Effect of the liquid crystal solute on the rotator phase transitions of n-alkanes

Abstract

Recent experimental studies have shown that the liquid crystal substance plays an important role in determining the structures and the phase transitions of the different rotator phases in binary mixtures of n-alkane and liquid crystals. The mixtures exhibit crystal and four different rotator phases R_I, R_II, R_III and R_V, although the R_III phase is absent in pure alkane. We described these experimental observations within phenomenological theory. The influence of the liquid crystal solute on the rotator phase transitions and the transition temperatures was discussed by varying the coupling between the concentration variable and the order parameters. The theoretical predictions were found to be in good qualitative agreement with the experimental results.

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I. Introduction

Normal alkanes, CH₃–(CH₂)_(n−2)–CH₃, are of physical interest because of their specific rotator phases and their intricate phase transition behavior. The pure n-alkanes have some peculiar features in phase equilibria, such as the odd–even effect, surface freezing effect¹ and nucleation kinetics.^2,3 Despite their simple molecular structures, long chain n-alkanes are the building blocks of many molecules, including liquid crystals, surfactants, lipids and polymers. Normal alkanes are also the main ingredient in many petroleum products, such as fuels and lubricants, as well as many pharmaceutical and petroderivative products. Special interest has been paid to the physical properties of the rotator phases in normal alkanes, with their specific anisotropic intermolecular and intramolecular interactions, which are of fundamental interest for the understanding of the phenomenological properties of other related macromolecular materials. Over the past few years, the rotator phases of n-alkanes have received considerable attention because of their unique and unusual properties. These include surface crystallization, anomalous heat capacity, negative thermal compressibilities, unusually high thermal expansions, and a vanishing nucleation barrier for the n-alkane rotator to crystal transformation.⁴ They also induce crystallization even beyond their range of thermal stability.⁵ It was observed that the rotator to crystal interfacial energy decreases as the entropy decreases. However, when the interfacial tension is extrapolated to zero, it disappears at a finite value of the entropy. This extrapolated disappearance of interfacial energy arises from the loss of the distinction between the rotator and X phases when the rotator phase is forced to a very low temperature.⁴ Another very interesting feature of the rotator phases is that they are very similar to Langmuir monolayers^6,7 in terms of the lattice distortion and the area/molecule. Thus, the rotator phases are interesting in their own right and we also expect that insights into their properties will help us to understand other condensed phases that exist in nature.

Rotator phases consist of layered structures with three-dimensional crystalline order in the positions of the molecular centers of mass, but no long-range orientational order in the rotation of the molecules around their long axes.^8–10 In alkanes (C_nH_2n+2), rotator phases have been observed for carbon numbers 9 ≤ n ≤ 40 interposed, in temperature, between the liquid and fully crystalline phases. Normal alkanes form a variety of different crystalline structures with triclinic, orthorhombic or hexagonal symmetry. The crystal structure of the even n-alkanes (triclinic) is different from that of the odd n-alkanes (orthorhombic). According to the literature,^11–15 in the crystalline form, odd n-alkanes exhibit a layered structure consisting of bilayer stacking of the lamellae (ABAB…). The low temperature crystalline phase (X) has orthorhombic (distorted-hexagonal) or triclinic packing within a layer, as well as long-range herringbone order of the rotational degrees of freedom of the backbones. The appearance of the long-range herringbone order in the low temperature X phase was determined by the presence of a Bragg reflection at the {2,1,0} position^6,11 (Fig. 3(c)). This peak is absent in the rotator phases and present in the X phase. In principle, herringbone order would appear as a diffusion peak at the {2,1,0} position, whose inverse width is related to the correlation length.⁶ The herringbone peak occurs at a position of q = 2.098 Å⁻¹ for the lattice parameters reported by Bohanon.¹⁶ Although such a diffuse peak is observed in liquid crystals^17,18 and in the crystalline phase of Langmuir monolayers,⁷ no such peak has been identified in alkanes due to the low intensities. In the case of liquid crystals, the centres of mass of the molecules are arranged at the nodes of a periodic rectangular lattice, the orientation of the molecules around their long axes being ensured by a glide-mirror placed in such a way that the molecular sections parallel to the smectic layers are arranged in a herringbone array. In the rigid crystal, most of the molecules contain all-trans configurations. However, when the temperature is increased a finite number of gauche bonds are known to occur near the chain ends. In the rotator phases, non-planar conformers begin to play an important role.¹⁹ The main type of such conformers is that containing the end-gauche modification, or gauche-trans-gauche sequence. The number of gauche defects is known to increase with increasing temperature in the rotator phases of longer chain lengths. Thus the gauche defects are known to strongly influence the transitions between rotator phases.^8,9

Five different rotator phases have been identified using X-ray techniques.^{8,9,11,12,15,20–23} These include the rotator-I (R_I) phase, rotator-II (R_II) phase, rotator-III (R_III) phase, rotator-IV (R_IV) phase and rotator-V (R_V) phase. These five rotator phases are distinguished by distortion, tilt and azimuthal order. In the rotator-I (R_I) phase the molecules are untilted with respect to the layers and there is a rectangularly distorted hexagonal lattice. The layers are stacked in an ABAB… bilayer stacking sequence (Fig. 1(a) and (b)). The rotator-II (R_II) phase is usually described as being composed of molecules that are untilted with respect to the layers that are packed in a hexagonal lattice. The layers are stacked in an ABCABC… trilayer stacking sequence (Fig. 1(c) and (d)). The R_III phase is triclinic, with molecules tilted in a non-symmetric azimuthal direction, in between the next nearest-neighbor (NNN) and the nearest-neighbor (NN) directions (Fig. 1(e) and (f)). The R_IV phase is monoclinic, with NNN-tilted molecules and end-to-end layer stacking (Fig. 1(g) and (h)). The distortion is small and in the same direction as the tilt. The finite distortion in this case is often considered to be induced by a finite coupling to the tilt. The R_V phase is the same as R_I phase, except that the molecules are tilted towards their next nearest neighbor (NNN) (Fig. 1(i) and (j)). The R_V phase contains a finite molecular tilt toward the NNN, the same direction as the direction of distortion. The transitions between the different rotator phases have been studied extensively experimentally by several authors,^8,9,24 using calorimetry and X-ray diffraction. Rotator phases pass through at least one rotator phase between crystalline solid and isotropic liquid. The transition between different phases corresponds to the breaking of some symmetry. Usually, the system does not re-enter the original state, unlike the liquid crystals. There is no pronounced even–odd effect within the rotator phases, unlike the liquid crystals.⁸ According to the calorimetry and X-ray diffraction measurements,^8,9 transitions from the R_V phase to the X phase (R_V–X) or from the R_I phase to the X phase (R_I–X) are first order. The R_I–X transition is known to involve the onset of long-range herringbone order.¹¹ The R_I–X and R_V–X transitions exhibit significant hysteresis due to the supercooling of the R_I and R_V phases. The R_II–R_I transition is weakly first order in nature, because only the distortion order parameter is lost at T_II–I and the heat of transition is only 0.1–1 kJ mol⁻¹.⁹ Here, T_II–I is the R_II–R_I transition temperature. The jump of the distortion order parameter at T_II–I is 0.003–0.03.⁹ The isotropic liquid to R_II (IL–R_II) transition is the strongest first order transition, with a supercooling value of less than 0.03 °C.

Fig. 1 Molecular arrangement of various rotator phases of n-alkanes. The unfilled circles represent the chain end positions in the second layer of the bilayer structure. The gray circles for the R_II phase represent the third layer in a trilayer structure. The arrow represents the tilt (θ) direction in the NNN tilted R_V phase.

Recently, investigations on binary mixtures of n-alkanes and nanoparticles, including porous matrices, have attracted increasing attention.^25–28 Experimental results^25–28 show that these substances can greatly enhance the physical properties of n-alkanes and rotator phases.^25–28 Very recently, Kumar et al.²⁹ carried out calorimetric and X-ray investigations on a binary system of n-tetracosane (C₂₄H₅₀) and the liquid crystalline compound butyloxybenzilideneoctylaniine (BBOA). According to the literature,^8,11,15 pure C₂₄H₅₀ exhibits the following four phases: an orthorhombic crystalline phase (X), a R_V phase, a R_I phase and a R_II phase. BBOA exhibits two liquid crystalline phases, namely the nematic (N) and smectic A (SmA) phases above the plastic phase referred to as the crystal B (CrB) phase. Owing to the chemical dissimilarity between the alkane and the liquid crystal molecule, the latter could act like an impurity and therefore try to destabilize the system. Kumar et al.²⁹ observed that if the alkane (C₂₄H₅₀) is present in a small concentration, the binary system (C₂₄H₅₀–BBOA) is nanophase segregated, whereas if the liquid crystal molecules are present in a small concentration, the layered structure merely gets roughened without any segregation. The more significant result of their experiments, at low liquid crystal concentration, is the induction of the R_III phase between the R_I and R_V phases. The phase diagram of the C₂₄H₅₀–BBOA system exhibits the following phase sequence: isotropic liquid (IL)–R_II–R_I–R_III–R_V–X. This is shown in Fig. 2. They pointed out that the appearance of the induced rotator phase R_III between the R_I and R_V phases could be due to the creation of additional gauche bonds at the termini of the alkane molecules. The addition of the liquid crystal component up to the alkane C₂₄H₅₀ leads to a decrease in the various transition temperatures and lattice distortion.

Fig. 2 Partial temperature–concentration phase diagram of C₂₄H₄₈–BBOA, showing the interesting feature that the R_III phase gets induced for a small concentration of BBOA and is bounded for a higher concentration. This results in two three-phase points in the vicinity of the concentration marked A and B.²⁹

The properties of rotator phases have been determined in a number of molecular simulation studies^30–36 and phenomenological studies.^37–44 Simulation studies showed that the structural properties of these phases can be captured by molecular models of n-alkane chains. Detailed atomistic simulations of shorter chains by Ryckaert et al.^30,31 confirmed the importance of translations in the rotator phases. Molecular dynamics simulations carried out by Ryckaert et al.³¹ showed that in the crystalline, orthorhombic phase, the chains remain all-trans and fully ordered, with a herringbone packing. By contrast, in the pseudohexagonal R_I phase, a dramatic increase in longitudinal chain motion is observed. Cao et al.³² studied the solid–fluid and solid–solid phase equilibria for binary mixtures of hard sphere chains, modelling n-alkanes, using Monte Carlo simulations and observed the first order character of the R_II–R_I transition. Marbeuf et al.³⁴ studied the pre-melting phenomena and molecular motions in the R_I and R_II phases of n-alkanes below the melting point using molecular dynamics simulations. Wentzel et al.^35,36 performed atomistic molecular dynamics simulations of ordered phases in pure C₂₃ alkane and mixed C₂₁–C₂₃ alkane systems, using several different all-atom potentials and confirmed the weak first order character of the R_II–R_I transition. Mukherjee^37–44 extensively studied the various rotator–rotator phase transitions within the framework of the Landau phenomenological model and renormalization group theory.

To the best of the author’s knowledge there are so far no detailed theoretical or simulation studies on the binary mixture of C₂₄H₅₀ and liquid crystals (BBOA). The purpose of the present paper is to develop a phenomenological theory to discuss some of the experimental results observed in binary mixtures of C₂₄H₅₀ and liquid crystals (BBOA) and to determine the influence of the concentration of the liquid crystal component on the various rotator phase transitions. The appearance of the induced rotator phase R_III in the binary system is discussed in great detail. It is shown how the coupling between the concentration and the order parameters shifts the transition temperatures.

II. Theory

In this section we apply the Landau theory of phase transitions, which has been quite successful^37–41 in explaining many features of rotator phases in n-alkanes. First we discuss the rotator phase transitions in pure alkane (C₂₄H₅₀). For simplicity, we neglect the weak interlayer interactions between the stacking layers in the X and rotator phases so that the problem becomes two-dimensional. The lattice distortion is the same in the X, R_I and R_V phases. The long-range herringbone order disappears in the rotator phases. Thus we take long-range herringbone order to be an order parameter for the R_V–X and R_I–X transitions. The herringbone ordering is a type of crystallization. Thus, the herringbone order parameter is described by^7,45 Φ = ϕ exp(i2η), where η is the angle between the local herringbone axis and a reference direction (Fig. 3(c)). The R_V phase differs from the R_I phase by the tilt angle (Fig. 3(b)), and so we take the tilt angle θ as an another order parameter for the R_I–R_V transition. The R_I phase differs from the R_II phase only in the distortion of the hexagonal lattice. Sirota et al.^8,9 define the lattice distortion parameter D = 1 − a/b based on an ellipse passing through all six nearest neighbors of a given molecule (Fig. 3(a)). a and b are the minor and major axes of an ellipse passing through all six nearest neighbors when viewed along the axis of the chains. However, Kaganer et al.^7,46 suggested the symmetrized form of the distortion parameter ξ = (a² − b²)/(a² + b²), where a and b are the major and minor axes of the ellipse. The distortion, D or ξ, must be defined with respect to a plane whose normal is parallel to the long molecular axes.⁶ We take ξ as the order parameter of the R_I–R_II transition, as it breaks the 6-fold symmetry of the R_II phase. It is zero in the high-symmetry R_II phase and finite in the low-symmetry R_I phase. Thus we take the herringbone order, tilt angle and lattice distortion as three primary order parameters involved in the various transitions involved in C₂₄H₅₀. Explicitly, we consider the ordering of a 2D hexagonal crystal. Thus the herringbone order, tilt angle, distortion and hexatic order are described by two-component order parameters. The components of the herringbone order can be expressed as amplitude ϕ and azimuth 2η. The tilt components can be expressed through a polar tilt angle θ and the tilt azimuth δ. The distortion components are expressed through the distortion amplitude ξ and the azimuth 2ω. The multiplier 2 comes from the fact that the distortion is a symmetric traceless tensor. Then the Landau free energy should be invariant η → η + π/3, δ → δ + π/3, ω → ω + π/3. Since the free energy is a scalar, the expansion only contains terms that are invariant combinations of the order parameters ξ, θ and ϕ. Expanding the total free energy F in terms of the abovementioned order parameters yields³⁸where F_II is free energy of the R_II phase. The coefficients p, α and a are temperature dependent. The other coefficients are assumed to be temperature independent. The first three terms in the free energy eqn (2.1) describe the herringbone order variation and first order R_V–X transition for θ = 0. The fluctuations and the cubic invariant of ξ in the free energy expansion give rise to the first order R_I–X and R_V–X transitions. The last two terms in the first line describe the tilt angle variation and the first term in the second line determines the tilt azimuth δ. For γ > 0, the only minimum of F is at δ = nπ/3 (n is an integer), i.e. the tilt occurs in the nearest neighbor (NN) direction. If γ < 0, the minimum is achieved at δ = π/6 + nπ/3, i.e. for tilt in the direction of the next nearest neighbor (NNN). Since tilt occurs in the NNN direction in the R_V phase, eqn (2.1) describes a second order R_I–R_V transition for β > 0, γ < 0 and ϕ = 0. For θ = 0 and ϕ = 0, the second, third and fourth terms in the third line describes a first order R_II–R_I transition for b > 0 and c > 0. In this case, the minimum free energy occurs at ω = 0 for b > 0 and at ω = π/2 for b < 0. According to the experimental observations, in the R_I phase distortion amounts to a compression in the NN bond direction, hence eqn (2.1) describes a first order R_II–R_I transition for b > 0, c > 0, θ = 0 and ϕ = 0. The ϕθ coupling term possesses a minimum at δ = 2η + π/2 for E > 0 and δ = 2η for E < 0. The coupling term ξϕ gives a minimum at ω = 2η for F > 0 and ω = 2η + π/2 for F < 0. In the X phase, distortion is in the same direction as the herringbone order. Since in the X phase ω = η = 0, we take F > 0 to describe the R_V–X transition. The θξ coupling term gives ω = δ for J > 0 and ω = δ + π/2 for G < 0. Since in the R_V phase δ = ω = 0, we take J > 0 for the description of the R_I–R_V transition.

Fig. 3 (a) Diagram showing the definition of the distortion order parameter ξ. (b) Diagram showing the tilt angle θ in the R_V phase. (c) Diagram showing possible herringbone orientations in the crystal (X) phase and the angle between the axis along which the nearest neighbors have parallel body orientations and the reference direction defining the herringbone angle η.

Now we consider the binary mixture of n-tetracosane (C₂₄H₅₀) and the liquid crystal compound (BBOA). When mixing C₂₄H₅₀ with BBOA the experiment confirmed the existence of the induced R_III phase between the R_I and R_V phases. To describe the anisotropy of the liquid crystal molecules, we introduced a scalar order parameter ψ, which models the orientation of the liquid crystal. The quantity defines the strength of the nematic ordering,⁴⁷ where θ₁ is the angle between the director and the long molecular axis. Furthermore, in the case of this mixture, the free energy must be expressed in terms of the symmetry-breaking order parameters and the concentration (weight percent) of the solute. Let x be the concentration (wt%) of BBOA in a mixture of C₂₄H₅₀–BBOA.

Consequently, the free energy for the binary mixture of C₂₄H₅₀–BBOA can be expressed as

where μ is the chemical potential difference between two components. The local coupling terms G₁, H₁ and K₁ guarantee that the anisotropy of the liquid crystal molecules remains small in the rotator phases where θ and ξ are large. The term in the free energy eqn (2.2) is the entropic cost of imposing anisotropy on liquid crystal molecules. G₂, K₂, H₂ and H₃ are coupling constants that describe the strength of the interactions between the liquid crystals and n-alkane. According to the experimental observations, for the R_III phase ω ≠ 0 and δ ≠ 0. The θξ coupling term gives ω = δ for J > 0 and ω = δ + π/2 for J < 0. Since in the R_V phase δ = ω = 0, we take J > 0 for the description of the R_III–R_V transition. The higher order terms like ξ²θ² cos(2δ + 4ω) and ξθ⁴ cos(4δ + 2ω) can be added, when necessary. We will see later that the higher order coupling terms d and η₁ are responsible for the description of the R_I–R_III and R_III–R_V transitions. Here, we assume a = a₀(T − T^*₁), α = α₀(T − T^*₂) and p = p₀(T − T^*₃). T^*₁, T^*₂ and T^*₃ are the hypothetical second order transition temperatures. a₀, p₀ and α₀ are positive constants. Furthermore, we assume b = b₀(x − x₀) and γ = γ₀(x − x₀). x₀ is the value of the concentration of BBOA for which the R_III phase begins to appear. b₀ and γ₀ are positive constants. In order of increasing concentration of the liquid crystal component, the relevant stable solutions of eqn (2.2) are as follows:

(i) R_II phase: ϕ = 0, θ = 0 and ξ = 0.

This phase exists for p > 0, α > 0, and a > 0.

(ii) R_I phase: ϕ = 0, θ = 0 and ξ ≠ 0.

This phase exists for p − Fξ > 0, α − Jξ > 0 and a < 0.

(iii) R_III phase: ϕ = 0, θ ≠ 0, and ξ ≠ 0 with tilt, and the distortion azimuth is directed along the tilt azimuth.

This phase exists for d ≠ 0, p + Eθ² > 0, η₁ ≠ 0, α − Jξ < 0 and a < 0.

(iii) R_V phase: ϕ = 0, θ ≠ 0, and ξ ≠ 0.

This phase exists for p + Eθ² > 0, α − Jξ < 0 and a < 0.

(iv) X phase: ϕ ≠ 0, θ = 0 and ξ ≠ 0.

This phase exists for p − Fξ < 0, α > 0 and a < 0.

All of these phases and their stabilities are listed in Table 1.

Table 1 Crystalline and rotator phases in the mixture of C₂₄H₅₀–BBOA

Phase type	Order parameters	Stability conditions
RII phase	ϕ = 0, θ = 0, ξ = 0	p > 0, α > 0, a > 0
RI phase	ϕ = 0, θ = 0, ξ ≠ 0	p − Fξ > 0, α − Jξ > 0, a < 0
RIII phase	ϕ = 0, θ ≠ 0, ξ ≠ 0, ω ≠ 0, δ ≠ 0	p + Eθ^2 > 0, η ≠ 0, α − Jξ < 0
RV phase	ϕ = 0, θ ≠ 0, ξ ≠ 0	p + Eθ^2 > 0, α − Jξ < 0, a < 0
X Phase	ϕ ≠ 0, θ = 0, ξ ≠ 0	p − Fξ < 0, α > 0, a < 0

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Here we must point out that the isotropic liquid phase (IL) must appear above the R_II phase. By lowering the temperature from the isotropic liquid phase, the above phases can appear sequentially. Thus it is clear from the solutions that four types of transition are possible: (i) R_II–R_I; (ii) R_I–R_III; (iii) R_III–R_V; and (iv) R_V–X, in addition to the IL–R_II transition. Experimental results²⁹ show that the above transitions are possible.

The elimination of ψ from eqn (2.2) and minimizing eqn (2.2) (after the elimination of ψ) with respect to x yields

Eqn (2.3) does not give ξ, ϕ and θ in the closed form. If , , , etc. are small in the region of interest, the exponential can be expanded, giving

where , , , and

From the relation eqn (2.4) it is clear that the additional terms δx₁, δx₂, δx₃ and δx₄ at the concentration x are related to the R_I, X, and R_V order parameters ξ, ϕ and θ and the azimuth ω and δ of the R_III phase. In the case of second order R_V–X, R_I–R_III, R_III–R_V and R_II–R_I transitions ϕ = 0, θ = 0, ξ = 0, δ = 0, and ω = 0, and δx₁ = 0, δx₂ = 0, δx₃ = 0, and δx₄ = 0. Hence only the R_II phase appears above the R_I phase. In the case of the first order R_V–X transition the jump δx₂ is proportional to that of ϕ. For the first order R_I–R_III and R_III–R_V transitions, the jumps δx₃ and δx₄ are proportional to those of θ, δ and ω. For the first order R_II–R_I transition the jump δx₁ is proportional to that of ξ. Thus, as long as δx₁ ≠ 0, δx₂ ≠ 0, δx₃ ≠ 0 and δx₄ ≠ 0 there is a possibility of two phase regions. Then the R_V + X and R_II + R_I phases can coexist, i.e. a two phase region appears. Thus, the addition of the liquid crystal component BBOA to the pure n-alkane C₂₄H₅₀ forms a two phase region. Experimental results²⁹ show the possibility of the formation of the R_V + X and R_II + R_I two phase regions. The above analysis agrees well with experimental phase diagrams (Fig. 2).

We will now discuss the effect of the concentration of BBOA on the different transitions in detail.

A. R_V–X transition

According to the experimental observations,⁶ the R_V–X transition is the onset of long-range herringbone order. Thus, to describe the first order R_V–X transition we now consider the free energy in terms of the order parameters ϕ, ξ and ψ and the concentration x. Expanding the free energy near the R_V–X phase transition to the lowest relevant order yields

F_V is the free energy of the R_V phase and a₁ is a positive constant.

Eliminating the equilibrium values of ξ and ψ from eqn (2.5) leads to the description of the free energy becoming

where

We notice from the renormalized coefficient q* that taking into account the coupling of ϕ, ψ and ξ with the concentration x leads to the renormalization of the coefficient q. Hence the coefficient q changes with the changes in the concentration x. Then the order of the R_V–X transition can also change. The quantitative determination of q* will reflect the strength of the first order character of the R_V–X transition. For a low concentration x of BBOA, q* > 0. Then the R_V–X transition becomes first order, which occurs in the C₂₄H₅₀–BBOA system. In this case, both the R_V and X phases can coexist, i.e. a two phase region appears. This two phase region is formed due to the presence of δx₂ in x in eqn (2.4). The R_V–X transition is accompanied by a jump of the herringbone order parameter Φ_V–X.

The renormalized R_V–X transition temperature T_V–X is given by

Eqn (2.7) can be rewritten in the expansion of x as

T_V–X(x) = A₁ − B₁x + C₁x² (2.8)

where , , , , and

Experimental findings²⁹ have shown that the R_V–X transition temperature decreases with the addition of the liquid crystal component BBOA to the pure alkane. The decrease in the R_V–X transition temperature is connected with the width of the two phase region and the entropy of the transition.

The jump of the enthalpy density at the transition point is

Eqn (2.9) can be rewritten in the expansion of x as

ΔH_V–X = A₀ − B₀x + C₀x² + D₀x³ (2.10)

where , , and .

B. R_I–R_III and R_III–R_V transitions

We will now discuss the effect of the liquid crystal component on the R_I–R_III and R_III–R_V transitions in the mixture of C₂₄H₅₀–BBOA. Then the total free energy can be written as

The R_III phase is an intermediate tilt where the tilt azimuth δ varies from 0° to 30°. Because the distortion itself is relatively small in the R_III phase, it was difficult to uniquely determine δ and ω independently. Hence for the R_III phase, we assume δ = ω ≠ 0. The distortion ξ in the R_III phase is very weak and is induced by a finite coupling to the tilt magnitude θ. Thus the tilt causes an induced distortion ξ ∼ θ². As we already pointed out, the lowest order angle dependent term for the ξθ coupling gives ω = δ for J > 0. Therefore J is constant. A positive value of η₁ and positive value of γ favors the R_III phase over the R_V and R_I phases and would describe the R_I–R_III and R_III–R_V transitions in the mixture of C₂₄H₅₀–BBOA. Thus, a positive value of η₁ and negative value of γ favor the R_V phase, corresponding to the mixture of n-tetracosane and an anisometric component. For α > 0, η₁ = 0 and γ = 0 describe the R_I phase.

The second order character of the R_III–R_V transition can be explained by taking into account the term ∼−dξ⁶ cos 12ω in the free energy expansion eqn (2.11). Since b is a function of the concentration, it can be positive as well as negative. Assuming d is positive, then for b = −2dξ³, i.e. for b < 0, the only minimum is at ω = 0°. Then the system is in the R_V phase. When b = 2dξ³, i.e. b > 0, the minimum at ω = 0° passes through 0° to 30°. We therefore have a second order transition from the R_III phase to the R_V phase at b = 2dξ³. As b increases from −2dξ³ to 2dξ³, the minimum in the R_III phase shifts from 0° to 30°.

Again, since γ is also a function of concentration, it can be positive as well as negative. Assuming η₁ is positive, then for γ = −2η₁θ⁶, i.e. for γ < 0, the only minimum is at δ = 0°. Then the system is in the R_V phase. When γ = 2η₁θ⁶, i.e. γ > 0, the minimum at δ = 0° passes through 0° to 30°. We therefore have a second order transition from the R_III phase to the R_V phase at γ = 2η₁θ⁶. As γ increases from −2η₁θ⁶ to 2η₁θ⁶, the minimum in the R_III phase shifts from 0° to 30°.

We will now discuss the R_I–R_III transition in the mixture of C₂₄H₅₀–BBOA.

After the substitution of ω = δ into the free energy eqn (2.11) and elimination of ψ from eqn (2.11), the free energy near the R_I–R_III transition reads

where F_I(ξ) is the free energy of the R_I phase.

The minimization of eqn (2.12) over δ gives

Suppose that η* is fixed at a negative value. γ decreases from a positive to negative value since γ is a function of x. When γ = 2η*θ⁶, i.e. for γ > 0, the only minimum is at δ = 0°. Then the system is in the R_I phase. When γ = −2η*θ⁶, i.e. for γ < 0, then the minimum at δ = 0° passes through 0° to 30°. Since in the R_III phase δ changes from 0° to 30°, we have the R_III phase for γ < 0. We therefore have a second order transition from the R_III phase to the R_I phase at γ = −2η*θ⁶, as observed in the mixture of C₂₄H₅₀–BBOA. As γ decreases from 2η*θ⁶ to −2η*θ⁶, the minimum in the R_III phase shifts from 0° to 30°as observed experimentally. The variation of the tilt azimuth δ with temperature in the R_III phase can be obtained by the expansion of cos 6δ and cos 12δ over the powers of δ.

Experimental findings²⁹ have also shown that a second order transition takes place between the R_I and R_III phases. Near the R_I–R_III transition the free energy reads

Here the tilt causes induced distortion ξ ∼ θ². Then the renormalized free energy reads

with and α₁ = α₀(T − T_C).

So tilt θ = 0 for T > T_C and θ ≠ 0 for T < T_C. So a second order transition takes place between the R_I and R_III phases. Here tilt is the order parameter for the R_I–R_III phase transition.

C. R_II–R_I transition

The R_II–R_I transition is accompanied by the finite jump of the distortion order parameter. This indicates the first order character of the R_II–R_I transition. Now the expansion of the free energy including the distortion order parameter ξ, anisotropic order parameter ψ and concentration x near the R_II–R_I transition can be written as

We now write down the free energy in terms of ξ only. So the elimination of ψ from the free energy eqn (2.16) yields

where

The variation of the distortion order parameter ξ(x,T) with x in the R_I phase is given by

where ξ⁺_II–I = b/2c. Eqn (2.18) shows that the distortion order parameter decreases with an increase in the concentration of the liquid crystal component in the R_I phase. To show more clearly the variation of the order parameter ξ with the concentration of the liquid crystal component as well as temperature in the R_I phase, we have plotted the order parameter ξ for three different concentrations (x) in Fig. 4. This is done for a set of phenomenological parameters for which the R_II–R_I transition is possible. Fig. 4 shows decrease of ξ with concentration x. This analysis clearly supports the experimental results.²⁹

Fig. 4 Temperature variation of the order parameter ξ in the R_I phase for the three different concentrations x = 5, x = 10 and x = 15.

The R_II–R_I transition temperature T_II–I is given by

Eqn (2.19) can be rewritten in the expansion of x as

where , and

The jump of the enthalpy density at the transition point is

Eqn (2.20) predicts that the R_II–R_I transition temperature decreases with the concentration x, which also agrees with experimental results. The jump of the order parameter ξ_II–I also changes with changing concentration. Since the Landau coefficient b changes with the concentration, b can become smaller. The smaller the value of b, the weaker the first order character of the R_II–R_I transition. Thus in the binary mixture of C₂₄H₅₀–BBOA, one can expect the weak first order character of the R_II–R_I transition, as confirmed experimentally.²⁹

D. IL–R_II transition

Zammit et al.²⁴ studied the IL–R_II transition in binary mixtures of alkanes. Over the IL–R_II transition, they observed that the single peak in both the specific heat and latent heat in the pure material splits into two features at different temperatures. This indicates the first order character of the IL–R_II transition. Kumar et al.²⁹ observed the weakening of the IL–R_II transition as the concentration of BBOA increased. The enthalpy jump became half at the IL–R_II transition with the addition of BBOA. Now, to describe the IL–R_II transition, we have to define the suitable order parameter for the IL–R_II transition. The R_II phase has a trilayer stacking sequence. The trilayer stacking can be represented as L₁L₂L₃L₁L₂L₃L₁L₂L₃… Following our previous work,⁴¹ we represent the probability of the k-th molecule of the system to occupy any of the j-th layer sequence as P_k(L_j), one of which is 1 and the other 0 in perfect layer ordering. Now we define a correlation factor β_k for the k-th molecule in a trilayer structure, i.e. in the R_II phase.

β_k = P_k(L₁)P_k(L₂) + P_k(L₂)P_k(L₃) + P_k(L₃)P_k(L₁). (2.22)

Its value is 0 (or almost zero) in the R_II phase. β_k does have a finite value in the liquid phase which can be calculated. In the IL phase the probability density of the k-th molecule is constant everywhere in space. So, P_k(L₁) = P_k(L₂) = P_k(L₃). In the IL phase the probability density of the k-th molecule is constant everywhere in space. So

where N_layer is the total number of layers. Hence β_k can be expressed as

Thus β_k is independent of k. Hence we define the correlation order parameter ζ as⁴¹

Here the average correlation factor is calculated for all molecules together.

Hence ζ is 0 in the IL phase and nonzero in the R_II phase. Thus, we take ζ and ψ as two order parameters involved in the IL–R_II transition in the mixture of C₂₄H₅₀–BBOA. Then the free energy near the IL–R_II transition can be expressed as

where F₀ is the free energy of the isotropic liquid phase. The material parameters u and v can be assumed as u = u₀(T − T^*₄) and v = v₀(x − x₀). T^*₄ is the virtual transition temperature. u₀ and v₀ are constant.

Again, we now write down the free energy in terms of ζ only. So the elimination of ψ from the free energy eqn (2.26) yields

where .

The variation of the correlation order parameter ζ(x,T) with x in the R_II phase is given by

where ζ⁺_II–I = v/2w. Eqn (2.28) shows that ζ changes with changes in the concentration x in the R_II phase. The IL–R_II transition temperature T_IL–II is given by

Eqn (2.29) can be rewritten in the expansion of x as

T_IL–II(x) = A₃ − B₃x + C₃x² (2.30)

where , and

The jump of the enthalpy density at the transition point is

Eqn (2.31) can be rewritten in the expansion of x as

ΔH_IL–II = (A₄ − B₄x + C₄x₃)(A₃ − B₃x + C₃x²) (2.32)

where , and

Eqn (2.30) predicts that the IL–R_II transition temperature decreases with the concentration x which also agrees with experimental results. The jump of the order parameter ζ_IL–II also changes with changing concentration. Since the Landau coefficients change with the concentration, v can become smaller. The smaller the value of v, the weaker the first order character of the IL–R_II transition. Thus, in the binary mixture one can also expect the weak first order character of the IL–R_II transition, as confirmed experimentally.²⁹

III. Comparison with experimental findings

In this section we will compare our theoretical results of the effect of the concentration of the liquid crystal solute on the IL–R_II, R_II–R_I and R_V–X transitions with the measurements of Kumar et al.²⁹ on the C₂₄H₅₀–BBOA mixture. These are, to the best of our knowledge, the only experimental results on the C₂₄H₅₀–BBOA mixture available in the literature.

For temperature versus concentration (T vs. x) phase diagrams for the IL–R_III, R_II–R_I and R_V–X transitions there exists only one experiment.²⁹ From eqn (2.18) and eqn (2.28), when ξ and ζ are fixed T vs. x should be non-linear, which agrees well with the experiment findings.²⁹ In principle, eqn (2.8) shows that the T_V–X vs. x phase diagram is also non-linear, which also agree with the experiment findings. According to eqn (2.8), eqn (2.20) and eqn (2.30), the R_V–X, R_II–R_I and IL–R_II transition temperatures decrease with an increasing concentration of the liquid crystal solute, as observed experimentally.²⁹ According to eqn (2.10) and eqn (2.32), the R_V–X and IL–R_II transition enthalpies also decrease with increasing concentrations of the liquid crystal solute, as observed experimentally.²⁹ In order to check eqn (2.8), eqn (2.20) and eqn (2.30), the measured T vs. x for the R_V–X, R_II–R_I and IL–R_III transitions of the C₂₄H₅₀–BBOA mixture of Kumar et al.²⁹ is plotted in Fig. 5. The fit yields A₁ = 42.52 °C, B₁ = 0.25 °C wt%⁻¹, C₁ = 0.0094 °C wt%⁻², A₂ = 46.39 °C, B₂ = 0.081 °C wt%⁻¹, C₂ = 0.0031 °C wt%⁻², A₃ = 49.65 °C, B₃ = 0.14 °C wt%⁻¹ and C₃ = 0.0018 °C wt%⁻². Furthermore, to check eqn (2.10) and eqn (2.32) the ΔH_V–X vs. x and ΔH_IL–II vs. x of Kumar et al.²⁹ are plotted in Fig. 6. The lines are the best fits from eqn (2.10) and eqn (2.32) to the data, resulting in the parameter values A₀ = 71.22 J g⁻¹, B₀ = 5.22 J g⁻¹ wt%⁻¹, C₀ = 0.54 J g⁻¹ wt%⁻², D₀ = −0.015 J g⁻¹ wt%⁻³, A₃ = 49.65 °C, B₃ = 4.66 °C wt%⁻¹, C₃ = 0.11 °C wt%⁻², A₄ = 3.23 J g⁻¹ °C⁻¹, B₄ = 0.91 J g⁻¹ °C⁻¹ wt%⁻¹ and C₄ = 0.061 J g⁻¹ °C⁻¹ wt%⁻². The fits to the measured values are good in Fig. 5 and 6. The agreement of the theory with the experiment is very good considering the scattered experimental data.

Fig. 5 The concentration dependence of the IL–R_II, R_II–R_I and R_V–X transition temperatures in the mixture of C₂₄H₅₀–BBOA. The measured data (points) are from ref. 29 and the lines are the best fits from eqn (2.8), eqn (2.20) and eqn (2.30).

Fig. 6 The concentration dependence enthalpy for the IL–R_II and R_V–X transitions in the mixture of C₂₄H₅₀–BBOA. The measured data (points) are from ref. 29 and the lines are the best fits from eqn (2.10) and eqn (2.32).

IV. Conclusions

We have presented here a mean-field analysis to describe the rotator–crystal and rotator–rotator phase transitions in the mixture of n-alkane and liquid crystals. The theory describes the effect of the liquid crystal component on the IL–R_II, R_II–R_I, R_I–R_III, R_III–R_V and R_V–X transitions. The coupling between the anisotropic order parameter of the liquid crystals and the macroscopic order parameters of the rotator phases of n-alkane is found to play a crucial role in determining the phase behavior and the order of the transitions. We found that although the R_III phase is absent in pure forms of C₂₄H₅₀, the R_III phase is induced with the increase of the concentration of the liquid crystal solute in the mixture of C₂₄H₅₀–BBOA. The R_V–X, R_II–R_I and IL–R_II transitions must always be first order, even in the mixture of C₂₄H₅₀–BBOA. The R_III–R_V and R_I–R_III transitions are second order in the mixture of C₂₄H₅₀–BBOA. Thus, our results are in qualitative agreement with experimental results. The suitable choice of the functional forms of the Landau coefficients, particularly the coefficients of γ and b will provide better ideas for the construction of the free energy and description of the R_III–R_V and R_I–R_III transitions. The values of the coefficients of γ and b are very sensitive for the description of the R_III phase. In order to gain a better understanding of the effect of the liquid crystal component on the rotator phase transitions in the mixture of C₂₄H₅₀–BBOA, further experimental and theoretical works are needed.

References

1. X. Z. Wu, B. M. Ocko, E. B. Sirota, S. K. Sinha, M. Deutsch, B. H. Cao and M. W. Kim, Science, 1993, 261, 1018–1021.
2. D. Turnbull and R. L. Cormia, J. Chem. Phys., 1961, 34, 820–831.
3. D. Uhlmann, G. Kritchevsky, R. Straff and G. Scherer, J. Chem. Phys., 1975, 62, 4896–4903.
4. A. B. Herhold, H. E. King Jr and E. B. Sirota, J. Chem. Phys., 2002, 116, 9036–9050.
5. E. B. Sirota and A. B. Herhold, Science, 1999, 283, 529–532.
6. E. B. Sirota, Langmuir, 1997, 13, 3849–3859.
7. V. M. Kaganer, H. Möhwald and P. Dutta, Rev. Mod. Phys., 1999, 71, 779–819.
8. E. B. Sirota, H. E. King Jr, D. M. Singer and H. H. Shao, J. Chem. Phys., 1993, 98, 5809–5824.
9. E. B. Sirota and D. M. Singer, J. Chem. Phys., 1994, 101, 10873–10882.
10. G. R. Strobl, B. Ewen, E. W. Fischer and W. J. Piesczek, J. Chem. Phys., 1974, 61, 5257–5264.
11. J. Doucet, I. Denicolo and A. Craievich, J. Chem. Phys., 1981, 75, 1523–1529.
12. J. Doucet, I. Denicolo, A. Craievich and A. Collet, J. Chem. Phys., 1981, 75, 5125–5127.
13. J. Doucet, I. Denicolo and A. Craievich, J. Chem. Phys., 1983, 78, 1465–1469.
14. G. Ungar and N. Masic, J. Phys. Chem., 1983, 89, 1036–1042.
15. G. Ungar, J. Phys. Chem., 1983, 87, 689–695.
16. T. M. Bohanon, B. Lin, M. C. Shih, G. E. Ice and P. Dutta, Phys. Rev. B: Condens. Matter Mater. Phys., 1990, 41, 4846R–4849R.
17. A. M. Levelut, J. Phys., Colloq., 1976, 37, 51–56.
18. R. Pindak, D. E. Moncton, S. C. Davey and J. W. Goodby, Phys. Rev. Lett., 1981, 46, 1135–1138.
19. M. Maroncelli, S. P. Qi, H. L. Strauss and R. G. Snyder, J. Am. Chem. Soc., 1982, 104, 6237–6247.
20. I. Denicolo, A. F. Craievich and J. Doucet, J. Chem. Phys., 1984, 80, 6200–6203.
21. E. B. Sirota Jr, H. E. King Jr, G. J. Hughes and W. K. Wan, Phys. Rev. Lett., 1992, 68, 492–495.
22. R. G. Snyder, G. Conti, H. L. Strauss and D. L. Dorset, J. Phys. Chem., 1993, 97, 7342–7350.
23. E. B. Sirota and X. Z. Wu, J. Chem. Phys., 1996, 105, 7763–7773.
24. U. Zammit, M. Marinelli and F. J. Mercuri, J. Phys. Chem. B, 2010, 114, 8134–8139.
25. H. Rensmo, A. Ongaro, D. Ryam and D. Fitzmaurice, J. Mater. Chem., 2002, 12, 2762–2768.
26. V. J. Kumar, S. K. Prasad and D. S. S. Rao, Langmuir, 2010, 26, 18362–18368.
27. K. Jiang, B. Xiae, D. Fu, F. Luo, G. Liu, Y. Su and D. Wang, J. Phys. Chem. B, 2010, 114, 1388–1392.
28. U. Zammit, M. Marinelli, F. Mercuri and F. Scudieri, J. Phys. Chem. B, 2011, 115, 2331–2337.
29. V. J. Kumar, S. K. Prasad, D. S. S. Rao and P. K. Mukherjee, Langmuir, 2014, 30, 4465–4473.
30. J. P. Ryckaert and M. L. Klein, J. Chem. Phys., 1986, 85, 1613–1620.
31. J.-P. Ryckaert, M. L. Klein and I. R. McDonald, Phys. Rev. Lett., 1987, 58, 698–701.
32. M. Cao and P. A. Monson, J. Chem. Phys., 2004, 120, 2980–2988.
33. M. Cao and P. H. Monson, J. Phys. Chem. B, 2009, 113, 13866–13873.
34. A. Marbeuf and R. J. Brown, J. Chem. Phys., 2006, 124, 054901.
35. N. Wentzel and S. T. Milner, J. Chem. Phys., 2010, 132, 044901.
36. N. Wentzel and S. T. Milner, J. Chem. Phys., 2011, 134, 224504.
37. P. K. Mukherjee and M. Deutsch, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 60, 3154–3162.
38. P. K. Mukherjee, J. Chem. Phys., 2002, 116, 10787–10793.
39. P. K. Mukherjee, J. Chem. Phys., 2007, 126, 114501.
40. P. K. Mukherjee, J. Chem. Phys., 2008, 129, 021101.
41. P. K. Mukherjee, J. Phys. Chem. B, 2010, 114, 5700–5703.
42. P. K. Mukherjee, J. Chem. Phys., 2011, 135, 134505.
43. P. K. Mukherjee, J. Phys. Chem. B, 2012, 116, 1517–1523.
44. P. K. Mukherjee and S. Dey, J. Mod. Phys., 2012, 3, 80–84.
45. R. Bruinsma and G. A. Aeppli, Phys. Rev. Lett., 1982, 48, 1625–1628.
46. V. M. Kaganer and E. B. Loginov, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1995, 71, 2599–2602.
47. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993.

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