Ferrofluid to ferronematic phase transition in lyotropic liquid crystals

MOLLIQ-04267; No of Pages 5 Journal of Molecular Liquids xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

Ferrofluid to ferronematic phase transition in lyotropic liquid crystals

2Q1

Prabir K. Mukherjee ⁎

3

Department of Physics, Government College of Engineering and Textile Technology, 12 William Carey Road, Serampore, Hooghly 712201, India

4

a r t i c l e

5 6 7 8 9

Article history: Received 15 February 2014 Received in revised form 16 March 2014 Accepted 2 May 2014 Available online xxxx

10 11 12 13 14

Keywords: Liquid crystal Micellar Ferrofluid Phase transition

O

a b s t r a c t

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i n f o

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25

1. Introduction

28

Ferrofluids are colloidal suspensions of magnetic nanoparticles, such as Fe3O4, in liquid. The nanoparticles typically have sizes of about 10 nm. Surfactants are added during the synthesis of ferrofluids to surround the small particles and overcome their attractive tendencies. Ferrofluids respond to an external magnetic field. When a strong magnet is placed near the ferrofluid, spikes are observed. Ferrofluids have a wide range of technological and biomedical applications [1]. Since its theoretical demonstration by Brochard and de Gennes [2], the ferrofluids has been found both in thermotropic liquid crystals (TLC) [3–6] and lyotropic liquid crystals (LLC) [7–12]. Although the stability of the thermotropic ferronematic state is far from being properly realized, lyotropic ferronematic is much more well realized. Lyotropic mesophases are formed by surfactant molecules in a suitable solvent. In this case the surfactant molecules aggregate into non-spherical micelles and nematic phases may appear. The micelles can be either rod-like or disk-like exhibiting long-range orientational ordering of their symmetry axes. The essential feature of the lyotropic ferronematic phase is the orientational coupling between the suspended ferrite nanoparticles and micelles of the lyotropic phase. Berejnov et al. [12] synthesized lyotropic ferrocolloid by mixing cationic ferrofluid and potassium laurate/1-decanol/water ternary solution. They observed that inside of the nematic zone the lyotropic ferrocolloid becomes ferronematic. They observed the first order character of isotropic ferrofluid to lyotropic ferronematic phase transition. They also measured the magnetization and magnetic susceptibility of lyotropic ferronematic phase depending on the concentration of dispersed magnetic particles.

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We present a mean-field analysis of the phase transitions of ferronematic in lyotropic liquid crystals. The model free energy is written in terms of the orientational order parameter, magnetization and micellar concentration. We find four different phases: (i) isotropic ferrofluid, (ii) isotropic micellar ferrofluid, (iii) lyotropic paranematic and (iv) lyotropic ferronematic. We put special emphasis on the phase transition from isotropic micellar ferrofluid to ferronematic phase. It was observed from the theoretical calculations that the transition from the isotropic micellar ferrofluid to lyotropic ferronematic phase is weakly first order. The temperature and concentration dependence of the magnetic birefringence and Cotton–Mouton constant are calculated. © 2014 Published by Elsevier B.V.

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⁎ Tel./fax: +91 33 26621058. E-mail address: [email protected].

On the theoretical sides, there is very little progress in this field. Based on a microscopic treatment of rod-like ferromagnetic grains Burylov and Raikher [13] studied ferronematic phase in TLC. In this study they treat the orientation of the magnetization and nematic ordering as separate degrees of freedom. Pleiner et al. [14] studied the ferrofluid to ferronematic phase transition in TLC within Landau–de Gennes theory. In a separate paper, Jarkova et al. [15] derived hydrodynamic equations for nematic ferrofluids in the limit that the magnetic degree of freedom has relaxed to its equilibrium value. To the best of the author's knowledge, there is practically no theoretical work on ferronematic in LLC. The purpose of the present paper is to develop a Landau model to describe the ferronematic state in LLC. Following the approaches [14,16], we developed a Landau model to investigate the key features of the lyotropic ferronematic (LFN) and various phase transitions with the special emphasis on the isotropic micellar ferrofluid (IMF) to LFN phase transition. We identified different order parameters involved in the IMF–LFN phase transition.

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2. Theory

71

We start by describing the order parameters involved in the ferronematic state for a micellar solution. The thermotropic and lyotropic nematic phases have the same symmetry. Hence the lyotropic nematic order parameter, originally proposed by de Gennes [17], is a
 symmetric, traceless tensor described as Q ij ¼ 2S 3ni n j −δij . The quantity S defines the strength of the nematic ordering. Following [16] we define another dimensionless parameter, ϕ = (x − xl), where xl is the molar fraction of the free surfactant molecules and x is the total molar fraction of the surfactant. Here x = nsr/(nsr + ns), where nsr and ns denote the numbers of surfactant and solvent molecules, respectively.

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http://dx.doi.org/10.1016/j.molliq.2014.05.002 0167-7322/© 2014 Published by Elsevier B.V.

Please cite this article as: P.K. Mukherjee, Ferrofluid to ferronematic phase transition in lyotropic liquid crystals, Journal of Molecular Liquids (2014), http://dx.doi.org/10.1016/j.molliq.2014.05.002

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

73 74 Q3 75 76 Q4 77 78 79 80 81



2

1 4 1 1 F ¼ F 0 þ aQ ij Q ij − bQ ij Q jk Q ki þ c1 Q ij Q ij þ c2 Q ij Q jk Q kl Q li 3 9 9 9 1 2 1 3 1 4 1 1 1 2 4 ð2:1Þ þ pϕ − qϕ þ rϕ þ α 1 M þ βM þ δQ ij Q ij ϕ 2 3 4 2 4 3 1 1 1 1 2 2 þ γ1 M i M j Q ij þ η1 M Q ij Q ij þ η2 M i M k Q ij Q kj þ ωM ϕ 2 3 2 2 91 92 93 94 95 96 97 98 99 100 101

where F0 is the nonsingular part of the free energy density. a = a0(T − T1∗), α1 = α0(T − Tf(x)) and p = p0(T − T∗2) with a0, α0 and p0 are positive constants. T∗1, T∗2 and Tf(x) are the virtual transition temperatures. Tf(x) is the function of concentration of ferro particles. To ensure thermodynamic stability, c1,2 N 0, b N 0, β N 0, q N 0 and r N 0. The parameters δ, γ1, η and ω are coupling constants. The negative values of γ, δ and ω ensure the ferromagnetic order induced by the nematic order and micellar concentration. The negative values of η1,2 favor the lyotropic ferronematic phase. Here we consider the phases in which both S and M are spatially ^ and m ^ make an angle θf homogeneous and the ordering directions n ^ m ^ ¼ cosθ f . Then the free energy density in Eq. (2.1) leads to i.e. n

102

where u, v ∈ {S, ϕ, M}. The LPN and LFN phases can arise from the high temperature IF phase along the curves IF–LPN, IF–LFN or along the curve LPN–LFN respectively. The LPN and LFN phases can also arise from the IMF phase along the curves IMF–LPN and IMF–LFN respectively. In general IF–IMF, IF–LPN, IF–LFN, IMF–LPN and IMF–LFN transitions are first order because of the cubic invariant in the free energy expansion. The LPN–LFN phase transition may be second order or first order. In the case of first order phase transitions, one can observe two triple points IF–IMF–LPN and IF–IMF–LFN. For the first order LPN–LFN phase transition, one can also observe the IF–LPN–LFN and IMF–LPN–LFN triple points.

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2.1. LPN–LFN phase transition

136

The LPN phase is in competition with the IF, IMF and LFN phases. The existence ranges of all four phases generally overlap. In competition with possible IF, IMF and LFN phases, the LPN phase will be the stable one only when it has lowest free energy in comparison to IF, IMF and LPN phases. In order to ensure the stability of the LPN phase it is required that

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F

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2

p−2qϕ þ 3rϕ N 0;

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R

(I) Isotropic ferrofluid phase (IF): S = 0, ϕ = 0, M = 0. This phase exists for a N 0, α1 N 0 and p N 0. (II) Isotropic micellar ferrofluid phase (IMF):

q 2r

  1=2 4pr 1 þ 1− 2 ; S ¼ 0; M ¼ 0: q

111

113

SLPN ¼ 115 116 117 118 119 120 121 122

   4ða þ δϕLPN Þc 1=2 1 þ 1− 2

b 2c

ð2:4Þ

!! 2   δ 2 2 −bS þ 2c c− p−2qϕ þ 3rϕ S N 0: 2c

ð2:5Þ

1

131 132 133 134 135

138 139 140 141 142

145 147 148

150

1 1 2 4 F ¼ F LPN þ α 2 M þ βM 2 4

ð2:6Þ

where FLPN is the free energy density of the LPN phase and α2 = 152 α1 + γ′SLPN + η′S2LPN + ωϕLPN. Hence M = 0 for T NT C    and M ≠ 0 for T b TC resulting 153 ¼ T f ðxÞ−α1 γ 0 SLPN þ η0 S2LPN þ ωϕLPN 0

0.8 φ =0.12 φ =0.52

0.6

S

S

0.4

b

where ϕLPN is defined by: pϕLPN −qϕ2LPN þ rϕ3LPN þ 12δS2LPN ¼ 0: This phase exists for a b 0, α1 N 0, − b + 2cS N 0 and p − 2qϕ + 3rϕ2 N 0. (IV) Lyotropic ferronematic phase (LFN): S ≠ 0, M ≠ 0, ϕ ≠ 0, θf = 0 or θ f ¼ π2. This phase exists for a + δϕ b 0, α1 + γ′S + η′S2 + ωϕ b 0 and p b 0. whereγ0 ¼ γ1 ðor−γ2 Þ for θf = 0 (or π2) and η′ = (η1 + η2) (or (η1 + η2/2)) for θf = 0 (or π2Þ.

129 130

The LPN–LFN phase transition is described by the free energy density

This phase exists for a N 0, α1 N 0 and p − 2qϕ + 3rϕ2 N 0. (III) Lyotropic paranematic phase (LPN): M = 0, ϕ ≠ 0,

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2

N

ϕI ¼

127 128

ð2:3Þ

−bS þ 2cS N 0

S, M 2

108 109

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105

where c = c1 + c2/2. Minimization of Eq. (2.2) with respect to S, ϕ, M and θf yields the following four stable solutions:

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104

126

144

T

1 2 1 3 1 4 1 2 1 3 1 4 1 2 F ¼ F 0 þ aS − bS þ cS þ pϕ − qϕ þ rϕ þ α 1 M 2 3 4 2 3 4 2   1 1 1 2 1 4 2 2 2 2 ð2:2Þ þ βM þ δS ϕ þ γ1 M S 3cos θ f −1 þ η1 M S 4 2 4 2   1 1 2 2 2 2 þ η2 M S 3cos θ f þ 1 þ ωM ϕ 8 2

123

∂2 F ∂2 F ∂2 F N 0; N 0; 2 N 0; 2 2 ∂S ∂M ∂ϕ !2 2 2 2 ∂ F ∂ F ∂ F  − N0; ∂u∂v ∂u2 ∂v2

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The necessary conditions for the different phases to be stable are:

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Therefore, ϕ measures the concentration of those surfactant molecules which are aggregated in micelles. The magnetic order is described by ^ whose modulus M is zero in paramagnetic the magnetization M ¼ Mm state and non-zero in a ferromagnetic state. Thus Qij, ϕ and M are three order parameters involved in the description of the ferronematic phase in LLC. To analyze the problem of ferronematic phase in LLC we consider the free energy density

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P.K. Mukherjee / Journal of Molecular Liquids xxx (2014) xxx–xxx

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2

0 298

M2

M2

0.2

298.5

299

299.5 T(K)

300

300.5

301

Fig. 1. Temperature variation of the order parameters S and M in the LFN phase for two different micellar concentrations ϕ = 0.12 and ϕ = 0.52.

Please cite this article as: P.K. Mukherjee, Ferrofluid to ferronematic phase transition in lyotropic liquid crystals, Journal of Molecular Liquids (2014), http://dx.doi.org/10.1016/j.molliq.2014.05.002

P.K. Mukherjee / Journal of Molecular Liquids xxx (2014) xxx–xxx

159 160

In this section we will discuss the IMF–LFN phase transition. S ≠ 0, ϕ ≠ 0, M ≠ 0 and θf = 0 or π2 are the equilibrium conditions realized in the LFN phase. In order to ensure the stability of the LFN phase it is required that

161 162

2

a2 −2b2 S þ 3c2 S þ δ1 ϕ N 0;

ð2:7Þ



F ¼ F IMF ðϕ0 Þ−

164 167 168 170 171

173

0

0 2

α 1 þ γ S þ η S þ ωϕ b 0;

ð2:8Þ

2

p−2qϕ þ 3rϕ N 0;

ð2:9Þ

2

a1 −2b1 S þ 3c1 S N 0;

α 1 η0 ; b2 β

b1 ¼ b þ

0 0

¼bþ

ð2:10Þ

3γ η ; c1 2β

γ 0 η0 ;c 2β 2

η02 3β

ω2 ;δ 2β 1

¼ c− ; p1 ¼ p−

ωη0 :a β 1

¼ δ−

b ¼bþ

02

η β

¼ c− :

183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199

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α1 ω γ 0ω 1 2 3 2 þ p1 ϕ−qϕ þ rϕ − S þ δ1 S ¼ 0 2β 2β 2

ð2:12Þ

ð2:13Þ

where p1 ¼ p−ω2β . It is clear from Eq. (2.11) that a nonzero real value of M exists only when (α1 + γ′S + η′S2 + ωϕ) b 0. Since there is a small temperature range where α1 N 0, γ′ b 0 and η′ b 0 in this region. The temperature variation of the order parameters (S and M) for two different micellar concentrations in the LFN phase is shown in Fig. 1. This is done for a set of phenomenological parameters for which the direct IMF–LFN phase transition is possible. Fig. 1 shows that the order parameters S and M jump simultaneously at the IMF–LFN transition point. For a fixed set of parameter values, we find that the values of the jump of the order parameter at T IMF − LFN are S IMF − LFN = 0.1 and M 2IMF − LFN = ∗ 0.023. We also obtained TIMF − LFN − TIMF − LFN = 0.17K. The low ∗ value of TIMF − LFN − T IMF − LFN = 0.17K and the low values of SIMF − LFN = 0.1 and M 2IMF − LFN = 0.023 indicate the weakly first order character of the IMF–LFN phase transition. Fig. 1 shows that the higher the value of micellar concentration, the higher are the order parameters S and M. We will now discuss the IMF–LFN phase transition in more details. Defining ϕ0 as the equilibrium value of the IMF phase at the IMF–LFN 2

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−

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T

γ 2 3 ðα þ ωϕÞ þ a1 S−b1 S þ c1 S þ δ1 Sϕ ¼ 0 2β 1

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0

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216

η2 β

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2

ω 2u



β ¼ β−



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δω : 2u

0

η ¼η−

224

The free energy density in Eq. (2.15) describes the direct IMF–LFN phase transition. The LFN phase appears only for a∗ b 0 i.e. 225  Tb







02

γ a0 T 1 −η αβ T þ η βωϕ þ 2β −δϕ0 0 

f



0



a0 −ηβα 



0

 :

ð2:16Þ 227

The cubic coefficient b∗ in the free energy density in Eq. (2.15) shows that the IMF–LFN phase transition must always be first order in mean 228 0.4

0.35

( ΔH n2 )−1

−

E

where the values of S and ϕ in the LFN phase can be calculated from the equations

R

176

ð2:11Þ

209 210



215

δ2 2u



 1 2 0 0 2 α 1 þ γ S þ η S þ ωϕ M ¼− β

02

3γ0 η 2β 



γ02 α 1 η0 − ; 2β β

¼ a−

The value of the magnetization in the LFN phase can be expressed as

207

γ ðα þ ωϕ Þη − 1  0 2β  β



a ¼ a þ δϕ0 −

c ¼ c− −

174

ð2:15Þ

ðα 1 þ ωϕ0 Þ2 ; 4β



F IMF ðϕ0 Þ ¼ F IMF ðϕ0 Þ−

where

a2 ¼ a−

γ0 1  2 1  3 1  4 ðα þ ωϕ0 ÞS þ a S − b S þ c S : 2 3 4 2β 1

The renormalized coefficients are

D

165

ð2:14Þ

where FIMF(ϕ0) is the corresponding free energy density of the IMF 203 phase and u = 1/χ, χ is the response function of the IMF phase. Elimination of ϕ and M from Eq. (2.14) leads to the free energy 204 density 205

F

2.2. The LFN phase and the IMF–LFN phase transition

1 1 2 1 3 1 4 1 2 2 F ¼ F IMF ðϕ0 Þ þ uðϕ−ϕ0 Þ þ aS − bS þ cS þ α 1 M 2 2 3 4 2 1 1 2 1 0 2 1 0 2 2 1 4 2 þ βM þ δS ϕ þ γ M S þ η M S þ ωM ϕ 4 2 2 2 2

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transition point, the free energy density in the neighborhood of the 200 IMF–LFN phase transition can be expressed as 201

R O

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in a second order transition between LPN and LFN phases. However, in the presence of pressure or for a particular value of micellar concentration or ferroparticles concentration, the LPN–LFN phase transition can become first order at the tricritical point.

P

154 155

3

0.3

0.25

4.75

4.8

4.85

4.9

4.95

5

5.05

5.1

T(a.u.) Fig. 2. The temperature dependence of the inverse of the Cotton–Mouton constant in the IMF phase of the IMF–LFN phase transition.

Please cite this article as: P.K. Mukherjee, Ferrofluid to ferronematic phase transition in lyotropic liquid crystals, Journal of Molecular Liquids (2014), http://dx.doi.org/10.1016/j.molliq.2014.05.002

238

β



   2  3  4 0 2 6a S −4b S þ 3c S −6γ ðα þ ωϕ0 ÞS−−3ðα 1 þ ωϕ0 Þ ¼ 0: ð2:17Þ

240

2.3. Magnetic birefringence and Cotton–Mouton constant 241 242 243 244 245 246 247 248 249 250

In this section we will calculate the Cotton–Mouton constant in the IMF phase of the IMF–LFN phase transition. An external magnetic field induces a finite magnetization in the ferrofluid by ordering the magnetic particles. In addition it also orients the mesogens due to the diamagnetic anisotropy effect. At the lowest order the magnetic field H couples quadratically with S through the susceptibility anisotropy and linearly with magnetization M. So the magnetic field H couples directly to S and M and indirectly to ϕ through the (S, ϕ) and (M, ϕ) couplings. Then the free energy density in the presence of an external magnetic field can be expressed as 1 F H ¼ F−M  H− ðΔχ Þmax H i H j Q ij 2

255 256

2

2

264 265 266 267 268

0

0

0

0

2

2

n∥ −n⊥ ¼ SðH Þ 270 271 272

0

0

O

0

R

0 max

0

C

262 Q8 263

where U 1 ¼ ðΔχaÞ , U 2 ¼ a γα , T 0 ¼ T 1 −δϕa , T 00 ¼ T f ðxÞ−ωϕ ′= α . (Δχ)max (Δχ)max for θf = 0 and −12ðΔχÞmax for θ f ¼ π2. While deriving Eq. (2.19), we have neglected the terms containing H4 order. For the temperature region T∗0 b T b TIMF − LFN, the free energy density in Eq. (2.18) gives two minimums. Under the action of field H, S = 0 and M = 0 are no longer solutions of the IMF state. S = 0 and M = 0 minimums are shifted to a small but non-zero value proportional to H2 (Eq. (2.19)). This phase is called superpara IMF phase. The second minimum corresponding to the LFN phase shifts also to a higher order value. However for the LPN–LFN phase transition, we get the superparanematic phase. An anisotropy optical property is proportional to the induced order

N

260 261

ð2:19Þ

U

259

R

U H U H  
2 SðHÞ ¼
1   −
T−T 0 T−T 0 T−T 00 ðxÞ 258

ð2:20Þ

where n∥ and n⊥ are the refractive indices parallel and perpendicular to the field. One can readily show [18] that a magnetic field applied to the IMF phase will induce a birefringence (Δn = n∥ − n⊥). Then the Cotton–Mouton constant in the IMF can be expressed as Δn U1 U
−
2 
: ¼ ðn∥ þ n⊥ Þ T−T 0 T−T 00 H2 ðn∥ þ n⊥ Þ T−T 0

ð2:21Þ

274 275

3. Conclusions

310

We have presented here a mean-field analysis to describe the IMF– LFN phase transition in LLC composed of surfactant and ferromagnetic molecules. The theory predicts a weakly first order character of the IMF–LFN phase transition because of the low value of TIMF − LFN − ∗ TIMF − LFN = 0.17K and second order character of the LPN–LFN phase transition. We have derived expressions of the conditions for which the direct IMF–LFN phase transition occurs. The magnetic birefringence and Cotton– Mouton constant are calculated and discussed. Although we focused on the IMF–LFN phase transition, a similar calculation can also be applied to the IF–LFN phase transition. We hope that the present theoretical analysis of the ferronematic phase in LLC will encourage researchers to take a fresh look at this problem.

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References

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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341

T

C

253 254

with F the free energy density in Eq. (2.14). (Δχ)max is the maximum possible value of the magnetic susceptibility anisotropy. Here M and H ^ = Hn ^ = cosθf. are parallel to each other. Hence M  n The induced nematic order produced by magnetic field H in the IMF phase can be calculated to a first approximation (b∗ = c∗ = 0) and can be expressed as

E

252

ð2:18Þ

F

236 237

O

235

276 277

R O

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anisotropy effect and the second part appears due to the orientation of the magnetic particles by the external field. The temperature dependence of the inverse of Cotton–Mouton constant in the IMF phase is shown in Fig. 2. This is done for a set of phenomenological parameter values. Units of the Cotton–Mouton constant and temperature are arbitrary. The right hand side of Eq. (2.21) diverges at T = T∗0. The diverging magnetically induced birefringence and Cotton–Mouton constant is analogous to the diverging susceptibility near a magnetic critical point or diverging compressibility near a fluid critical point. Rosenblatt et al. [19,20] measured the concentration and temperature dependences of the Cotton–Mouton constant and the magnetic birefringence in the isotropic phase of the I–N phase transition in LLC. They showed that the I–N transition temperature decreases strongly with the decrease of micellar concentration and measured the low value of TI − N − T ∗ b 20mK which indicates the closeness of the second order character of the I–N phase transition in LLC. The present analysis of the IMF–LFN phase transition also indicates the same result similar to the I–N phase transition in LLC. However the IMF–LFN transition temperature decreases with micellar concentration as well as the magnetic nanoparticles concentration. The Cotton–Mouton constant and the magnetic birefringence also changes with the change of micellar concentration as well as the magnetic nanoparticles concentration. The Cotton– Mouton constant expression in Eq. (2.21) gives the exponent γ = 1. The value of the exponent γ = 1 indicates the fluid like analogy in the IMF phase of the IMF–LFN phase transition similar to the I–N phase transition in TLC as well as in LLC. The mean-field result predicts the exponent α = 0. We need a fluctuation theory for the better estimation of the value of the exponent α. The heat capacity measurement or non linear dielectric effect (NDE) measurement offers an adequate description for the critical exponent α. Here we should point out that dielectric measurement by Rzoska et al. [21] gave a clear evidence for the critical exponent α = 0.5 in the isotropic phase of the I–N phase transition in TLC. One can expect that the same NDE measurement can also be tested for the IMF–LFN phase transition.

P

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field approximation. Lower the value of b∗, weakly the first order character of the IMF–LFN phase transition. The IMF–LFN phase transition must occur before or simultaneously with the LPN–LFN phase transition. The first order IMF–LFN phase transition line emerges either from the IF–IMF–LFN or IMF–LPN–LFN triple points. The IMF–LFN phase transition is accompanied by simultaneous jump of SIMF − LFN and MIMF − LFN as shown in Fig. 1. In that case there is a first order IMF–LFN phase transition possible within the framework and assumptions of the model. The first order IMF–LFN phase transition line is given by

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P.K. Mukherjee / Journal of Molecular Liquids xxx (2014) xxx–xxx

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The expression of Cotton–Mouton constant in Eq. (2.21) shows that it consists two parts. The first part appears due to the diamagnetic

[12] [13] [14] [15]

C. Scherer, A.M. Figueiredo Neto, Braz. J. Phys. 35 (2005) 718–727. F. Brochard, P.G. de Gennes, J. Phys. (Fr.) 31 (1970) 691–708. J. Rault, P.E. Cladis, J.P. Burger, Phys. Lett. A 32 (1970) 199–200. S.H. Chen, S.H. Chang, Mol. Cryst. Liq. Cryst. 144 (1987) 359–370. S.H. Chen, N.M. Amer, Phys. Rev. Lett. 51 (1983) 2298–2301. C.F. Hayes, Mol. Cryst. Liq. Cryst. 36 (1976) 245–253. L. Liebert, A. Martinet, J. Phys. (Fr.) Lett. 40 (1979) 363–368. L. Liebert, A. Martinet, IEEE Trans. Magn. 16 (1980) 266–269. A.M.F. Neto, M.M.F. Saba, Phys. Rev. A 34 (1986) 3483–3485. T. Kroin, A.M.F. Neto, Phys. Rev. A 36 (1987) 2987–2990. V.V. Berejnov, V. Cabuil, R. Perzynski, Yu.L. Raikher, J. Phys, J. Phys. Chem. B 102 (1998) 7132–7138. V. Berejnov, V. Cabuil, R. Perzynski, Yu.L. Raikher, Colloid J. 62 (2000) 414–423. S.V. Burylov, Yu.L. Raikher, Mol. Cryst. Liq. Cryst. 258 (1995) 107–122. H. Pleiner, E. Jakrova, H.W. Muller, H.R. Brand, Magnetohydrodynamics 37 (2001) 254–260. E. Jarkova, H. Pleiner, H.W. Muller, A. Fink, H.R. Brand, Euro. Phys. J. E 5 (2001) 583–588.

Please cite this article as: P.K. Mukherjee, Ferrofluid to ferronematic phase transition in lyotropic liquid crystals, Journal of Molecular Liquids (2014), http://dx.doi.org/10.1016/j.molliq.2014.05.002

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[16] P.K. Mukherjee, Liq. Cryst. 29 (2002) 863–869. [17] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993. [18] T.W. Stinson, J.D. Litster, N.A. Clark, J. Phys. (Paris) 33 (1972) 69–72.

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[19] C. Rosenblatt, S. Kumar, J.D. Litster, Phys. Rev. A 29 (1984) 1010–1013. [20] C. Rosenblatt, Phys. Rev. A 32 (1985) 1924–1926. [21] A. Drozd-Rzoska, S.J. Rzoska, J. Ziolo, Phys. Rev. E. 54 (1996) 6452–6455.

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Please cite this article as: P.K. Mukherjee, Ferrofluid to ferronematic phase transition in lyotropic liquid crystals, Journal of Molecular Liquids (2014), http://dx.doi.org/10.1016/j.molliq.2014.05.002

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