DTIC ADA591463: Stability of Equilibria of Nematic Liquid Crystalline Polymers

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Acta Mathematica Scientia 2011,31B(6):2289–2304
http://actams.wipm.ac.cn
STABILITY OF EQUILIBRIA OF NEMATIC
∗
LIQUID CRYSTALLINE POLYMERS
Dedicated to Professor Peter D. Lax on the occasion of his 85th birthday
Hong Zhou
Department of Applied Mathematics
Naval Postgraduate School, Monterey, CA 93943, USA
E-mail: [email protected]
Hongyun Wang
Department of Applied Mathematics and Statistics
Universityof California, Santa Cruz, CA 95064, USA
E-mail: [email protected]
Abstract We provide an analytical study on the stability of equilibria of rigid rodlike
nematic liquid crystalline polymers (LCPs) governed by the Smoluchowski equation with
the Maier-Saupe intermolecular potential. We simplify the expression of the free energy
of an orientational distribution function of rodlike LCP molecules by properly selecting
a coordinate system and then investigate its stability with respect to perturbations of
orientational probability density. By computing the Hessian matrix explicitly, we are
able to provethehysteresis phenomenon of nematic LCPs: when the normalized polymer
concentration b is below a critical value b∗ (6.7314863965), the only equilibrium state is
isotropicanditisstable; when b∗ <b<15/2, twoanisotropic (prolate) equilibriumstates
occur together with a stable isotropic equilibrium state. Here the more aligned prolate
stateisstablewhereasthelessalignedprolatestateisunstable. Whenb>15/2, thereare
threeequilibriumstates: astableprolatestate,anunstableisotropicstateandanunstable
oblate state.
Key words equilibria; stability; nematic liquid crystalline polymers; hysteresis phe-
nomenon
2000 MR Subject Classification 35Kxx; 70Kxx
1 Introduction
Since its discovery by Austrian botanical physiologist Friedrich Reinitzer in 1888, liquid
crystals have spurted intensive experimental, theoretical and numerical studies [1, 5, 6, 18, 31,
39]. One notable example is that in 1991Pierre-Gillesde Gennes wasawardedthe Nobel Prize
∗Received September 30, 2011. This work is partially supported by the National Science Foundation and
bytheOfficeofNavalResearch.
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Stability of Equilibria of Nematic Liquid Crystalline Polymers
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14. ABSTRACT
We provide an analytical study on the stability of equilibria of rigid rodlike nematic liquid crystalline
polymers (LCPs) governed by the Smoluchowski equation with the Maier-Saupe intermolecular potential.
We simplify the expression of the free energy of an orientational distribution function of rodlike LCP
molecules by properly selecting a coordinate system and then investigate its stability with respect to
perturbations of orientational probability density. By computing the Hessian matrix explicitly, we are able
to prove the hysteresis phenomenon of nematic LCPs: when the normalized polymer concentration b is
below a critical value b&#8727; (6.7314863965), the only equilibrium state is isotropic and it is stable;
when b&#8727; < b < 15/2, two anisotropic (prolate) equilibrium states occur together with a stable
isotropic equilibrium state. Here the more aligned prolate state is stable whereas the less aligned prolate
state is unstable. When b > 15/2, there are three equilibrium states: a stable prolate state, an unstable
isotropic state and an unstable oblate state.
15. SUBJECT TERMS
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2290 ACTAMATHEMATICASCIENTIA Vol.31Ser.B
in Physics “for discovering that methods developed for studying order phenomena in simple
systems can be generalized to more complex forms of matter, in particular to liquid crystals
and polymers”.
Liquidcrystalsrefertoastateofmatterthathaspropertiesbetweenthoseofasolidcrystal
and those of an isotropic liquid. For example, a liquid crystalmay flowlike an ordinary liquid,
but its molecules may be orientedin some directions. Examples of liquid crystalscan be found
both in nature (e.g. solutions of soap and detergents) and in technological applications (e.g.
electronic displays). There are many different types of liquid crystal phases, among which one
ofthe mostcommonphasesis the nematic. Ina nematic phase,the rod-shapedmoleculeshave
no positional order but they display long-range directional order with their long axes roughly
aligned parallel to a common axis (which is called a “director”). Most nematics are uniaxial,
meaning that they have one axis that is longer and preferred while the other two axes are
equivalent. However, some nematics are biaxial which implies that the molecules also orient
along a secondary axis in addition to orienting their long axis. Nematics can be easily aligned
by applying an external magnetic or electric field. The optical properties of aligned nematics
make them extremely useful in liquid crystal displays.
Liquid crystallinity in polymers may occur either by heating a polymer above its melting
point (thermotropic LCPs) or by dissolving a polymer in a solvent (lyotropic LCPs). Liquid
crystalline polymers combine the properties of liquid crystals with the properites of polymers.
As a result, it is easy to guide LCPs by applying small external forces such as flows and it is
easyto controltheir chemicaland materialparameters. LCPs exhibit certainanisotropyofthe
electrical, mechanical and magnetic properties with high flexibility or elasticity. They are also
exceptionallyunreactiveandinert,andhighly resistantto fire. Due to theirvariousproperties,
LCPs have found more and more applications, including electrical and mechanical parts, food
containers,andanyotherapplicationsrequiringchemicalinertnessandsuperstrength. Amain
example of LCPs in solid form is high strength fibers which have been made into bullet-proof
vests and airbags.
The theoretical studies of liquid crystals traced back more than 60 years ago. In 1949
Onsager [29] developed a statistical theory showing that simple exclude-volume repulsions be-
tween long rods are sufficient to create a liquid crystal. Onsager’s theory is valid for dilute
liquid crystals. Flory [9, 10] presented a lattice model and his theory yields good approxima-
tion for dense and highly ordered liquid crystals. Both the Onsager and Flory theories predict
an isotropic-nematic phase transition when the concentration is high enough. In three pub-
lications Maier and Saupe [26–28] proposed a mean-field theory to explain the occurrence of
a liquid crystal phase in low molecular weight materials based on the assumption that only
induced dipolar forces are relevant for orientational ordering. An impressive signature for the
popularity of the Maier-Saupe theory is its high number of citations (almost 4000 up to now)
overtheyears[30]. In1965LandauandGinzburg[22]developedaphenomenologicaltheoryfor
liquid crystals. Eventhoughit has success inmost cases,Landau-Ginzburgtheory is not exact
anditsuffersfailureinsomecases[18]. Oneoftheearliestconstitutivetheoriesofliquidcrystals
isthe Leslie-Ericksenvectortheory[7,23]in1970s. TheLeslie-Ericksentheoryisattractivefor
studying low molar-mass liquid crystals.
AfterMaierandSaupeandFlory’swork,manynewtheorieshavebeendevelopedforliquid
No.6 H.Zhou&H.Y.Wang: STABILITYOFEQUILIBRIAOFNEMATICLIQUID 2291
crystalline polymers. One of the most famous kinetic models for polymer liquid crystals was
developedbyDoiandEdwards[6]in1980sandHess[19]. ThecrucialideaoftheDoitheoryisto
modelthepolymericmoleculesasasuspensionofrigidrodlikenematogensandthendescribethe
ensemblewithanorientationalprobabilitydistributionfunction. Theorientationaldistribution
functionisgovernedbyanonlinearSmoluchowski(orFokker-Planck)equationwhichtakesinto
account the hydrodynamic, Brownian and intermolecular forces.
In recent years, the Doi theory has attracted enormous mathematical attentions (for ex-
ample, [2–4, 8, 11–17, 21, 24, 25, 32–38]). One mathematical advance in the understanding
of the equilibrium states of the Doi model is the rigorous proof on that the equilibria of the
Smoluchowski equation with the Maier-Saupe potential are uniaxial [8, 24, 35]. However, it
seems more challenging to obtain a solid proof on the stability of the equilibria. The purpose
of this paper is to circumvent this difficulty and seek a mathematical proof on the stability of
theequilibria. Themainresultprovenhereis: allstableequilibriaareeitherisotropicorhighly
aligned prolate uniaxial.
2 Background and Formulation
For reader’s convenience, we start with the briefest glimpse of the Doi kinetic theory.
Let us denote the orientational direction of each rodlike molecule by a unit vector m,
the probability density function by ρ(m). The evolution of the probability density function is
described by the Smoluchowski equation [6]:
(cid:2) (cid:3)
∂ρ ∂ 1 ∂U ∂ρ
=D · ρ+ , (1)
∂t ∂m kBT ∂m ∂m
where ∂/∂m is the orientational gradient operator [1], D the rotational diffusivity, kB the
Boltzmann constant, and T the absolute temperature. For simplicity, we set kBT =1 which is
equivalent to normalizing U by kBT.
For nematic polymers, the total potential consists of only the Maier-Saupe interaction
potential
U(m)=−b(cid:3)mm(cid:4):mm, (2)
where the tensor product mm and tensor double contraction A:B are defined as
⎡ ⎤
m m m m m m
1 1 1 2 1 3
⎢ ⎥
mm≡ ⎢⎣m2m1 m2m2 m2m3⎥⎦,
m m m m m m
3 1 3 2 3 3
A:B ≡ trace(AB).
In (2) the constant b is proportional to the normalized polymer concentration and it describes
the strength of inter-molecular interactions, and (cid:3)mm(cid:4) denotes the second moment of the
orientation distribution: (cid:10)
(cid:3)mm(cid:4)≡ mmρ(m)dm,
(cid:2)m(cid:2)=1
where ρ(m) is the orientational probability density function of the ensemble of polymer rods,
i.e., the probability density that a polymer rod has direction m.
2292 ACTAMATHEMATICASCIENTIA Vol.31Ser.B
The Maier-Saupe potential in a properly selected coordinate system:
Let us select a coordinate system such that the second moment (cid:3)mm(cid:4) is diagonal:
⎡ ⎤
1
⎡(cid:3)m2(cid:4) 0 0 ⎤ ⎢⎢s1+ 3 0 0 ⎥⎥
⎢ 1 ⎥ ⎢ ⎥
(cid:3)mm(cid:4)=⎢⎣ 0 (cid:3)m22(cid:4) 0 ⎥⎦=⎢⎢⎢ 0 s2+ 13 0 ⎥⎥⎥,
0 0 (cid:3)m23(cid:4) ⎣ 1⎦
0 0 s3+
3
where sj ≡(cid:3)m2j(cid:4)− 13 satisfies
s1+s2+s3 =(cid:3)m21+m22+m23(cid:4)−1=0.
In this coordinate system, the Maier-Saupe potential takes the following expression:
U(m)= −b((cid:3)m2(cid:4)m2+(cid:3)m2(cid:4)m2+(cid:3)m2(cid:4)m2)
1 1 2 2 3 3
b
= − −b(s1m21+s2m22+s3m23). (3)
3
The constant (−b/3) does not affect the dynamics or the stability of the nematic polymer
ensemble. Forsimplicity,wedroptheconstant(−b/3). Froms1+s2+s3 =0andm21+m22+m23 =
1, we have
−s s −s
s1 = 3 − 2 1,
2 2
−s s −s
s2 = 3 + 2 1,
2 2
m2+m2 =1−m2.
1 2 3
Using these equations, we can rewrite the Maier-Saupe potential (3) as
(cid:2)(cid:2) (cid:3) (cid:2) (cid:3) (cid:3)
−s s −s −s s −s
U(m)=−b 3 − 2 1 m21+ 3 + 2 1 m22+s3m23
2 2 2 2
(cid:2) (cid:3)
=−b3s3 m2− 1 −b(s2−s1)(m2−m2)
3 2 1
2 3 2
(cid:2) (cid:3)
1
=−η1 m23− −η2(m22−m21), (4)
3
where
(cid:2) (cid:3)
η1 ≡b3s3 =b3 (cid:3)m23(cid:4)− 1 , (5)
2 2 3
η2 ≡b(s2−s1) =b1(cid:3)m22−m21(cid:4). (6)
2 2
Note that for any orientationaldistribution ρ(m), by selecting a proper coordinate system, we
canalwaysexpress the Maier-Saupe potential U(m) in the form of (4) with η1 and η2 givenby
(5) and (6). In particular, ρ(m) does not have to be an equilibrium distribution.
The Boltzmann distribution:
No.6 H.Zhou&H.Y.Wang: STABILITYOFEQUILIBRIAOFNEMATICLIQUID 2293
Intheabsenceofexternalfield,theequilibriumdistributionsofanematicpolymerensemble
are described by the Boltzmann distribution [6]:
exp[−U(m)]
ρ(BZ)(m;η1,η2)= (cid:11) exp[−U(m)]dm
S (cid:12) (cid:13)
= (cid:11) exp η1m23+η2(m22−m21) , (7)
Sexp[η1m23+η2(m22−m21)]dm
where S represents the unit sphere. It is clear that an equilibrium distribution is completely
specified by (η1,η2). To find an equilibrium distribution, we need to determine (η1,η2). The
governingequationfor(η1,η2)isobtainedbycombiningthedefinitionof(η1,η2)andtheBoltz-
mann distribution (7):
(cid:10) (cid:2) (cid:3)
3 1
η1 =b m23− ρ(BZ)(m;η1,η2)dm,
2 3
S
(cid:10)
(cid:14) (cid:15)
1
η2 =b m22−m21 ρ(BZ)(m;η1,η2)dm. (8)
2
S
Note thatwhile the Maier-Saupepotential(4)isalwaystrue,the Boltzmanndistribution(7)is
valid only for equilibrium distributions. It is very important to notice this difference between
(4) and (7) especially when we consider the free energy of orientational distributions in the
stability discussion.
Equilibrium states of a nematic polymer ensemble:
It has been shown that in the absence of external field, all equilibrium states of nematic
polymersareaxisymmetric[8,24,35]. Withoutlossofgenerality,we assumethe m3-axisis the
axis of symmetry. The axisymmetry implies s1 = s2 and consequently we have η2 = 0. Thus,
an equilibrium state of nematic polymers is completely specified by η1.
Tofacilitatethediscussion,weuser(b)todenotethevalueofη1 intheequilibriumstate(s)
corresponding to parameter b. More specifically, we adopt the convention that η, η1 and η2
denote general variables while r(b) denotes specific value of η1 in the equilibrium state(s).
By definition r(b) is the solution of equation (8). With η2 = 0 and η1 simply denoted by
η, equation (8) becomes
(cid:11) (cid:14) (cid:15) (cid:14) (cid:15)
η =b3 S m(cid:11)23− 13 exp ηm23 dm. (9)
2 exp(ηm2)dm
S 3
It is now convenient to introduce the function f(η) defined by
(cid:11) (cid:14) (cid:15) (cid:14) (cid:15)
f(η)≡ 3 S m(cid:11)23− 13 exp ηm23 dm. (10)
2η exp(ηm2)dm
S 3
Then it follows at once that equation (9) can be written as [35]
(cid:2) (cid:3)
1
η −f(η) =0. (11)
b
First, observe that η = r0(b) = 0 is always a solution of equation (11), which corresponds to
U(m)= constant(independent of m), the isotropic equilibrium. Next, anisotropic equilibrium
states are the solutions of the equation
1
f(η)= . (12)
b
2294 ACTAMATHEMATICASCIENTIA Vol.31Ser.B
Using the spherical coordinates,
m1 =sinφcosθ,
m2 =sinφsinθ,
m3 =cosφ, (13)
followedbythesubstitutionu=cosφandthenapplyingintegrationbyparts,wewritefunction
f(η) as
(cid:11) (cid:14) (cid:15) (cid:14) (cid:15)
3 π cos2φ− 1 exp η cos2φ sinφdφ
f(η)= 0 (cid:11) 3
2η πexp(η cos2φ)sinφdφ
(cid:11) (cid:14) 0 (cid:15) (cid:14) (cid:15)
3 1 u2− 1 exp ηu2 du
= −1 (cid:11) 3
2η 1 exp(ηu2)du
−1
(cid:11) (cid:14) (cid:15) (cid:14) (cid:15)
1 u2−u4 exp ηu2 du
= −1 (cid:11)
1 exp(ηu2)du
(cid:11) (cid:14) −1 (cid:15) (cid:14) (cid:15)
m2−m4 exp ηm2 dm
= S 3(cid:11) 3 3 . (14)
exp(ηm2)dm
S 3
In[35],wehaveshownthatthefunctionf(η)expressedin(14)satisfiesthefollowingproperties:
1. f(0)= 2 ;
15
2. lim f(η)=0 and lim f(η)=0; in other words, f(η) tends to zero as |η| goes to
η→+∞ η→−∞
infinity.
3. f(η) attains its maximum at η∗ = 2.1782879748 > 0 where the maximum is f(η∗) =
0.14855559992254>0;
4. f(cid:6)(η)>0 for η <η∗ and f(cid:6)(η)<0 for η >η∗.
The graph of the function f(η) is depicted in Figure 1.
Fig. 1 Graph of thefunction f(η). A keyfeature of thefunction f(η) is thatf (η) is strictly
increasing for η<η∗ (i.e. f(cid:3)(η)>0) and strictly decreasing for η>η∗ (i.e., f(cid:3)(η)<0)
From these properties of the function f(η), we draw conclusions listed below related to
equation (12):
• b∗ = 1 =6.7314863965is a critical value for parameter b.
f(η∗)
No.6 H.Zhou&H.Y.Wang: STABILITYOFEQUILIBRIAOFNEMATICLIQUID 2295
• Forb<b∗,equation(12)hasnosolution;thatis,thereisnoanisotropicstateforb<b∗.
• At b=b∗, equation (12) has one solution r(b∗)=η∗ >0.
• For b∗ <b< 125, equation (12) attains two solutions: rU(b)> η∗ >0 and 0< rM(b)<
η∗. The equilibrium states corresponding to positive order parameters rU(b) and rM(b) are
called prolate states.
• At b= 125, equation (12) possesses two solutions: rU(125)>η∗ >0 and r(125)=0.
• For b > 125, equation (12) has two solutions: rU(b) > η∗ > 0 and rL(b) < 0. The
equilibrium state with negative order parameter rL(b) is called oblate state.
Hereweuse the subscript“U”toreferto the “Upper”partofthe phasediagramwherer >η∗,
the subscript “M” to represent the “Middle” part of the phase diagram where 0<r <η∗, and
the subscript “L” to denote the “Lower” part of the phase diagram where r < 0. The phase
diagram for nematic polymers is shown in Figure 2.
Fig. 2 Phase diagram of nematic polymers
In the above, we have used b as the independent variable and treated r as a function of
b. However, r(b) is a multi-valued function of b and for b < b∗ the function r(b) is not even
defined. When we study the relation between b and r, it is mathematically more convenient
if we use r as the independent variable and treat b as a function of r instead. Then, b(r) is a
single-valued function of r and is defined for all values of r in (−∞,+∞). The function b(r)
can be easily determined from equation (12) as
1
b(r)= . (15)
f(r)
Once we know the function b(r), the branch rU(b) is simply the inverse function of b(r) for
r >η∗; the branchrM(b) is the inverse function of b(r) for 0<r <η∗; and the branchrL(b) is
the inverse function of b(r) for r<0.
Free energy of an orientational distribution:
The free energy of the orientational distribution ρ(m) is
(cid:10) (cid:16) (cid:17)
1
G([ρ])= logρ(m)+ U(m,[ρ]) ρ(m)dm. (16)
2
S
Recall that for any ρ(m), by selecting a proper coordinate system, we can always write the
2296 ACTAMATHEMATICASCIENTIA Vol.31Ser.B
Maier-Saupe potential U(m,[ρ]) as
(cid:2) (cid:3)
1
U(m,[ρ])=−η1 m23− −η2(m22−m21),
3
(cid:14) (cid:15)
where η1 ≡b23 (cid:3)m23(cid:4)− 13 and η2 ≡b12(cid:3)m22−m21(cid:4). But in general, ρ(m) is not the same as the
Boltzmann distribution
(cid:12) (cid:13)
ρ(m)(cid:7)=ρ(BZ)(m;η1,η2)≡ (cid:11)Seexxpp[ηη11mm2323++ηη22((mm2222−−mm2121))]dm.
It is important to point out that even if we restrict our consideration to probability density
ρ(m) that has the form of
(cid:12) (cid:13)
ρ(m)= (cid:11) exp q1m23+q2(m22−m21) ,
Sexp[q1m23+q2(m22−m21)]dm
we still have (q1,q2)(cid:7)=(η1,η2) unless ρ(m) is an equilibrium distribution, which means q2 =0
and q1 =r(b). It is tempting to replace ρ(m) by ρ(BZ)(m;η1,η2) in (16) and formally define a
function of (η1,η2)
(cid:10) (cid:16) (cid:2) (cid:2) (cid:3) (cid:3)(cid:17)
1 1
F(η1,η2)≡ logρ(BZ)(m;η1,η2)+ −η1 m23− −η2(m22−m21)
2 3
S
×ρ(BZ)(m;η1,η2)dm.
Unfortunately, the assertion F(η1,η2) = G([ρ]) is valid only when ρ(m) is an equilibrium
distribution. In other words, at a fixed value of b, the assertion F(η1,η2) = G([ρ]) is true
only at a few (at most, three) equilibrium states that are well separated from each other.
Therefore, while we can try to analyze the stability of F(η1,η2) with respect to perturbations
in(η1,η2),thestabilityofF(η1,η2)doesnotinformusaboutthestabilityofG([ρ])withrespect
to perturbations in ρ(m).
3 Stability Analysis
In this section we present a detailed stability analysis on the equilibria.
Tomakeourpresentationeasytofollow,wedivideourapproachintofivesteps. Inthefirst
step, we minimize the free energy functional G([ρ]) over a set B of functions with the second
moments fixed, specified by two parameters (η1,η2). The resulting free energy is a function of
twovariablesG(η1,η2). The stability ofG([ρ])is equivalentto the stabilityofG(η1,η2). Inthe
second step, we show that the minimum of G([ρ]) is attained at a probability density ρ∗ of the
Boltzmann form. The coefficients (q1,q2) in the exponent of ρ∗ define a one-to-one mapping
between(η1,η2)and(q1,q2). Thus,we canrewriteG(η1,η2)asH(q1,q2). The importantpoint
to note here is that (q1,q2) is in general different from (η1,η2) unless ρ∗ is an equilibrium
state, which is animproper assumption in stability analysis. In the third step, we compute the
Hessian matrix of H(q1,q2) with respect to (q1,q2), which is fortunately a diagonal matrix. In
the fourth step, we consider the stability of the isotropic equilibrium state. At the laststep we
focus our attention on the stability of the anisotropic equilibrium states.
No.6 H.Zhou&H.Y.Wang: STABILITYOFEQUILIBRIAOFNEMATICLIQUID 2297
Step 1 We consider the constrained minimum over the set
(cid:18) (cid:19)(cid:2) (cid:3) (cid:22)
(cid:19) (cid:20) (cid:21)
B(η1,η2)= ρ(m)(cid:19)(cid:19) (cid:3)m23(cid:4)[ρ]− 13 = 32bη1, m22−m21 [ρ] = 2bη2 ,
where the probability density ρ(m) is involved in the average in the following way:
(cid:10)
(cid:3)f(m)(cid:4)[ρ] ≡ f(m)ρ(m)dm.
S
The minimum of G([ρ]) over the set B(η1,η2) is a function of (η1,η2):
G(η1,η2)≡ min G([ρ]). (17)
ρ(m)∈B(η1,η2)
FromthedefinitionofG(η1,η2),weseethatthestabilityofG([ρ])withrespecttoρisthesame
as the stability of G(η1,η2) with respect to (η1,η2).
Step 2 We study where G([ρ]) attains the minimum over the set B(η1,η2) and use the
result to calculate G(η1,η2). For ρ(m)∈B(η1,η2), we have
(cid:2) (cid:3)
1
U(m,[ρ])= −η1 m23− −η2(m22−m21),
3
(cid:10) (cid:23) (cid:24) (cid:25) (cid:26)
1 1 1
U(m,[ρ])ρ(m)dm= −η1 m23− −η2(cid:3)m22−m21(cid:4)[ρ]
2 2 3
S [ρ]
(cid:2) (cid:3)
1 1
= − η2+ η2 .
3b 1 b 2
That is, the second term in the free energy G([ρ]) given in (16) is actually a constant over the
set B(η1,η2). It follows that we only need to look at the first term of G([ρ]) in the constrained
minimization problem. Let us denote the first term by
(cid:10)
G1([ρ])≡ logρ(m)ρ(m)dm. (18)
S
The constrained minimization problem then becomes
argmin G1([ρ]).
ρ(m)∈B(η1,η2)
Supposetheconstrainedminimumisattainedatρ∗(m). Weconsidertheperturbedprobability
density ρ(m)=ρ∗(m)+sΔρ(m) where Δρ(m) satisfies the following properties:
(cid:10)
Δρ(m)dm=0,
(cid:10)S(cid:2) (cid:3)
1
m2− Δρ(m)dm=0,
3
3
(cid:10)S
(m2−m2)Δρ(m)dm=0. (19)
2 1
S
(cid:19)
G1([ρ∗+sΔρ]) attaining the minimum at s=0 implies dG1([ρd∗+ssΔρ])(cid:19)(cid:19) =0, which gives us
s=0
(cid:10)
logρ∗(m)Δρ(m)dm=0. (20)
S