Orientation-Insensitive Methodology for Second Harmonic

These two functions, shown graphically in Figure 1, can vary considerably even for .... For the solid line, angles between 0° and 50° only were cons...
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Anal. Chem. 2000, 72, 3399-3406

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Orientation-Insensitive Methodology for Second Harmonic Generation. 1. Theory Garth J. Simpson and Kathy L. Rowlen*

Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309

The use of second harmonic generation as a means to probe either adsorption isotherms or the kinetics of adsorption/desorption is limited by the fact that the detected signal is dependent upon both surface coverage and molecular orientation. Thus, the second harmonic intensity must be tediously corrected if either the mean orientation angle or the width of the orientation distribution changes as a function of surface coverage. In this study, a new mathematical view of the second harmonic intensity as a function of excitation polarization is developed. This new approach predicts an experimental geometry that is relatively insensitive to molecular orientation, such that appropriate choice of the excitation polarization rotation angle allows for direct measurement of surface coverage. The theory is presented for the three common dominant hyperpolarizability tensor elements, βz′z′z′, βz′x′x′, and βx′x′z′.

By nature of the experimental simplicity, surface selectivity, and submonolayer sensitivity afforded by second harmonic generation (SHG), its applications as a powerful and versatile probe for surface coverage at dielectric solid and liquid interfaces are rapidly growing.1-4 In the simplest cases, the detected SHG intensity is proportional to the square of the surface number density, allowing for interface-selective adsorption isotherm and interfacial kinetics measurements in situ for systems that would be otherwise difficult to study (e.g., liquid/solid and liquid/liquid interfaces). * Corresponding author: (e-mail) [email protected]; (phone) (303) 492-5033.; (fax) (303) 492-5894. (1) Corn, R. M.; Higgins, D. A. Chem. Rev. 1994, 94, 107-25. (2) Eisenthal, K. B. Chem. Rev. 1996, 96, 1343-60. (3) Eisenthal, K. B. Annu. Rev. Phys. Chem. 1992, 43, 627-61. (4) Shen, Y. R. Nature 1989, 337, 519-25. 10.1021/ac000346s CCC: $19.00 Published on Web 07/07/2000

© 2000 American Chemical Society

A potential complication often overlooked in adsorption isotherm and kinetics measurements by SHG is that the intensity measured under a given polarization condition is dependent on both the surface number density and the molecular orientation. In general, the SHG intensity scales with the square of combinations of the two functions, and , in which θ is the tilt angle of the molecular orientation axis with respect to the surface normal and the brackets indicate an average. These two functions, shown graphically in Figure 1, can vary considerably even for small changes in orientation. Consequently, measurements of the SHG intensity are only directly related to the surface coverage if the molecular orientation distribution remains unchanged as a function of surface coverage. Although a constant orientation with coverage has been observed experimentally in several cases,5-18 in many other SHG studies, the apparent orientation angle has been observed to change significantly as a function of surface coverage.19-31 Additionally, changes in orientation with surface density have been (5) Bae, S.; Haage, K.; Wantke, K.; Motschmann, H. J. Phys. Chem. B 1999, 103, 1045-50. (6) Paul, H. J.; Corn, R. M. J. Phys. Chem. B 1997, 101, 4494-7. (7) Vogel, V.; Mullin, C. S.; Shen, Y. R. Langmuir 1991, 7, 1222-4. (8) Naujok, R. R.; Higgins, D. A.; Hanken, D. G.; Corn, R. M. J. Chem. Soc., Faraday Trans. 1995, 91, 1411-20. (9) Higgins, D. A.; Corn, R. M. J. Phys. Chem. 1993, 97, 489-93. (10) Bae, S.; Harke, M.; Goebel, A.; Lunkenheimer, K.; Motschmann, H. Langmuir 1997, 13, 6274-8. (11) Lehmann, S.; Busse, G.; Kahlweit, M.; Stolle, R.; Simon, F.; Marowsky, G. Langmuir 1995, 11, 1174-7. (12) Naujok, R. R.; Paul, H. J.; Corn, R. M. J. Phys. Chem. 1996, 100, 10497-507. (13) Sarkar, N.; Das, S.; Nath, D.; Bhattacharyya, K. J. Chem. Soc., Faraday Trans. 1995, 91, 1769-73. (14) Vogel, V.; Mullin, C. S.; Shen, Y. R.; Kin, M. W. J. Chem. Phys. 1991, 95, 4620-5. (15) Xu, Z.; Li, J.; Dong, Y. Langmuir 1998, 14, 1183-8. (16) Karpovich, D. S.; Ray, D. J. Phys. Chem. B 1998, 102, 649-52. (17) Morgenthaler, M. J. E.; Meech, S. R. J. Phys. Chem. 1996, 100, 3323-9. (18) Zhuang, X.; Lackritz, H. S.; Shen, Y. R. Chem. Phys. Lett. 1995, 246, 279-84.

Analytical Chemistry, Vol. 72, No. 15, August 1, 2000 3399

Figure 1. Behavior of sin2 θ cos θ and cos3 θ as a function of the molecular orientation angle, θ. The experimentally detected SHG intensity generally scales with a squared combination of the two functions and with the squared number density of chromophores at the interface. Changes in either orientation or number density can influence the detected SHG intensity.

absorbance with photoacoustic detection demonstrated a change in orientation with number of layers for a zirconium phosphatephosphonate multilayer system.35 Despite evidence for the coverage dependence of molecular orientation, changes in molecular orientation are often either not considered or not reported in SHG adsorption isotherm measurements45-50 and SHG kinetics measurements,51-58 presumably due to the relatively few methods available to obtain orientation-independent surface density measurements by SHG. In this paper, theory is developed describing a novel approach allowing for acquisition of SHG measurements in an experimental configuration insensitive to changes in orientation. Review of Adsorption Isotherm and Kinetics Measurements by SHG. For a uniaxial distribution about the surface normal the s- and p-polarized components of the second harmonic intensity are related to the three independent surface second-order nonlinear tensor elements, χZZZ, χZXX, and χXXZ by59-61 2 ω 2 I 2ω s (γ) ) C|s1 sin(2γ)χXXZ| (I )

(1a)

reported for monolayer and multilayer films using techniques other than SHG, including sum frequency generation,32-34 absorbance linear dichroism,35-42 and fluorescence.43,44 For example, a linear dichroism study in our laboratory utilizing angle-resolved

2 I 2ω p (γ) ) C|s5χZXX + cos γ(s2χXXZ + s3χZXX +

(19) Yitzchaik, S.; Roscoe, S. B.; Kakkar, A. K.; Allan, D. S.; Marks, T. J.; Xu, Z.; Zhang, T.; Lin, W.; Wong, G. K. J. Phys. Chem. 1993, 97, 6958-60. (20) Inoue, T.; Moriguchi, M.; Ogawa, T. Thin Solid Films 1999, 350, 238-44. (21) Rasing, Th.; Shen, Y. R.; Kim. M. W.; Valint, P.; Bock, J. Phys. Rev. A 1985, 31, 537-9. (22) Enderle, Th.; Meixner, A. J.; Zschokke-Granacher, I. J. Chem. Phys. 1994, 101, 4365-72. (23) Zhang, T.; Feng, Z.; Wong, G. K.; Ketterson, J. B. Langmuir 1996, 12, 2298-302. (24) Marowsky, G.; Steinhof, R.; Reider, G. A.; Erdmann, D.; Dorsch, D. Opt. Commun. 1987, 63, 109-13. (25) Inoue, T.; Moriguchi, M.; Ogawa, T. Anal. Chim. Acta 1996, 330, 11721. (26) Tsukanova, V.; Harata, A.; Ogawa, T. Langmuir 2000, 16, 1167-71. (27) Lin, S.; Meech, S. R. Langmuir 2000, 16, 2893-8. (28) Nakamo, T.; Yamada, Y.; Matsuo, T.; Yamada, S. J. Phys. Chem. B 1998, 102, 8569-73. (29) Hill, W.; Werner, L.; Marlow, F. Ber. Bunsen-Ges. Phys. Chem. 1991, 95, 1453-8. (30) Xu, Z.; Li, J.; Dong, Y. Langmuir 1998, 14, 1183-8. (31) Zimdars, D.; Eisenthal, K. B. J. Phys. Chem. 1999, 103, 10567-70. (32) Watry, M. R.; Richmond, G. L. J. Am. Chem. Soc. 2000, 122, 875-83. (33) Allen, H. C.; Gragson, D. E.; Richmond, G. L. J. Phys. Chem. B 1999, 103, 660-6. (34) Wang, H.; Borguet, E.; Yan, E. C. Y.; Zhang, D.; Gutow, J.; Eisenthal, K. B. Langmuir 1998, 14, 1472-7. (35) Doughty, S. K.; Simpson, G. J.; Rowlen, K. L. J. Am. Chem. Soc. 1998, 120, 7997-8. (36) Neivandt, D. J.; Gee, M. L.; Hair, M. L.; Tripp, C. P. J. Phys. Chem. B 1998, 102, 5107-14. (37) Buscher, C. T.; McBranch, D.; Li, DeQ. J. Am. Chem. Soc. 1996, 118, 2950-3. (38) Sukhishvili, S. A.; Granick, S. J. Phys. Chem. B 1999, 103, 472-9. (39) Lee, J. E.; Saavedra, S. S. Langmuir 1996, 12, 4025-32. (40) Moon, J. H.; Kin, J. H.; Kim, K. J.; Kang, K. J.; Kim, B.; Kim; C. H.; Hahn, J. H.; Park, J. W. Langmuir 1997, 13, 4305-10. (41) Ando, H.; Nakahara, M.; Yamamoto, M.; Itoh, K. Langmuir 1996, 12, 6399-403. (42) Vallant, T.; Kattner, J.; Brunner, H.; Mayer, U.; Hoffmann, H. Langmuir 1999, 15, 5339-46. (43) Anfinrud, P. A.; Hart, D. E.; Hedstrom, J. F.; Struve, W. S. J. Phys. Chem. 1986, 90, 3116-23. (44) Guryev, O.; Dubrovsky, T.; Chernogolov, A.; Dubrovskaya, S.; Usanov, S.; Nicolini, C. Langmuir 1997, 13, 299-304.

where I2ω is the detected second harmonic intensity in which the subscript describes the polarization state of the second harmonic, γ is the polarization rotation angle of the incident fundamental (with γ ) 0° corresponding to p-polarized light), and the si terms are fitting coefficients dependent on the experimental geometry. Expressions for the fitting coefficients are given by59,60

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Analytical Chemistry, Vol. 72, No. 15, August 1, 2000

s4χZZZ - s5χZXX)|2(I ω)2 (1b)

ω ω ω s1 ) L2ω YY LYY LZZ sin θ

s2 )

L2ω XX

LωXX

LωZZ

ω

sin(2θ ) cos θ

(2a) 2ω

(2b)

(45) Huston, A. L.; Reimann, C. T. Chem. Phys. 1991, 149, 401-7. (46) Tohda, K.; Umezawa, Y.; Yoshiyagawa, S.; Hashimito, S.; Kawasaki, M. Anal. Chem. 1995, 67, 570-7. (47) Crawford, M. J.; Haslam, S.; Probert, J. M.; Gruzdkov, Y. A.; Frey, J. G. Chem. Phys. Lett. 1994, 229, 260-4. (48) Rinuy, J.; Brevet, P. F.; Girault, H. H. Biophys. J. 1999, 77, 3350-5. (49) Tsukanova, V.; Slyadneva, O.; Inoue, T.; Harata, A.; Ogawa, T. Chem. Phys. 1999, 250, 207-15. (50) Roscoe, S. B.; Yitzchaik, S.; Kakkar, A. K.; Marks, T. J.; Xu, Z.; Zhang, T.; Lin, W.; Wong, G. K. Langmuir 1996, 12, 5338-49. (51) Olenik, I. D.; Kim, M. W.; Rastegar, A.; Rasing, Th. Appl. Phys. B 1999, 68, 599-603. (52) Srivastava, A.; Eisenthal, K. B. Chem. Phys. Lett. 1998, 292, 345-51. (53) Wijekoon, W. M. K. P.; Ho, Z. Z.; Mull, M. W.; Padmabandu, G. G.; Hetherington, W. M. J. Phys. Chem. 1992, 96, 10450-3. (54) Geiger, F. M.; Tridico, A. C.; Hicks, J. M. J. Phys. Chem. B 1999, 103, 8205-15. (55) Dannenberger, O.; Wolff, J. J.; Buck, M. Langmuir 1998, 14, 4679-82. (56) Hsiung, H.; Beckerbauer, R.; Rodriquez-Parada, J. M. Langmuir 1993, 9, 1971-3. (57) Morgenthaler, M. J. E.; Meech, S. R. J. Phys. Chem. 1996, 100, 3323-9. (58) McAloney, R. A.; Goh, M. C. J. Phys. Chem. B 1999, 103, 10729-32. (59) Simpson, G. J.; Westerbuhr, S. G.; Rowlen, K. L. Anal. Chem. 2000, 72, 887-98. (60) Dick, B.; Gierulski A.; Marowsky, G.; Reider, G. A. Appl. Phys. B 1985, 38, 107-16. (61) Higgins, D. A.; Abrams, M. B.; Byerly, S. K.; Corn, R. M. Langmuir 1992, 8, 1994.

ω 2 2 ω 2ω s3 ) L2ω ZZ (LXX) cos θ sin θ

(2c)

ω 2 2 ω 2ω s4 ) L2ω XX(LZZ) sin θ sin θ

(2d)

ω 2 2ω s5 ) L2ω ZZ (LYY) sin θ

(2e)

where θω is the internal angle of incidence of the fundamental beam at the total internal reflection interface, θ2ω is the internal angle of reflection of the second harmonic, and Lω and L2ω are linear and nonlinear Fresnel factors, respectively. The Fresnel factors relate the electric field components at the interface (indicated by the subscripts) to the incident and detected fields. Explicit expressions for the Fresnel factors appropriate for the total internal reflection flow cell used in these investigations (see part 2) and a description of the model used to determine them can be found in ref 59. Evaluation of the three tensor elements can be performed from measurement of the entire polarization response curves (asin eq 1) or from the SHG intensities acquired for the following three polarization conditions: 2ω I s45 ) C|s1χXXZ|2(I ω)2

(3a)

2 ω 2 I 2ω ps ) C|s5χZXX| (I )

(3b)

2 ω 2 I 2ω pp ) C|s2χXXZ + s3χZXX + s4χZZZ| (I )

(3c)

χXXZ ) χXZX )

1 N [〈cos θ sin2 θ〉βz′z′x′ + 〈cos θ〉βx′x′z′ 2 s

〈cos θ sin2 θ sin2 Ψ〉(βz′x′x′ + 2βx′x′z′)] (4c)

where Ψ is the Euler rotation angle about the molecular z′-axis. Since the X- and Y-axes in the surface plane are equivalent, χYYZ ) χYZY ) χXXZ and χZYY ) χZXX. The expressions in eq 4 are greatly simplified if only a single tensor element of β(2) is dominant, which is often the case experimentally. Provided that both the orientation distribution and β(2) remain constant, each of the χ(2) tensor elements is directly proportional to the surface number density, Ns. From eq 4, the direct proportionality between the tensor elements and Ns leads to a quadratic relationship between the detected SHG intensity and Ns (eqs 1 and 3). However, as discussed previously, the simple quadratic relationship between Ns and the SHG intensity is only valid provided the orientation distribution does not change significantly during the course of the measurement. In several previous investigations, adsorption isotherm measurements by SHG23,29,30 and by linear dichroism42 have been corrected for observed changes in orientation by first determining the apparent orientation angle of the surface chromophore and then rescaling the SHG response accordingly. For example, in the case of a dominant βz′z′z′ molecular hyperpolarizability (characteristic of rodlike chromophores), evaluation of eq 4 yields the following surface second-order nonlinear tensor elements:

χZZZ ) Ns〈cos3 θ〉βz′z′z′ in which the first subscript describes the polarization state of the second harmonic and the second subscript designates the polarization state of the fundamental (e.g., the subscript “s45” corresponds to the s-polarized second harmonic generated for a fundamental polarization rotation angle of γ ) 45°). If the constant C is not known, relative values of the tensor elements may be determined from combined measurements under different polarization conditions. In general, two solutions satisfy the above set of equations, one in which χZXX and χXXZ are of like sign and another in which they are of opposite sign. The individual surface second-order nonlinear tensor elements of χ(2) are related to the molecular second-order nonlinear tensor, β(2), through orientational averages. Only three independent molecular second-order nonlinear tensor elements are present for SHG, βz′z′z′, βz′x′x′, and βx′x′z′.62 For SHG measurements with an orientation distribution that is isotropic within the surface plane, the three nonzero, independent tensor elements of χ(2) are given by62

χZZZ ) Ns[〈cos3 θ〉βz′z′z′ + 〈cos θ sin2 θ sin2 Ψ〉(βz′x′x′ + 2βx′x′z′)] (4a) χZXX )

1 N [〈cos θ sin2 θ〉βz′z′x′ + 〈cos θ〉βz′x′x′ 2 s

〈cos θ sin2 θ sin2 Ψ〉(βz′x′x′ + 2βx′x′z′)] (4b) (62) Heinz, T. F. Nonlinear Surface Electromagnetic Phenomena; NorthHolland: New York, 1991; Chapter 5.

χZXX ) χXXZ )

1 N 〈cos θ sin2 θ〉βz′z′z′ 2 s

(5a) (5b)

For this system, the SHG orientation parameter, D, is given by63

D≡

χZZZ 〈cos3 θ〉 ) χZZZ + 2χZXX 〈cos θ〉

(6)

The apparent orientation angle is given by θ* = cos-1(D1/2), where the asterisk on θ indicates that the calculated value is evaluated by assuming a narrow orientation distribution (depending on the form of the orientation distribution, the true mean orientation angle and the apparent orientation angle can potentially be very different59,64). Once the apparent orientation angle has been determined, it may be substituted into 〈sin2 θ cos θ〉 and 〈cos3 θ〉 in eq 5, and the value of Ns calculated by simple rearrangement of the expressions in eq 3: 1 2

(7a)

1 2

(7b)

2 -1 Ns,ps = C‚I 1/2 ps | s5(sin θ* cos θ*)|

1/2 | s1(sin2 θ* cos θ*)|-1 Ns,s45 = C‚I s45

(63) Dick, B. Chem. Phys. 1985, 96, 199-215. (64) Simpson, G. J.; Rowlen, K. L. J. Am. Chem. Soc. 1999, 121, 2635-6.

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1 2

2 3 -1 Ns,pp = C‚I 1/2 (7c) pp | (s2 + s3)(sin θ* cos θ*) + s4(cos θ*)|

in which the additional subscripts on Ns indicate the polarization conditions under which the intensity was measured (as per eq 3). Ideally, all three measured values for Ns should be identical after correcting for the apparent orientation angle. The orientation angle correction (or OAC) approach described above has the advantage of being applicable regardless of molecular orientation angle, but suffers from some practical limitations. First, the OAC approach requires a minimum of three SHG measurements at each surface coverage for evaluation of all three tensor elements, tripling the time required for acquisition of adsorption isotherm measurements. Additionally, the calculated value of Ns is only as reliable as the orientation measurements, such that the accuracy of each corrected data point is limited by the signal-to-noise ratio of the smallest SHG response detected, rather than the largest. As a final caveat, the OAC approach relies on correcting the measured response by substitution of the apparent orientation angle, calculated by assuming a narrow orientation distribution. The validity of this substitution may not be justified for broad distributions, which suggests the OAC approach is only strictly reliable for narrow orientation distributions. Artifacts in evaluation of the surface coverage related to changes in molecular orientation can often be minimized experimentally in linear measurements (such as absorbance and fluorescence) by judicious choice of an experimental geometry. An experimental configuration that orients the excitation polarization vector locally at an angle of 54.7° with respect to the surface normal (i.e., the magic angle) leads to absorbance measurements dependent only on the number of species probed and independent of molecular orientation.59,65 Recent work in our laboratory demonstrating the existence of a magic angle for SHG measurements (39.2°)64 led us to consider the possibility of an analogous orientation insensitive geometry for nonlinear optical measurements of surface coverage. THEORY: ORIENTATION-INSENSITIVE DETECTION Case 1. βz′z′z′. To explore the possibility of an orientationinsensitive experimental geometry for SHG, theoretical p-polarized SHG response curves were generated for several different molecular orientations (shown in Figure 2 for a dominant βz′z′z′ molecular hyperpolarizability tensor element). The curves were calculated using eq 1 with fitting coefficients appropriate for a total internal reflection cell consisting of a BK7 glass prism in contact with methylene chloride (eq 2). Each curve corresponds to a different apparent molecular orientation angle, in increments of 10°. From the figure, it is apparent that several of the curves cross around a polarization rotation angle of ∼63° for apparent orientation angles less than ∼50°. This trend suggests that measurement of the p-polarized second harmonic for a fundamental polarization rotation angle of ∼63° using the TIR cell should yield SHG intensities that are fairly insensitive to changes in molecular orientation for apparent orientation angles between 0° and ∼50°. (65) Michl, J.; Thulstrup, E. K. Spectroscopy with Polarized Light; Solute Alignment by Photoselection, in Liquid Chrystals, Polyners, and Membranes; VCH Publishers: New York, 1995, Chapter 5.

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Figure 2. Theoretical p-polarized SHG response curves calculated for several apparent orientation angles for a dominant βz′z′z′ molecular hyperpolarizability tensor element. Curves were calculated using the fitting coefficients for a total internal reflection cell described previously59 and employed in part 2. Each curve corresponds to a different apparent orientation angle, indicated on the left margin. The two solid vertical lines are the values of γ* calculated for the real and imaginary components of the fitting coefficients using eq 14.

Figure 3. The rms deviation of the p-polarized SHG response for a given range of apparent molecular orientation angles (θ*) shown as a function polarization angle by assuming a dominant βz′z′z′. Calculations were performed by evaluating the mean and the rms deviation from the mean for theoretical curves similar to those shown in Figure 2. For the solid line, angles between 0° and 50° only were considered. For the dotted line, all orientation angles were included. For orientation angles less than 50°, the potential errors associated with change orientation are minimized for a fundamental polarization rotation angle of 63°.

The value of the orientation-insensitive polarization rotation angle can be more rigorously quantified by evaluating the rootmean-square (rms) deviation as a function of polarization rotation angle, as shown in Figure 3. The rms deviation is a convenient value to quantify the potential for errors associated with changes in orientation for a given experimental geometry. Essentially, the average result of several of the curves shown in Figure 2 is evaluated, and the rms deviation about that mean is plotted as a function of the polarization rotation angle, γ. In the top curve, all

orientation angles are included in the calculation (i.e., 0°-90° in increments of 1°), while in the bottom curve, only orientation angles between 0° and 50° are included. From the minimum in the lower curve, the error associated with changes in orientation is greatly minimized for a polarization rotation angle of 63°, provided that the apparent molecular orientation angle is between 0° and 50°. In contrast, measurements of only I 2ω pp (i.e., for γ ) 0°), which is most routinely done, effectively maximizes the potential for errors associated with changes in orientation. The origin of the orientation-insensitive polarization rotation angle (hereby designated by γ*) can be clarified by a fairly straightforward mathematical treatment. Taylor series expansion of sin2 θ cos θ and cos3 θ about θ ) 0° yields the following infinite series:

χZXX ) χXXZ )

1 N 〈sin2 θ cos θ〉βz′z′z′ ) 2 s 1 5 91 6 N θ2 - θ4 + θ - ... βz′z′z′ (8a) 2 s 6 360

〈

〉

χZZZ ) Ns〈cos3 θ〉βz′z′z′ )

〈

3 2

7 8

Ns 1 - θ2 + θ4 -

〉

61 6 θ + ... βz′z′z′ (8b) 240

Keeping only the lowest order terms:

1 χZXX ) χXXZ ≈ Ns〈θ2〉βz′z′z′ 2

(

3 2

(9a)

)

χZZZ ≈ Ns 1 - 〈θ2〉 βz′z′z′

(9b)

Substitution into eq 1b yields the following expressions for the p-polarized second harmonic intensity: 2 2 2 2 I 2ω p (γ) = C′|s5〈θ 〉 + cos γ(s2〈θ 〉 + s3〈θ 〉 +

s4(2 - 3〈θ2〉) - s5〈θ2〉)|2(I ω)2 (10) Collection of terms yields 2 I 2ω p (γ) = C′|2s4 cos γ +

〈θ2〉[s5 + cos2 γ(s2 + s3 - 3s4 - s5)]|2(I ω)2 (11) If the polarization rotation angle, γ, is chosen such that the term in brackets following 〈θ2〉 is equal to zero, the measured intensity will be independent of the molecular orientation angle (in the limit that θ is small). This condition is achieved for

γ* ) cos-1

(

)

s5 3s4 + s5 - s2 - s3

1/2

(12)

where γ* is the theoretical orientation insensitive polarization rotation angle. The value of γ* given in eq 12 is only strictly valid if the fitting coefficients are all real numbers (or all imaginary). In general, the fitting coefficients are complex, and necessarily so for the total internal reflection geometry employed, such that the

detected intensity may be separated into the real and imaginary contributions:

I 2ωp = C′{Re(2s4 cos2 γ + 〈θ2〉[s5 + cos2 γ(s2 + s3 - 3s4 - s5)])2 + Im(2s4 cos2 γ + 〈θ2〉[s5 + cos2 γ(s2 + s3 - 3s4 - s5)])2}(I ω)2 (13)

For this expression, two values of γ* will be obtained, one from the minimum in the real contribution and one from the minimum in the imaginary contribution.

[ [

γ/Re ) cos-1 γ/Im ) cos-1

Re(s5)

] ]

1/2

Re(3s4 + s5 - s2 - s3) Im(s5)

(14a)

1/2

Im(3s4 + s5 - s2 - s3)

(14b)

In general, an orientation-insensitive polarization rotation angle will only be observed if one of the two contributions in eq 14 dominates the detected intensity or if the two contributions both yield similar values for γ*. In standard reflection and transmissionexperiments, the fitting coefficients are purely real in most treatments, such that γ*Im can be ignored and γ*Re will yield the appropriate orientation-insensitive polarization rotation angle. In the total internal reflection geometry employed here, the opposite is true with the imaginary components of the fitting coefficients being on average ∼1 order of magnitude greater than the real components. Using the fitting coefficients for the TIR cell described previously,59 composed of a BK7 glass prism in contact with methylene chloride, γ*Re is equal to 51° and γ*Im is equal to 61°, both of which are shown by vertical lines in Figure 2. As predicted on the basis of the relative magnitudes of the real and imaginary contributions to the fitting coefficients, the value of γ*Im obtained from the Taylor series expansion reliably reproduces the value obtained by minimizing the rms deviation (i.e., 63°) to within a few degrees. It is interesting to compare the range of validity of the orientation-insensitive methodology with the OAC approach described in the introduction. The OAC approach is only strictly valid provided that the orientation distribution is sufficiently narrow (such that 〈cos3 θ〉 = cos3〈θ〉 and 〈sin2 θ cos θ〉 = sin2〈θ〉 cos〈θ〉). In contrast, the orientation-insensitive methodology has no restrictions on the width of the orientation distribution and is limited instead by the requirement that the mean tilt angle be reasonably small (such that 〈θ4〉 is significantly less than 〈θ2〉, as per eq 9). It should be emphasized that the orientation-insensitive polarization rotation angle, γ*, given by eq 12 (and eq 14) should not be confused with the SHG magic angle of 39.2° mentioned in the introduction. The SHG magic angle is a constant describing the apparent orientation angle obtained by SHG if a broad orientation distribution is erroneously assumed to be narrow64 and is distinctly different from the phenomenon investigated in this report. As can be seen by inspection of eq 12, γ* is not a constant at all, changing with changes in the experimental configuration (e.g., with the Analytical Chemistry, Vol. 72, No. 15, August 1, 2000

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angle of incidence and refractive indices of the substrate and ambient media). Case 2. βz′x′x′. Determination of γ* for a dominant βz′x′x′ molecular hyperpolarizability can be performed in a similar fashion as for βz′z′z′. For a dominant βz′x′x′, simplification of eq 4 yields the following expressions for the independent surface nonlinear tensor elements:

χZZZ ) -2χXXZ ) Ns〈cos θ sin2 θ sin2 Ψ〉βz′x′x′ (15a) χZXX )

1 N [〈cos θ〉 - 〈cos θ sin2 θ sin2 Ψ〉]βz′x′x′ (15b) 2 s

If a uniform distribution about the orientation axis is assumed (e.g., from free rotation about the orientation axis), such that 〈sin2 ψ〉 ) 1/2, the expression can be rewritten (assumption of a uniform distribution in ψ not necessary, but leads to a unique relationship between β(2), χ(2), and D):63

χZZZ ) -2χXXZ ) χZXX )

1 N 〈sin2 θ cos θ〉βz′x′x′ 2 s

(16a)

1 N (〈cos3 θ〉 + 〈cos θ〉)βz′x′x′ 4 s

(16b)

Figure 4. Theoretical p-polarized SHG response curves calculated for several apparent orientation angles for a dominant βz′x′x′ molecular hyperpolarizability tensor element. Curves were calculated using the fitting coefficients for a total internal reflection cell described previously59 and employed in part 2. Each curve corresponds to a different apparent orientation angle, indicated on the right margin. The solid vertical line represents the values of γIm* calculated for the imaginary components of the fitting coefficients using eq 19.

leading to the following expression for the orientation parameter, D:63

2χZXX - χZZZ D) 2χZXX + χZZZ

(17)

The p-polarized SHG curves for several apparent molecular orientation angles are shown in Figure 4. The curves were calculated using eq 1 with fitting coefficients appropriate for a total internal reflection cell consisting of a BK7 glass prism in contact with methylene chloride (see eq 2). After substitution of the approximate expressions for 〈sin2 θ cos θ〉 and 〈cos3 θ〉 as per eq 8 and collection of terms, the p-polarized SHG intensity in the limit of 〈θ2〉 being small is given by

|

[

)] |

1 s + s3 - s4 - s5 2 2

(

2 (18)

γ* ) cos

(

s5

s4 + s5 -

1 s - s3 2 2

)

1/2

(19)

The designations for the real and imaginary components have not been explicitly included. For the TIR cell employed in these investigations, γ*Re is equal to 35° and γ*Im is equal to 46°. By minimizing the rms width for orientation angles between 0° and 50° (such as was done to generate Figure 3), a value for γ*rms of 42° is obtained. Again, the value calculated from the Taylor series 3404

χXXZ )

1 N [〈cos θ〉βx′x′z′ - 2〈cos θ sin2 θ sin2 Ψ〉βx′x′z′] (20b) 2 s

If a uniform distribution about the orientation axis is assumed

χXXZ )

1 N 〈cos3 θ〉βx′x′z′ 2 s

(21a) (21b)

for which the orientation parameter, D, can be shown to be63

which leads to the following expression for γ*: -1

χZZZ ) -2χZXX ) 2Ns〈cos θ sin2 θ sin2 Ψ〉βx′x′z′ (20a)

χZZZ ) -2χZXX ) Ns〈sin2 θ cos θ〉βx′x′z′

2 I 2ω p = C′ s5 + (s3 - s5) cos γ -

〈θ2〉 s5 + cos2 γ

expansion is within a few degrees of the value obtained by minimizing the rms deviation. Case 3. βx′x′z′. For a molecular hyperpolarizability dominated by βx′x′z′, simplification of eq 4 generates the following relations for the χ(2) tensor elements:

Analytical Chemistry, Vol. 72, No. 15, August 1, 2000

D≡

χXXZ 〈cos3 θ〉 ) χXXZ - χZXX 〈cos θ〉

(22)

The calculated p-polarized SHG responses curves generated for several apparent molecular orientation angles are shown in Figure 5. In contrast to the response curves for dominant βz′z′z′ and βz′x′x′ tensor elements, no significant p-polarized response was predicted for an apparent molecular orientation angle of 0°. The largest response was obtained instead for apparent orientation angles between 50° and 60°. Additionally, a minimum in

Figure 5. Theoretical p-polarized SHG response curves calculated for several apparent orientation angles for a dominant βx′x′z′ molecular hyperpolarizability tensor element. Curves were calculated using the fitting coefficients for a total internal reflection cell described previously59 and employed in part 2. Each curve corresponds to a different apparent orientation angle, indicated on the right margin. The solid vertical line represents the values of γIm* calculated for the imaginary components of the fitting coefficients using eq 24.

the p-polarized response occurred around a polarization rotation angle of 56° for essentially all orientation angles. To understand the observed behavior of the p-polarized response curved for a dominant βx′x′z′ tensor element, it is useful to first evaluate the calculated value of γ*. Substitution of the lowest-order terms in the Taylor series expansions of 〈sin2 θ cos θ〉 and 〈cos3 θ〉 as per eq 8 and collection of terms yields the following approximate expression for the p-polarized SHG intensity:

|

[ 3 cos γ( s + s - 2s - s )] |2(I ) 2

2 2 I 2ω p = C′ s2 cos γ - 〈θ 〉 s5 + 2

ω 2

2

3

4

From eq 23, γ* is given by

5

γ* ) cos-1

(

)

s5

2s4 + s5 -

(23)

1/2

3 s - s3 2 2

(24)

in which the designations for the real and imaginary components have again not been explicitly included. The value of γ*Im (55.5°) is indicated in Figure 5 by the vertical line. While the SHG response curves in Figure 5 appear to cross near the calculated value of γ*Im, this point also corresponds to a minimum in the overall measured intensity. Consequently, this value of γ* is not particularly useful for orientation-insensitive measurements. In fact, the minimum around 57° is not limited to small values of θ (for which the Taylor series expansion is reasonably accurate). This trend can be rationalized by nature of the near

Figure 6. Combined s- and p-polarized SHG response curves calculated for several apparent orientation angles for a dominant βx′x′z′ molecular hyperpolarizability tensor element. Each curve corresponds to a different apparent orientation angle, indicated on the left margin. The solid vertical lines represent the values of γIm(+)* and γIm(-)* calculated from the roots in eqs 28 and 30, respectively.

grazing angle internal reflection geometry employed (the internal angle of incidence at the surface is ∼70°). In this geometry, both s2 and s3 (which are dependent largely on the x-polarized surface electric field component) are much smaller than s4 and s5. Substitution of -2χZXX for χZZZ, together with the assumption that s2 and s3 are much smaller than the other fitting coefficients in eq 1b, yields the following approximate expression for the p-polarized SHG intensity: 2 2 I 2ω p = |χZXX[s5 - cos γ(2s4 + s5)]|

(25)

The p-polarized intensity exhibits a null value for cos2γ ) s5/(2s4 + s5), or 55.8° for our experimental geometry. The null calculated using eq 25 is identical to the expression for γ* in eq 24 in the limit that s2, s3 , s4, s5. While this phenomenon can be a valuable check to confirm the accurate treatment of the interfacial optics and a dominant βx′x′z′ hyperpolarizability, it is not particularly useful in obtaining SHG measurements insensitive to changes in orientation. One potential means to overcome the limitation of negligible signal at γ* is to utilize the total SHG intensity (i.e., the combined s- and p-polarized intensities). The subsequent polarization response curves are shown in Figure 6. Two minimums are present in the rms plot (Figure 7) for angles less than 50°, one for a polarization rotation angle of 27° and another for 76°. An explanation for these minimums is fairly straightforward. In this geometry, the total SHG response is given by 1/2 (I 2ω ∝ |s1 sin(2γ)χXXZ| + tot )

|s5χZXX + cos2 γ(s2χXXZ + s3χZXX + s4χZZZ - s5χZXX)| (26) Since the phase difference between the s- and p-polarized comAnalytical Chemistry, Vol. 72, No. 15, August 1, 2000

3405

Table 1. Calculated Orientation-Insensitive Polarization Rotation Angles (γ*, deg)a dominant β(2) tensor element βz′z′z′ (p-pol 2ω) βz′x′x′ (p-pol 2ω) βx′x′z′ (p-pol 2ω) βx′x′z′ (p+s-pol 2ω)

γRe*

γIm*

γrms*

50.7 35.3 44.6 17.2 72.8

60.9 46.2 55.5 31.2 74.4

63 42 56 27 76

a The calculations were performed for a total internal reflection geometry (see text).

and, in the limit of small orientation angles Figure 7. The rms deviation of the combined s- and p-polarized SHG response for a given range of apparent molecular orientation angles (θ*) shown as a function polarization angle. Calculations were performed by taking the rms deviation between theoretical curves similar to those shown in Figure 3, assuming a dominant βx′x′z′ molecular hyperpolarizability tensor element. In the solid line, angles between 0° and 50° only were considered. In the dashed line, all orientation angles were included. For orientation angles less than 50°, the potential errors associated with changes orientation are minimized for fundamental polarization rotation angles of 27° and 76°.

ponents does not influence the overall intensity, it has been neglected. If both terms inside the two absolute values are of like sign, then 1/2 (I 2ω ∝ {[s1 sin(2γ)χXXZ] + [s5χZXX + tot )

cos2 γ(s2χXXZ + s3χZXX + s4χZZZ - s5χZXX)]} (27) substitution for the tensor elements yields

{

1/2 ∝ [s1 sin(2γ) + s2 cos2 γ] (I 2ω tot )

[

)]} (28)

3 1 〈θ2〉 s1 sin(2γ) + s5 - cos2 γ 2s4 + s5 - s2 - s3 2 2

(

The root of the coefficient of 〈θ2〉 in eq 28 was determined numerically to yield a value of γ*Im of 31° for our experimental geometry (see Table 1). The calculated value of γ*Im corresponds well with the first of the two minmums observed in Figure 7, of 27°. An approximation for the second minimum can be evaluated by considering the case in which the two terms inside the absolute values in eq 26 are of opposite sign, such that 1/2 (I 2ω ∝ {[s1 sin(2γ)χXXZ] - [s5χZXX + tot )

cos2 γ(s2χXXZ + s3χZXX + s4χZZZ - s5χZXX)]} (29)

3406

Analytical Chemistry, Vol. 72, No. 15, August 1, 2000

{

1/2 (I 2ω ∝ [s1 sin(2γ) - s2 cos2 γ] tot )

[

)]} (30)

3 1 〈θ 〉 s1 sin(2γ) - s5 + cos2 γ 2s4 + s5 - s2 - s3 2 2 2

(

In this case, a value of 74° is calculated for γIm, in good agreement with the second minimum of 76° in Figure 7. A summary of the values for γ* generated both by minimizing the rms deviations and from the Taylor series expansions is provided in Table 1 for each case investigated. In all cases, the rms-minimized value of γ* corresponds well with the value of γ* calculated using the imaginary component of the fitting coefficients, γ*Im, as expected for the TIR geometry employed. Summary. The theory necessary to determine the appropriate experimental geometry for orientation-insensitive SHG measurements was presented for the three most common cases (βz′z′z′, βz′x′x′, βx′x′z′.) of dominant hyperpolarizability tensor elements. For apparent orientation angles between 0 and 50° with respect to the surface normal in a total internal reflection geometry, and a dominant βz′z′z′, the sensitivity of SHG to molecular orientation may be minimized by using a polarization rotation angle 63° for the fundamental beam under our experimental conditions. Under the same conditions for a molecule with a dominant βx′x′z′, the sensitivity of SHG to molecular orientation may be minimized by using a polarization rotation angle of either 27° or 76° for the fundamental beam. In part 2 of this work, the theory for these two molecular systems is tested experimentally. ACKNOWLEDGMENT The authors gratefully acknowledge funding from the National Science Foundation. Received for review March 23, 2000. Accepted June 14, 2000. AC000346S