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Critical heat capacity of octylcyanobiphenyl (8CB) near the nematic-smectic A transition
G.B. Kasting, C.W. Garland, K.J. Lushington
To cite this version:
G.B. Kasting, C.W. Garland, K.J. Lushington. Critical heat capacity of octylcyanobiphenyl (8CB) near the nematic-smectic A transition. Journal de Physique, 1980, 41 (8), pp.879-884.
�10.1051/jphys:01980004108087900�. �jpa-00208909�
Critical heat capacity of octylcyanobiphenyl (8CB)
near the nematic-smectic A transition
G. B. Kasting, C. W. Garland and K. J. Lushington
Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
(Reçu le 11 février 1980, accepté le 14 avril 1980)
Résumé. 2014 La technique ac a été utilisée pour mesurer la capacité calorifique près de la transition nématique- smectique A (N-SmA) du composé 8CB le long d’isobars à 1, 750 et 1 500 bar. La grandeur du pic Cp associé à
cette transition décroît quand la pression augmente mais la forme du pic reste essentiellement inchangée. La capacité calorifique en excès à 1 atm. est conforme à une transition N-SmA du second ordre, caractérisée par
un exposant critique effectif 03B1 = 0,30 ± 0,05.
Abstract. 2014 An ac technique has been used to measure the heat capacity near the nematic-smectic A (N-SmA)
transition in 8CB along isobars at 1, 750 and 1 500 bar. As the pressure is increased, the magnitude of the Cp peak associated with this transition decreases but the shape of the peak remains essentially unchanged. The excess
heat capacity at 1 atm. is consistent with a second-order N-SmA transition characterized by an effective critical exponent 03B1 = 0.30 ± 0.05.
Classification Physics Abstracts 64.60 - 64.70E
1. Introduction. - There has been considerable interest in the nematic-smectic A (N-SmA) transition
in liquid crystals, especially since McMillan’s mean-
field prediction [1] of possible second-order character.
A more realistic model proposed by de Gennes [2]
placed this transition in the d = 3, n = 2 universality
class (XY model in three dimensions) along with the superconducting-normal transition in metals and the lambda transition in ’He. Subsequent analysis of
de Gennes’ hamiltonian showed that the N-SmA transition should be weakly first-order [3] and that
the observed quasicritical behaviour could be much
more complicated than that of the simple X Y model.
In particular, anisotropic behaviour with correlation-
length exponents v jj > vl is possible, as are crossover
sequences from mean-field to anisotropic critical to isotropic critical (XY) and finally to a first-order transition [4].
Several materials have been investigated in which
the N-SmA transition is second-order to within the resolution of careful experiments [5, 8]. Among these
materials are p-cyanobenzilidene-n-octyloxyaniline
(CBOOA), its biphenyl analog octyloxycyanobiphe-
nyl (80CB), and octylcyanobiphenyl (8CB) which is
the subject of the present study. All three of these materials are bilayer smectics; i.e., each smectic
layer is composed of pairs of oppositely oriented
molecules with their aromatic portions overlapping [9].
CBOOA and 80CB show a reentrant nematic phase
at high pressures [10], whereas 8CB does not (at
least below 7 kbar) [10, 11].
The critical behaviour at the N-SmA transition in these compounds is inconsistent with the simple XY interpretation of the de Gennes model. According
to X-ray scattering results in the nematic phase of
all these materials [6], the divergence of the correlation-
length parallel to the incipient smectic density wave
is generally consistent with v Il VHe = 0.67, but the corresponding exponent for the perpendicular corre- lation-length is somewhat smaller (v 1. 0.5 - 0.6).
Light scattering measurements of the K2 and K3
elastic constants in the N phase yield v Il values that
are generally consistent with the X Y model [6, 12]
and vl values that are somewhat uncertain but smal- ler [8] ; furthermore, the elastic constants B and D in the SmA phase are both anpmalous [6]. Calori-
metric studies of CBOOA and 80CB have shown that the specific heat divergence at the N-SmA tran-
sition is best characterized with a fairly large positive
exponent (in the range oc 0.15 for CBOOA [13]
and a = 0.25 for 80CB [7]) rather than the almost
logarithmic (a = - 0.026) singularity of the XY
model.
We report here the results of an ac calorimetric
investigation of the N-SmA transition in 8CB over
the range 1-1 500 bar. The experimental method and
the procedure for reducing the data to Cp values
have been described elsewhere [7]. The results are
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004108087900
880
similar to those reported for 80CB in that (1) a large positive exponent a = a’ = 0.30 ± 0.05 is required
to represent the data at 1 atm. ; (2) the decrease in the magnitude of the Cp peak with increasing pressure has the same dependence on TNAI TNr in both cases;
and (3) the shape of the Cp peak is not sensitive to
the pressure.
2. Results. - We have measured the heat capacity
of 8CB along isobars at 1, 750, and 1 500 bar over the température ranges shown in figure 1. Two samples
Fig. l. - P-T phase diagram for 8CB (MW = 291.44). The dashed lines indicate the range of our data, the triangles indicate
the observed transition temperatures, and the crosses represent the melting point of the crystal on warming. The solid phase lines are
from reference [10].
of material from BDH Chemicals were investi-
gated [14]. The temperature dependence of Cp for sample A at 1 atm. and at 750 bar is shown in figure 2.
As in 80CB, the magnitude of the Cp peak associated
with the nematic-isotropic (N-I) transition is almost
independent of pressure, while the magnitude of the
N-SmA peak is very sensitive to the pressure. This behaviour suggests that the energy fluctuations res-
ponsible for the excess heat capacity at the N-SmA
transition are largely associated with fluctuations in the nematic order parameter S, which are expected
to decrease with increasing TNI - TNA [15].
The excess heat capacity ACp in the vicinity of the
N-SmA transition at 1 atm. is shown in figure 3.
The smooth background shown in figure 2 has been
subtracted from the observed Cp values, and the resulting values of àc,IR=C,(obs)IR-Cp(bkgd)IR
have been plotted as a function of the reduced tempe-
rature t = (T - TNA)/TNA. The data shown in figure 3
have been corrected for a constant drift in the value of TNA ( + 5.6 mK per day) during the 18 days required
Fig. 2. - Heat capacity of 8CB (sample A) at 1 atm. and 750 bar.
Background curves used to obtain ACp at the N-SmA transition
are shown.
Fig. 3. - Critical heat capacity near the N-SmA transition in 8CB
(sample A) at 1 atm. The data for N = 3 and N = 4 have been shifted upward by 10 and 20 units, respectively, for clarity. The
arrows indicate the positions of 1 t Imin for the least-squares fits reported in tables 1 and II.
to obtain this set of data points. This drift rate was
determined by observing the variation of Cp at
constant temperature during three consecutive over-
night (- 12 h) periods when the temperature was
very close to TNA.
The fact that TNA increased slowly with time suggests that this drift may have been caused by the segre-
gation of an impurity from the bulk of the sample during the run. The heat capacity peak shown in figure 3 is also considerably more rounded close to
TNA than that observed by Thoen et al. [16] in an
adiabatic experiment or that observed by us in ,80CB [7]. This round-off region is much too broad
to be explained as the result of finite ac temperature oscillations, which are 7 mK peak-to-peak near TNA [17].
High-pressure data near the N-SmA transition are
shown in detail in figure 4. These data are less reliable
Fil. 4. - Critical heat capacity near the N-SmA transition in 8C3 at 750 bar and 1 500 bar, multiplied by the scale factors
z = 2.18 and 5.45, respectively, to allow direct comparison with
the 1 atm. data (solid line, z = 1). The data for N = 3 has been shifted upward by 30 units for clarity.
than those obtained at 1 atm. for several reasons :
(1) Hnhole leaks in the silver cells for both samples
allowed argon gas to slowly dissolve in the liquid crystà during the course of the pressure runs ; (2) Poor
pressure stability during the 750 bar run caused TNA
to drcp 90 mK in the 18 h period during which
the data very close to the transition (j t 1 10-3)
were obtained, leading to a large correction in the AT values; (3) Several abrupt changes in the Cp
values (see Ref. [7]) led to slightly different results for each of four separate passes through the N-SmA
transition at 750 bar ; (4) The 1 500 bar data were
quite scattered due to problems with the high-pressure
electrical leads during this run.
The dissolving of argon into the samples at high
pressures was our principal experimental difficulty.
The présence of dissolved argon was detected in two ways. First, the value of TNI was observed to drop by several degrees between the beginning and the
end of each high pressure run (a period of approxi- mately 30 days). The corresponding drop in ÎNA
was only - 0.2 K. Second, the sample cells blew up upon depressurization from 1 500 bar due to the release of dissolved gas. The cell containing sample A
was repaired and studied again at 1 atm. The Cp peaks were very similar to those observed in the initial 1 atm. run (Fig. 2), and TNI was lower by only
0.3 K. Hence, the large shifts in TNI observed at high
pressures were a reversible consequence of dissolved argon rather than a result of thermal decomposition
or some other chemical reaction.
We believe that the high-pressure I1Cp values
shown in figure 4 are not greatly influenced by dis-
solved gas since these data were taken near the
beginning of the first pressure run on each sample.
In particular, the decrease in the magnitude of ACp
with pressure cannot be caused by dissolved gas.
When sample A was definitely influenced by dissolv-
ed argon after being held at 750 bar for 35 days and
1 500 bar for 15 days more, it gave a ACp peak at
1 500 bar that was considerably larger than the sample B peak shown in figure 4. Furthermore, the trend of decreasing ACp peaks was also observed in 80CB samples sealed in cells that were free of leaks [7]. Thus the major features - a ACp peak
that preserves its shape but diminishes in magni-
tude upôn increasing the pressure - are felt to be intrinsic properties of 8CB. Nonetheless, it seems likely that the linear Cp variation between
at 750 bar corresponds to a two-phase region induced by dissolved argon and is not a property of pure
8CB [18].
3. Discussion. - 3.1 N-SmA ENTHALPY. - The critical enthalpy ÔHNA = ACP dT obtained from
our 1 atm. Cp data near the N-SmA transition is 228 J . mol-1. This value may be compared with
total enthalpy values of 126 [11], 130 [19], and
200 [20] J . mol-1 determined by means of differential
scanning calorimetry. It is common practice to report such DSC results as the latent heat AH, although
the rapid sweep rates used in this technique do not
allow one to distinguish between a true latent heat and a rapid but continuous variation of enthalpy through the transition. The ac method does not give
reliable latent heats either, but it does give an accurate
measure of the equilibrium pretransitional enthalpy
associated with a transition. Thus a comparison
between our value for ÔHNA and the DSC results indicates that the latent heat at this transition must be very small. An estimate of I1H = 0 - 10 J . mol-1
is supported by a recent adiabatic study of 8CB [16].
Thus, the best calorimetric data are consistent with a second-order N-SmA transition in 8CB.
Leadbetter et al. [20] reported a volume change
(i.e., AV/V = 5 x 10-4) over a 0.1 K range near this transition. Although they interpreted this result
as indicating a very weak first-order transition, it
seems quite possible in light of our data that the volume change at the transition could also occur
continuously. We therefore feel that there is no
convincing experimental evidence of first-order cha- racter at this transition.
3.2 CRITICAL EXPONENTS. - We have used the
Marquardt algorithm to fit the 1 atm. data at the
882
N-SmA transition to simple power laws of the form
In view of the very small error in the temperature measurements, the data points have been given equal weights [7] (constant standard deviation ui = 0.12).
The first step was to determine the minimum reduced temperature at which these data still followed a
power-law divergence. To do this we fit the data above and below TNA separately over the range
for values of 1 t Imin between 1 x 10-5 and 3 x 10-4.
TNA and TNA were fixed at Tm = 33.540 °C, the temperature of the Cp maximum [21]. The results of these fits are summarized in figure 5. The inclusion of data at t values less than those indicated by the
arrows led to a marked deterioration in the quality
of the fits and introduced a systematic pattern of deviations. Above TNA these deviations could possibly
be associated with the finite Tac amplitude [17], but
the deviations below TNA extend too far away from the transition to be explained as finite amplitude
effects. We believe that positive deviations at small
1 t in the SmA phase and the region of rounding (negative deviations) at even smaller 1 t in both phases are related to impurities. For all subsequent fits, the values Of 1 t Im;n are those indicated by the
arrows in figures 3 and 5.
Having established 1 t Lin. we then fit the data above and below TNA over the ranges shown in table I. These fits are very stable and there is no
evidence that TNA =1= TNA’ However, the effective critical exponent in the N phase does not equal that
in the SmA phase. The fits listed in table 1 are all
Fig. 5. - Critical exponents and reduced chi-squares for leat-
squares fits of eq. (1) to the 1 atm. data in 8CB near the N-SnA transition over the range 1 t Imin - 1 t 3 x 10- 3. The arrcws
indicate the values ouf 1 t Imin used in the fits reported in tabIJs I
and II.
based on ACp values determined from the smooth background shown in figure 2. In order to test the effect of a different choice of background, we idded
a term Et to eq. (1 a) and E’ t to eq. (1b) and refit
the data with TNA = TNA = Tm. The maximum change in a(a’) resulting from this procédure was
0.01 and the largest improvement in xÿ was 10 %.
Hence the background shown in figure 2, which
seems very reasonable on physical grounds, is also
close to the optimum choice.
To test the compatibility of the data witk scaling (a = a’, B = B’) and allow for a possible confluent singularity, we have fit the data in both phases simul-
Table 1. - Results of least-squares fits »’ith eq. (1) to the 1 atm. data near the N-SmA transition in 8CB over
the range 1 t Imin 1 t t max· A bracket indicates that the parameter was fixed at the indicated value. Error bounds are 95 % confidence limits based on the F test.
taneously with the equation
The values of 1 t Imin were taken to be the same as
those in table 1 and TNA was set equal to T’NA. (Allowing TNA -# T’NA led to no improvement in X, ’.) The results
of these fits are shown in table II. It is evident that a
simple power law (D = D’ = 0) is inadequate to explain the data except over a very restricted range
(approx. one decade of reduced temperature). The corrections-to-scaling form, on the other hand, allows
a good fit all the way out to 1 t max = 1 x 10-2. It
is interesting to note that the values a = 0.26 - 0.29 obtained in these fits agree well with those obtained in similar fits on 80CB, but the coefficients D and D’
have the opposite sign [7]. The latter fact raises some
questions about their physical significance.
Due to the experimental problems discussed in
section 2 we have refrained from fitting the high-
pressure data at the N-SmA transition. Instead, figure 4 is presented as an indication that the shape
of the Cp peak is not affected by pressure or by the
ratio TNA/TNI. This conclusion agrees with our
result for 80CB [7] but does not agree with the inter-
pretation of Cp measurements near the N-SmA transition in the homologous series nS . 5 [22].
3.3 INTERPRETATION. - The critical exponents obtained from scaling fits of the heat-capacity data
near the N-SmA transitions in 80CB and 8CB
(a = 0.25 ± 0.05 and a = 0.30 ± 0.05, respectively)
present certain problems in terms of our current understanding of critical phenomena. These results
strongly rule out the nearly logarithmic (a - 0.026) singularity expected for a simple XY model. Moreover, they do not agree with any of the familiar exponents for n-vector models or multicritical points. It is possible that they represent effective exponents result- ing from the anisotropic fixed point or one of the complex crossover sequences described by Lubensky
and Chen [4] or from crossover from tricritical to
X Y behaviour [22]. If such crossover explanations
are correct, however, the insensitivity of the a values
to range shrinking and to changes in pressure is
quite surprising.
A feature of the N-SmA transition which may
complicate the observed quasicritical behaviour is
the fact that the lower marginal dimensionality d°
is 3 for the SmA phase [23]. For spatial dimensio-
nalities below d° thermal fluctuations are sufficiently
strong to destroy the phase transition and prevent the establishment of long-range order. Although the divergent phase fluctuations in the SmA order para-
meter associated with marginal dimensionality
effects are effectively removed in the Lubensky-Chen analysis, they are still expected to destroy long-
range order in a real SmA phase. This feature has been invoked to explain the unusual temperature dependence of the SmA elastic constants B and D [6, 23] and could possibly lead to some unexpected
Cp effects.
One way to assess the role of the anisotropic fixed point in the Lubensky-Chen model [4] is to consider
their modified version of hyperscaling :
This equation enables us to relate the calorimetric results with experimental measurements of the lon-
gitudinal and transverse correlation-lengths near the
N-SmA transition. The X-ray scattering results for 80CB, 8CB and CBOOA are ail consistent with
while the best values of vl are 0.58 ± 0.04 for 80CB,
0.51 ± 0.03 for 8CB, and 0.62 ± 0.05 for CBOOA [6].
Using v = 0.67 together with the a values for these materials [7, 13], one obtains from eq. (3)
for 80CB, 0.515 ± 0.025 for 8CB and 0.59 for
CBOOA, in good agreement with the measured values.
Another comparison of our data with the scat- tering results may be made in terms of the concept of two-scale-factor universality [24]. This hypothesis (allowing for correlation-length anisotropy) predicts
Table II. - Results of simultaneous fits with eq. (2) to the 1 atm. data above and below the N-SmA transition in 8CB. The minimum values of 1 tiare shown by the arrOM’S in figures 3 and 5.
