Thermodynamic Stability Analysis of Liquid-Crystalline Polymer Fibers

Feb 15, 1997 - Classical theories of liquid-crystalline materials are used to develop a ... identifies the new contributions to elastic storage due to...
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Ind. Eng. Chem. Res. 1997, 36, 1114-1121

Thermodynamic Stability Analysis of Liquid-Crystalline Polymer Fibers Alejandro D. Rey* Department of Chemical Engineering, McGill University, Montreal, Quebec, Canada H3A 2A7

Classical theories of liquid-crystalline materials are used to develop a new model that describes the thermodynamic stability of nematic liquid-crystal cylindrical fibers that arise in the fabrication of in-situ liquid crystal polymer (LCP) composites. The thermodynamic model identifies the new contributions to elastic storage due to the nematic orientational order present in LCP fibers. It is shown that the additional nematic surface and bulk elastic storage mechanisms tend to promote fiber stability when compared with isotropic fibers. The theory predicts that elastic storage due to orientational deformations within the fiber may be able to overcome the classical capillary (Rayleigh) instability present in isotropic fibers. The parametric conditions that lead to fiber stability are smaller fibers, low interfacial tensions, and large nematic elastic constants. Nevertheless, using estimates typical of the actual in situ LCP-polymer composites, it is found that LCP fibers are unstable and will break up, in agreement with existing experimental studies on the stability of liquid-crystalline polymer fibers in a thermoplastic elastomeric matrix when subjected to annealing at high temperatures. 1. Introduction Thermotropic liquid-crystalline polymers (TLCP’s) are polymeric materials containing rigid-rodlike mesogenic molecules that display orientational order (de Gennes and Proust, 1993; Donald and Windle, 1992; Larson, 1988). The orientational order provides an additional microstructural mechanism to be used for optimizing processing and product property profiles (Donald and Windle, 1992). Since at sufficiently high temperatures TLCP’s flow as conventional themoplastics, they can be processed and shaped with standard polymer-processing equipment, such as injection molding. Given this compatibility with themoplastics technology and the tendency of TLCP’s to form fibrous structures in shear and elongational flows, new types of polymeric composites consisting of blends of thermoplastic polymers and TLCP’s have been widely studied and characterized (Acierno and La Mantia, 1993). One of the major advantages of these LCP polymer blends is the formation of in situ generated reinforcing LCP fibers in a matrix of a conventional themoplastic. The in situ generated LCP polymer composite avoids the difficulties and disadvantages of other fiber composites. Several recent reviews on processing, properties, fabrication, and performance of LCP polymer composites are available in the literature (Acierno and La Mantia, 1993; Handlos and Baird, 1995; Jin and Li, 1995; Qin, 1996). In the fabrication of in situ LCP composites, the morphology of interest is that of LCP fibrils embedded in the thermoplastic polymer matrix. Other morphologies such as particles, droplets, and interpenetrating networks are possible, as in other immiscible two-phase systems, but these do not provide the enhancement of mechanical properties sought in reinforced composites (Handlos and Baird, 1995; Jin and Li, 1995; Qin, 1996). The in situ formation of the LCP fiber morphology is developed during shear and elongation flow deformations, and some of the parameters that control the morphology are the rheological properties of the two phases, the temperature, the concentration, and the * E-mail: [email protected]. Fax: (514) 398-6678. Telephone: (514) 398-4196. S0888-5885(96)00463-0 CCC: $14.00

processing geometry (Handlos and Baird, 1995). The formation step is a poorly understood nonequilibrium process that involves droplet formation, breakup, and coalescence among other phenomena. One fundamental question of importance to the fabrication of LCP polymer composites is to identify the mechanisms that promote the stability of the LCP fibers during and after flow deformations. This paper starts this unexplored area of theoretical investigation by focusing on the latter case, since this is naturally the point of departure for such a program. In addition, comprehensive experimental data have been reported on the stability of liquid-crystalline polymer fibers in a thermoplastic elastomeric composite (Verhoogt et al., 1993). It was found that isolated LCP fibers surrounded by a thermoplastic elastomer disintegrate when annealed above the melting point of the LCP. The mode of fiber breakup is found to be similar to that of Newtonian threads (Verhoogt et al., 1993). The frame of reference used in this work is the instability of rodlike morphologies by interfacial surface energy reduction, first considered by Rayleigh and known as the capillary instability (Rayleigh, 1879). Rayleigh predicted that an infinitely long liquid cylinder will eventually break up into spherical droplets if an infinitesimal sinusoidal perturbation of wavelength greater than the fiber’s perimeter is introduced. The instability of rodlike morphologies has been and continues to be widely studied in the material science and engineering of composites structures (see, for example, Qian et al. (1994)). A central theme in this area is to determine the stability and mechanisms that lead to coarsening and instability of rodlike morphologies, so as to avoid deterioration under exposure to elevated temperatures and stresses. It is expected that a similar analysis is equally important to the development of LCP thermoplastic composites. The scope of this work is to establish the relevant features that describe the thermodynamic stability of LCP fibers embedded in a polymer matrix. The LCP is assumed to have nematic ordering, where the rodlike molecules are more or less parallel to each other but otherwise free to translate past each other (de Gennes and Prost,1993). The theory and analysis are based on © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1115

free-energy calculations, and kinetic nonequilibrium phenomena are not considered in this paper. Thus, questions such as final breakup morphology, time scales, and kinetic effects are outside the scope of this paper. The particular objectives of this paper are (1) to formulate a thermodynamic model that describes the elastic free energy of LCP fibers, (2) to identify the energy contributions due to orientational nematic order to the morphological stability of LCP-polymer composite structures, (3) to determine the parametric envelopes for LCP fiber stability, and (4) to qualitatively compare the theoretical predictions with the experimental data on LCP fiber stability given by Verhoogt et al. (1993). The organization of this paper is as follows. In section 2, we define the coordinate system and the state variables, derive the thermodynamic model, and present the fiber characteristics used to discuss and classify the representative cases. In section 3, we present and discuss the LCP fiber stability analysis for the isotropic and nematic phases and compare the results with Rayleigh’s capillary instability. Finally, the parameter values arising in the theory are estimated and the LCP fiber stability predictions are calculated and validated with existing experimental data. 2. Governing Equations 2.1. Definitions of Fiber Microstructure, Deformations, and Geometry. In this paper, we study the thermodynamic stability of a infinitely long cylindrical thermotropic liquid-crystal polymer (LCP) fiber subjected to an infinitesimal periodic surface shape perturbation. The single fiber is embedded in a polymer melt of infinite extension, and incompressible and isothermal conditions are assumed. We use Cartesian tensor notation, repeated indices are subjected to the summation convention, and partial differentiation with respect to the jth spatial coordinate is denoted by a comma (i.e., Qik, j ) ∂Qik/∂xj). We use a cylindrical coordinate system (r,φ,z), with the z axis being collinear with the fiber axis (see Figure 1A). The microstructure of the cylindrical nematic liquidcrystalline polymer fiber is characterized by the tensororder parameter Qij (de Gennes and Prost, 1993). This second-order tensor is traceless and symmetric. For rodlike nematics close to equilibrium, the tensor Qij is uniaxial (i.e., two equal eigenvalues) and given by

Qij ) S(ninj - δij/3)

(1a)

where the following restrictions apply:

Qij ) Qji; Qii ) 0; -1/2 e S e 1; nini ) 1

(1b)

and δij is the unit tensor. The magnitude of the scalar order parameter S is a measure of the molecular alignment along the director n and is given by S ) 3(niQijnj)/2. Equation 1a gives a proper description of the order in a nematic phase if we identify the director n to be the average orientation of the rodlike molecules. In thermotropic LCP’s, the scalar order S of the nematic phase is positive and a function of temperature T. At temperatures higher than the nematic-isotropic transition temperature T* and in the absence of thermal decomposition, the polymer becomes isotropic, the scalarorder parameter vanishes (S ) 0), and the molecular orientation is random. Thus, Q or equivalently (S,n) are able to describe both the isotropic and nematic phase: (i) for T > T*, the phase is isotropic, S ) 0, and

Figure 1. Definition of fiber geometry and characteristic LCP fiber transformations. The fiber is defined in a cylindrical coordinate system (r,φ,z), and its axis is along z. (A) Isotropic fiber of constant radius R0 is periodically deformed at a temperature T above the nematic-isotropic transition temperature T*. The shape transformation is denoted by Φ(Σ0,Mi) f Φ(Σp,Mi); see text. (B) LCP fiber of constant radius R0 is periodically deformed into an LCP fiber, with constant director orientation, at a temperature T below the nematic-isotropic transition temperature T*. The shape transformation is denoted by Φ(Σ0,M0) f Φ(Σp,M0); see text. The rodlike segments represent the average molecular orientation, n. (C) LCP fiber of constant radius R0 is periodically deformed into an LCP fiber, with a periodic director orientation, at a temperature T below the nematic-isotropic transition temperature T*. The shape transformation is denoted by Φ(Σ0,M0) f Φ(Σ0,Ma); see text.

n is undefined; for T < T*, the phase is nematic, S ) S(T), and n is determined by a balance of bulk and surface torques. To identify a nematic polymer fiber, we need to define, as for isotropic fibers, its shape. Here we restrict the analysis to infinitely long fibers with circular cross sections of radius F(z). Thus, the shape Σ is defined by the possible variation of the radius F with axial distance z. For an undeformed fiber of constant radius R0, the shape Σ is

Σ0 ) {F ) R0; -∞ < z < +∞}

(2)

In this paper, we study periodic variations in the fiber radius. A fiber with a periodic surface modulation of wavelength l has a shape Σp given by

Σp ) {F ) F(z) ) F(z + l); -∞ < z < +∞, l > 0} (3) The surface with a sinusoidal modulation of amplitude A and wavelength l is described by

(2πl z)

F(z) ) R + A sin

(4)

where R is the average fiber diameter given by

x

R ) R0

1-

A2 2R02

(5)

In this paper, we study distortions of small amplitudes (A/R0 , 1, A/l , 1), and in this instance, the average fiber radius R simplifies to

1116 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

[

R ) R0 1 -

A2 4R02

]

(6)

In the rest of this paper, we always use eq 6 and analyze the two shapes, Σ0 and Σp. To identify a nematic polymer fiber, we need to define, in addition to its shape, the microstructure M. The microstructure M of the fiber is given by

M ) {S ) (S(r,z), n(r,z)), 0 e F e F, -∞ < z < +∞} (7) In the isotropic phase (T > T*), the fiber microstructure is given by M ) Mi ) {S ) 0, 0 e r e F, -∞ < z < +∞}. In the nematic phase (T < T*) the microstructure M is specified by S and n. In the fabrication of in-situ LCP fiber composites, the director n aligns parallel to the fiber axis. Thus, we consider as a reference fiber a nematic fiber with a microstructure given by S ) S(T) and a director field n ) (0,0,1), for 0 < r < R0. In this paper, we shall assume that S is unchanged by the fiber shape deformation, that it is only a function of the temperature (S ) S(T)), and that this value is equal to that observed in bulk samples. This is a reasonable assumption in the absence of flow and defects (de Gennes and Prost, 1993). The two representative nematic microstructures considered in this paper are (a) M0, where the director is parallel to the fiber axis (n ) (0,0,1)), and (b) Ma, where the director is affinely deformed. In the former case (M0), the director is constant and independent of the fiber shape. The director orientation along the fiber axis in fiber-forming flow processes is readily achieved for the rodlike molecules considered here (Rey and Denn, 1988a,b; Warner, 1994). The microstructure M0 is given by

M0 ) {S ) S(T), n ) (0,0,1), 0 e r e F, -∞ < z < +∞} (8) In this case, any surface distortion (i.e., p * R0) away from the z axis introduces a deviation between the surface and the constant director. In the latter case (Ma), the director is affinely deformed by periodic shape distortions:

Ma ) {S ) S(T), n ) n(r,z), nr(r ) F,z) ) ∂F/∂z, 0 e r e F(z), -∞ < z < +∞} (9) where the expression for nr makes the surface orientation tangent to the surface of the fiber. Finally, the nematic fiber Φ(Σ,M) is defined by its shape Σ and microstructure M:

Φ ) {F ) F(z), -∞ < z < +∞; (S,n) ) (S(r,z), n(r,z)), 0 e r e F, -∞ < z < +∞} (10) The reference fiber with constant radius R0 in the isotropic phase is Φ(Σ0,Mi). The reference fiber with constant radius R0 in the nematic phase with a director aligned along the fiber axis is Φ(Σ0,M0). In this paper, we study the thermodynamic stability of the different reference fibers by computing the elastic energy difference present in the three characteristic transformations, as shown in Figure 1 and given by (i) periodic shape transformations in the isotropic phase (T > T*) (Φ(Σ0,Mi) f Φ(Σp,Mi), shown in Figure 1A); (ii) periodic shape transformations in the nematic phase (T < T*) at

constant director orientation (Φ(Σ0,M0) f Φ(Σp,M0), shown in Figure 1B), and (iii) periodic shape and affine microstructure transformations in the nematic phase (T < T*) (Φ(Σ0,M0) f Φ(Σp,Ma), shown in Figure 1C). 2.2. Free Energy of Nematic Liquid-Crystal Polymer Fibers. The total free energy density Λ of a nematic LCP fiber of radius F is given by (de Gennes and Prost, 1993; Donald and Windle, 1992)

(11)

Λ ) ΛH + ΛB + ΛSδ(r - F)

where ΛH is the homogeneous, ΛB is the Frank (bulk), and ΛS is the surface free-energy density, respectively, and δ is the delta Dirac function. Following standard notation, the specific expressions for these free-energy densities are (de Gennes and Prost, 1993; Nobili and Durand, 1992)

ΛH ) Λ(0) +

(43AQ Q ij

ji

3 + BQmlQlkQkm + 2 9 C(QlkQkl)2 (12a) 16

)

ΛB )

L1 L2 (Q )2 + (Qij,i)2 2 ij,k 2

(12b)

β ΛS ) γ + (Qij - Q0ij)2 2

(12c)

Q0ij ) S0(kikj - δij/3)

(12d)

where Λ(0) is the free energy for Q ) 0; A, B, and C are the phenomenological parameters (energy/volume); L1 and L2 are elastic constants (energy/length), γ is the interfacial energy (energy/area); β is a phenomenological parameter (energy/area); Q0 is the preferred tensororder parameter; k is a unit vector that defines the preferred surface orientation known as the easy axis; and S0 is the preferred surface scalar-order parameter. In the uniaxial approximation, ΛH becomes

ΛH ) Λ(0) +

[A2 S

2

B C + S3 + S4 3 4

]

(13)

Using the classical Doi theory of nematic polymers (Doi and Edwards, 1986), it is possible to reduce the number of parameters from three to two and write ΛH as

[(

)

2 U 1 1 1 ΛH ) Λ(0) + νkT 1 - S2 - US3 + US4 3 2 3 9 6

]

(14)

where 1/ν is a molecular volume, k is Boltzmann’s constant, T is the absolute temperature, and U ) 3T*/T is the nematic potential. When T < T*, the scalar order parameter of the nematic phase is given by

S(T) )

x1 - 98 T*T

1 3 + 4 4

(15)

such that S(T*) ) 1/2. In the isotropic phase, T > T*, and the Doi theory predicts that the scalar order parameter vanishes, S ) 0. The bulk free energy ΛB is due to spatial gradients in the tensor order parameter. Assuming uniaxiality, equal elastic constants, and a spatially homogeneous S, ΛB simplifies to

ΛB )

9LS2 [(∇‚n)2 + (∇×n)2] 4

(16)

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1117

The surface free-energy density ΛS has an isotropic component (γ) and an orientation component (terms containing Q). Although other similar expressions for ΛS involving the surface unit normal are possible [see, for example, Alexe-Ionescu et al. (1992)], expression 12c in terms of an easy tensor order parameter has been proved to be accurate and useful in the past (Nobili and Durand, 1992), and will be used in the present analysis. Details of the mathematical transformations using the surface free energy density in terms of k with those in terms of the unit normal are given by Alexe-Ionescu et al. (1992), for example. In the uniaxial approximation and up to second order in Q, ΛS simplifies to the classical Rapini-Papoular equation (de Gennes and Prost, 1993):

ΛS ) γ + GS2 - GS2(n‚k)2

(17)

where S0 has been taken to be equal to S(T). This simplifying assumption neglects the nematic elastic storage modes that arise from deviations of S from S0. In the isotropic phase (T > T*), the scalar-order parameter S is zero and ΛS ) γ, as in isotropic materials. For G > 0, the surface energy is a minimum when n is parallel to k, and we denote this surface orientation as planar. For LCP nematic fibers, the orientation is expected to be along the fiber axis, and thus, k is parallel to the surface and to the fiber axis. When G is infinitely large, the tensor-order parameter adopts the preferred value Q ) Q0, and the only surface energy storage mode is due to γ. Having introduced the terminology, concepts, and equations required to describe the shape and microstructural transformations in isotropic and nematic LCP fibers with cylindrical symmetry, we next present and discuss the main results of this paper. 3. Results and Discussion In this section, we present the thermodynamic fiber stability analysis for shape transformations in the isotropic phase, for shape transformations in the nematic phase with constant director, and for shape transformations in the nematic phase with variable director fields. It is instructive to define the parametric ranges for the three cases. The three relevant parameters of the problem are the (1) operating condition (fiber temperature with respect to the nematic-isotropic transition temperature), (2) geometries (fiber radius, R0), and (3) physical property (molecular weight of the nematic polymer). For temperatures higher than the nematic-isotropic transition temperature, the fiber is in the isotropic phase and nematic ordering effects are negligible. At temperatures below the nematic-isotropic transition temperature, nematic ordering must be included in a thermodynamic analysis. In addition to temperature, the molecular weight and fiber radius have to be considered. For large-diameter fibers, the bulk energy is insignificant in relation to the surface energy, and strong anchoring is present, thus leading to shape transformations in the nematic phase with variable director. On the other hand, for sufficiently small-diameter fibers, the anisotropic surface energy is insignificant with respect to the bulk energy, and weak anchoring is therefore present, leading to shape transformations in the nematic phase with constant director. The above comments follow directly from the fact that nematic bulk elasticity per unit volume scales with R0-2, while the nematic surface elasticity per unit surface is

independent of R0. The role of molecular weight can only be discussed in a tentative manner since there is no recognized body of experimental data on the effect of the molecular weight on the surface anchoring energies of nematic polymers. On the other hand, Lee and Meyer (1991) have established the principles that govern the effect of molecular weight on the bulk elastic constants. Since it is shown (Lee and Meyer, 1991) that the bulk elastic constants increase with increasing molecular weight, it is possible to make two observations. First, for larger fibers, where the fiber instability involves director deformations, a strong sensitivity to molecular weight will be present. Second, fibers made of lower molecular weight nematics will be more easily deformed than fibers made of higher molecular weight nematics. 3.1. Shape Transformations in the Isotropic Phase (T > T*): Φ(Σ0,Mi) f Φ(Σp,Mi). The shape transformation Φ(Σ0,Mi) f Φ(Σp,Mi) is shown in Figure 1A. For temperatures higher than the nematicisotropic transition temperature T*, the scalar order parameter is zero, S ) 0, and the fiber microstructure is Mi. If the fiber radius changes from R0 to F(z), the shape transformation is Σ0 f Σp. In a length l, the total free energy of the fiber Φ(Σ0,Mi) is Fi ) 2πΣ0l. On the other hand, in a length l, the total free energy of the fiber Φ(Σp,Mi) is

∫0lFx1 + (dF/dz)2 dz

F ) 2πγ

(18)

Using eq 18, we find that the net change in the total free energy ∆ for the transformation Φ(S0,Mi) f Φ(Σp,Mi) is

∆)

(

)

2 πA2lγ 4πR0 -1 2R0 l2

(19)

Setting ∆ ) 0, we find that the critical wavelength lR* for which there is no free-energy change is lR* ) 2πR0. For wavelengths l greater than lR*, l > lR*, the total free energy decreases, and thus constant radius fibers are unstable and eventually lead to the transformation of the fiber into an array of droplets. This is the thermodynamic analysis of the classical Rayleigh capillary instability. 3.2. Shape Transformations in the Nematic Phase (T < T*) with Constant Director Orientation: Φ(Σ0,M0) f Φ(Σp,M0). The shape transformation Φ(Σ0,M0) f Φ(Σp,M0) is shown in Figure 1B. Here we consider the case of a nematic LCP fiber that undergoes a shape transformation Σ0 f Σp, with a constant director field, n ) (0,0,1). For the fiber Φ(Σp,M0), the relevant free energy densities are ΛH and ΛS. For a given constant temperature T < T*, the expressions for ΛH and ΛS can be found from eqs 14 and 17. Since for the present case n‚k ) 1, the surface free-energy density is constant along the interface and is given by ΛS ) γ. The total free energy for the fiber Φ(Σ0,M0) is then given by

F ) 2πγR0l + πR02ΛHl

(20)

For the fiber Φ(Σp,M0), the relevant free energies are ΛH and ΛS. The homogeneous energy and the order parameter remain unchanged since the transformation Φ(Σ0,M0) f Φ(Σp,M0) is isothermal. The additional phenomenon to be included is the storage of surface elastic energy that arises because the unit tangent to

1118 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

the interface k is not parallel to the director. Let n‚k ) cos θ. The surface energy density ΛS of the fiber Φ(Σp,M0) is

ΛS ) γ + GS2 - GS2(cos θ)2

(21)

For a surface of shape Σp and microstructure M0, the expression for (cos θ)2 is found from

tan θ )

( )

dF 2π 2π ) A cos z dz l l

(22a)

and to second order in A, it is given by

cos2 θ ) 1 -

(2πA l )

2

(2πl z)

cos2

(22b)

To find the total free energy of the fiber Φ(Σ,Mp) in a length l, we integrate ΛH over the volume and ΛS over the surface. The net total free-energy change ∆ for the transformation Φ(Σ0,M0) f Φ(Σp,M0) is

∆)

(

)

2 R0 πA2l 4πR0 γ -1 + 4π3GS2A2 2 2R0 l l

(23)

To find the wavelength l* at which the free energy of the two fibers are equal, we set ∆ ) 0 and find

l* ) l*Rx1 + P

(24a)

l*R ≡ 2πR0

(24b)

2GS2 γ

(24c)

P≡

where the parameter P is the ratio of the nematic anchoring energy (2GS2) to the isotropic interfacial energy γ. For wavelengths l larger than l*, the nematic fiber decreases the total energy through surface undulations, and for wavelengths l shorter than l*, the nematic fiber increases the total energy. Equation 24a thus establishes the stability criterion for nematic liquid crystalline fibers. The result (eq 24a) is the product of the classical Rayleigh capillary instability criteria l*R ) 2πR0 times a square-root factor arising from the two surface elastic storage modes. The additional elastic storage due to the nematic ordering increases the value of the classical critical wavelength, l* > l*R, introducing a new stabilization mechanism for the fiber morphology. As expected, the Rayleigh criterion l*R ) 2πR0 is recovered when P f 0. The two conditions that lead to this limit are when S f 0 and when G/γ f 0. The former may be present when the temperature T is higher than the nematic-transition temperature T*. The latter may be present when the fiber-matrix interface does not induce a preferred orientation or when the interfacial tension γ is much greater than the anchoring energy G. The dimensionless surface energy ratio P is a function of the chemistry of the fiber-polymer matrix system and temperature. To estimate the dependence of the critical wavelength l* on temperature, we use the Doi expression for S(T) (eq 15) and for simplicity assume that γ and G are constants. We find

l* ) l*R

x1 + 2Gγ ( / + / x1 - 8T/9T*) 1

3

4

2

4

(25)

Figure 2. Dimensionless critical wavelength l*/2πR0 as a function of dimensionless temperature T/T* for three values of the surface energy ratio 2G/γ: 0.1 (dashed line), 1 (dashed dot line), and 10 (dashed triple-dot line). The vertical line corresponds to the nematic-isotropic transition. As the temperature T increases toward the nematic-isotropic transition temperature T*, the critical wavelength decreases and gets closer to 2πR0. For T > T*, the fiber is isotropic (S ) 0) and l* ) l*R ) 2πR0. The drop of the critical wavelength between the nematic and isotropic fiber at T* is given by eq 26 and increases with increasing G/γ. For T > T*, the critical wavelength is l* ) 2πRo.

Figure 2 shows a plot of the dimensionless critical wavelength l*/l*R as a function of dimensionless temperature T/T*, for three values of 2G/γ: 0.1 (dashed line), 1 (dashed-dot line), and 10 (dashed triple-dot line). The vertical line corresponds to the nematic-isotropic transition. As the temperature T increases toward the transition temperature T*, the critical wavelength gets closer to 2πR0. For T > T*, the fiber is isotropic (S ) 0), and the Rayleigh capillary instability criterium is recovered, l* ) l*R ) 2πR0. The drop of the critical wavelength between the nematic and isotropic fiber at T* is

l*(T)T*) - l*R ) l*R[x1 + G/2γ - 1]

(26)

which increases with increasing G/γ. 3.3. Shape and Microstructure Transformations in the Nematic Phase (T < T*) with Variable Director Orientation: Φ(Σ0,M0) f Φ(Σp,Ma). The transformation Φ(Σ0,M0) f Φ(ΣP,Ma) is shown in Figure 1C. Here we consider the case of a nematic LCP fiber that undergoes a shape transformation Σ0 f Σp, with a change in director field M0 f Ma such that the director surface orientation is always parallel to the surface (i.e., planar orientation). The free energy of the fiber Φ(Σ0,M0) was computed in the previous section, and it is given by eq 20. For the fiber Φ(Σp,Ma), the relevant free energies are ΛH, ΛS, and ΛB. The homogeneous energy and the order parameter remain unchanged since the transformation Φ(Σ0,M0) f Φ(Σp,Ma) is isothermal. Since the director surface orientation is always parallel to the unit tangent k, the surface free-energy density is found from eq 17 with θ ) 0 and is ΛS ) γ. The additional phenomenon to be included is the storage of bulk elastic energy that arises due to director distortions for r e F(z). Here we consider the case where the shape Σp produces an affine deformation of the original director field, n ) (0,0,1). The new director field that results from the periodic surface distortion is given by n ) (θ,0,1), where θ is an infinitesimal perturbation. It is instructive to note that the infinitesimal director deformation is assumed to be planar, and in the language of liquid-crystal elasticity,

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1119

it is known as a splay-bend deformation (de Gennes and Prost, 1993). A more general case would be to consider three-dimensional orientation, corresponding to twist distortions, but the planar deformation mode assumed here is an appropriate starting point and has been very useful in similar situations in the past (Rey and Denn, 1988a,b; Rey, 1991). To satisfy the constraint of planar surface director orientation (i.e., n‚k ) 1), we find that θ(r,z) is given by

( )

A 2π θ(r,z) ) 2π cos z f(r) l l

(27a)

f(F) ) 1

(27b)

f(0) ) 0

(27c)

where a separation of variables has been assumed. For simplicity and without loss of the essential physics for the radial dependence of the director angle, we use f(r) ) r/F. To find the total free energy of the fiber Φ(Σp,Ma) in a length l, we integrate ΛH and ΛB over the volume and ΛH over the surface. The net total free-energy change ∆ for the transformation Φ(Σ0,M0) f Φ(Σp,Ma) is

(

[

)

Figure 3. Scaled critical wavelength l*/2πR0 as a function of the dimensionless elastic energy ratio V (V is the ratio of bulk nematic energy to isotropic interfacial energy). The vertical line corresponds to the nematic-isotropic transition. As V tends to one, the critical wavelength diverges, indicating that the LCP fiber is stable since energy savings due to the surface energy are canceled by energy cost due to the director gradients. As V grows infinitesimally small, the director bulk energy becomes negligible when compared to the surface energy and the Rayleigh capillary instability is recovered.

]

2 2 π2R04 πA2l 4πR0 2 3A l ∆) γ - 1 + 18LS π 2 1 + 2R0 l2 R0 2l4

(28) To find the wavelength l* at which the free energies of the two fibers are equal, we set ∆ ) 0 and find

l* ) l*R

x

1 + x1 - (V - 1)V/2 2(1 - V)

V)

36π2LS2 R0γ

(29a) (29b)

where the parameter V is the characteristic ratio of the bulk nematic elastic energy (36π2LS2/R0) to the isotropic interfacial energy (γ). The result (eq 29a) is the product of the classical Rayleigh criterium lR* ) 2πR0 times a square-root factor arising from the isotropic surface elastic storage modes (γ) and the bulk nematic elastic storage mode (L). The additional elastic storage due to the nematic ordering increases the value of the classical critical wavelength, l* > l*R, introducing a new stabilization mechanism for the fiber morphology. From (29a), we see that a fiber may reduce its total energy only if V < 1. On the other hand, if V > 1, the fiber will not reduce its total energy by shape fluctuations. Thus, following the classical Rayleigh fiber instability criteria, we find that for this particular regime

01 R0γ

stable nematic fibers

(30b)

The stability threshold of the nematic LCP fibers is given by V ) 1. Figure 3 shows the scaled critical wavelength l*/2πR0 as a function of the dimensionless elastic energy ratio V. As V tends to one, the critical wavelength diverges, indicating that the fiber is stable

Figure 4. Dimensionless critical wavelength as a function of the dimensionless temperature T/T* for three values of 36Lπ2L/γR0: 0 (full line), 1 (dashed line), and 1.5 (dash-dot line). For 36π2L/ γR0 ) 0, the fiber is isotropic and we recover Rayleigh’s capillary instability. For 36π2L/γR0 ) 1, the nematic fiber is always unstable but the critical wavelength decreases with increasing temperature. As T increases toward T*, S decreases toward 0.5 and the difference between critical wavelength of a nematic fiber and an isotropic fiber diminishes. For 36π2L/γR0 ) 1.5, the nematic fiber is unstable for T/T* > 0.48 and stable for T/T*< 0.48. At sufficiently low (high) temperatures, S increases (decreases) and the nematic stabilization mechanism is able (unable) to overcome the capillary instability driven by the surface tension. For T > T*, S ) 0 and the model predicts l* ) 2πR0.

since energy savings due to the surface energy are canceled by energy cost due to the director gradients. As V grows infinitesimally small, the director bulk energy becomes negligible when compared to the surface energy and the Rayleigh result is recovered. The dimensionless surface energy ratio V is a function of the chemistry of the fiber-polymer matrix system and temperature. To identify the dependence of the stabilization of the nematic LCP fibers to surface distortions on temperature, we replace the Doi expression for S(T) in eq 29a and for simplicity assume that L, γ, and G are constants. Figure 4 shows the dimensionless critical wavelength as a function of the dimensionless temperature T/T*, for three values of 36Lπ2L/γR0: 0 (full line), 1 (dashed line), and 1.5 (dashdot line). For 36π2L/γR0 ) 0 the fiber is isotropic and

1120 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

we recover the Rayleigh instability result l* ) 2πR0. For 36π2L/γR0 ) 1 the nematic fiber is always unstable, but the critical wavelength decreases with increasing temperature. As T increases toward T*, S decreases toward 0.5 and the difference between a nematic fiber and an isotropic fiber diminishes. For 36π2L/γR0 ) 1.5, the nematic fiber is unstable for T/T* > 0.48 and stable for T/T* < 0.48. At sufficiently low (high) temperatures, S increases (decreases) and the nematic stabilization mechanism is able (unable) to overcome the capillary instability driven by the surface tension. For T > T*, S ) 0 and the model predicts l* ) 2πR0. 3.4. Stability Parameters for Nematic LCP Fibers. The above sections identified the main stabilizing and destabilizing mechanisms in nematic fibers. In isotropic fibers, the interfacial energy can decrease by periodic shape fluctuations longer than the fibers perimeter, thus producing the capillary instability. The presence of nematic order tends to stabilize the fiber due to increases either in bulk or in surface energies. Bulk energies are able to overcome the capillary instability and stabilize a nematic fiber. The strength of this stabilization mechanism increases with decreasing fiber radius (R0V), and with increasing orientation elasticity (Lv). On the other hand, the nematic surface energy is not able to stabilize a nematic fiber by overcoming the capillary instability, but it will increase the critical wavelength of the fluctuations that produce a decrease in the total energy. Although no experimental data seem to be available for all the parameters that determine the fiber stability, we can use estimates of the properties and fiber sizes that are relevant to in-situ LCP composites and, thus, predict if nematic fibers are ever stable under the assumed parametric conditions. For typical LCP fibers, we shall assume that R0 ) 1 × 10-6 m (Basset and Yee, 1990), γ )10-2 N/m (Doi and Kuzuu, 1985), and L ) 10-11 J/m (Donald and Windle, 1992). Recalling that the nematic surface energy will not stabilize the fiber, we only need to consider the nematic bulk stabilization effect. Using these values, we can estimate the ratio of nematic bulk energy to the isotropic surface energy V at various temperatures. At the nematic-isotropic transition temperature T* (S ) 1/ ), we find that V ) 0.089 < 1, and thus, the LCP fiber 2 is unstable to sinusoidal perturbations. At the other temperature extreme, assuming perfect ordering (S ) 1) we find V ) 0.356 < 1, and again, the LCP fiber is unstable. Using the critical threshold equation V ) 1 and the above parameters, we find that LCP fibers are stable for Rc < 0.089 µm. Since most reports (Acierno and La Mantia, 1993; Basset and Yee, 1990) indicate that the typical LCP fiber radius is greater than Rc ∼ 0.1 µm, we can conclude that nematic LCP fibers undergo capillary instability. This result is in agreement with experimental observations (see Figures 2, 4, and 5 of Verhoogt et al. (1993)), obtained using Vectra 900, a thermotropic liquid-crystalline polymer, blended with Kraton G1650, which is a thermoplastic elastomer. The experiments show that annealing of isolated LCP fibers as well as fibers embedded in the thermoplastic elastomer matrix, at temperatures above the melting temperature of the LCP, results in fiber breakup behavior that resembles that of Newtonian fibers (Verhoogt et al., 1993). The fiber breakup leads to irregular shapes, which have also been observed during slow cooling in LCP-polystyrene blends.

4. Conclusions This paper used classical theories of liquid-crystalline materials to develop a new model that describes the thermodynamic stability of nematic liquid-crystal cylindrical fibers, like those arising during the fabrication of in situ LCP polymer composites. In addition to isotropic interfacial tension, the model identifies two new elastic storage modes arising from the nematic order present in the LCP fibers. A new nematic surface mode arises due to misalignment between the average molecular orientation and the surface. The new nematic surface mode can store elastic energy and, therefore, tends to stabilize the cylindrical fiber but is not able to overcome the capillary instability. The net effect of the surface stabilization mechanism due to nematic ordering is to increase the minimum critical wavelength beyond the value of the isotropic case. In addition, a nematic fiber can store elastic energy by bulk orientation distortions. This bulk energy storage may be able to overcome the capillary instability if the fiber radius is sufficiently thin, if the interfacial LCP-polymer tension is sufficiently low, or if the orientation elastic constant is sufficiently high. Estimating the model parameters using published data, it is found that for typical LCP fibers arising in in situ LCP-polymer composites, the fibers break-up in agreement with experiments (Verhoogt et al., 1993). Acknowledgment This work is supported by a grant from the Natural Sciences and Engineering Research Council of Canada. Literature Cited Acierno, D., La Mantia, F. P., Eds. Processing and Properties of Liquid Crystalline Polymers and LCP based Blends; ChemTec: Toronto, 1993. Alexe-Ionescu, A. L.; Barberi, R.; Barbero, G.; Beica, T.; Moldovan, R. Surface Energy for Nematic Liquid Crystals: A New Point of View. Z. Naturforsch. 1992, 47a, 1235-1240. Basset, B. R.; Yee, A. F. A Method of Forming Composite Structures Using In Situ-Formed Liquid Crystal Polymer Fibers in a Thermoplastic Matrix. Polym. Comp. 1990, 11, 10-18. de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Oxford University: London, 1993. Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, 1986. Doi, M.; Kuzuu, N. Structure of the Interface between the Nematic Phase and the Isotropic Phase in the Rodlike Molecules. J. Appl. Polym. Sci.: Appl. Polym. Symp. 1985, 41, 65-68. Donald, A. M.; Windle, A. H. Liquid Crystalline Polymers; Cambridge University: Cambridge, 1992. Handlos, A. A.; Baird, D. G. Processing and Associated Properties of In Situ Composites Based on Thermotropic Liquid Crystalline Polymers and Thermoplastics. Rev. Macromol. Chem. Phys. 1995, C35 (2), 183-238. Jin, X.; Li, W. Correlation of Mechanical Properties with Morphology, Rheology, and Processing Parameters for Thermotropic Liquid Crystalline Polymer-Containing Blends. Rev. Macromol. Chem. Phys. 1995, C35 (1), 1-13. Larson, R. G. Constitutive Equations for Polymer Melts and Solutions; Butterworths: Stoneham, 1988. Lee, S. D.; Meyer, R. B. Elastic and Viscous Properties of Lyotropic Polymer Nematics. In Liquid Crystalinity in Polymers; Ciferri, A., Ed.; VCH: New York, 1991. Nobili, M.; Durand, G. Disorientation-induced Disordering at a Nematic-Liquid-Crystal-Solid Interface. Phys. Rev. A 1992, 46, R6174-6177. Qian, M.; Baicheng, L.; Runqi, L. Thermodynamic Considerations of the Equilibrium Shape of an Infinitely Long Rod. Acta Metall. Mater. 1994, 42, 4083-4086.

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1121 Qin, Y. A. Literature Review on the in situ Generation of Reinforcing Fibers. Polym. Adv. Technol. 1996, 7, 151-159. Rayleigh, L. On the Instability of Jets. Proc. London Math. Soc. 1879, 10, 4-13. Rey, A. D. Periodic Textures of Nematic Polymers and Orientational Slip. Macromolecules 1991, 24, 4450-4456. Rey, A. D.; Denn, M. M. Jeffrey-Hamel Flow of Leslie-Ericksen Nematic Liquids. J. Non-Newt. Fluid Mech. 1988a, 27, 375401. Rey, A. D.; Denn, M. M. Analysis of Transient Periodic Textures in Nematic. J. Non-Newt. Fluid Mech. 1988b, 27, 375-401. Verhoogt, H.; Willems, C. R. J.; Langelaan, H. C.; van Dam, J.; de Boer, A. P. Formation and Stability of Liquid Crystalline Polymer Fibres in a Thermoplastic Elastomeric Matrix. In Processing and Properties of Liquid Crystalline Polymers and

LCP based Blends; Acierno, D; La Mantia, F. P., Eds.; ChemTec: Toronto, 1993; pp 209-225. Warner, S. B. Fiber Science; Prentice Hall: Englewood Cliffs, NJ, 1994.

Received for review July 29, 1996 Revised manuscript received September 16, 1996 Accepted September 16, 1996X IE9604637

X Abstract published in Advance ACS Abstracts, February 15, 1997.