Novel biaxial nematic phases of side-chain liquid crystalline polymers
Akihiko Matsuyama
Citation: J. Chem. Phys. 137, 224906 (2012); doi: 10.1063/1.4771592
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THE JOURNAL OF CHEMICAL PHYSICS 137, 224906 (2012)

Novel biaxial nematic phases of side-chain liquid crystalline polymers
Akihiko Matsuyamaa)
Department of Bioscience and Bioinformatics, Faculty of Computer Science and Systems Engineering,
Kyushu Institute of Technology, Kawazu 680-4, Iizuka, Fukuoka 820-8502, Japan(Received 4 October 2012; accepted 21 November 2012; published online 14 December 2012)
We present a mean field theory to describe biaxial nematic phases of side-chain liquid crystalline
polymers, in which rigid side-chains (mesogens) and rigid-backbone chains favor mutually perpendicular orientations. Taking into account both excluded volume and attractive interactions between
rigid rods, novel biaxial nematic phases are theoretically predicted. We calculate uniaxial and biaxial
orientational order parameters as a function of temperature and the length of backbone chains. We
find a first-order biaxial-biaxial phase transition and a first (or second)-order uniaxial-biaxial one,depending on the length of mesogens and backbone chains. © 2012 American Institute of Physics.
[http://dx.doi.org/10.1063/1.4771592]
I. INTRODUCTION

Biaxial nematic phase has been first theoretically predicted by Freiser.1 Since then, it has been the subject of much
experimental,2–10 computational,11–13 and theoretical13–22
work. Biaxiality occurs if anisotropic particles orient along
a second axis perpendicular to a main director of theparticles.23, 24 Such biaxiality is expected significant advantages in display applications with a fast response.25 Biaxial phases of side-chain liquid crystalline polymers (LCPs)
have been studied in nematic and smectic A phases with sideon or end-on attachment of mesogenic group to the polymer
backbone.26–30
When we consider a nematic phase of side-chain LCPs,
there are three uniaxial nematic phases, which were first considered by Wang and Warner.31 The three nematic phases canbe defined by two orientational order parameters: one is Sm
of nematogenic side-chains (mesogens) and the other is Sb ofa rigid-backbone chain. When one order parameter is positive, the other can be positive or negative.32, 33 Figure 1 shows
three principal uniaxial nematic phases for a side-chain LCP.
The N1 phase (Sm > 0 and Sb < 0) is defined as that the mesogens are aligned along to the ordering direction (z) and the
backbone chains are randomly distributed on the plane perpendicular to the director Dm . The N2 phase (Sm < 0 and
Sb > 0) is defined as the backbone chain is aligned along
to the ordering direction Db and the mesogens are randomly
distributed on the plane perpendicular to the backbone chain.
The third N3 phase is defined by Sm > 0 and Sb > 0, where
the backbone and mesogens are oriented along to the ordering
direction Dmb . In the N1 phase, the backbone chain adopts an
oblate shape. In the N2 and N3 phases, a prolate shape of the
backbone is obtained.34–40
In these uniaxial nematic phases (N1 , N2 ), we can expect
biaxial nematic phases. Figure 1 schematically shows novel
biaxial nematic phases (N1b , N2b ) of side-chain LCPs, where
the mesogens and backbone chains favor mutually perpendica) Electronic mail: matuyama@bio.kyutech.ac.jp. URL: http://iona.bio.

kyutech.ac.jp/~aki/.
0021-9606/2012/137(22)/224906/6/$30.00

ular orientations.41 In the uniaxial nematic N1 phase, we can
expect additional ordering of backbone chains in the direction
db (minor director) perpendicular to the director Dm (major
director). This corresponds to biaxial ordering (N1b ). In theN2 phase, we may have a biaxial nematic (N2b ) phase, where
the additional ordering of mesogens appears in the direction dm (minor director) perpendicular to the major director
Db . Recently, such mutually perpendicular orientations havebeen theoretically predicted in mixtures of a liquid crystal anda nanotube,32, 42 in liquid crystalline elastomers,43, 44 and incombined main-chain/side-chain liquid crystalline polymers
using the self-consistent field theory.45
In this paper, we present a mean field theory to describe
the biaxial nematic phases of side-chain LCPs. Taking into
account both excluded volume and attractive interactions between rigid rods, the novel biaxial nematic phases are theoretically predicted in side-chain LCP melts, where side-chains(mesogens) and rigid-backbone chains favor mutually perpendicular orientations. We calculate uniaxial and biaxial orientational order parameters and find first (or second)-orderuniaxial-biaxial phase transitions, depending on the length ofthe mesogen and of the backbone chain.

II. FREE ENERGY

We consider a melt of a side-chain LCP, consisting of aliquid crystalline backbone chain and rigid side-chains (mesogens). The repeating unit on the side-chain LCP consists of amesogen with the axial ratio nm (= Lm /d) and a rigid-backbone
chain of the axial ratio nb (= Lb /d), where Lm and Lb are thelengths and d is the diameter of the mesogen and the backbone chain on the repeating unit. The total number of segments on the polymer is given by n = (nm + nb )t, where t
is the number of a repeating unit on the LCP. The volumeof the mesogen and that of the backbone chain are givenby vm = (π/4)d 2 Lm and vb = (π/4)d 2 Lb , respectively. The
volume per polymer molecule is given by V = (vm + vb )t= (π/4)d 3 n and the number density of mesogens and backbone chains is given by c = t/V .

137, 224906-1© 2012 American Institute of Physics

224906-2

Akihiko Matsuyama

J. Chem. Phys. 137, 224906 (2012)Db

Dm

angle between the local orientations u and u . Here we use
the additional theorem of spherical harmonics:

Dmb

P2 (cos γ ) = P2 (cos θ )P2 (cos θ  ) + 22(2 − k)!
k=1

N2

N1

×P2k (cos θ )P2k (cos θ  ) cos[k(ϕ − ϕ  )], (4)

N3

Db

Dm
db

z

dm

y
x

N1b

N2b

FIG. 1. Three possible uniaxial nematic phases (N1 , N2 , N3 ) and two biaxial nematic phases (N1b , N2b ) of side-chain liquid crystalline polymers.
Depending on two orientational order parameters: one is that Sm of a sidechain (mesogen) and the other is Sb of a rigid-backbone chain, we can definea nematic N1 phase with Sm > 0 and Sb < 0, a N2 phase with Sm < 0 and
Sb > 0, and a N3 phase with Sm > 0 and Sb > 0. Biaxial nematic N1b phaseis defined as a minor director db appears perpendicular to the major director Dm . Biaxial nematic N2b phase is defined as a minor director dm appears
perpendicular to a major director Db .

The volume fraction of the mesogens is given by
xm = nm /(nm + nb ),(1)

and that of the backbone chain is xb = nb /(nm + nb ).
To describe the nematic ordering, we take into accountboth the excluded volume interactions47 and the orientationaldependent anisotropic interactions between liquid crystalline
molecules.48 Let ν mm (>0) be the anisotropic (Maier-Saupe)
interactions between the mesogens, ν mb be that between themesogen and the rigid-backbone chain, and ν bb (> 0) be thatbetween the rigid-backbone chains. These anisotropic interaction parameters are dimensionless parameters: ν ij ≡ Uij /kB T,
where T is the absolute temperature, kB is the Boltzmann constant, Uij is the interaction energy between the components
i and j (i, j = m, b). Let fm (u) and fb (u) be the orientational
distribution function of the mesogen and the backbone chain
with the orientational unit vector u = {θ , ϕ}, defined by a polar angle (0 ≤ θ ≤ π ) and an azimuthal angle (0 ≤ ϕ ≤ 2π ),or solid angle d = sin θ dθ dϕ. The nematic free energy is
given by46
 
βFnem /V =c fi (u) ln[4πfi (u)]di,j =m,b+(2 + k)!


1  2
c
fi (u)fj (u )βij (u, u )dd ,2 i,j =m,b(2)

where β ij is the Mayer-Mayer function between two components i and j. For the interaction between rigid rods, wetake into account both the excluded volume and the attractive
interaction:46–50
8
|u × u | − νij P2 (cos γ ) ,βij = vij
(3)
πwhere vij = (π/4)Li Lj d is the average excluded volume between the molecules i and j in an isotropic phase, γ is thewhere the terms proportional to the associated Legendre polynomials (P2k ) will vanish in a uniaxial nematic phase, which
does not depend on the azimuthal angle. In a biaxial nematic
phase, however, the terms of the Legendre polynomials (P22 )have a finite contribution.
The orientational order parameters Si (i = m, b) of a uniaxial nematic phase is given by
Si = P2 (cos θ )fi (θ, ϕ)d,(5)where P2 (cos θ ) = (3/2)(cos 2 θ − 1/3) and the biaxial orderparameter is given by=D(θ, ϕ)fi (θ, ϕ)d,(6)i√where D(θ, ϕ) ≡ ( 3/2) sin2 θ cos(2ϕ). Using the tensor order parameter Si, αβ = (3/2)Si (nα nβ − δ αβ /3), (α, β = x, y,z), we have i = Si, yy − Si, xx and Si = Si, zz . Here Si, zz describes alignment of molecules along the z axis (major director), whereas the nonzero value of i describes ordering along
the x or y axis. Then Eq. (4) can be given by
P2 (cos γ ) = P2 (cos θ )P2 (cos θ  ) + D(θ, ϕ)D(θ  , ϕ  ).

(7)

Substituting Eqs. (5)–(7) into (2), the nematic free energy is
given by xi 
βFnem /n =fi (u) ln[4πfi (u)]dni,j =m,b i1− (νmm + 5/4)xm2 Sm2 + 2m
2−(νmb + 5/4)xm xb (Sm Sb + m b )1− (νbb + 5/4)xb2 Sb2 + 2b ,(8)2where the term 5/4 corresponds to the excluded volume interactions.
Minimizing (8) with respect to fi (u) with the normalization condition fi (u)d = 1, we obtain the orientational distribution functions,fm (u) =1
exp[ m,u P2 (cos θ ) +Zm(9)m,b D(u)],for a mesogen andfb (u) =1
exp[ b,u P2 (cos θ ) +
Zb

(10)

b,b D(u)],for a backbone chain, where we definem,u ≡ nm [(νmm + 5/4)xm Sm + (νmb + 5/4)xb Sb ],m,b ≡ nm [(νmm + 5/4)xm

m + (νmb + 5/4)xb

(11)

b ],

(12)

Akihiko Matsuyama

J. Chem. Phys. 137, 224906 (2012)b,u ≡ nb [(νbb + 5/4)xb Sb + (νmb + 5/4)xm Sm ],b,b ≡ nb [(νbb + 5/4)xb

b + (νmb + 5/4)xm

m ].

1.1

(13)

(14)

T / TNI,1

224906-3

The constants
 Zm and Zb can be obtained by the normalization
condition: fi (u)d = 1, as

0

0

× exp

N3

(15)

where the function Ii (i = m, b) is defined as
 1
 1
Ii [p, q] ≡dx
dy[P2 (x)]p [D(x, y)]q

1.0

0.9

0.8
1.0

√
3 2 1
3
(1−x 2 ) cos(4πy) ,i,u (x − )+ i,b
2
3
2

nb

3.0

with p, q = 0, 1, 2, . . . .
The orientational order parameters can be numericallycalculated from Eqs. (5) and (6), using the distribution functions (9) and (10):Sm = Im [1, 0]/Im [0, 0],(17)m = Im [0, 1]/Im [0, 0],(18)

Sb = Ib [1, 0]/Ib [0, 0],(19)

b = Ib [0, 1]/Ib [0, 0].

(20)

Substituting Eqs. (9) and (10) into Eq. (8), the nematic
free energy is given by1(νmm + 5/4)xm2 Sm2 + 2m
2+(νmb + 5/4)xm xb (Sm Sb + m b )1+ (νbb + 5/4)xb2 Sb2 + 2b
2
xm
xb
−
ln Im [0, 0] −ln Ib [0, 0].
(21)
nm
nb

III. PHASE DIAGRAMSFor numerical calculations of Eqs. (17)–(20), we here defined the anisotropic interaction parameters
cmb = νmb /νmm ,(22)

cb = νbb /νmm ,

(23)

and

where cmb and cb are constants. The larger values of cmb show
the stronger attractive interaction between the mesogen andthe backbone chain. When cmb > 0, the backbone chains favor
to be parallel to the mesogens, while when cmb < 0 the backbone chains favor to be perpendicular to the mesogens.32, 33, 43
The positive and negative values of ν mb are related to the attractive and repulsive anisotropic interactions, respectively,

4.0
(b)

Sm
<S>

0.4

Sb

0.2

(16)

βFnem /n =

2.0

0.6
Order parameters

Zi = 4π Ii [0, 0],

(a)
nm=2

I

0.0
0.8

nb=2
nm=2

Δb Δm

0.9

1.0
T / TNI,2

1.1

FIG. 2. (a) Phase diagram on (T/TNI, 1 , nb ) plane for cmb = 0.1 and nm = 2.
(b) The orientational order parameters plotted against the temperature T/TNI, 2
for nm = nb = 2.

which eventually lead to the parallel or perpendicular orientations. In the following, we set cb = 0.5. The reduced temperature is defined as T̃ = 1/νmm (= kB T /Umm ). When xm = 1,
the nematic-isotropic phase transition (NIT) of a pure mesogen takes place at nm (ν mm + 4/5) = 4.54,49, 50 and then the
NIT temperature is given by
T̃NI =

nm,
4.54 − 1.25nm(24)where the term 1.25nm corresponds to the excluded volume
effect between mesogens. The NIT temperature increases
with nm and diverges to the high temperatures at nm > 3.64
because of the excluded volume interactions.50
Figure 2(a) shows the phase diagram on (T/TNI, 1 , nb )plane for cmb = 0.1 and nm = 2. The temperature is normalized by the NIT temperature TNI, 1 of nb = 1. The solidline shows the first-order phase transition (N3 IT) between an
isotropic and a nematic N3 phase. The N3 IT temperature hasa minimum as a function of nb . For large nb , the anisotropic
coupling between mesogens and backbone chains prevails andwe have N3 phase. Figure 2(b) shows the orientational orderparameters Sm and Sb plotted against the temperature T/TNI, 2
for nm = nb = 2, where TNI, 2 is the N3 IT temperature of nb= 2 in Fig. 2(a). The orientational order parameters Sm andSb jump from zero to positive values at the N3 IT temperature, while the biaxial order parameters m and b are zero.
This corresponds to the uniaxial nematic N3 phase, where
the mesogens and backbone chains are parallel to each other.
The orientational order parameter Sb of the backbone chain issmaller than Sm , because we take cb = 0.5, which means theattractive interaction ν bb between backbone chains is weaker
than the ν mm between mesogens. When cb = 1 and nm = nb ,we have Sm = Sb . The mean value of the uniaxial order parameters is given by
S = xm Sm + xb Sb(25)

224906-4

Akihiko Matsuyama

J. Chem. Phys. 137, 224906 (2012)

1.0
nm=2

TP

Order parameters

T/ TNI,1

I

BP

0.9

N2

N1
0.8
N1b
0.7
1

N2b

2

3

0.5

(a)

0.0

Δm
Δb

<Δ >

–0.5
0.7

4

Sm

<S>

Sb

0.8

FIG. 3. Phase diagram of a side-chain LCP, plotted against the axial ratio
nb of the backbone chain with cmb = −1.5 and nm = 2. The solid (dotted)
lines show first (second)-order phase transitions. The red closed circle showsa triple point (TP), where there phases coexist. The black closed circle showsa bicritical point (BP), where two distinct second-order lines meet at the firstorder line.

m + xb(26)

b.

Figure 3 shows the phase diagram on the (T/TNI, 1 , nb )plane for nm = 2 and cmb = −1.5. The solid (dotted) lines
show first (second)-order phase transitions. The red closed
circle shows a triple point (TP), where I, N1 , and N2 phases
coexist. The black closed circle shows a bicritical point (BP),where two distinct second-order lines meet at the first-orderline.51 For short backbone chains of nb < 2.5, we find thefirst-order nematic N1 -isotropic phase transition (N1 IT) andthe second-order uniaxial N1 -biaxial N1b phase transitions(N1 N1b T). For long backbone chains of nb > 2.5, we findthe first-order nematic N2 -isotropic phase transition (N2 IT)
and the second-order uniaxial N2 -biaxial N2b phase transitions (N2 N2b T). At nb  2.5 we have the first-order N1 N2 T
and N1b N2b T. The biaxial nematic phases N1b and N2b appear
between the N1 and N2 phases and are broadened with decreasing temperature. A similar phase diagram has been predicted by Rusakov and Shliomis.40
Figure 4 shows the orientational order parameters plotted against nb at T/TNI, 1 = 0.8 in Fig. 3. The solid curves
show the uniaxial order parameters Sm and Sb and the broken
curves are the biaxial order parameters m and b . We havethe first-order N1b N2b T at nb  2.5, where the order parameters jump. We also find the second-order N1 N1b T at nb  2.2
and N2 N2b T at nb  3, where the order parameters m and b
are continuously changed. The N1 and N2 phases are differ1.0
Order parameters

T/TNI=0.8
Sb
Sm

0.5

Δm

0.0

Δb

Δb

Δm

Sb

–0.5
1

Sm

2

nb

3

4

FIG. 4. Orientational order parameters plotted against nb at T/TNI, 1 = 0.8 in
Fig. 3.

1.1

(b)

0.5
Sb
Δb

0.0

–0.5
0.7

and the mean value of the biaxial order parameters is  = xm

Order parameters

nb

0.9
1.0
T/TNI,2.3

<Δ >

0.8

<S>

Δm
Sm

0.9
1.0
T/TNI,2.6

1.1

FIG. 5. (a) Order parameters plotted against temperature for nb = 2.3. (b) nb= 2.6 in Fig. 3.

ent in a sign of order parameters Sm and Sb and have the same
symmetry, or uniaxial nematic phase. However, the biaxial nematic phase is less symmetric than the uniaxial phase and thetransition form the uniaxial to the biaxial state is of second
order.1, 15 The biaxial N1b and N2b phases are also different
in a sign of order parameters and have the same symmetry,
resulting the first-order phase transition.
Figure 5 shows order parameters plotted against temperature for nb = 2.3 (Panel (a)) and nb = 2.6 (Panel (b)) in
Fig. 3. The temperature is normalized by the NIT temperature(TNI,nb ) of nb = 2.3 (Panel (a)) and nb = 2.6 (Panel (b)). The
uniaxial order parameters Sm and Sb jump at the first-order
N1 IT (Panel (a)) and N2 IT (Panel (b)) temperatures. The meanvalue of the uniaxial order parameters is small, however, theSm and Sb have large values. The absolute value of the biaxial order parameter is determined by the difference between
the two elements of the tensor order parameters: | | = |Sxx
− Syy |, and the biaxial order parameter | | continuously increases at the second-order N1 N1b T (Panel (a)) and N2 N2b T(Panel (b)). In the N1b (Panel (a)), the biaxial order parameterm is almost zero and the | b | increases with decreasing temperature, because the backbone chains start to orient along to
the second director. On the other hand, in the N2b (Panel (b)),
the biaxial order parameter b is almost zero and the | m |increases with decreasing temperature, because the mesogens
start to orient along to the second director.
Figure 6 shows the phase diagram on the (T/Tc , nb )plane for nm = 4 and cmb = −1.5. The temperature is normalized by the temperature Tc at the BP. As discussed in
Eq. (24), when nm > 3.56, the NIT temperature disappears,
or an isotropic phase disappears, because of the excluded
volume interactions between long mesogens, and then we
have the uniaxial N1 and N2 phases at high temperatures. On
decreasing temperature we have the second-order uniaxialbiaxial phase transitions: N1 N1b T and N2 N2b T. We also find

224906-5

Akihiko Matsuyama

J. Chem. Phys. 137, 224906 (2012)

1.2
Order parameters

T / Tc

nm=4
N2

N1

BP

1.0

0.8
N1b
2

3

4

nb

N2b
5

6

Sb

0.8

<S>

Δm

Sm

0.4
<Δ >

0.0

Sm
Δb

–0.4
0.8

7

Sm

0.9

1.0

1.1

T / TcFIG. 6. Phase diagram for cmb = −1.5 and nm = 4. The solid (dotted) lines
show first (second)-order phase transitions.

FIG. 8. Order parameters plotted against temperature for nm = 6 and
nb = 11 in Fig. 7.

the first-order N1 N2 T and N1b N2b T. The region of the biaxial
phases is broadened on decreasing temperature.
Figure 7 shows the phase diagram on the (T/Tc , nb ) plane
for nm = 6 and cmb = −1.5. On increasing the length of themesogen, the uniaxial nematic N3 phase appears between N1
and N2 phases. The black closed circle shows a tricriticalpoint (TCP), where the first-order phase transition line meetsthe second-order one. The first-order N1 N3 T and N2 N3 T lines
terminate at a liquid-gas like critical point (CP). We also find
three TPs, where three different nematic phases coexist. For
the long mesogens, we find many multi-critical points appear
on the phase diagram. In Figs. 6 and 7, we calculate for nm= 4 and nm = 6, respectively. The lengths of mesogens arelarger than 3.64 and then the isotropic phase disappears dueto the excluded volume effects. In the nematic phase, however, the anisotropic interaction terms ν ij exist and then thephase diagrams depend on temperature.
Figure 8 shows order parameters plotted against temperature for nm = 6 and nb = 11 in Fig. 7. We have two firstorder phase transitions: N3 N2 T and N2 N2b T, at T/Tc = 0.99
and 0.95, respectively, where the value of Sm and the biaxial
order parameters jump. In the N2b , the biaxial order parameter b is almost zero and the | m | increases with decreasing
temperature, because the mesogens start to orient along to thesecond director. The mean value S of the uniaxial order parameter decreases with decreasing temperature, because themesogens align perpendicular to the first director.
In this paper we have calculated four order parameters as
a function of temperature. To observe such uniaxial and biaxial nematic phases introduced here, we need to experimentally determine orientational order parameters of mesogens
and backbone chains, individually. The perpendicular orien-

tations may be realized side-on side-chain liquid crystalline
polymers. Our results demonstrate that the combinations ofa long mesogen and a backbone chain are capable of forming biaxial nematic phases. We here focus on biaxial nematic
phase. Biaxiality in smectic A phase will also be discussed byapplying our model.

CP

1.0
T / Tc

N1
0.9

CP

N3

ACKNOWLEDGMENTS

This work was supported by Grant-in Aid for Scientific Research (C) (Grant No. 23540477) from the Ministryof Education, Culture, Sports, Science and Technology ofJapan.
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TP

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TCP

TP

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We have presented a theory to describe novel biaxial nematic phases of a side-chain liquid crystalline polymer by taking into account both excluded volumes and attractive interactions between rigid rods. We calculate uniaxial and biaxial orientational order parameters, depending on temperatureand the length of mesogens and backbone chains. Three different uniaxial nematic phases (N1 , N2 , and N3 ) and two biaxial nematic phases (N1b and N2b ) are predicted. For shortmesogens and backbone chains, we predict the first-orderuniaxial-biaxial phase transitions: N1 N1b T and N2 N2b T. For
long mesogens with excluded volume effects, uniaxial-biaxial
phase transitions can be a first or a second-order phase transitions, depending on the length of backbone chain. The mutually perpendicular alignment between mesogens and backbone chains can be a major cause of the novel biaxial nematic
phases. We hope that these results encourage further experimental and theoretical studies for biaxial nematic phases ofliquid crystalline polymers.

N2

TCP

0.7
4

IV. SUMMARY

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TP

8

nb

12

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224906-6

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