Macroscopic dynamics near the isotropic-smectic-A phase transition.

Abstract

The hydrodynamic theory for the smectic-A phase and the isotropic phase is generalized to the macroscopic dynamics in the vicinity of the isotropic-smectic-A phase transition. The macroscopic dynamic equations are presented on the isotropic side as well as on the smectic-A side of the phase transition, incorporating the effect of an external electric field. Specific experiments to test some of the effects contained in the macroscopic dynamic equations are suggested.

Full text

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PHYSICAL REVIEW E, VOLUME 63, 061708
Macroscopic dynamics near the isotropic–smectic-A phase transition
Helmut R. Brand1, Prabir K. Mukherjee1and Harald Pleiner2
1Theoretische Physik III, Universit¨ at Bayreuth, D-95440 Bayreuth, Germany
2Max-Planck-Institut f¨ ur Polymerforschung, D-55021 Mainz, Germany
(Received 31 March 2000; revised manuscript received 4 December 2000)
The hydrodynamic theory for the smectic-A phase and the isotropic phase is generalized to the
macroscopic dynamics in the vicinity of the isotropic–smectic-A phase transition. The macroscopic
dynamic equations are presented on the isotropic side as well as on the smectic-A side of the phase
transition, incorporating the effect of an external electric field. Specific experiments to test some of
the effects contained in the macroscopic dynamic equations are suggested.
DOI: 10.1103/PhysRevE.63.061708
PACS numbers : 64.70.Md, 05.70.Ln, 61.30-v
I. INTRODUCTION
The hydrodynamic description of liquid crystals has
attracted physicists since the 1970s [1–6]. It turns out
that the anisotropy of liquid crystals has a number of in-
teresting implications for their hydrodynamic behavior.
Over the last two decades the hydrodynamic approach
has been applied to a number of liquid crystalline phases,
including biaxial nematics [7,8] and hexatic phases [9].
Following Khalatnikov’s work [10] near the λ transi-
tion in4He, Liu [11] and Brand [12] discussed how to
incorporate the modulus of the order parameter near
the nematic–smectic-A and the nematic–columnar tran-
sitions. In this approach one takes into account not only
the truly hydrodynamic variables, which have an infinite
relaxation time in the long wavelength limit, but also so-
called macroscopic variables, which relax on a long, but
finite time scale.
This macroscopic dynamics near a phase transi-
tion should not be confused with a (time-dependent)
Ginzburg-Landau description of the phase transition.
The latter involves a free energy functional that contains
an expansion in the order parameter usually to fourth or
sixth order depending on whether the phase transition
under consideration is of second or weakly first order.
The approach of macroscopic dynamics, however, deals
with the dynamics of the deviations of the order param-
eter modulus from its equilibrium value. It is valid near
the phase transition, since it includes the order parame-
ter modulus as a variable (in addition to those variables
already present far from the phase transition), but not at
the phase transition, where nonlinear and critical effects
must be considered. Therefore there are two different
(and not directly connected) macroscopic dynamic de-
scriptions below and above the phase transition, because
the symmetries of the phases are different.
The concept of macroscopic dynamics is also useful in
macroscopic complex systems far from phase transitions,
but with variables that relax slowly in space and time.
One group of such systems encompasses polymer melts
and solutions in their isotropic and nematic phases to
which the approach of macroscopic dynamics was applied
in [13,14] and [15,16], respectively. For a review of these
recent developments we refer to [17].
In the present paper we study the macroscopic dy-
namic behavior near the isotropic–smectic-A transition
on both sides of the phase transition. In Sec. II we give
the macroscopic equations on the smectic-A side of the
transition. First we discuss the static properties and then
we investigate reversible and dissipative dynamic effects.
In Sec. III we present the macroscopic equations on the
isotropic side of the phase transition followed in Sec. IV
by suggestions for experiments, by which one could de-
tect some of the cross-coupling terms introduced here.
We also close by brief conclusions.
II. MACROSCOPIC EQUATIONS IN THE
SMECTIC-A PHASE BELOW THE
ISOTROPIC–SMECTIC-A TRANSITION
Throughout we shall focus our discussion on macro-
scopic aspects.This means that we are interested in
length scales and time intervals that are large compared
with molecular lengths and collision times.
words, the characteristic frequencies ω and wave vectors
k must satisfy the inequalities ωτc?1, klc?1. Here
1/τcand lcare microscopic frequencies and microscopic
length scales. The discussion of the macroscopic dynam-
ics of liquid crystals proceeds in three steps. It is first
necessary to identify the macroscopic and hydrodynamic
variables that describe the macroscopic state of the sys-
tem. Secondly one derives a set of macroscopic equations
for the macroscopic variables, for the conserved quanti-
ties (mass, energy and linear momentum [18]) and for the
variables associated with spontaneously broken continu-
ous symmetries. Finally these equations must be solved
for specific geometries relevant to experiment.
In other
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For the smectic-A (smA) phase we have the hydrody-
namic variables density ρ (conservation of mass), entropy
density σ, density of linear momentum g (conservation
of linear momentum) and displacement uzof the smectic
layers along the z axis associated with the density wave
parallel to the layer normal. (Conservation of energy is
taken care of by the Gibbs relation below.) In addition to
these hydrodynamic variables we have additional macro-
scopic variables, namely the modulus S of the nematic
order parameter Qij = (S/2)(3ninj− δij) (a symmet-
ric traceless tensor characterizing the orientational order)
and the real modulus of the smectic order parameter W.
To characterize the smectic order, we use the magnitude
of the smectic order parameter [11], i.e., the real quantity
W. As an alternative one could use, instead of W and
uz, the complex scalar ψ (compare the discussion below).
Throughout this paper we assume that the nematic di-
rector ˆ n and the smectic layer normalˆk are parallel to
each other. Thus we have ˆ n ?ˆk. We note, however, that
it has recently become clear that there might be situa-
tions in smectic liquid crystals, for example under shear
flows, where this restriction might no longer strictly ap-
ply [19]. Throughout our analysis of the smectic-A side
of the isotropic–smectic-A phase transition we will focus
on the linearized macroscopic equations. With the help
of Euler’s relations the Gibbs relation takes the form
Tdσ = df − µdρ − vidgi− Ψid∇iuz− PdS − MdW (1)
where f is the (conserved) energy density of the system
in the laboratory frame. The summation over repeated
indices is always implied. S and W denote the deviations
of the nematic and smectic order parameter moduli from
their equilibrium values S0and W0. In Eq. (1) the tem-
perature T, the chemical potential µ, the velocity field vi,
the field Ψiand the order parameter fields P and M are
called thermodynamic conjugates. We note that both S
and W are scalar quantities under all symmetry opera-
tions, while uzand Ψichange sign, when ˆ n is replaced by
−ˆ n. Equation (1) gives a relation between the changes
in the macroscopic variables and the entropy density σ.
The above mentioned thermodynamic conjugates and
the static properties of the smA phase can be obtained
from the expansion of the generalized energy F = F0+
?fdτ. Hence, for the generalized energy (in quadratic
F = F0+
and bilinear order), we find
?
+(cρδρ + cσδσ)W +1
dτ[1
2aS2+ (bρδρ + bσδσ)S +1
2B(∇zuz)2+1
+(dρδρ + dσδσ + dWW)∇zu + γ(∇zuz)S]
where F has the standard form of the smectic free energy
[1] supplemented by quadratic and bilinear terms in the
moduli. As usual for the smectic-A phase, terms con-
taining ∇⊥uzare not allowed due to the spontaneously
broken rotational symmetry. F0=?f0dτ is the general-
2αW2+ hSW
2K(∇2
⊥uz)2
(2)
ized energy of isotropic liquids with f0= (1/2)Aρρ(δρ)2+
Aρσ(δρ)(δσ) + (1/2)Aσσ(δσ)2+ (1/2ρ)g2, where Aρρ=
(∂µ/∂ρ)σ, Aσσ= (∂T/∂σ)ρ, and Aρσ= (∂T/∂ρ)σ. 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 nematics [1]. The transverse Laplacian
is defined as ∇2
that the linearized theory [and hence (2)] is not rota-
tionally invariant. For nonlinear invariant treatments of
smectic phases, see Refs. [20,21].
The thermodynamic conjugates can easily be obtained
from the generalized energy by taking the variational
derivative of F with respect to one variable while keeping
all other variables fixed, resulting in
⊥= (δij− ninj)∇i∇j. It is well known
Ψi=
δF
δ(∇iuz)= (B∇zu + dσδσ + dρδρ + dWW
+γS)δiz− K(δij− ninj)∇j∇2
P =δF
δS= aS + bσδσ + bρδρ + hW + γ∇zuz
M =δF
δW= αW + cρδρ + cσδσ + hS + dW∇zuz
δT =δF
δσ= Aσσδσ + Aσρδρ + bσS + cσW + dσ∇zuz (6)
δµ =δF
δρ= Aρρδρ + Aσρδσ + bρS + cρW + dρ∇zuz
vi=1
ρgi
⊥uz
(3)
(4)
(5)
(7)
(8)
The resulting dynamic equations for the conserved fields
are
∂ρ
∂t+ ∇igi= 0
∂gi
∂t+ ∇jσij= 0
(9)
(10)
where σijis the stress tensor. The above two equations
are the conservation laws for the density and the density
of the linear momentum. The balance equations for the
nonconserved fields take the form
∂σ
∂t+ ∇ijσ
∂uz
∂t
∂S
∂t+ Y = 0
∂W
∂t
i=R
T
(11)
+ X = 0(12)
(13)
+ Z = 0(14)
Here jσ
the quasicurrents associated with the density wave and
changes of the nematic and smectic order parameters, re-
spectively. The quantity R/T is called the entropy pro-
duction, and R the dissipation function. The dissipation
function is zero for reversible processes and positive for
irreversible processes.
iis the entropy current, and X, Y , and Z are
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We now evaluate the currents σij, and jσ
sicurrents X, Y , and Z. To obtain them we split all
currents and quasicurrents into reversible (R = 0) and
irreversible (R>0) contributions. Using general symme-
try and Galilean invariance arguments, for the reversible
parts of the currents, we obtain
iand the qua-
gR
σR
XR= vz
YR= µijAij
ZR= βijAij
jσR
i
= 0
i= ρvi
ij= pδij− Ψjδiz+ µijP + βijM
(15)
(16)
(17)
(18)
(19)
(20)
where p is the hydrostatic pressure, Aijis the symmetric
velocity gradient Aij = 1/2(∇ivj+ ∇jvi), and βij and
µijtake the uniaxial form βij= β?ninj+β⊥(δij−ninj).
The hydrostatic pressure p is like in an isotropic fluid and
can be expressed as
p = −f + µρ + Tσ + v · g
(21)
For the dissipative parts we will now give R, which rep-
resents the dissipative work that must be done on the
system by external forces if the thermodynamic varia-
tions were sustained. Within linear irreversible thermo-
dynamics the dissipation function is a bilinear form of
the forces. Using standard symmetry arguments [3,17],
like time reversal symmetry, rotational invariance, etc.,
we find
?
+m
2KMM2+ λij(∇iT)(∇jM)
+cij(∇iT)(∇jP) + bnj(∇mΨm)(∇jP) + αPM
+dnk(∇jΨj)(∇kM) + enk(∇jΨj)(∇kT)
R =
dτ
?1
2κij(∇iT)(∇jT) +1
2(∇iΨi)(∇kΨk) +1
2ηijkl(∇ivj)(∇kvl) +τ
2P2
?
(22)
where λij, cij, and the thermal conductivity tensor κij
have uniaxial forms. The viscosity tensor ηijkl has five
independent coefficients [3] in the smectic-A phase.
Then the dissipative parts of the currents and quasi-
currents are obtained by taking variational derivatives of
R with respect to one thermodynamic force, while keep-
ing all other forces fixed. Thus the dissipative parts read
jσD
i
gD
σD
XD= −m∇kΨk− bnj∇jP − dnk∇kM − enk∇kT (26)
YD= τP + αM − cij∇i∇jT − bnj∇j∇kΨk
ZD= KMM + αP − λij∇j∇iT − dnj∇j∇kΨk
= −κij∇jT − λij∇jM − cij∇jP − eni∇kΨk
i = 0
ij= −ηijkl∇kvl
(23)
(24)
(25)
(27)
(28)
III. MACROSCOPIC EQUATIONS IN THE
ISOTROPIC PHASE ABOVE THE
SMECTIC-A–ISOTROPIC TRANSITION
In the isotropic phase above the isotropic–smectic-A
transition there are patches with transient positional as
well as orientational order characteristic of smectic clus-
ters and those with only transient orientational order
characteristic of nematic clusters. Both types of clus-
ters vary as a function of space and time and do not give
rise to a nonvanishing value of either type of order pa-
rameter. Thus, as macroscopic variables, we have the
nematic tensor order parameter Qijas well as the com-
plex scalar smectic order parameter ψ, whose modulus
characterizes the (time dependent) strength of the smec-
tic order and whose phase (or rather the gradient of it) is
related to the wave vector of the smectic patches. For the
hydrodynamic variables in the isotropic phase, we have
ρ, σ, and g, respectively. For mixtures one has, in ad-
dition, the concentration c as a conserved quantity. We
will also include electric field effects and take the electric
displacement field Dias dynamic variable. For a descrip-
tion of the macroscopic dynamics in the isotropic phase
we proceed in a way similar to that in Sec. II.
Thus the Gibbs relation takes the form
Tdσ = df − µdρ − vidgi− PijdQij
−(µψdψ + µψ∗dψ∗) + EidDi
(29)
where the asterisk denotes complex conjugation.
thermodynamic forces are temperature T, chemical po-
tential µ, velocity vi, electric field Ei, Pij, µψand µψ∗,
respectively. For the generalized energy in the isotropic
phase we find
?
+a
2Lijklmn(∇iQjk)(∇lQmn) + O(Q3)
+α
2|(∇2+ q2
+(bρδρ + bσδσ)Q2
−1
+GijklQij(∇kψ)(∇lψ∗)
The
F =
dτ
?Aρρ
2
(δρ)2+ Aρσ(δρ)(δσ) +Aσσ
2
(δσ)2+
1
2ρg2
2QijQij+1
2|ψ|2+γ
4|ψ|4+C
0)ψ|2+ hQ2
ij|ψ|2
ij+ (cρδρ + cσδσ)|ψ|2
i− χ1DiDjQij− χ2Di∇jQij− χ3DiDi|ψ|2
?
The first line contains the contributions familiar from a
simple fluid, where the abbreviations Aρρ, Aσσ, and Aρσ
are the same as Sec. II. The second line contains all the
terms characteristic of the isotropic–nematic transition
[22]. Line 4 lists the static terms coupling the order pa-
rameters to density ρ and entropy density σ, respectively,
and the fifth line represents the coupling of the order pa-
rameters Qijand ψ to electric fields, where ε is the dielec-
tric constant of the isotropic phase. In the third line we
2εD2
(30)
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have given the contributions of the smectic order parame-
ter ψ and its coupling terms to the nematic order param-
eter Qij. We note that the gradient term ∼C is isomor-
phic to the corresponding term in the Swift-Hohenberg
equation describing the onset of convection in two di-
mensions [23].In both cases (the isotropic–smectic-A
phase transition and the nonequilibrium transition heat-
conduction-state–convection) one has a transition from a
completely isotropic state to a state which is character-
ized by a mean wavelength (the wavelength of the con-
vective rolls and the layer spacing in the smectic-A phase,
respectively). Thus - from the point of view of symme-
try considerations - there is a close structural analogy for
this contribution between an equilibrium phase transition
(isotropic–smectic-A) and a nonequilibrium phase tran-
sition (heat-conduction-state–convection). In the sixth
line the contribution ∼Gijklrepresents the lowest order
coupling of the nematic and the smectic order parame-
ters containing spatial gradients. In the isotropic phase
it takes the structure Gijkl= G(δikδjl+ δilδjk) and thus
contains one independent coefficient.
contribution has not been discussed in the literature be-
fore. We note, however, that in a very recent paper we
investigated [24] a Ginzburg-Landau description of the
isotropic–smectic-A transition including such a coupling
term.
In the following we will write down the macroscopic
equations in terms of real quantities. We therefore re-
place the smectic order parameter ψ by its modulus,
W = ψ0, and its phase φ. Since the energy does not
depend on the phase (”gauge invariance” meaning trans-
lational invariance in the smectic case) only gradients of
φ enter the Gibbs relation. This is very similar in spirit
to the situation in superfluid4He above the λ transition
[10].
For the modified Gibbs relation we then have
Apparently this
Tdσ = df − µdρ − vidgi− PijdQij− MdW
−Ωid∇iφ + EidDi
where M = µψeiφ+ µψ∗e−iφand ∇iΩi= ψ0[µψ∗e−iφ−
µψeiφ], and for the thermodynamic forces we find (ne-
glecting gradients of W and of Qij)
(31)
δT = Aσσδσ + Aσρδρ + bσQ2
δµ = Aρρδρ + Aσρδσ + bρQ2
vi=1
ρgi
Pij= aQij+ O(Q2) + 2(bσδσ + bρδρ + hW2)Qij
−χ1DiDj+ 2GW2(∇iφ)(∇jφ)
M = αW + γW3+ 2W(cρδρ + cσδσ + hQ2
+4WGQij(∇iφ)(∇jφ) − 2χ3WDiDi
+CW([∇2φ]2+ ([∇iφ]2− q2
Ei=1
εDi+ 2χ1DjQij+ 2χ3DiW2
Ωi= 4GW2Qij∇jφ
ij+ cσW2
ij+ cρW2
(32)
(33)
(34)
(35)
ij)
0)2)(36)
(37)
−CW2(∇2+ 2q2
0− 2[∇iφ]2)∇iφ
(38)
In writing down Eqs.(32–38) we have concentrated on
spatially homogeneous terms with respect to W and Qij,
and we have also only kept the coupling terms to low-
est order in the corresponding variables. The balance
equations are
∂ρ
∂t+ ∇igi= 0
∂gi
∂t+ ∇jσij= 0
∂σ
∂t+ ∇ijσ
(39)
(40)
i=R
T
(41)
∂Qij
∂t
+ vk∇kQij+ Yij= 0
∂W
∂t
∂φ
∂t+ vi∇iφ + Iφ= 0
(42)
+ vi∇iW + Z = 0 (43)
(44)
∂Di
∂t
+ vk∇kDi+ (D × ω)i+ je
i= 0(45)
with the vorticity ωi= (1/2)(curlv)i. Equation (45) ex-
presses conservation of the electric charge density ρe[21]
with ρe= ∇iDiin suitable units.
We now evaluate the reversible and dissipative contri-
butions to the currents jσ
Z, and Iφ. For the reversible parts of the currents we
obtain
i, σij, je
iand quasicurrents Yij,
gR
σR
YR
ZR= βAkk
IR
jeR
i
= 0
i= ρvi
ij= pδij+ λPij+ βδijM +1
ij= λAij
(46)
(47)
(48)
(49)
(50)
(51)
2(DjEi− EjDi)
φ= 0
where we have used the symmetrized velocity gradient
Aij= (1/2)(∇ivj+ ∇jvi).
In a similar way as in Sec. II we obtain the dissipation
function
?
+λ(∇iT)(∇iM) + αijklPijPklM +1
+σE
2EiEi+ κ?Ei(∇iT) +˜λEi(∇iM)
+τ?
+ζ
2(∇iΩi)(∇jΩj)]
R =
dτ [κ
2(∇iT)(∇iT) +1
2ηijklAijAkl+1
2KMM2
2τijklPijPkl
ijkl(∇iT)(∇jPkl) + ˜ τijklEi(∇jPkl)
(52)
where αijkl, τijkl, ˜ τijkl, and τ?
(α/2)(δikδjl+ δilδjk), and where ηijkl has a structure
ijklare of the form αijkl=
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familiar from the hydrodynamics of simple liquids [18].
Hence the dissipative parts of the currents read
jσD
i
gD
σD
YD
= −κ∇iT − λ∇iM − τ?∇jPij− κ?Ei
i = 0
ij= −ηijklAkl
ij= τPij+ 2αPijM − τ?∇i∇jT
−˜ τ
ZD= KMM + αPijPij− λ∇i∇iT −˜λ∇iEi
ID
= σEEi+ κ?∇iT +˜λ∇iM + ˜ τ∇jPij
(53)
(54)
(55)
2(∇iEj+ ∇jEi)(56)
(57)
(58)
(59)
φ= ζ∇iΩi
jeD
i
IV. SUGGESTIONS FOR EXPERIMENTS AND
CONCLUSIONS
To evaluate the influence of the cross-coupling between
the smectic and the nematic order parameter, we investi-
gate its influence on the electric birefringence induced by
an external electric field in the isotropic phase above the
isotropic–smectic-A transition. From Eqs. (35–37), in a
static situation for constant density and under adiabatic
conditions in lowest order, neglecting all nematic-smectic
cross couplings and taking E ? ˆ z (with Qzz = S) and
∇zφ = q0, we obtain
0 = aS − χ1D2
0 = α − 2χ3D2
Dz= εE
z
z+ γW2
(60)
(61)
(62)
Thus in this approximation we find that an external
field can induce both nematic and smectic orders, but
these two types of order are decoupled to lowest order.
The classical result above the nematic–isotropic transi-
tion [22]
S =χ1
aε2E2
(63)
is obtained, showing a divergent nematic order at the
temperature for a hypothetical isotropic to nematic sec-
ond order phase transition, where the system would order
spontaneously already without an external field. For the
smectic order we obtain
W2=1
γ(−α + 2χ3ε2E2)(64)
indicating a possible smectic ordering for positive χ3, if
the external field exceeds the threshold E2
This threshold is zero at the temperature T∗
thetical isotropic to smectic A second order phase tran-
sition. If E<Ec, there is no smectic order (W = 0). For
a given external field E>Ecthe induced smectic order is
largest for T = T∗
c= α/(2χ3ε2).
AIfor a hypo-
AI, but does not diverge. The difference
from the nematic order [Eq. (63)] is due to the fact that
an external field breaks rotational (but not translational)
invariance externally.
Next we investigate the influence of the contribution
∼h, which couples smectic and nematic orders. In this
case, we obtain
0 = aS + 2hSW2− χ1D2
−γW2= α + 2hS2− 2χ3D2
εE = Dz(1 + 2εχ1S + 2εχ3W2)
z
(65)
(66)
(67)
z
Eliminating W2we obtain to lowest order in h (for
E>Ec)
?
where ∆2
erwise. Thus a negative h increases the nematic order.
The nematic order (68), and therefore the electric bire-
fringence, acquires a correction ∼E4from the coupling to
the smectic order parameter above the isotropic–smectic-
A transition for E >Ec. We note that the temperature
dependence of S(E2) predicted here is completely dif-
ferent from that observed above the nematic–isotropic
transition.
The effect of the parameter G, which couples smec-
tic and nematic order through a finite wavelength q0of
the transient smectic layer structure, is rather different.
With the same procedure as above, to lowest order in G,
and for fields exceeding Ec, the nematic order is
aS = χ1D2
z
1 −4hχ3
aγ
∆2
D
?
(68)
D= D2
z− α/(2χ3), if positive, and zero oth-
aS = χ1D2
z−4χ3
γ
Gq2
0∆2
D
(69)
showing no E4correction, but a change of slope in S/E2
above Ec[below Ecthere is no G correction to the clas-
sical result (63)].
It also seems worthwhile from emphasize the differ-
ence to the case of the strain birefringence observed
experimentally above the isotropic–smectic-A transition
in liquid crystalline elastomers [25]. In the latter case
the induced amount of birefringence ∆n is found to be
∆n∼σ (here σ is the applied mechanical stress). Further-
more the temperature dependence is qualitatively differ-
ent from the one discussed above.
To test one of the dynamic cross-coupling effects we in-
vestigate the consequences associated with the coefficient
β in Eqs.(47) and (49): the divergence of the velocity field
couples to the modulus of the smectic order. This leads
to suggest the following setup. A sample of a material
above an isotropic to smectic-A transition is exposed to a
stationary sound wave. This sound wave is longitudinal
and thus connected to ∇ivi?= 0. This in turn gives rise
to a nonvanishing value of W: using Eqs.(43) and (49)
we obtain
W =β
cA exp[i(ωt − ck)](70)
5