Macroscopic dynamics near the isotropic micellar to lamellar phase transition

Chemical Physics 430 (2014) 56–61

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Macroscopic dynamics near the isotropic micellar to lamellar phase transition Prabir K. Mukherjee ⇑ Department of Physics, Government College of Engineering and Textile Technology, 12 William Carey Road, Serampore, Hooghly 712201, India

a r t i c l e

i n f o

Article history: Received 29 August 2013 In final form 21 December 2013 Available online 8 January 2014 Keywords: Lamellar phase Hydrodynamic Liquid crystal Phase transition

a b s t r a c t We present the hydrodynamic equations for the lamellar phase in lyotropic liquid crystals. The hydrodynamic equations are investigated to the vicinity of the isotropic micellar to lamellar phase transition. To derive the hydrodynamic equations we make use of symmetry arguments and irreversible thermodynamics. Besides the usual order parameters to describe the lamellar phase we also keep the concentration of the surfactant molecules which are aggregated in micelles as a variable in order to describe the correct macroscopic behavior of the lamellar phase. The macroscopic dynamic equations are presented on both sides of the transition. We discuss possible experiment were our theory can be tested. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Lyotropic liquid crystals (LLC) are mixtures of amphiphilic molecules and solvents at given temperature and relative concentrations. An important feature of LLC is the self-assembly of the amphiphilic molecules as supermolecular structures. Lyotropic mesophases are very similar to the mesophases that exist in thermotropic liquid crystals (TLC) in terms of the orientational and translational ordering. Smectic-A (SmA) phase in TLC is composed of parallel liquid layers. In general the lamellar phase is a SmA liquid crystal in which the layers are composed of surfactants molecules and the gap between them is occupied by water. This phase is designated as LD. Thus the LD phase exhibits the positional ordering of micelles into planes arranged periodically along the nematic ^ . In many systems, such as lyotropic liquid crystals [1–4] director n and block copolymers [5–10] transition is observed from a uniform isotropic to a lamellar phase upon lowering the temperature. Alexandridis et al. [9] studied the phase behavior and structure of binary amphiphilic polymer-water systems as a function of polymer concentration and temperature for three poly (ethylene oxide)-bpoly (propylene oxide)- b-poly (ethylene oxide) (PEO-PPO-PEO) copolymers of different composition by using 2H-NMR and small-angle X-ray scattering. They observed three different cubic, hexagonal, and lamellar LLC phases. The characteristic lattice parameter of the lamellar structure decreased with increasing polymer content; the bilayers swell less as water is removed. Wanka et al. [10] also studied the phase diagrams and aggregation ⇑ Tel./fax.: +91 3326621058. E-mail address: [email protected] 0301-0104/$ - see front matter Ó 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chemphys.2013.12.012

behavior of triblock copolymers in aqueous solutions. They also confirmed three different cubic, hexagonal, and lamellar LLC phases. Experimental results [1–4] and theoretical predictions [11–14] support the evidence of the presence of a transition from the isotropic micellar (I) to a lamellar phase transition in LLC system. In experimental studies [1–4], the binary mixtures of cesium perfluoro-octanoate (CsPFO)-water/heavy water (H2O/D2O) and APFO/D2O systems were found to produce the I, ND and LD phases and the I-LD transition via a I - ND - LD triple point. The I-LD phase transition is quite analogous to the isotropic-smectic-A (I-SmA) phase transition in TLC. Experimentally I-LD phase transition is found to be strongly first order. In a recent paper [14], we theoretically studied the key features of the I-LD phase transition in detail. It was pointed out that the same pretransitional behavior occurs for the I-LD phase transition similar to the I-SmA transition in TLC. The purpose of the present paper is to study the hydrodynamic properties near the I-LD phase transition. Liquid crystalline phases have interesting hydrodynamic properties which is known for over four decades both in TLC [15–23] and LLC [24–28]. It turns out that the anisotropy of TLC has a number of interesting implications for their hydrodynamic behavior. The macroscopic dynamics deal with the dynamics of the deviations of the order parameter modulus from its equilibrium value. It is valid only near the phase transition. Although the hydrodynamic and macroscopic dynamics study in TLC have been a topic of active theoretical and experimental studies over the four decades, reports of the macroscopic dynamics study in LLC are comparatively scarce. It is the goal of this paper to study the macroscopic dynamics behavior near the I-LD phase transition. Brochard and de Gennes [24] extensively studied the hydrodynamic

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properties of fluid lamellar phases of lipid/water system. The observations of the collective modes in the lamellar phases shows details of the interactions between layers and also on specific dissipative process in the lipid region. Bary-Soroker and Diamant [28] studied the relaxation modes of an interface between a lyotropic lamellar phase and a gas or a simple liquid. They found that surfaces of lyotropic lamellar phases can relax via a much slower, over- damped diffusive mode over a wide range of wavelengths. To the best of the author knowledge there is so far no theoretical work on the macroscopic dynamics study near the I-LD phase transition. Thus it is interesting to study the macroscopic dynamics behavior near the I-LD phase transition. In the present paper we study the macroscopic dynamic behavior near I-LD phase transition. We adopt the general framework developed by us in our previous work [29] for the isotropic to Smectic-A (I-SmA) phase transition in TLC and for dynamics of binary liquid crystal mixtures [23,30,31]. We identify different hydrodynamic and macroscopic dynamic variables near the I-LD phase transition. 2. Derivation of macroscopic equations

R

where F 0 ¼ f0 ds is generalized energy of the isotropic liquids with i R h1 where A ðdqÞ2 þ Aqr ðdqÞðdrÞ þ 12 Arr ðdrÞ2 þ 21q g 2 , 2 qq

f0 ¼

Aqq ¼ ð@ l=@ qÞr ; Arr ¼ ð@T=@ rÞq , and Aqr ¼ ð@T=@ qÞr . Here B is the compressional modulus of the smectic layers and the layer bending modulus K is close in magnitude to the splay modulus in nematic. The transverse Laplacian is defined as r2? ¼ ðdij  ni nj Þri rj . Now the conjugated variables in terms of the hydrodynamic and macroscopic variables are expressed as

dF q r ¼ aS þ b dq þ b dr þ hW þ kd/ þ c1 rz uz ; dS

P¼

M¼

dF W ¼ aW þ cq dq þ cr dr þ hS þ md/ þ d rz uz ; dW

Wi ¼

dF q r W ¼ ðBrz u þ d dq þ d dr þ d W þ c1 S þ c2 d/Þdiz dðri uz Þ

N¼

2.1. Macroscopic equations in the LD phase below the I-LD transition We start by describing the hydrodynamic and macroscopic variables in the LD phase that describes the macroscopic state of the system. The hydrodynamic variables for the lamellar phase are density q, entropy density r, density of linear momentum g, displacement uz of the smectic layers along the z axis associated with the density wave parallel to the layer normal and / which measures the concentration of those surfactant molecules which are aggregated in micelles. Here / ¼ ðx  xl Þ, where xl is the molar fraction of the free surfactant molecules and x is the total molar fraction of the surfactant. Here x ¼ nsr /ðnsr þ ns Þ, where nsr and ns denote the numbers of surfactant and solvent molecules, respectively. The LD phase has the same symmetries as the smectic-A (SmA) phase in TLC. So the macroscopic variables for the LD phase are the modulus S of the nematic order parameter Q ij ¼ ðS=2Þð3ni nj  dij Þ, the real modulus of the smectic order parameter W similar to SmA phase in TLC. We also assume that ^ are parallel ^ and the smectic layer normal k the nematic director n to each other. Here we will focus only on the linearized macroscopic equations. To describe the static properties of the LD phase, we find the Gibbs relation in terms of the hydrodynamics and macroscopic variables

de ¼ ldq þ Tdr þ v i g i þ Wi dri uz þ Nd/ þ PdS þ MdW

ð2:1Þ

where e is the energy density. In Eq. (2.1) the thermodynamic quantities chemical potential (l), temperature (T), the velocity field v i , the field Wi , the concentration field N and the order parameter fields S and W are defined as partial derivatives of the thermodynamic potential with respect to the appropriate variables. N is a conjugate quantity of /. So N is analogous to relative chemical potential l/ . Equation (2.1) gives a relation between the changes in the macroscopic variables and the entropy density r. R Now the generalized energy (F ¼ F 0 þ fds) of the lamellar phase can be written as

F ¼ F0 þ

Z

"

ds

1 2 1 q r aS þ ðb dq þ b drÞS þ aW 2 þ ðcq dq þ cr drÞW 2 2

1 þ p1 ðd/Þ2 þ ðf q dq þ f r drÞd/ þ hSW þ kSd/ þ mWd/ 2 2 1 1 q r w þ Bðrz uz Þ2 þ Kðr2? uz Þ þ ðd dq þ d dr þ d WÞrz uz 2 2 # þc1 ðrz uz ÞS þ c2 ðrz uz Þd/ ;

ð2:2Þ

ð2:3Þ

ð2:4Þ

 Kðdij  ni nj Þrj r2? uz ;

ð2:5Þ

dF ¼ p1 d/ þ f q dq þ f r dr þ kS þ mW þ c2 rz uz ; dðd/Þ

ð2:6Þ

dT ¼

dF r r ¼ Arr dr þ Aqr dq þ b S þ cr W þ f r d/ þ d rz uz ; dr

ð2:7Þ

dl ¼

dF q q ¼ Aqq dq þ Aqr dr þ b S þ cq W þ f q d/ þ d rz uz ; dq

ð2:8Þ

vi ¼

1

q

gi :

ð2:9Þ

To determine the dynamics of the variables we have two class of variables (1) the variables that contains conserved quantities and (2) the variables that contains nonconserved quantities. Then the resulting dynamic equations for the conserved fields are

@q þ ri g i ¼ 0; @t

ð2:10Þ

@g i þ rj rij ¼ 0: @t

ð2:11Þ

@/ / þ v i ri / þ ri ji ¼ 0 @t

ð2:12Þ /

where rij is the stress tensor and ji is the concentration current. There is also another mass conservation equation for the LD phase appears because of the surfactant migration in the layers plane relative to water i.e. of slip. As Brochard and de Gennes [24] pointed out that there is a constraint relation between /; @u @z and the relative variation of the surface per polar head (d) i.e. / ¼ @u þ d. Then the mass conservation equation due to slipping @z is given by Brochard and de Gennes [24]

@/ þ ri v iL ¼ 0; @t

ð2:13Þ

@ð Þ where v L is velocity of the lipid fraction and @/ ¼ @t þ @d . @t @t We will now write down the boundary conditions for Eqs. (2.10) and (2.11). At the interface, the differential form of dynamic Eqs. (2.10) and (2.11) are not valid anymore. Adopting the procedure as outlined by Bary-Soroker et al. [28,32], we consider an elementary box in the interface. Then at the interface Eq. (2.10) can be written as @u @z

d dt

Z V

qdV ¼ 

Z S

^ dS: gn

ð2:14Þ

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P.K. Mukherjee / Chemical Physics 430 (2014) 56–61

Eliminating the volume integral for a suitable box height, the boundary condition can be written as

^ ¼ 0; ððg 1  g 1s Þ  ðg 2  g 2s ÞÞ  n

ð2:15Þ

where g 1 ð¼ q1 v 1 Þ and g 2 ð¼ q2 v 2 Þ are the momentum densities on the two sides of the interface. g 1s ¼ ðq1 v s Þ and g 2s ¼ ðq2 v s Þ. v s is the interface velocity. Considering the effect of surface tension at the interface, the boundary condition for the dynamic Eq. (2.11) can be written as [32,33]

^¼C ðr1  r2 Þn

! @2f @2f ^: þ n @x2 @y2

 1 1 1 1 ds jij ðri TÞðrj TÞ þ gijkl ðri v j Þðrk v l Þ þ sP2 þ K M M2 2 2 2 2 1 1 þ K ij ðri NÞðrj NÞ þ m1 ðri Wj Þðrk Wl Þ þ kij ðri TÞðrj MÞ 2 2 þcij ðri TÞðrj PÞ þ fij ðri TÞðrj NÞ þ a1 PM þ a0ij ðri NÞðrj PÞ

R¼

ð2:16Þ

Z

þa00ij ðri NÞðrj MÞ þ bnj ðrm Wm Þðrj PÞ þ dnk ðrj Wj Þðrk MÞ  þgnk ðrj Wj Þðrk NÞ þ enk ðrj Wj Þðrk TÞ ; ð2:24Þ where kij ; cij ; f ij ; a0ij , a00ij and the thermal conductivity tensors jij ; K ij have uniaxial forms. The viscosity tensor gijkl has five independent coefficients in the LD phase. The tensors jij , K ij and fij describe heat conduction, diffusion and thermodiffusion respectively. The cross coupling terms fij between temperature variation and micellar concentration can expressed in uniaxial form as

C is surface tension coefficient and f is the interface deviation from the x-y plane. Eq. (2.16) shows that stress tensor components rxz and ryz are equal in both sides of the interface and the element rzz is discontinuous because of surface tension. The balance equations for the nonconserved fields take the forms [23]

fij ¼ f? d?ij þ fk ni nj :

@r R r þ r i ji ¼ ; T @t

rD ij ¼ gijkl rk v l ;

ð2:27Þ

X D ¼ m1 rk Wk  bnj rj P  dnk rk M  enk rk T  gnk rk N;

ð2:28Þ

Y D ¼ sP þ a1 M  a0ij ri rj N  cij ri rj T  bnj rj rk Wk ;

ð2:29Þ

Z D ¼ K M M þ a1 P  a00ij ri rj N  kij rj ri T  dnj rj rk Wk ;

ð2:30Þ

ð2:17Þ

@uz þ X ¼ 0; @t

ð2:18Þ

@S þ Y ¼ 0; @t

ð2:19Þ

@W þ Z ¼ 0; @t

ð2:20Þ

r

rRij ¼ pdij  Wj diz þ lij P þ bij M; /R

ji ¼ 0;

ð2:21Þ ð2:22Þ

where p is the hydrostatic pressure and bij and lij take the uniaxial form bij ¼ bk ni nj þ b? ðdij  ni nj Þ. The components of rRij can be written as

rRxx ¼ p þ K rx r2? uz þ l? P þ b? M rRyy ¼ p þ K ry r2? uz þ l? P þ b? M rRzz ¼ p  ðBrz u þ dq dq þ dr dr þ dW W þ c1 S þ c2 d/Þ þ lk P þ bk M rRiz ¼ K ri r2? uz :

Then the dissipative parts of currents and quasicurrents in the LD phase read rD

ji

/D

Here ji is the entropy current and X; Y and Z are quasicurrents associated with the density wave and changes of the order parameters S and W. R is the dissipation function and R=T is the entropy production. There are two classes of dynamics, one is reversible and the other is irreversible part. Using general symmetry and Galilean invariance arguments, one can obtain the reversible currents /R rR g Ri ; rRij ; X R ; Y R ; Z R , ji and ji in the LD phase of the I-LD transition. However, the expressions of g Ri ; X R ; Y R , and Z R are same as for the I-SmA transition in TLC [29]. The new expressions for the reversible rR currents rRij and ji in the LD phase up to the linear order in the thermodynamic force can be written as

ð2:23Þ

Rest of the expressions of the reversible current are same as in the SmA phase of the I-SmA transition [29]. For the derivation of the irreversible currents and quasicurrents, the dissipation function R can be expressed as a Liapunov functional

ð2:25Þ

ji

¼ jij rj T  kij rj M  cij rj P  fij rj N  eni rk Wk ;

¼ K ij rj N  fij rj T  a0ij rj P  a00ij rj M  gni ri rk Wk :

ð2:26Þ

ð2:31Þ

Viscosity in LD phase is anisotropic. So the viscous stress tensor in the incompressible LD phase is given by Brochard and de Gennes [24]

@v z @x  z  @v z @v i ¼ gM þ @xi @xz   @v j @v i ; ¼ gT þ @xi @xj

rDzz ¼ 2gV rDiz rDij

ð2:32Þ

with i; j – z; gV ¼ ðg1  g2 þ g4  2g5 Þ; gM ¼ g3 and gT ¼ g2 . gM is related to relative sliding of layers, and is therefore comparable to the viscosity of the solvent in LD phase, the other two are associated with distortions of the membranes themselves. 2.2. Macroscopic equations in the isotropic micellar phase of the I-LD transition In the isotropic micellar phase of the I-LD transition there are patches of both transient positional as well as orientational characteristics of ND and LD clusters. Both types of clusters vary as a function of space and time. In addition there is also concentration of surfactant molecules which are aggregated in the I phase. Thus we have two macroscopic variables nematic tensor order parameter Q ij and the complex scalar smectic order parameter w. For the hydrodynamic variables in the I phase, we have q, r; / and g, respectively. For a description of the macroscopic dynamics in the I phase we proceed in a way similar to that in subSection 2.A. Thus the Gibbs relation in the I phase takes the form

de ¼ ldq þ Tdr þ v i g i þ P ij dQ ij þ MdW þ Xi dri U þ Nd/

ð2:33Þ

Where W ¼ w0 ; M ¼ ðlw eiU þ lw eiU Þ and ri Xi ¼ w0 ðlw eiU lw eiU Þ. w0 is the modulus of the smectic order parameter w and

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P.K. Mukherjee / Chemical Physics 430 (2014) 56–61

U is its phase. The asterisk denotes complex conjugation. The thermodynamic forces are T; l; v i ; /; Pij and W. Now the generalized energy in the I phase can be expressed as

Z



1 1 1 2 Aqq ðdqÞ2 þ Aqr ðdqÞðdrÞ þ Arr ðdrÞ2 þ g 2 2 2q 1 1 1 1 þ p1 ðd/Þ2 þ aQ ij Q ij þ aW 2 þ Cjðr2 þ q20 Þwj2 2 2 2 2 1 q r þðb dq þ b drÞQ 2ij þ Lijklmn ðri Q jk Þðrl Q mn Þ þ ðcq dq þ cr drÞW 2 2 2 þhQ ij W 2 þ k1 Q 2ij d/ þ m2 W 2 d/ þ ðf q dq þ f r drÞd/  þGijkl Q ij ðrk wÞðrl w Þ : ð2:34Þ

F¼

ds

Gijkl is the lowest order coupling of the Q ij , and w. In the isotropic phase Gijkl ¼ Gðdik djl þ dil dkl Þ. The thermodynamic forces can be derived as

j

/R

¼ 0;

ð2:50Þ

Similar to the preceding section, we obtain the dissipation function

R¼

 1 1 1 1 ds jij ðri TÞðrj TÞ þ gijkl Aij Akl þ sijkl Pij P kl þ K M M 2 2 2 2 2

Z

1 þ K ij ðri NÞðrj NÞ þ kðri TÞðri MÞ þ s0ijkl ðri TÞðrj Pkl Þ 2 þf ðri TÞðri NÞ þ aijkl Pij Pkl M þ a0ijkl ðri NÞðrj Pkl Þ þ a3 ðri NÞðri MÞ  1 ð2:51Þ þ fðri Xi Þðrj Xj Þ 2 where aijkl ; a0ijkl ; sijkl , and s0ijkl are of the form aijkl ¼ ða=2Þðdik djl þ dil djk Þ, and where gijkl has a structure familiar from the hydrodynamics of simple liquid [33]. Hence the dissipative parts of the currents are rD

r

ð2:35Þ

ji

dl ¼ Aqq dq þ Aqr dr þ b Q 2ij þ cq W 2 þ f q d/;

q

ð2:36Þ

g Di ¼ 0;

ð2:53Þ

N ¼ pd/ þ f q dq þ f r dr þ k1 Q 2ij þ m2 W 2 ;

ð2:37Þ

rDij ¼ gijkl Akl ;

ð2:54Þ

ð2:38Þ

ji

dT ¼ Arr dr þ Aqr dq þ b Q 2ij þ cr W 2 þ f r /

vi ¼

1

q

/D

gi ; q

r

2

Pij ¼ aQ ij þ 2ðb dq þ b dr þ hW þ k1 d/ÞQ ij þ 2GW 2 ðri UÞðrj UÞ;

ð2:39Þ 2

M ¼ aW þ 2ðcq dq þ cr dr þ hQ ij þ m2 d/ÞW 2

þ 4WGQ ij ðri UÞðrj UÞ þ CWð½r2 U ; þð½ri U2 

Xi ¼ 4GW 2 Q ij rj U  CW 2 ðr2 þ 2q20  2½ri U2 Þri U:

2 q20 Þ Þ

ð2:40Þ ð2:41Þ

In writing down Eqs. (2.25)–(2.31) we have concentrated on spatially homogeneous terms with respect to W and Q ij . The balance equations are

@q þ ri g i ¼ 0; @t

ð2:42Þ

@g i þ rj rij ¼ 0: @t

ð2:43Þ

@r R r þ r i ji ¼ ; T @t

ð2:44Þ

@/ / þ v i ri / þ ri ji ¼ 0; @t

ð2:45Þ

@Q ij þ v k rk Q ij þ Y ij ¼ 0; @t

ð2:46Þ

@W þ v i ri W þ Z ¼ 0; @t

ð2:47Þ

@U þ v i ri U þ IU ¼ 0: @t

ð2:48Þ

¼ K ri N  f ri T  a0 rj Pij  a3 ri M;

ð2:52Þ

ð2:55Þ

Y Dij ¼ sPij þ 2aPij M  s0 ri rj T  a0 ri rj N;

ð2:56Þ

Z D ¼ K M M þ aPij Pij  krj ri T  a3 ri rj N;

ð2:57Þ

IDU ¼ fri Xi :

ð2:58Þ

By inspection of Eqs. (2.12), (2.13) and (2.35), we see two new hydrodynamic equations for the I-LD phase transition in LLC compared to the I-SmA transition in TLC. 3. Experimental consequences Electric field are often used to study the electric birefringence in the isotropic phase of the isotropic to nematic (I-N) and I-SmA phase transitions [34–37]. The electric birefringence in the isotropic micellar phase of the I-LD transition is only related to S; w and /. This leads to the coupling of the order parameters S; w and / with an uniform external electric field. Then the electric field terms like  21e D2i , v1 Di Dj Q ij ; v2 Di Dj d/ and v3 Di Dj jwj2 should be added in the free energy (2.34). Here D ¼ E þ 4pP. Then for constant density and under adiabatic condition, we obtain (with Ek^z) 2

aS þ 2hSW þ 2k1 Sd/  v1 D2z ¼ 0;

ð3:1Þ

a þ cW 2 þ 2hS2 þ 2m2 d/  v3 D2z ¼ 0;

ð3:2Þ

p1 d/ þ k1 S2 þ m2 W 2  v2 D2z ¼ 0;

ð3:3Þ

eE þ Dz ð1 þ 2ev1 S þ 2ev3 W 2 þ 2ev2 d/ ¼ 0:

ð3:4Þ

Eliminating W 2 and d/, we obtain

Again we obtain the same expressions for the reversible currents rR g Ri ; X R ; Y R ; Z R , and ji in the isotropic micellar phase in LLC similar to the isotropic phase of the I-SmA transition [29]. The only new expression for the reversible current in the I phase up to the linear order in the thermodynamic force can be written as

rRij ¼ pdij þ kPij þ bdij M þ k1 dij N;

¼ jri T  kri M  s0 rj Pij  f ri N;

S¼

  V 2 2 ; D 1  D a z a z

v1

where 4m2 ah a ¼ a þ 2kp1mc2 a  2hca  c22p , 1 1    2v3 m2 4m2 h 1 þ 2vc 3 , V ¼ 2k  v    2 p c p c 1

ð2:49Þ

p1

¼ p1 

1

2m22

c

.

ð3:5Þ

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For the negative value of aV , the nematic order increases. The nematic order (3.5) and therefore the electric birefringence and non linear dielectric effect (NDE) acquires a correction  E4 from the coupling to the smectic order parameter and surfactant concentration in the isotropic micellar phase of the I-LD transition. This can be checked experimentally. The material parameters a; a and p1 can be assumed as a ¼ a0 ðT  T 1 Þ; a ¼ a0 ðT  T 2 Þ and p1 ¼ p0 ðT  T 3 Þ. T 1 ; T 2 and T 3 are virtual transition temperatures. a0 ; a0 and p0 are positive constants. The temperature dependence of the Kerr constant in the vicinity of the I-LD transition to a first approximation can be written as [15]

K¼

Dn0 SðEÞ U Dn0 ¼ ; ðT  T  Þ E2

ð3:6Þ

where Dn0 is the optical birefringence.

U¼

v1 a0

T ¼

e1 ¼

e þ p  Tr : q

ð3:8Þ

Then the static equations of state are

  1

qd

q

  N

qd

; 2

micellar phase is also shown in Fig.1. Units of the eNDE and temperature are arbitrary. The parameter value used for the NDE is UðDef Þmax ¼ 0:38. Inspection of Eqs. (2.6)–(2.8), we noticed that we have used mass and entropy density are variables rather than experimentally more favorable variables temperature and pressure. In order to get the pressure variables instead of chemical potential, we consider following Legendre transformation

q

¼ jT dp þ ap dT þ f p d/;

¼ p01 d/ þ f p dp þ f T dT;

ð3:9Þ

ð3:10Þ

The thermal expansion coefficient ap , isothermal compressibility jT that enter in Eq. (3.9) can be measured experimentally. So Eq. (3.9) can be verified experimentally.

3

16 2k1 m2 a0 T 2 2ha0 T 2 4m22 ha0 T 2 7    5;  a T   2  4 0 1 2m 2m2 a0 c p0 c T 3 þ c 2 p0 c2 T 3 þ c 2

4. Conclusion

2k1 m2 a0 2ha0 4m22 ha0    : a0 ¼ a0   2m22 2m2  c p0 c T 3 þ c p0 c2 T 3 þ c 2 One can readily see that Eq. (3.6) enables us to explain the qualitatively the behavior of K as a function of temperature. Eq. (3.6) shows that the Kerr constant of melt changes proportionality of 1=ðT  T  Þ. This behavior is similar to the case of the I-N and ISmA transitions [34–37]. The temperature dependence of the Kerr constant (K) is shown in Fig. 1. This is done for a set of phenomenological parameter U Dn0 ¼ 0:62. Units of the K and temperature are arbitrary. Following the approach of Mukherjee and Rzoska [38], the NDE in the isotropic micellar phase of the I-LD transition can be expressed as

eNDE ¼

UðDef Þmax ðT  T  Þ

ð3:7Þ

where ðDef Þmax being the maximum anisotropy of the dielectric permittivity at the frequency f of the measuring field. Eq. (3.7) predicts that in the isotropic micellar phase eNDE decreases with increasing temperature. The NDE expression (3.7) gives the fluid-like critical exponent c0 ¼ 1. This gives a clear evidence of the pretransitional anomaly in the isotropic micellar phase of the I-LD transition similar to the I-N and I-SmA phase transitions in TLC. The temperature dependence of eNDE in the isotropic

Fig. 1. The temperature dependence of the Kerr constant (K) and NDE (eNDE ) in the isotropic micellar phase of the I-LD transition.

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