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J. Phys. Chem. B 2010, 114, 5700–5703
Tricritical Behavior in the Rotator Phases of Normal Alkanes Prabir K. Mukherjee† Department of Physics, Presidency College, 86/1 College Street, Kolkata 700 073, India ReceiVed: January 4, 2010; ReVised Manuscript ReceiVed: March 5, 2010
The first theoretical observation of the tricritical point for the rotator-I to rotator-V phase transition of normal alkanes is reported. The influence of pressure on the rotator-II-rotator-I-rotator-V phase sequence in normal alkanes has been studied within the Landau phenomenological theory. Our results show that for a particular value of pressure the first-order rotator-I to rotator-V phase transition becomes second-order at a tricritical point. We outline how the novel phase diagram could be detected experimentally. Introduction During the past several years rotator phases of normal alkanes have been receiving considerable attention1-11 because of their unique and unusual properties. Normal alkanes CH3-(CH2)n-2-CH3 are of great physical interest because of their specific rotator phases and their intricate phase transition behavior. Normal alkanes are the building blocks of many molecules including liquid crystals, surfactants, lipids, and polymer. Normal alkanes are also the main ingredient in many petroleum products such as fuels and lubricants. 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. 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. 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 about their long axes. These phases have a number of unique and unusual properties, which include surface crystallization, negative thermal compressibilities, anomalous high heat capacities and unusually high thermal expansions. The barrier for nucleation from the liquid into these phases is anomalously low, and they induce crystallization even beyond their range of thermal stability. Another unique feature of the rotator phases is that they are very similar to both Langmuir monolayers at liquid surfaces1,12 and liquid crystals. In alkanes, the rotator phases have been observed for carbon numbers 9 e n e 40 interposed, in temperature, between the liquid and fully crystalline phases. Five different rotator phases have been identified by X-ray techniques.13-20 These include the rotator-I (RI) phase, rotator-II (RII) phase, rotator-III (RIII) phase, rotator-IV (RIV) phase, and rotator-V (RV) phase. In the present study we will be concerned with the RII, RI, and RV phases. In the RI phase the molecules are untilted with respect to the layers and there is a rectangular distorted hexagonal lattice. The layers are stacked in an ABAB... bilayer stacking sequence. The RI phase is similar to the S phase of Langmuir monolayers. The rotator-II (RII) phase is usually described † E-mail:
[email protected].
Figure 1. Pressure-temperature phase diagram for C23. The measured data (points) are from ref 21, and the lines are the best theoretical fit from refs 4 and 11.
as 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. The RII phase is similar to the LS phase of Langmuir monolayers. The RV phase is the same as the RI phase except that the molecules are tilted toward their next nearest neighbor (NNN). This phase is similar to the L′2 phase of Langmuir monolayers. The transitions between the different rotator phases have been studied extensively experimentally by Sirota et al.,17,19 using calorimetry and X-ray diffraction at room pressure. Sirota et al.21 have also carried out measurements on the pressure dependence of these phases for alkanes Cn with n ) 21, 23, and 25 carbons. The pressure-temperaure phase diagram for C23 is shown in Figure 1. They studied the pressure and temperature dependences of the lattice distortion, tilt, and other structural properties. Among other things, they find that the RI-RV phase transition changes from first- to second-order with pressure, and that there is a strong coupling between the order parameter of the RI phase, the lattice distortion, and that of the RV phase, the tilt. In their high pressure studies, Sirota et al.21 did not show any specific P-T diagram where the RI-RV transition changes from first- to second-order. Rather they only pointed out the possibility of the second-order RI-RV transition under high pressure. The order of the RI-RV transition is somewhat ambiguous. The X-ray data show a possible discontinuity at the transition in the layer spacing, and hence also in the tilt angle, which argues for a first-order transition. However, the limited resolution of the measurements could not exclude a continuous transition, i.e., a second-order one. Theoretical studies of rotator phases follow two main lines. The first approach consists of the Monte Carlo and molecular
10.1021/jp1000495 2010 American Chemical Society Published on Web 04/14/2010
Rotator Phases of Normal Alkanes dynamics simulations,22 which confirms the two types of rotator phases. The second approach is pursued by Wurger5 and Mukherjee.4,6-11 In our previous work4 we studied theoretically the influence of pressure on the RI-RV phase transition. We also pointed out that the possibility of the first- to second-order RI-RV phase transition at elevated pressure. The tricritical behavior of phase transition, where a line of phase transitions goes from first- to second-order, has been studied extensively in various systems as metamagnets, He3-He4 mixtures, superconductors, and liquid crystals. Rotator phases are also expected to show the tricritical behavior. Naturally, one expects the tricritical behavior of the RI-RV phase transition. The tricritical point (TCP) can be observed by varying the concentration in binary mixtures and increasing pressure. Thus it is interesting to see under what conditions the RI-RV phase transition shows the tricritical behavior. To the best of the author’s knowledge, there is so far practically no experimental as well as theoretical study of the tricritical behavior near the RI-RV phase transition. The purpose of the present paper is to investigate the tricritical behavior of the RI-RV phase transition within Landau phenomenological theory. Model We present our analysis with the goal of finding the pressure dependent tricritical point at the RI-RV phase transition line within the Landau theory. First we derive the conditions for the various phase transitions involved. This is followed by plotting and discussing the new topology in the phase diagram. We present a detailed analysis of the different phases that can occur and analyze the question under which conditions the RI-RV transition is first- or second-order. To analyze our result, we first briefly describe various order parameters involved in the RII-RI-RV phase transition. For simplicity we neglect the weak interlayer interaction between the stacking layers in different rotator phases so that the problem becomes two-dimensional. The RI phase differs from the RII phase only in the distortion of the hexagonal lattice. Following1,12 we define the lattice distortion parameter ξ ) (a2 - b2)/(a2 + b2), where a and b are the major and minor axes of an ellipse drawn through the six nearest neighbors. The distortion ξ is defined with respect to a plane whose normal is parallel to the long molecular axes. ξ ) 0 for the hexagonal phase. Thus we take ξ as an order parameter for the RII-RI phase transition. The RV phase differs from the RI phase by tilt angle, we take tilt angle θ as another order parameter for the RI-RV phase transition. The tilt angle and distortion are described by two-component order parameters. 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 δ f δ + π/3 and ω f ω + π/3. Expanding the total Gibbs free energy gr in terms of the above-mentioned order parameters yields
1 1 1 1 gr ) gII + R′θ2 + β′θ4 - γ′θ6 cos 6δ + a'ξ2 2 4 6 2 1 3 1 b'ξ cos 6ω + c'ξ4 - G'θ2ξ cos 2(δ - ω) + 3 4 1 (1) H'ξ2θ2 cos (2δ + 4ω) 2
J. Phys. Chem. B, Vol. 114, No. 17, 2010 5701 where gII is the free energy of the RII phase. The coefficients a′ and R′ are assumed to vary strongly with an external parameter. The first two terms in eq 1 describe the tilt angle variation and the third term determines the tilt azimuth δ. In the absence of the third term and for ξ ) 0, the above free energy (1) describes a second-order transition. The third term is small in comparison with other ones but is the lowest order term depending on the angle δ. For γ′ > 0, the only minimum of gr is at δ ) nπ/3 (n is an integer); i.e., the tilt occurs to the nearest neighbor (NN) direction. If γ′ < 0, the minimum is achieved at δ ) π/6 + nπ/3, i.e., for tilt toward the next nearest neighbor (NNN) direction. Since in the RV phase tilt occurs toward the NNN direction, eq 1 describes a secondorder transition for the RI-RV and RII-RV transition for β′ > 0 and γ′ < 0. For θ ) 0, free energy (1) describes a firstorder RI-RII 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/3 for b′ < 0. According to the experimental observations, in the RI phase, ω ) 0. The last two terms are the lowest order coupling terms. The first coupling term gives ω ) δ for G′ > 0 and ω ) δ + π/2 for G′ < 0. Since in the RV phase δ ) ω ) 0, we take G′ > 0 for the description of the RI-RV transition. The second coupling term H′ > 0 represents a saturation and is chosen to be positive. b′, c′, β′, γ′, G′, and H′ are constants. G′ is chosen to be positive to favor the RV phase. Since the term γ′ is small in comparison to R′ and β′, we will neglect this term for the simplification of the calculation. To compute the phase diagram predicted by this model, it is convenient to introduce a rescaled variable to reduce the number of parameters. We define θ′ ) β′1/4θ and ξ′ ) c′1/4ξ. In terms of these variables, the rescaled free energy becomes
1 1 1 1 1 g ) gII + Rθ′2 + θ′4 + aξ′2 - bξ′3 + ξ′4 2 4 2 3 4 1 Gθ′2ξ′ + Hξ′2θ′2 2
(2)
where a ) a′/c′1/2, b ) b′/c′3/4, R ) R′/β′1/2, G ) G′/c′1/4β′1/2, and H ) H′/c′1/2β′1/2. The material parameters a and R can be assumed as a′ ) a0(T - T*1 (P)) and R′ ) R0(T - T*2 (P)). T*1 and T*2 are virtual transition temperatures. a0 and R0 are constants. From the experimental phase diagrams21 one observes at elevated pressure P, T*1 (P), and T*2 (P) can be portrayed as T*1 (P) ) T10 + uP and T*2 (P) ) T20 + VP. T10 is the lowest temperature (at P ) 0) at which the RII phase is metastable and T20 is the lowest temperature (at P ) 0) at which the RI phase is metastable. a0, R0, u, and V are positive constants. We assume G ) G(P) and H ) H(P). The equilibrium condition ∂g/∂θ′ ) 0 gives
θ′2 ) (R - 2G(P)ξ′ + H(P)ξ′2) g 0
(3)
which for θ′ ) 0 defines the limits of existence of the RV phase. In eq 3, θ′2 is negative for small ξ′. If P becomes strong enough, ξ′ should attain a value ξ′θ′ ) 0 for which eq 3 has a solution θ′2 * 0 and the RV phase should appear. The equilibrium condition ∂g/∂ξ′ ) 0 gives
G(P)R + [a - (2G2(P) + H(P)R)]ξ′ - [b 3G(P) H(P)]ξ′2 + [1 - H2(P)]ξ′3 ) 0
(4)
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Figure 2. Possible pressure (P)-temperature (T) phase diagram in the vicinity of the RI-RV tricritical point (TCP). The solid lines represent a line of first-order phase transitions while the dashed line represents a second-order phase transition.
which defines ξ′(T, P) for θ′ g 0. The spinodal line of the coupled system is given by
[a - (2G2(P) + H(P)R)] - 2[b - 3G(P) H(P)]ξ′ + 3[1 - H2(P)]ξ′2 ) 0
(5)
The RII, RI, and RV phases can arise either directly from the isotropic liquid phase or from one another along the curves RII-RI, RII-RV, and RI-RV. In the spirit of the Landau theory RII-RI and RII-RV transitions are first-order. The RI-RV transition can be either first- or second-order. If all the phase transitions involved are first-order ones, then one can observe the triple point. Otherwise, one can observe a TCP. The conditions for the first-order RII-RI transition can be obtained as
gI(ξ′) ) 0
gI′(ξ′′) ) 0
gI′′(ξ′′) g 0
(6)
The conditions for the first-order RI-RV transition are given by
gV(ξ′) ) 0
gV′ (ξ′) ) 0
gV′′(ξ′) g 0
(7)
The conditions for the second-order RI-RV transition read θ′2 ) R - 2G(P)ξ′ + H(P)ξ′2 ) 0
gV′ (ξ′) ) 0 gV′′(ξ′) g 0
(8)
Solving eqs 6-8 simultaneously will determine the various phase transition lines. The TCP is located at the intersection of the RV line eq 3 and the RV spinodal eq 5. These conditions together with eq 4 define the coordinates of the TCP. Results and Discussion Figure 2 summarizes the new topology of phase diagram associated with the free energy g, the conditions eqs 6-8 and the different values of the material parameter. Units of pressure and temperature are arbitrary; we assume that the temperature and density region is small so that linear dependence of phase transition temperatures on pressure is justified. Figure 2 shows
a novel phase diagram for the RII-RI-RV phase sequence. To draw Figure 2, we chose the following model parameters: For the RII-RI transition line we use values a0 ) 0.1, b ) 0.54, T10 ) 0.9, and u ) 0.49. The parameters corresponding to the firstorder RI-RV transition are chosen as T20 ) 0.8, R0 ) 0.1, V ) 0.4, G ) 1.41, and H ) 0.9 and using the values of a0, b, T10, and u obtained for the RII-RI transition. To obtain the secondorder RI-RV transition line, we chose the values G ) 0.32 and H ) 1.20. Other parameter values R0, T20, and V, are kept fixed at the values listed for the first-order RI-RV transition. There are three phases: RII, RI, and RV. For this system there are only two possible phase sequences: RII-RI and RI-RV. At low pressure both RII-RI and RI-RV transitions are first-order. As the pressure increases, the first-order RI-RV transition becomes second-order at the TCP. Thus the RI and RV phases are separated by the first-order as well as by the second-order transition lines. Then the stability conditions are broken at the TCP. The coordinates of the TCP are PTCP ) 2.6 and TTCP ) 1.35. However, the RII-RI transition is still first-order even at high pressure. Equations 6-8 are very sensitive to the values of G and H. Thus it is necessary to fine-tune these parameter values to get the topology of Figure 2. The other values of these parameters lead to different and unphysical results. The high values of G and H show that there is a strong coupling between the order parameters in the RV phase. The positive signs for G and H indicate that these coefficients indeed drive the system toward the RV phase. What transpires from the analysis above is that the coupling between the order parameters can be very important in determining the nature of the RI-RV transition and the stability of the different phases. Conclusion In conclusion, we have found a novel phase diagram for the rotator phases of normal alkanes, including the possibility of the TCP near the RI-RV transition. Figure 2 is a new kind of phase diagram in normal alkanes. The theory predicts the second-order character of the RI-RV transition at high pressure. In the presence of pressure, the first-order RI-RV transition can become second-order above a TCP. The coordinates of the TCP are calculated. The theory also predicts a first-order character of the RII-RI transition even at high pressure. A better understanding of how G and H vary with P and T would undoubtedly improve the qualitative and quantitative aspects of this description. Ideally, experimental observation should resolve whether the TCP near the RI-RV transition actually occur. Clearly more detailed experiments on the high resolution pressure-temperature study near the RI-RV transition of normal alkanes are highly desirable to determine if the novel phase diagrams presented here are actually accessible experimentally. Furthermore, a detailed density study near the RI-RV transition can also check the tricritical behavior for this transition. Finally, we point out that the theoretical confirmation of a RI-RV TCP at elevated pressure would suggest the existence of the rotator phases of alkanes with a second-order transition. If so, then a similar TCP should exist in appropriate mixtures of alkanes. We hope the present model stimulates further experimental studies in this direction. Acknowledgment. The author thanks the Alexander von Humboldt Foundation for a book grant. References and Notes (1) Sirota, E. B. Langmuir 1997, 13, 3849–3859.
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