Appendix
Absorption spectroscopy with linearly polarized light is a powerful tool to evaluate the preferential
orientation of adsorbed molecules in ordered assemblies. Indeed, the absorption intensity depends
on the relative orientation between the plane of the polarized light and the transition moment of the
absorbing molecule. Consequently, from the anisotropy effect of the adsorbed molecules in the
ordered systems, the preferential orientation angle can be evaluated.
First of all, it is necessary to relate the laboratory axes (X, Y, Z) with the sample axes (x, y, z)
by means of Eulerian angles α, β and γ. The Euler angles are defined as (Fig.A-1): γ is the rotation
around the z-axis that brings the y-axis in the XY-plane; β angle is the rotation around the y-axis,
which translates the z-axis to the Z-axis; and α is the rotation angle around the new z-axis that
makes x- coincident with X-axis and y- with Y-axis.
(insert Figure A-1)
Figure A-2 shows the schematic arrangement used in the present work to record the response of
the absorption spectrum of R6G dye absorbed in Lap films to incident linearly polarized light. The
laboratory Z-axis is the direction of the incident light, which can be X- and Y-polarized. The sample
z-axis is the normal to the film and the x- and y-axes define the clay film plane. Absorption spectra
for X- and Y-polarized light are recorded for different twisting δ angles around the y-axis sample
(see Scheme 2). With this arrangement, the corresponding Euler angles are: α = 0º, β = 180º-δ and γ
= 0º.
(insert Figure A-2)
In light absorption, the electric field vector of incident plane-polarized light interacts with the
transition moments of the dye molecules and the probability that the molecule absorbs light is
proportional to the square of the projection of its absorbing transition moment onto the polarization
axis of the incident light (i.e. the square of their scalar product).1-4 The probability that the system
absorbs light polarized along X-direction will be proportional to the square of scalar product
r
r
between the transition moment vector ( M ) of the dye and the electric field ( E ) of the incident light.
r
r
Because E has only the X-axis component (EX), only the projection of M into the X-axis (MX) has
r
to be considered in the scalar products. So, the projection of the x-, y- and z-component of M (Mx,
My and Mz) into the x-axis (through the corresponding Eulerian angles) has to be considered and the
absorbance of the sample for X-polarized incident light (AX) is given by:
AX ∝ [EX MX]2 = KX EX2 [Mx cos(xX) + My cos(yX) + Mz cos(zX)]2
(A-1)
where the proportional constant KX includes the response of the instrument to the X-polarized light
The values of these cosines are related to the Euler angles by means of:2
cos(xX) = cos α cos β cos γ – sin α sin γ
(A-2)
cos(yX) = -sin α cos γ - cos α cos β sin γ
(A-3)
cos(zX) = cos α sin β
(A-4)
In the present case:
cos(xX) = - cos δ
(A-5)
cos(xY) = 0
(A-6)
cos(xZ) = sin δ
(A-7)
r
On the other hand, Mx, My and Mz are related to the polar angles between the vector M and the x-,
y- and z-axes, by means of (see Fig. A-2):
Mx = M sin ψ cos φ
(A-8)
My = M sin ψ sin φ
(A-9)
Mz = M cos ψ
where
(A-10)
r
ψ is the angle between the M and the z-axis, the normal to the clay films in the present
r
paper, and φ is the angle formed by the projection of M in the xy-plane (Mxy = M sin ψ) and the yaxis. Applying eqs. from (A-5) to (A-10) into eq.(A-1), AX is converted to:
AX = KX EX2 [- M sin ψ cos φ cos δ + M cos ψ sin δ]2
(A-11)
Expanding this equation, we obtain:
AX = KX EX2 [M2 sin2 ψ cos2 φ cos2 δ + M2 cos2 ψ sin2 δ - 2 M2 sin ψ cos φ cos δ cos ψ sin δ](A-12)
Taking into account that the clay layers do not present any anisotropy through their xy-plane, all
orientations in the xy-plane of the clay film are equally probable, i.e. the φ angle can take any value
between 0 and 360º. Consequently cos2φ and cos φ have to be substituted by their respective
average values:
2π
cos 2φ
cosφ
0
2π
0
2π
= ∫ cos 2φ dφ = ½
0
2π
= ∫ cosφ dφ = 0
0
(A-13)
(A-14)
Applying both equations in eq.(A-12), the following expression is finally derived:
AX = KX EX2 [½ M2 sin2ψ cos2δ + M2 cos2ψ sin2δ]
(A-15)
In the same way the probability that the system absorbs Y-polarized light is given by:
AY ∝ [EY MY]2 = KY EY2 [Mx cos(xY) + MY cos(yY) + MZ cos(zY)]2
(A-16)
with cosine values of:
cos(xY) = cos α sin γ + sin α cos β cos γ = 0
(A-17)
cos(yY) = cos α cos γ – sin α cos β sin γ = 1
(A-18)
cos(zY) = sin α sin β = 0
(A-19)
and combining eqs.(A-8 - A-10) and eqs.(A-13 and A-14):
AY = KY EY2 [M sin ψ sin φ]2 = ½ KY EY2 M2 sin2 ψ
(A-20)
The dichroic ratio (DX,Y), defined as the relation between the absorbances recorded for X- and Ypolarized incident light will be given by:
A X K X E X ½ M 2 sin 2 ψ cos 2 δ + M 2 cos 2 ψ sin 2 δ
=
AY KY EY2
½ M 2 sin 2 ψ
2
D X,Y =
(A-21)
The first fraction in eq.(A-21) is related to the instrumental response to the X- and Y-polarized
light. It includes, among other factors, the efficiency of the X- and Y-polarization, the response of
the monochromator and all the optical components to the X- and Y-polarization components. The
instrumental response can be corrected by recording the response to the x- and y-polarized light of
an isotropic system. Thus, DX,Yis = (AX/AY)is and the real dichroic ratio of the sample (DX,Ycor) is
given by:
D X,Y
cor
= D X,Y /D X,Y =
is
½ sin 2 ψ cos 2 δ + cos 2 ψsin 2 δ
½ sin 2 ψ
(A-22)
which proposes a linear relationship between the dichroic ratio of the sample and the experimental
sin2δ angle by means of
D X,Y
cor
A A
= X  Y
AY  AX
from this slope the tilted




is
= 1+
2 − 3sin 2 ψ 2
sin δ
sin 2 ψ
(A-23)
ψ angle of the transition moment and the normal to the surface can be
evaluated. Since the transition moment of R6G is oriented along the long-molecular axis, this
ψ
angle directly provides the orientation of the dye molecules with respect to the normal of the Lap
surface.
(1) Lakowicz, J.R. In Principles of Fluorescence Spectroscopy, 2ª Ed., Kluwer Academic: New
York, 1999.
(2) Milch, J.; Thulstrup, E.W. In Spectroscopy with Polarized Light, VCH Publishers: New York,
1986.
(3) Kliger, D.S.; Lewis, J.W.; Randall, C.E. In Polarized Light in Optics and Spectroscopy,
Academia Press: San Diego, 1990.
(4) Valeur, B. In Molecular Fluorescence. Principles and Applications, Wiley-VCH: Weinheim,
2002.
Figure A-1. Euler angles.
Figure A-2. Experimental setup to record absorption spectra with horizontally (X-axis) and
vertically (Y-axis) polarized light for different twisted δ angles between the normal to the film (N)
and the incident beam (Z-axis). The polar ψ and φ angles defining the orientation of the transition
moment (M) of the dye with respect to the x-, y- and z-axes of the film are also included.
Figure A-1.
M
X
Incident
beam •
ψ
•
Y Z
δ
N
z
N
film
ψ
•
y
To detector
φφ
x
polarizer:
X(H),Y(V)
M
z
x
depolarizer
Figure A-2
z
y