366
Langmuir 1993,9, 356-361
Second Harmonic Gemeration from Composite
(a)
Filmr of Spheroidal Metal Particler
z
k
medium 1
G.Berkovic Department of Materials and Interfaces, Weizmann Znetitute of Science, Rehovot, Israel 76100
W
medium 2
S. Efrima' Department of Chemistry, Ben Curion University, P.O.Box 653, Beer Sheva, Israel 84105 Received August 31, 1992
Introduction Recently second harmonic generation (SHG)was used to investigate structural parameters of dense interfacial suspensions of silver colloids.1 T h e system which was investigated was that of a silver metal liquid-like film (MELLF).2 These colloidal films are formed in the interface of water and an organicliquid (dichloromethane, for instance)and are stabilized by surfactantsand organic additives (such as p-methoxybenzoicacid). These films have the general structure of composites or cermets, i.e. metal particles dispersed in a nonconductive dielectric medium. For metal particles which are much smallerthan the wavelength of light and for high concentrations(as in MELLFs), these films show high specular reflectivity. In the SHG study the polarization intensity ratio (G,,! Gap) and the enhancement relative to a smooth silver surface, f, were used to obtain information regarding the volume fraction of the metallic component in the colloid and the shape of the particles. Here G i j is the intensity of the SHG signal with i-polarized incident light and j-polarized reflected light, divided by the square of the power of the incident light. The analysis was based on a theoreticalmodelwhich was developedassumingspherical particles, and the "polarization sheet" model for the nonlinear behavior? utilizing dielectric data for the composite film, rather than that of the individual silver particles. In thisletter we explorethe behavior of spheriodalmetal particles with respect to the polarization ratio and the enhancement in SHG. Simple expressions are derived and are applied to model interfacialf h of silver colloids. Thismodel againconsidersparticles that are much smaller (e100nm)than the wavelength of light, and arranged as an array, e.g. at the interface between two media-see Figure la. Figure l b describes the coordinates which are convenientfor treating an ellipsoid of revolution given by x2
+ y2 + (z/a)2 = 1
(1)
wherex , y, andz are normalized to the radius of the circular projection of the ellipsoid on the xy plane and a is the eccentricity. The radius vector to the surface of the ellipsoid is given by (1) Bavli, R.;Yogev,D.;Efrima, S.;Bezkovic, G. J. Phys. Chem. 1991, 95,7422. (2) (a)Efrima,S. In. CRCCrit.Reo. Surf. Sci. 1991, I, 167. (b)Yogev, D.; Efrima, S.J. Phyr. Chem. 1988,92,6764. (c) Yogev,D.; Efrima, 5. J. Phya. Chem.1988,92,6761. (d) Yogev,D.; Efrima, S.;Kefri, 0.Opt. Lett. 1988,13,934. (e)Yogev,D.;Deutsch,M.;Etrima,S. J.Phys. Chem. 1989,9S, 4174. (0Yogev,D.; Shtutina, 5.; Efrim,5.J. PhyS. Chem. 19)0,94,752. (g)Yogev, D.;Efrima, S.Longmurr 1991,7,267. (h) Yogev, D.; Kuo,C. H.;" m a n , R.D.;Efrima, 5. J. Chem.phyS.1989,91,3222. (i) Yogev,D.;EiXnm,S.J. Colloidlnterface Sci. 1991,147,88. cj) Yogev, D.; Rortltier-EdeLtein,D.;Efrima, S. J. Colloid Interface Sci. 1991,147, 78. (3) Mierahi, V.; Sip, J. E. J. Opt. Soc. Am. B 1988,6,660.
Figure 1. (a)Schematic view of the interfacial colloidal system. (b) The coordinate syetem wed in the analysis.
r = (p sine COS 4, p sin e sin 4, p c-8)
(2)
with
p(e) = [sin2e + a-2 cos2
(3)
We adopt a parametric representation in terms of the angle a defined by
cos a = (p/a)cos 8, or alternatively by sin a = p sin 0 (4) In this representation the normal (N)and two tangenta to the surface (T1,T2) at any arbitrary point are given by expressions reminiscent of those of a sphere'
N = (a sin a cos 4,a sin asin 4, cosa)/[a2sin2a+ cos2a p
Tl = (cos a cos 4, cos a sin 4, -a sin a)/[a2sin2a + a1112
T2= (-sin 4, cos 4,O) (6) The second-order nonlinear electric susceptibility of a metallic surface, such as that of silver, is dominated by the X I II~interaction.' Thus the average susceptibilities, Xijk, of e particlea are calculated by projecting X T ~ N T+ ~ X T ~ N TOnto ~ x i j k , integrating over the hemiellipsoid (#e[0,2r3,ae[O,r/23) and normalizing to the projected area, w. T h e surface element, dA,is given by
h
dA = sin a[a2sin2a + cos2
da d4 = a-2p4 [a' sin20 + cos2 sine dB d4 (6) The result of the averaging is Xyrr
= Xryr
--
Xyyy
=0
(7)
and XlZ2
= '/2X4C(a)
xryy = -'/4xoC(a) X y r y = '/&J2 - C(a)l
(8)
with xo being the intrinsic nonlinear susceptibility per
unit area of curved surface of a silver colloid, and
(4) (a) Boyd,G. T.; Whg,Th.;bib, J. R. R, Shen, Y.R. Phys. Reu. B 1984,30,619. (b) Chen, C. K.;Heinz, T. F.;heard, D.;S h a , Y. R Phys. Rev. B 1983,27,1965.
0143-1463/93/2409-0366$04.00/0Q 1993 American Chemical Society
356 Langmuir, Vol. 9, No. 1, 1993
Notes
1000000
'Ooo
.1 I .01
1
.1
0
1
10
eccentricity Figure2. The polarization ratio (G P/G,,) and enhancement cf, 88 a function of the eccentricity, Capculated for a model system of a silver metal liquid-like f i b . Incidence angle of 37O. C(a) = 2a2[a2- 1 - 2 In a1/(a2- 1)'
- -
(9)
The subscript y is either x or y. It is easily seen that for a 1, C(a) 1 and the results for a sphere' are retrieved. Note that there is a difference by a factor of 2 due to the changeof convention whereby x is expressed per unit area of the projection of the hemisphere,rather than the curved surface area. Following the derivation in ref 1, we now obtain for the polarization ratio
G,JG,,
we found that the observed polarization ratios spanned a somewhat larger range-three IWLLFs had polarization ratios in very close correspondenceto that of the spherical model; one sample had a rather large ratio (2.7) and the remainingMEuFs had polarizationraticm slightlysmaller than unity. By using eqs 10 and 11 with the appropriate dielectric data, we calculate that the highest observed polarization ratio of 2.7 correspondsto eccantricity 0.76 (i.e. an oblate ellipsoid),while the lowest polarization valuescorrespond to more prolate ellipsoids of eccentricity around 3. It should be pointed out that for eccentricities greater than 1, the ratio only varies very weakly with eccentricity, and thus this ratio is not as sensitive a probe for prolate eccentricity as it is on the oblate side. As previously shown' one can obtain an estimate of the volume fraction of the metal in the film, a, from the calculated enhancements, f, and the measured enhancement factors,fspp For an array of ellipsoids,assuming a regular square arrangementwith equal surface-to-surface interparticle distances at the major and minor 8188, and uniform orientation (Le. the z axis is normal to the surface of the film),we obtain @(a)/@(l)= a / { l + 2(a - 1)[(3/4)W)/d13) (12) which reduces to the previous result of a sphere' for a = 1. Here the case of spherical particles is given by
= )'/&,[2 - C(a)l + '/&,C(a) '/4k3C(a)12/l'/4k4C(a)~ (10)
and for the enhancement
= 11/&1[2- C(a)l + '/&2C(a)- '/4k3C(a)12/lkl(smooth)12 (11) The coefficients kip which are functions of the laser light angle of incidence and the complex refractive indices of the media on both sides of the interface, are calculated according to Mizrahi and Sipe3 and Shen,b and explicit expressions were given previous1y.l In order to examinethe SHG response of a "real" system of dispersed particles, we plot in Figure 2 the polarization ratio and enhancement as given by eqs 10 and 11. We have used the dielectric data of a typical silver MELLF (sample 3 in the previous report'), with 41064) = 15.90 + 6.87i and 4632) = -5.13 + 9.991. Experimentally, this MELLF exhibited a polarization ratio of 1.6, which is shown in the f i i e to correspond exactly and uniquely to a system of spherical particles (a = 1). The polarization ratio is extremely large for small eccentricities,as is expectedfor flat structures. In practice, other componentsof the nonlinear susceptibilitybecome important,andtheratioisexperimentallylimitedto-22.' For elongated shapes the polarization ratio converges asymptoticallyto a constant value of the order of unity. This value depends on the angle of incidence as discussed below. In our previous publication' we examined nine different MELLFs, which varied in their chemical constituents. These different samples showed different dielectric constants, which apparently arise from differencesin size and concentration of the silver particles.6 A consequence of these differences is that the calculated spherical particle polarization ratio is not identical for all MELLFs, but neverthelese all fell in the range 1.4-1.8. Experimentally, f = G,(system)/G,(smooth)
(6) Shen, Y. R. Annu. Rev. Phys. Chem. 1989,40, 327. (6) (a) Farbman, I.; Lsvi, 0.;Efrima, S.J. Chem. Phys. 1992, W, 6477. (b) Farbman, I.; Efrima, S.J. Phya. Chem. 1992, W, 8489.
These relationships are important for extracting values for the metal loading from the SHG measurements. It was shown' that assuming spherical particles, the experimental values for the nine different samples of MELLFs are in the range 6-21%. This is in agreement with the current model of MELLFs, consisting of nanometer size silver particles, enshrouded with a -1-2 nm protective layer of adsorbed molecules. Using the spheriodalmodel and the eccentricitiesderivedfrom the polarizationratios, as described above, we obtain for the MELLF samples metal loading values in the range 6-17%, comparable to what the sphericalmodel gives, and physicallyreasonable. This work appears to be the fmt application of SHG to such a "composite" interface which is macroscopically uniform, but composed of nanometer sized particles suspended h a continuousphase of very differentdielectric and nonlinear opticalproperties. In our analysis,we have adopted the usual model of a "polarization sheet" in the lateral d ~ e c t i o n .Hence, ~ our model incorporates the "average" dielectric constant of the MJ3LLF layer, which is consistent with the observed linear reflection from the layer.s However, one could question the use of the composite dielectric data rather than that of pure silver for the analysis of the SHG from silver MELLFs. After all, the silver particles are the active center in the SHG. One argument in favor of our choice is that the average dielectricdata incorporatethe electromagneticinteraction between the particles. For a dense colloidal suspension, such as a MELLF, the local fields and the response of the particles are very different thanfor a dilute system,where the particles do not affect each other. This choice is also in keeping with the "polarization sheet" model and addresses the linear and nonlinear properties within the same general scheme. Furthermore, the results of our analysisare sensitive to the dielectricparameters, as seen in the followingcalculations. Applying the dielectric data of pure silver to eqs 10 and 11 produces the predicted polarization and enhancementbehavior of Figure 3. (The complex dielectric constants are, ((1064) = (-68.1?,0.61)
Langmuir, Vol. 9, No.1, 1993 367
Notes 100000
100
1 0000
10
1000
1
100 10
1
0
C
0
. I
c)
3 2
. I
E Q)
.01
'
.001
.1
g a g Q)
I
g
E
,0001
.01
0
.001 .01
.1
1
10
eccentricity Figure 3. Polarization ratio (C, Cap)and enhancement V, as a function of the eccentricity, c culatd for a model system of a f i of eilver particles having the dielectric properties of bulk silver. Incidence angle of 37O.
Bp/
and 4532)= (-11.69,0.34).7)It is assumed that the silver particles form a film which is probed through its interface with water and that it is diluted so that the dielectric response is that of the isolated metal particles. T h e behavior at very small eccentricitiesis a result of the neglect of small components of x , as diecussed above. At an eccentricity of 0.2 there is a singularity in the behavior which is manifested ale0 in the enhancement factors. At this point the sp component becomes much larger than the pp reflection, though this is accompanied by a strong reduction of the total intensity (by more than 4 orders of magnitude). Thus at the immediate region around a = 0.2one expects extremely weak SHG. The enhancements (normalized to the area occupied by the metal particles) are considerable for spheres and the elongated shapes (a > 1) and, as expected, converge to unity for the oblate ellipsoids. Though the global behavior is rather similar to that seen ai>ove for a MELLF, there are some significant changes. First, the prominentsingularits at a = 0.2appears when usingthe dielectric data for bulk silver rather than that of a MELLF. Also the values of the enhancements are about 1 order of magnitude smaller, though they vary with eccentricity over a larger range than previously. Similarly the polarization ratios are about 1 order of magnitude larger for MELLFs than for bulk silver. This demonstrates the significant effect of the dielectric propertiea of the film on the SHG behavior. Figure 4 shows the dependenceof the polarization ratio on the angle of incidence for a sphere (a = 1.0). Interestingly, for each of the ellipsoids with a = 0.1,0.2,5.0, and 10,one obtainsexactlythe samefunctionaldependence of the polarization ratio on the angle of incidence, except for a constant factor. This scale factor depends on the eccentricity, and we obtained the values of 8.6,0.082,1.0, 1.3,and 1.3 for u -.l, 0.2, 1.0,5.0,and 10.0 respectively. It seemsthat the scale factor converges to a value of -1.3 for large eccentricities. Larger angles favor the pp component, which is expected if one considers that the parallel-to-surface component of the electric field of the ~
(7) J o h n , P.B.;Christy, R.W.Phye. Rev.
B 1972,6, 4370.
0
20
40
60
80
angle, deg Figure 4. Dependence of the polarization ratio on the angle of incidencefor a model f i of spherical silver particleshaving the dielectric properties of bulk silver.
incidentradiation decreaseswith the increase of the angle. Unlike the polarization ratios, the enhancements are only very weakly dependent on the angle. When one appliea these resulta to the experimental measurements from the MELLFs, in a fashion similar to the analysis described above, one ends up with oblate particles (a in the range 0.14-0.7) and impossible volume fractions (typically much larger than 100% or even negative). Thus we establish the validity of the polarization sheet model and the need to use dielectricfunctions of the complex MELLF system. The plausibility of the results also indicates that the model and the estimated dielectric parameters are reasonable.
Conclusion
We derived here expressions relevant to SHG studies of colloidal filma composed of spheriodal particles. We showed that the eccentricity is a sensitive function of the measured polarization ratio and the input dielectric parameters used in the analysis. The loadingis dependent on the enhancements of the SHG signals and the eccentricitiesused in the analysis,but not excessivelyso. Thus more reliable values for this structural parameter of silver MELLFs have been obtained. Our method of analysis has usedas an input parameter the observed h e a r constants of the composite met& dielectric. In doing so, we automatically include the consequences of particle shape, size, and density on the local field in the nonlinear medium. We have given predictions for the behavior of disperse silver colloidal films (at low concentrations)and analyzed the SHG resulte for various samplesof silver metal liquidlike films to demonstrate the viability of this approach to the study of colloidal films. We have demonstrated the importance of using an appropriate set of dielectric functions. We would like to point out that, in principle, with a predetermined structure one could invert the procedure employed here and obtain the dielectric constants from SHG measurements.
Acknowledgment. S.E.acknowledgespartial support by the United States-Israel Binational Science Foundation.
