Transmittance and Reflectance of an Imprinted Cholesteric Elastomer During a Segregation Process

We have calculated the reflectance and transmittance of obliquely incident circularly polarized light on a distorted cholesteric elastomer slab distorted. The distortion is achieved by variations of the chiral order parameter, which is defined as the imprinting efficiency of the volume fraction of the absorbed molecules from the racemic solvent, having the preferably absorbed chirality. We have found a right-handed band reflection whose bandwidth increases and blue-shifts as incidence angle increases. A thinner left-handed band reflection also is observed. The slab be used in a chemically controlled optical filter or optically traced chiral pump.


INTRODUCTION
The interest in the separation of enantiomers of organic compounds has been recently increased in the food, medicine and cosmetic industries. As it is well know, the chirality is extremely important for living systems, because most of the biomolecules are chiral ones. For example, glucose, a sugar exists in one conformation; while the opposite enantiomer cannot be used as a food source. Biochemistry uses only one hand and not the other however the reason for this has not been understood, so far. Over half of the organic compounds in drugs are chiral whose different enantiomers have different effects. For instance, Ritalin prevents hyperactivity in children in one enantiomer, whereas it has not effect when the opposite enantiomer is used. Another example is thalidomide: one enantiomer of thailomide can cure morning sickness of pregnant women, whereas the other causes birth defects. The sense of smell is also sensitive to the chirality of the odor molecules. For instance, limonene has an orange smell whereas the opposite enantiomer smells like lemon [1]. This exemplifies the fact that separation of molecules by chirality is an important process in biochemistry and in the pharmaceutical industry. In this paper we suggest a method to optically monitoring the chiral sorting of molecules performed by an elastomer liquid crystal.
Liquid crystal elastomers combine the rubbery elastic properties with the liquid crystals (LC) anisotropy [2]. When these stiff molecules are joined to a large and flexible polymer chain, they leave their muddy liquid macroscopic form and acquire the appearance of a rubbery network. Particularly, imprinted cholesteric elastomers acquire its chirality from an induced chiral structure instead of chiral molecules. Upon cross-linking nematic polymers in a chiral solvent forms a gel. The chiral solvent, which causes a natural twist to the nematic is thus removed but nonetheless leaves behind an imprinted cholesteric elastomer [3]. A spectacular property of imprinted networks is their capacity to preferentially absorb and retain right or left molecules from a racemic solvent which renders them chiral pumps. Moreover the elastomers are photonic materials in the sense that exhibit an optical band structure just as the one shown by electrons in metals and semiconductors. Elastomers provide a large spectrum of applications that exploit the possibility of externally handle a band structure by various types of agents like mechanical stress, electric fields and chemical solvents for designing devices such as optical sensors, mirrors, laser and filters [2,4].
The purpose of this paper is to propose an optical method to determine the volume fraction of preferentially absorbed molecules by measuring the transmittance and reflectance spectra during a preferentially absorbing process. Specifically, we are interested in calculate the reflectance and transmittance of circularly polarized light for oblique incidence due to a cholesteric elastomer, assuming arbitrary values of chiral order parameter, a. We have thoroughly discussed the solution of the boundary value problem for cholesteric elastomers in sections 4 and have determined the optical spectra of the system in Sec. 5.

ELASTIC FORMULATION
A cholesteric elastomer has locally a nematic structure. Consequently, the director vectorn n ¼ ðcos /ðzÞ; sin /ðzÞ; 0Þ is at x À y plane and /ðzÞ is the angle between director vector and x axis (see Fig. 1). The director vector has a continuous rotation as function of z, which describes a periodic helical structure characterized by a pitch p (or equivalent by wave number q 0 ¼ 2p=p) [5,6]. The chiral order parameter a ¼ nq 0 is related with the imprinting efficiency of the cholesteric phase, where n ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the nematic penetration depth in rubbery networks, K 2 is the twist elastic constant and D 1 is local anchoring of the director to the rubbery network. Variations in the chiral order parameter imply changes in the robustness of chiral imprinting, which can be controlled by varying the density of cross-links in the network (affecting D 1 ) [7].
The energy for an elastomer formed under the presence of a chiral solvent which is subsequently replaced with an achiral one is given by [8], The Frank free energy to twist the material is given by associated with the first term of the Eq. (1) tend to unwind the cholesteric helix but it finds resistance to rotate in the anchoring of the rubbery network-associate with the second term of Eq. (1). D 1 is proportional to rubbery elasticity modulus and the rubbery elasticity modulus is related with cross-linked density [7]. The scale length or nematic penetration depth rubbery at which the two energy contributions are comparable is usually small n ' 10 À8 m for a typical K 2 ' 10 À11 J=m, D 1 ' lðr À 1Þ 2 =r, where r $ 0:9 is the anisotropy chain number or cross-linked number and l ' 10 5 J=m 3 is the elasticity rubber modulus. The pitch is 10 3 larger than nematic penetration depth n, then There are two forms of modifying this relation, (i) by increasing the nematic penetration depth which amounts to have a weaker gel or one with smaller anisotropy, (ii) by increasing the Frank free energy.
The chiral imprinting will be lost if D 1 diminish, i.e., so that the director vector anchoring to rubbery will be almost null. Similarly, if the wave number q 0 or twist elastic constant K 2 are larger the chiral imprinting will be also lost. The imprinting efficiency is reached for values of chiral parameter order a c ¼ nq o lower than 2=p.
Equation (1) can be expressed in terms of chiral parameter order as, The equilibrium configuration can be obtained by minimizing F with respect to the angle, then we have got the Euler-Lagrange equations, Transmittance and Reflectance of an Imprinted Cholesteric Elastomer 237=[589] which in this case turns out to be Notice that if a ¼ 0 the solution of Eq. (6) is / 0 ðzÞ ¼ q 0 z which corresponds to an ideal or undistorted cholesteric helix. By introducing the variable w ¼ q 0 z À / þ p=2, Eq. (6) transforms in the expression which is the simple pendulum equation. Hence, for large values of a, small harmonic oscillations of frequency q 0 =a around / ¼ q 0 z þ p=2 are expected. Nevertheless, for smaller values of a anarmonic oscillations take place until w abandons its oscillatory dependence and grows without limit following the dependence corresponding to a rotatory motion in a simple pendulum. This behavior can be inferred by using the general solution of Eq. (6) which is given by where Amðz; mÞ is the Jacobian amplitude and c is a constant related with the reduced elastic energy [8] g ¼ 2nF=TD 1 . In the Figure 2 we show the director vector angle /ðzÞ against position for different values of chiral order parameter a. It is interesting to note that by increasing n from zero, /ðzÞ presents oscillations without changing its amplitude, around the undistorted solution / 0 ðzÞ whose spatial period increases, until n reaches the critical values: n c ¼ 3. After this value /ðzÞ deviate considerably from / 0 ðzÞ and enlarges its effective q-value for n ! n c which is equivalent to have an untwisted helix than that of / 0 ðzÞ.

OPTICAL DESCRIPTION
As any chiral material whose axis of chiral nonhomogeneity is along the z-axis, the optical relative permittivity matrix of the chiral elastomer may be stated as where E a ¼ E k À E ? is the dielectric anisotropy with E k and E ? the dielectric constants in the optical regime parallel and perpendicular to the directorn n, respectively. Maxwell curl equations (in MKS units) for media whose dielectric tensor is only z-dependent, are given by r Â Eðx; y; zÞ ¼ ixl o Hðx; y; zÞ r Â Hðx; y; zÞ ¼ ÀixE o E EðzÞ Á Eðx; y; zÞ where E o and l o are the permittivity and the permeability of free space (i.e., vacuum). If a plane wave were to be incident obliquely on a Cholesteric Elastomer slab of thickness L, we can describe the fields in terms of the column vector [9] w wðx; y; zÞ ¼ exp½ijðx cos u 1 þ y sin u 1 Þ e x ðzÞ e y ðzÞ h x ðzÞ h y ðzÞ where u 1 the angle formed by the propagation vector and the y-axis in the x À y plane. Thus, the essential part of Maxwell curl equations can then be written as [10]. d dz w wðx; y; zÞ ¼ i A 0 A 0 ðzÞ Á w wðx; y; zÞ; 0 < z < L; ð12Þ Transmittance and Reflectance of an Imprinted Cholesteric Elastomer 239= [591] In this equation, the 4 Â 4 matrix can be factorized as Equation (12) can be solved by a variety of techniques. We chose to implement the piecewise homogeneity approximation method. The final result is the computation of a transfer matrix

BOUNDARY VALUE PROBLEM
Suppose that plane waves are obliquely incident on a cholesteric of finite thickness from both half-spaces z 0 and z ! L, which are taken to be vacuous. As a result, the total electric and magnetic fields in these half-spaces may be written as eðzÞ ¼ ða Lû u þ þ a Rû u À Þ exp½ik 0 z þ ijðx cos u 1 þ y sin u 1 Þ þ ðr Lû u À þ r Rû u þ Þ exp½Àik 0 z þ ijðx cos u 1 þ y sin u 1 Þ hðzÞ ¼ iðÀa Lû u þ þ a Rû u À Þ exp½ik 0 z þ ijðx cos u 1 þ y sin u 1 Þ þ iðÀr Lû u À þ r Rû u þ Þ exp½Àik 0 z þ ijðx cos u 1 þ y sin u 1 Þ and whereû and ðû u x ;û u y ;û u z Þ is the triad of Cartesian unit vectors. In these equations, the complex-valued amplitudes a L;R and b L;R of the left-and right-circularly polarized (LCP and RCP) components of incident plane waves are assumed to be known, whereas the reflection amplitudes r R;L and transmission amplitudes t L;R are not known. Continuity of the tangential components of the electric and the magnetic fields across the planes z ¼ 0 and z ¼ L leads to the prescriptions of the boundary values w wð0Þ where the matrix Q Q ¼ By virtue of Eqs. (16), (17) and (19), we obtain the matrix algebraic equation where the so called transfer matrix is given by and the symbol y denotes the conjugate transpose. The result of solving Eq. (21) is best arranged as S ¼ t RRa t RLa t RRb t RLb r LRa r LLa r LRb r LLb t RRa t RLa t RRb t RLb r LRa r LLa r LRb r LLb where t LRa and t LRb , etc., are the transmission coefficients and r LRa and r LRb , etc., are the reflection coefficients. The co-polarized transmittances engendered by the waves coming from the left-hand side are denoted by T LLa ¼ jt LLa j 2 and T RRa ¼ jt RRa j 2 , and the cross-polarized ones by T LRa ¼ jt LRa j 2 and T RLa ¼ jt RLa j 2 ; and similarly for the reflectances R RRa ¼ jr RRa j 2 , etc. similar definitions are valid for the waves generated by the waves coming from the right-hand side T LLb ¼ jt LLb j 2 , T RRb ¼ jt RRb j 2 and T LRb ¼ jt LRb j 2 . The matrix involved S in Eq. (23) is known as the scattering matrix and contains the same information of the transfer matrix but order in different way. It can be straightforwardly shown by writing explicitly the algebraic equations defined by Eq. (21) and solving them for t R , t L , r R and r L that both matrices are related by the expression where P f and P b are the forward and backward projectors given by In what follows we shall restrict our calculation to the case in which the incident waves are coming only from the left hand side that is we take b R ¼ b L ¼ 0.

SPECTRA
In our calculations we have chosen the physical amounts of a real cholesteric elastomer material ðp 0 ¼ 218 nmÞ [5,11]. Figure 3 shows that in the absence of solvent (a ¼ 0) there is only a band reflection for right-circularly copolarized light R RR with the bandwidth of '50 nm FIGURE 3 Circularly polarized reflectance R RR and R LL as function of the wavelength and angle incidence for the ideal helix case a ¼ 0 (left column) and a ¼ 0:4 (right column). Other parameters are: E ? ¼ 1:91, E k ¼ 2:22, h ¼ 1 and q 0 ¼ 218 nm. and maximum amplitude. The center of the reflection band blue-shift whereas the width and amplitude band remain constant as the incidence angle h increases. The corresponding reflectances R RL and R LL do no exhibit practically any band. In contrast, when a ¼ 0:4 there appear bands for both left-and right-copolarized reflectances R RR and R LL whereas the cross-polarized reflectance R RL is almost zero. In this case the R RR is not as well formed as in the absence of solvent because is amplitude present various peaks and the bandwidth varies as a function of the incidence angle, reaching its smaller value for angles around 45 degrees. Differently, R LL is very thin since its bandwidth is around 15 nm and contained in the interval where the R RR band lies. Figure 4 presents the corresponding trasmittances to Figure 3. The case without solvent (a ¼ 0) confirms consistently the complemetary behavior of an undistorted cholesteric. The co-polarized T RR and T LL exhibit troughs where the transmittance diminish, the one for T RR is well defined whereas that of T LL changes is thickness a deep. The blue-shift of the trough for large incident angles is also observed.

CONCLUSIONS
We have obtained the circularly polarized reflectances and transmittances as function of the chiral order parameter of a cholesteric elastomer immersed in a racemic solvent. We have found considerably changes in the bandwidth of the reflectance for left-and right-copolarized light under the presence of the solvent which are susceptible to be detected experimentally. We have obtained a R RR bandwidth larger than that for an undistorted elastomer (a ¼ 0) for a ¼ 0:4 and obliquely incidence for angles larger than 60 . Also in this value we have observed a thin band reflection for both polarizations R RR and R LL for oblique incidence.
The modifications of the transmittance and reflectance spectra during a preferentially absorbing process suggest to utilize the optical spectra as an indirect method to determine the concentration of preferentially absorbed molecules during a segregation process.