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]:
2652 J. Yan, A.D. Rey / Carbon 40 (2002)2647-2660 coherence length( defect core size)[4], L,=L,/L, is the dimensionless elastic coefficient, R is the geometry length scale (i.e. the fiber radius), Z is the short-range contribu- 6D江[(1-3)-(Q-3) tion, and L is the long-range distribution. The coherence length has units of length [6, and is the internal lengt +U(2. 2)2+6Dr Q scale of the material that arises due to the short-range energy. The coherence length s is the defect core radius In a model with no short-range energy defect cores cannot *(v(. 2)+[v(vg) be described. This is the reason why the Landau-de Gennes theory is widely used to describe defect nucleation 2t[v(v·Q (19a) processes [10]. In the Landau-de gennes model the tensor order parameter is defined all the way up to, and including, Dr≈Dr the center of the defect core. as discussed below in (1-(3/2)QQ2 (19b) conjunction with Eq (41). The energy of the defect core is found by integrating the total energy density(see Fig. 6 ckT (19c) for an example of the total long-range density as a function of time, which captures the splitting of an s=+ I defect where Dr is the microstructure-dependent rotational dif- into two s=+ 1/2 defects) Here we discuss the main effects arising from z and L isotropic diffusivity, which is independent of e, and? or (see Eq.(21). As mentioned above, Z represents the viscosity. Non-dimensioning Eq (19) yields [10] short-range elastic contribution, which governs the iso- tropic-nematic phase transition and tends to keep the molecular order(S, P)equal to that of the equilibrium state in a local domain. The second term L is the long-range 元计[1-2o)[(-3o)g order elastic effect of the molecular field to impose an energetic penalty for any spatial gradients(v2#0).As (g-3o)+ mentioned before. the long-range effect is known as the Frank elasticity. The dimensionless parameters of the nodel, which arise in Eq(20), are U,y=R/s, and L2 1-30: Q 920 The nematic potential U is a dimensionless temperature that controls the equilibrium order parameter Sa at the phase transition. According to the Doi-Edwards uniaxial y(·Q+[(·Q刀 nematic theory [14] t(Q川 (22a) To facilitate the discussion we define where T* is an absolute reference temperature just below 1-(QQ) the isotropic-nematic phase transition temperature as defined before. For U 1, long-range elasticity is where I*=A(3ckT/n) is dimensionless time, U=3T /T insignificant with respect to short-range elasticity and is dimensionless temperature s=(L,/ckT") defects may proliferate, since spatially, non-homogeneous
2652 J. Yan, A.D. Rey / Carbon 40 (2002) 2647–2660 ˜ coherence length (defect core size) [4], L 5 L /L is the dQ 2 21 ]5 dimensionless elastic coefficient, R is the geometry length dt scale (i.e. the fiber radius), Z is the short-range contribu- ] 31 1 2 ] ] ] tion, and L is the long-range distribution. The coherence 6Dr FS1 2 UDQ 2USQ ? Q 2 (Q ? Q)ID U 3 3 length has units of length [6], and is the internal length ] L1 2 scale of the material that arises due to the short-range 1U(Q ? Q)QG 1 6DrF]]\ Q ckT * energy. The coherence length j is the defect core radius. L In a model with no short-range energy defect cores cannot ] 2 T 1 ]S\(\ ? Q) 1 [\(\ ? Q)] * be described. This is the reason why the Landau–de 2ckT Gennes theory is widely used to describe defect nucleation 2 2 ]tr[\(\ ? Q)]IDG (19a) processes [10]. In the Landau–de Gennes model the tensor 3 order parameter is defined all the way up to, and including, ] 1 the center of the defect core, as discussed below in Dr ¯ Dr]]]]] (19b) 2 conjunction with Eq. (41). The energy of the defect core is (1 2 (3/2)Q:Q) found by integrating the total energy density (see Fig. 6e ckT ] for an example of the total long-range density as a function Dr 5 (19c) 6h of time, which captures the splitting of an s 5 1 1 defect ] into two s 5 1 1/2 defects). where Dr is the microstructure-dependent rotational dif- Here we discuss the main effects arising from Z and L fusivity, D is the preaveraged rotational diffusivity or r (see Eq. (21)). As mentioned above, Z represents the isotropic diffusivity, which is independent of Q, and h is short-range elastic contribution, which governs the iso- viscosity. Non-dimensioning Eq. (19) yields [10] tropic–nematic phase transition and tends to keep the dQ molecular order (S,P) equal to that of the equilibrium state ]5 dt* in a local domain. The second term L is the long-range 13 3 1 22 order elastic effect of the molecular field to impose an 2 ] ]U U F1 2 ] ] (Q:Q)G FS1 2 UDQ energetic penalty for any spatial gradients (\Q ± 0). As 2 3 mentioned before, the long-range effect is known as the 1 2USQ ? Q 2 ](Q:Q)ID1U(Q:Q)QG Frank elasticity. The dimensionless parameters of the 3 model, which arise in Eq. (20), are U ˜ , C 5 R/j, and L . 2 2 22 j 1 3 2 The nematic potential U is a dimensionless temperature 1 ]2 ]F1 2 ](Q:Q)G H\˜ Q R U 2 that controls the equilibrium order parameter S at the eq phase transition. According to the Doi–Edwards uniaxial L˜ 2 T 1 ]F\˜˜ ˜˜ (\ ? Q) 1 [\(\ ? Q)] nematic theory [14]: 2 ]]] 2 13 8 2 ]tr[\˜ ˜ (\ ? Q)]IGJ (20) S 5 1] ] S1 2 ]D (22a) 3 eq 44 3 œ U To facilitate the discussion, we define 3T * U 5 ]] (22b) T Z 5 13 3 1 22 where T * is an absolute reference temperature just below 2 ] ]F1 2 ] ] (Q:Q)G FS1 2 UDQ U U 2 3 the isotropic–nematic phase transition temperature as defined before. For U , 8/3 the stable phase is isotropic, 1 2USQ ? Q 2 ](Q:Q)ID1U(Q:Q)QG (21a) for 8/3 #U # 3 there is biphasic equilibrium, and for 3 higher values of U the phase is uniaxial nematic. In this 2 22 j 1 3 2 work, we have used 2.8 #U # 6.55. The parameter C 5 L 5 ]2 ]F1 2 ](Q:Q)G H\˜ Q R U 2 R/j is the ratio of the fiber radius (R) to the coherence length scale (j ). As mentioned above, the coherence L˜ 2 T 1 ]F\˜˜ ˜˜ length represents the characteristic size of a defect core (\ ? Q) 1 [\(\ ? Q)] 2 and for micron-sized carbon fibers at typical processing 2 temperatures (300 8C), it is usually much smaller than the 2 ]tr[\˜ ˜ (\ ? Q)]IGJ (21b) 3 system size R. In this work, we have used 0 ,C , 150. When C <1, long-range energy dominates, spatial gra- dQ ]5 Z 1 L (21c) dients are costly and homogeneous states are selected. On dt* the other hand, when C 41, long-range elasticity is where t** * 5 t(3ckT /h) is dimensionless time, U 5 3T /T insignificant with respect to short-range elasticity and 1/2 is dimensionless temperature, j 5 (L /ckT *) is the defects may proliferate, since spatially, non-homogeneous 1
J. Yan, A.D. Rey / Carbon 40 (2002)2647-2660 653 states are energetically not costly. The dimensionless scale and the second term is the o-dependent anisotropic L, =l,/L is a measurement of elastic anisotropy When contribution. The symbol y, represent the anchoring L2=0, the system is isotropic and the splay, bend, and coefficient and 2. is the preferred order parameter that twist elastic modes have the same elastic modulus. The minimizes the surface energy. The extrapolation length EL thermodynamic restrictions(Eq.(16) yield [11 is the ratio of the characteristic bulk elastic energy (Li and L the surface anchoring energy (y)[6 In addition, since for discotic nematics inequalities(3a, b) EL= hold,we further restrict L,as and has units of length since L, has units of energy /length 30,F=1/2,Q=Q (25a) always minimized and 2s=g. Since we are modelling a real material fiber with known and fixed surface director 0=S(aa-1 (25b) orientation along the azimuthal direction we adopt bound ary condition(25). To complete the discussion on bound- The symbol F is the dimensionless radial distance(F=r/ ary conditions for liquid crystalline materials we note that R), and F=0 is the center of the computational domain when the fiber radius R is much smaller than the extrapola- (i.e. fiber axis). The Dirichlet boundary condition(20)sets tion length LE, the more costly bulk energy is minimized the eigenvalues of the uniaxial tensor order parameter by adopting a spatially homogeneous tensor order parame equal to its equilibrium value (S=Sea), and the disting ter and storing less costly surface energy by eigenvector n parallel to the azimuthal(a)direction of the deviations of the surface tensor order parameter @sfrom ylindrical coordinates system(F, a). The symbol a repre- the preferred value @o; in this case 0.+Q sents the unit vector along the azimuthal a direction. the The initial conditions are boundary condition(20)restricts o to be uniaxial (P=0) with its unique eigenvector n along the azimuthal direction I"=0, Q cx. Since the actual fiber textures [3 we wish to simulate Sini (nin nini-)+Pini(min mini-Iin /ini)(28) have a director orientation n parallel to a, we impose this where S_: and p andom and s.≈0.P≈0.and do not show any significant biaxiality we restrict the tensor ", minis and Iini are the three corresponding random order parameter g to its uniaxial state(P=0). Biaxiality eigenvectors. The initial conditions represent an isotropic only plays a minor role in the defect core region. Far from state(S=0, P=0) with thermal fluctuations in order(S,P) the defect core the tensor order parameter is uniaxial and orientation (n, m, /) The initial conditions are spatially because the disc-like molecules are uniaxial and the inhomogeneous, since at each mesh point the amplitudes omogeneous equilibrium state is therefore uniaxial(see (S, P)and the phases(n, m, /)of Q are randomly selected. Eq (22a)and Ref [ 6). Boundary condition(25) is known in the liquid crystal literature as a strong anchoring condition [6]. The tensor order parameter at a free surface 3. Geometric analysis for planar polar textures or an interface depends chemo-mechanical interac- tions of the liquid crystal and the surrounding material In this section, we present a geometric analysis of the The free surface (or interfacial) free energy density of a lanar polar texture, with the objective of deriving an liquid crystal surface (or interface) is given by [15,16] expression for the distance d between the two s=+1/2 disclinations. The analytical results for the defect sepa- y=ys+0(2-2) (26) ration distance d of this section will be compared with the numerical results obtained by solving the full model (eq where iso is the isotropic (o-independent)contribution (20). The comparison will be presented in Section 4. It
J. Yan, A.D. Rey / Carbon 40 (2002) 2647–2660 2653 states are energetically not costly. The dimensionless scale and the second term is the Q-dependent anisotropic L˜ 2 21 5 L /L is a measurement of elastic anisotropy. When contribution. The symbol gan represent the anchoring L˜ 5 0, the system is isotropic and the splay, bend, and coefficient and Q is the preferred order parameter that 2 o twist elastic modes have the same elastic modulus. The minimizes the surface energy. The extrapolation length EL thermodynamic restrictions (Eq. (16)) yield [11] is the ratio of the characteristic bulk elastic energy (L ) and 1 the surface anchoring energy (g ) [6]: an ˜ 3 L . 2 ] (23) 2 5 L1 EL 5 ] (27) In addition, since for discotic nematics inequalities (3a,b) gan hold, we further restrict L˜ as 2 and has units of length since L1 has units of energy/length 3 ˜ 2 ,] 5 L2 # 0 (24) and gan of energy/area. The extrapolation length is an interfacial intrinsic length scale that may be larger, equal, ˜ In this paper, we have used L 5 2 0.5 throughout. 2 or smaller than the characteristic system size, which in our Since we are modelling mesophase fiber of circular case is the fiber radius R. For a given shape, when the cross-section the selected computational domain must be system size R is much larger than the extrapolation length circular. As mentioned above, flow is absent (v 5 0) and EL the minimization of the total (bulk1surface) energy is the fiber’s cross section is always circular, since it is achieved by introducing less costly bulk distortions and initially circular; as mentioned above, the tensor order minimizing the more costly surface energy. In this case the parameter is independent of the velocity even under spatio- tensor order parameter at the surface Q is always equal to s temporal changes. The governing Eq. (20) is solved in a the preferred tensor order parameter Q : Q 5 Q . Typical- os o circle of dimensionless radius r˜ 5 1/2, with the following ly, EL is in the nanometer scale and thus for the mesoph- boundary conditions: ase fibers of interest in this paper the surface energy is t* . 0, r˜ 5 1/2, Q 5 Q (25a) always minimized and Q 5 Q . Since we are modelling a o s o real material fiber with known and fixed surface director Q 5 S (aa 2 ] 1 I) (25b) orientation along the azimuthal direction we adopt bound- o eq 3 ary condition (25). To complete the discussion on boundThe symbol r˜ ˜ is the dimensionless radial distance (r 5 r/ ary conditions for liquid crystalline materials we note that, R), and r˜ 5 0 is the center of the computational domain when the fiber radius R is much smaller than the extrapola- (i.e. fiber axis). The Dirichlet boundary condition (20) sets tion length LE, the more costly bulk energy is minimized the eigenvalues of the uniaxial tensor order parameter by adopting a spatially homogeneous tensor order parameequal to its equilibrium value (S 5 S ), and the distinct eq ter and storing less costly surface energy by introducing eigenvector n parallel to the azimuthal (a) direction of the deviations of the surface tensor order parameter Qs from cylindrical coordinates system (r˜,a). The symbol a repre- the preferred value Q ; in this case Q ± Q . o so sents the unit vector along the azimuthal a direction. The The initial conditions are boundary condition (20) restricts Q to be uniaxial (P 5 0) with its unique eigenvector n along the azimuthal direction t* 5 0, Qini a 1 1 . Since the actual fiber textures [3] we wish to simulate 5 S (n n 2 ] ] I) 1 P (m m 2 l l ) (28) ini ini ini ini ini ini ini ini 3 3 have a director orientation n parallel to a, we impose this where S and P are random and S ¯ 0, P ¯ 0, and orientation to capture reality. Since the actual fiber textures ini ini ini ini do not show any significant biaxiality we restrict the tensor nini ini ini , m , and l are the three corresponding random eigenvectors. The initial conditions represent an isotropic order parameter Q to its uniaxial state (P 5 0). Biaxiality state (S 5 0, P 5 0) with thermal fluctuations in order (S,P) only plays a minor role in the defect core region. Far from and orientation (n,m,l). The initial conditions are spatially the defect core the tensor order parameter is uniaxial inhomogeneous, since at each mesh point the amplitudes because the disc-like molecules are uniaxial and the (S,P) and the phases (n,m,l) of Q are randomly selected. homogeneous equilibrium state is therefore uniaxial (see Eq. (22a) and Ref. [6]). Boundary condition (25) is known in the liquid crystal literature as a strong anchoring condition [6]. The tensor order parameter at a free surface 3. Geometric analysis for planar polar textures or an interface depends on the chemo-mechanical interactions of the liquid crystal and the surrounding material. In this section, we present a geometric analysis of the The free surface (or interfacial) free energy density of a planar polar texture, with the objective of deriving an liquid crystal surface (or interface) is given by [15,16] expression for the distance d between the two s 5 1 1/2 disclinations. The analytical results for the defect sepa- gan 2 g 5g 1 ](Q 2 Q ) (26) ration distance d of this section will be compared with the iso o 2 numerical results obtained by solving the full model (Eq. where g is the isotropic (Q-independent) contribution (20)). The comparison will be presented in Section 4. It iso
2654 J. Yan, A.D. Rey / Carbon 40 (2002)2647-2660 will be shown that the analytical and numerical results are texture studied in this paper(see Fig 3b), with two defects in perfect agreement with each other. The close form of equal strength s=+ 1/2, the director field is analytical results for the defect separation distance provide insight into the planar polar texture structure and, equally 0==(a, +a2)+c important, they also provide a consistency check for the numerical results. The director field of the planar polar This director field describes the orientation state that is texture cannot be obtained analytically, and a numerical independent of the radial coordinate in the absence of any solution to the full Eq(20)is necessary boundaries. Since in the pp texture the director is a To find a close form expression for d, we must assume function of the radial coordinate, Eq (34)is not a solution isotropic elasticity, L,=0, and uniaxiality (i.e. P=O). The to the pp texture. As mentioned above. since there is no director n orientation in the pp texture is best analyzed analytical solution to the planar polar texture, numerical using a polar cylindrical coordinate system (, a). To solutions(presented in Section 4)to the governi satisfy the unit length restriction (nn= 1), we parame- tions are necessary Next, we use the generally valid Eq. (32)to analyze the n,=cos Aa) planar polar texture for a fiber of radius R. Fig. 4a shows a schematic of the fiber cross-section, and the coordinate n,=sin e(a) assuming that, outside the defect cores, the scalar order parameter is constant and equal to its equilibrium value, S=S,(see Eq (22). The long- range free energy density f=(v6) where we have used Eqs. (14)and(15). Replacing this last equation in the governing equation(20), assuming steady state, we obtain the classical Laplace's equation of orienta- tion elasticity that governs the steady planar(2-D)director field of a nematic liquid crystal in any geometry [6] A general singular defect solution of lace equation in polar (r, a) coordinates to the director angle 0 is [6] (32) here c is an arbitrary constant and s is the strength of the defect. This singular solution is independent of the radial coordinate 6. These singular defect solutions are known as wedge disclination lines and are always observed nematic liquid crystals [6]. The name nematic means thread and refers to the disclination lines observed under cross polars [6]. Since the director orientation angle 0 is governed by the linear Laplace operator(V ), the princi- le of superposition can be used to describe textures with two or more defects. The general solution to the Laplace presence of an arbitrary number n of defects of strength s,, at a point N, is [6 Fig. 4. Schematic of fiber geometry dots I and ll are two s,a+c (33) fects of strength s=+1/2 in the computational domain and Ill and iv are two image defects of s=+1/2.P is an arbitrary point on the surface, on which the director n is tangential where a is the polar angle of the ray originating at the to the surface. (b) The distance between defect and coordinate defect of strength s, and ending at point N, c is a constant origin is x, the distance between image and coordinate origin is t and 0 is the director angle at point N. For the planar polar and the fiber radius is r
2654 J. Yan, A.D. Rey / Carbon 40 (2002) 2647–2660 will be shown that the analytical and numerical results are texture studied in this paper (see Fig. 3b), with two defects in perfect agreement with each other. The close form of equal strength s 5 1 1/2, the director field is analytical results for the defect separation distance provide 1 insight into the planar polar texture structure and, equally u 5 ] (a 1 a ) 1 c (34) 2 1 2 important, they also provide a consistency check for the numerical results. The director field of the planar polar This director field describes the orientation state that is texture cannot be obtained analytically, and a numerical independent of the radial coordinate in the absence of any solution to the full Eq. (20) is necessary. boundaries. Since in the PP texture the director is a To find a close form expression for d, we must assume function of the radial coordinate, Eq. (34) is not a solution ˜ isotropic elasticity, L 5 0, and uniaxiality (i.e. P 5 0). The to the PP texture. As mentioned above, since there is no 2 director n orientation in the PP texture is best analyzed analytical solution to the planar polar texture, numerical using a polar cylindrical coordinate system (r,a). To solutions (presented in Section 4) to the governing equasatisfy the unit length restriction (n ? n 5 1), we parame- tions are necessary. terize the director as follows: Next, we use the generally valid Eq. (32) to analyze the planar polar texture for a fiber of radius R. Fig. 4a shows a n 5 cos u(a) r schematic of the fiber cross-section, and the coordinate (29) n 5 sin u(a) a assuming that, outside the defect cores, the scalar order parameter is constant and equal to its equilibrium value, S 5 S (see Eq. (22)). The long-range free energy density eq then becomes K 2 f 5 ](\u ) (30) l 2 where we have used Eqs. (14) and (15). Replacing this last equation in the governing equation (20), assuming steady state, we obtain the classical Laplace’s equation of orientation elasticity that governs the steady planar (2-D) director field of a nematic liquid crystal in any geometry [6]: \2 u 5 0 (31) A general singular defect solution of the Laplace equation in polar (r,a) coordinates to the director angle u is [6] u(a) 5 sa 1 c (32) where c is an arbitrary constant and s is the strength of the defect. This singular solution is independent of the radial coordinate [6]. These singular defect solutions are known as wedge disclination lines and are always observed in nematic liquid crystals [6]. The name nematic means thread and refers to the disclination lines observed under cross polars [6]. Since the director orientation angle u is 2 governed by the linear Laplace operator (\ ), the principle of superposition can be used to describe textures with two or more defects. The general solution to the Laplace equation in the presence of an arbitrary number n of defects of strength s , at a point N, is [6] i n Fig. 4. Schematic of fiber geometry. (a) The dots I and II are two defects of strength s 5 1 1/2 in the computational domain and III u 5O s a 1 c (33) i i i51 and IV are two image defects of strength s 5 1 1/2. P is an arbitrary point on the surface, on which the director n is tangential where a is the polar angle of the ray originating at the i to the surface. (b) The distance between defect and coordinate defect of strength s and ending at point N, c is a constant origin is x, the distance between image and coordinate origin is ,, i and u is the director angle at point N. For the planar polar and the fiber radius is R
J. Yan, A.D. Rey / Carbon 40 (2002)2647-2660 system. The dots denoted I and II are the two disclination R sin a lines of streng t 1/2 parallel to the fiber texture tan axis. The x-axis is defined by a=0 in our polar coordi- nates. The geometric analysis consists of finding the dimensionless distance d* between two defects that mini- mizes the long-range energy. According to Fig. 3b, the director angle at the fiber surface is tangential. Thus at any sina +cos'a=l arbitrary point n on the surface, the director angle 8 measured with respect to the x-axis is cOS a= 0 (35) we finally (39b) Since Eq.(34)describes the director angle for a two which relates the fiber radius r to the defect distance x, defects texture without constraints the effect of fixed boundary conditions given by Eg. (35) needs to be quation is needed The additional equation needed to find boundary constraints is best obtained using the method of (r)is the defect-defect force balance equation. We next corporated. A multiple defects solution in the presence of images in which a surface orientation constraint is captured discuss defect-defect interactions in nematic liquid crys- tals following the classical treatment [6]. The energy per by an image defect. The method of images is widely used unit length of an isolated wedge disclination of strength s in a circular layer of radius R is given by [6] materials, and full details can be found in the literature [6] Fig 4a shows the two image defects, denoted Ill and IV, for this particular problem. The strength s of the two image f=(v0)2 defects is again +1/2. We locate the two image defects at a distance from the fiber axis such that the director W=W+ da rdr=W+mks In(Ry (33)to take into account the contribution from the two defects, we find that the director orientation at an arbitrary wherer is a cut-off radius of molecular size and w is the point N lying on the fiber surface is core energy.AsR→∞ the energy diverges,W→∞, logarithmically. Since W scales with s", defects of strength greater than +1/2 will tend to dissociate into +1/2 defects. Following the previous single-defect energy calcu- ()+B)+d)+(王+a)+c rocedure(Eqs. (40a)and (40b), the energy per unit associated with two defects of strength s, and s2 where a, B and are defined in Fig. 4a, and where c is by a distance ru2, is readily found [6] arbitrary constant. Eq. (31)states that the superposition of the four defect (I, Il, I, IV)solutions satisfies the ondition given by Eq (35) W=W+ da fre shows how the defect and image distances are related to the fiber axis. The two defects (l, II)are located at distance +x from the fiber axis and the two images (Ill 呢++()-2m8两( are located at a distance +t from the fiber axis. It (41) turns out that to find x and define the geometry of the Pp texture, t must also be known. In our case, s=+1/2, where rs <r <R. Then the interaction force F per c=0, and Eq(36)then becomes unit length between two defects is [6 (37) According to Eq.(37), the relation between the angular which shows that the force Fu is inversely proportional to coordinates is tan a= tan(+b)= 1女 sign repel and defect pairs of the same sign attract. Note (38) that the interaction force is valid when the defect sepa- ration distance is macroscopic. For length scales of the Using Eq.(38)in conjunction with the following tri- order of the core size(r)numerical integration of the full gonometric relations Eq.(20)is necessary, since at the core the eigenvalues of
J. Yan, A.D. Rey / Carbon 40 (2002) 2647–2660 2655 system. The dots denoted I and II are the two disclination R sin a tan f 5 ]] lines of strength s 5 1 1/2 parallel to the fiber texture , 1 x axis. The x-axis is defined by a 5 0 in our polar coordi- R sin a tan a 5 ]] nates. The geometric analysis consists of finding the x dimensionless distance d* between two defects that mini- R sin a (39a) tan b 5 ]] mizes the long-range energy. According to Fig. 3b, the 2x 2 2 director angle at the fiber surface is tangential. Thus at any sin a 1 cos a 5 1 arbitrary point N on the surface, the director angle u x cos a 5 ] measured with respect to the x-axis is R p we finally get u 5 1 ] a (35) 2 2 xR 5 , (39b) Since Eq. (34) describes the director angle for a two which relates the fiber radius R to the defect distance x, defects texture without constraints, the effect of fixed and image position ,. Since , is unknown, another boundary conditions given by Eq. (35) needs to be equation is needed. The additional equation needed to find incorporated. A multiple defects solution in the presence of ,(x) is the defect–defect force balance equation. We next boundary constraints is best obtained using the method of discuss defect–defect interactions in nematic liquid crys- images in which a surface orientation constraint is captured tals following the classical treatment [6]. The energy per by an image defect. The method of images is widely used unit length of an isolated wedge disclination of strength s to obtain analytical defect solutions in liquid crystalline in a circular layer of radius R is given by [6] materials, and full details can be found in the literature [6]. Fig. 4a shows the two image defects, denoted III and IV, K 2 f 5 ](\u ) (40a) l for this particular problem. The strength s of the two image 2 defects is again 11/2. We locate the two image defects at 2p R a distance , from the fiber axis such that the director 2 R W5Wc lc 1E da E f r dr 5W 1 pKs ln (40b) S D] boundary condition is tangential (see Eq. (36)). Using Eq. rc 0 r c (33) to take into account the contribution from the two defects, we find that the director orientation at an arbitrary where r is a cut-off radius of molecular size, and W is the c c point N lying on the fiber surface is core energy. As R → ` the energy diverges, W → `, 2 logarithmically. Since W scales with s , defects of strength p uN 5 1 ] a greater than 61/2 will tend to dissociate into 61/2 2 defects. Following the previous single-defect energy calcu- p p 5 sSD S D ] ] 1 s(b) 1 s(f) 1 s 1 a 1 c (36) lation procedure (Eqs. (40a) and (40b)), the energy per unit 2 2 length W associated with two defects of strength s and s , 1 2 separated by a distance r , is readily found [6]: where a, b and f are defined in Fig. 4a, and where c is an 12 arbitrary constant. Eq. (31) states that the superposition of 2p R the four defect (I, II, II, IV) solutions satisfies the W5W 1E da E f r dr c l tangential boundary condition given by Eq. (35). Fig. 4b 0 r c shows how the defect and image distances are related to R r the fiber axis. The two defects (I, II) are located at a 2 12 5Wc 1 2 12 1 pK(s 1 s ) lnS D] 2 2pKs s lnS D ] distance 6x from the fiber axis and the two images (III, r 2r c c IV) are located at a distance 6, from the fiber axis. It (41) turns out that to find x and define the geometry of the PP texture, , must also be known. In our case, s 5 1 1/2, where r <r , R. Then the interaction force F per c 12 12 c 5 0, and Eq. (36) then becomes unit length between two defects is [6] dW s s a 5 f 1 b 1 2 (37) F 52 5 ] 2pK], r 4r (42) 12 12 c dr r 12 12 According to Eq. (37), the relation between the angular which shows that the force F12 is inversely proportional to coordinates is their separation distance r , and that defect pairs of like 12 tan f 1 tan b sign repel and defect pairs of the same sign attract. Note tan a 5 tan(f 1 b) 5 ]]]] (38) 1 2 that the interaction force is valid when the defect sepa- tan f ? tan b ration distance is macroscopic. For length scales of the Using Eq. (38) in conjunction with the following tri- order of the core size (r ) numerical integration of the full c gonometric relations: Eq. (20) is necessary, since at the core the eigenvalues of
J. Yan, A.D. Rey / Carbon 40 (2002)2647-2660 o vary(see Fig. 5). In the case of 2s=+ 1/2 defects and o at the surface(F= 1/2)is exactly separated by a distance 2x the repulsion force between the Eq. (20)). Convergence and mesh ind two defects is established in all cases Convergence spati relization was judiciously selected taking into account the len scale of our model. As mentioned above. the landau-de Gennes model for nematic liquid crystals has an external We can now calculate the total force per unit length length scale Le and an internal length scale r. as follows exerted on defect I by the presence of:(a)a defect (D) of strength s=+1/2 at a distance 2, (b)an image defect of L=R (46a) strength +1/2 at a distance t-x(lll) and(c) another image defect of strength 1/2 at a distance ( +x(v), as (46b) shown in Fig. 4b. All forces are repulsive because all the defects and images are of the same type: s=+1/2. The where R is the fiber radius, and rs=s is the defect core repulsive force between I and III is to the left in the size introduced in Section 3. For typical liquid crystals the selected coordinate system(see Fig. 4)and the repulsive defect core size is in the molecular size range [6], and its forces between I and Il. and between I and IV to the right characteristic size (radius) is given by the square root (see Fig. 4). Employing Eq. (42 ), the force balance on the ratio of the characteristic long-range energy(L,) defect I exerted by Il, Ill and Iv is density and the characteristic short-range energy (ckT") The length scale obeys L=R≥r。[6]. To find a numerical value for re, the Landau bare elasticity modulus Li and the thermal energy factor ckT are needed. The Coupling Eqs. (39b)and (44)we finally obtain the Landau bare elasticity modulus L, [4] has been measured ollowing defect distance equation: for many liquid crystalline materials 6, while the value or ckT* is found from the isotropic-nematic phase transition temperature T*4]. If defects are present, the (45) mesh size has to be smaller than re, which produces severe computational problems. The computational problem aris- which means that when the system reaches steady state, the ing from disparate length scales(Le =R>r)is a topic of two defects lie on a circle of radius 1/V5R. Eq. (45)is current research and is far from solved. It should be noted one of the main results of this paper and will be validated the directors'orienta- (see Section 4 and Figs. 8 and 9)by the numerical tion(n, m, /)while the internal length scale governs the tegration of Eq (20). In the absence of elastic anisotropy scalar order parameter(S, P). In addition, care should be L,=0), the distance between two defects is fixed only by the fiber radius. The number 1/5 arises due to the fiber taken to select an appropriate time integration technique to overcome the intrinsic stiffness of the system. The model geometry. In the rectangular geometry this factor is 0.5 equations contain an internal time scale T and an external but in the circular domain it is 0.66. Generalization of this time scale T. The internal time scale governs the evolution alysis to other fiber tries of industrial relevance, of the scalar order parameters(S, P)and is given by uch as elliptical, is possible. Next we shall establish the vance theoretical results by com- (47) umerical solutions of the full non-linear system of lic partial deferential equations(21)with the theo- A much longer external time scale T. controls the evolution retical prediction given by Eq. (45) of the directors and is given by 4. Modeling fiber texture structure The selected adaptive time integration scheme is able to 4.1. Computational model efficiently take into account the stiffness that arises due to the disparity between time scales The model equation(21) is a set of five coupled non- linear parabolic partial differential equations, solved in a 4.2. Results and discussion circle, subject to the auxiliary conditions(see Eqs. (25) and(28). The equations are solved using Galerkin Finite To efficiently Elements with Lagrangean linear basis functions for spatial represent o by a cuboid whose axes are the director discretization and a fifth-order Runge-Kutta-Cash-Karp (n, m /)and whose axes are proportional to its eigenvalues time adaptive method. Since we use Galerkin Finite Since g has negative eigenvalues, we visualize M=e Elements, the boundary conditions are exactly satisfied, I instead of g
2656 J. Yan, A.D. Rey / Carbon 40 (2002) 2647–2660 Q vary (see Fig. 5). In the case of 2s 5 1 1/2 defects and Q at the surface (r˜ 5 1/2) is exactly equal to Q (see o separated by a distance 2x the repulsion force between the Eq. (20)). Convergence and mesh independence were two defects is established in all cases. Convergence spatial discretization was judiciously selected taking into account the length pK F 5 ] (43) scale of our model. As mentioned above, the Landau–de 12 x Gennes model for nematic liquid crystals has an external We can now calculate the total force per unit length length scale L and an internal length scale r as follows: e c exerted on defect I by the presence of: (a) a defect (II) of L 5 R (46a) e strength s 5 1 1/2 at a distance 2x, (b) an image defect of ]] strength 11/2 at a distance , 2 x (III) and (c) another ] L1 image defect of strength 11/2 at a distance , 1 x (IV), as r 5 j 5 ] (46b) c œckT * shown in Fig. 4b. All forces are repulsive because all the defects and images are of the same type: where R is the fiber radius, and r 5 j is the defect core s 5 1 1/2. The c size introduced in Section 3. For typical liquid crystals the repulsive force between I and III is to the left in the defect core size is in the molecular size range [6], and its selected coordinate system (see Fig. 4) and the repulsive characteristic size (radius) is given by the square root of forces between I and II, and between I and IV to the right the ratio of the characteristic long-range energy (L ) (see Fig. 4). Employing Eq. (42), the force balance on 1 defect I exerted by II, III and IV is density and the characteristic short-range energy (ckT *). The length scale obeys L 5 R4r [6]. To find a e c 111 numerical value for r , the Landau bare elasticity modulus ]] 5 1 ] ]] c (44) , 2 x 2x , 1 x L and the thermal energy factor ckT * are needed. The 1 Landau bare elasticity modulus L1 [4] has been measured Coupling Eqs. (39b) and (44) we finally obtain the for many liquid crystalline materials [6], while the value following defect distance equation: for ckT * is found from the isotropic–nematic phase 1 transition temperature T * [4]. If defects are present, the x 5 ]R (45) Œ 4 ] mesh size has to be smaller than r , which produces severe 5 c computational problems. The computational problem ariswhich means that when the system reaches steady state, the ing from disparate length scales (L 5 R4r ) is a topic of ] e c Œ 4 two defects lie on a circle of radius 1/ 5R. Eq. (45) is current research and is far from solved. It should be noted one of the main results of this paper and will be validated that the external length scale governs the directors’ orienta- (see Section 4 and Figs. 8 and 9) by the numerical tion (n,m,l) while the internal length scale governs the integration of Eq. (20). In the absence of elastic anisotropy scalar order parameter (S,P). In addition, care should be (L 5 0), the distance between two defects is fixed only by 2 taken to select an appropriate time integration technique to 4 ] the fiber radius. The number 1/ 5 arises due to the fiber Œ overcome the intrinsic stiffness of the system. The model geometry. In the rectangular geometry this factor is 0.5, equations contain an internal time scale t and an external i but in the circular domain it is 0.66. Generalization of this time scale te. The internal time scale governs the evolution analysis to other fiber geometries of industrial relevance, of the scalar order parameters (S,P) and is given by such as elliptical, is possible. Next we shall establish the h accuracy and relevance of the theoretical results by com- t 5 ]] (47) i ckT * paring numerical solutions of the full non-linear system of parabolic partial deferential equations (21) with the theo- A much longer external time scale te controls the evolution retical prediction given by Eq. (45). of the directors and is given by 2 ] hLe t 5 (48) e L1 4. Modeling fiber texture structure The selected adaptive time integration scheme is able to 4 .1. Computational modeling efficiently take into account the stiffness that arises due to the disparity between time scales: t <t . i e The model equation (21) is a set of five coupled nonlinear parabolic partial differential equations, solved in a 4 .2. Results and discussion circle, subject to the auxiliary conditions (see Eqs. (25) and (28)). The equations are solved using Galerkin Finite To efficiently visualize the solution vector Q, we Elements with Lagrangean linear basis functions for spatial represent Q by a cuboid whose axes are the directors discretization and a fifth-order Runge–Kutta–Cash–Karp (n,m,l) and whose axes are proportional to its eigenvalues. time adaptive method. Since we use Galerkin Finite Since Q has negative eigenvalues, we visualize M 5 Q 1 1 Elements, the boundary conditions are exactly satisfied, ] I instead of Q. 3