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Second-order susceptibility tensor of a monolayer at the liquid–air interface: SHG spectroscopy by compression
4 August 2000
Chemical Physics Letters 325 Ž2000. 545–551 www.elsevier.nlrlocatercplett
Second-order susceptibility tensor of a monolayer at the liquid–air interface: SHG spectroscopy by compression Mitsumasa Iwamoto a
a,)
, Chen-Xu Wu a , Ou-Yang Zhong-can
b,c
Department of Physical Electronics, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan b Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China c Center of AdÕanced Study, Tsinghua UniÕersity, Beijing 100084, China Received 27 March 2000; in final form 25 May 2000
Abstract The complete expression for the macroscopic second-order susceptibility ŽSOS. tensor x Ž2. for a monolayer with C` symmetry at the air–liquid interface is derived as functions of the molecular SOS tensor a Ž2. and the orientational order parameters. The chiral Žnon-chiral. terms of x Ž2. are distinguished by their association with even Žodd. order parameters, by which a second-harmonic generation ŽSHG. circular-dichroism effect can be better understood. An SHG experiment of a 4-cyano-4X-5-alkyl-biphenyl monolayer on an air–water interface by monolayer compression is also discussed. q 2000 Elsevier Science B.V. All rights reserved.
A variety of experimental methods including scattering, spectroscopic, and electrical techniques have been developed to study the molecular structure and orientational phase transition of monolayers at the air–water interface w1–4x. However, there has been a lack of detailed information at the molecular level, especially the symmetries of the molecules. On the other hand, the second-harmonic generation ŽSHG. from a monolayer at the air–water surface possesses an obvious advantage as it is forbidden in centrosymmetric air or water w5x. Therefore, SHG, sum-, and difference-frequency generations ŽSFG and DFG, respectively. from a monolayer have been successfully investigated by Shen’s group w6–9x. The SFG
)
Corresponding author.
generated from a monolayer ™ is governed by a surface non-linear polarization P N in the form Pi N Ž v 1 q v 2 . s x i jk y Ž v 1 q v 2 . , v 1 , v 2
™
=Ej Ž v 1 . Fk Ž v 2 . ,
™
Ž 1.
where EŽ v 1 . and F Ž v 2 . are the optical fields at frequency v 1 and v 2 , respectively. The SHG and DFG are, respectively, the cases of v 1 s v 2 and v 2 yv 2 . The second-order susceptibility ŽSOS. x Ž2. ' w x i jk x is related to the molecular SOS tensor :almn . Ns is the sura Ž2. ' w almn x by x i jk s Ns²Tilmn jk face density of the molecules, Tilmn describes the jk coordinate transformation between the molecular Ž z ,h , j . system and the lab Ž x, y, z . system and ² : denotes a thermodynamics average over the molecular orientations. For the expression Ž1., for simplicity no account is taken of the local field correction
™
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 6 5 8 - 8
546
M. Iwamoto et al.r Chemical Physics Letters 325 (2000) 545–551
factors such as Lorentz factor which arise due to the surface, but also, significantly, due to dipole–dipole interaction with neighboring monolayer molecules, where the latter will be obviously concentration dependent. This simplification does not lose physics underlying here because one way of local field correction is introducing a factor to the right-hand side of Ž1., which will not influence the present analysis and the following discussion w10x. From the SHG measurements, one can deduce x Ž2., but to obtain : which a Ž2. one has to have knowledge of ²Tilmn jk has so far been a task in mathematics. Due to l m n Ž . w l x being the Tilmn jk s R i R j R k and R f , b , c s R i usual Euler rotation matrix between Ž x, y, z . and Ž z ,h , j . with Euler angles Ž f , b , c . defined in Ref. w11x, one has to perform a lengthy average calculation for ²T : of 9 3 components with a distribution f Ž f , b , c .. Unfortunately, such calculation has not yet been done, but only in a special case when a Ž2. is dominated by a single component, i.e. ajjj along a molecular main axis j w6x. The independent nonvanishing elements of x Ž2. then reduce to two w6x. Obviously, such a simplification can deal only with a C`Õ symmetric monolayer, a non-chiral uniaxial surface. Recently, the SHG circular-dichroism ŽCD. has been found from a monolayer composed of oriented chiral molecules of R- or S-2, 2X-dihydroxyl-1, 1’binaphthyl ŽR- or S-BN. w12x and explained theoretically by the electric dipole-allowed x Ž2. terms for an isotropic surface invariant with rotations about the perpendicular z axis w13x, i.e., the 27 elements of the tensor x Ž2. can be reduced to four non-zero and independent elements w14x; x 1Žs x z z z ., x 2 Žs x z x x s x z y y ., x 3 Žs x x z x s x x x z s x y z y s x y y z ., and x4Žs x x z y s x x y z s yx y z x s yx y x z .. Here x4 characterizes the chirality of the monolayer. These elements are derived from the assumption that the input photons are degenerated Ži.e., almn s alnm . w14x. From Ž1., such an assumption is not completely used for SFG and DFG, both of which give rise at least to an additional independent element, since x x z x s x y z y / x x x z s x y y z . In other words, this is still not a general case for a uniaxial surface and, moreover, not related to the molecular orientation. The latter is remarkably lacking in physics. For example, the chiral term x4 is believed not to be related to polar orientational order by imaging a helix having no difference be-
tween its up and down geometries. Thus, the general question can be posed as follows: What is the general expression of x Ž2. associated with a Ž2. and molecular orientation for a monolayer, and can we use it to distinguish chiral and achiral monolayers and then to make a spectroscopy to measure the molecular SOS a Ž2. ? In this Letter, we show the complete expression for the macroscopic SOS tensor x Ž2. of a monolayer with C` symmetry on a material surface as functions of the molecular SOS tensor a Ž2. and the orientational order parameters Sn ' ² PnŽcos u .: w15x. Here u is the tilt angle of the molecular axis j from the normal z of the monolayer, and Pn is the nth Legendre polynomial. With this definition, the chiral Žnon-chiral. terms of x Ž2. are clearly distinguished by their association with S0 , S2 Ž S1 and S3 ., and the SHG-CD experiment reported in Refs. w12,13x is better understood. It has been shown that the uniaxially orientational order parameters can be calculated or measured as functions of the molecular area A w15x. Therefore, upon the variation of the S1Ž A., S2 Ž A. and S3 Ž A. by monolayer compression, the SHG spectroscopy from the monolayer allows the study of the molecular SOS tensor a Ž2. and its structural symmetry. With this hypothesis, an SHG experiment of a 4-cyno-4X-5-alkyl-biphenyl Ž5CB. monolayer is discussed. Phenomenologically, the general expression for x Ž2. of C` -symmetry has been given by Giordmaine w16x. By changing Ž1. from tensor notation to matrix notation w17x, we obtain Pi N s si j f j q a i j f j
Ž i s 1 y 3, j s 1 y 6 .
with si1 s x i11 , si2 s x i22 , si3 s x i33 , si4 s x i23 q x i32 , si5 s x i31 q x i13 , si6 s x i12 q x i21 ,
ai j s 0
for j s 1–3 , a i4 s x i23 y x i32 , a i5 s x i31 y x i13 , a i6 s x i12 y x i21 , f s 12 Ž 2 E1 F1 ,2 E2 F2 ,2 E3 F3 , E2 F3 qE3 F2 , E3 F1 q E1 F3 , E1 F2 q E2 F1 . , f s 12 Ž 0,0,0, E2 F3 y E3 F2 , E3 F1 yE1 F3 , E1 F2 y E2 F1 . .
Ž 2.
M. Iwamoto et al.r Chemical Physics Letters 325 (2000) 545–551
Here x Ž2. with C` symmetry is given by the following two matrices w16x 0 0 ss s31
0 as 0 0
0 0 s31 0 0 0
0 0 s33 0 0 0
s14 s15 0
a14 ya15 0
s15 ys14 0 a15 a14 0
0 0 a 36
0 0 , 0
0
0
2 Ns
s15 s
5 q
s31 s
Ns 10 q
Ž S1 y S3 . Ž 2 s 33 y s 32 y s 31 . NS
s33 s
Ns 5 q
NS
3 q
Ns
a 36 s
Ns 2 Ns 3 q
Ž 3S1 q 2 S3 . s 33 ,
Ž l14 q l25 q l36 . Ns 6
a15 s
Ž 4S1 q S3 . Ž s 32 q s 31 . ,
Ž S1 y S3 . Ž s 32 q s 31 q s 24 q s 15 . 5
Ns
Ž 3S1 q 2 S3 . Ž s 24 q s 15 . ,
Ž S1 y S3 . Ž 2 s 33 y s 24 y s 15 . 10
Ž 4.
Ž f , b , c . has been defined in the beginwhere Tilmn jk ning and the orientation distribution function f Ž b . is independent of f and c specifically in the case of C` symmetry, otherwise, C` symmetry can be broken Že.g. f Ž b , c . must result in biaxial ordering w18x.. Here angle b is just u , the tilt angle of the molecular axis j from surface normal z w11x. The generalized method for calculating Ž4. has been presented by Andrews et. al w19–21x, and the result for monolayers at the water surface has been reported, assuming the phase and Boltzmannweighted rotational averages. However, as our main interest here is to show explicitly the general physics of x Ž2. associated with the molecular SOS tensor a Ž2. for chiral and achiral monolayers, which has not yet been discussed before, it is important to reconstruct x Ž2.. And if neglecting the intermolecular
S2 Ž s 14 y s 25 . ,
10
a14 s
H
H
Ns
s14 s
²Tilmn : s Tilmn jk jk Ž f , b , c . f Ž b . sin b d f d b d c r f Ž b . sin b d f d b d c ,
interaction without loss of physics underlying here, :almn and i.e., f Ž b . s 1, calculating x i jk s Ns²Tilmn jk substituting the results into the terms si j and a i, j in Ž2., we do show that Ž4. leads exactly to the same form as Ž3.. More specifically, after calculating the number of 9 3 components of x i jk 1 , the seven independent non-zero elements in Ž3. are found to be expressed as the following important relations:
Ž 3.
with seven independent non-zero elements. Here and after we refer suffixes Ž1,2,3. to Ž x, y, z . in lab system and Ž z ,h , j . in molecular system. Our task is to prove that for a uniaxial chiral monolayer, x Ž2. does lead to Ž3. without any additional assumption on a Ž2. because in the electric-dipole approximation a Ž2. concerns only with the electronic structure of the molecule and should not be supposed to have some symmetry. On the other hand, the uniaxial property of the monolayer at the interface is related to the repulsive interaction between molecules of the monolayer and the interaction between dipolar molecules and the interfaces w15x. In other words, the : for a uniaxial surface is written average of ²Tilmn jk as
547
S2 Ž l14 q l25 y 2 l36 . ,
S1 Ž l15 y l24 . ,
Ž l14 q l25 q l36 . Ns 3
S2 Ž 2 l36 y l14 y l 25 . ,
Ž 5.
where the two 3 = 6 matrices, Ž si j . and Ž l i j ., are defined from the molecular SOS tensor a Ž2. in the same way as the conventional contracted notation of Ž si j . and Ž a i j . defined from x Ž2., i.e., s 11 s a 111 ,
1
In the calculation of the number of 9 3 components, MATHEw22x was used. However, we can easily check the seven elements given by Ž5. by hand calculation.
MATICA
548
M. Iwamoto et al.r Chemical Physics Letters 325 (2000) 545–551
s 14 s a 123 q a 132 , l14 s a 123 y a 132 , etc. Here S1 , S2 , and S3 are the orientational order parameters. As the first application of the expression Ž5., we can discuss the specific properties of chiral and achiral monolayers and their association with molecular chirality. From the association with the order parameters, the seven elements in Ž5. can be distinguished by two groups, the chiral elements Ž s14 ,a14 ,a 36 . and the non-chiral elements Ž s15 , s31 , s33 ,a15 .. The reason has been roughly pointed out in the beginning on the feature of a helix, but Ž5. can provide more intrinsic mechanism from molecular level. Using tensor notation we find s14 s Ž Nsr2. S2 Ž a 123 q a 132 y a 213 y a 231 ., in which all terms of a i jk are obviously molecularly chiral, i.e. Ž a 123 , a 132 , a 213 , a 231 . changes to Žya 123 ,y a 132 ,y a 213 ,y a 231 . whenever the molecular system ˆ ˆ ˆ . changes to Žy1,2,3 ˆ ˆ ˆ . or Ž1,y ˆ 2,3 ˆ ˆ . or Ž1,2,y ˆ ˆ 3ˆ .. Ž1,2,3 On the other hand, we find s15 s Ž Nsr5.Ž S1 y S3 . Ž2 a 333 y a 322 y a 311 . q Ž Ns r 10. Ž3S1 q 2 S3 . Ž a 113 qa 223 q a 131 q a 232 ., in which all a i jk are molecularly non-chiral, i.e. the components a 333 , a 322 , etc. are unchanged under the operation from molecular ˆ ˆ ˆ . to Žy1,2,3 ˆ ˆ ˆ .. Given these, now we can system Ž1,2,3 discuss the symmetry of x Ž2. by the molecular structures, i.e. a Ž2.. If molecules are non-chiral Ži.e. s14 s a14 s a36 s 0., one can find that the two matrices given in Ž3. reduce precisely to those for the case of C`Õ symmetry Žsee the complete tabulations in w16x. as what it should be. Eq. Ž5. can also serve to discuss the symmetry of x Ž2. with molecular orientation. For instance, even for molecules with aliphatic long alkylchains, e.g., fatty acids lacking of up-down symmetry, dielectrically S1 s S3 s 0 as they possess no permanent dipole moments, and thence the non-chiral elements in Ž5. are vanishing Ži.e. s15 s s31 s s33 s a15 s 0.. In this case, the textures of Ž3. reduce to D` symmetry as shown in the mentioned tabulations. The D` symmetry is that for a film of cholesteric liquid crystals. From Ž5., one can also find that chiral molecules always possess optical non-linearity even when randomly distributed Ži.e. S1 s S2 s S3 s 0.. In this case Ž3. leads to the optically active isotropic case Žsee w`,`x tabulation, ibid... Let us now turn to consider SHG-CD effect of a chiral monolayer reported in Refs. w12,13x. Although the general expression Ž5. cannot result precisely in as in Ref. w14x without assuming the Kleinman’s rule
w23x, i.e., a i jk s a i k j Že.g. Ž5. gives no relations of x x x z s x x z x , etc.., by viewing the air-quartz interface as C` -symmetric interface and with the similar derivation process as shown in Ref. w13x, we obtained from Ž5. the same appearance of SHG-CD effect, e.g. with the same geometry and symbolism as in Ref. w13x, we obtained the intensity for spolarized component of the reflected SHG wave from circularly polarized incident light as Žsee Ž7a. in Ref. w13x. I2s v s
8p 3v 2 c3
tan2 u i N "i s15 y s14 cos u i N 2 Iv2 ,
Ž 6.
where upper Žlower. sign represents right Žleft. circularly polarized incident light. From the definition of SHG SHG . Ž SHG SHG-CD, ISHG - CD s 2Ž I left y Iright r I left q SHG . Iright , where suffixes left and right represent the handedness of circularly polarized incident light Žsee Ref. w13x for details., we can from Ž5. and Ž6. verify immediately two effects: ŽI. For the two molecular enantiomers of left- and right-handed chirality Ždenoted as L- and R-enantiomers., we have ISHG - CD ŽL-enantiomer. s yISHG - CD ŽR-enantiomer. due to s14 ŽR-enantiomer. s ys14 ŽL-enantiomer. and s15 ŽR-enantiomer. s s15 ŽL-enantiomer.. The latter two relations can be seen from their associations with a Ž2., i.e. s14 being chiral and s15 achiral. ŽII. For a monolayer of the same handed enantiomer, ISHG - CD must change sign when the light incident from up-side changes to that from down-side. The reason is apparent from Ž5. that the changes in two geometries are equivalent to the changes of the orientation, S1 , S3 yS1 , yS3 but S2 is still unchanged, and thence up down up down s14 s s14 and s15 s ys15 , which leads to up down ISHG - CD s yISHG - CD . Both effects have been verified by the experimental results shown in tables I and II in Ref. w13x. Although some explanations were given in Ref. w13x, the current one contains more physics in association with chirality and orientational orders for such kind of monolayers. The advantage of the general expression Ž5. also leads to a potential use to determine molecular SOS a Ž2. from the SHG spectroscopy from a monolayer at the air–water interface. With the geometry of a C` -symmetric monolayer shown in the inset of Fig. 1, in our previous work w15x molecular orientational orders have been investigated based on the hardcore
™
M. Iwamoto et al.r Chemical Physics Letters 325 (2000) 545–551
repulsive interaction between monolayer-molecules Žrepresented by 0 ( u s b ( u Ž A. ' arcsinŽ ArA 0 ., where A 0 s p l 2 and l is the partial length of the molecules along the long axis above the water surface. and the interaction W Ž u . between dipolar molecules and water ŽA yP 2 , P is the molecular dipole moment.. With the apparent distribution function f Ž u . s expwyW Ž u .rk B TA x, k B being the Boltzmann’s constant, and TA the absolute temperature, any-order orientational order parameter Sn can be deduced w15x. It is shown that Sn are mainly attributed to the repulsive interaction and weakly depend on the dipolar interaction Žsee figures 1 and 2 in Ref. w15x.. Therefore, as a first approximation, we may let P 0 and have the analytic results of
(
S1 s
1 2
™
™
™
N Pach s Ž s33 y s15 y s31 . Ž ™ nPE .
™
q 12 s15
S2 s cos u Ž A . 1 q cos u Ž A . ,
™™
™ ™
q 12 a15 Ž E = F . =™ n,
Ž 8.
where ™ n is the unit vector normal of the monolayer Ži.e. ™ n s z .. From Ž7., one can see that s14 , a14 , and ™ a 36 must be pseudo-scalar because all vectors of P N , ™ ™ ™ E, F and n are polar vectors. For SHG from ™ ™a non-chiral monolayer, by letting s14 s 0 and F s E, Ž7. and Ž8. reduce to the following form
™2
™™
q B Ž™ n P E . E q CE 2™ n
S3 s 18 5cos 2u Ž A . y 1 1 q cos u Ž A . . Substituting these into Ž5. leads x Ž2. to be an apparent function of A and a Ž2. only. Therefore, we reach that by a best fitting on the variation curve of the SHG intensity with molecular area A, we can determine the molecular SOS a Ž2.. As an example, in what follows, we analyze our recent SHG experiment. The achiral 4-cyano-4X-5-alkyl-biphenyl Ž5CB. molecules were spread on the surface of water in a Langmuir trough. With a linear polarized light ŽNd:YAG laser from BIG SKY LASER TECH. with l s 1.064 mm, maximum-50 mJ pulses of half width 7 ns and 10 Hz. incident from the air side onto the monolayer, the reflected SHG was observed. The upper inset in Fig. 1 presents an SHG intensity spectrum of pp p polarized generation by com˚ 2 w15x to pressing molecular area from A s A 0 s 90 A 2 ˚ . To use the above theoretical prediction to 36 A Ž1. analyze the spectrum, we ™ first use ™ Ž5.™toN rewrite ™ as a vectorial form of P N s PchN q Pach with ™ PchN N associated to the chirality of the monolayer, and Pach to the non-chirality:
™
™ ™ Ž E P™n .Ž F =™n . ™ n E™=™n q 1 a E™= F™ q Ž F P™ .Ž . 2 14 Ž . ™ ™ n, q 12 Ž a36 y a14 . ™ n P Ž E = F .™
™ ™
n P F . E q Ž™ nPE . F Ž™
P N s Ž A y B . Ž™ nPE . ™ n
1 2
™
™™
= Ž™ n P F .™ n q s31 Ž E P F . ™ n
™
1 q cos u Ž A . ,
549
PchN s 12 s14
Ž 7.
Ž 9.
with A s s33 y s31 , B s s15 , and C s s31. Ž9. is identical to the phenomenological equation Ž1. in Ref. w24x special for the surface of an isotropic medium. This gives evidence again for Ž5. and also provides a fortunate chance for the current task: For the 5CB SHG experiment, we do not need to calculate afresh. By substituting f s 0 into the rigorous results shown in equations Ž13. and Ž14. in Ref. w24x Ž f is defined in Ref. w24x., we find the optical electric field of pp p Žinputroutput. polarization combination, which is just the case in our experiment,
™
E p Ž2 v . s
32pv E02 sin u i cos 2 u i c
(e
s
2 g OS g OT
= A sin2 u i e T r e S q C e S e T
(
(
yB cos u S cos u T ,
(
Ž 10 .
where u i is the angle of incident, u S and u T are those of the reflected and transmitted secondharmonic waves, respectively, and g OS and g OT are p the Fresnel corrections w24x. From the measured ISHG pŽ 2 . Ž AN E 2 v N , we can determine A, B, and C i.e., s15 , s33 , and s31 .. Here, as an instructive example, we only use it to check the one-component model for a Ž2. w6–9x. By assuming a Ž2. dominated by the
M. Iwamoto et al.r Chemical Physics Letters 325 (2000) 545–551
550
Fig. 1. The calculated SHG intensity as a function of area per molecule for 5CB monolayer at the air–water interface. Each 5CB molecule is assumed to be a rod-like molecule with a length l and its rotation motion restricted within the region depicted by cones due to the hard-core repulsive interaction between molecules w15x. The molecular area A is given by A s A 0 sin2 uŽA. with A 0 s p l 2 . The polarization combination is chosen to 608-inrp-out. The upper inset shows the experimental result and lower is the geometry of the orientation for the monolayer.
single component a 333 , from Ž5. and the relations of S1 y S3 with molecular area A mentioned above, we have As Bs
Ns 8 Ns 4
Ž 1 q x . Ž 1 q 3 x 2 . a 333 , Ž 1 y x 2 . Ž 1 q x . a 333 , C s
B 2
,
where x 2 ' cos 2u Ž A. s 1 y ArA 0 and NsŽs 1rA.. Substituting the above results into Ž10., and letting e S f 1 Žat air., e T s 1.33 Žat water., u i s 608 Žfrom ˚ 2 Žsee Ref. w15x., our experiment., and A 0 s 90 A we obtain the SHG-A spectrum of 5CB monolayer at the air–water interface
(
(
p ISHG Ž 2 v , ArA 0 .
same figure. In the above calculation the minimum ˚ 2 with A minrA 0 s 0.4 is molecular area A min s 36 A obtained by experiment. When A is reduced below A min , the monolayer is destroyed as it is composed of a monolayer plus some interdigitated bilayers w25x. The bilayers possess centrosymmetry, therefore, SHG from the multilayers does not increase Žsee Fig. 1.. In summary, we have derived the general expression of the relation between macroscopic SOS x Ž2. and molecular SOS a Ž2. as well as their orientational order parameters for a monolayer. Based on the result, we have distinguished the chiral and non-chiral terms of x Ž2., which can sereve to study the SHG-CD effect for a chiral monolayer. Furthermore, we presented a SHG spectroscopy by compressing the monolayer to estimate the molecular SOS a Ž2..
p ISHG Ž 2 v ,0.4 .
s
2.54 y 1.54 Ž ArA 0 .
(
1 y 1 y ArA 0
2
ž
1 y '0.6 1.924
2
/
References .
Ž 11 .
Fig. 1 shows the numerical result, of which the profile is similar to the experiment shown in the
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Chemical Physics Letters 325 Ž2000. 545–551 www.elsevier.nlrlocatercplett
Second-order susceptibility tensor of a monolayer at the liquid–air interface: SHG spectroscopy by compression Mitsumasa Iwamoto a
a,)
, Chen-Xu Wu a , Ou-Yang Zhong-can
b,c
Department of Physical Electronics, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan b Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China c Center of AdÕanced Study, Tsinghua UniÕersity, Beijing 100084, China Received 27 March 2000; in final form 25 May 2000
Abstract The complete expression for the macroscopic second-order susceptibility ŽSOS. tensor x Ž2. for a monolayer with C` symmetry at the air–liquid interface is derived as functions of the molecular SOS tensor a Ž2. and the orientational order parameters. The chiral Žnon-chiral. terms of x Ž2. are distinguished by their association with even Žodd. order parameters, by which a second-harmonic generation ŽSHG. circular-dichroism effect can be better understood. An SHG experiment of a 4-cyano-4X-5-alkyl-biphenyl monolayer on an air–water interface by monolayer compression is also discussed. q 2000 Elsevier Science B.V. All rights reserved.
A variety of experimental methods including scattering, spectroscopic, and electrical techniques have been developed to study the molecular structure and orientational phase transition of monolayers at the air–water interface w1–4x. However, there has been a lack of detailed information at the molecular level, especially the symmetries of the molecules. On the other hand, the second-harmonic generation ŽSHG. from a monolayer at the air–water surface possesses an obvious advantage as it is forbidden in centrosymmetric air or water w5x. Therefore, SHG, sum-, and difference-frequency generations ŽSFG and DFG, respectively. from a monolayer have been successfully investigated by Shen’s group w6–9x. The SFG
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Corresponding author.
generated from a monolayer ™ is governed by a surface non-linear polarization P N in the form Pi N Ž v 1 q v 2 . s x i jk y Ž v 1 q v 2 . , v 1 , v 2
™
=Ej Ž v 1 . Fk Ž v 2 . ,
™
Ž 1.
where EŽ v 1 . and F Ž v 2 . are the optical fields at frequency v 1 and v 2 , respectively. The SHG and DFG are, respectively, the cases of v 1 s v 2 and v 2 yv 2 . The second-order susceptibility ŽSOS. x Ž2. ' w x i jk x is related to the molecular SOS tensor :almn . Ns is the sura Ž2. ' w almn x by x i jk s Ns²Tilmn jk face density of the molecules, Tilmn describes the jk coordinate transformation between the molecular Ž z ,h , j . system and the lab Ž x, y, z . system and ² : denotes a thermodynamics average over the molecular orientations. For the expression Ž1., for simplicity no account is taken of the local field correction
™
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 6 5 8 - 8
546
M. Iwamoto et al.r Chemical Physics Letters 325 (2000) 545–551
factors such as Lorentz factor which arise due to the surface, but also, significantly, due to dipole–dipole interaction with neighboring monolayer molecules, where the latter will be obviously concentration dependent. This simplification does not lose physics underlying here because one way of local field correction is introducing a factor to the right-hand side of Ž1., which will not influence the present analysis and the following discussion w10x. From the SHG measurements, one can deduce x Ž2., but to obtain : which a Ž2. one has to have knowledge of ²Tilmn jk has so far been a task in mathematics. Due to l m n Ž . w l x being the Tilmn jk s R i R j R k and R f , b , c s R i usual Euler rotation matrix between Ž x, y, z . and Ž z ,h , j . with Euler angles Ž f , b , c . defined in Ref. w11x, one has to perform a lengthy average calculation for ²T : of 9 3 components with a distribution f Ž f , b , c .. Unfortunately, such calculation has not yet been done, but only in a special case when a Ž2. is dominated by a single component, i.e. ajjj along a molecular main axis j w6x. The independent nonvanishing elements of x Ž2. then reduce to two w6x. Obviously, such a simplification can deal only with a C`Õ symmetric monolayer, a non-chiral uniaxial surface. Recently, the SHG circular-dichroism ŽCD. has been found from a monolayer composed of oriented chiral molecules of R- or S-2, 2X-dihydroxyl-1, 1’binaphthyl ŽR- or S-BN. w12x and explained theoretically by the electric dipole-allowed x Ž2. terms for an isotropic surface invariant with rotations about the perpendicular z axis w13x, i.e., the 27 elements of the tensor x Ž2. can be reduced to four non-zero and independent elements w14x; x 1Žs x z z z ., x 2 Žs x z x x s x z y y ., x 3 Žs x x z x s x x x z s x y z y s x y y z ., and x4Žs x x z y s x x y z s yx y z x s yx y x z .. Here x4 characterizes the chirality of the monolayer. These elements are derived from the assumption that the input photons are degenerated Ži.e., almn s alnm . w14x. From Ž1., such an assumption is not completely used for SFG and DFG, both of which give rise at least to an additional independent element, since x x z x s x y z y / x x x z s x y y z . In other words, this is still not a general case for a uniaxial surface and, moreover, not related to the molecular orientation. The latter is remarkably lacking in physics. For example, the chiral term x4 is believed not to be related to polar orientational order by imaging a helix having no difference be-
tween its up and down geometries. Thus, the general question can be posed as follows: What is the general expression of x Ž2. associated with a Ž2. and molecular orientation for a monolayer, and can we use it to distinguish chiral and achiral monolayers and then to make a spectroscopy to measure the molecular SOS a Ž2. ? In this Letter, we show the complete expression for the macroscopic SOS tensor x Ž2. of a monolayer with C` symmetry on a material surface as functions of the molecular SOS tensor a Ž2. and the orientational order parameters Sn ' ² PnŽcos u .: w15x. Here u is the tilt angle of the molecular axis j from the normal z of the monolayer, and Pn is the nth Legendre polynomial. With this definition, the chiral Žnon-chiral. terms of x Ž2. are clearly distinguished by their association with S0 , S2 Ž S1 and S3 ., and the SHG-CD experiment reported in Refs. w12,13x is better understood. It has been shown that the uniaxially orientational order parameters can be calculated or measured as functions of the molecular area A w15x. Therefore, upon the variation of the S1Ž A., S2 Ž A. and S3 Ž A. by monolayer compression, the SHG spectroscopy from the monolayer allows the study of the molecular SOS tensor a Ž2. and its structural symmetry. With this hypothesis, an SHG experiment of a 4-cyno-4X-5-alkyl-biphenyl Ž5CB. monolayer is discussed. Phenomenologically, the general expression for x Ž2. of C` -symmetry has been given by Giordmaine w16x. By changing Ž1. from tensor notation to matrix notation w17x, we obtain Pi N s si j f j q a i j f j
Ž i s 1 y 3, j s 1 y 6 .
with si1 s x i11 , si2 s x i22 , si3 s x i33 , si4 s x i23 q x i32 , si5 s x i31 q x i13 , si6 s x i12 q x i21 ,
ai j s 0
for j s 1–3 , a i4 s x i23 y x i32 , a i5 s x i31 y x i13 , a i6 s x i12 y x i21 , f s 12 Ž 2 E1 F1 ,2 E2 F2 ,2 E3 F3 , E2 F3 qE3 F2 , E3 F1 q E1 F3 , E1 F2 q E2 F1 . , f s 12 Ž 0,0,0, E2 F3 y E3 F2 , E3 F1 yE1 F3 , E1 F2 y E2 F1 . .
Ž 2.
M. Iwamoto et al.r Chemical Physics Letters 325 (2000) 545–551
Here x Ž2. with C` symmetry is given by the following two matrices w16x 0 0 ss s31
0 as 0 0
0 0 s31 0 0 0
0 0 s33 0 0 0
s14 s15 0
a14 ya15 0
s15 ys14 0 a15 a14 0
0 0 a 36
0 0 , 0
0
0
2 Ns
s15 s
5 q
s31 s
Ns 10 q
Ž S1 y S3 . Ž 2 s 33 y s 32 y s 31 . NS
s33 s
Ns 5 q
NS
3 q
Ns
a 36 s
Ns 2 Ns 3 q
Ž 3S1 q 2 S3 . s 33 ,
Ž l14 q l25 q l36 . Ns 6
a15 s
Ž 4S1 q S3 . Ž s 32 q s 31 . ,
Ž S1 y S3 . Ž s 32 q s 31 q s 24 q s 15 . 5
Ns
Ž 3S1 q 2 S3 . Ž s 24 q s 15 . ,
Ž S1 y S3 . Ž 2 s 33 y s 24 y s 15 . 10
Ž 4.
Ž f , b , c . has been defined in the beginwhere Tilmn jk ning and the orientation distribution function f Ž b . is independent of f and c specifically in the case of C` symmetry, otherwise, C` symmetry can be broken Že.g. f Ž b , c . must result in biaxial ordering w18x.. Here angle b is just u , the tilt angle of the molecular axis j from surface normal z w11x. The generalized method for calculating Ž4. has been presented by Andrews et. al w19–21x, and the result for monolayers at the water surface has been reported, assuming the phase and Boltzmannweighted rotational averages. However, as our main interest here is to show explicitly the general physics of x Ž2. associated with the molecular SOS tensor a Ž2. for chiral and achiral monolayers, which has not yet been discussed before, it is important to reconstruct x Ž2.. And if neglecting the intermolecular
S2 Ž s 14 y s 25 . ,
10
a14 s
H
H
Ns
s14 s
²Tilmn : s Tilmn jk jk Ž f , b , c . f Ž b . sin b d f d b d c r f Ž b . sin b d f d b d c ,
interaction without loss of physics underlying here, :almn and i.e., f Ž b . s 1, calculating x i jk s Ns²Tilmn jk substituting the results into the terms si j and a i, j in Ž2., we do show that Ž4. leads exactly to the same form as Ž3.. More specifically, after calculating the number of 9 3 components of x i jk 1 , the seven independent non-zero elements in Ž3. are found to be expressed as the following important relations:
Ž 3.
with seven independent non-zero elements. Here and after we refer suffixes Ž1,2,3. to Ž x, y, z . in lab system and Ž z ,h , j . in molecular system. Our task is to prove that for a uniaxial chiral monolayer, x Ž2. does lead to Ž3. without any additional assumption on a Ž2. because in the electric-dipole approximation a Ž2. concerns only with the electronic structure of the molecule and should not be supposed to have some symmetry. On the other hand, the uniaxial property of the monolayer at the interface is related to the repulsive interaction between molecules of the monolayer and the interaction between dipolar molecules and the interfaces w15x. In other words, the : for a uniaxial surface is written average of ²Tilmn jk as
547
S2 Ž l14 q l25 y 2 l36 . ,
S1 Ž l15 y l24 . ,
Ž l14 q l25 q l36 . Ns 3
S2 Ž 2 l36 y l14 y l 25 . ,
Ž 5.
where the two 3 = 6 matrices, Ž si j . and Ž l i j ., are defined from the molecular SOS tensor a Ž2. in the same way as the conventional contracted notation of Ž si j . and Ž a i j . defined from x Ž2., i.e., s 11 s a 111 ,
1
In the calculation of the number of 9 3 components, MATHEw22x was used. However, we can easily check the seven elements given by Ž5. by hand calculation.
MATICA
548
M. Iwamoto et al.r Chemical Physics Letters 325 (2000) 545–551
s 14 s a 123 q a 132 , l14 s a 123 y a 132 , etc. Here S1 , S2 , and S3 are the orientational order parameters. As the first application of the expression Ž5., we can discuss the specific properties of chiral and achiral monolayers and their association with molecular chirality. From the association with the order parameters, the seven elements in Ž5. can be distinguished by two groups, the chiral elements Ž s14 ,a14 ,a 36 . and the non-chiral elements Ž s15 , s31 , s33 ,a15 .. The reason has been roughly pointed out in the beginning on the feature of a helix, but Ž5. can provide more intrinsic mechanism from molecular level. Using tensor notation we find s14 s Ž Nsr2. S2 Ž a 123 q a 132 y a 213 y a 231 ., in which all terms of a i jk are obviously molecularly chiral, i.e. Ž a 123 , a 132 , a 213 , a 231 . changes to Žya 123 ,y a 132 ,y a 213 ,y a 231 . whenever the molecular system ˆ ˆ ˆ . changes to Žy1,2,3 ˆ ˆ ˆ . or Ž1,y ˆ 2,3 ˆ ˆ . or Ž1,2,y ˆ ˆ 3ˆ .. Ž1,2,3 On the other hand, we find s15 s Ž Nsr5.Ž S1 y S3 . Ž2 a 333 y a 322 y a 311 . q Ž Ns r 10. Ž3S1 q 2 S3 . Ž a 113 qa 223 q a 131 q a 232 ., in which all a i jk are molecularly non-chiral, i.e. the components a 333 , a 322 , etc. are unchanged under the operation from molecular ˆ ˆ ˆ . to Žy1,2,3 ˆ ˆ ˆ .. Given these, now we can system Ž1,2,3 discuss the symmetry of x Ž2. by the molecular structures, i.e. a Ž2.. If molecules are non-chiral Ži.e. s14 s a14 s a36 s 0., one can find that the two matrices given in Ž3. reduce precisely to those for the case of C`Õ symmetry Žsee the complete tabulations in w16x. as what it should be. Eq. Ž5. can also serve to discuss the symmetry of x Ž2. with molecular orientation. For instance, even for molecules with aliphatic long alkylchains, e.g., fatty acids lacking of up-down symmetry, dielectrically S1 s S3 s 0 as they possess no permanent dipole moments, and thence the non-chiral elements in Ž5. are vanishing Ži.e. s15 s s31 s s33 s a15 s 0.. In this case, the textures of Ž3. reduce to D` symmetry as shown in the mentioned tabulations. The D` symmetry is that for a film of cholesteric liquid crystals. From Ž5., one can also find that chiral molecules always possess optical non-linearity even when randomly distributed Ži.e. S1 s S2 s S3 s 0.. In this case Ž3. leads to the optically active isotropic case Žsee w`,`x tabulation, ibid... Let us now turn to consider SHG-CD effect of a chiral monolayer reported in Refs. w12,13x. Although the general expression Ž5. cannot result precisely in as in Ref. w14x without assuming the Kleinman’s rule
w23x, i.e., a i jk s a i k j Že.g. Ž5. gives no relations of x x x z s x x z x , etc.., by viewing the air-quartz interface as C` -symmetric interface and with the similar derivation process as shown in Ref. w13x, we obtained from Ž5. the same appearance of SHG-CD effect, e.g. with the same geometry and symbolism as in Ref. w13x, we obtained the intensity for spolarized component of the reflected SHG wave from circularly polarized incident light as Žsee Ž7a. in Ref. w13x. I2s v s
8p 3v 2 c3
tan2 u i N "i s15 y s14 cos u i N 2 Iv2 ,
Ž 6.
where upper Žlower. sign represents right Žleft. circularly polarized incident light. From the definition of SHG SHG . Ž SHG SHG-CD, ISHG - CD s 2Ž I left y Iright r I left q SHG . Iright , where suffixes left and right represent the handedness of circularly polarized incident light Žsee Ref. w13x for details., we can from Ž5. and Ž6. verify immediately two effects: ŽI. For the two molecular enantiomers of left- and right-handed chirality Ždenoted as L- and R-enantiomers., we have ISHG - CD ŽL-enantiomer. s yISHG - CD ŽR-enantiomer. due to s14 ŽR-enantiomer. s ys14 ŽL-enantiomer. and s15 ŽR-enantiomer. s s15 ŽL-enantiomer.. The latter two relations can be seen from their associations with a Ž2., i.e. s14 being chiral and s15 achiral. ŽII. For a monolayer of the same handed enantiomer, ISHG - CD must change sign when the light incident from up-side changes to that from down-side. The reason is apparent from Ž5. that the changes in two geometries are equivalent to the changes of the orientation, S1 , S3 yS1 , yS3 but S2 is still unchanged, and thence up down up down s14 s s14 and s15 s ys15 , which leads to up down ISHG - CD s yISHG - CD . Both effects have been verified by the experimental results shown in tables I and II in Ref. w13x. Although some explanations were given in Ref. w13x, the current one contains more physics in association with chirality and orientational orders for such kind of monolayers. The advantage of the general expression Ž5. also leads to a potential use to determine molecular SOS a Ž2. from the SHG spectroscopy from a monolayer at the air–water interface. With the geometry of a C` -symmetric monolayer shown in the inset of Fig. 1, in our previous work w15x molecular orientational orders have been investigated based on the hardcore
™
M. Iwamoto et al.r Chemical Physics Letters 325 (2000) 545–551
repulsive interaction between monolayer-molecules Žrepresented by 0 ( u s b ( u Ž A. ' arcsinŽ ArA 0 ., where A 0 s p l 2 and l is the partial length of the molecules along the long axis above the water surface. and the interaction W Ž u . between dipolar molecules and water ŽA yP 2 , P is the molecular dipole moment.. With the apparent distribution function f Ž u . s expwyW Ž u .rk B TA x, k B being the Boltzmann’s constant, and TA the absolute temperature, any-order orientational order parameter Sn can be deduced w15x. It is shown that Sn are mainly attributed to the repulsive interaction and weakly depend on the dipolar interaction Žsee figures 1 and 2 in Ref. w15x.. Therefore, as a first approximation, we may let P 0 and have the analytic results of
(
S1 s
1 2
™
™
™
N Pach s Ž s33 y s15 y s31 . Ž ™ nPE .
™
q 12 s15
S2 s cos u Ž A . 1 q cos u Ž A . ,
™™
™ ™
q 12 a15 Ž E = F . =™ n,
Ž 8.
where ™ n is the unit vector normal of the monolayer Ži.e. ™ n s z .. From Ž7., one can see that s14 , a14 , and ™ a 36 must be pseudo-scalar because all vectors of P N , ™ ™ ™ E, F and n are polar vectors. For SHG from ™ ™a non-chiral monolayer, by letting s14 s 0 and F s E, Ž7. and Ž8. reduce to the following form
™2
™™
q B Ž™ n P E . E q CE 2™ n
S3 s 18 5cos 2u Ž A . y 1 1 q cos u Ž A . . Substituting these into Ž5. leads x Ž2. to be an apparent function of A and a Ž2. only. Therefore, we reach that by a best fitting on the variation curve of the SHG intensity with molecular area A, we can determine the molecular SOS a Ž2.. As an example, in what follows, we analyze our recent SHG experiment. The achiral 4-cyano-4X-5-alkyl-biphenyl Ž5CB. molecules were spread on the surface of water in a Langmuir trough. With a linear polarized light ŽNd:YAG laser from BIG SKY LASER TECH. with l s 1.064 mm, maximum-50 mJ pulses of half width 7 ns and 10 Hz. incident from the air side onto the monolayer, the reflected SHG was observed. The upper inset in Fig. 1 presents an SHG intensity spectrum of pp p polarized generation by com˚ 2 w15x to pressing molecular area from A s A 0 s 90 A 2 ˚ . To use the above theoretical prediction to 36 A Ž1. analyze the spectrum, we ™ first use ™ Ž5.™toN rewrite ™ as a vectorial form of P N s PchN q Pach with ™ PchN N associated to the chirality of the monolayer, and Pach to the non-chirality:
™
™ ™ Ž E P™n .Ž F =™n . ™ n E™=™n q 1 a E™= F™ q Ž F P™ .Ž . 2 14 Ž . ™ ™ n, q 12 Ž a36 y a14 . ™ n P Ž E = F .™
™ ™
n P F . E q Ž™ nPE . F Ž™
P N s Ž A y B . Ž™ nPE . ™ n
1 2
™
™™
= Ž™ n P F .™ n q s31 Ž E P F . ™ n
™
1 q cos u Ž A . ,
549
PchN s 12 s14
Ž 7.
Ž 9.
with A s s33 y s31 , B s s15 , and C s s31. Ž9. is identical to the phenomenological equation Ž1. in Ref. w24x special for the surface of an isotropic medium. This gives evidence again for Ž5. and also provides a fortunate chance for the current task: For the 5CB SHG experiment, we do not need to calculate afresh. By substituting f s 0 into the rigorous results shown in equations Ž13. and Ž14. in Ref. w24x Ž f is defined in Ref. w24x., we find the optical electric field of pp p Žinputroutput. polarization combination, which is just the case in our experiment,
™
E p Ž2 v . s
32pv E02 sin u i cos 2 u i c
(e
s
2 g OS g OT
= A sin2 u i e T r e S q C e S e T
(
(
yB cos u S cos u T ,
(
Ž 10 .
where u i is the angle of incident, u S and u T are those of the reflected and transmitted secondharmonic waves, respectively, and g OS and g OT are p the Fresnel corrections w24x. From the measured ISHG pŽ 2 . Ž AN E 2 v N , we can determine A, B, and C i.e., s15 , s33 , and s31 .. Here, as an instructive example, we only use it to check the one-component model for a Ž2. w6–9x. By assuming a Ž2. dominated by the
M. Iwamoto et al.r Chemical Physics Letters 325 (2000) 545–551
550
Fig. 1. The calculated SHG intensity as a function of area per molecule for 5CB monolayer at the air–water interface. Each 5CB molecule is assumed to be a rod-like molecule with a length l and its rotation motion restricted within the region depicted by cones due to the hard-core repulsive interaction between molecules w15x. The molecular area A is given by A s A 0 sin2 uŽA. with A 0 s p l 2 . The polarization combination is chosen to 608-inrp-out. The upper inset shows the experimental result and lower is the geometry of the orientation for the monolayer.
single component a 333 , from Ž5. and the relations of S1 y S3 with molecular area A mentioned above, we have As Bs
Ns 8 Ns 4
Ž 1 q x . Ž 1 q 3 x 2 . a 333 , Ž 1 y x 2 . Ž 1 q x . a 333 , C s
B 2
,
where x 2 ' cos 2u Ž A. s 1 y ArA 0 and NsŽs 1rA.. Substituting the above results into Ž10., and letting e S f 1 Žat air., e T s 1.33 Žat water., u i s 608 Žfrom ˚ 2 Žsee Ref. w15x., our experiment., and A 0 s 90 A we obtain the SHG-A spectrum of 5CB monolayer at the air–water interface
(
(
p ISHG Ž 2 v , ArA 0 .
same figure. In the above calculation the minimum ˚ 2 with A minrA 0 s 0.4 is molecular area A min s 36 A obtained by experiment. When A is reduced below A min , the monolayer is destroyed as it is composed of a monolayer plus some interdigitated bilayers w25x. The bilayers possess centrosymmetry, therefore, SHG from the multilayers does not increase Žsee Fig. 1.. In summary, we have derived the general expression of the relation between macroscopic SOS x Ž2. and molecular SOS a Ž2. as well as their orientational order parameters for a monolayer. Based on the result, we have distinguished the chiral and non-chiral terms of x Ž2., which can sereve to study the SHG-CD effect for a chiral monolayer. Furthermore, we presented a SHG spectroscopy by compressing the monolayer to estimate the molecular SOS a Ž2..
p ISHG Ž 2 v ,0.4 .
s
2.54 y 1.54 Ž ArA 0 .
(
1 y 1 y ArA 0
2
ž
1 y '0.6 1.924
2
/
References .
Ž 11 .
Fig. 1 shows the numerical result, of which the profile is similar to the experiment shown in the
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