#### Equilibrium Phase Diagrams

It is well known that the Flory–Huggins (FH) interaction parameter, χ depends not only on temperature, but also on composition.1, 13 While the corresponding state, the equation of state or the lattice fluid theories are capable of satisfactorily explaining the possible coexistence of both UCST and LCST, an alternative approach in describing the phase diagrams of real polymer systems is to treat the interaction parameter to be a more general function of both temperature and concentration, χ(*T*,ϕ). Hence, the FH free energy of mixing may be expressed as:

- (1)

where *r*_{1} and *r*_{2} represents statistical segment lengths of the constituent LCP and polymer, respectively, and ϕ is the volume fraction of LCP in the blends. *N* is the number of moles, *k*_{B} Boltzmann constant, and *T* the absolute temperature. χ(*T*,ϕ) is the generalized interaction parameter, but it cannot be determined explicitly. Hence, it has been expressed empirically as a product of two functions: temperature-dependent term and concentration-dependent term13 as

- (2)

In a simpler system, χ may be taken as a function of temperature only, thereby leading to

- (3)

Although both eqs 2 and 3 are empirical, the physical interpretation for each term of eq 3 may be conjectured in what follows. The first coefficient, *A* is the athermal (temperature independent) term which may be attributed to the entropic contribution to the intermolecular interaction, whereas the second term is the enthalpic in origin. The third term may be associated with the free volume that expands with temperature in a logarithmic manner. In principle, various types of phase diagrams depend on the sign of *C*. When *C* is positive, χ goes through a minimum, and a combined UCST and LCST phase diagram emerges. On the other hand, when *C* is negative, the closed loop phase diagram is obtained. When *C* is equal to zero, either UCST or LCST phase diagram can emerge depending on the sign of *B*. An hourglass phase diagram of polymer–polymer mixture can be established by increasing the positive χ, i.e., the UCST and LCST boundaries coalesce showing limited miscibility gaps only at the extreme compositions.

The total free energy density for MCLCP/polymer mixtures may be expressed by a simple addition of the FH free energy for isotropic mixing, the MS free energy for nematic ordering, and the Flory free energy for chain stiffening. As demonstrated by Matsuyama et al.,16, 17 the aforementioned model for the semirigid polymer systems is applicable over a wide range of chain rigidities. In their model, the rigidity of the system is controlled by a parameter, *x*, which denotes the fraction of bond angles which exist in an extended conformation with respect to a flexible conformation. Crystallization of the MCLCP was excluded as it is beyond the scope of this work. The free energy expression then includes entropic terms for rigid and bent bonds and an overall term that describes the free energy change upon switching from a bent to a rigid conformation. These terms are then added to the MS free energy of nematic ordering and the FH free energy of mixing to obtain the following free energy expression:17

- (4)

in which

- (5)

where *g*_{0} is the molar free energy difference between rigid and bent bonds, and the κ term represents the nonlocal contribution to the bulk free energy due to interfacial effects. χ_{a} signifies the anisotropic interaction parameter, representing the nematic–nematic LC interaction. The subscripts ϕ, *x*, *S* refer to concentration, fraction of rigid segments, and orientation order parameter fields. In addition, the parameter *I*_{m} is given by the integral expression involving,

- (6)

where η = χ_{a}*nx*^{2}ϕ. Here ϕ denotes the volume fraction of the LCP polymer whose molar volume is equal to *n* times the molar volume of the solvent, *S* denotes the uniaxial orientational order parameter, which may be defined as *I*_{1}/I_{0}, and θ denotes the angle that a polymer chain segment makes with respect to a reference direction (in this case, perpendicular to the plane of the 2-dimensional (2-D) system under consideration).

#### Phase Diagrams of MCLCP/Polymer Mixtures

Figures 2(a,b) depict a hypothetical combined LCST and UCST phase diagram and an hourglass phase diagram, respectively, for a given MCLCP/polymer mixture with a critical volume fraction ϕ_{c} = 0.5 and the ratio of the numbers of segments for each components, *r*_{1}/*r*_{2} = 1, keeping the constants in the FH interaction parameter χ_{FH} the same as those in Figure 1 setting *A* = −1.18 and *B* = 135 for the former case of the combined LCST/UCST, whereas for the latter case, we set *A* = −1.01 and *B* = 60.

A combined teapot and LCST phase diagram was established on the basis of the parameters of liquid–liquid phase separation used in Figure 2(a) with the following conditions for the nematic ordering: ν = 0.715/*T*, ω = *s*_{0}/*k*_{B} = 0.01, and ε = ε_{0}/*U*_{a} = 10, where ω and ε are the entropy and the internal energy related to the LCP chain rigidity, respectively. As shown in Figure 2(a), there are two critical temperatures, which represent LCST and UCST, respectively. The calculated phase diagram consists of two parts; one is an LCST corresponding to the liquid–liquid phase separation, and the other is an overlap of an UCST and a NI transition line, exhibiting liquid–liquid phase separation at high temperatures, the pure isotropic region at intermediate temperatures, pure nematic phases, and a variety of coexistence regions in a lower part of phase diagram such as liquid–liquid and nematic–liquid phases. The narrow NI phase transition line is bound between the liquidus line and the solidus line. Beyond this solidus line, the pure nematic phase exists. Following the methodology of Shen and Kyu,18 the nematic spinodal line is calculated as depicted by the dash lines in Figure 2(a). The nematic spinodal curve is the demarcation line between the nematic metastable and the nematic unstable regions.17, 19, 20 As labeled in the phase diagram, the nematic metastable region is bound by the pure nematic and the nematic spinodal line.

In the latter case, the NI transition temperature of the MCLCP melt, the *T*_{NI} was determined by self-consistently solving eqs 4–6 with the following numerical parameters: the NI transition temperature *T*_{NI} = 300 °C, the constants in anisotropic nematic interaction parameter, ν = 0.5/*T*, ω = *s*_{0}/*k*_{B} = 0.01, and ε = ε_{0}/*U*_{a} = 15. As shown in Figure 2(b), there are at most three solutions at a wide range of temperatures, which may characterized by the liquid spinodal lines in the unstable liquid–liquid region, and the nematic spinodal line in the unstable nematic region that separates the metastable nematic from the unstable nematic state. The calculated phase diagram is basically an overlap of an hourglass phase diagram and a NI transition, exhibiting only the liquidus line as the solidus line virtually coincided with the axis of the neat MCLCP.

#### Dynamics of Phase Separation

We shall now focus our attention on the spatio-temporal growth of thermal-quench induced phase separation (TIPS) in MCLCP/polymer mixtures. The spatio-temporal emergence of the LCP domains in MCLCP/polymer mixtures is generally governed by the interplay between liquid–liquid phase separation and the nematic ordering. The time evolutions of the LCP domains and the growth of structure factor may be described in the framework of the coupled time dependent Ginzburg–Landau equations (TDGL Model C) pertaining to the conserved compositional order parameter and the nonconserved orientational order parameter. We incorporated the local free energy densities of the FH theory for isotropic mixing, the MS theory for nematic ordering, and the Flory theory for chain stiffening in the coupled TDGL model C equations since *S* and *x* are interrelated such that21

- (7)

- (8)

and

- (9)

where the functional derivative is defined as: in which is the standard spatial operator in a vector form. ϕ(*r*,*t*), *S*(*r*,*t*), and *x*(*r*,*t*) represent the volume fraction, the orientational order parameter, and the chain rigidity of MCLCP at position *r* and time *t*, respectively. η_{ϕ} (*r*,*t*) represents thermal noise that satisfies the fluctuation–dissipation theorem.

The total free energy functional of such a system, *G* is given by the integration of the local free energy density over all volume, viz.,

- (10)

where *g*^{i} is the free energy density of isotropic mixing, *g*^{n} is the free energy density due to the anisotropic ordering, and *g*^{r} is the free energy density due to the chain stiffening and the tilda sign represents the vector operation. The term represents the free energy contribution of the concentration gradient in which κ_{ϕ} is a coefficient related to the segmental correlation length and the local concentration according to κ_{ϕ} = (*a*/ϕ^{2} − *a*/(1 − ϕ)^{2})/36. and represent the gradient of the free energy of the orientational order parameter and the chain rigidity, respectively. Since the MS theory has the inherent coupling between *S* and ϕ as described in the quadratic coupling form in eq 5 i.e., (1/2)χ_{a}*x*^{2}ϕ^{2}*S*^{2}, and eq 6, i.e., ln(*I*_{m}[η*S*]), it seems adequate for the present case. The additional coupling between the interface gradients required by others22, 23 because there is no inherent coupling (unlike the scalar order parameter of MS theory) between the two order parameters in their cases of the vector orientational order parameter. The coupled TDGL equations given in eqs 7–9 become

- (11)

- (12)

- (13)

where, Λ is the mutual diffusion coefficient that satisfies the Onsager reciprocal relationship, viz., Λ = Λ_{1}Λ_{2}/(Λ_{1} + Λ_{2}) with Λ_{1} = ϕ*D*_{1} and Λ_{2} = (1 − ϕ)*D*_{2}. *D*_{i} is the translation diffusion coefficients of the constituents. *R*_{s} and *R*_{x} are related to the rotational mobility of chain orientation and of chain rigidity, respectively. The chemical potentials with respective to individual order parameters may be described in what follows:

- (14)

- (15)

- (16)

- (17)

- (18)

For simplicity, it is assumed that the rotational mobility, *R*_{s} and the interfacial gradient coefficient, κ_{s} are taken as constants. Moreover, the formation of rigid bonds is considered as instantaneous, i.e., the polymer chains can reach the equilibrium state instantaneously. Since *s* and *x* are interrelated, the coupled TDGL equations may be reduced to two coupled equations with the conserved compositional order parameter ϕ and the nonconserved orientational order parameter *S*, i.e.,

- (19)

- (20)

The time evolution of LCP domain in MCLCP/polymer mixtures may be investigated by numerically by solving eqs 11 and 13 with a periodic boundary condition. The coupled TDGL equations were solved on a 256 × 256 square grid by the explicit method for temporal steps and the central difference scheme for spatial steps.

A critical *T* quench was triggered in an unstable nematic region of the hourglass type phase diagram at *T* = 240 °C and ϕ = 0.5 [denoted by point (a) in Figure 2(b)] to investigate the time evolutions of LCP domain structure and the growth of the structure factors in MCLCP/polymer mixtures. The parameters used were: diffusivities of polymer as well as MCLCP are comparable, i.e., *D*_{1} = *D*_{2} = *D* = 15,000 (nm^{2}/s); the correlation length *a* = 1.25 (assuming *a*_{1} and *a*_{2} are equal); Δ*t* = 0.02; *R*_{s} = 1.0/s and κ_{s} = 0.001. This point (a) is located above the NI line and thus the system is unstable state against the compositional order parameter, but it is stable against the nematic ordering before phase separation. As can be seen in Figure 3, the percolated interconnected SD like textures develop without accompanying any development of the nematic domains (see 40,000 time step). At 57,000 time step, the nematic ordering begins to develop with the preformed LC rich domains suggesting that the concentration of the LC-rich phase has reached the NI transition line. This in turn suggests that the nematic ordering lags behind the composition field. At 70,000 time steps, one can discerned the break-up of the interconnected SD domains. By that time, the nematic ordering caught up with the composition field showing the same morphology. These domains structures remains virtually remained the same with continued elapsed time, indicative of reaching an asymptotic equilibrium.

As shown in Figure 4, the development scattering halo can be discerned in Fourier space of concentration at 40,000 time step, but there was no scattering in the orientation field in the corresponding time steps. The scattering halo emerges in the orientation field only after 57,000 time steps. Through-out the time frame investigated, the diameter of the scattering ring virtually remains unchanged as the liquid–liquid phase separation dictates the nematic ordering to take place within these preformed domains. The invariance of this scattering peak should not be interpreted as the early stages of spinodal decomposition, because the transformation of interconnected SD structure to droplet morphology already occurs implying that the phase separation may be already in the very late stages of SD. The slope is rather low, i.e., approximately −1/4 which is smaller than the classical 1/3 growth. Since the nematic ordering itself is an energy minimization process thus the preformed domains do not to grow appreciably in the structural relaxation process. It may be inferred that the morphology emerging process in this region (a) of the hourglass phase diagram is different from that observed in the conventional thermal quenching from a single phase into the UCST two-phase SD region.21 In the previous case, a slope of −2/5 was obtained in the liquid–liquid separating region of a UCST envelope. In this region, the liquid crystal molecules have yet to reach the critical concentration to form nematics, but due to the mutual alignment of LCP molecules, thus the viscosity would be low or conversely the LCP mobility would be fast that may expedite the growth process. To clarify the role of nematics in the growth behavior, the thermal quenching was performed to a lower temperature below the NI line, i.e., *T*-quench to point (b) in the phase diagram.

Thermally induced phase separation can be triggered at point (b) at 200 °C of the hour-glass phase diagram that corresponds to the unstable nematic state with respect to both the orientational and the compositional order parameter fields. The parameters used were: diffusivities of polymer as well as MCLCP are comparable, i.e., *D* = 10,000 (nm^{2}/s), the correlation length *a* = 1.25; Δ*t* = 0.02; *R*_{s} = 1.0/s; and κ_{s} = 0.001. As depicted in Figure 5, the development of the interconnected structure appears in the compositional order parameter in the left column within 200 time steps (picture not shown) that grows rapidly with elapsed time of 1000 steps. The nematic ordering takes place almost simultaneously with the liquid–liquid phase separation, and thus the percolated network domains are somewhat rugged having sharp edges, which is reminiscent of a so-called amoeba structure. The development of amoeba structures in the MCLCP/polymer blends may be unique to phase separation via nematic spinodal decomposition as the system crosses the nematic spinodal line almost spontaneously. From 1000 steps through 10,000 steps, the interconnected channels of the percolated network structure are disrupted through coalescence. At 50,000, the percolated structure transforms to the cluster domains in the compositional order parameter, suggestive of the percolation-to-cluster transition. This crossover process is discernible in the orientational order parameter field in the right column in which the emerging patterns are almost identical to those in the composition field.

Having investigated the time evolutions of both compositional and orientational order parameters in real space, the next logical step is to mimic the temporal evolution of scattering patterns by taking the 2-D fast Fourier transformation of the domain structures of the concentration field of Figure 5 (the middle column). At the early period, the broad and weak scattering halo is developed at 200 steps (picture not shown), then the size of this scattering halo is reduced to a smaller diameter corresponding to the growth of nematic spinodal structure due to coalescence (1000 steps). During 10,000–50,000 time steps, the scattering halos show minute change in average diameter. Subsequently, the scattering halo collapses in diameter slightly with further progression of time to 200,000 steps.

The scaling behavior of the growth dynamics has been analyzed in terms of the temporal evolution of the scattering maxima in both compositional and orientational order parameters. As depicted in Figure 6, the scattering maxima (*q*_{max}) for both order parameters are similar, and thus only that of the concentration field is plotted as a function of time to demonstrate the growth curves for the temperature quench into the region (b). The growth dynamics seemingly follow a power law exponent close to –1/4, corresponding to the emergence and growth of nematic spinodal structure during the initial process. It should be cautioned that the time interval cover here is too small to make any claim. However, it may be speculated that the formation of multi nematic domains may have hampered the growth process in comparison with those of the liquid–liquid phase separation. In the intermediate region, the growth somewhat slows down showing a plateau-like trend which probably corresponds to the loss of the interconnectivity of the percolated structure. The amoeba structure gets rounded without significant change in the domain size, and thus the interdomain distances remain virtually unchanged. Again the surface smoothing of the amoeba structures to the rounded domains itself is already the surface energy minimization process. Hence, the growth of the domains may slow down since the growth of domain size is not the only process to occur in the structural relaxation process. Subsequently, the growth resumes again with a larger slope of −1/3 suggesting that the growth of the nematic LCP-rich droplet morphology. In view of the very small time intervals covered, these slopes should not be over-interpreted. The growth process following the *T* quench to region (b) may be characterized in three stages: (i) the instantaneous appearance of interconnected domain structure in the first stage (1000 time steps), (ii) the formation of amoeba-type structure in the intermediate stage (10,000–50,000 time steps), and the transformation of amoeba morphology to the anisotropic cluster domains and (iii) the onset of the coalescence of the domains (50,000–200,000 time steps). The temporal growth trend of Figure 6 is very similar to that reported by Nakai et al.,15 except that they captured the longer growth of the liquid–liquid phase separation with the growth exponent of 1/3. The present work may be seeing the tail end of this 1/3 regime which is influenced by the nematic ordering leading to a smaller slope of −1/4. The transformation of the amoeba to the anisotropic cluster seen in the experiment of Nakai et al.15 is captured in the present simulation. Although the present calculation was done for a hypothetical MCLCP/polymer blend, the agreement in respect of the emerged morphological pattern of the LCP domains to those experimental findings by Nakai et al.15 is promising. It should be cautioned that the growth exponent was limited to a very small time interval and thus the analysis on the dynamical scaling behavior should be regarded as tentative. Nevertheless, the predicted trends suggest that the present model simulation may have a wider validity to other systems such as rigid-rod polymer mixtures. The present model may be improved by changing the scalar orientational order parameter to the vector order parameter as demonstrated by Fukuda22 for the small molecule LC/polymer systems.