Langmuir 1997, 13, 7135-7140
7135
Effect of Surface-Influenced Order on Thermal Expansivity of Polymer Thin Films C. Thompson* Department of Physics, Polytechnic University, Brooklyn, New York 11201
R. F. Saraf and J. L. Jordan-Sweet IBM Thomas J. Watson Research Laboratory, P.O. Box 218, Yorktown Heights, New York 10598 Received December 5, 1996. In Final Form: September 30, 1997X Variations of the thermal expansion coefficient of thin polyimide films as a function of film thickness (40-260 nm) have been measured by monitoring film thickness as a function of temperature using X-ray reflectivity from samples of poly(pyromellitic dianhydride oxydianiline) (PMDA-ODA) films spin-cast onto silicon wafers. The effective expansion coefficient of the film decreases with decreasing thickness and is consistent with a bilayer model in which an interfacial region of constant thickness is formed for each film. The expansion coefficient of this surface or interfacial region is less than one-third the coefficient of the interior of the films. A simple method is described whereby the thermal expansion coefficient may be determined by reflectivity.
Introduction The role of interface-enhanced structural-anisotropy effects on the physical properties of polyimide films presents a problem of both fundamental and technological interest. For rigid, aromatic polyimides that order, such as poly(pyromellitic dianhydride-oxydianiline) (PMDAODA) and poly(biphenyl dianhydride-p-phenylenediamine) (BPDA-ODA), the presence of a film substrate interface tends to orient the chains of polyimide along the film plane. In PMDA-ODA, the polyimide studied in this paper, the chain orientation propagates throughout the film with thicknesses as high as 100 µm.1 The resulting structure exhibits a planar texture, with the c-axis (i.e., chain axis) oriented parallel to the film surface, with random orientation of a- and b-axes about the c-axis, and with random orientation of c-axis about the surface normal. This is a bulk-orientation influence caused by the presence of a substrate. Additionally, the interface between the air and film also influences the film structure. Grazing incidence X-ray scattering (GIXS) studies have shown that, within ∼10 nm to the air/film interface, the coherence length along the chain increases and the chain conformation is more extended compared to the bulk.2,3 Furthermore, GIXS studies have shown that this enhanced order results from a fiber texture limited to this ∼10 nm region, with the b-axis tending to align perpendicular to the film plane and the a- and c-axes randomly oriented within the film plane. This is referred to as surface ordering. The degree of surface ordering and its extent can be dependent on processing. However, this nearsurface ordering is primarily due to increased mobility3 at the surface and not due to the (suspected) complexedsolvent-induced mobility that is released during the imidization cycle. * Current address: Department of Physics, Northern Illinois University, DeKalb, IL 60115. X Abstract published in Advance ACS Abstracts, November 15, 1997. (1) Saraf, R. F.; Roldan, J. M.; Derderian, T. IBM J. Res. Dev. 1994, 38, 441. (2) Factor, B. J.; Russell, T. P.; Toney, M. F. Phys. Rev. Lett. 1991, 66, 1181. (3) Saraf, R. F.; Dimitrakopoulos, C.; Toney, M. F.; Kowalczyk, S. P. Langmuir 1996, 12, 2802.
S0743-7463(96)02073-2 CCC: $14.00
The goal of this paper is to study the behavior of submicron-thick PMDA-ODA films and to explore the differences of the physical properties of the interfacial film compared to the bulk. The property we chose for study is the coefficient of thermal expansion along the thickness direction. Since conventional bulk techniques4,5 and special optical and capacitance methods for thin-film thermal expansion measurement methods6,7 are not suitable for characterizing ultrathin film, we measure the coefficient of thermal expansion by X-ray reflectivity. Using an X-ray reflectivity technique to monitor film thickness, we have measured the thermal expansion coefficients for thin films of poly(pyromellitic dianhydride oxydianiline) (PMDA-ODA). Recently, there have been several reports probing the glassy nature of the interface between a polymer film and a substrate or at the free surface, by measuring thermal expansion as a function of thickness.8-12 Since the measurements are similar, in the Discussion section we will compare our results to the literature, noting that the studies referenced were on amorphous polymers and the material used in this study is a semicrystalline polymer. The advantage of X-ray reflectivity for this study is that it quite sensitive to small changes in the total film thickness during the thermal cycle.13 A straightforward method is described by which the variation in the film total thickness with temperature may be easily measured, (4) Tiwary, H. V.; Sao, G. D. J. Phys. E: Sci. Instrum. 1981, 14, 1378. (5) Elsner, G.; Kempf, J.; Bartha, J. W.; Wagner, H. H. Thin Solid Films 1990, 185, 189. (6) Tong, H. M.; Hsuen, H. K. D.; Saenger, K. L.; Su, G. W. Rev. Sci. Instrum. 1991, 62, 422. (7) Pottinger, M. T.; Coburn, J. C. In Polymers for Microelectronics; Wilson, C. G., Thompson, L. J., Eds.; ACS Symposium Series 537; American Chemical Society: Washington, DC, 1993. (8) Beaucage, G.; Composto, R.; Stein, R. S. J. Polym. Sci., Part B: Polym. Phys. 1993, 31, 319. (9) Orts, W. J.; vanZanten, J. H.; Wu, W.-L.; Satija, S. K. Phys. Rev. Lett. 1993, 71, 867. (10) Keddie, J. L.; Jones, R. A. L.; Cory, R. A. Europhys. Lett. 1994, 27, 59. (11) Wu, W.-L.; vanZanten, J. H.; Orts, W. J. Macromolecules 1995, 28, 771. (12) vanZanten, J. H.; Wallace, W. E.; Wu, W.-L. Phys. Rev. 1996, E53, R2053. (13) Wallace, W. E.; vanZanten, J. H.; Wu, W.-L. Phys. Rev. 1995, E52, R3329.
© 1997 American Chemical Society
7136 Langmuir, Vol. 13, No. 26, 1997
Thompson et al.
even under conditions in which an unknown internal structure is present. This method does not require profile fits of the X-ray reflectivity measurements. Experimental Section Samples of three thicknesses were prepared by spin casting onto silicon substrates a 5-7% (weight) solution of the ethyl ester precursor of PMDA-ODA in N-methylpyrrolidinone (NMP). The substrates were 51 mm in diameter and 5 mm (3/16 in.) thick. A fourth sample was prepared from a 4.98% solution of a 1:1 blend of poly(amic acid) (PAA) and ester precursor (PAETE) in NMP. In a forming gas environment, the spin-cast samples were subjected to a curing cycle which consisted of 5 °C/min ramps between 30 min anneals at the following temperatures: 70, 170, 210, and 410 °C. The samples were cooled to room temperature. The thermal expansion measurements were performed on the cured spin-cast samples. In the regime well below the discontinuities at the polymer glass transition temperature, the thermal expansion coefficient is approximately constant. It is appropriate, therefore, to calculate a thermal expansion coefficient, , from measurements of film thickness, d, at two temperatures, using the formula
)
d2 - d1 d1(T2 - T2)
(1)
For the reflectivity measurements, the samples were mounted in a vacuum furnace and X-ray reflectivity patterns were taken at two temperatures, 25 and 200 °C. These temperatures are more that 150 °C below the PMDA-ODA glass transition temperature.14 Prior to the measurements, the samples were stored in a desiccator to minimize moisture absorption. However, it was found necessary to cycle once to the target temperature of 200 °C for subsequent reflectivity profiles to be reproducible and reversible upon temperature cycling. It was assumed that this was due to moisture absorbed during sample handling or the removal of residual solvent left from the initial curing of the film. Therefore, for the films discussed in this paper, each sample was first taken to the target temperature of 200 °C and the high temperature reflectivity scans were initiated. Room-temperature measurements followed the high-temperature measurements on a sample. There was no evidence for additional sample outgassing or film changes during the isothermal 200 °C scans or the room-temperature scans (which would quite clearly be shown by changes in the reflectivity profile during repeats of portions of the scans). All reflectivity experiments were performed less than 3 days after the preparation of the sample. The furnace was mounted on the high-resolution X-ray scattering diffractometer of the IBM/MIT beam line X-20C at the National Synchrotron Light Source. This beam line has horizontal and vertical focusing optics upstream of the monochromator in which two silicon (111) crystals were set to give a photon energy of 8.05 keV. The scattering plane was vertical. Slits were used to give an incident beam size of 0.155 mm vertical by 2 mm horizontal. The incident beam intensity was monitored in the conventional fashion by measuring the scattering from a piece of Kapton film placed after the beam-defining slits. A copper foil absorber wheel was placed following the monitor and before the sample. The incident beam divergence in the scattering plane was 0.12 mrad (0.007°). The defining exit beam slit in front of the scintillation detector was set to accept the full width of the specularly scattered beam. Additional slits were set in the beam paths to eliminate parasitic scattering. Figure 1 shows a typical experimental X-ray reflectivity profile for a film at room temperature. These profiles have been corrected for the fraction of the incident beam subtended by the sample. The background scattering, which was measured with a longitudinal scan, offset by +0.03° in θ from the specular condition, has been subtracted from the data.
Figure 1. Experimental X-ray reflectivity pattern for the 41nm film at 25 °C. As discussed in the text, the data have been corrected for background scattering and for the fraction incident beam subtended by the sample.
a function of the symmetric glancing angle, θ. The scattering wavenumber is q ) 4π sin(θ)/λ. With this definition of θ, Snell’s law takes the form cos θ ) n cos θ1, where n is the index of refraction in the film. For X-rays, the real part of the refractive index, n, of the films is less than unity by the small amount δ ) 1 - n ≈ 10-6, which is proportional to the total electron density of the material.15 Since n is less than unity, total external reflection occurs for angles smaller than the critical angle, given by sin2 θcrit ≈ 2δ. For a single film of homogeneously dense polyimide on a silicon substrate, the following equation would hold for the refraction-corrected position of the successive minima, Qm,
xqm2 - qcrit2 ) Qm(q) ) 2πm/d
(2)
where d is the thickness of the film, qcrit is the scattering vector at the critical angle, and m is an integer index for the minima. For each sample, in the region of the oscillations, the reflectivity curves, R(q) × q4 (∼R/RF, as noted in the appendix), were plotted, and the positions of the successive minima in the experimental reflectivity curves, qm, were graphically determined. The scattering vector at the critical angle, qcrit, was also determined graphically. Its value was consistent with qcrit calculated from the expected composition and bulk density of PMDA-ODA. The refraction-corrected minima positions, Qm, were determined using eq 2. In Figure 2, these successive minima positions, Qm, versus minima index, m, are shown for the films at 25 °C. For a single film morphology, it is clear from eq 2 that
dQm 2π ) dm d
(3)
Analysis The X-ray reflectivity curve indicates the ratio of the specularly reflected intensity to the incident intensity, as
and thus the total thickness of the films may be found by the slopes of the fits. The total film thicknesses at 25 °C determined by this analysis are given in Table 1. The exact shape of an experimental reflectivity profile, such as is shown in Figure 1, is determined not only by the total film thickness but also by the roughness of the
(14) Okamoto, K.-I.; Tanihar, N.; Watanabe, H.; Tanaka, K.; Kita, H.; Nakamura, A.; Kusuki, Y.; Nakagawa, K. J. Polym. Sci., Part B: Polym. Phys. 1992, 30, 1223.
(15) International Tables for Crystallography; Wilson, A. J. C., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992; Vol. C.
Thermal Expansivity of Polymer Thin Films
Langmuir, Vol. 13, No. 26, 1997 7137
Figure 2. Refraction-corrected minima position, Qm, versus minima index for the films at 25 °C. Fits, shown by these lines, give film thickness values of 41, 106, and 256 nm; the thickness of the blended film was 116 nm. Table 1 samples spin-cast on silicon wafer
thickness D (nm)
constrained (10-6 °C-1)
thin sample medium sample thick sample precursor blend sample bulk polyimide9
41 106 256 116 25 000
52 ( 5 104 ( 10 126 ( 10 88 ( 5 157 ( 10
silicon and polymer interfaces and the density profile variations in the film and at the interfaces, such as those which may arise due to the surface being slightly denser and more crystalline than the interior. A full analysis of the reflectivity profile would give details of the density profile across the thickness of the sample. It is possible that additional interfaces between more dense and less dense layers in a film can affect the minima spacing. Nevertheless, because of the relatively small X-ray scattering contrast expected in the polyimide film between the less crystalline and more crystalline regions in the polyimide film, we expect that the spacing of the intensity oscillations is approximately independent of the internal structure of the film. This is discussed in greater detail in the appendix. Discrepancies would show up as nonlinearities in a plot of Qm versus minima index, m. The linearity of these results in Figure 2 supports this simple procedure for determining the total thickness of the films, which neglects the subtle effects on shifts in minima spacing as a function of q due to small density variations within the polyimide film. The quality of the linear fits to the high-temperature results was indistinguishable from these room temperature results. Ignoring the variations in the minima spacing that are expected to arise when the film is not homogeneously dense throughout its thickness, the thermal expansion coefficient is given by
Figure 3. Difference in position of the oscillation minima at the two temperatures, ∆Qm ) Qm(200 °C) - Qm(25 °C), plotted as a function of the minima position at 200 °C. The determined by these data are plotted in the inset, where the value estimated for bulk polyimide (1/D ) 0) is shown for comparison. The bulk value was not used in the fit by which I was determined for the homogeneous samples. The open circle in the inset is the of the blended sample. Numerical values are listed in Table 1.
Qm(25 °C) - Qm(200 °C), is plotted as a function of the minima position at 200 °C. From eq 4, one can see that (T2 - T1) is equal to the slope of the linear fits in Figure 3. The values in Table 1 were calculated using the slopes of the straight line fits shown in Figure 3 and noting that the temperature difference, T2 - T1, is equal to 175 °C. As seen in Figure 3, the data exhibit some deviations from linearity. These discrepancies indicate that these films may possess internal structure. These measurements determine the overall film thickness with respect to the silicon/polyimide interface. Therefore, relative thickness changes of the silicon wafer due to its thermal expansion have no effect on the experimental values of . In the geometry of the experiment, the thin films are constrained from expanding or contracting laterally due to their adhesion to the silicon substrate. Thick silicon wafers were used in order to ensure rigidity and eliminate bending of the system due to expansion mismatch of the film to the substrate. The film is constrained in the plane to expand the same amount as the substrate. Assuming isotropic elasticity in the film but allowing the unconstrained linear thermal expansion coefficient, R, to be anisotropic, the effective thermal expansion coefficient, , is given by the equation
) R⊥ +
(1 2ν- ν)(R - R |
substrate
|
)
(5)
dQm(T1) dQm(T2) d∆Qm dm dm ) (4) dQm(T2) dQm(T2) dm
where ν is Poisson’s ratio for the film, R| and R⊥ are the unconstrained linear thermal expansion coefficients parallel and perpendicular to the substrate surface. These R| and R⊥ linear coefficients for a 25-µm spin-coated film are reported to be 33 × 10-6 and 126 × 10-6 °C-1, respectively.16 Using these values, the constrained linear expansion coefficient, , for bulk polyimide is estimated to be 157 × 10-6 °C-1. This value has been estimated using eq 5 and the following assumptions: an isotropic Poisson’s ratio of 0.3417 and an R for the silicon18 substrate of 3 × 10-6 °C-1. This value is consistent with the trend
In Figure 3, the difference in the position of the oscillation minima for the two temperatures, ∆Qm )
(16) Pottinger, M. T.; Coburn, J. C.; Edman, J. R. J. Polym. Sci., Part B: Polym. Phys. 1994, 32, 825.
(T2 - T1) )
d2 - d1 2π/d1 - 2π/d2 ) ) d1 2π/d2
7138 Langmuir, Vol. 13, No. 26, 1997
Thompson et al.
seen in Table 1, where decreases with thickness. Note the significantly lower for the film made from a 1:1 PAA/ PAETE blend compared to the film of comparable thickness. Discussion It is evident from Table 1 that varies with thickness for these thin polymer films. As indicated earlier, owing to interface effects, PMDA-ODA films will have planar texture. It is known that there is a bulk-orientation influence throughout the film from the presence of a substrate and additionally a surface layer with better chain packing with fiber texture.2,3 We refer to the latter structure as a surface or interfacial region. We refer to the former structure as an interior structure. The blend sample, for which an additional phase separation occurs during curing because of the immiscibility of the precursors, will be discussed later in this section. Each sample was assumed to have interfacial or surfacetype regions of effective thickness dS and expansion coefficient S. The interior region thickness is dI, and its expansion coefficient is I. The total thickness of a film is D ) dI + dS. The measured coefficient expt is related to the surface and interior regions by the following equation:
expt )
SdS + IdI 1 ) {(S - I)dS} + I dS + dI D
(6)
If we assume that the mechanisms controlling the creation of the surface or interfacial regions in the unblended films are unchanged for each film, then the thickness of this region, dS, is a constant. This model and fit does not distinguish whether a surface region is at a vacuum/polymer interface, substrate/polymer interface, or both. The interior region thickness, dI, is allowed to vary for each film thickness. Under these assumptions, fitting expt for the nonblended films as a function of 1/D (inset of Figure 3) gives
I ) 139 ((10) × 10-6 °C-1
(7a)
S ) 139 ((10) × 10-6 °C-1 3596 ((250) × 10 dS
-6
nm/°C
e 52 × 10-6 °C-1 (7b)
This value of I is reasonable for a PMDA-ODA film. The expected value from the literature for a bulk film, quoted earlier, is ∼157 ((10) × 10-6 °C-1. Our estimate for I is closer to the lower bound. This may be attributed to a higher order in thin films due to the larger fraction of the film being affected by the interface than in thicker films. From GIXS studies, the nominal thickness of the surface region is 10 nm.2,3,19 It is obvious from eq 7b that, for S to remain positive, dS g 25 nm.20 This study strongly suggests the presence of an interfacial region at the silicon/ polymer interface which is approximately of the same thickness as the top layer. There exist some grazing incidence X-ray scattering studies on 40-nm PMDA-ODA films which suggested that a structurally distinct region (17) Bauer, C. L.; Farris, R. J. In Polyimides: Materials, Chemistry and Characterization; Feger, C., Khojasteh, M. M., McGrath, J. E., Eds.; Elsevier Science Publishers B. V.: Amsterdam, The Netherlands, 1989; p 551. (18) Thermal ExpansionsNonmetallic Solids; Touloukian, Y. S., Kirby, R. K., Taylor, R. E., Lee, T. Y. R., Eds.; Thermophysical Properties of Matter; Plenum Publishing Corp.: New York, 1977; Vol. 13, p 154. (19) Factor, B. J.; Russell, T. P.; Toney, M. F. Macromolecules 1993, 26, 2847.
at the silicon/polymer interface would be very limited in extent;19 however, this lower interface is difficult to probe in the GIXS geometry. We now interpret the results for the sample which was created from a 1:1 blend of acid (PAA) and PAETE precursors of PMDA-ODA in the NMP solvent. At the end of the curing cycle, these samples are chemically identical with the sample formed from a homogeneous solution of PAA precursor. However, during the curing cycle, several mechanisms occur which can lead to a different internal microstructure in the film. As the solvent evaporates, the blend system phase separates into PAA-rich and PAETE-rich phases of lateral dimension on the order of 102-103 nm, as observed by atomic force microscopy.21 During curing, the two regions imidize at different rates;22 the PAA-rich regions convert to a rigid polyimide phase, at which time the PAETE-rich regions are still unconverted and soft. When the PAETErich regions convert to polyimide, the heterogeneous regions include not only the air/film and film/substrate interfaces but also the interfaces between the PAA-rich regions already imidized and the PAETE-rich phase. With an increased number of internal interfaces upon which the PAETE-rich region imidizes, these samples may have more material which exhibits interfacial properties. Due to random orientation of the internal interface region, the overall orientation of the chains is more isotropic than that of films from one precursor solution. Noting, however, that the values are coupled strongly to the linear expansion coefficients in all directions, we may estimate the ratio of interfacial-region material produced in the blended sample with respect to the samples prepared from a homogeneous precursor. If we assume, first, that the I and S values are the same for the blend sample as in the other film samples and, second, that the expansion perpendicular to I and S is small, then by inspection of eq 5 and using the values in Table 1 it is straightforward to note that exptl dSBLEND (BLEND - I)DBLEND ) ≈ 1.6 exptl dS ( - )D FILM
I
(8)
FILM
This implies that the amount of interfacial type material in the blend sample is nearly twice the amount in the conventional films and that internal interfaces also affect the ordering of the polyimide film during its processing. The decrease in thermal expansivity with decreasing thickness is consistent with the observations for poly(methyl methacrylate)11 and poly(2-vinylpyridine).12 An opposite effect with thickness10 in thin polystyrene films was interpreted as evidence of a liquid film at the interface. The absence of the observation of a liquid film may be due to two reasons: (i) We are 200 K below the polymer glass transition, Tg, in contrast to other experiments that probed less than 100 K below Tg.8-12 (ii) PMDA-ODA is a semicrystalline polymer; therefore, due to improved mobility the chains at the surfaces have crystallized from the glassy/liquid state. Explanation ii is quite consistent with the GIXS studies that indicated higher order in (20) It is conceivable that the linear coefficient of thermal expansion of an ordered polymer can be negative in certain directions. This occurs along the c-axis where the ordered conformation of the chain tends to shrink. This is because the chain increases its entropy by this conformational change. See: Lusignea, R. W. In MRS Proceedings; Adams, W. W., Eby, R. K., McLemore, D. E., Eds.; MRS Society: Pittsburgh, 1989; Vol. 134, p 265. (21) Saraf, R. F. Macromolecules 1993, 26, 1993, 3623. (22) Snyder R. W.; Painter, P. C. In Polymeric Materials for Electronics Packaging and Interconnection; Lupinski, J., Moore, R., Eds.; ACS Symposium Series 407; American Chemical Society: Washington, DC, 1989; p 49.
Thermal Expansivity of Polymer Thin Films
polyimide films at the surface in the presence2 and absence3 of solvent during film casting and curing. Accordingly, the relative contribution from surface-mobility-induced ordered interphasal regions will increase with decreasing thickness, causing the corresponding thermal expansivity to increase. Our results indicate that at all interfacial discontinuities there is significant mobility well below Tg to allow the chains to order below Tg. The density profile cannot be measured precisely enough to determine whether the amorphous phase is liquid or glassy. Alternatively, one can also ask the following: Will the free volume of the glass/liquid state also decrease due to this enhanced mobility at the surface? This is difficult to infer. More than 50% of the polymer is amorphous at the surface. If the surface layer has a tendency to liquify, then the expansivity could indeed increase with decreasing thickness.10 This could be the case in spite of the higher crystallinity at the interphasal regions because the crystalline chains of PMDA-ODA at the surface are oriented with their largest expansivity direction (i.e., the b-axis) parallel to the surface normal.3 Our opposite observation suggests that the following cases may occur: (i) There is a strong Poisson effect due to restriction in the x-y plane that reduces the z-axis expansion. (ii) Substrate-film interactions reduce the mobility.12 (iii) There is an overall densification of the glassy state at the surface. Reason i is certainly true, as noted in eq 5. It ensures that the measured expansion coefficient couples the expansivity of all orientations. Reason ii is valid because the surfaces are treated to ensure wetting. Reason iii is particularly possible for PMDA-ODA because, during the curing process, which initiates more than 200 K below the bulk Tg, the mobility decreases abruptly as the chain imidizes. This may lead to frustrated packing. However, the surface-induced mobility can cause significant aging, leading to a denser glass at a lower energy state. Summary and Conclusions Using X-ray reflectivity the thermal expansion coefficient of polyimide is found to be a function of thickness for very thin films. This is attributed to the differences between thermal expansion coefficients of more ordered regions at the interfaces and of the interior region. Using this model, the interior is found to be 139 ((10) × 10-6 °C-1, which is consistent with and slightly lower than a bulk value of 157 × 10-6 °C-1 for films less than 103 nm. The surface or interfacial is a third of this value or even less. For films as thin as ∼40 nm, the low suggests that the total interfacial region at the air/film and film/ substrate interface is g25 nm. Since GIXS studies show that the air/film interfacial thickness is ∼10 nm, this study implies a film/substrate interface is of comparable thickness. The polyimide films from precursor blends are studied. Apart from the interfaces with air and substrate, “interphasal” interfaces are also formed. This leads to a remarkable decrease in their expansion coefficient. Acknowledgment. The experiments were performed at the National Synchrotron Light Source, Brookhaven National Laboratory, which is supported by U.S. Department of Energy Contract DE-AC02-76CH00016. Appendix: The Effect of Small Density Variations within a Film on the Simple Determination of the Thickness from the Minima Spacing In the kinematic limit, that is, when the reflectivity is much less than 1, the reflectivity for an arbitrary density
Langmuir, Vol. 13, No. 26, 1997 7139
profile, F(z), is given by
r ) rF
∫-∞∞F-1 ( -∞
)
dF(z) -iqz e dz ) η dz
(A1)
where the observed reflectivity, R ) |r|2. The rF is the Fresnel reflectivity calculated for a single step-function interface between substrate density, F-∞, and vacuum. The positions of the maxima and minima of the reflectivity curve, R/RF, occur at
d(R/RF) dη dη* ) η* +η )0 dq dq dq
(A2)
where η* is the complex conjugate of η. For q/qcrit . 1, RF ) |rF|2 ∝ q-4. Let the surface of the substrate be at z ) 0, and a single layer extends to z ) z1. Assume stepfunction, H(q), density changes of magnitude ∆F0 between the substrate and the film and ∆F1 between the film and vacuum. Note that dH(q)/dq ) δ(q), the Dirac delta function. Then
η)
[∆F0δ(z) + ∆F1δ(z-z1)]e-iqz ∫-∞∞F-1 -∞
1
dz )
-∆F0 -∆F1e-iqz1 + (A3) F-∞ F-∞ dη 1 [∆F1iz1e-iqz1] ) dq F-∞
(A4)
Using the above relationships, for a single film,
d(R/RF) ) -∆F0∆F1z1 × 2 sin(qz1) ) 0 dq
(A5)
which implies the maxima and minima occur when qz1 ) mπ or, since the maxima and minima alternate, the difference between positions of minima is the expected 2mπ/z1, as used in the analysis of the data. Extend this to two interfaces, where the substrate ends at z0 ) 0, the first film ends at z1, and the second film extends to z2. Note that z2 is the total thickness of the sample. The density change between the substrate and adjacent film is ∆F0, between this first film and the top film, ∆F1, and between the top film and vacuum, ∆F2. The ∆F values may be negative or positive.
η)
[∆F0δ(z) + ∆F1δ(z-z1) + ∫-∞∞F-1 -∞
∆F2δ(z-z2)]e-iqz dz )
-∆F0 -∆F1e-iqz1 -∆F2e-iqz2 + + F-∞ F-∞ F-∞ (A6a)
dη 1 [∆F1iz1e-iqz1 + ∆F2iz2e-iqz2] ) dq F-∞
(A6b)
d(R/RF) ) ∆F0∆F1z1 sin(qz1) + dq ∆F1∆F2(z2 - z1) sin(q(z2 - z1)) + ∆F0∆F2z2 sin(qz2) ) 0 (A7) z2, the total film thickness, is always larger than z1 and z2 - z1. If |∆F1| is small compared to the magnitude of the density change between the substrate and film, |∆F0|, and between the film and vacuum, |∆F2|, so that |∆F0∆F1| and |∆F1∆F2| are much less than |∆F0∆F2|, then we may assume
7140 Langmuir, Vol. 13, No. 26, 1997
that the amplitudes of the first two terms are very small compared to the third term. The zeros of eq A7 will make small oscillatory deviations about the approximate zeros at q ) mπ/z2, calculated from the third term. Therefore, under the assumption of small density changes within the film, the slope of Qm versus m still
Thompson et al.
gives 2π/D, where D is the total thicknesss of the film. The Qm are the refraction-corrected positions of the minima determined from the experimental reflectivity curve, R × q+4, which approximates R/RF. LA962073V
