Linear Viscoelasticity of Textured Carbonaceous Mesophases

Fibras de carbono de alto desempenho podem ser produzidas a partir de piche de petróleo anisotrópico usando o processo de extrusão-fiação. O piche mesomórfico é um cristal líquido discótico nemático termotrópico, que possui ordem na orientação das moléculas. Cristais líquidos são materiais que possuem textura, anisotropia e visco-elasticidade. Um fenômeno importante que é encontrado em cristais líquidos é a interação entre a textura devido à orientação das moléculas e o comportamento reológico, no qual a textura ocorre devido à presença de defeitos na orientação das moléculas. O processo de extrusão-fiação usado na fabricação de fibras de carbono a partir de precursores nemáticos usa uma cascata de escoamentos que incluem fluxo capilar convergente e extensional, os quais promovem a formação de uma variedade de estruturas, incluindo radial, concêntrica, randômica e bipolar, que darão propriedades especificas para as fibras de carbono resultantes. Portanto, a compreensão fenomenológica da interação entre a textura e a reologia do precursor nemático é necessária para o controle do processo e para a otimização do produto. Neste artigo, a interação entre a reologia e a textura de cristais líquidos discóticos nemáticos será enfocada e um modelo anisotrópico será desenvolvido para descrever o regime linear visco-elástico de piches mesofásico com textura. O modelo é capaz de prever todas as características observadas experimentalmente, incluindo a dependência dos módulos de armazenamento e de perda com a freqüência de oscilação e a sua dependência com a temperatura, assim como promover uma melhor compreensão do papel dos defeitos e da anisotropia no comportamento visco-elástico do piche anisotrópico.


Introduction
High performance carbon fibers can be produced from liquid crystalline petroleum pitches, also known as carbonaceous mesophase, using the melt spinning process, which was introduced in the 1970s to produce carbon fibers with ultra-high modulus and very high thermal and electrical conductivities.][3][4] Improvements in the high performance carbon fiber quality require a better understanding of the interaction between the rheology and texture of carbonaceous mesophase precursors.
2][3][4] This material exhibits orientational order and positional disorder and is a discotic nematic thermotropic liquid crystal. 5,6The description of complex fluids systems such as discotic nematic liquid crystals by molecular dynamic or other microscopic approach is a complex task due to difficulties of computational implementation and the correct definition of the interaction potentials.A convenient and widely used approach to describe anisotropic liquids employs the average description of the molecular orientation.6][7] The latter is basically a torque balance in which viscous torques are balanced by elastic torques.The torque balance equation describes the spatio-temporal evolution of the average molecular orientation under the action of externally imposed flow fields.Figure 1b shows a schematic of the discotic nematic liquid crystalline phase, where the unit normals to the discs-like molecules (u) orient more or less parallel to the average molecular orientation, known as the director n, which is a unit vector: n.n=1.
][7] These materials are also elastic, such that spatial gradients of the average molecular orientation increases the energy.The three basic elastic storage modes for discotic nematics are splay, twist, and bend, and are shown in Figure 2. [5][6][7] Therefore, any imposed processing flow creates through the balance between flow-induced orientation and elastic torques a unique texture that is defined by spatial characteristics of the director field such as orientation gradients, boundary layers, topological defects, and defect networks.][20][21][22][23] Small amplitude oscillatory shear flow is a main rheological tool used to characterize the viscoelaticity in terms of storage modulus (G'), loss (G") modulus, and loss angle (δ) as a function of frequency (ω).][27] Figure 4 shows an schematic of a 2D view of a textured carbonaceous mesophases displaying the polydomain state; 7 H is the average domain size within which a more or less uniform n is found.The key observations that can be extracted from experimental data (for the reduced frequency less than 100(rad/sec)) are: (i) The loss modulus scales with G"~ω and is independent of temperature; (ii) the storage modulus has a temperaturesensitive terminal zone (small ω) exhibiting solid-like behavior followed by a temperature insensitive viscoelastic, where G'~ω 2 .
Previous theoretical work 31 on linear viscoelasticty was able to predict the frequency scalings (G'~ω 2 , G"~ω) using a single mode (i.e.pure splay) anisotropic model.5][26][27] Hence in this paper we present a mesoscopic model that includes the required information of director orientation 〈n〉 and texture length scale H on the linear viscoelastic moduli.It should be mentioned that twist distortions have contribution to G' and their contribution to G" has the same frequency dependency as splay and bend. 31Hence there is no need to include the twist mode.
This paper is organized as follows: Section 2 presents the governing equations used to describe liquid crystalline system.The section 3 presents the results of the linear viscoelasticity for anisotropic textured carbonaceous mesophase.Section 4 presents the conclusions.

Governing equations
The continuum theory of elasticity of liquid crystals takes into account external forces and torques that introduce deformations in the relative molecular orientations and can distort the equilibrium configurations of the molecules.The elastic free energy density, F d, a nematic uniaxial liquid crystal material is given by: [5][6][7] (1) where the three basic modes of elastic storage are the splay (K 11 ), twist (K 22 ) and bend (K 33 ) modes, as mentioned above (see Figure 2).
The continuum theory of uniaxial nematic liquids consists of the linear and angular momentum balances, and  constitutive equations for the stresses, viscous and elastic torques that takes into account external forces distort the spatially uniform equilibrium configurations of liquid crystals molecules.For incompressible isothermal conditions the general conservation of linear and angular momentum are given by the follows equations: [5][6][7] (2) (3 where ρ is the density, v is the velocity vector, f is the body force per unit volume vector, σ σ σ σ σ is the total stress, ρ 1 is the moment of inertia per unit volume, G is the external director body force vector, g is the intrinsic director body force vector, and π π π π π is the director stress tensor.The following constitutive equations for the stress tensor and the director body force that describes anisotropic liquids was found using transversely isotropic tensor coefficients, which reflect the material symmetry: [5][6][7] (4) (5) (6)   where, (7a,b,c) (8a,b,c) p is the pressure, I is the unit tensor, {α i : i=1, 2, 3, 4, 5, and 6}, are the Leslie viscosity coefficients that describes an anisotropic liquid, A is the rate of deformation tensor, N is the corotational derivative of the director vector, β β β β β is an arbitrary vector, "a" is an arbitrary scalar used to constrain the director (n) to be a unit vector, F d is the elastic free energy density known as Frank elasticity, γ 1 is the rotational viscosity, γ 2 is the irrotational torque coefficient, W is the vorticity tensor, and λ is the reactive parameter.
Experiments show that the average molecular orientation of mesophase pitch on solid surfaces can be either planar (edge-on) or homeotropic (face-on) depending of the nature of the substrate. 32,33The edge-on anchoring (see Figure 5) is the more frequently found, and is the result of strong non-covalent interactions involving aromatic π clouds. 32,33[36] A detailed discussion about the viscoelastic properties involved in the small amplitude oscillatory shear of discotic nematics can be found in our previous papers, 20,31,[34][35][36] here we briefly present the essential features of the material properties.The main properties include: (i) the Frank elastic moduli with the three distortion modes, splay (K 11 ), twist (K 22 ), and bend (K 33 ) (see equation 1); (ii) the Miesowicz shear viscosities that characterize viscous anisotropy and are measured in a steady simple shear flow between parallel plates with fixed director orientations along three characteristic orthogonal directions (η 1 : when the director is parallel to the velocity direction, η 2 : when it is parallel to the velocity gradient, and η 3 : when it is parallel to the vorticity axis); (iii) the reactive parameter (see equation 8c) and the flowalignment angle (θ s = 1/2 cos -1 (1/λ)) that obtains when λ<-1; (iv) the director re-orientation viscosities associated with splay, twist, and bend deformations (η tiwst = γ 1 , η splay = γ 1 -α 3 2 / η 1 , and η bend = γ 1 -α 2 2 / η 2 ), which are given by the rotational viscosity (γ 1 ) decreased by a factor introduced by the backflow effect. 7Backflow is re-orientation driven flow and is essentially the reverse effect to flow-induced orientation. 7Note that twist is the only mode that creates no backflow. 7

Linear viscoelasticity of monodomain and polydomain carboanceous mesophase
The solution of equations 1 to 8c for simple shear flow and capillary Poiseuille flow, assuming linear behavior that happens at small deformation regime as in the case of small amplitude oscillatory flows, allows the derivation of viscoelastic functions, namely the storage modulus (G'), the loss modulus (G"), and loss tangent ( tan δ = G ~"/G ~'): 31,[34][35][36]  (9) (10) (11)   where U 1 and U 2 are frequency-dependent functions, and P 1 and P 2 are the material properties ratios that assume distinct values for the distinct distortion modes (i.e.splay, bend, and twist) as shown in Table 1.As mentioned above, the twist mode is not included because this mode does not contribute to elastic storage.Since the material parameters {P 1 (T,n), P 2 (T,n)} are functions of (T,n) (see Table 1), 31,36 the linear viscoelastic functions (equations 9 to 11) are also functions of (T,n).In particular, when n is along the flow direction the distortion mode bend and when is along the shear rate direction, the distortion mode is pure splay.
][22][23] Figures 6a and 6b shows the dynamic moduli (G" and G') and the phase lag as a function of the frequency (ω) for a single distortion mode (splay and bend, respectively) of a monodomain (defect free).In both splay and mode cases, the low frequency (terminal) regime is classic of a viscous fluid, and the loss modulus is always greater than the storage modulus.The characteristic slopes in both cases are: Table 1.Material properties, scaling, viscoelastic parameters, and frequency functions for bend and splay deformation modes 31,35,36 Bend Splay* ω ~ :

G ~' :
G ~'' : ][36] These single mode results capture features of the experimental data shown in Figure 3 but miss the temperature dependence of G', as well as the solid-like terminal frequency plateau of G'.On the other hand, the single mode model reproduces the experimentally observed G" response.Since the slope of G" is insensitive to the deformation mode, there is no need to developed a multimode equation.Hence we conclude from the present theory (Figures 6a and 6b) and from experiments (Figure 3) that G" is insensitive to the distortion mode and to textural processes.Next we focus attention to G' and develop a model based on liquid crystal physics that reproduces the features of G' shown in Figure 3.
In this study the storage modulus G' of textured carbonaceous mesophase, with splay, and bend deformation domains is estimated using the aggregate model, which is widely used in stress analysis of anisotropic materials. 27In the present paper the polydomain material behavior is assumed to be given by weighted sum of both modes and a defect contribution: (14)   where G' texture is the storage modulus of the textured sample, G' bend is the storage modulus for the monodomain in bend distortion, G' splay is the storage modulus for the monodomain in splay distortion, G' defect is the elasticity stored by the orientation defects (disclinations) that create the texture, and ϕ is the volume fraction of each distortion mode.Equation 14 is derived using the uniform stress principle of textured materials, and it is reasonably convenient to represent the mesophase pitch in the multidistortion mode.In this study we assume that the mesophase pitch an that will be distorted in bend and splay modes, therefore ϕ splay = ϕ bend = 0.5.
3][44] The amount of elastic storage depends on the domain size (H), the defect charge (S), and the temperature, as follows: 5,6,38 (15) where c is a constant, K(T) is the characteristic Frank elastic constant or average of splay, bend, and twist.In the present study we take into account the defect elastic storage for three temperatures, T 1 <T 2 <T 3 .As well known, K(T) is a decreasing function of increasing temperature, equation 15 predicts that G' defect decreases with increasing temperature.No exact temperature is indicated in this paper because no comprehensive data exist for the temperature dependency of K for carbonaceous mesophases.The selected K values are equal to 10 -7 Pa (T=T 3 ), 10 -6 Pa (T=T 2 ), and 10 -5 Pa (T=T 1 ), respectively, which imply that domain size is in the micron range, as observed experimentally. 44igure 7 shows the computed storage modulus ( as a function of frequency (ω (rad/sec)) of a textured  carbonaceous mesophase in the small frequency regime at three temperatures (T 1 <T 2 <T 3 ), computed using equation 14.
The figure shows a plateau in the terminal zone (ω < 0.1), whose amplitude decreases with increasing temperature, in perfect agreement with Figure 3.The solid like behavior is due to the domain texture, whose elastic solid-like nature is due to the presence of orientational defects. 44

Conclusions
This paper extends the theoretical methods used to describe the linear viscoelasticity of anisotropic textured carbonaceous mesophases.Previous models focused on single deformation modes in the absence of defects.It is well known that carbonaceous mesophases are textured materials, that when sheared store elasticity due to orientation gradients and orientational defects.A linear viscoelasticity model based on the constant stress principle used to described anisotropic textured materials is developed based on liquid crystal physics.The model predict that at low frequencies the loss modulus is insensitive to temperature, textures and deformation modes, since dissipative mechanisms are insensitive to the presence of defects.On the other hand, the model predicts that the storage modulus has a strong sensitivity to temperature, and texture.The solid like behavior observed in the terminal zone is attributed to topological defects.

Figure 1 .
Figure 1.a) Typical disk-like molecule found in mesophase pitch (Adapted from references 2, 3, 4).(b) Definition of the director orientation of a uniaxial discotic nematic liquid crystalline material; the director vector n is the average orientation of the unit normal vector u to the disklike molecules.

Figure 2 .
Figure 2. Schematics of the elastic splay, twist, and bend distortion modes for uniaxial discotic nematics, and the associated elastic constants, K 11 , K 22 and K 33 , respectively.

Figure 3 .
Figure 3. Schematics showing the general features of the master curves for dynamic storage and loss moduli versus reduced frequency for mesophase pitch at three temperatures (T 1 <T 2 <T 3 ) (Adapted from references 24, 25, 26, 27).

Figure 4 .
Figure 4. Schematics of a polydomain texture showing regions with different orientation of average size H.

Figure 5 .
Figure 5. Schematics representation of planar (edge-on) surface anchoring of disc-like molecules on a substrate (Adapted from references 32, 33).
The model predictions are in excellent agreements with experiments.This work provides a new way to incorporate textural information into linear viscoelasticity and hence provides new tools to understand and use experimental linear viscoelastic data.As shown by Equation 15 the solid like behavior displayed by G' can easily be converted into a texture scale (H) or into a Frank elastic constant.

Figure 7 .
Figure 7. Computed storage modulus (G' mix ) for mesophase pitch as a function of the frequency (ω) for multi distortion mode and polydomain with elasticity stored in the defects (G' defect ) and three temperatures (T 1 <T 2 <T 3 ).