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

By P. Castro-Garay and Jesus Manzanares-Martinez

Submitted: November 2nd 2011Reviewed: June 18th 2012Published: September 12th 2012

DOI: 10.5772/50747

Downloaded: 1211

1. 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 to form a gel, the 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 elastomersare photonic materialsin 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 chemical solvents for designing devices such as optical sensors, mirrors, laser and filters [2] [4].

The purpose of this chapter 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, Ϗ. 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.

2. Elastic formulation

A cholesteric elastomer has locally a nematic structure. Consequently, the director vector n̠=(cosϳ(z),sinϳ(z),0)is at x-yplane and ϳ(z)is the angle between director vector and xaxis (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 q0=2Ϟ/p) [5] [6].

Figure 1.

Schematic representation of a liquid crystal cholesteric elastomer. The circular components of the obliquely incident beam as well as the components of the transmitted an reflected parts are shown.

The chiral order parameter Ϗ=Ϝq0is related with the imprinting efficiency of the cholesteric phase, where Ϝ=K2D1is the nematic penetration depth in rubbery networks, K2is the twist elastic constant and D1is local anchoring of the director to the rubbery network. Variations in the chiral order parameterimplychanges in the robustness of chiral imprinting, which can be controlled by varying the density of cross-links in the network (affecting D1) [7].

The energy for an elastomer formed under thepresence of a chiral solvent which is subsequently replaced with an achiral one is given by [8],

F=12K2dz2+D1sin2ϳ-q0zdz.E1

The Frank free energy to twist the material is given by

Ff=12Kn̠×n̠2E2

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). D1is 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 Ϝ10-8m for a typical K210-11J/m, D1Ϛ(r-1)2/r,where r~0.9is the anisotropy chain number or cross-linked number and Ϛ105J/m3is the elasticity rubber modulus. The pitch is 103bigger than nematic penetration depth Ϝ, then

Ϝp=Ϟq0.E3

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 enlarging the Frank free energy. The chiral imprinting will be lost if D1diminish, i.e. so that the director vector anchoring to rubbery will be almost null. Similarly, if the wave number q0or twist elastic constant K2are bigger the chiral imprinting will be also lost. The imprinting efficiency is reached for values of chiral parameter order Ϗc=Ϝqolower than 2/Ϟ.

Eq. (1) can be expressed in terms of chiral parameter order as,

F=12Ϗ2q02dz2+sin2ϳ-q0zE4

The equilibrium configuration can be obtained by minimizing Fwith respect to the angle, then we have got the Euler-Lagrange equations,

-zFzϳ=0,E5

which in this case turns out to be

Ϗ2q02d2ϳdz2+sin2q0z-ϳ=0.E6

Notice that if Ϗ=0the solution of eq. (6) is ϳ0(z)=q0zwhich corresponds to an ideal or undistorted cholesteric helix. By introducing the variable w=q0z-ϳ+Ϟ/2, Eq.(6) transforms in the expression

d2wdz2+q02Ϗ2sin2w=0.E7

which is the simple pendulum equation. Hence, for large values of Ϗ, small harmonic oscillations of frequency q0/Ϗaround ϳ=q0z+Ϟ/2are expected. Nevertheless, for smaller values of Ϗanarmonic oscillations take place until wabandons 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

ϳ(z)=q0z-Am(cz/Ϝ,1/c2)+Ϟ/2,E8

where Am(z,m)is the Jacobian amplitude and cis a constant related with the reduced elastic energy [8] g=2ϜF/TD1. In the Fig. 2 is shown the director vector angle ϳ(z)against position for different values of the parameter c. In this case, the director vector changes the amplitude of its oscillations around the ideal case solution ϳ0(z), while the spatial period is maintained constant to increase the values of parameter cfrom zero until to reach the critical value in c=1. Schematic representation of the director vector is shown in the the Fig. 3. In the Fig. 4 is shown the director vector angle ϳ(z)against position for different values of chiral order parameter Ϗ. It is interesting to note that by increasing Ϝfrom zero, ϳ(z)presents oscillations without changing its amplitude, around the undistorted solution ϳ0(z)whose spatial period increases, until Ϝreaches the critical values: Ϝc=3. After this value ϳ(z)deviate considerably from ϳ0(z)and enlarges its effective q-value for ϜϜcwhich is equivalent to have an untwisted helix than that of ϳ0(z)as is shown in the Fig. 5.

Figure 2.

Director vector angle as function of the position for different values of the parameter,c.

Figure 3.

Schematic representation of director vector for different values of parameter, c, a) ideal case, b) c=0.25 and c) c=0.75.

Figure 4.

Director vector angle as function of the position for different values of the chiral order parameter, Ϗ.

Figure 5.

Schematic representation of director vector for different values of chiral order parameter, Ϗ, a) ideal case, b) Ϗ=0.3 and c) Ϗ=2/Ϟ.

3. 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

ϓ̣(z)=000ϓ(0)+ϓacos2ϳϓasinϳcosϳ0ϓsinϳcosϳϓ(0)+ϓasin2ϳϓ(0)E9

where ϓa=ϓ-ϓis the dielectric anisotropy with ϓand ϓthe dielectric constants in the optical regime parallel and perpendicular to the director n̠, respectively.

Maxwell curl equations (in MKS units) for media whose dielectric tensor is only z-dependent, are given by

×E(x,y,z)=iϧϚoH(x,y,z)×H(x,y,z)=-iϧϓoϓ̣(z)E(x,y,z),0<z<L,E10

where ϓoand Ϛoare 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]

Ϧ̣x,y,z=exp[iϘ(xcosu1+ysinu1)]ex(z)ey(z)hx(z)hy(z).E11
(

where u1the angleformed by the propagation vector and they-axisin the x-yplane. Thus, the essential part of Maxwell curl equations can then be written as [10].

ddzϦ̣(x,y,z)=iẠ'(z)Ϧ̣(x,y,z),0<z<L,E12

In this equation, the 4×4matrix can be factorized as Ạ'=B(z)ẠB(-z)where the 4×4 matrix

Ạ(z)=000ϧϚo00-ϧϚo00-ϧϓoϓ2(0)00ϧϓoϓd000

+Ϙ2ϧϓoϓ1(0)00-sinucosu-cos2u00sin2usinucosu00000000

+Ϙ2ϧϚo00000000sinucosucos2u00-sin2u-sinucosu00,E13

with

u=hϞzχ-u1E14
and

B(z)=cosϳ(z)sinϳ(z)00-sinϳ(z)cosϳ(z)0000cosϳ(z)sinϳ(z)00-sinϳ(z)cosϳ(z).E15

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 Ụ'such that

Ϧ̣(L)=ṂT(L)Ụ'Ṃ(0)Ϧ̣(0)ỤϦ̣(0).E16

4. Boundary value problem

Suppose that plane waves are obliquely incident on a cholesteric of finite thickness from both half-spaces z0and zL, which are taken to be vacuous. As a result, the total electric and magnetic fields in these half-spaces may be written as

e(z)=(aLu^++aRu^)exp[ik0z    +iκ(xcosu1+ysinu1)]                +(rLu^+rRu^+)exp[ik0z    +iκ(xcosu1+ysinu1)]h(z)=i(aLu^++aRu^)exp[ik0z    +iκ(xcosu1+ysinu1)]                +i(rLu^+rRu^+)exp[ik0z    +iκ(xcosu1+ysinu1)]},z0,E17
and
e(z)=(tLu^++tRu^)exp[ik0(zL)+iκ(xcosu1+ysinu1)]+(bLu^++bRu^)exp[ik0(zL)    +iκ(xcosu1+ysinu1)]h(z)=i(tLu^++tRu^)exp[ik0(zL)+iκ(xcosu1+ysinu1)]+i(bLu^++bRu^)exp[ik0(zL)    +iκ(xcosu1+ysinu1)]},zL .E18

where

u̠±=u̠x±iu̠y2,E19
and (u̠x,u̠y,u̠z)is the triad of Cartesian unit vectors. In these equations, the complex–valued amplitudes aL,Rand bL,Rof the left– and right–circularly polarized (LCP and RCP) components of incident plane waves are assumed to be known, whereas the reflection amplitudes rR,Land transmission amplitudes tL,Rare not known.

Continuity of the tangential components of the electric and the magnetic fields across the planes z=0and z=Lleads to the prescriptions of the boundary values

Ϧ̣(0)=Q̣aRaLrRrL,Ϧ̣(L)=Q̣tRtL00,E20

where the matrix

Q̣=1111-iii-ii-ii-i11-1-1.E21

By virtue of Eqs. (16), (17) and (19), we obtain the matrix algebraic equation

tRtLbRbL=TaRaLrRrL,E22

where the so called transfer matrix is given by

T=14Q̣ỤQ̣E23

and the symbol denotes the conjugate transpose.

The result of solving Eq. (21) is best arranged as

tRtLrRrL=SaRaLbRbLE24
S=tRRatRLatRRbtRLbrLRarLLarLRbrLLbtRRatRLatRRbtRLbrLRarLLarLRbrLLbE25

where tLRaand tLRb, etc., are the transmission coefficients and rLRaand rLRb, etc., are the reflection coefficients. The co–polarized transmittances engendered by the waves coming from the left-hand side are denoted by TLLa=|tLLa|2and TRRa=|tRRa|2, and the cross–polarized ones by TLRa=|tLRa|2and TRLa=|tRLa|2; and similarly for the reflectances RRRa=|rRRa|2, etc. similar definitions are valid for the waves generated by the waves coming from the right-hand side TLLb=|tLLb|2, TRRb=|tRRb|2and TLRb=|tLRb|2. The matrix involved Sin 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 tR, tL,rRand rLthat both matrices are related by the expression

S=(T.Pf-Pb)-1.(Pf-T.Pb),E26

where Pfand Pbare the forward and backward projectors given by

Pf=1000010000000000,Pb=0000000000100001.

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 bR=bL=0.

5. Spectra

In our calculations we have chosen the physical amounts of a real cholesteric elastomer material p0=218nm[5] [11]. Fig. 6 shows that in the absence of solvent (Ϗ=0)there is only a band reflection for right-circularly copolarized light RRRwith the bandwidth of 50nm and maximum amplitude. The center of the reflection band blue-shift whereas the width and amplitude band remain constant as the incidence angle ϖincreases. The corresponding reflectances RRLand RLLdo no exhibit practically any band. In contrast, when Ϗ=0.4there appear bands for both left- and right-copolarized reflectances RRRand RLLwhereas the cross-polarized reflectance RRLis almost zero. In this case the RRRis 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, RLLis very thin since its bandwidth is around 15nm and contained in the interval where the RRRband lies.

Figure 6.

Circularly polarized reflectance RRR and RLL as function of the wavelength and angle incidence for the ideal helix case Ϗ=0 (left column) and Ϗ=0.4 (right column). Other parameters are: ϓ⊤=1.91, ϓ∤=2.22, h=1 and q0=218nm.

Fig. 7 presents the corresponding trasmittances to Fig.6. The case without solvent (Ϗ=0)confirms consistently the complemetary behavior of an undistorted cholesteric. The co-polarized TRRand TLLexhibit troughs where the transmittance diminish, the one for TRRis well defined whereas that of TLLchanges is thickness a deep. The blue-shift of the trough for large incident angles is also observed. In the Fig. 8 is shown the band reflection for left and right circularly polarized ligh RRRand RLLfor different values of parameter, c.In both cases, the band reflection for left circularly polarized light is thinner than the band reflection for right circularly polarized light.

Figure 7.

Circularly polarized transmittance TRR and TLL as function of the wavelength and angle incidence for Ϗ=0 (left column) and Ϗ=0.4 (right column).

In the Fig. 9 is shown the reflectance of left and right circularly polarized light as a function of wavelenght and the chiral order parameter to oblique incidence. Complementary, In the Fig. 10 is shown the same of the Fig.9 as function of wavelenght and the parameter c.

Figure 8.

Circularly polarized reflectance RRR and RLL as function of wavelength and angle incidence for c=0.3 (left column) and c=1 (right column).

Figure 9.

Circularly polarized reflectance RRR and RLL as function of wavelength and the chiral order parameter to oblique incidence angle of 45o and c=0.5.

Figure 10.

Circularly polarized reflectance RRR and RLL as function of wavelength and the c parameter to oblique incidence angle of 45o and Ϗ=0.2.

6. 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 RRRbandwidth larger than that for an undistorted elastomer (Ϗ=0) for Ϗ=0.4and obliquely incidence for angles larger than 60.Also in this value we have observed a thin band reflection for both polarizations RRRand RLLfor 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.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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P. Castro-Garay and Jesus Manzanares-Martinez (September 12th 2012). Transmittance and Reflectance of an Imprinted Cholesteric Elastomer During a Segregation Process, Advanced Elastomers - Technology, Properties and Applications, Anna Boczkowska, IntechOpen, DOI: 10.5772/50747. Available from:

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