Study of the Critical Behavior and Scaling Relationships at the N-to-I

de Catalunya, Diagonal, 647 08028 Barcelona, Spain, and Departamento de Física Aplicada II, Facultad de ... Publication Date (Web): November 12, ...
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J. Phys. Chem. B 2009, 113, 15967–15974

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Study of the Critical Behavior and Scaling Relationships at the N-to-I Phase Transition in Hexyloxycyanobiphenyl J. Salud,† P. Cusmin,† M. R. de la Fuente,‡ M. A. Pe´rez-Jubindo,‡ D. O. Lo´pez,*,† and S. Diez-Berart† Grup de Propietas Fı´siques dels Materials (GRPFM), Departament de Fı´sica i Enginyeria Nuclear, E.T.S.E.I.B. UniVersitat Polite`cnica de Catalunya, Diagonal, 647 08028 Barcelona, Spain, and Departamento de Fı´sica Aplicada II, Facultad de Ciencia y Tecnologı´a, UniVersidad del Paı´s Vasco, Apartado 644, E-48080 Bilbao, Spain ReceiVed: June 30, 2009; ReVised Manuscript ReceiVed: October 7, 2009

An exhaustive analysis of the critical behavior of the nematic to isotropic (N-to-I) phase transition on the liquid crystal hexyloxycyanobiphenyl (6OCB) has been performed. To do so, the accurate evolution of various physical magnitudes (static dielectric permittivity data together with specific heat and volumetric determinations) around the N-to-I transition has been required. The specific heat data with the isobaric thermal expansion coefficient and the derivative of the static dielectric permittivity with temperature have been proven to be related to each other by a scaling relationship. However, some discrepancies have been observed for the dielectric data in relation to such a scaling relationship and the critical behavior of the N-to-I phase transition. All information has been used to get some insight on the strength of the first-order N-to-I phase transition of the 6OCB in relation to the other counterparts in the nOCB series of compounds. 1. Introduction Liquid crystalline states have been and are interesting condensed states of matter, not only from the technological but also from the basic research point of view. Most of the liquid crystal materials are composed of asymmetric rod-shaped molecules and exhibit a variety of liquid crystalline states or mesophases, the properties of which are intermediate between those of an ordered crystal and a disordered liquid. These mesophases are basically defined by the orientational and spatial ordering of the molecules. One of the most common mesophases is the uniaxial nematic (N) phase in which the centers of mass of the molecules are randomly distributed and do not exhibit long-range positional order, but they are characterized by longrange orientational order as a result of the alignment of the molecular axes along a preferred direction, the so-called molecular director. The N-to-isotropic (I) phase transition in liquid crystals has attracted considerable attention, both theoretically1–11 and experimentally,11–18 but there are still no clear answers to some key questions about its nature. Taking into account that in the isotropic phase there is a random orientation of the symmetry axes, whereas in the nematic phase there is orientational order of the symmetry axes, the N-to-I phase transition seems to be an order-disorder phase transition. According to the MaierSaupe theory,1–3 its origin results from the competition between thermally excited fluctuating forces tending to destroy the orientational order and “mean-field” molecular forces tending to align the molecular axes. The simplest and best-known description of the N-to-I phase transition is given by the Landau-de Gennes theory,7 which corresponds to a mean-field approach in which the thermodynamic properties of a system are based on treating the order * Corresponding author. E-mail: [email protected]. † E.T.S.E.I.B. Universitat Polite`cnica de Catalunya. ‡ Universidad del Paı´s Vasco.

parameter (denoted as SN assuming an optically uniaxial liquid crystal) as spatially constant and constitutes a useful description of the system if spatial fluctuations are not important. In this theory, the expansion of the free energy density in powers of the nematic order parameter up to the fourth order contains a cubic term B responsible for the weakly first-order character of the N-to-I transition, TNI being the corresponding transition temperature. The Landau-de Gennes theory leads to the same specific-heat critical exponent R and the order parameter critical exponent β in the N phase (RN ) βN ) 0.5), while in the isotropic phase RI is set equal to zero. These theoretical predictions are noncompatible with experimental results11–18 that seem to point out a nearly tricritical behavior in which β ) 0.25 and RN ) RI ) 0.5. This tricritical behavior was suggested by Keyes5 and is reflected by the Landau-de Gennes theory when the free energy density is expanded up to the sixth-order term E in the nematic order parameter. One of the most recent models proposed to describe the properties at the N-to-I phase transition in the context of the Landau-de Gennes theory is the fluidlike model,8,10 according to which TNI lies on a branch of a hypothetical coexistence (binodal) curve; i.e., TNI is a fluidlike critical temperature, and the corresponding spinodal temperatures are T* and T** and correspond to the metastable limits of the I and N phases, respectively. There exists, thus, a hypothetical critical region around the TNI first-order transition temperature at which the Landau-de Gennes theory does not apply. Mujherjee10 showed that those values for (TNI - T*) observed for low molecular weight liquid crystals tend to be more consistent with tricritical behavior. Small values of (TNI - T*) of about 200 mK18 or even less are rare in low molecular weight liquid crystals, but in such a case, the cubic coefficient B has to be very small and the N-to-I phase transition very weakly first-order. Fluctuations of the nematic order parameter seem to provide a reason for the smallness of B.11 Even so, there still remain some aspects concerning the characteristic temperatures, T*, T**, and TNI,

10.1021/jp906105u CCC: $40.75  2009 American Chemical Society Published on Web 11/12/2009

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and their relative differences which seem to depend on the used experimental technique. Critical fluctuation effects appear to be significant in nonlinear studies associated to sufficiently strong applied external fields. Fluctuations of the order parameter also play a relevant role in physical properties of strongly polar mesogens in the vicinity of the N-to-I phase transition when the external field is weak, as in the case of linear static dielectric permittivity, or null, as in the case of specific heat or molar volume studies. Unfortunately, powerful methods of fluctuation theory such as the renormalization group approach4,7,11 can be strictly applied to second-order phase transitions. It should be stressed that such kinds of calculations, made by Priest and Lubensky,4 in the vicinity of the so-called Landau point showed that strong fluctuation effects may lead to critical exponents for the N-to-I phase transition corresponding to the O(5) Universality Class (n ) 5 and d ) 3). In this general approach, the nematic order parameter is considered to display five independent components: one component is the so-called SN, two components account for the biaxiality, and the other two account for the director fluctuations. In liquid crystals which are uniaxial, SN is the usual nematic order parameter, and the other components are zero on average. In this paper, a detailed investigation of the N-to-I phase transition of 6OCB, to test its critical behavior by means of several thermodynamic properties which are likely to be influenced by pretransitional effects, is made. This study has been carried out on the basis of specific heat, molar volume, and static dielectric permittivity as a function of temperature around the transition, in a manner similar to that already made for other compounds of the nOCB series.16,19 It is important to realize that, as far as we know, no specific heat data around the N-to-I transition of the 6OCB have been reported up to date. As for molar volume data, only very few values for N and I phases were reported in the literature,20 as it has been already evidenced in a previous work by us.21 Molar volume data of 6OCB published in our previous work,21 completed with new values, will be used in the current study to obtain information about the N-to-I critical behavior. As far as the static dielectric permittivity around the N-to-I phase transition of 6OCB is concerned, many data can be found in the literature.14,22–24 However, some discrepancies are observed, and even, in certain cases, few data are reported at temperatures close to the transition making an accurate analysis to extract information about the N-to-I critical behavior very difficult. For that purpose, new data are presented in the current study. From all this experimental information, new insights related to the scaling relationships predicted by theoretical models25,26 are possible to obtain. Moreover, information about the strength of the firstorder N-to-I phase transition will be extracted following a comparative procedure taking into account several nOCB liquid crystal homologues. Such information is usually addressed through latent heat and volume jump determinations. The difference between the spinodal temperatures (T** - T*), as earlier proposed by Zywocinski,15 will be tested as a parameter to explore the strength of a first-order transition. It is important to realize that spinodal temperatures are difficult to find in experiments and are important parameters from a technical point of view. To cite an example, phase separation during spinodal decomposition, important in the technology of polymers or the well-known polymer-dispersed liquid crystals (PDLC), occurs beyond the spinodal T* temperature. The study of phase transitions in a pure component, for instance, liquid crystals, deserves a comparable situation and contributes greatly in

Salud et al. streamlining the study of such parameters. On the other hand, the very small value of (TNI - T*) of about 30 mK obtained for 6OCB in the present paper indicates that the coefficient B is very small and the N-to-I phase transition approaches to second order. The possibility of a nearby Landau point and the more than likely absence of biaxial character could be of importance in further studies. The paper is organized as follows. In section 2, the experimental details are described. In section 3, the results at the N-to-I phase transition of 6OCB coming from specific heat, static dielectric permittivity, and volume measurements are presented. On the basis of these results, an analysis of the critical behavior at the N-to-I phase transition is made. Finally, an overall discussion and a summary of the main conclusions are presented in sections 4 and 5, respectively. 2. Experiment 2.1. Material. The mesogenic liquid crystal 6OCB was synthesized by Professor Dabrowsky in the Institute of Chemistry, Military University of Technology, Warsaw, Poland. The purity was stated to be higher than 99.9% and was used without any further purification. 2.2. Specific Heat Measurements. Specific heat data at normal pressure were obtained by means of a commercial differential scanning calorimeter DSC-Q100 from TA-Instruments working in modulated mode (MDSC) to obtain the static specific-heat data. It is important to realize that similar to an AC-calorimeter, the MDSC-technique, besides specific heat data, simultaneously provides phase shift data (δ) which allow determining the coexistence region in weakly first-order transitions. In such a case, the experimental conditions were adjusted in such a way that the imaginary part of the complex specific heat data vanished. Likewise, by means of a special calibration procedure in which very precise latent heat data measured from other homologous compounds through adiabatic calorimetry are considered, the MDSC-technique is also suitable for quantitative measurements of latent heats of first-order transitions, even if they are weak. A more detailed description of the MDSC technique can be found somewhere else.16,19,27 The measurements consisted of heating and cooling runs around the N-to-I phase transition at 0.01 K · min-1, with a modulation temperature amplitude of (0.035 K and a period of 25 s. The sample masses (chosen between 1 and 2 mg) were selected to ensure a uniform thin layer within the aluminum pans. 2.3. Static Dielectric Measurements. The dielectric permittivity in the N and I phases of 6OCB was obtained using two impedance analysers: the HP 4192A (frequencies up to 13 MHz) and the Agilent 4291A (frequencies up to 1.8 GHz). The cell consists of two gold-plated brass electrodes (diameter 5 mm) separated by silica spacers making a plane capacitor. A modified HP16091A coaxial test fixture was used as the sample holder. It was held in a cryostat from Novocontrol, and both temperature and dielectric measurements were computer controlled. Two different alignments of the sample in the plane capacitor were considered: parallel (director parallel to the probing electric field) and perpendicular (director perpendicular to the probing electric field). For the former, no treatment of the electrodes was necessary to get a correct alignment. As it will consider later, the 6OCB exhibits a positive dielectric anisotropy, and so, the parallel alignment is promoted by applying a dc-bias voltage. However, no increase in permittivity was observed upon application of dc-bias voltage up to 40 V, allowing us to think that the metallic electrodes induce a quite good parallel

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Figure 1. Specific heat data as a function of temperature for 6OCB in a temperature range from the crystalline phase (Cr) to the isotropic phase (I). The inset shows the N-to-I phase transition in a zoom window.

alignment. To obtain perpendicular alignment, both metal electrodes were, beforehand, spin coated with PI2555 polyamide (HD Microsystems) following the procedure described in the manufacturer recipe and next rubbed uniaxially to obtain the required alignment of the liquid crystal molecules. It is important to realize that the polyamide layers, although very thin, contribute to the measured capacity in such a way that a correction procedure was performed. Additional details of such a procedure and also about the technique can be found somewhere else.16,19 The perpendicular alignment was tested, comparing the results with those obtained using 5 µm thickness Linkam cells. From the different test experiments, the frequency of 105 Hz could be assumed as the static dielectric permittivity is lower than the relaxation frequency of the molecular modes and high enough to reduce the low-frequency electrode effect. 2.4. Molar Volume Measurements. An Anton Paar DMA5000 density meter coupled to a homemade filling syringe system was used to obtain molar volume measurements. The sample temperature was controlled with a precision of ( 1 mK, and the acquisitions of data were made by steps of 0.01 K in the region of the N-to-I phase transition and 0.02 K otherwise, with stabilization periods of 300 s. The device was initially calibrated by using bidistilled water and octylcyanobiphenyl (8CB), for which high-resolution density data coming from dilatometric measurements can be found in the literature.15,28 3. Results and Data Analysis 3.1. Specific Heat Study. Specific heat data of 6OCB as a function of temperature are shown in Figure 1, where two main features must be remarked: the presence of only one mesophase (the N phase) between the ordered crystalline phase (Cr) and the I phase and a nematic range (defined as TNI - TCrN, where TCrN is the Cr-to-N transition temperature) of about 20 K. The temperature associated to the N-to-I phase transition (see the inset of Figure 1) which is a relevant value for the present analysis,21 together with those values coming from the literature, is gathered in Table 1. As has been commented in the Introduction, appreciable pretransitional (or so-called critical fluctuation) phenomena are observed around the weakly first-order N-to-I phase transition, especially on the N-side (see inset of Figure 1). The total enthalpy change associated to any transition, in this case the N-to-I phase transition (∆HNITOT), can be written as

∆HNITOT ) ∆HNI +

∫ ∆CpdT

(1)

where the second term of the right-hand of eq 1 is the pretransitional fluctuation contribution (∆Cp being the difference

Figure 2. Specific heat data (open and filled circles) as a function of temperature near the N-to-I phase transition of the 6OCB. δ-shift data (open-up triangles) are only included to delimit the specific heat coexistence region (shaded area). Solid lines are fittings according to eqs 2a and 2b.

TABLE 1: Latent Heat and Transition Temperature Corresponding to the N-to-I Phase Transition of 6OCB TNI (K)

∆HNI (kJ · mol-1)

refs

349.50 349.37 349.54 349.20 350.20 349.20 348.70 348.70

------0.31 0.50 0.63 0.8

this worka this workb 21c 29 30 31 32 33

a Data from dielectric measurements. b Data from molar volume measurements. c Data from specific heat measurements.

Cp - Cp,background due to the change of orientational order intrinsic to this transition) and the latent heat is ∆HNI which vanishes for second-order transitions. Both right-hand side terms in eq 1 can be individually obtained by means of the MDSC technique.27 The resulting ∆HNI for 6OCB, by using this method, is reported in Table 1 together with the values obtained from the literature. As can be clearly observed from Table 1, although the temperature seems to be well established in a short range of no more than 1 K between our reported value and those reported by other authors, a serious discrepancy exists in the latent heat, in some cases being twice our value or even more. It is important to realize that our value for the latent heat has been carefully tested through the Clausius-Clapyeron equation,21 combining values from pressure-temperature phase diagrams and molar volume jumps at the phase transition. As far as we know, a quantitative analysis of the specific heat critical behavior at the N-to-I phase transition is lacking for 6OCB. Figure 2 shows the specific heat and the δ-phase shift data in a region of about ( 2 K around the N-to-I phase transition. The sharp peak in the δ-phase shift data will be used to delimit the coexistence region of the first-order N-to-I phase transition, shown as a shaded area in Figure 2. The standard expressions11,16,34 used to proceed with such an analysis in the N and I phases, considered to be valid in a region of no more than ( 3 K around TNI, but excluding all the points in the coexistence regions are

Cp,I ) BC + DC

[ T*T - 1] + A | T*T - 1| C,I

-R

for T >

TNI ) T* + ∆T*

(2a)

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TABLE 2: Results of the Fittings for the N-to-I Phase Transition of 6OCB specific heat

dielectric constant

molar volume

(Cp)

(εmean; εiso)

(ln V)

fitting to eqs 2a and 2b 2.121 ( 0.002 BC (J · K-1 · g-1) -0.3 ( 0.5 DC (J · K-1 · g-1)

fitting to eqs 3a and 3b ε** 10.213 ( 0.004 ε* 10.280 ( 0.005 aN (K-1) 0.0084 ( 0.0003 -0.028 ( 0.002 aI (K-1)

fitting to eqs 4a and 4b Ev0,N 5.623 ( 0.009 Ev0,I 5.624 ( 0.001 105 × Ev1(K-1) 80 ( 10 105 × Ev2,N (K-2) -0.11 ( 0.1 105 × Ev2,I (K-2) -0.10 ( 0.1 Av,N/Av,I -3.2 ( 0.9 T** (K) 349.38 ( 0.03 T* (K) 349.16 ( 0.02 R 0.51 ( 0.02 1011 × χ2N 1 1011 × χ2I 1.5

3.0 ( 0.9 349.58 ( 0.05 349.34 ( 0.09 0.50 ( 0.05 1 1

AC,N/AC,I T**(K) T*(K) R 103 × χ2N 103 × χ2I

-0.77 ( 0.14 349.29 ( 0.04 346.66 ( 0.09 0.51 ( 0.03 3 5

Aε,N/Aε,I T** (K) T* (K) R 106 × χ2N 106 × χ2I

Figure 3. Static dielectric permittivity behavior in the I and N phases of 6OCB (open up and down triangles stand for ε| and ε⊥, respectively; open diamonds and filled circles stand for εmean and εiso, respectively). Other data from the literature are also included as shown.

Cp,N ) BC + DC 1

|

[ T**T - 1] + A | T**T -

-R

C,N

for T < TNI ) T** - ∆T**

(2b)

where T** and T* are, respectively, the temperatures toward which the nematic and isotropic behaviors diverge, being always T** > T*. Both BC and DC terms concern to the so-called specific heat background, being identical at both sides of the transition; the last term in both equations is the critical power law divergence, and AC,N and AC,I are the corresponding amplitudes. For the critical exponent R, the same at the I and N phases, a value of 0.5 is expected to be obtained, according to the meanfield Landau-de Gennes theory and tricritical behavior.11,35 Both ∆T* and ∆T**, or even better, the difference (T** - T*) represent the width of the metastable region at the N-to-I phase transition. Common parameters in both phases (BC, DC, and R) have been simultaneously refined after a previous independent fitting. All the parameters are collected in Table 2 and represent well enough the measured specific heat data around the N-to-I phase transition, as indicated by χ2 values and as is seen in Figure 2 where both eqs 2a and 2b are drawn. 3.2. Static Dielectric Permittivity Study. In Figure 3, the static dielectric permittivity of 6OCB for both alignments, parallel (ε||) and perpendicular (ε⊥), in the N phase and the static dielectric permittivity (εiso) in the I phase, obtained in this study along with the ones coming from the literature,14,22–24 are presented as a function of temperature. There is a certain discrepancy, mainly referred to as εiso values and ε|| data. The latter also implies a certain discrepancy in those values corresponding to the so-called mean dielectric permittivity, εmean ) 1/3(ε|| + 2ε⊥). It seems to be that those discrepancies could

Figure 4. Behavior of the mean static dielectric permittivity (open diamonds) in the N phase and the static dielectric permittivity (filled circles) in the I phase for 6OCB. Solid lines are fittings according to eqs 3a and 3b.

be related to the purity of the analyzed samples and in the N-mesophase, additionally, to alignment deficiencies, especially in the parallel component ε||. An important part of our work has been devoted to perform the sample alignment and also to get many data close to the N-to-I phase transition. As has been previously cited, our parallel alignment was checked by applying electric dc-fields, in which the axis of the higher dielectric permittivity is aligned. Other authors apply magnetic fields and then align the axis of the higher magnetic susceptibility. Details of the evolution of the dielectric permittivity data (εmean and εiso) around the N-to-I phase transition are shown in Figure 4. On the I side, εiso exhibits a maximum, this behavior usually being observed for other mesogenic liquid crystals with an important dipole moment,14,16 although its location in temperature may vary from one compound to another. The data analysis at the N-to-I phase transition has been carried out according to the following equations14,16,19

εiso ) ε* + aI |T - T*| + Aε,I |T - T*| 1-R for T > TNI ) T* + ∆T* (3a) εmean ) ε** + aN |T - T**| + Aε,N |T T**| 1-R

for T < TNI ) T** - ∆T**

(3b)

where R is the specific heat critical exponent; T*, T**, ∆T*, and ∆T** have the same meaning as in eqs 2a and 2b; and ε* and ε** are, respectively, the extrapolated values of εiso and εmean at T* and T**. Both aI and aN are the static dielectric permittivity background terms,and Aε,I and Aε,N are the corresponding dielectric amplitudes. All these parameters can be obtained by fitting the dielectric data, and their values are consigned in Table 2. Both eqs 3a and 3b with the corresponding fitting parameters portray almost perfectly our experimental data as shown in Figure 4.

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J. Phys. Chem. B, Vol. 113, No. 49, 2009 15971 exhibit a critical behavior at the N-to-I phase transition comparable to specific heat, and thus, this behavior should be well-portrayed by equations similar to eqs 2a and 2b. However, following the same strategy as shown in other work,16 the critical behavior analysis will be done on the original experimental data in the form of logarithm of molar volume, instead of Rp-values in which the scattering of the data as a consequence of the distortion-sensitive derivative method prevents a rigorous analysis. The logarithm of molar volume around the transition N-to-I, as shown in Figure 6, has been analyzed by means of the following equations (assuming to be valid in a region of no more than ( 3 K around the N-to-I phase transition)

Figure 5. Molar volume (left y-axis and open right-triangles) and the logarithm of the molar volume (right y-axis and open circles) as a function of temperature for 6OCB. The left y-axis of the inset shows a zoom window of the logarithm of the molar volume near the N-to-I phase transition (open and filled circles stand for heating and cooling runs, respectively). The right y-axis of the inset shows the calculated isobaric thermal expansion coefficient data (gray open and filled triangles).

v v ln VI ) E0,I + Ev1[T - T*] + E2,I [T - T*]2 + Av,I |T -

for T > T* + ∆T*

T*| 1-RI ln VN )

v E0,N

+

Ev1[T - T**] 1-RN

Av,N |T - T**|

+

V E2,N [T

3.3. Molar Volume Study and Scaling Relationship. The results obtained from volumetric measurements around the N-to-I phase transition are shown in Figure 5, where both the molar volume and its logarithm are presented as a function of temperature in a heating run from the nematic phase. In the inset, heating and cooling runs of the logarithm of the molar volume are shown in a zoom window. The thermal hysteresis is clearly visible at the phase transition. The last presentation is appropriate to obtain the isobaric thermal expansion coefficient (Rp ) [∂ ln V/∂T]p) by means of the distortion-sensitive derivative analysis of the logarithm of the molar volume data. From Figure 5, the volume change associated to the N-to-I phase transition can be read leading to a value of 0.28 cm3 · mol-1 which has been proven to be compatible with the associated latent heat through the Clausius-Clapeyron equation.21 The analysis at the N-to-I phase transition of the volume data shown in Figure 5 requires us to know if a scaling relationship between specific heat and isobaric thermal expansion coefficient data exists. It is important to realize that such a scaling has been predicted theoretically25,36 and for some liquid crystal-like compounds evidenced experimentally.16 The inset of Figure 6 shows both kinds of data (Cp and Rp) in a region of ( 1 K around the N-to-I phase transition. From this figure, and without taking into account the coexistence region, a nearly perfect scaling between both kinds of data must be remarked, which means that the isobaric thermal expansion coefficient should

- T**] +

for T < T** - ∆T**

(4b)

where R, T*, and T** have the same meaning as in eqs 2 and 3. Taking into account that eqs 4a and 4b come from an integration process of equations similar to eqs 2 that are considered to be valid to describe the critical behavior of the isobaric thermal expansion coefficient, it should be stressed that some conditions over certain parameters of eqs 4a and 4b must be satisfied. In particular, we have two independent terms, Ev0,I and Ev0,N, their values being different in both phases. One Ev1 term, the same above and below TNI, the terms Ev2,I and Ev2,N which are related between them as v v E2,I ) E2,N

Figure 6. Logarithm of the molar volume near the N-to-I phase transition of the 6OCB along with the fittings according to eqs 4a and 4b. The inset evidences the scaling relationship between isobaric thermal expansion coefficient data (left y-axis and open triangles) and specific heat values (right y-axis and gray symbols).

(4a)

2

T** T*

(5)

and finally, the amplitudes Av,I and Av,N which should be identified with AC,I and AC,N of eqs 2a and 2b through

Av,I AC,I )Av,N AC,N

(6)

which means that taking the absolute value, both amplitude ratios (|Av,I/Av,N| ) |AC,I/AC,N|) should be set equal. The results of fittings are consigned in Table 2, with extremely low χ2 values. It may be underlined that both eqs 5 and 6 are fulfilled. In Figure 6, both eqs 4a and 4b are drawn along with the experimental data of the logarithm of the molar volume. 3.4. Nematic Order Parameter Study. Another significant quantity to test the critical behavior at the N-to-I phase transition is the nematic order parameter, SN, which can be experimentally obtained from the anisotropy of macroscopic quantities such as diamagnetic susceptibility,37 refractive index,38 dielectric constant,14 or thermal conductivity,39 among others, as well as from very precise measurements of the specific heat.19,34 In this work, the nematic order parameter has been estimated through the dielectric anisotropy, ∆ε (SN ∝ ∆ε ) ε|| - ε⊥). The data can be extracted from Figure 3 and are shown in Figure 7 as open diamonds. From specific heat data, the procedure based on a mean-filed approach carried out by Iannacchione et al.34 and later by us19 allows us to define SN2(T) )

2M aV

∫

T

TCN

∆HNI ∆Cp dT + + T TNI

∫

TCI

To

∆Cp dT T

(7)

where M and V are, respectively, the mass and the volume of the liquid crystal; the constant a (taken as 0.13 J · K-1 · cm-3, the standard value used for 7OCB) is the first coefficient in the Landau-de Gennes free density energy expansion (f ) f0 +

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Figure 7. Measured dielectric anisotropy ∆ε (left y-axis and open diamonds) and calculated average nematic order parameter SN (right y-axis and filled gray circles) against their respective (T - T**) for 6OCB. The inset shows a double logarithmic plot of (SN - S**) (left y-axis) and (∆ε - ∆ε**) (right y-axis) vs reduced temperatures. Solid lines are fittings according to eq 8.

1/2a(T - T*)SN2 + Θ(SN), where Θ(SN) contains higher-order terms in SN with temperature-independent coefficients). The integration must be numerically done on cooling from T0 in the I-phase, well above TNI, at which SN could be considered zero down to one temperature T in the N-mesophase. Both temperatures TCI and TCN are the limit temperatures of the coexistence region in the I- and N-phases, respectively. The calculated data are shown in Figure 7 as filled circles, in an adequate scale to observe the agreement between both sets of values, the dielectric anisotropy, and the calculated SN through eq 7. The critical behavior of the nematic order parameter at the N-to-I phase transition is usually parametrized, according to the Landau-de Gennes theory11,14 as a generic expression X ) X** + B|T - T**| β

for T < T** - ∆T**

(8)

where X means either SN or ∆ε as appropriate. The temperature T** has the same meaning as in eqs 2b, 3b, and 4b, and β is the critical exponent. Fittings according to eq 8 of both sets of values consigned in Figure 7 lead to comparable T** (349.57 K from SN and 349.78 K from ∆ε) and nearly the same value of β (0.21 ( 0.05) as can be observed from the inset of Figure 7 in which either (∆ε - ∆ε**) or (SN - SN**) vs (T - T**) in a double logarithmic scale exhibit lines with nearly the same slope. 4. Overall Discussion 4.1. Critical Behavior at the N-to-I Phase Transition of 6OCB. Our first glimpse is addressed to the critical exponents, possibly the most important parameters to describe the critical behavior of a phase transition. In the present study, two different critical exponents are provided: the specific heat critical exponent, R, and the nematic order parameter critical exponent, β. As for the former, there exists a full unanimity (see Table 2) irrespective the experimental technique, R being 0.5 or very close. As far as β is concerned, the same value (0.21 ( 0.05) is obtained either from specific heat data or from dielectric ones (see Figure 7). Our β-value, taking into account the margin of error, is fully compatible with the tricritical hypothesis (β ) 0.25) and roughly compatible with the value reported by Rzoska et al.14 (β ) 0.3 ( 0.1). Let us now consider the dielectric amplitude ratio (Aε,N/Aε,I). Our value read from Table 2 is -0.77 which is very close to the value reported by Rzoska et al.14 (-0.73). However, as can be deduced from the comparison of both sets of eqs 2 and 3, the absolute value of the dielectric amplitude ratio (0.77) should be similar to the specific heat amplitude ratio (AC,N/AC,I ) 3, read from Table 2) and also to the absolute value of the molar

Figure 8. Derivative of the static dielectric permittivity with temperature (left y-axis and filled diamonds) and specific heat data (right y-axis and gray solid line) as a function of (T - Tpeak) for 6OCB.

volume amplitude ratio (Av,N/Av,I ) -3.2, read from Table 2) as imposed by eq 6. Compatibility between (AC,N/AC,I) and the absolute value of (Av,N/Av,I) is expected to exist if one recalls the scaling relationship exhibited in the inset of Figure 6. It shows that, although all exhibit compatible critical exponents, dielectric data on one side and specific heat and molar volume data on the other side lead to very different amplitude ratios. It is important to realize that the value of about 3 for the amplitude ratio (AC,N/AC,I) seems to be in agreement with the values obtained for the N-to-I phase transition of other liquid crystallike compounds,16,19,34,40 although it is significantly lower than 7, the value found by Islander and Zimmermann41 for the tricritical behavior of the mixtures He3 + He4. It is important to recall now the scaling relationship between the derivative of the static dielectric permittivity (dεmean/dT and dεiso/dT) and the specific heat, theorized by Mistura26 and evidenced experimentally by Rzoska et al.14 for the 7OCB and also by us for the homologous 9OCB.16 Figure 8 shows both magnitudes (dεmean/dT and dεiso/dT) and Cp as a function of (T - Tpeak) around the N-to-I phase transition for the 6OCB. It may be underlined that such a derivative has been obtained numerically by distortion-sensitive derivative analysis. At first glance, the scaling between both magnitudes seems to be fulfilled in the I-phase and at the N-phase but for temperatures lower than about 0.5 K below phase transition. Probably, this discrepancy could explain the differences in the amplitude ratio (Aε,N/Aε,I) and (AC,N/AC,I), although, as has been shown above, two independent determinations of (Aε,N/Aε,I) give rise to rather similar values (-0.77 and -0.7314). Let us now consider the characteristic temperatures of the phase transition. From Table 1, the transition temperature TNI seems to be well established. As for the spinodal temperatures T* and T**, again dielectric determinations exhibit discordant results with regard to specific heat and volumetric determinations. In addition, it is verified that T**ε e TNI, which has no physical sense, and the most probable explanation should be due to experimental problems (see Figure 8). As for the width of the metastable region, expressed as the difference of the spinodal temperatures (T** - T*), the corresponding results, obtained by means of specific heat, molar volume, and static dielectric permittivity measurements, are consigned in Table 3. It is clear that there is a very good agreement between the specific heat and molar volume measurements, but they are again different from the dielectric static permittivity ones. To sum up, it seems to be clear that the critical behavior at the N-to-I phase transition exhibited by the specific heat measurements mimics almost perfectly that observed by volumetric determinations. Static dielectric data lead to some

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TABLE 3: Transition Temperatures (TNI), Latent Heats (∆HNI), Volume Jumps (∆WNI), and the Width of the Metastable Regions (T** - T*) for Some nOCB Liquid Crystals (n ) 6 to 9)a compound 6OCB 7OCB 8OCB 9OCB

TNIb (K)

∆HNIb (kJ · mol-1)

∆VNIc (cm3 · mol-1)

(T** - T*)C (K)

(T** - T*)vf (K)

347.54 347.31 353.39 353.22

0.31 0.33 0.49 0.84

0.28 0.29 0.42 0.71

0.24 0.05d 0.55e 0.43d

0.22 0.11 0.49g 0.46

(T** - T*)ε (K)

h

2.63 1.24 1.45e 0.91

(T** - T*)C, (T** - T*)v, and (T** - T*)ε account for specific heat, volumetric, and dielectric results, respectively. b Data from specific heat measurements in ref 21. c Data from molar volume measurements in ref 21. d Data from specific heat measurements in ref 42. e Data from ref 43. f Data from molar volume measurements in ref 42. g Unpublished result from molar volume data in ref 21. h Data from static dielectric measurements in ref 42. a

Figure 9. (A) Latent heat (left y-axis and filled circles) and volume jump (right y-axis and open circles) associated to the N-to-I phase transition as a function of the n-chain length in the nOCB series of liquid crystals. (B) The width of the metastable region of the N-to-I phase transition (left y-axis; open and filled diamonds stand for specific heat and volume data, respectively) and the specific heat normalized metastable region by TNI (right y-axis and gray cross symbols) as a function of the n-chain length in the nOCBs. The inset shows the evolution with the n-chain length of the width of the metastable region extracted from dielectric data.

differences in the amplitude ratio (Aε,N/Aε,I) and spinodal temperatures, which seems to be inherent to the experimental technique. Probably, the additional presence of an external electric field in the dielectric permittivity measurements could be the main justification of these differences. In addition, there are some problems of scaling, as is evidenced in Figure 8. 4.2. N-to-I Phase Transition in Some Liquid Crystals of the nOCB Series. Let us now consider in a comparative way the N-to-I phase transition of 6OCB in relation to its counterparts in the nOCB series of compounds. The main idea is to obtain comparative information that may give an indication of how first order the transition is in these compounds. Table 3 offers some characteristic values of certain magnitudes associated to the N-to-I phase transition which seem to be the most adequate for that purpose. Let us first consider the latent heat (∆HNI) and volume jump (∆VNI) data. Figure 9A shows both magnitudes for the nOCB series of liquid crystals (n ) 6-9) that increase, with the n-chain length, following the same trend. It seems that the strength of the first-order N-to-I phase transition is weaker for the 6OCB than for the 9OCB, or in other words, the firstorder character seems to be n-chain length-dependent. Another way to measure the strength of this first-order transition could be the width of the metastable region (T** T*) which, in the same way as ∆HNI and ∆VNI, should be set equal to zero in second-order phase transitions. In Figure 9B (left side), this magnitude (T** - T*) is represented as a function of the n-chain length. In fact, only the values of (T**

- T*) obtained from specific heat data and volumetric determinations have been drawn (open and filled diamonds, respectively). Values from static dielectric data ((T** - T*)ε) are shown in the inset. The results show that the behavior of (T** - T*) with the n-chain length of the nOCBs is, surprisingly, very different from that exhibited by the latent heat and volume jump. In fact, a certain alternation of values with the n-chain length is observed, which is usually known as the odd-even effect. At this point, it is convenient to remember that the odd-even effect is usually observed for certain magnitudes and phase transitions in liquid crystals. In the nOCB series, an odd-even effect is known to exist in the transition temperature TNI.21 In Figure 9B (right-side), the width of the metastable region normalized by TNI is also shown as a function of the n-chain length. In a qualitative look, both width magnitudes ((T** - T*) and (T** - T*)/TNI) exhibit a comparable evolution with the n-chain length, and thus, an odd-even effect on the metastable width could be concluded, similar to what is observed for the transition temperature. 5. Concluding Remarks An exhaustive analysis of the N-to-I phase transition of 6OCB compound through several physical magnitudes (dielectric, specific heat, and volumetric data) has been reported. It has been experimentally proven that, around the N-to-I phase transition, Rp-data, Cp-values, and the derivative of the static dielectric permittivity with temperature (dεmean/dT and dεiso/dT) are related to each other by scaling relationships. However, some discrepancies are observed for the dielectric data on the N-side (dεmean/dT), at least about 0.5 K below the transition peak (see Figure 8). It has been found, irrespective the physical magnitude, that the N-to-I phase transition can be described by a specific heat critical exponent of about 0.5 (the tricritical value) in both the N and I sides. Likewise, the critical exponent of the nematic order parameter has been found to be 0.21 ( 0.05 (nearly the tricritical value of 0.25, as well). According to Table 2, a comparable absolute value of the amplitude ratio (AN/AI) has been obtained from specific heat and molar volume of about 3. However, the absolute value of the dielectric amplitude ratio has been found to be 0.77, a value rather different. Dielectric data from Rzoska et al.14 give rise to a comparable value (of 0.73) which leads us to think that ought to be inherent of the experimental technique. It should be stressed that the width of the metastable region (T** - T*) also portrays these differences between dielectric data on the one side and the specific heat as well as volumetric determinations on the other side. To gain more insight about the strength of the first-order N-to-I phase transition of 6OCB, in relation to the counterparts of the nOCB series, a comparative study through the latent heat,

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volume jump, and the width of the metastable region has been performed. Concerning the first two (∆HNI and ∆VNI), the same trend with the n-chain length is observed which serves to conclude that the first-order character is enhanced as n increases. However, as for the width of the metastable region, surprisingly only an odd-even effect characteristic of the transition temperature can be inferred. Acknowledgment. The authors want to express their gratitude to the MICINN (project MAT2008-01372/MAT) and the MEC (project MAT2006-13571-C02-02) of Spanish Government and also to the Gobierno Vasco (project GI-C07-40-IT-484-07). References and Notes (1) Maier, W.; Saupe, A. Z. Naturforsch. A 1958, 13, 564. (2) Maier, W.; Saupe, A. Z. Naturforsch. A 1959, 14, 882. (3) Maier, W.; Saupe, A. Z. Naturforsch. A 1960, 15, 287. (4) Priest, R. G.; Lubensky, T. C. Phys. ReV. B 1976, 13, 4159. (5) Keyes, P. H. Phys. Lett. A 1978, 67, 132. (6) Tao, R.; Sheng, P.; Lin, Z. F. Phys. ReV. Lett. 1993, 70, 1271. (7) de Gennes, P. G. The Physics of Liquid Crystals; Oxford Science Publications: Oxford, 1994. (8) Mukherjee, P. K.; Mukherjee, T. B. Phys. ReV. B 1995, 52, 9964. (9) Wang, Z. H.; Keyes, P. H. Phys. ReV. E 1996, 54, 5249. (10) Mukherjee, P. K. J. Phys.: Condens. Matter 1998, 10, 9191. (11) Anisimov, M. A. Critical Phenomena in Liquids and Liquid Crystals; Gordon and Breach Science Publishers: Amsterdam, 1991. (12) Thoen, J.; Marynissen, H.; van Dael, W. Phys. ReV. A 1982, 26, 2886. (13) Drozd-Rzoska, A.; Rzoska, S. J.; Ziolo, J. Phys. ReV. E 1996, 54, 6452. (14) Rzoska, S. J.; Ziolo, J.; Sulkowski, W.; Jadzyn, J.; Czechowski, G. Phys. ReV. E 2001, 64, 052701. (15) Zywocinski, A. J. Phys. Chem. B 2003, 107, 9491. (16) Cusmin, P.; de la Fuente, M. R.; Salud, J.; Pe´rez-Jubindo, M. A.; Diez-Berart, S.; Lo´pez, D. O. J. Phys. Chem. B 2007, 111, 8974. (17) Sridevi, S.; Prasad, S. K.; Rao, D. S. S.; Yelamaggad, C. V. J. Phys. Condens. Matter 2008, 20, 465106. (18) Li, J. F.; Percec, V.; Rosenblatt, C. Phys. ReV. E 1993, 48, R1.

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