PHYSICAL REVIEW E 82, 011709 共2010兲

Critical behavior at transitions from uniaxial to biaxial phases in a smectic liquid-crystal mixture
Y. Sasaki and K. Ema*
Department of Physics, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro, Tokyo 152-8551, JapanK. V. Le and H. Takezoe
Department of Organic and Polymeric Materials, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro, Tokyo 152-8552, JapanS. Dhara
School of Physics, University of Hyderabad, Hyderabad 500046, India

B. K. Sadashiva
Raman Research Institute, C. V. Raman Avenue, Bangalore 560080, India
共Received 17 March 2010; published 29 July 2010兲
We report results of calorimetric and optical investigations of binary mixtures of rodlike and bent-shaped
molecules. We find that the observed critical heat anomaly associated with the smectic-A2 to biaxial
smectic-A2b transition is well described with a Fisher-renormalized form of the usual scaling expression. The
effect of renormalization is large in this system in part because of the moderately steep slope of the phaseboundary 共dTc / dX ⬃ 100 K, where X is the mole fraction of the bent-core molecules兲 and in part because of
the proximity to the tricritical point. The magnitude of heat anomaly at the smectic-A2–smectic-A2b transition
showed a drastic decrease as X becomes smaller. Moreover, the nematic–smectic-A2 transitions investigatedturned out to be always first order and the transition enthalpy showed only weak dependence on the concentration X. The results imply that the energy fluctuations around the smectic-A2–smectic-A2b transition are very
sensitive to the underlying ordering of the smectic-A2 background.
DOI: 10.1103/PhysRevE.82.011709

PACS number共s兲: 64.70.M⫺, 64.60.F⫺, 65.40.Ba

I. INTRODUCTION

Liquid-crystal mixtures formed from bent-shaped molecules and rodlike molecules have been extensively studied
in the last two decades. Remarkable numbers of new phases
have been found 共see, for instance, Refs. 关1–3兴兲. One of therecent significant researches 关1兴 reported the discovery of anorientational transition of bent-shaped molecules in a background medium of smectic 共Sm兲 liquid crystal formed fromrodlike molecules. The structures of bent-shaped molecule
1,3-phenylene-bis关4-共3-methylbenzoyloxy兲兴-4⬘ - n-dodecylbiphenyl-4⬘-carboxylate 共BC12兲 and rodlike molecule
4-biphenylyl-4⬙ - n-undecyloxybenzoate 共BO11兲, used intheir work, are shown in Fig. 1. When the mole fraction of
BC12 molecules 共denoted as X兲 is small, the phase sequenceof mixtures has been reported as I共isotropic兲 – N共nematic兲 –Sm共smectic兲-A2 – SmA2b – Cr共crystal兲. Here, the SmA2b phasehas been found as a novel smectic phase with biaxiality induced by an orientational ordering of the minor amounts ofbent-core molecules in the mixture. Such an ordering phenomenon of bent-shaped molecules embedded in the SmA2
background is unusual and, therefore, of special interest. Although the work by Pratibha et al. 关1,2兴 included extensive
studies of the ordering in the system by optical, x-ray, andother methods, thermal properties were not studied in detail
except some differential calorimetric analyses. In particular,
based on the reasons described below, this mixture system*Author to whom correspondence should be addressed.
kema@phys.titech.ac.jp1539-3755/2010/82共1兲/011709共6兲

deserves attention from two points of view: 共A兲 a very steep
X dependence of the SmA2-SmA2b transition temperature and
共B兲 the existence of the N-SmA2 transition for a substantiallywide X range.
According to the phase diagram 关1兴, the SmA2-SmA2bphase boundary has a very steep slope of dTc / dX ⬃ 100 K.
Generally, such cases are remarkable because we can observe the so-called Fisher renormalization 关4兴. Usually, measurements are performed as a function of tX ⬅ T / Tc共X兲 − 1along a path with constant X. Here, Tc共X兲 is the measured
transition temperature. Keeping the chemical potential difference ␾ constant in the mixture can significantly influence theunderlying critical behavior of the system considered as athermodynamic constraint. It is well known that C pX values
measured along a constant composition path should exhibit adifferent behavior from C p␾, the heat capacity at a constantchemical potential difference. Obtaining true critical behavior requires the measurement of C p␾ as a function of t␾⬅ T / Tc共␾兲 − 1. As a consequence of this, as Fisher 关4兴

FIG. 1. Structures of 共a兲 bent-core molecule 共BC12兲
1,3-phenylene-bis关4-共3-methylbenzoyloxy兲兴-4⬘-n-dodecylbiphenyl4⬘-carboxylate
and共b兲
rodlikemolecule
共BO11兲
4-biphenylyl-4⬙-n-undecyloxybenzoate.

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PHYSICAL REVIEW E 82, 011709 共2010兲

SASAKI et al.
40
X=0.08
nas
ac

−1 −1

Cp (JK g )−1 −1

Cp (JK g )

3.6
3.2

20

2.8
383

384

385

386

0.5

382

384
T(K)

X=0.08
X=0.07
X=0.065
X=0.06
X=0.045

372

386

376

380

384

T (K)

FIG. 2. 共Color兲 Heat capacity near the N-SmA2 and the
SmA2-SmA2b transitions at X = 0.08. Here, X is the mole fraction of
BC12 molecules.

pointed out, the critical exponent has been renormalized togive ␣X = −␣ / 共1 − ␣兲. Since the critical exponent ␣ characterizes the leading singularity of the heat anomaly, this corresponds to a crossover from divergent power-law behavior
共␣ ⬎ 0兲 to finite cusplike renormalized behavior 共␣X ⬍ 0兲 in
C pX. In the heat-capacity measurements, this phenomenon
has been observed, for example, in 3He-4He mixtures 关5兴,several rodlike liquid-crystal mixtures 关6–8兴, and liquid mixtures 关9兴. Furthermore, the extent of such a renormalization
significantly depends on the values of ␣ and 共T−1
c dTc / dX兲关9,10兴. In particular, when ␣ = 0.5, the magnitude of renormalized exponents is “doubled” as ␣X = −1, giving rise to a
drastic change.
Another issue of special interest here is the behavior of
the N-SmA transition. So far, a series of N-SmA transitions
has been studied through theoretical and experimental approaches, particularly relating to the so-called McMillan ratio TNA / TIN 关11,12兴, with TNA and TIN being the N-SmA
transition and the I-N transition temperatures. The coupling
between the smectic density-wave amplitude ␺ and the magnitude S of the nematic order parameter causes a crossover
toward a tricritical point and eventually leads to a first-ordertransition as TNA / TIN → 1. According to McMillan 关13兴, thetricritical point takes place at the value of 0.87, although thisvalue is not universal and experimental data report slightly
higher values compared with the expectation 关14兴. Somefirst-order N-SmA transitions were found 关15,16兴 when thenematic range becomes very narrow, i.e., TNA / TIN ⬎ 0.99. It
has been noticed that the tricriticality occurs at different McMillan ratios depending on different types of SmA phases
关7,17兴, although experimental studies are quite limited withregard to SmA2 phase.
In this paper, we report extensive calorimetric and opticalinvestigations on pure BO11 and mixtures of BO11 andBC12 with X = 0.045– 0.10. The obtained data indicate theexistence of a tricritical point along the SmA2-SmA2b phaseboundary. The heat anomaly associated with the
SmA2-SmA2b transition showed cusplike behavior near thetricritical point and can be well understood as the appearanceof Fisher-renormalized tricritical behavior in the vicinity of
Tc. It was also found that the magnitude of the heat anomaly
drastically diminishes with decrease in X. On the other hand,
the N-SmA2 transition was found to be first order for pure
BO11 as well as for mixtures, while the transition enthalpy
showed only weak dependence on X. These results imply

FIG. 3. Heat-capacity data obtained for five mixture ratios. The
arrows indicate the position of the SmA2-SmA2b transition.

that the energy fluctuations around the smectic-A2–
smectic-A2b transition are very sensitive to the underlying
ordering of the smectic-A2 background.
II. METHODS AND RESULTS

The calorimetric investigation has been performed in a
low-frequency ac mode 关18兴 as well as a nonadiabatic scanning 共nas兲 mode 关19兴. Nas mode allows one to measure theenthalpy as a function of the temperature at first-order transitions, while ac mode can determine precise pretransitionalchanges in heat anomalies. Samples of about 7 mg wereplaced in hermetically sealed gold cells. Temperature scanrate was about 0.03 K/h near the transition region.
Figure 2 shows the temperature dependence of the heat
capacity at X = 0.08. A large anomaly is evident at a temperature of about 385.8 K, which corresponds to the N-SmA2
transition. The heat-capacity anomaly obtained by nas modeis much larger than that obtained by ac mode. Another small
peak around 384.1 K corresponds to the SmA2-SmA2b transition where ac and nas modes gave essentially the same
result, in contrast to the N-SmA2 transition. This indicatesthat the transition is second order with no measurable latentheat. In addition, the heat anomaly at the SmA2-SmA2b transition exhibits cusplike behavior. For 0.045⬉ X ⬉ 0.08, heatcapacity data showed almost the same trend as described for
X = 0.08. However, the magnitude of the SmA2-SmA2b heatanomaly decreased sharply with the decrease in X as shownin Fig. 3 while the transition enthalpy associated with
N-SmA2 did not show strong dependence on concentration.
For example, the N-SmA2 transition enthalpies measured at

FIG. 4. 共Color兲 Coexistence of SmA2 and SmA2b phases in ahomeotropically aligned cell for a sample with X = 0.09.

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CRITICAL BEHAVIOR AT TRANSITIONS FROM …

PHYSICAL REVIEW E 82, 011709 共2010兲

X = 0.045 and X = 0.08 are ⬃2.2 and ⬃2.6 kJ/ mol, respectively. Furthermore, the calorimetric measurement on pure
BO11, which has a nematic range of 14.5 K, turned out to bealso first order with the enthalpy of ⬃2.1 kJ/ mol.
When X = 0.09, the heat-capacity data at the SmA2-SmA2btransition measured by nas mode showed a slightly larger
peak than the one by ac mode. The existence of latent heat
seems plausible. However, since N-SmA2 and SmA2-SmA2btransitions approach each other, they cannot be distinguishedclearly. Optical measurement at the composition has been
performed to obtain solid evidence. As shown in Fig. 4, aclear coexistence of the two phases has been observed in ahomeotropically aligned cell 共alignment layer: polyimide
JALS-204 from JSR, Japan兲 where the black domain is SmA2
and the bluish one is SmA2b. Judging from calorimetric andoptical investigations, the transition at X = 0.09 is first order.
The phase diagram based on the above results is shown in
Fig. 5. A tricritical point is located between X = 0.08 and 0.09
on the moderately steep SmA2-SmA2b transition linedTc / dX ⬃ 100 K. The overall appearance of the phase diagram is in good agreement with the original phase diagram关1兴, while somewhat small differences in transition temperatures have been noticed. This is probably because of the slow
temperature scan rate in ac calorimetry as compared withcommercial differential scanning calorimetric and opticalmeasurements.
To estimate the excess heat capacity ⌬C pX accompanying
the SmA2-SmA2b transition, the pretransitional backgroundrelated to the N-SmA2 transition has been determined andsubtracted from the observed C pX data. Then, the ⌬C pX data
have been analyzed with the following renormalizationgroup expression in terms of the reduced temperature tX including corrections to scaling 关20,21兴:␪⌬C pX = A⫾兩tX兩−␣共1 + D⫾
1 兩tX兩 兲 + Bc ,共1兲

where ⫾ indicates above and below Tc. Theoretically expected correction-to-scaling exponent ␪ values are approximately 0.5 for three-dimensional 共3D兲 cases 关22兴 and fixed to
0.5 in the present fittings. Bc is a constant term that arises
from the critical background. In general, when one carries
out critical exponent analysis, data points close to Tc should
be excluded from the fit due to inhomogeneities of the mixture and/or instrumental errors associated with the finite sizeof the temperature oscillations ⌬Tac. The limit of the round-

FIG. 5. 共Color online兲 Phase diagram of BO11/BC12 binary
mixtures. Solid and dashed lines show second- and first-order transitions, respectively. The temperature width of two-phase coexistence region at the first-order transitions is quite narrow in this scale,
being on the order of 0.1 K, or at most 0.5 K, so that it is not shownhere.

ing region has been estimated according to the proceduredescribed in 关23兴 by examining ␹␯2,
1
␹␯2 =

n

冉

冊

1

fit
2
兺 关⌬Cobs
pi − ⌬C pi 共Ti兲兴 ,␯ i ␴2i

共2兲fit
where ⌬Cobs
pi and ⌬C pi denote the measured and calculated
heat capacities, respectively. ␴i is a standard deviation of⌬Cobs
pi and ␯ is the number of degrees of freedom.
Table I lists the result of the fits with Eq. 共1兲 for X= 0.08. Here, 兩t兩max is the maximum value of 兩tX兩 used in thefit. When the first correction-to-scaling terms 共D⫾
1 terms兲 are
included, ␣ values are clearly negative and sensitive to thedata range shrinking. The amplitude ratio A− / A+ also depends significantly on 兩t兩max. Moreover, the ratio D−1 / D+1 islarge negative values, violating the theoretical expectation
D+1 = D−1 关24兴. For comparison, imposing D⫾
1 = 0 gives fits ofpoor quality, indicating that D⫾
1 terms play an important role
anyway. These results show that Eq. 共1兲 does not describe theobserved heat anomaly adequately. Since the slope of the
SmA2-SmA2b transition temperatures is steep, the above features suggest the appearance of Fisher renormalization.

TABLE I. Results of fits to the heat-capacity data at X = 0.08 using Eq. 共1兲. The minimum reduced
temperatures showing the rounding region for T ⬎ Tc and T ⬍ Tc are determined as 兩t兩+min = 2.60⫻ 10−6 and兩t兩−min = 6.25⫻ 10−5, respectively. The units are J K−1 g−1 for A⫾ and B, and K for Tc. The parameters inbrackets were held fixed at the shown value in the fits.
兩t兩max␣Tc

A+A− / A+D+

D− / D+B

␹2␯

0.00050.00050.00100.00100.00300.0030−0.2520.013−0.1840.088−0.0740.193

384.099384.097384.100
384.096384.099384.089−5.6276
7.0174−3.41630.5498−2.9143
0.1272

0.23481.04140.34131.27520.72211.4348

−9.31
关0兴
−4.88
关0兴
−1.46
关0兴

−3.87
关1兴
−4.89
关1兴
−2.72
关1兴

0.8041−7.60190.8965−0.9259
1.7531−0.3970

1.24
6.33
1.34
20.4
4.35
165

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SASAKI et al.

TABLE II. Results of fits to the heat-capacity data at X = 0.07 using Eq. 共3兲. The minimum reduced
temperatures showing the rounding region for T ⬎ Tc and T ⬍ Tc are determined as 兩t兩+min = 2.61⫻ 10−6 and兩t兩−min = 7.83⫻ 10−6, respectively. The units are J K−1 g−1 for Ā⫾ and B̄, and K for Tc. The parameters inbrackets were held fixed at the shown value in the fits.
兩t兩max

␣X

B̄

Tc

Ā+

Ā− / Ā+D̄+D̄− / D̄+

␹2␯

0.00040.00050.00100.00300.00040.00050.00100.0030−0.804−0.779−0.628−0.558
关−1兴
关−1兴
关−1兴
关−1兴

0.45970.45970.4672
0.48760.45310.45040.44190.4316383.280383.280383.280383.280383.279
383.279
383.278383.274918.288703.682
155.88485.3385880.95
5654.694257.582889.86

0.10840.1036
0.12470.1622
0.09340.08540.08650.10622273.41714.3326.48160.47
16097
15449
10993
7041.6

0.11270.07910.02970.05790.13620.10390.07810.0838

2.59
2.42
2.39
2.39
3.22
3.30
6.39
11.2

−4.17⫻ 10−5 ⬍ tX ⬍ 5.21⫻ 10−6 for X = 0.08. If divergent behavior occurs, the rounding region should expand owing to
the renormalized expression of Eq. 共3兲. We believe this result
reflects that the shape of the heat anomaly very close to Tc is
cusplike rather than divergent. Tables II and III list the results of the fits with Eq. 共3兲 at X = 0.07 and 0.08, respectively.
In both cases, effective ␣X values approach the fully renormalized tricritical value of ␣X = −1 as 兩t兩max becomes small asshown in these tables. To verify this tendency, fits were also
tried with ␣X fixed at −1, and the results are included inTables II and III. It is seen that these fits are statisticallyworse than those with free ␣X values especially for wider
data ranges. The ␹␯2 value becomes less for smaller 兩t兩max fits
and approaches the value obtained for the free ␣X fits. Such abehavior is just expected as the result of the crossover fromdivergent behavior far from Tc to a Fisher-renormalized cusp
near Tc. Figure 6 shows the ⌬C p data and fitting curves for
X = 0.07. The data close Tc illustrated above in Fig. 6 shows
clearly cusplike behavior. The dashed line in the figure,showing the result of the fit obtained for data range 兩t兩max= 0.0004, deviates considerably from the data far from Tc.
This is also in accordance with the range dependence of ␣X
described above. Nevertheless, the obtained ␣X is not closer

Consequently, on the basis of the relationship between⌬C pX and ⌬C p␾, we have attempted the use of the following
expression 关7兴:⌬C pX = B̄ −Ā⫾兩tX兩−␣X
1 + D̄⫾兩tX兩−␣X

共3兲

,where the parameters Ā, B̄, and D̄ can be regarded as constant in the vicinity of Tc although they have slight temperature dependence. It should be noted that the D̄ term is not acorrection-to-scaling term, unlike the D1 term in Eq. 共1兲.
Before analyzing the data, the rounding region has been determined again using Eq. 共3兲 instead of Eq. 共1兲. This process
plays a particularly nontrivial role when nondivergent behavior is the intrinsic physical property. Since Fisher renormalization is the very asymptotic behavior near Tc, eliminating
the data around Tc causes the loss of important information.
Like our system under consideration, when the renormalized
region is expected to appear only in the immediate vicinityof Tc, the effect is distinct. As a result of employing Eq. 共3兲,the new temperature region omitted becomes narrower thanthat estimated by using Eq. 共1兲. The rounding region determined is −7.83⫻ 10−6 ⬍ tX ⬍ 2.61⫻ 10−6 for X = 0.07 andTABLE III. Results of fits to the heat-capacity data at X = 0.08 using Eq. 共3兲. The minimum reduced
temperatures showing the rounding region for T ⬎ Tc and T ⬍ Tc are determined as 兩t兩+min = 5.21⫻ 10−6 and兩t兩−min = 4.17⫻ 10−5, respectively. The units are J K−1 g−1 for Ā⫾ and B̄, and K for Tc. The parameters inbrackets were held fixed at the shown value in the fits.
兩t兩max

␣X

B̄

Tc

Ā+

Ā− / Ā+D̄+D̄− / D̄+

␹2␯

0.00040.00050.00100.00300.00040.00050.00100.0030−0.965−0.843−0.646−0.550
关−1兴
关−1兴
关−1兴
关−1兴

0.67690.68660.71770.76810.67440.67290.66740.6448384.090384.091384.092
384.092
384.090384.089384.086
384.0785904.60
1880.88307.652140.3038083.04
7656.06
6148.383969.100.0876
0.09830.13370.18130.08590.08890.10700.13949880.12918.5402.81160.7313733
12810
9780.26035.90.07190.05520.04070.08190.07660.0776
0.09400.1138

2.87
3.01
2.82
3.82
2.87
3.44
6.99
16.5

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CRITICAL BEHAVIOR AT TRANSITIONS FROM …
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383.3

PHYSICAL REVIEW E 82, 011709 共2010兲

383.4

over appears from the underlying nonrenormalized values tothe fully renormalized ones. The temperature range where
Fisher renormalization is visible experimentally is given by
关10兴

-1 -1

∆Cp (JK g )

0.4
0.2
0.4

冋

X=0.07

␶ = A0X共1 − X兲

0.2
0
383
T (K)

384

FIG. 6. 共Color online兲 Excess heat-capacity ⌬C p data of the X= 0.07 sample shown for two temperature ranges with 兩t兩max= 0.0004 共upper data set兲 and 兩t兩max = 0.003 共lower data set兲. Solidand dashed lines represent fits to the data with Eq. 共3兲. The solidline is for 兩t兩max = 0.003, and the dashed line is for 兩t兩max = 0.0004.

to the fully renormalized exponent than that obtained for X= 0.08.
We also note that, if we pay attention to numerical aspects
without considering physical meanings, ␹␯2 values shown in
Tables II and III with Eq. 共3兲 are comparable with the result
including D⫾
1 in Table I. However, the violation of theoretical expectation D−1 / D+1 = 1 mentioned above indicates that thefits with Eq. 共1兲 and nonzero D⫾
1 are artificial and therefore
less reliable.
III. DISCUSSION
A. N-SmA2 transition

The present investigation provides a rare example where
the N-SmA2 transition has been studied. The measurement
revealed that N-SmA2 phase transitions in this system are
first order with large C p peaks. Judging from the fact that thenematic range as well as the N-SmA2 transition temperatures
is essentially independent of X, it is expected that the natureof the first-order transitions does not arise from the effect ofthe mixing of bent-core molecules. This means that the SmA2
structure plays a role of relatively stable background forBC12 molecules. The obtained nematic range of BO11 isabout 15 K and larger than that of earlier reports of firstorder N-SmAd transitions 关15,16兴. The TNA / TIN value of
BO11 is 0.964 and, for example, almost the same as thatobserved in 4-n-octyloxy-4⬘-cyanobiphenyl 共8OCB兲 exhibiting a second-order N-SmAd transition with ␣ ⬃ 0.2 关14兴. A
second-order N-SmA2 transition has also been reported for acompound with a large nematic range of 65 K with the accompanying heat anomaly being described well by the
3D-XY behavior 关25兴. Therefore, we can expect the existenceof N-SmA2 tricritical behavior as the nematic region grows
as was expected in 关25兴. In particular, our data serve as aclear evidence that the TNA / TIN value at a tricritical point isnonuniversal or that the dependence on TNA / TIN values exhibited for N – SmA2 transitions differs from other types of
N-SmA transitions.
B. SmA2-SmA2b transition

It must be emphasized that the fully renormalized criticalexponents are visible only close to Tc. In this sense, a cross-

冉 冊册
1 dTc
Tc dX

1/␣

.

共4兲

Here, A0 is the amplitude of the nonrenormalized heatanomaly in units of R. For small critical exponent values like
Ising critical behavior, the observation of Fisher renormalization is expected to be difficult. However, from an experimental point of view, Fisher renormalization has been observed
over a wide range compared to the theory 关7兴. On the other
hand, the effect becomes more pronounced in the case of thetricritical behavior. A suitable parameter for that case to char2
acterize the dependence in Eq. 共4兲 is Z ⬅ 共T−1
c dTc / dX兲 ,2 2
where ␣ is fixed at 0.5. Then Eq. 共4兲 becomes ␶ = A0X 共1− X兲2Z2. A fully renormalized tricritical behavior over a substantial temperature range has been observed at several systems with Z ranging from 0.74 to 0.2 关7,8兴. In our system
considered, it turned out that Z = 0.067. This value is small
but accessible experimentally. In fact, if one assumes A0⬃ 0.1, this yields ␶ ⬃ 10−4, which does not seem to be an
unrealistic estimation.
To check the validity of the analyses, both the criticalexponent ␣ and the universal amplitude ratio A− / A+ are important factors because they are highly correlated with eachother. The results of the fits with Eq. 共1兲 give an asymmetricand nonuniversal feature of heat anomaly. The possibility of
the 3D-XY 共A− / A+ ⬃ 0.97 关26兴兲 or the 3D-Ising 共A− / A+
⬃ 1.85 关27兴兲 behavior is ruled out. Tables II and III also shownonuniversal values. If we adopt the tricritical behavior,
these features of A− / A+ ratios are not an issue of concern
because tricritical A− / A+ ratios are not universal. Our result
is also supported by the report in 关1兴, where an exponent
associated with the refractive indices has been estimated as␤ ⬃ 0.67. This large value can be explained by taking Fisher
renormalization into account, because critical exponents
should be doubled at a tricritical point.
Furthermore, it deserves attention that the magnitude of
the heat anomaly diminishes drastically as X decreases. As
mentioned in Sec. III A, the transition is caused essentiallyby BC12 molecules floating in the “sea” of SmA2 structure
formed from BO11 molecules. Therefore, the ⌬C p amplitude
has a natural dependence proportional to X since the data areshown in the unit of mass of total mixture, although theobserved decrease is obviously much faster. From the concept of the two-scale-factor universality 关28,29兴, such a decrease in heat anomaly implies a rapid increase in the correlation length. With the hyperscaling relation 2 − ␣ = d␯, it isexpected that the specific heat amplitude A and the correlation volume satisfy the following relation in smectic phases:
2A␰储␰⬜
= const.

共5兲

If we assume that the correlation length grows simply geometrically in accordance with the distance between correlated bent-core molecules as X decreases, we obtain from Eq.
共5兲 another multiplying factor proportional to X. This gives

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rise to an X dependence being proportional to X2 since wealready have the above-mentioned factor that is proportional
to X coming from the measuring unit. Nevertheless, the observed decrease is clearly still faster. The third and seemingly
most effective factor for the C p anomaly diminishing is the
suppression of the energy fluctuation coming from the saturation of the underlying order parameter, as observed in thecase of N-SmA transitions 关30–33兴. In those cases the widthof the N phase is very sensitive to parameters such as thehomolog number 关30兴, mixture ratio 关31,32兴, pressure 关33兴,
etc. As the underlying order for the present case, we canthink of the extent of how well the BC12 molecules sit in the
SmA2 framework of BO11 molecules. Our data give the
SmA2 phase width being ⬃3 K for X = 0.07 and ⬃14 K for
X = 0.045, differing by a factor of ⬃4.5. From Fig. 4 of Ref.
关31兴, this amount of difference will yield the C p anomaly
magnitude differ by several tens of times. Although this estimation may be rather crude, the observed reduction in the
C p anomaly in our Fig. 3 seems to be explained. In any case,the drastic decrease in A⫾ also implies that the renormalized
region depends on the concentration. Therefore, the heatanomaly of X = 0.07 should have a narrower renormalized
region than that of X = 0.08, which leads to the result that theeffective critical exponent value ␣X of X = 0.08 is closer to␣X = −1.

In conclusion, high-resolution calorimetric studies established a crossover phenomenon from underlying behavior far
away from Tc to Fisher-renormalized tricritical behavior near
Tc. Moreover, the measurements of the transitional enthalpy
revealed a clear evidence of an example of first-order
N-SmA2 transition with a wide nematic temperature range.
The results support the idea that the SmA2 structure serves asa relatively stable background for the bent-shaped molecules.
In addition, the dramatic reduction in the heat anomaly at thesmectic-A2–smectic-A2b transition as X decreases can be explained assuming that the energy fluctuations around thistransition are very sensitive to the ordering of the smectic-A2
background.

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ACKNOWLEDGMENTS

We are grateful to D. Miyajima and Professor T. Aida forsynthesizing additional pure BO11 samples. This research
was supported by the Global Center of Excellence Programby MEXT, Japan through the Nanoscience and Quantum
Physics Project of the Tokyo Institute of Technology. This
work was partly supported by the G-COE program 共Education and Research Center for Material Innovation兲 at the Tokyo Institute of Technology. B.K.S. thanks the Indian National Science Academy, New Delhi for financial support.

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