PHYSICAL REVIEW E 83, 061701 (2011)

Calorimetric study of the effect of bent-shaped dopant molecules on the critical behavior
at the nematic-smectic- Ad phase transition
Y. Sasaki and K. Ema*
Department of Physics, Tokyo Institute of Technology, O-okayama, Meguro, 152-8551 JapanK. V. Le and H. Takezoe
Department of Organic and Polymeric Materials, Tokyo Institute of Technology, O-okayama, Meguro, Tokyo 152-8552, JapanS. Dhara
School of Physics, University of Hyderabad, Hyderabad-500 046, India

B. K. Sadashiva
Raman Research Institute, C. V. Raman Avenue, Bangalore 560 080, India(Received 23 December 2010; revised manuscript received 29 March 2011; published 8 June 2011)
We report results of calorimetric studies for the binary mixture of rodlike host n-alkyloxy-cyanobiphenyl(nOCB, n = 8,9) and bent-shaped guest 1,3-phenylene-bis[4-(3-methylbenzoyloxy)]-4 -n-dodecylbiphenyl-4 carboxylate (BC12). The effect of bent-shaped dopant molecules on the critical behavior associated with thenematic-smectic-Ad phase transition has been studied in detail. The transition temperature for the nematicsmectic-Ad phase sharply decreases as the increase of the mole fraction of the dopant concentration (denoted X
for the BC12/9OCB mixture and Y for the BC12/8OCB mixture). The dependence of the critical exponent α on
X and Y is well explained in terms of the McMillan ratio. A nearly tricritical exponent has been obtained for theX = 0.01 mixture. X = 0.02 − 0.03 mixtures, pure 8OCB, and Y = 0.01 − 0.03 mixtures exhibit nonuniversal
behaviors with effective exponents lying between the 3D-XY and tricritical exponents. The heat capacity anomaly
for Y = 0.05 has been well described with the 3D-XY exponent. The critical amplitude ratio A− /A+ is close to 1
and insensitive to the dopant concentration. No Fisher renormalization of the critical exponent has been observed
even for nearly tricritical compositions, which indicates the smallness of the concentration plays a decisive rolerather than the steepness of the N-SmAd phase boundary.
DOI: 10.1103/PhysRevE.83.061701

PACS number(s): 64.70.M−, 64.60.F−, 65.40.Ba

I. INTRODUCTION

Recent research into liquid crystalline binary mixtures
composed of bent-shaped and rodlike molecules has shownvarious interesting phenomena such as novel phase transitions[1,2] and nanophase segregations [3–5]. On the other hand,the exact treatment of phase transitions in such mixture
systems is complicated because measurements are usually
performed along with a constant concentration x, while fixing
the chemical potential difference in the mixture significantly
acts as a thermodynamic constraint. As Fisher discussed [6,7],
the heat capacity exponent α is modified as αx = −α/(1 − α)in the vicinity of the transition temperature Tc , which is known
as the Fisher renormalization. So far, this phenomenon has
been observed in rodlike liquid crystalline mixtures at thenematic (N)-smectic-A (SmA) phase transition [8–10] andsmectic-A1 -smectic-A2 phase transition [11,12]. In addition,
recently, the present authors have carried out high-resolution
calorimetric studies on a novel orientational phase transition ofbent-shaped molecules in a smectic liquid crystal matrix [13].
It has been found that the heat capacity anomaly can be
described with Fisher-renormalized form in the vicinity of
Tc and that the magnitude of α plays a significant role as foundin several systems [6,8,14].

*

kema@phys.titech.ac.jp1539-3755/2011/83(6)/061701(9)A detailed estimation [15] regarding the crossover temperature between the Fisher renormalization and the underlying
critical behavior can be written as 1/α1 dTc 2
τ = Ax(1 − x)
,
(1)
Tc dx
where A is the unrenormalized critical amplitude of specific heat in units of R. This indicates that the extent of
the renormalization strongly depends on the magnitude ofseveral parameters, not only Tc−1 · dTc /dx, but also α, etc.
In particular, it can also be noticed from Eq. (1) that theconcentration x itself plays an important role. The crossover
temperature becomes small with decreasing concentration, sothat it can happen that no renormalization can be observed
experimentally, even for the tricritical exponent. Somehow,
almost all earlier reports mainly discussed the observation ofFisher renormalization with respect to the steepness of thephase boundary. Thus the importance of the concentration still
remains ambiguous and should be examined experimentallyto obtain a quantitative conclusion.
In this paper we report the results of intensive calorimetricinvestigations on the N-SmAd phase transition with bentshaped dopant molecules. Molecular structures of the hostand dopant materials are shown in Fig. 1. Rodlike liquid
crystalline molecules, n-alkyloxy-cyanobiphenyl (8OCB) and(9OCB), have been used as host compounds. Bent-shaped

061701-1©2011 American Physical Society

SASAKI, EMA, LE, TAKEZOE, DHARA, AND SADASHIVAPHYSICAL REVIEW E 83, 061701 (2011)0.3

φ (rad)

9OCB

0.2

0.1

FIG. 1. Structures of (upper) bent-core dopant molecule
1,3-phenylene-bis[4-(3-methylbenzoyloxy)]-4 -n-dodecylbiphenyl(BC12)
and(lower)rodlikemolecule4 -carboxylaten-alkyloxy-cyanobiphenyl (nOCB, n = 8,9).

II. METHODS AND RESULTS

The measurements have been done using two techniques,an ac calorimetry and a heat-flux differential scanningcalorimetry (DSC). The details of the ac calorimetry usedin the present study are found elsewhere [18]. Basically,
the precise specific heat capacity data have been obtainedby measuring the magnitude of an oscillatory temperature
response Tac exp i(2πf t + π/2 + φ) to an input heater
power P exp (i2πf t). Here f is the frequency of the ac
heating, which has been set to 0.03125 Hz in the present study.
The phase shift φ between the input heater power and thetemperature oscillation of the sample has also been measured,
which provides qualitative information for the nature of
the transition [19]. Particularly, in the case of a first-ordertransition, it is known that φ exhibits an anomalous increase
near the transition, as will be discussed later. The temperaturescan rate in the ac calorimetry was about ±0.03 K/h near thetransition region. Another method used here, DSC, has been
employed for the determination of the transition enthalpy.
Semiconducting thermoelectric modules have been used forthe measurement of the temperature difference between asample cell and a reference cell. A temperature scan rate of−1 −1

Cp (JK g )molecules (1,3-phenylene-bis[4-(3-methylbenzoyloxy)]-4 -ndodecylbiphenyl-4 -carboxylate; BC12) has been used as adopant in the rodlike molecules. The overall phase diagramof the BC12/8OCB system has been reported in Ref. [2]. We
particularly focus on the region where the concentration of thebent-shaped molecules is small. In both systems, it has beenfound that doping of the bent-shaped molecules gives rise toa sharp decrease of the N-SmAd transition temperature TNA
and a gradual increase of the isotropic (I)-N phase transition
temperature TI N . For clarity, we use hereafter the notation X
for the BC12/9OCB and Y for the BC12/8OCB mixtures todenote the mole fraction of BC12. The detailed analysis for
the N-SmAd critical behavior revealed that the dependenceof the critical exponent on X and Y has been well describedin terms of the McMillan ratio r = TNA /TI N . The crossoverof α from tricritical (αTCP = 0.5) to 3D-XY (αXY = −0.0066[16,17]) values has been observed by mixing a small amount ofbent-shaped molecules. On the other hand, no sign of the Fisher
renormalization of the critical exponent has been found, evenfor the nearly tricritical cases. This indicates that, despite thesteep phase boundary, the smallness of the concentration plays
a decisive role and hinders the observation of the renormalized
critical exponent.

30
20
10

345

350
T(K)

355

FIG. 2. Temperature dependence of the phase shift and heat
capacity for pure 9OCB near the I-N and N-SmAd phase transitions.
The temperature scan rate used is 0.3 K/h for the overall range.

±0.05 K/min has been used. Samples of about 7 mg wereplaced in hermetically sealed gold cells.
The specific heat Cp of the liquid crystal sample is obtainedwith the use of the following expression:
Cp = (Cpobs − Cpempty )/m.

(2)
empty

Here Cpobs is the observed heat capacity of the filled cell, Cp
is the heat capacity of the empty cell, and m is the mass of theliquid crystal sample.
The temperature variation of Cp and φ obtained for pure
9OCB is presented in Fig. 2. The Cp anomalies associatedwith the N-SmA and I-N phase transitions are evident at
350.84 and 353.03 K, respectively. The abrupt anomalous
increase of the phase shift φ, which is characteristic of thefirst-order transition [20], has been observed. Such behavior
is also in agreement with other N-SmA phase transitions
with narrow nematic ranges [21]. The two-phase coexistence
regions estimated from the φ anomalies are ∼ 0.04 K for the
N-SmAd and ∼ 0.09 K for the I-N phase transitions.
Figure 3 displays the temperature dependence of Cp
and φ for the X = 0.01 mixture. The I-N phase transition
exhibited essentially the same results, indicating the first-ordertransition, whereas, in contrast to pure 9OCB, φ shows a small
dip near the N-SmAd transition. Although this behavior isusually understood to indicate that the transition is second
order, it is valuable to take a closer look at this because theanomaly in Cp is rather large and sharp. As discussed inRef. [22], a linear relationship between φ and Cp is expectedin the absence of two-phase coexistence. In Fig. 4 data near the
N-SmAd transition for pure 9OCB, X = 0.01, and X = 0.03
mixtures are plotted on a φ-Cp plane. In the case of pure
9OCB, data points in the immediate vicinity of the transition
break the linear relation, indicating the first-order nature of
the transition. On the other hand, all the data points fall on astraight line in the case of the X = 0.03 mixture, which ensures
the absence of the two-phase coexistence. Similar results were
found for mixtures with X = 0.02 and Y = 0.00 − 0.05. For
the X = 0.01 mixture, shown in the middle portion of Fig. 4,061701-2

CALORIMETRIC STUDY OF THE EFFECT OF BENT- . . .

PHYSICAL REVIEW E 83, 061701 (2011)0.09

X=0.01

0.2

0.2

0.08
0.09
0.08

0.1

348

350

0.07

φ (rad)φ (rad)

0.1

352

9OCB
0.22

−1 −1

Cp (JK g )

10

0.23

0.24

0.25

0.1

7.5

0.3
−1
Cp (JK )5

0.4

X=0.01

2.5
350
T(K)

0.09

355

φ (rad)345

FIG. 3. Temperature dependence of the phase shift and heat
capacity for the X = 0.01 mixture near the I-N and N-SmAd phasetransitions. The dashed curve represents the background heat capacity
associated with the N-SmAd phase transition. The temperature scanrates used are 0.03 K/h near the N-SmAd and 0.3 K/h near the I-N
phase transitions.

Cp = Cp − Cp (background).

(3)

A similar procedure has been also done for data on othersamples.
The transition enthalpy δH in the N-SmAd phase transition
has been estimated asδH = HL + Cp dT .
(4)Here HL and Cp dT indicate the latent heat and theenthalpy change due to pretransitional fluctuations,
respectively. Whereas Cp is measured by the ac calorimetric

0.22

0.24

−1

0.26

0.28

Cp (JK )X=0.030.07

φ (rad)a slight but noticeable deviation is seen near the transition.
This shows that the transition is very weakly first order inthis mixture. Two-phase coexistence as determined from theabove-mentioned deviation becomes rather narrow, in a rangeof about 0.030 K.
Figure 5 shows the N-SmAd heat capacity anomalies
obtained for various BC12/9OCB and BC12/8OCB mixtures.
The heat capacity anomalies associated with the N-SmAd
transition are evident, accompanying a decrease of the transition temperatures with the increase of the concentration.
As seen from the phase diagram displayed in Fig. 6, theTNA value decreases sharply with increase of the dopant
concentration with phase boundaries of dTNA /dX ∼ −150 K
for BC12/9OCB and dTNA /dY ∼ −130 K for BC12/8OCB,
which are shown as solid lines. Contrary to the N-SmAd
transition, TI N monotonically increases with respect to thedopant concentration, resulting in the expansion of the nematic
temperature range.
The dashed curve shown in Fig. 3 represents the estimationof the background heat capacity associated with the I-N phasetransition. The excess specific heat capacity for the N-SmAd
phase transition is obtained by subtracting the contribution
from the I-N phase transition Cp (background):

0.08

0.065
0.23

−1

0.24

Cp (JK )FIG. 4. (Color online) Plot of the phase lag φ against Cp
for (upper) pure 9OCB, (middle) X = 0.01 mixture, and (lower)
X = 0.03 mixture near the N-SmAd phase transition obtained byac calorimetry. Open circles (red) and squares (blue) represent data
above and below the transition temperature.

technique, DSC directly measures δH . When the transitionis second order, HL is absent, and the transition enthalpy
obtained by DSC should be the same as that by ac calorimetry.
For pure 9OCB, the δH value obtained by DSC shows a larger
value compared with that of ac calorimetry, indicating thepresence of the latent heat. Figure 7 displays the heat capacity
anomalies for X = 0.01 and Y = 0.02 obtained by DSC and ac
calorimetry. For the Y = 0.02 case, both results agree with one
another on the whole and indicate a second-order transition.
On the other hand, the heat capacity anomaly by DSC for
X = 0.01 is slightly larger than that by ac calorimetry in thevicinity of Tc . This agrees with the very weak first-ordernature of the transition accompanying a small latent heat,
which has been expected from the φ-Cp plot mentioned above.
Table I summarizes phase-transition temperatures TNA andTI N , two-phase coexistence width δTNA and δTI N , transition
enthalpy δHNA , latent heat HL,NA , and the McMillan ratio.

061701-3

SASAKI, EMA, LE, TAKEZOE, DHARA, AND SADASHIVA

335

340

10

Y=0.020.4
Y=0.030.2

Y=0.05−1 −1

Cp (JK g )

6
−1 −1

ΔCp (JK g )349.8

X=0.01

4
X=0.02

X=0.01

8
6
4

5

3

0

2

349.75

350

10

8OCB

−1 −1

−1 −1

ΔCp (JK g )Y=0.01

349.5

345

Cp (JK g )

330
0.6

PHYSICAL REVIEW E 83, 061701 (2011)

337.05337.1

Y=0.02
2.8

2.8

2.6

2.6

X=0.03

337

337.5

T (K)

0
346

348

350

FIG. 7. (Color online) Temperature dependence of the heat
capacity for mixtures X = 0.01 (upper) and Y = 0.02 (lower) near the
N-SmAd phase transitions. The closed (red) and open circles (black)
represent the data obtained by ac calorimetry and DSC, respectively.

352

T (K)
FIG. 5. Temperature variation of the excess heat capacity Cpnear the N-SmAd phase transition for BC12/8OCB mixtures (upper)and for BC12/9OCB mixtures (lower) obtained by ac calorimetry.

rates. The fits to the Cp have been performed with the following expression, including corrections-to-scaling terms [16,23]:
(5)
Cp = A± |t|−α 1 + D1± |t|θ + D2± |t|2θ + Bc .

We also add that any noticeable hysteresis indicating phaseseparation was not observed in all cases.
For the critical exponent analyses, the data obtained byac calorimetry have been used. As described above, the DSC
measurements also provide the high-resolution data. However,since the temperature scan rate used in our DSC measurements
is faster than that of the ac calorimetric measurement, the DSC
data in the very vicinity of Tc may be smeared by the finite scan

Here t = (T − Tc )/Tc is the reduced temperature, and ±indicates above and below Tc . D1± and D2± are the coefficientsof the first- and second-order corrections-to-scaling terms,
respectively. The corrections-to-scaling exponent θ is taken
to be 0.5 [16,23]. The constant term Bc is the criticalcontribution. The data in the vicinity of the transition
temperature are not used due to the instrumental limitations
and the inhomogeneity of the sample. The fits to the data
have been performed for |t|min  |t|  |t|max , with |t|max
being the maximum reduced temperature used in the fits. The|t|min values have been determined according to the method
described in Ref. [24]. The rounding regions thus determined
are described in the caption of the tables shown below.
Table II shows the results for the X = 0.01 mixture.
Although the transition was found to be first order for thismixture, the critical exponent analyses with Eq. (5) can bejustified because the first-order nature is very weak. Indeed,
two-phase coexistence width only amounts to about 30 mK,
comparable to the typical rounding region found for secondorder phase transitions. The fits have been carried out forseveral |t|max values to check the dependence of the adjustableparameters on the data range. From the simple power-law fit
without corrections-to-scaling terms, the critical exponent α
is in the range 0.54–0.57, which is independent of the data
range shrinking and slightly larger than the tricritical value
αTCP = 0.5. The amplitude ratio A− /A+ is smaller than 1.

Iso

Iso

N

350

T (K)

350

N

340

340
SmAd
SmAd

0

0.02
X

0

0.02 0.04
Y

FIG. 6. Partial phase diagrams for BC12/9OCB (left) andBC12/8OCB (right). The transition temperatures have been determined by ac calorimetry.

061701-4

CALORIMETRIC STUDY OF THE EFFECT OF BENT- . . .

PHYSICAL REVIEW E 83, 061701 (2011)

TABLE I. Values of transition temperature TNA and TI N , two-phase coexistence width δTNA and δTI N ,transition enthalpy δHNA , latent heat HL,NA , and McMillan ratio. Subscripts N A and I N represent
N -SmAd and I -N transitions, respectively. The unit is K for TNA , TI N , δTNA , and δTI N , and J/g for δHNAand HL,NA .
Sample

TNA

δTNA

δHNA

HL,NA

TI N

δTI N

McMillan ratio

X = 0.00
X = 0.01
X = 0.02
X = 0.03
Y = 0.00
Y = 0.01
Y = 0.02
Y = 0.03
Y = 0.05

350.83349.79347.85346.22340.18338.65337.81
335.66333.76

0.038
0.030

1.47
1.18
0.86
0.68
0.40
0.33
0.31
0.29
0.27

0.18
0.12

353.03353.63353.72354.33353.38354.08354.78355.45355.96

0.09
0.13
0.17
0.14
0.06
0.07
0.11
0.14
0.56a

0.994
0.989
0.983
0.977
0.963
0.956
0.952
0.944
0.938

a
This value is less reliable because only data with relatively fast scan-rates (∼ 4 K/h) are available near the
I -N transition for this mixture.

However, this is rather artificial because A− /A+ values are
dependent on the data range. The quality of the fit has been
improved in the sense of χ 2 values after inclusion of thecorrections-to-scaling terms, while α remains insensitive to thedata range. As displayed in Fig. 8, the fit with |t|max = 0.003
shows a good agreement with the experimental data. Thus
we see that the D1± and D2± terms play an important role.
A− /A+ ∼ 1 has been obtained by taking the corrections-toscaling terms into account. The first corrections-to-scaling
terms show a good agreement with the theoretical expectationthat D1− /D1+ ∼ 1. As described above, α values are close to0.5, indicating that there is no Fisher renormalization of thecritical exponent. This also implies that the first correctionsto-scaling terms A± D1± |t|θ−α reduce to be almost temperature
independent. Hence, we rewrite Eq. (5) as the following
expression:Cp = A± |t|−α 1 + D2± |t| + B ± .
(6)
The obtained adjustable parameters are summarized inTable III. Values of α and Tc are more or less the same as
those obtained using Eq. (5). The fits with D2± = 0 show poor
χ 2 values, especially for large |t|max fits, while the quality of
the fit has been improved by taking the D2± term into account.
The results of the fits for the X = 0.03 mixture are shownin Table IV. The critical exponent lies in α = 0.23 − 0.30.

These α values are nonuniversal, being between αXY andαTCP , and clearly small compared to those of the X = 0.01
mixture. The amplitude ratio A− /A+ still lies ∼ 1. For theX = 0.03 mixture, the fits with the corrections-to-scaling
terms sometimes exhibit unphysical results for which the D1±
and D2± values are very large. In those cases the Tc values
slightly differ from the fits in which the leading exponent term
is dominant or the D1± are fixed to zero. To avoid such artifacts,
fits were made with Tc value fixed at 346.226 K. As the result,the D1± and D2± terms become small and relatively stable.
When D1± terms are included, D1− /D1+ values are close to 1,
and the quality of the fit is improved. Moreover, it is to be notedthat the magnitude of the D1± terms becames smaller than thatof the X = 0.01 mixture. For other BC12/9OCB mixtures, thevalues of α decreased monotonically with the increase in X,
although the detailed results are not shown. With decreasing X,
it was also found that the corrections-to-scaling terms equallyplay an important role while their magnitude becomes smaller.
Table V shows the results for the Y = 0.01 mixture. For thefits with D2± = 0, the obtained α values range in 0.064–0.21,
which is smaller than that for 8OCB. It is seen that themagnitude of D1± decreases further compared to the X = 0.03
sample. On the other hand, A+ /A− is still close to 1, implying
the symmetric character of the heat capacity anomaly. The last
two lines in the table show results with nonzero D2± . Quality

TABLE II. Results of the fits to the heat capacity data for the X = 0.01 mixture using Eq. (5). The−5
and |t|−minimum reduced temperatures used in the fits for T > Tc and T < Tc are |t|+min = 6.6 × 10
min =
−1 −1
−5
+
4.6 × 10 , respectively. The units for A and Bc are JK g .
|t|maxαTc

A+A− /A+

0.00050.00100.00300.00500.00300.00500.00700.00500.0070

0.574
0.545
0.536
0.535
0.542
0.559
0.561
0.474
0.453

349.804
349.804
349.803349.803349.806
349.806
349.805349.807349.8070.01120.01520.01700.01720.01440.01190.01210.03240.0416

0.955
0.951
0.923
0.921
1.074
1.071
1.018
1.118
1.117

D1+

D1− /D1+

19.45
30.19
23.13
149.4098.37

061701-5

0.67
0.77
0.84
0.84
0.82

D2+

−99.14−103.99

D2− /D2+

Bc

χ2

0.24
0.16

−0.124−0.207−0.232−0.234−0.524−0.642−0.549−4.349−3.352

3.36
5.13
13.47
12.56
1.02
2.33
6.31
1.46
1.79

SASAKI, EMA, LE, TAKEZOE, DHARA, AND SADASHIVAPHYSICAL REVIEW E 83, 061701 (2011)
0.2

6
-1 -1

ΔCp (JK g )-1 -1
ΔCp (JK g )

8

4

0.1

2
0
349

349.5

350

0

350.5

333

334

335

T (K)

T (K)
FIG. 8. (Color online) Temperature variation of the anomalous
heat capacity near the N-SmAd phase transition for X = 0.01. The
solid line shows a fit to Cp with Eq. (5). Open circles show the data
omitted from the least-squares-fitting procedure.

of the fits has been improved in the χ 2 sense. The α values areclose to 0 and seem reasonable because they agree with thosefor D2± = 0 over narrower data ranges such as |t|max = 0.003.
The results for the Y = 0.05 mixture are summarized inTable VI. For the simple power-law fits with the corrections toscaling terms fixed to zero, the α values range between −0.019
and 0.138. It should be noted that the magnitude of the excess
heat capacity is small compared to other data, as displayedin Fig. 5, and therefore the obscurity for determination of thecritical exponent is inevitable. Because of this, fits with Tc
fixed at 333.777 K have been tried. Fits were also made withthe 3D-XY critical exponent α = αXY = −0.0066. Figure 9shows the result obtained with Tc = 333.777 K and α = αXY .
The quality of the fits with both α and Tc fixed are comparablewith the fits with parameters adjusted freely. On the other hand,no clear evidence of the 3D-XY critical amplitude ratio has
been found, and the A− /A+ values still remain close to 1.
III. DISCUSSION

Our calorimetric studies revealed that the N-SmAd phasetransition is first order for pure 9OCB, which agree with theresult obtained by means of a modulated DSC technique [25].
For the X = 0.01 mixture, it has been found that the transitionis very weakly first order. Other mixtures exhibit second-ordertransitions. The behavior of the phase signal φ in the actechnique has been used for the distinction between first- and

FIG. 9. (Color online) Temperature variation of the anomalous
heat capacity near the N-SmAd phase transition for Y = 0.05. The
solid line shows a fit to Cp with Eq. (5) with α and Tc fixed at−0.0066 and 333.777 K. Open circles show the data omitted fromthe least-squares-fitting procedure.

second-order transitions. However, it should be stressed thatthe ac calorimetric technique cannot determine the transitional
enthalpy, and the response of φ is rather insensitive to thesmall latent heat. The Cp data obtained by ac calorimetry
sometimes show a remarkable difference from those of
DSC [26]. Thus, the direct measurement of the transition
enthalpy would be reliable information. So far, the adiabatic
calorimetry is suitable for this requirement [27,28]. Comparedwith this, as far as the the earlier DSC measurements areconcerned, because of the dynamic temperature scan incapableof detecting the heat flow difference instantaneously, it wasdifficult to clearly distinguish between first- and second-ordertransitions. In the present study, our high-sensitivity DSC
allowed us to give a fairly good agreement of DSC and ac
calorimetry data, providing a reliable evidence to distinguishbetween first- and second-order transitions.
Nearly tricritical behavior has been observed for theX = 0.01 mixture. The value of McMillan ratio r = 0.99
for this mixture is in a good agreement with that of thepreviously reported value at the tricritical point of the
N -SmAd phase transition. The obtained α values, slightly
larger than αTCP = 0.5, may be due to the weakly first-ordercharacter. From the viewpoint of α values, the observationof the Fisher renormalization in the present work should be
the most pronounced for X = 0.01 due to the doubling of
the critical exponent at the tricritical point. The value of

TABLE III. Results of fits to the heat capacity data for the X = 0.01 mixture using Eq. (6). The−5
andminimum reduced temperatures used in the fits for T > Tc and T < Tc are |t|+min = 6.6 × 10
−1 −1
−
−5
+
±
|t|min = 4.6 × 10 , respectively. The units for A and B are JK g .
|t|maxαTc

A+A− /A+

0.00050.00300.00500.00700.00300.00500.0070

0.554
0.529
0.528
0.533
0.537
0.523
0.522

349.805349.806
349.806
349.805349.806
349.807349.8070.01340.01690.01710.01670.01550.01730.0174

1.002
1.062
1.052
1.006
1.063
1.142
1.158

D2+

−19.50−38.51−49.91061701-6

D2− /D2+

B+

B − /B +

χ2

0.98
−1.23
−1.30

−0.155−0.193−0.200−0.208−0.159−0.155−0.146

1.24
1.52
1.44
1.27
1.62
2.37
2.69

2.14
1.02
2.56
6.60
1.00
1.49
1.94

CALORIMETRIC STUDY OF THE EFFECT OF BENT- . . .

PHYSICAL REVIEW E 83, 061701 (2011)

TABLE IV. Results of fits to the heat capacity data for the X = 0.03 mixture using Eq. (5). The−5
andminimum reduced temperatures used in the fits for T > Tc and T < Tc are |t|+min = 9.5 × 10
−1 −1
−4
+
=
1.4
×
10
,
respectively.
The
units
for
A
and
B
are
JK
g
.
Quantities
inbrackets
are
|t|−
c
min
held fixed at the given values.
|t|maxα0.00100.00100.00500.00500.00300.00500.0070Tc

A+

0.292 [346.226] 0.0675
0.260 346.224 0.0977
0.269 [346.226] 0.0891
0.252 346.222 0.1076
0.302 346.226 0.0552
0.241 [346.226] 0.1252
0.231 346.226 0.1439

A− /A+
1.017
1.006
1.006
0.993
1.044
1.034
1.032

D1+

−5.5391.885
2.391

Z ≡ (Tc−1 · dTc /dx)2 in Eq. (1) is a suitable measure to characterize the observability of the renormalized exponent. Previously, fully Fisher-renormalized tricritical behaviors show Z
values ranging from 0.74 to 0.2 [9,10,29]. In the present study,Z ∼ 0.18 for X = 0.01 seems sufficient for the appearance ofthe crossover temperature. However, according to the analysis
described above, no sign of the Fisher renormalization has
been obtained. By substituting the parameters to Eq. (1) using
the critical amplitude A± in Table II, the range of the crossoverin the reduced temperature becomes τ ∼ 10−6 . This indicatesthat the observable region is quite narrow, in accordance with
our analysis. The present result can be compared with ourrecent study in a different system, which revealed that a
Fisher-renormalized behavior actually occurs in the vicinityof Tc [13]. In that case the system also included a tricriticalpoint and a steep phase boundary, which is apparently similar
to those in the present study. The crucial difference lies in the
dilute composition of the bent-shaped molecules used here.
Hence, we see that the concentration is also a very importantfactor to determine the crossover temperature.
The present results confirm that the width of the nematic
temperature range plays an important role for deciding thenature of the N-SmA phase transition. First, the magnitude

D1− /D1+ D2+ D2− /D2+

1.17
0.48
0.61

Bc

χ2

−0.229−0.310−0.293−0.331−0.087−0.422−0.484

1.76
1.29
5.00
2.96
1.43
1.44
1.38

of the heat capacity anomaly decreases significantly as
the nematic range grows, showing that thermal fluctuations
associated with the heat capacity anomaly are very sensitive
to the saturation of the nematic order, as pointed out in ourprevious work [13] and also in earlier works cited therein (seeRef. [30], for instance).
The trend of the critical behavior as a function of thenematic range also deserves attention. The critical behavior for
X = 0.02 and 0.03 mixtures, pure 8OCB, and Y = 0.01–0.03
mixtures exhibit nonuniversal exponents between the 3D-XY
and tricritical values. For X = 0.05, the excess heat capacity
anomaly is well described with the 3D-XY critical exponent.
The amplitude ratio showing the symmetric feature is moreanalogous to that of the nCB series [27,28] than that of the
nonpolar mixtures exhibiting nonuniversal tricritical amplitude [29]. The relation between α and TNA /TI N is shown in
Fig. 10. The overall behavior approaching the 3D-XY value is inremarkable agreement with the previous reports on the N-SmA
phase transitions [31], which are shown in the figure as inverted
triangles and open circles. We also mention here that thiscrossover of α gives rise to characteristic behavior regarding
the magnitude of the first-order corrections-to-scaling terms
D1± . Their values clearly diminish as the decrease of α values

TABLE V. Results of fits to the heat capacity data for the Y = 0.01 mixture using Eq. (5). The minimum−5−6
and |t|−reduced temperatures used in the fits for T > Tc and T < Tc are |t|+min = 7.4 × 10
min = 5.9 × 10 ,respectively. The units for A+ and Bc are JK−1 g−1 . Quantities in brackets are held fixed at the given values.
|t|maxα0.00050.153
0.00100.135
0.00300.151
0.00500.158
0.00700.162
0.00300.124
0.00300.064
0.00500.167
0.00500.109
0.00700.211
0.00700.147
0.00700.032
0.0070 −0.021

Tc

A+

338.663
0.1237338.6640.1576338.665
0.1159338.665
0.1041
338.665
0.0980[338.663]
0.2147[338.664]
0.7147[338.663]
0.1035[338.664]
0.2668[338.663]
0.0526[338.664]
0.1351[338.663]
2.1828[338.664] −5.3621

A− /A+
1.027
1.038
1.062
1.069
1.071
1.003
1.006
1.011
1.014
1.023
1.027
0.997
1.000

D1+

1.654
1.201
0.807
1.354
−1.5890.848
1.075−0.753061701-7

D1− /D1+

1.76
1.37
3.06
1.54
−0.30
2.00
1.50
1.35

D2+

−3.2392.629

D2− /D2+

Bc

χ2

2.69
2.06

−0.267−0.312−0.239−0.220−0.210−0.444−1.067−0.248−0.502−0.125−0.294−2.720
4.626

1.22
1.36
1.56
1.52
1.47
1.05
1.06
1.07
1.09
1.17
1.18
0.95
0.97

SASAKI, EMA, LE, TAKEZOE, DHARA, AND SADASHIVAPHYSICAL REVIEW E 83, 061701 (2011)

0.6
αTCP = 0.5

α

0.4

0.2
αXY = −0.0066

0.92

0.94

0.96
0.98
TNA / TIN

1

FIG. 10. Specific heat critical exponent α plotted against theMcMillan ratio. The α values displayed here have been obtained fromfits without D2± terms. Results reported in Ref. [31] for N -SmAm
transitions (inverted triangles) and for N -SmAd transitions (opencircles) are also shown.

as shown in Table II–VI. It is seen that D1± values for theX = 0.01 mixture are much larger than those for the Y = 0.05
mixture. Although it is known that D1± values are nonuniversal,
we can speculate that this trend reflects the change in α. If α iscloser to 0.5, the distinction between D1± |t|θ−α and Bc becomesdifficult because they are almost constant terms. Contrary to
the tricritical case, the heat capacity anomaly for X = 0.05
with smaller α can exhibit the clear temperature dependence of
D1± |t|θ−α . Moreover, the magnitude of D1± terms for Y = 0.05
is in reasonable agreement with other liquid crystal samples
exhibiting the 3D-XY critical behavior [32].
In contrast to the crossover of the critical exponent, thecritical amplitude ratio shows no clear evidence that A− /A+
agrees with the 3D-XY value of 0.971 or the inverted value of
1.027. Experimentally, the well-described XY -like behaviors
were observed for the cases with a large nematic range ofTNA /TI N < 0.90 [31]. There, heat capacity anomalies aboveand below Tc show clearly asymmetric character. Contrary
to those previous reports, the present data are symmetricand inconsistent with both A− /A+ < 1 and A− /A+ > 1. A
plausible reason may be attributable to an interaction between
director fluctuations δn and the smectic order parameter [33].
This anisotropic effect gives rise to deviations from isotropic
XY behavior when the nematic range is small. In this sense
the TNA /TI N = 0.937 is still not sufficient to produce theasymmetric heat capacity anomaly.
The fact that the nematic range is widened significantly by
inclusion of bent-shaped molecules is quite remarkable, and a
deeper discussion is warranted. In the present work, since themole fractions of the doped bent-shaped molecules are quite
small, they can be regarded as impurities. Apparently similar
widening of nematic phase was reported by Denolf et al. [34].
They explained the linear temperature dependence for the
N-SmA and I-N phase transitions by taking the coupling between the orientational order parameter S and the mole fractionof the impurities. However, they always observed a downward
shift of transition temperatures in contrast to the present result,where the I -N transition increases significantly with X andY . Such a difference seems natural because nonmesogenic
impurities have been used in their work, while in our casethe dopant molecules have the capability of forming liquid
crystal phases by themselves. It is of particular interest that theinclusion of BC12 stabilizes the nematic phase and destabilizes
the Sm-Ad phase. It is possible that the more rigid core of
the BC12 dopant dampens director fluctuations. The question
arises if the bent-core nature of BC12 plays a role, or if a linear
rigid core molecule of similar character produces equal effects.
A linear-core version of BC12 may not stabilize the Sm-A
phase to the same extent but may perhaps stabilize the nematic
phase more, leading to a more or less alike nematic range. Such
possibilities will be interesting candidates for future studies.

TABLE VI. Results of fits to the heat capacity data for the Y = 0.05 mixture using Eq. (5). The−6
andminimum reduced temperatures used in the fits for T > Tc and T < Tc are |t|+min = 2.6 × 10
−1 −1
−
−5
+
|t|min = 6.3 × 10 , respectively. The units for A and Bc are JK g . Quantities in brackets areheld fixed at the given values.
A+

A− /A+

0.0010 [−0.0066] 333.777 −5.196
0.0010 −0.019 333.777 −1.991
0.0030 [−0.0066] 333.783 −4.762
0.0030 0.138
333.777
0.080
0.0050 0.129
333.778
0.091
0.0070 0.115
333.779
0.113
0.0010 [−0.0066] 333.775 −5.074
0.0030 [−0.0066] [333.777] −6.369
0.0030 [−0.0066] 333.775 −6.443
0.0050 [−0.0066] [333.777] −6.005
0.0050 0.004 [333.777] 9.037
0.0050 [−0.0066] 333.775 −6.039
0.0070 [−0.0066] [333.777] −5.767
0.0070 0.073 [333.777] 0.262
0.0070 [−0.0066] 333.776 −5.785
0.0070 [−0.0066] [333.777] −6.522
0.0070 [−0.0066] 333.775 −6.675

1.002
1.006
1.001
0.974
0.979
0.985
1.004
1.003
1.003
1.002
0.999
1.003
1.002
0.974
1.002
1.003
1.004

|t|maxαTc

D1+

D1− /D1+ D2+ D2− /D2+

0.052 −0.42−0.094 1.53
−0.093 1.63
−0.075 1.51
0.043 1.55
−0.073 1.60
−0.061 1.54
0.209 3.01
−0.061 1.58
−0.120 1.81
−0.121 1.97

061701-8

0.289
0.273

3.58
4.42

Bc

χ2

5.036
1.817
4.620
−0.137−0.152−0.180
4.926
6.137
6.208
5.794
−9.2335.827
5.570−0.366
5.587
6.279
6.424

1.03
1.03
1.33
1.22
1.24
1.26
1.00
1.10
1.09
1.16
1.16
1.16
1.20
1.19
1.20
1.16
1.14

CALORIMETRIC STUDY OF THE EFFECT OF BENT- . . .

PHYSICAL REVIEW E 83, 061701 (2011)

In summary, we have measured the heat capacity near
the N-SmAd phase transition in mixtures with a small
amount of the bent-shaped molecules. The critical exponent analyses revealed that the tricritical exponent crosses
over toward the XY -like value with the increase of thenematic temperature range. On the other hand, despite the
sharp change of the transition temperatures with the mixing ratio, no feature of the Fisher renormalization of thecritical exponent has been seen. This result confirms the
expectation that the smallness of the concentration plays

The authors are grateful to D. Miyajima and T. Aida forkindly supplying the liquid crystal samples. This research
was supported by the G-COE programs (Nanoscience andQuantum Physics, Education and Research Center for Material
Innovation) at the Tokyo Institute of Technology.

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a significant role in the observability of the renormalized
region.
ACKNOWLEDGMENTS

061701-9