The purpose of this paper is to study the extendability of equilibrium states of
rodlike nematic polymers with the Maier-Saupe intermolecular potential. We formulate equilibrium states as solutions of a nonlinear system and calculate the determinant of the Jacobian matrix of the nonlinear system. It is found that the Jacobian matrix is nonsingular everywhere except at two equilibrium states. These two special equilibrium states correspond to two points in the phase diagram. One point is the folding point where the stable prolate branch folds into the unstable prolate branch; the other point is the intersection point of the nematic branch and the isotropic branch where the unstable prolate state becomes the unstable oblate state. Our result establishes the existence and uniqueness of equilibrium states in the presence of small perturbations away from these two special equilibrium states.

1. Introduction

Liquid crystal
polymers (LCPs) consisting of rigid rodlike macromolecules in a viscous solvent
have wide technological applications [1–4]. The most common theoretical framework for modeling
nematic polymers is to represent the nematogenic molecules as rigid rods and
to describe the ensemble with an orientational probability density function (pdf)
[5]. Equilibrium
orientational distribution is related to the interaction potential by the
Boltzmann relation. Recently, rigorous mathematical analysis on the equilibrium
states of nematic polymers has garnered serious attention from mathematicians
led by Constantin and Titi [6–22]. For example, various proofs for the axisymmetry of
equilibrium states with only the Maier-Saupe interaction were given in
[8, 19] for the 2-D case and in
[9, 18, 20] for the 3-D case; the
equilibrium states for the case where dipole-dipole interaction is coupled to
the Maier-Saupe interaction were studied in [17, 21]. Not surprisingly, many mathematical issues of
polymers are still unexplored. In particular, to our knowledge, the
extendability of the equilibrium states has never been attempted.

In our previous study [21], it was revealed that for
large values of the nematic strength and small perturbations, there exists at
least one solution near the corresponding unperturbed pure nematic solution. In
[21], we concluded the
existence using complicated free energy arguments.
Moreover, in [21] only
existence was established whereas the uniqueness was not addressed at all. Our
goal in the current study is to establish rigorously the extendability, both
existence and uniqueness, of the equilibrium states of nematic polymers in the
presence of small perturbations. Notice that all equilibrium states of nematic
polymers are axisymmetric [9, 18, 20]. To facilitate the discussion of extendability, we
introduce two new variables for the nonlinear system governing the equilibrium
states: one variable is proportional to the order parameter of the pdf; the
other variable is proportional to the biaxial order parameter, which measures
the deviation of the pdf from being axisymmetric. One advantage of this
approach is that the Jacobian matrix is diagonal at equilibrium states. We will
show that the Jacobian determinant of the nonlinear system is nonzero
everywhere except at two equilibrium states. These two equilibrium states
correspond to two points in the phase diagram: one point is the folding point
where the stable prolate branch folds into the unstable prolate branch; the
other point is the intersection point of the nematic states and the isotropic
states where the unstable prolate state becomes the unstable oblate state. The
extendability of the equilibrium states except these two states follows
immediately from the implicit function theorem.

2. Nonlinear System for Equilibrium States, Jacobian Matrix, and Its Determinant

We now briefly
recall the mathematical description for equilibrium states of rigid rodlike
nematic polymers. The orientation direction of each polymer rod is denoted by a
unit vector m.
In this study,
by “pure nematic polymers,” we mean the case where the Maier-Saupe
interaction potential is the only potential. The Maier-Saupe potential is given
by
U(m)=−b〈mm〉:mm,
where the tensor product mm and the tensor double contraction A:B are defined asmm≡[m1m1m1m2m1m3m2m1m2m2m2m3m3m1m3m2m3m3],A:B≡trace(AB). In (2.1), b=3N/2,N is the normalized polymer concentration
describing the strength of intermolecular interactions, and 〈mm〉 is the second moment of the orientation
distribution:〈mm〉≡∫∥m∥=1mmρ(m)dm,where ρ(m) is the orientational probability density
function (pdf) of the ensemble of polymer rods. For convenience, potential (2.1)
has been normalized with respect to kBT, where kB is the Boltzmann constant and T the absolute temperature. For pure nematic
polymers, equilibrium states are described by the Boltzmann distribution
[5]:ρ(m)=1Zexp[−U(m)],where Z=∫Sexp[−U(m)]dm is the partition function and S is the unit sphere.

We choose the coordinate system such that the second
moment 〈mm〉 is diagonal:〈mm〉=[〈m12〉000〈m22〉000〈m32〉].As a consequence, the
Maier-Saupe potential can be written asU(m)=−b(〈m12〉m12+〈m22〉m22+〈m32〉m32).The most significant conclusion
for pure nematic polymers is that all equilibrium states are axisymmetric
[9, 18, 20]. Since not all equilibrium
states of a perturbed nematic polymer ensemble are necessarily axisymmetric, to
study the extendability, we formulate the problem without using the axisymmetry
so that non-axisymmetric perturbations are allowed. The axisymmetry will be
useful in our analysis because the extendability is determined by the Jacobian
determinant evaluated at the unperturbed equilibrium state, which is
axisymmetric. The definition and derivation of the Jacobian matrix, however,
require non-axisymmetric formulation. For pure nematic polymers, the total
potential is completely specified by the second moment 〈mm〉.
As a result of the Boltzmann relation (2.4), an equilibrium state is completely
specified by the second moment 〈mm〉.
Because of the constraint m12+m22+m32=1,
an equilibrium state is completely specified bys1≡〈m12〉,s2≡〈m22〉.At a pure nematic equilibrium
state without perturbation, we have s1=s2 or s1=1−s1−s2 (which means s1=s3) or s2=1−s1−s2 (which means s2=s3). At an equilibrium state, we select the
coordinate system such that the equilibrium satisfies s1=s2.
In terms of (s1,s2),
the Maier-Saupe potential can be expressed asU(m)=−b(s1m12+s2m22+(1−s1−s2)m32)=−b[12(s2−s1)(m22−m12)+(1−32(s1+s2))m32]+const.Consequently, the equilibrium
pdf is given byρ(m)=1Zexp{b[12(s2−s1)(m22−m12)+(1−32(s1+s2))m32]}.At an
equilibrium state, (s1,s2) satisfies the nonlinear
system:s1−〈m12〉=0,s2−〈m22〉=0.The form of the equilibrium pdf
(2.9) motivates us to introduce (η1,η2):η1≡b[1−32(s1+s2)],η2≡b(s2−s1)2.Here η1 is proportional to the order parameter of the
pdf while η2 is proportional to the biaxial order
parameter, which measures the deviation of the pdf from being axisymmetric. As
we will see, one advantage of using (η1,η2) is that the Jacobian matrix is diagonal at
equilibrium states. In terms of (η1,η2),
the pdf has the expressionρ(m,η1,η2)=exp[η2(m22−m12)+η1m32]∫Sexp[η2(m22−m12)+η1m32]dm.The second advantage of using (η1,η2) is that the pdf does not depend on b explicitly. Later on, this property will
enable us to write b as a function of r which is the equilibrium value of η1.
At an equilibrium state, the nonlinear system for (η1,η2) isF1(η1,η2;b)≡η1b−12(3〈m32〉−1)=0,F2(η1,η2;b)≡η2b−12〈m22−m12〉=0.In this paper, we study the
extendability of equilibrium states of nematic polymers in the presence of
small perturbations. We consider a perturbed version of system
(2.13):F1(η1,η2;b)=ϵ1,F2(η1,η2;b)=ϵ2.Mathematically, we study the
existence and uniqueness of solution of system (2.14) near a solution of system
(2.13) for small ϵ1 and ϵ2.
According to the implicit function theorem, the existence and uniqueness are
determined by the Jacobian determinant of (F1(η1,η2),F2(η1,η2)).
To calculate the Jacobian matrix, we first calculate the derivatives of pdf
(2.12):∂∂η1ρ(m,η1,η2)=(m32−〈m32〉)ρ(m,η1,η2),∂∂η2ρ(m,η1,η2)=(m22−m12−〈m22−m12〉)ρ(m,η1,η2).Recall that at a pure nematic
equilibrium state without perturbation, we have selected our coordinate system
such that η2=0.
Evaluating the partial derivatives of F1 and F2 at the equilibrium (η1=r,η2=0) yields∂∂η1F1(η1,η2;b)|η1=r,η2=0=1b−32〈m32(m32−〈m32〉)〉=1b−32var(m32),∂∂η2F1(η1,η2;b)|η1=r,η2=0=−32〈m32(m22−m12−〈m22−m12〉)〉=0,∂∂η1F2(η1,η2;b)|η1=r,η2=0=−12〈(m22−m12)(m32−〈m32〉)〉=0,∂∂η2F2(η1,η2;b)|η1=r,η2=0=1b−12〈(m22−m12)(m22−m12−〈m22−m12〉)〉,=1b−12〈(m22−m12)2〉.Here we point out that the
Jacobian matrix of system (2.13) evaluated at an equilibrium state is diagonal,
which is caused by the axisymmetry of the equilibrium state.

It follows that the determinant of Jacobian matrix of
system (2.13) at the equilibrium isdet(∂(F1,F2)∂(η1,η2))|η1=r,η2=0=(1b−32var(m32))⋅(1b−12〈(m22−m12)2〉).In (2.17), all averages are
evaluated using the equilibrium pdfρ(m,η1,η2)|η1=r,η2=0=1Zexp(rm32).For pure nematic polymers, an
equilibrium state is completely specified by r.
The governing equation for r is the first equation of (2.13): F1(r,0;b)=0,
whereas the second equation of (2.13) is satisfied automatically when η2=0.
We introduce a new function:F(η;b)≡F1(η,0;b)=ηb−12(3〈m32〉−1).The governing equation for r is F(r;b)=0.
It follows directly from the definition that function F(η;b) is related to the (1,1) element of the Jacobian matrix at equilibrium
as∂∂η1F1(η1,η2;b)|η1=r,η2=0=∂∂ηF(η;b)|η=r.As we will find later, this
relation is a key tool for determining the sign of the (1,1) element of the Jacobian matrix at equilibrium.
In the above, we have used variables η and r more or less interchangibly. Later on when
necessary, we will use η to denote the independent variable of
functions and use r to denote the particular value of η that satisfies F(r;b)=0.
For that purpose, we will continue using these two variables.

3. Extendability of Equilibria of Nematic Polymers

To obtain more
specific properties of equilibrium states of nematic polymers, we rewrite the
governing equation for r using spherical coordinates. We select the m3-axis as the pole of the spherical coordinate
system. The equilibrium pdf given in (2.18) becomesρ(ϕ,θ;r)=exp(rcos2ϕ)2π∫0πexp(rcos2ϕ)sinϕdϕ,where ϕ is the polar angle and θ the azimuthal angle. Here we need to point out
that r is not the radius in the spherical coordinate
system. It is a parameter of the equilibrium state. Substituting equilibrium
pdf (3.1) into (1/2)(3〈m32〉−1),
using a change of variable u=cosϕ,
and integrating by parts, we have12(3〈m32〉−1)=r⋅∫01(u2−u4)exp(ru2)du∫01exp(ru2)du.Thus, F(η;b) defined in (2.19) becomes F(η;b)=η[1/b−f(η)],
where f(η) is defined asf(η)≡∫01u2(1−u2)exp(ηu2)du∫01exp(ηu2)du.The equation for r, F(r;b)=0,
becomesr[1b−f(r)]=0.Equation (3.4) has a trivial
solution r=0 for arbitrary value of b,
which corresponds to the isotropic branch of the nematic polymer phase diagram.
At r=0,
the equilibrium pdf is uniform, ρ(ϕ,θ;0)=1/4π.
It follows that〈mj2〉=13,〈mj4〉=〈m34〉=∫01u4du=15,〈mi2mj2〉=〈m12m32〉=12∫01u2(1−u2)du=115,i≠j,var(m32)=〈m34〉−〈m32〉2=15−19=445,〈(m22−m12)2〉=〈m24〉−2〈m12m22〉+〈m14〉=25−215=415.Substituting these results into
(2.16), we obtain∂∂η1F1(η1,η2;b)|η1=0,η2=0=1b−32var(m32)=1b−215,∂∂η2F2(η1,η2;b)|η1=0,η2=0=1b−12〈(m22−m12)2〉=1b−215.Therefore, when b≠15/2,
the Jacobian matrix is nonsingular and thereby the isotropic equilibrium is
extendable. At b=15/2,
the isotropic branch intersects with the nematic branch.

The remaining
solutions of (3.4), if any, satisfy1b−f(r)=0.
In [20], it has been shown that the
function f(η) has the following properties:

f(0)=2/15, limη→+∞f(η)=0 and limη→−∞f(η)=0;

f(η) attains its maximum at η=η*=2.1782879748>0, where the maximum is f(η*)=0.14855559992254>0;

f′(η)>0 for η<η* and f′(η)<0 for η>η*.

A graph of f(η) is shown in Figure 1(a). Thus, we draw
conclusions below for (3.7) as follows.

The critical value of b is b*=1/f(η*)=6.7314863965.

For b<b*,
(3.7) has no solution.

At b=b*,
(3.7) has one solution r(b*)=η*>0.

For b*<b<2/15,
(3.7) has two solutions: rU(b)>η*>0 and 0<rM(b)<η*.

At b=2/15,
(3.7) has two solutions: rU(2/15)>η*>0 and r(2/15)=0.

For b>2/15,
(3.7) has two solutions: rU(b)>η*>0 and rL(b)<0.

As mentioned before, we use subscript “U” to denote to the
“Upper” part of the phase diagram where r>η*,
subscript “M” to denote to the “Middle” part of the phase diagram, where 0<r<η*,
and subscript “L” to denote to the “Lower” part of the phase diagram, where r<0.
The phase diagram for nematic polymers is shown in Figure 1(b). In the
terminologies of nematic polymers, curve segment rU(b) is the stable part of the prolate branch; rM(b) is the unstable part of the prolate branch;
and rL(b) is the unstable oblate branch.

Recall that b is the strength of the Maier-Saupe
interaction, which is proportional to the normalized polymer concentration and
is inversely proportional to the temperature. r,
the solution of F(r;b)=0,
has the expression r=b⋅(1/2)(3〈m32〉−1), where (1/2)(3〈m32〉−1) is the order parameter. Thus, from physical
considerations, it is desirable to use b as the independent variable and treat r as a function of b.
However, r(b) is a multivalue function of b and for b<b* function r(b) is not even defined. Mathematically, it is
much more convenient if we use r as the independent variable and treat b as a function of r. b(r) is a single-value function of r and is defined for all values of r in (−∞,+∞).
Below, we will adopt this new formulation of viewing b as a function of r.
In the new formulation, function b(r) is determined from (3.7) as b(r)=1/f(r).
In terms of this new formulation, the stable part of the prolate branch rU(b) can be simply represented as b(r) for r>η*;
the unstable part of the prolate branch rM(b) as b(r) for 0<r<η*;
and the unstable oblate branch rL(b) as b(r) for r<0.
Now we discuss the extendability of these branches. Using relation (2.20), F(η;b)=η[1/b−f(η)],
and 1/b−f(r)=0,
we arrive at∂∂η1F1(η1,η2;b(r))|η1=r,η2=0=∂∂ηF(η;b(r))|η=r=[1b(r)−f(η)]|η=r−ηf′(η)|η=r=−rf′(r)={>0,forr>η*<0,for0<r<η*>0,forr<0.To study the (2,2) element of Jacobian matrix, we first rewrite (1/2)〈(m22−m12)2〉 by substituting equilibrium pdf (3.1) into (1/2)〈(m22−m12)2〉 and using substitution u=cosϕ,12〈(m22−m12)2〉=14∫01(1−u2)2exp(ru2)du∫01exp(ru2)du.Using (2.16), b(1)=1/f(r),
the expression of f(r) given in (3.3), and result (3.9), we
have∂∂η2F2(η1,η2;b(r))|η1=r,η2=0=1b(r)−12〈(m22−m12)2〉=14∫01(1−u2)(5u2−1)exp(ru2)du∫01exp(ru2)du≡g(r).It is straightforward to verify
that g(0)=0.
Below, we want to show that g(r)>0 for r>0 and g(r)<0 for r<0.
To facilitate the analysis below, we write g(r) as an averageg(r)=14〈(1−u2)(5u2−1)〉,where the average is with
respect to the pdf ρ(u;r)=exp(ru2)/∫01exp(ru2)du.

Lemma 3.1.

Function g(r) given in (3.11) has the property that g(r)=0 implies g′(r)>0.

Proof.

We first calculate the derivative of the pdf: (∂/∂r)ρ(u;r)=(u2−〈u2〉)ρ(u;r), which leads to
g′(r)=14〈(1−u2)(5u2−1)(u2−〈u2〉)〉=120〈(1−u2)(5u2−1)2〉+15(1−5〈u2〉)g(r).
Thus, whenever g(r)=0, we have g′(r)=(1/20)〈(1−u2)(5u2−1)2〉>0.

Lemma 3.1
together with g(0)=0 leads to g(r)>0 for r>0 and g(r)<0 for r<0.
Combining this result on g(r) with result (3.8) on −rf′(r),
we arrive atdet(∂(F1,F2)∂(η1,η2))|η1=r,η2=0=(−rf′(r))g(r)={>0,forr>η*,<0,for0<r<η*,<0,forr<0.Therefore, we conclude that all
nematic equilibrium states (stable or unstable) are extendable except for the
equilibrium state at (r=0,b=15/2) and the equilibrium state at (r=η*,b=b*).

4. Conclusions

In this work,
we studied the extendability of equilibrium states of nematic polymers with the
Maier-Saupe intermolecular potential. We found that the Jacobian matrix of the
nonlinear system is nonsingular except at two special equilibrium states. The
significance of this result is its implication on the existence and uniqueness
of equilibrium states of a perturbed system, in the neighborhood of the
unperturbed equilibrium states.

(a) Graph of function f(η); (b) phase diagram of nematic polymers.

Acknowledgments

H. Wang was partially supported by the National
Science Foundation. H. Zhou was supported in part by the Air Force Office of
Scientific Research under Grant no. F1ATA06313G003.

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