerge so as to minimize the elastic energy associated tions. Section 3 presents an analytic geometric analysis of with defect distortions. In addition, defects of equal sign the planar polar texture that yields closed form results. repel each other, while defects of different sign attract. As Section 4 shows the numerical solutions of our model that shown below, in the PP texture defect–defect interaction verify the analytical predictions made in Section 3, and plays a critical role in the geometry of the texture. also discusses the characteristics of the texture evolution Fig. 3. Schematics of transverse textures of actual mesophase carbon fibers. (a) The planar radial (PR) texture, in which the pure bend mode (K ) exists with one defect in the center of strength 11. (b) The planar polar (PP) texture, in which two modes of deformation, splay (K ) 3 1 and bend (K ), couple in the system with two defects of strength 11/2. 3
2650 J. Yan, A.D. Rey / Carbon 40 (2002)2647-2660 and the texture phase diagram. Finally, conclusions are harmonics are orthogonal completely symmetric surface tensors, given by 后=1, 2. Theory and governing equations In this section, we present the Landau-de gennes theory 4-元(5+山++,叫 for nematic liquid crystals [6], and the parametric equa- tions used to describe mesophase fiber texture formation +H4}+3×76+5,+A As mentioned above, this classical [6] liquid crystal theory is well suited to simulate texture formation since defects Expanding f(m) as a Fourier series are non-singular solutions to the governing equations 3×5 Q 3×5×7×9 4丌×2×3×4 2. 1. Definition of orientation and alignment The Landau-de gennes quid crystals [6] where the coefficients of the Fouri ner ex describes the viscoelastic of nematic liquid 2, 2,... are symmetric and traceless tensors, and where rystals using the second of the orientation the numerical coefficients(1/4,.)are used to normal- distribution function, known as the tensor order parameter ize the @'s. The coefficients are found using the principle @x,n), and the velocity field u(x ) The tensor order of orthogonality, as used in any Fourier series expan parameter field @(r, n)and the velocity field v(x, n) have For example, to find 2- we dot f(u) with/ to obtain independent origins. In the absence of macroscopic flow, U=0, the viscoelasticity of liquid crystals is described by 2=0= fu dA= u( Qlr, t ). This means that spatio-temporal changes in the order parameter may exist even in the absence of flow. this paper macroscopic flow does not occur, U=0, and the (8) state of the liquid crystal is defined solely by o(x, 1) Numerous examples of viscoelastic property measurements where for simplicity we define 0=0. In the Landau-de and phenomena involving spatio-temporal changes in the Gennes theory the description of the nematic microstruc- absence of filow are found in the liquid crystal literature ture is limited to the second-order term 0, while the [6 higher-order terms are neglected. Thus this theory contains We next explain the nature, origin, and physical signifi- an approximation since information residing in higher cance of the tensor order parameter @. To characterize the order terms is not accounted for rientation in a discotic nematic liquid crystal we use the The second-order symmetric and traceless tensor order orientation distribution function(ODF)f(m), which gives parameter e [6] is efficiently expressed as the probability of finding a disc unit normal I with orientation between u and u +du. Since u is a unit vector, 2=S(nn-38)+P(mm-) all its possible orientations are contained in the unit sphere, where the following restrictions apply fu)=fu)dA=I tr(o)=0 Since u is equivalent to -l, to describe f() we must use even products: uu,uuul,.... To expand a function f(u) of a unit vector u we car Fourier series of orthogonal basis functions fo u,..., known as surface spherical harmonics. The products obey n·n=m·m=l·l=1 (9f dA=4,audA=8, Huu dA nn+mm+ll=s 3X1(68+I+I) Equivalently, the symmetric traceless tensor order parame- where (Oo)mn=8.S, I=8.8, and I ter Q can be written as an expansion of its eigenvectors which are used in expanding f(u). The surface spherical Q=A, nn+u mm+All 10a)
2650 J. Yan, A.D. Rey / Carbon 40 (2002) 2647–2660 and the texture phase diagram. Finally, conclusions are harmonics are orthogonal completely symmetric surface presented. tensors, given by dij 2 4 f 5 1, f 5 u u 2 ], f o ij i j ijkl 3 2. Theory and governing equations 1 5 uuuu i j k l ij k l ik j l il j k jk i l 2 ]hd u u 1d u u 1d u u 1d u u 7 In this section, we present the Landau–de Gennes theory for nematic liquid crystals [6], and the parametric equa- 1 1d u u j 1 ]]hd d 1d d 1d d j (6) jl i k ij kl ik jl il jk tions used to describe mesophase fiber texture formation. 5 3 7 As mentioned above, this classical [6] liquid crystal theory Expanding f(u) as a Fourier series: is well suited to simulate texture formation since defects are non-singular solutions to the governing equations. 1 3 3 5 3 3 5 3 7 3 9 f(u) 5 ]1 ? f 1 ]]Q f 22 44 1 ]]]]Q f o ij ij ij ij 4p 4p 3 2 4p 3 2 3 3 3 4 2 .1. Definition of orientation and alignment 1??? (7) The Landau–de Gennes theory of liquid crystals [6] where the coefficients of the Fourier expansion, 2 4 describes the viscoelastic behavior of nematic liquid Q ,Q , . . . are symmetric and traceless tensors, and where crystals using the second moment of the orientation the numerical coefficients (1/4p, . . . ) are used to normaldistribution function, known as the tensor order parameter ize the Q’s. The coefficients are found using the principle Q of orthogonality, as used in any Fourier series expansion. (x,t), and the velocity field v(x,t). The tensor order 2 2 parameter field Q(x,t) and the velocity field v(x,t) have For example, to find Q we dot f(u) with f to obtain independent origins. In the absence of macroscopic flow, 2 2 d v 5 0, the viscoelasticity of liquid crystals is described by Q ; Q 5E f(u)f dA 5E f(u)Suu 2 ]D dA 3 Q(x,t). This means that spatio-temporal changes in the 2 2 Z Z order parameter may exist even in the absence of flow. In d this paper macroscopic flow does not occur, v 5 0, and the 5 K L uu 2 ] (8) 3 state of the liquid crystal is defined solely by Q(x,t). Numerous examples of viscoelastic property measurements 2 where for simplicity we define Q ; Q . In the Landau–de and phenomena involving spatio-temporal changes in the Gennes theory the description of the nematic microstrucabsence of flow are found in the liquid crystal literature ture is limited to the second-order term Q, while the [6]. higher-order terms are neglected. Thus this theory contains We next explain the nature, origin, and physical signifi- an approximation since information residing in highercance of the tensor order parameter Q. To characterize the order terms is not accounted for. orientation in a discotic nematic liquid crystal we use the The second-order symmetric and traceless tensor order orientation distribution function (ODF) f(u), which gives parameter Q [6] is efficiently expressed as the probability of finding a disc unit normal u with 1 1 orientation between u Q 5 S(nn 2 ] ] d ) 1 P(mm 2 ll) (9a) and u 1 du. Since u is a unit vector, 3 3 all its possible orientations are contained in the unit sphere, 2 where the following restrictions apply denoted by Z . The ODF is normalized: T Q 5 Q (9b) f(u) 5E f(u) dA 5 1 (4) 2 tr(Q) 5 0 (9c) Z Since u is equivalent to 2 u, to describe f(u) we must use 1 2 #] S # 1 (9d) 2 even products: uu,uuuu, . . . . To expand a function f(u) of a unit vector u we can use a Fourier series of orthogonal 3 3 2 4 2 #] ] P # (9e) basis functions 2 2 f , f , f , . . . , known as surface spherical o ij ijkl harmonics. The products obey n ? n 5 m ? m 5 l ? l 5 1 (9f) 4p E dA 5 4p, E uu dA 5 ]d, E uuuu dA 100 3 ZZ Z 22 2 nn 1 mm 1 ll 5d 5F G 0 1 0 (9g) 001 ] 4p 1 5 ](dd 1 I 1 I ), . . . (5) 3 3 15 Equivalently, the symmetric traceless tensor order parame- 1 ter Q can be written as an expansion of its eigenvectors: where (dd ) 5d d , I 5d d , and I 5d d , mnpq mn pq mnpq mq np mp nq which are used in expanding f(u). The surface spherical Q 5 m nn 1 m mm 1 m ll (10a) nm l
J. Yan, A.D. Rey / Carbon 40 (2002)2647-2660 Hn+μn+p=0 (10b) where U is the nematic potential, which is related to the where the uniaxial director n corresponds to the maximum temperature in a thermotropic liquid crystal, and c, k, T are the number density of the discs, the boltzmanns eigenvalue p-3s, the biaxial director m corresponds the constant, and an absolute reference temperature just below second largest eigenvalue 3(S-P), and the sec- the isotropic-nematic phase transition temperature,respec- ond biaxial director /(=X m) corresponds to the tively. The bare elastic constants L, and i, are known Landau coefficients. To relate the Landau coefficients defined completely by the orthogonal director triad (n, m/ appearing in Eq. (8)to the previously defined Frank elastic The magnitude of the uniaxial scalar order parameter S is constants for uniaxial liquid crystals appearing in Eq (1), the molecular alignment along the uniaxial director n, and we restrict the tensor order parameter Q to its uniaxial scalar order parameter P is the molecular alignment in a form @=Sea(nn-173)with S equal to its equilibrium plane perpendicular to the direction of the uniaxial director spatially homogeneous value Seq(see Eq.(22)). In the n,and is given by P=,(m@ m-1 2. 1.On the principal axes, the tensor order parameter g is represented M =Se(L+3L,(v n)+L,(n X v. n) (L1+L2)m S+P)0 Comparing equal terms in Eqs. (1)and (14) the relations between L, and L and the frank's constants of uniaxial LCs are [13] where both S and P are positive for normal disc-like The Landau-de Gennes model uses the tensor order (11), the model is able to describe biaxial (S+0, P+0), 42=2 (15b) uniaxial(S+0, P=0), and isotropic (S=0, P=0)states IsotropIc state is the zero tensor: 0=0. Defects The Landau coefficients L, and L, are bare elastic con- regions of molecular size in which orientational order(S, P stants, independent of temperature. On the other hand, the sharply decrease. These localized disordered regions are in principle captured by mesoscopic models since Q remains Frank elastic constants are temperature dependent, since well behaved the scalar order parameter S is a function of temperature [6]. Eq.(14)implies that, when using Eq.(13b), the elastic anisotropy restrictions K=K,=K*K, apply. Thermo- 2. 2. Landau-de gennes mesoscopic model for liquid dynamic stability restrictions impose the following crystalline materials According to the Landau-de Gennes model, the bulk L1>0 (16a) energy density of nematic liquid crystals(NLC)in the absence of external fields is given by [10] (16b) f=+ geometry involved in the discotic ne where is the short-range energy density, which is that [12] responsible for the nematic-isotropic phase transition, and L2<0 f is the elastic free energy density [6], which contains range gradient contributions to the system. The Using the classical relaxation of free energy model, the nsionless free energy densities f and f of thermot- time-dependent equation in terms of o and v@ is found liquid crystals, in terms of the second-order tensor 2, to be [14] are given by (18) 1=((1-3)g-3g where [s indicates symmetric and traceless, yo)is a phenomenological kinetic coefficient, and 8F/8Q is the +0(0. 0) (13a) functional derivative of the total energy F. Eq (18)is five coupled non-linear parabolic reaction-diffusion equations f-2ckTsl 2 (v@)]+ ckt=(v2). (v 2) for the five independent components of 0: @xx,Owm,@. O,O. Substituting Eq(13)into Eq.(18) yields following governing equations of o(x, 1)[10
J. Yan, A.D. Rey / Carbon 40 (2002) 2647–2660 2651 m 1 m 1 m 5 0 (10b) where U is the nematic potential, which is related to the nml temperature in a thermotropic liquid crystal, and c, k, T * where the uniaxial director n corresponds to the maximum are the number density of the discs, the Boltzmann’s 2 eigenvalue m 5 ] S, the biaxial director m corresponds the n 3 constant, and an absolute reference temperature just below 1 second largest eigenvalue m 5 2 ] (S 2 P), and the sec- m 3 the isotropic–nematic phase transition temperature, respec- ond biaxial director l (5 n 3 m) corresponds to the tively. The bare elastic constants L and L are known as 1 2 1 smallest eigenvalue m 5 2 ] (S 1 P). The orientation is l 3 Landau coefficients. To relate the Landau coefficients defined completely by the orthogonal director triad (n,m,l). appearing in Eq. (8) to the previously defined Frank elastic The magnitude of the uniaxial scalar order parameter S is constants for uniaxial liquid crystals appearing in Eq. (1), the molecular alignment along the uniaxial director n, and we restrict the tensor order parameter Q to its uniaxial 3 is given by S 5 ] (n ? Q ? n). The magnitude of the biaxial 2 form Q 5 Seq(nn 2 I/3) with S equal to its equilibrium scalar order parameter P is the molecular alignment in a spatially homogeneous value S (see Eq. (22)). In the eq plane perpendicular to the direction of the uniaxial director uniaxial state, Eq. (13b) then becomes 3 n, and is given by P 5 ] (m ? Q ? m 2 l ? Q ? l). On the 2 2 22 1 principal axes, the tensor order parameter Q is represented f 5 S h(L 1 ]L )(\ ? n) 1 L (n 3 \ ? n) n eq 1 2 1 2 as 1 2 1 (L 1 ]L )un 3\ 3 nu j (14) 1 2 2 1 2 ] (S 2 P)0 0 3 1 Comparing equal terms in Eqs. (1) and (14) the relations Q 5 0 2 ] (S 1 P) 0 (11) 3 3 4 between L and L and the Frank’s constants of uniaxial 1 2 2 0 0 ] S3 LCs are [13] where both S and P are positive for normal disc-like ] K2 uniaxial nematic liquid crystals. L 5 ] (15a) 1 2 The Landau–de Gennes model uses the tensor order 2Seq parameter to describe nematic ordering. According to Eq. ] K 2 K2 (11), the model is able to describe biaxial (S ± 0, P ± 0), L 5 ] (15b) 2 2 uniaxial (S ± 0, P 5 0), and isotropic (S 5 0, P 5 0) states. Seq The isotropic state is the zero tensor: Q 5 0. Defects are The Landau coefficients L and L are bare elastic con- 1 2 regions of molecular size in which orientational order (S,P) stants, independent of temperature. On the other hand, the sharply decrease. These localized disordered regions are in Frank elastic constants are temperature dependent, since principle captured by mesoscopic models since Q remains the scalar order parameter S is a function of temperature well behaved. [6]. Eq. (14) implies that, when using Eq. (13b), the elastic anisotropy restrictions K 5 K 5 K ± K apply. Thermo- 132 2 .2. Landau–de Gennes mesoscopic model for liquid dynamic stability restrictions impose the following crystalline materials inequalities [11]: According to the Landau–de Gennes model, the bulk L . 0 (16a) 1 energy density of nematic liquid crystals (NLC) in the absence of external fields is given by [10] 3L 1 5L . 0 (16b) 1 2 fbsl 5 f 1 f (12) In addition, the molecular disc-like geometry involved in the discotic nematic phase requires that [12] where f is the short-range energy density, which is s L , 0 (17) responsible for the nematic–isotropic phase transition, and 2 f is the elastic free energy density [6], which contains l Using the classical relaxation of free energy model, the long-range gradient contributions to the system. The time-dependent equation in terms of Q and \Q is found dimensionless free energy densities f and f of thermot- s l to be [14] ropic liquid crystals, in terms of the second-order tensor Q, [s] [s] are given by dQ dF ≠f ≠f s l 2g(Q)]5F G ] 5 2 S D ] \ ?]] (18) dt dQ ≠Q ≠\Q 31 1 1 f 5 ]S S]] ] 1 2 UDQ ? Q 2 UQ ? (Q ? Q) s U 23 3 where [s] indicates symmetric and traceless, g(Q) is a 1 phenomenological kinetic coefficient, and dF/dQ is the 2 1 ]U(Q ? Q) D (13a) functional derivative of the total energy F. Eq. (18) is five 4 coupled non-linear parabolic reaction-diffusion equations L L 1 2 T for the five independent components of Q: Q , Q , Q , ]] ]] xx yy xy f 5 [\Q ? (\Q) ] 1 (\ ? Q)? (\ ? Q) l 2ckT * * 2ckT Q , Q . Substituting Eq. (13) into Eq. (18) yields the xz yz (13b) following governing equations of Q(x,t) [10]: