Nonlinear optical studies of liquid crystal alignment on a rubbed polyvinyl alcohol surface
Xing Wei, Seok-Cheol Hong, Xiaowei Zhuang,* Tomohisa Goto,† and Y. R. Shen
Department of Physics, University of California, and Materials Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720
共Received 5 January 2000兲
Sum-frequency vibrational spectroscopy and second-harmonic generation have been used to measure the
orientational distributions of the polymer chains and adsorbed 8CB liquid crystal molecules on a rubbed
polyvinyl alcohol surface. Results show that the polymer chains at the surface appear to be well aligned by
rubbing, and the adsorbed liquid crystal molecules are aligned, in turn, by the surface polymer chains. Strong
correlation exists between the orientational distributions of the polymer chains and the liquid crystal molecules,
indicating that the surface-induced bulk alignment of a liquid crystal film by rubbed polymer surfaces is via an
orientational epitaxylike mechanism.
PACS number共s兲: 61.30.⫺v, 68.35.Bs, 78.30.Jw, 42.65.Ky
The alignment of liquid crystal 共LC兲 molecules on rubbed
polymer surfaces was discovered in 1911 关1兴. In recent years,
this phenomenon has been studied extensively not only because of the basic interest in understanding the underlying
mechanism, but also because of its relevance to LC display
technology. Today, rubbed polymer films are widely used in
industry to obtain homogeneous bulk LC alignment for LC
displays 关2兴. Different mechanisms have been proposed for
LC alignment on rubbed polymer surfaces. One assumes that
rubbing creates microgrooves or scratches on polymer surfaces which then align LC along the grooves to minimize the
energy of elastic distortion 关3兴. Another suggests that rubbing aligns surface polymer chains, which in turn align LC’s
through intermolecular interaction 关4兴. The latter is believed
to be operative when LC molecules anchor strongly to polymer surfaces, as is commonly the case in the LC industry.
To study LC alignment by rubbed polymer surfaces, a
number of experimental techniques have been used. Optical
second-harmonic generation 共SHG兲 showed that a rubbed
polymer surface can align an adsorbed LC monolayer which
then aligns the LC bulk by molecular correlation 关5–8兴, providing convincing evidence that molecular interaction between a LC and a polymer at the surface is responsible for
the LC alignment. Attempts to study the rubbed polymer
itself with SHG have also been made, but the structural information obtained so far from SHG is rather limited 关9兴.
Atomic force microscopy could provide images of rubbed
polymer surfaces showing an overall anisotropy, but was unable to resolve the surface polymer chains 关10–12兴. Ellipsometry 关13兴 and infrared spectroscopy 关14–17兴 can measure
rubbing-induced anisotropy and other structural changes in
the polymer film. However, because of their lack of surface
specificity, it is unclear whether these results indeed represent the real surface structure of the rubbed polymer. Grazing
incidence x-ray scattering could probe a surface region of
*Present address: Department of Physics. Stanford University,
Stanford, CA 94035-4060.
Present address: Organic Material Research Laboratory, NEC
Corporation, Kanagawa 216, Japan.
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⬃5 nm thick 关18,19兴. A higher surface sensitivity has been
achieved by near-edge x-ray-absorption fine-structure
共NEXAFS兲 spectroscopy which could probe a surface layer
of ⬃1 nm thick 关20–23兴.
In this paper, we describe the use of infrared-visible sumfrequency generation 共SFG兲 vibrational spectroscopy to
probe the structure of a rubbed polymer surface. SFG has
been developed into a powerful surface analytical tool 关24兴.
Similar to SHG, it is forbidden in media with inversion symmetry under the electric dipole approximation, but allowed at
interfaces, where the inversion symmetry is broken. Therefore, it is ideally suited as a probe to study interfacial structure between two centrosymmetric media. Being a nonlinear
optical process involving three optical waves, in principle,
SFG can yield more detailed structural information than all
the linear optical techniques including NEXAFS. We have
applied SFG vibrational spectroscopy to rubbed polyvinyl
alcohol 共PVA, 关 ⫺CH2 -CHOH⫺ 兴 n ). Rubbed PVA is known
to align LC molecules in a way similar to rubbed polyimide
关6兴, but has a much simpler monomer unit. We focus on the
stretch vibrational modes of the CH2 groups of PVA at the
surface. The CH2 groups directly associated with the PVA
backbone are oriented perpendicularly to the local PVA
chains, as shown in Fig. 1. From the measured SFG spectra,
FIG. 1. Molecular structure of PVA and orientational geometry
of a CH2 group on a rubbed PVA surface. Axis x is along the
rubbing direction, and z is along the surface normal of the polymer
film. Axis ␰ is normal to the CH2 plane and along the PVA chain,
␨ is along the symmetry axis of CH2 , and ␩ is orthogonal to ␰
and ␨ .
©2000 The American Physical Society
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Equations 共2.1兲–共2.3兲 also apply to second-harmonic generation with ␻ 1 ⫽ ␻ 2 . In SHG, often only one input laser
beam at frequency ␻ 共usually in the visible兲 is used and the
SH signal at ␻ s ⫽2 ␻ is collected.
B. SFG vibrational spectroscopy for surface molecular groups
The nonlinear susceptibility tensor ␹(2) ( ␻ s ⫽ ␻ 1 ⫹ ␻ 2 ) for
SFG is expected to be resonantly enhanced when ␻ 2 approaches a surface vibrational resonance. Scanning over such
resonances yields a surface SFG vibrational spectrum. We
can express ␹(2) in terms of the resonant nonlinear polarizability ␣R(2) for the surface molecular groups,
FIG. 2. Molecular structure and orientational coordinates of an
8CB molecule deposited on a rubbed PVA surface. ␨ is along the
long axis of the cyanobiphenyl core of the 8CB molecule. ␪ LC and
␾ LC are the polar and azimuthal angles of ␨ . The x axis stands for
the rubbing direction.
we can deduce an orientational distribution for the CH2
groups at the surface. This then directly yields an orientational distribution for the PVA chains on the rubbed surface.
We found that the PVA chain orientation is indeed strongly
affected by rubbing. Results of this work were briefly reported in an earlier paper 关25兴. Here we present a detailed
analysis of the SFG results together with the SHG study of
an 8CB 共4-n-octyl-4’-cyanobiphenyl兲 LC monolayer deposited on rubbed PVA 共Fig. 2兲, showing how a rubbed PVA
surface aligns the LC monolayer.
Section II provides the theoretical background for the
work, and Sec. III describes the experimental arrangement.
Experimental results and data analysis are presented in Sec.
IV, and discussions of the results in Sec. V.
␹(2) ⫽ ␹NR
⫹N S具 ␣R(2) 典 f ⫽ ␹NR
⫹N S
␣R(2) 共 ␻ 2 兲 ⫽ 兺
P(2) 共 ␻ s 兲 ⫽ ⑀ 0 ␹(2) :E共 ␻ 1 兲 E共 ␻ 2 兲 ,
where ␹(2) denotes the surface nonlinear susceptibility tensor
关26兴. It can be shown that in MKS units the SFG output
intensity in the reflected direction is given by 关27兴
I共 ␻s兲⫽
␻ s2
8 ⑀ 0 c 3 cos2 ␤ s
(2) 2
兩 ␹ eff
兩 I共 ␻1兲I共 ␻2兲.
Here ␤ s is the exit angle of the SF output, I( ␻ i ) is the beam
is the effective surface nonlinear
intensity at ␻ i , and ␹ eff
susceptibility defined as
␹ eff
⫽ 关 L共 ␻ s 兲 •es 兴 • ␹(2) : 关 L共 ␻ 1 兲 •e1 兴关 L共 ␻ 2 兲 •e2 兴 , 共2.3兲
with ei being a unit polarization vector of the optical field at
␻ i , and L( ␻ i ) the tensorial Fresnel factor 共See Appendix A
for details兲.
␻ 2 ⫺ ␻ q ⫹i⌫ q
where aq , ␻ q , and ⌫ q are the amplitude, resonant frequency
and damping constant of the qth molecular vibrational mode.
Insertion of Eq. 共2.5兲 into Eq. 共2.4兲 gives
A. Surface sum-frequency and second-harmonic generation
Surface SFG results from a second-order nonlinear polarization induced at an interface by two input fields E( ␻ 1 ) and
E( ␻ 2 ) at frequencies ␻ 1 共visible兲 and ␻ 2 共infrared兲, respectively,
R 共 ⍀ 兲 f 共 ⍀ 兲 d⍀,
describes the nonresonant contribution, N S is the
where ␹NR
surface density of molecules, ⍀ denotes a set of orientational
angles ( ␪ , ␾ , ␺ ) defined in Fig. 1, and 具 典 f represents an orientational average over the orientational distribution function
f (⍀).
We assume that ␣R(2) is composed of Lorentzian resonant
␹(2) 共 ␻ 2 兲 ⫽ ␹NR
Aq ⫽N S
兺q ␻ 2 ⫺ ␻ qq⫹i⌫ q ,
aq 共 ⍀ 兲 f 共 ⍀ 兲 d⍀.
Being a rank-3 tensor, ␹(2) has in general 27 elements.
Surface symmetry, however, can make some elements vanish
and some become mutually dependent. In many cases, we
can determine all the independent nonvanishing ␹(2) ele(2)
with various beam geometries and
ments by measuring ␹ eff
polarization combinations. From the observed dispersion of
␹(2) we can deduce Aq in Eq. 共2.6兲. We can also obtain aq in
Eq. 共2.5兲 for the molecular nonlinear polarizability ␣R(2) from
other measurements or theoretical calculations. Then, Eq.
共2.7兲 will allow us to obtain an approximate orientational
distribution function f (⍀). Note that the microscopic localfield correction 关28兴 is not included in Eqs. 共2.4兲 and 共2.7兲.
As discussed in Appendix B, such correction has been partially included in the Fresnel factor L ZZ by an empirical surface dielectric constant ⑀ ⬘ .
Since we shall be interested in the stretch vibrations of the
CH2 groups of PVA, here we present a simple theoretical
model to calculate the resonant mode amplitude aq for CH2 .
There are two stretch vibrational modes for a CH2 group,
namely, the symmetric 共s兲 stretch and the antisymmetric 共a兲
stretch. The theory of sum-frequency vibrational spectros-
copy shows that aq is related to the infrared and Raman
characteristics of a vibrational mode through the equation
共 a q 兲 lmn ⫽⫺
1 ⳵ ␮ n ⳵ ␣ lm
2 ⑀ 0␻ q ⳵ Q q ⳵ Q q
the values of the dipole and polarizability derivatives of a
single C-H bond. Here we use the single bond dipole derivative deduced from Ref. 关31兴 共the negative sign is based on
the argument in Ref. 关32兴兲,
are the infrared dipole dewhere ⳵ ␮ n / ⳵ Q q and
rivative and the Raman polarizability tensor of the qth vibrational mode, and Q q is the classical normal coordinate. From
Eq. 共2.8兲 we can see some important properties of the tensor
aq . First, the Raman polarizability tensor is symmetric for
CH2 , and therefore (a q ) lmn is also symmetric in l and m, i.e.,
(a q ) lmn ⫽(a q ) mln . Second, since all the ⳵ ␮ n / ⳵ Q q and
⳵ ␣ lm
/ ⳵ Q q elements are real, (a q ) lmn must also be real, although ␹(2) can be complex due to the damping constant ⌫ q
in Eq. 共2.6兲.
We can use Eq. 共2.8兲 to calculate (a q ) lmn for the two
stretch modes of CH2 . If we assume that the carbon atom is
fixed in position, the normal coordinates of the s and a
stretch modes are
⬇⫺0.86 De / Å ⬇⫺2.9⫻10⫺20 C,
⳵ ⌬r
⳵ ␣ lm
Q s⫽
Q a⫽
⳵ ␣ (1)
⬇3.0⫻10⫺30 mC/V,
⳵ ⌬r
⳵ ␣⬜(1)
⳵ ␣ (1)
⬇0.4⫻10⫺30 mC/V.
⳵ ⌬r
⳵ ⌬r
After inserting these numbers into Eq. 共2.8兲, we obtain all
the nonvanishing aq elements for the CH2 stretch modes,
共 a s兲 ␰␰␨ ⬇0.16 a 0 ,
共 a s兲 ␩␩ ␨ ⬇0.82 a 0 ,
共 a s兲 ␨␨␨ ⬇0.49 a 0 ,
共 ⌬r 1 ⫺⌬r 2 兲 ,
共 a a兲 ␩ ␨ ␩ ⫽ 共 a a兲 ␨ ␩␩ ⬇0.66 a 0 ,
where ⌬r 1 and ⌬r 2 are the stretch distances of the two C-H
bonds 共Fig. 1兲, and m H is the mass of a hydrogen atom. The
dipole derivative ⳵ ␮ n / ⳵ Q q and the Raman polarizability
⳵ ␣ lm
/ ⳵ Q q of these two modes can be calculated by assuming that the total dipole moment 共or polarizability兲 is the sum
of the dipole moments 共or polarizabilities兲 of the two individual C-H bonds, and that the dipole moment 共or polarizability兲 of each individual C-H bond depends only on its own
coordinate. This bond additivity model has been used by
other authors in similar calculations 关30兴. Using this model,
we find
⳵ ␮1
⳵ ␮2
⳵ Q s 冑2m H ⳵ ⌬r 1 ⳵ ⌬r 2
⳵ ␮1
⳵ ␮2
⳵ Q a 冑2m H ⳵ ⌬r 1 ⳵ ⌬r 2
and the single bond polarizability derivative from Ref. 关33兴,
共 ⌬r 1 ⫹⌬r 2 兲 ,
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⳵ ␣(1)
⳵ ␣(1)
⳵ ␣(1)
⳵ Q s 冑2m H ⳵ ⌬r 1 ⳵ ⌬r 2
⳵ ␣(1)
⳵ ␣(1)
⳵ ␣(1)
⳵ Q a 冑2m H ⳵ ⌬r 1 ⳵ ⌬r 2
By symmetry, for the s stretch there are three independent
nonvanishing (a s) lmn elements (a s) ␰␰␨ , (a s) ␩␩ ␨ , and
(a s) ␨␨␨ , and for the a stretch there is only one: (a a) ␩ ␨ ␩
⫽(a a) ␨ ␩␩ . Here ( ␰ , ␩ , ␨ ) are the molecular coordinates defined in Fig. 1. To carry out numerical calculations, we need
where a 0 is a constant defined for a single C-H bond:
a 0 ⫽⫺
⳵ ␮ ⳵ ␣ (1)
⬇5.3⫻10⫺27 m4 V⫺1 sec⫺1 .
2 ⑀ 0 ␻ q m H ⳵ ⌬r ⳵ ⌬r
C. SHG for a liquid crystal monolayer
SHG has been used successfully to measure the molecular
orientational distributions of liquid crystal monolayers. A detailed description of the theory and technique can be found in
Ref. 关6兴. Similar to SFG, the surface SHG nonlinear susceptibility is also an orientational average of the nonlinear polarizability of surface molecular groups.
For 8CB, the SHG nonlinear polarizability mainly originates from highly delocalized electrons in the cyanobiphenyl
along the
group, and has one dominant tensor element ␣ ␨␨␨
long molecular axis ␨ 共Fig. 2兲. As a good approximation 关6兴,
we can neglect other tensor elements of ␣(2) , and express
␹ (2)
i jk as
␹ (2)
i jk ⫽N S具 共 î • ␨ 兲共 ĵ• ␨ 兲共 k̂• ␨ 兲 典 g ␣ ␨␨␨
⫽N S
g 共 ⍀ 兲 d⍀,
共 î • ␨ˆ 兲共 ĵ• ␨ˆ 兲共 k̂• ␨ˆ 兲 ␣ ␨␨␨
where N S is the surface number density of 8CB molecules; î ,
ĵ, and k̂ are the lab coordinates defined on the substrate; and
具 典 g denotes an average over the orientational distribution
function g(⍀). By measuring all the nonvanishing independent ␹ (2)
i jk elements we can deduce an approximate g(⍀).
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FIG. 3. Geometry of the incidence plane with respect to the
sample surface in the SFG experiment. The rubbing direction is
along x.
Polyvinyl alcohol 共Scientific Polymer Products, Inc.,
M.W.⫽14,000, 100% hydrolyzed兲 was dissolved in water
共1.5% weight兲 and spin coated on fused quartz plates 共hydrophilic兲, followed by baking and rubbing with velvet cloth.
The film thickness was about 30 nm, and the rubbing
strength used was at a saturation level, i.e., stronger rubbing
would not improve the chain alignment further 关6兴.
The SFG experimental setup has been described elsewhere 关24兴. In this experiment, a visible beam at 532 nm and
an IR beam tunable from 2.6 to 3.7 ␮ m to cover the CH
stretch region 共with a linewidth ⬇6 cm⫺1 ), both having a
15-ps pulse width and a 20-Hz repetition rate, were overlapped at the sample with incidence angles ␤ 1 ⫽45° and
␤ 2 ⫽57°, respectively 共Fig. 3兲. The SFG output was detected
in the reflected direction. The sample was mounted on a
360° rotational stage, and the SFG spectra were taken with
various input-output polarization combinations and different
sample orientations specified by the azimuthal angle ␥ between the incidence plane and the rubbing direction 共Fig. 3兲.
In the SHG study of LC alignment on PVA, 8CB was
deposited on a rubbed PVA surface by evaporation, and a
visible laser beam at 532 nm was directed onto the sample
with an incidence angle of 67° 共see Ref. 关6兴 for details兲. The
SHG output was detected in the reflected direction. Four different input-output polarization combinations (s in-s out , s inp out , p in-s out , and p in-p out) were used. The azimuthal variation of SHG was also measured.
FIG. 4. SFG spectra of a rubbed PVA surface in the CH stretch
range for different polarization combinations. Only spectra at ␥
⫽0° and 90° are shown for ssp, sps, and ppp. The spectra at ␥
⫽90° for sss, spp, and pps are dominated by noise. Solid curves
are fits from Eq. 共2.6兲.
sp p, and p ps. For comparison, the spectra of ssp and sps
from an unrubbed PVA surface are shown in Fig. 5.
All the measured SFG intensities have been calibrated
with a reference z-cut quartz crystal 共see Appendix C for
details兲, yielding for each polarization combination and
(2) 2
sample orientation a spectrum of 兩 ␹ eff
兩 in MKS units ac(2)
cording to Eq. 共2.2兲. All the 兩 ␹ eff ( ␻ 2 ) 兩 2 spectra can then be
fit by
A. SFG spectra and mode amplitudes
Surface SFG spectra in the CH stretch region of a rubbed
PVA sample were taken with six different polarization combinations, each with different azimuthal orientations 共typically every 45°). The spectra were found to be reproducible
for different PVA samples prepared under the same conditions. Shown in Fig. 4 are the SFG spectra with the azimuthal angle ␥ ⫽0° 共parallel to rubbing兲 and ␥ ⫽90° 共perpendicular to rubbing兲 for the ss p 共s-, s- and p-polarized SF
output, visible input and infrared input, respectively兲, sps
and ppp polarization combinations, and ␥ ⫽90° for sss,
␹ eff
共 ␻ 2 兲 ⫽ ␹ NR,eff
兺q ␻ 2 ⫺ ␻q,eff
q ⫹i⌫ q
assuming the presence of three resonant modes at
␻ q /(2 ␲ c)⫽ 2882, 2907, and 2940 cm⫺1 , each with a damping constant ⌫/(2 ␲ c)⫽16 cm⫺1 . The first one, which is
rather weak, probably comes from the stretch mode of the
CH group on the PVA chain; the last two, highly prominent
except for some polarization combinations and sample orientations, can be identified with the symmetric 共s兲 and antisymmetric 共a兲 stretch modes of CH2 关34兴. The azimuthal polar
plots of the mode amplitudes A s,eff and A a,eff deduced by
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FIG. 5. SFG spectra of unrubbed PVA for different polarization
combinations 共a兲 ssp and 共b兲 sps. The solid and dashed lines are
theoretical predictions assuming different orientational distributions
共see Sec. V B兲.
fitting of the spectra are presented in Fig. 6. Some additional
data points of A s,eff(ss p) and A a,eff(s ps) in Fig. 6 were deduced from the measured SFG peak intensity for every 5°
with the infrared input frequency fixed on the resonant peak.
A s,eff(sps) is below the noise level, and not shown.
Note that the measurements described above could not
determine the relative signs of A q,eff for different polarization
combinations. These relative signs, however, can be determined by measuring interference between different A q,eff
components. For example, the SFG intensity with
p-polarized infrared input, mixed (m-, partially s- and partially p-兲 polarized visible input and m-polarized SFG output
is proportional to the absolute square of the linear combination of A q,eff(ss p) and A q,eff(ppp), which can interfere constructively or destructively depending on their relative sign.
From such interferences, we were able to determine the relative signs of all A q,eff . As shown in Fig. 6, if we choose the
sign of A s,eff(ss p) to be positive, we find A a,eff(sps) and
A a,eff(ppp) positive, and A a,eff(ss p) and A s,eff(pp p) negative. Later, we will use these measured A q,eff including their
signs to deduce the tensor Aq through Eq. 共2.3兲.
B. Surface specificity
FIG. 6. Polar plots of the effective mode amplitudes 兩 A q,eff兩 of
the CH2 symmetric 共circles兲 and antisymmetric 共squares兲 stretch
modes as functions of ␥ for different polarization combinations 共a兲
ssp, 共b兲 sps, and 共c兲 ppp. The relative signs of A q,eff are also
shown in the figure. Symbols are values deduced from the measured
SFG spectra. Lines are obtained from fits 共see Sec. IV D兲.
In surface sum-frequency spectroscopy, there is always
the question whether the SFG signal indeed comes from the
surface under investigation 关35兴. In order to deduce surface
structure from SFG spectra, one has to make sure that the
SFG signal is dominated by the surface contribution.
For the rubbed PVA sample, the SFG signal might come
from the bulk through electric-quadrupole and magneticdipole contributions, or from the interface between PVA and
the fused quartz substrate. However, the fact that the SFG
spectra of PVA are comparable in intensity to those observed
from a closely packed monolayer of alkyl chains 关37兴 indicates that the SFG signal of PVA originates from a monolayer of CH2 pointing out of the polymer. The electricquadrupole and magnetic-dipole contributions from CH2 in
the bulk are much weaker 关36兴, and significant contribution
from the polymer-quartz interface with CH2 pointing toward
the quartz side is unlikely because the quartz surface is hydrophilic 关38兴. Chemical studies indicate 关39兴 that the monomer units in PVA ‘‘prefer’’ a head-to-tail arrangement, i.e.,
the OH groups are on alternate carbon atoms, and therefore
all the CH2 groups on a straight PVA chain should be on the
same side of the chain, as illustrated in Fig. 1. Being hydrophobic, the CH2 groups like to point out of the polymer
surface. It is also known from infrared absorption studies
关34兴 that strong hydrogen bonding exists between adjacent
PVA chains. In order to maximize the number of hydrogen
bonds to lower the surface free energy, the top layer of PVA
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FIG. 7. The SFG spectra of two PVA samples measured through
the fused quartz substrates. The SFG signal from 共a兲 the thick PVA
sample on fused quartz is dominated by the PVA–fused quartz
interface, while the signal from 共b兲 the thin spin-coated PVA film
on fused quartz is dominated by the PVA-air interface. The polarization combination used is ssp. Note that the intensity of the spectrum from the PVA–air interface is higher than that in Fig. 5共a兲,
because the Fresnel factors are different for the facing-down geometry.
chains would orient their OH bonds into the bulk, and leave
the CH2 groups more or less polar ordered and pointing into
To check whether SFG from the PVA-air interface indeed
dominated, we prepared a thick PVA sample (⬃500 ␮m兲 on
a fused quartz substrate, and measured the reflected SFG
spectrum through the fused quartz substrate with the PVA
layer facing down. From this geometry, the SFG signal from
the PVA-air interface is negligible because 共1兲 the PVA surface on the air side is rough, and 共2兲 the infrared input is
completely attenuated by the thick PVA layer. The result is
shown in Fig. 7, in comparison with the SFG spectrum from
a thin (⬃30 nm兲 spin-coated PVA film 共unrubbed兲 which
was also measured through the fused quartz substrate. It
clearly shows that the SFG contribution from the PVA–
fused quartz interface is much weaker than that from the
PVA–air interface.
We also measured the SFG spectrum of the thin PVA film
through the fused quartz substrate, as we put the sample in
contact with water, and found that the strong SFG signal
disappeared. After drying the film, the spectrum reappeared,
indicating that the PVA film had not been dissolved in water.
This is in agreement with our understanding of the strong
SFG signal from the PVA-air interface. With the sample in
contact with water, strong hydrogen bonding between water
and PVA would randomize the orientation of the surface
CH2 groups, causing a drastic decrease of the SFG intensity.
Interestingly, this test experiment may also provide an example of environment-induced surface structural change of
polymers, another important topic in polymer science and
technology 关40兴.
FIG. 8. Sketches showing that with the given CH2 orientation,
the beam geometry ␥ ⫽0° in 共a兲 allows observation of the CH2 s
stretch mode by the ssp polarization combination, and the CH2 a
stretch mode by sps, in the SFG spectra, and the beam geometry
␥ ⫽90° in 共b兲 allows observation of only the CH2 s-stretch mode by
the ssp polarization combination.
C. Qualitative analysis of the SFG spectra
Without any calculation, we can already obtain some
qualitative information about the CH2 orientation on the
rubbed PVA surface. This is illustrated in Fig. 8.
First, the excitation of the CH2 symmetric stretch requires
an IR polarization component along the CH2 symmetry axis
␨ defined in Fig. 1. The fact that the mode is very strong for
the ss p polarization combination 共with the last index p being
the infrared polarization兲, but very weak for the sps for all
␥ , indicates that the CH2 axis ␨ must be nearly along the z
axis. Second, the excitation of the CH2 antisymmetric stretch
requires an IR component along the axis ␩ in the CH2 plane.
Since this mode is very strong for sps at ␥ ⫽0° 关Fig. 8共a兲兴,
but very weak at ␥ ⫽90° 关Fig. 8共b兲兴, the CH2 plane must be
nearly the y-z plane. Correspondingly, the PVA chains must
be oriented nearly parallel to the surface along the x axis, the
rubbing direction. Finally, as seen from the plot of
A s,eff(ss p) in Fig. 6, there is a small forward-backward
asymmetry for ␥ ⫽0° and 180°. As will be shown below,
this indicates that the average chain orientation has a slight
upward tilt along the rubbing direction.
D. Quantitative analysis of the SFG data
As mentioned earlier, the surface nonlinear susceptibility
␹ (2)
i jk „correspondingly, (A q ) i jk for each vibrational mode… has
27 matrix elements, but symmetry may greatly reduce the
number of independent nonvanishing elements. First, because the rubbed PVA surface has C 1 v symmetry with the
x-z plane being a mirror plane, all the ␹ (2)
i jk elements with
indices i jk containing an odd number of y should vanish.
Second, as discussed in Sec. II B, we have the symmetry
␹ (2)
i jk ⫽ ␹ jik . Combining these two, we find that (A q ) i jk for
each CH2 mode has only ten independent nonvanishing elements with the following indices:
y yz,
yzy⫽zy y,
y yx,
xy y⫽yxy,
Furthermore, we note that the forward-backward asymmetry shown in Fig. 6 is very small, and that the SFG spectra
for the sss, s pp, and pps polarization combinations are too
weak to distinguish from noise 共Fig. 4兲. These suggest that
we may first use C 2 v as an approximation for the symmetry
of the rubbed PVA surface, and neglect the last five matrix
elements listed above which contain an odd number of x.
With this approximation, we can deduce the five major (C 2 v
allowed兲 (A q ) i jk elements for both s and a stretches of CH2 .
Later, the observed forward-backward asymmetry will be
used to deduce some of the remaining (C 2 v forbidden兲 elements if possible.
From the theory described in Sec. II, we find that A q,eff in
Eq. 共4.1兲 is related to (A q ) i jk through
A q,eff⫽ 关 L共 ␻ s 兲 •es 兴 •Aq : 关 L共 ␻ 1 兲 •e1 兴关 L共 ␻ 2 兲 •e2 兴 . 共4.2兲
Neglecting the forward-backward asymmetry, we can write
A q,eff for the three polarization combinations in terms of the
five major (A q ) i jk elements:
A q,eff共 ␥ ,ss p 兲 ⫽sin ␤ 2 L Y Y 共 ␻ s 兲 L Y Y 共 ␻ 1 兲 L ZZ 共 ␻ 2 兲
⫻ 关共 A q 兲 y yz cos2 ␥ ⫹ 共 A q 兲 xxz sin2 ␥ 兴 ,
A q,eff共 ␥ ,sps 兲 ⫽sin ␤ 1 L Y Y 共 ␻ s 兲 L ZZ 共 ␻ 1 兲 L Y Y 共 ␻ 2 兲
TABLE I. Measured and calculated nonvanishing tensor elements A i jk for the CH2 symmetric 共s兲 and antisymmetric 共a兲 stretch
modes. All values are in unit of 10⫺10 m2 V⫺1 sec⫺1 .
s stretch
measured calculated
a stretch
A xxz
A y yz
A zzz
A xzx ⫽A zxx
A yzy ⫽A zy y
A xxx
A y yx
A zzx
A xy y ⫽A yxy
A xzz ⫽A zxz
共 A q 兲 xzx
⑀ ⬘共 ␻ 1 兲
共 A q 兲 zxx
⑀ ⬘共 ␻ 1 兲
共 A q 兲 yzy
⑀ ⬘共 ␻ 1 兲
共 A q 兲 zy y
⑀ ⬘共 ␻ 1 兲
instead of (A q ) i jk . Here, we have neglected the dispersion of
⑀ ⬘ in the visible so that ⑀ ⬘ ( ␻ s )⫽ ⑀ ⬘ ( ␻ 1 ). Using these five
‘‘reduced’’ (A q ) i jk elements as independent fitting parameters we can fit the experiment data well, as shown by the
solid lines in Fig. 6, except for the forward-backward asymmetry of A s,eff(ss p) which has been neglected so far.
In general we cannot separate ⑀ ⬘ ( ␻ i ) from (A q ) i jk purely
by SFG measurement. However, there exist the following
equations relating some of the (A q ) i jk that are specific to the
CH2 stretch vibrations:
共 A s兲 xxz ⫹ 共 A s兲 y yz ⫹ 共 A s兲 zzz
⫽N S具 ␨ˆ •ẑ 典 f 关共 a s兲 ␰␰␨ ⫹ 共 a s兲 ␩␩ ␨ ⫹ 共 a s兲 ␨␨␨ 兴 ,
⫻ 关共 A q 兲 yzy cos ␥ ⫹ 共 A q 兲 xzx sin ␥ 兴 ,
PRE 62
共 A s兲 xzx ⫹ 共 A s兲 yzy ⫹ 共 A s兲 zzz ⫽N S具 ␨ˆ •ẑ 典 f 共 a s兲 ␨␨␨ ,
A q,eff共 ␥ ,ppp 兲
⫽⫺cos ␤ s cos ␤ 1 sin ␤ 2 L XX 共 ␻ s 兲 L XX 共 ␻ 1 兲 L ZZ 共 ␻ 2 兲
⫻ 关共 A q 兲 xxz cos2 ␥ ⫹ 共 A q 兲 y yz sin2 ␥ 兴
⫹sin ␤ s sin ␤ 1 sin ␤ 2 L ZZ 共 ␻ s 兲 L ZZ 共 ␻ 1 兲 L ZZ 共 ␻ 2 兲
⫻ 共 A q 兲 zzz .
Here, for A q,eff( ␥ ,p pp), we have neglected the contributions
from (A q ) xzx , (A q ) zxx , (A q ) yzy , and (A q ) zy y , which nearly
cancel out themselves simply because ␤ s ⬇ ␤ 1 . All the
Fresnel factors L ii in Eqs. 共4.3兲–共4.5兲 can be calculated 共Appendix A兲 except L ZZ , in which the surface dielectric constant ⑀ ⬘ is unknown 共Appendix B兲. Therefore, with Eqs.
共4.3兲–共4.5兲, we can only determine the quantities
共 A q 兲 xxz
⑀ ⬘共 ␻ 2 兲
共 A q 兲 y yz
⑀ ⬘共 ␻ 2 兲
共 A q 兲 zzz
⑀ ⬘共 ␻ 1 兲 2⑀ ⬘共 ␻ 2 兲
共 A a兲 xxz ⫹ 共 A a兲 y yz ⫹ 共 A a兲 zzz ⫽0,
共 A a兲 xzx ⫹ 共 A a兲 yzy ⫹ 共 A a兲 zzz ⫽N S具 ␨ˆ •ẑ 典 f 共 a a兲 ␩ ␨ ␩ .
The derivation of Eq. 共4.6兲 can be found in Appendix D.
These additional equations of constraint allow us to deduce
⑀ ⬘ ( ␻ i ) from experiment without knowing the actual orientational distribution of the CH2 groups. Applying Eq. 共4.6兲 to
the five ‘‘reduced’’ (A q ) i jk elements deduced from experiment, we find ⑀ ⬘ ( ␻ 1 )⫽2.1 and ⑀ ⬘ ( ␻ 2 )⫽1.5. We can then
obtain the five corresponding (A q ) i jk elements; their values
are listed in Table I and labeled as ‘‘measured.’’
Finally we include the forward-backward asymmetry observed in the SFG spectra for the ss p polarization combination. Including contributions from (A q ) y yx and (A q ) xxx , Eq.
共4.3兲 becomes
PRE 62
A q,eff共 ␥ ,ss p 兲 ⫽sin ␤ 2 L Y Y 共 ␻ s 兲 L Y Y 共 ␻ 1 兲 L ZZ 共 ␻ 2 兲
⫻ 关共 A q 兲 y yz cos2 ␥ ⫹ 共 A q 兲 xxz sin2 ␥ 兴
⫹cos ␤ 2 L Y Y 共 ␻ s 兲 L Y Y 共 ␻ 1 兲 L XX 共 ␻ 2 兲
⫻ 关共 A q 兲 y yx cos3 ␥ ⫹ 共 A q 兲 xxx cos ␥ sin2 ␥ 兴 .
It turns out that (A s) y yx is the only additional element that
can be determined with sufficient accuracy. The dashed line
in Fig. 6共a兲 is the fit with a non-zero (A s) y yz . The deduced
value of (A s) y yx is listed in Table I.
With these measured values of (A q ) i jk , we can then use
Eqs. 共2.7兲 and 共2.14兲 to obtain an approximate orientational
distribution function f ( ␪ , ␾ , ␺ ) for the CH2 groups. Knowing
that the PVA chains are quite well aligned, we can assume a
Gaussian distribution
f 共 ␪ , ␾ , ␺ 兲 ⫽C exp ⫺
共 ␪⫺␪0兲2
2 ␴ 2␪
共 ␾⫺␾0兲2
2 ␴ ␾2
共 ␺⫺␺0兲2
2 ␴ ␺2
where C is a normalization constant; ␾ 0 ⫽ ␺ 0 ⫽0° by symmetry; and ␪ 0 , ␴ ␪ , ␴ ␾ , and ␴ ␺ are parameters to be determined. For this calculation, the distribution function
f ( ␪ , ␾ , ␺ ) is defined such that the probability of finding a
CH2 group oriented at ( ␪ ⬘ , ␾ ⬘ , ␺ ⬘ ) in the range ␪ ⬍ ␪ ⬘ ⬍ ␪
⫹d ␪ , ␾ ⬍ ␾ ⬘ ⬍ ␾ ⫹d ␾ , and ␺ ⬍ ␺ ⬘ ⬍ ␺ ⫹d ␺ is equal to
f ( ␪ , ␾ , ␺ )d ␪ d ␾ d ␺ . We find, for the best fit,
␪ 0 ⫽2.5°⫾0.7°,
␴ ␪ ⫽26°⫾5°,
␴ ␾ ⫽27°⫾5°,
␴ ␺ ⫽35°⫾5°.
These values, when used with Eq. 共4.8兲 in Eq. 共2.7兲 to calculate (A q ) i jk , reproduce almost all the measured (A q ) i jk
values within the experimental error, as shown in the column
‘‘calculated’’ in Table I. One may notice that the number of
experimentally deduced (A q ) i jk far exceeds the number of
input parameters 关 ␪ 0 , ␴ ␪ , ␴ ␾ , ␴ ␺ , ⑀ ⬘ ( ␻ 1 ), ⑀ ⬘ ( ␻ 2 ), and
N S ] used for this calculation. The fact that we can still consistently reproduce all the measured (A q ) i jk values indicates
that Eq. 共4.8兲 is a good representation of the orientational
The above results focus on the surface CH2 groups. Since
the CH2 molecular plane is perpendicular to the PVA chain
locally, the same set of parameters ␪ and ␾ also describe the
orientation of the polymer chains. The values listed above
indicate that the PVA chains lie almost flat and are well
aligned on the surface with an average 2.5° upward tilt along
the rubbing direction.
E. SHG study of 8CB monolayer on rubbed PVA
As described in Sec. II C, the molecular orientational distribution of an 8CB monolayer adsorbed on a rubbed polymer surface can be determined from SHG measurements.
FIG. 9. Polar plots of SHG intensities 共arbitrary units兲 from an
8CB monolayer on a rubbed PVA surface. 0° and 180° are the
rubbing and antirubbing directions, respectively. Circles are the experiment data and solid lines are the theoretical fits. The inputoutput polarization combinations are 共a兲 s in-s out ; 共b兲 s in-p out , 共c兲
p in-s out , and 共d兲 p in-p out .
Deposition of the 8CB monolayer was monitored by SHG as
an in situ probe 关41兴. In Fig. 9 we present the SHG intensities from the 8CB monolayer on rubbed PVA as a function
of the sample azimuthal angle ␥ for different input-output
polarization combinations.
The 8CB monolayer on rubbed PVA also has a macroscopic C 1 v symmetry, which restricts the number of nonvanishing independent ␹ (2)
i jk elements to six under the approximation that ␣ ␨␨␨ dominates over other nonvanishing
elements of ␣(2) of 8CB. As listed in Ref. 关6兴, they are
␹ zzz
␹ xxx
␹ (2)
xy y ⫽ ␹ yxy ⫽ ␹ y yx ,
␹ xzz
⫽ ␹ zxz
⫽ ␹ zzx
␹ zxx
⫽ ␹ xzx
⫽ ␹ xxz
␹ (2)
zy y ⫽ ␹ yzy ⫽ ␹ y yz ,
and can be deduced by fitting the data in Fig. 9 using Eqs.
共2.2兲 and 共2.3兲. The fit is plotted as solid lines in Fig. 9, and
the deduced nonvanishing ␹ (2)
i jk elements are presented in
Table II.
As in Ref. 关6兴, we can assume for 8CB molecules in the
monolayer an orientational distribution of the form
g 共 ␪ LC , ␾ LC兲 ⫽C exp ⫺
共 ␪ LC⫺ ␪ LC,0兲 2
关 1⫹d 1 cos ␾ LC
⫹d 2 cos共 2 ␾ LC兲 ⫹d 3 cos共 3 ␾ LC兲兴 ,
PRE 62
TABLE II. Nonvanishing independent ␹ i(2)
jk elements of the 8CB
monolayer on rubbed PVA deduced from the SHG experiment, with
the zzz component normalized to 1.
(2) ⬘ 3
␹ zzz
␹ xxx
␹ xy
(2) ⬘ 2
␹ xzz
␹ zxx
␹ zy
FIG. 10. Proposed polymer chain distribution 共top view兲 on a
PVA surface 共a兲 before and 共b兲 after rubbing.
where ␪ LC and ␾ LC are the polar and the azimuthal angles
defined in Fig. 2 , and ␪ LC,0 , ␴ , d 1 , d 2 , and d 3 are five
independent parameters to be determined from the five measured ratios in Table II using Eq. 共2.16兲 with a given value of
⑀ ⬘ . The results are presented in Table III. In this case, ⑀ ⬘
cannot be determined separately, and the assumption ⑀ ⬘ ( ␻ )
⫽ ⑀ ⬘ (2 ␻ ) used here also may not be true because of the
electronic resonance of 8CB molecules at the secondharmonic frequency. Nevertheless, as shown in Table III,
varying ⑀ ⬘ from 1 to 2.25 mainly changes the deduced values
of ␪ LC,0 and ␴ , and has little effect on the parameters d 1 , d 2 ,
and d 3 which describe the azimuthal distribution.
A. Surface density of CH2 groups
As seen in Eq. 共2.7兲, N S , the surface density of CH2
groups, is a parameter needed in our quantitative analysis of
the SFG data. To obtain the best-fit values of (A q ) i jk listed in
the ‘‘calculated’’ column in Table I, we used
N S⫽ 共 1.7⫾0.2兲 ⫻1015 cm⫺2 ,
which seems too large considering that the CH2 surface density calculated from the PVA crystalline structure 关42兴 is
only about 7⫻1014 cm⫺2 . This is presumably because in our
calculation we have neglected the factor l 储 ( ␻ s )l 储 ( ␻ 1 )l 储 ( ␻ 2 )
resulting from the microscopic local-field effect. As discussed in Appendix B, N S in Eq. 共2.7兲 should be replaced by
N S l 储 ( ␻ s )l 储 ( ␻ 1 )l 储 ( ␻ 2 ). References 关43,44兴 show that l 储 is
usually larger than 1, which makes the value of N S closer to
the expected one.
B. Effect of rubbing on PVA surface structure
We have presented the SFG spectra of the ss p and sps
polarization combinations for the unrubbed PVA sample
TABLE III. Deduced parameters in g( ␪ LC , ␾ LC) for various surface dielectric constant ⑀ ⬘ of the 8CB monolayer.
␪ 0 共deg兲
␴ 共deg兲
共Fig. 5兲. The spectra from the rubbed and unrubbed samples
look similar except that there is no azimuthal spectral variation for the unrubbed one. The dashed lines in Fig. 5 are
predicted spectra for the unrubbed sample assuming the
same Gaussian distribution for ␪ and ␺ deduced earlier for
the rubbed one, but a uniform distribution in ␾ . The absolute
intensities of the measured spectra are slightly lower than
predicted. A somewhat broader distribution in ␪ and ␺ ( ␴ ␪
⫽35°, ␴ ␺ ⫽45°) fits the spectra well 共solid lines兲.
Based on these results, a possible scenario of rubbing induced PVA chain ordering is proposed in Fig. 10. Before
rubbing, some sections of the PVA chains were lying more
or less flat on the surface and isotropic in the plane, with
their ends presumably buried in the bulk. During rubbing,
fibers on the rubbing material would grab the surface polymer chains, stretch them in the rubbing direction and even
pull some chain sections out from the subsurface, resulting in
stacked elongated half loops one on top of another. This
would explain not only the azimuthal chain ordering in the
rubbing direction, but also the forward-backward asymmetry.
C. Molecular interaction between 8CB and PVA
From the SFG and SHG studies we have obtained the
orientational distribution functions f ( ␪ , ␾ , ␺ ) for the surface
PVA chains and g( ␪ LC , ␾ LC) for the molecules in the 8CB
monolayer independently. It is important to find the correlation between them. In order to do so, we calculate a grand
azimuthal distribution function F( ␾ ) for PVA chains by integrating f ( ␪ , ␾ , ␺ ) over ␪ and ␺ , and also G( ␾ LC) for the
8CB molecules by integrating g( ␪ LC , ␾ LC) over ␪ LC . However, we note that unlike an 8CB molecule, a section of PVA
chain has no polarity, i.e., ( ␪ , ␾ ) and (⫺ ␪ , ␾ ⫹180°) describe the same chain orientation. To define f ( ␪ , ␾ , ␺ ) over
all orientations, we can limit ␪ between 0° and 90° and vary
␾ over the entire 360°. We naturally use the same limiting
ranges for ␪ LC and ␾ LC to define g( ␪ LC , ␾ LC) for polar 8CB
A polar plot of 冑F( ␾ ) and 冑G( ␾ LC) are presented in Fig.
11. The correlation between the two is remarkable. As we
expected, the rubbed PVA surface appears to be more ordered in the azimuthal distribution than the adsorbed 8CB
monolayer. This suggests that the rubbed polymer surface
indeed serves as a molecular template to align LC molecules
through short-range molecular interaction 关45兴.
The forward-backward asymmetry of the 8CB orientational distribution 共represented by the positive coefficient
d 1 ⫽0.07) indicates that the 8CB molecules prefer to align in
the forward direction. This must be somehow related to the
average upward tilt angle ( ␪ 0 ⫽2.5°) of the PVA chains.
PRE 62
TABLE IV. Some parameters of the three beams and the calculated Fresnel factors for the air–fused quartz interface 共applicable to
the top surface only兲.
FIG. 11. Polar plot of the grand azimuthal distribution functions
of the PVA chains 共dashed line兲 and 8CB molecules 共solid line兲 on
a rubbed PVA surface. Square root values are used, so that the total
areas inside the two curves remain constant.
Similar results have also been found from other rubbed polymers 关16,21,22兴, yet no theoretical model is available to correlate these two tilt angles quantitatively. There are, however, some qualitative explanations. For example, it was
assumed that rubbing induces a saw-tooth-like polymer surface which leads to a homogeneous LC alignment with a
forward pretilt angle 关46兴. This is consistent with the scenario we proposed for the rubbed PVA surface 共Fig. 10兲, in
which the 8CB molecules adsorbed on the back-slanted terraces would appear to align more in the forward direction.
As demonstrated in Ref. 关8兴, the LC monolayer then governs
the forward pretilt angle of a bulk LC film.
We have used SFG surface vibrational spectroscopy to
determine a quantitative orientational distribution of the
polymer chains at the very top surface of a rubbed PVA
sample. We have also used SHG to determine the orientational distribution of a monolayer of 8CB molecules adsorbed on rubbed PVA. Comparison of the two in the azimuthal plane shows that they are well correlated. This
strongly supports the belief that ‘‘orientational epitaxy’’ is
the mechanism responsible for the surface-induced LC bulk
alignment by rubbed polymer surfaces. We have proposed a
possible scenario on how rubbing changes the polymer chain
conformation at the surface, which is subject to future experimental tests. This work is also a demonstration to show
that SFG vibrational spectroscopy can be an effective tool to
probe quantitatively the surface structure of a polymer, with
or without external perturbation.
The authors gratefully acknowledge Dr. A. I. Lvovsky’s
help in the numerical calculations. This work was supported
by the NSF through Grant No. DMR-9704384.
The Fresnel factors of an interface between two continuous media with dielectric constants ⑀ 1 and ⑀ 2 are 关24兴
460 nm
532 nm
3.4 ␮ m
1.07/⑀ s⬘
1.08/⑀ 1⬘
0.98/⑀ 2⬘
e X L XX (p)
e Y L Y Y (s)
e Z L ZZ (p)
0.78/⑀ s⬘
0.77/⑀ 1⬘
0.82/⑀ 2⬘
L XX 共 ␻ i 兲 ⫽
2 ⑀ 1 共 ␻ i 兲 k 2Z 共 ␻ i 兲
⑀ 2 共 ␻ i 兲 k 1Z 共 ␻ i 兲 ⫹ ⑀ 1 共 ␻ i 兲 k 2Z 共 ␻ i 兲
LY Y共 ␻i兲⫽
L ZZ 共 ␻ i 兲 ⫽
2k 1Z 共 ␻ i 兲
k 1Z 共 ␻ i 兲 ⫹k 2Z 共 ␻ i 兲
2 ⑀ 1 共 ␻ i 兲 ⑀ 2 共 ␻ i 兲 k 1Z 共 ␻ i 兲
⑀ 2 共 ␻ i 兲 k 1z 共 ␻ i 兲 ⫹ ⑀ 1 共 ␻ i 兲 k 2Z 共 ␻ i 兲 ⑀ ⬘ 共 ␻ i 兲
where ⑀ ⬘ ( ␻ i ) is an empirical dielectric constant of the surface monolayer at ␻ i . The physical meaning of ⑀ ⬘ ( ␻ i ) will
be discussed in Appendix B.
The Fresnel factors for the surface of a thin film coated on
a substrate are slightly more complicated. Details can be
found in the appendixes of Ref. 关6兴. In this experiment, the
coated PVA film turns out to have little effect on the Fresnel
factors because the thickness of the PVA film is only ⬃30
nm, much less than an optical wavelength, and the refractive
index of PVA is not too different from that of the fused
quartz substrate. Therefore we can use the calculated Fresnel
factors listed in Table IV for the air–fused quartz interface as
a good approximation.
In the theory of surface nonlinear optical spectroscopy we
have introduced a dielectric constant ⑀ ⬘ for the surface layer,
which appears in the Fresnel factor L ZZ in Eq. 共A1兲. From
the theoretical point of view, the dielectric constant is not
well defined for a monolayer because it is a macroscopic or
mesoscopic property. However, we realize that ⑀ ⬘ can be
interpreted as a result of the microscopic local-field correction in a monolayer 关43,44兴, which needs to be addressed
We consider a surface monolayer of molecules at an interface between two media with dielectric constants ⑀ 1 and
⑀ 2 . The local-field components experienced by these molecules are
E X(Loc) ⫽l XX L XX E X ,
⫽l Y Y L Y Y E Y ,
E (Loc)
E Z(Loc) ⫽l ZZ L ZZ E Z ,
where l ii denotes the microscopic local-field correction factor, L ii is the Fresnel or macroscopic local-field factor 关defined by Eq. 共A1兲 but without the factor 1/⑀ ⬘ ], and E i is the
electric field component of the incoming and outgoing optical plane waves. If l XX ⫽l Y Y ⫽l 储 and l ZZ ⫽l⬜ , the total localfield factors including both macroscopic and microscopic effects are
F XX ⫽l XX L XX ⫽l 储
F ZZ ⫽l ZZ L ZZ ⫽l⬜
2k 1Z
k 1Z ⫹k 2Z
2 ⑀ 1 ⑀ 2 k 1Z
⑀ 2 k 1Z ⫹ ⑀ 1 k 2Z
It has been shown that the value of ⑀ ⬘ defined this way is
usually between 1 and the bulk dielectric constant ⑀ 关44兴.
The physical meaning of ⑀ ⬘ now becomes clear; it is simply
the ratio of l 储 and l⬜ . By introducing the factor 1/⑀ ⬘ in Eq.
共A1兲, we have partially included the microscopic local-field
correction. To have it fully included, Eq. 共2.4兲 should be
changed to
and Eq. 共2.7兲 becomes
Aq ⫽N Sl 储 共 ␻ s 兲 l 储 共 ␻ 1 兲 l 储 共 ␻ 2 兲
aq 共 ⍀ 兲 f 共 ⍀ 兲 d⍀.
460 nm
532 nm
3.4 ␮ m
␹ (2)
xyz ⫽⫺ ␹ yxz ,
␹ (2)
xzy ⫽⫺ ␹ yzx ,
␹(2) ⫽ ␹NR
⫹N Sl 储 共 ␻ s 兲 l 储 共 ␻ 1 兲 l 储 共 ␻ 2 兲 具 ␣R(2) 典 f ,
n⫽n o
⫽⫺ ␹ (2)
␹ xxx
xy y ⫽⫺ ␹ y yx ⫽⫺ ␹ yxy ,
which should be used in Eq. 共2.3兲 instead of L ii . We notice
that F ii differs from L ii in Eq. 共A1兲 only by a common factor
l 储 if we define
⑀ ⬘⫽
TABLE V. Parameters used to calculated ␹ eff
of a z-cut quartz
crystal. For convenience the birefringence of the crystal is neglected, and the refractive index of the ordinary wave n o is used for
all polarizations.
from the bulk, which has D 3 symmetry with the nonvanishing ␹ (2)
i jk elements
2 ⑀ 1 k 2Z
⑀ 2 k 1Z ⫹ ⑀ 1 k 2Z
F Y Y ⫽l Y Y L Y Y ⫽l 储
PRE 62
Such additional correction modifies the surface density N S
by a constant l 储 ( ␻ s )l 储 ( ␻ 1 )l 储 ( ␻ 2 ), but has no effect on our
deduction of the distribution function f (⍀).
(2) 2
Equation 共2.2兲 shows that we can measure 兩 ␹ eff
兩 of a
sample by comparing its SFG intensity with that from a stan(2)
. In this experiment,
dard reference sample with a known ␹ eff
we used a z-cut ␣ -SiO2 共quartz兲 crystal as our reference
sample. The SFG signal from crystalline quartz is mainly
␹ (2)
zxy ⫽⫺ ␹ zyx ,
共defined as ␹ (2)
among which ␹ xxx
q below兲, and those equal
to ␹ xxx are much larger than the others 关47兴. In the following
calculation we neglect the weaker ones. Formal solution of
the reflected SFG amplitude from a medium with bulk nonlinear susceptibility can be found in Ref. 关48兴. Applying it to
a z-cut quartz crystal shows that the SFG intensity is maximized when the x axis of the crystal is in the incidence plane,
and the absolute values of the effective surface nonlinear
susceptibilities are
兩 ␹ eff
共 ss p 兲 兩 ⫽g cos ␤ 2 L Y Y 共 ␻ s 兲 L Y Y 共 ␻ 1 兲 L XX 共 ␻ 2 兲 ␹ (2)
q lc ,
兩 ␹ eff
共 sps 兲 兩 ⫽g cos ␤ 1 L Y Y 共 ␻ s 兲 L XX 共 ␻ 1 兲 L Y Y 共 ␻ 2 兲 ␹ (2)
q lc ,
兩 ␹ eff
共 p p p 兲 兩 ⫽g cos ␤ s cos ␤ 1 cos ␤ 2
⫻L XX 共 ␻ s 兲 L XX 共 ␻ 1 兲 L XX 共 ␻ 2 兲 ␹ (2)
q lc .
Here, ␤ i and L ii are the incidence angle and Fresnel factor,
both listed in Table V, l c is the effective coherence length for
the reflected SFG, and g⫽2 is a degeneracy constant which
arises from the number of distinguishable permutations of
the input frequencies 关47兴. We should mention here that in
our usual definition of surface nonlinear susceptibility ␹ (2)
i jk
for sum-frequency vibrational spectroscopy, this factor of g
⫽2 has already been included. In other words, we do not
distinguish between ␹ (2)
i jk ( ␻ s ⫽ ␻ 1 ⫹ ␻ 2 ) and ␹ ik j ( ␻ s ⫽ ␻ 2
⫹ ␻ 1 ). In the present experiment,
PRE 62
l c⫽
k 2z 共 ␻ s 兲 ⫹k 2z 共 ␻ 1 兲 ⫹k 2z 共 ␻ 2 兲
冑n 共 ␻ s 兲 2 ⫺sin2 ␤ s 冑n 共 ␻ 1 兲 2 ⫺sin2 ␤ 1 冑n 共 ␻ 2 兲 2 ⫺sin2 ␤ 2
Since in our experiment, SFG from crystalline quartz is far
from resonances, we may neglect dispersion and take
␹ (2)
q ⫽2d 11⬇8.0⫻10
兩 ␹ eff
共 ss p 兲 兩 ⫽1.08⫻10⫺20 m2 /V,
兩 ␹ eff
共 s ps 兲 兩 ⫽1.12⫻10⫺20 m2 /V,
兺i j ␦ i j 共 A s兲 i jk ⫽N S␭兺␮ ␯ ␦ ␭ ␮共 a s兲 ␭ ␮ ␯ 冕 共 ␯ˆ •k̂ 兲 f 共 ⍀ 兲 d⍀.
For the CH2 symmetric stretch there are only three independent nonvanishing (a s) ␭ ␮ ␯ elements (a s) ␰␰␨ , (a s) ␩␩ ␨ ,
and (a s) ␨␨␨ . Therefore, we have
␦ ␭ ␮ 共 a s兲 ␭ ␮ ␯ ⫽ ␦ ␯␨ 关共 a s兲 ␰␰␨ ⫹ 共 a s兲 ␩␩ ␨ ⫹ 共 a s兲 ␨␨␨ 兴 .
Insertion of Eq. 共D4兲 into Eq. 共D3兲 yields
The proof of Eq. 共4.6兲 is quite straightforward. Let us
consider the first equation in Eq. 共4.6兲 as an example. We
can express Eq. 共2.7兲 in the form
we find
兩 ␹ eff
共 ppp 兲 兩 ⫽0.94⫻10⫺20 m2 /V.
⬇27 nm.
兺i j ␦ i j 共 ␭ˆ • î 兲共 ␮ˆ • ĵ 兲 ⫽ ␦ ␭ ␮ ,
where d 11 refers to the nonlinear coefficient for SHG, and its
value for ␭⫽1.064 ␮m found in Ref. 关47兴 was used.
We then find, from Eq. 共C1兲, for the z-cut quartz in our
共 A s兲 i jk ⫽N S
兺i j ␦ i j 共 A s兲 i jk ⫽N S关共 a s兲 ␰␰␨ ⫹ 共 a s兲 ␩␩ ␨ ⫹ 共 a s兲 ␨␨␨ 兴
ˆ • ĵ 兲共 ␯ˆ •k̂ 兲 f 共 ⍀ 兲 d⍀.
共 a s兲 ␭ ␮ ␯ 共 ␭ˆ • î 兲共 ␮
共 ␨ˆ •k̂ 兲 f 共 ⍀ 兲 d⍀,
which is identical to the first equation in Eq. 共4.6兲 if k̂⫽ẑ.
Similarly, the other three equations in Eq. 共4.6兲 can be
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PRE 62
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