Equation ellipsometry

Smith [113] studied the adsorption of n-pentane on mercury, determining both the surface tension change and the ellipsometric film thickness as a function of the equilibrium pentane pressure. F could then be calculated from the Gibbs equation in the form of Eq. ni-106, and from t. The agreement was excellent. Ellipsometry has also been used to determine the surface compositions of solutions [114,115], as well polymer adsorption at the solution-air interface [116]. 

The Fresnel equations predict that reflexion changes the polarization of light, measurement of which fonns the basis of ellipsometry [128]. Although more sensitive than SAR, it is not possible to solve the equations linking the measured parameters with n and d. in closed fonn, and hence they cannot be solved unambiguously, although their product yielding v (equation C2.14.48) appears to be robust. 

All electromagnetic phenomena are governed by Maxwell s equations, and one of the consequences is that certain mathematical relationships can be determined when light encounters boundaries between media. Three important conclusions that result for ellipsometry are ... 

The measurement of 4 and A gives rP/rs via equation (2.74), for the coated electrode system, with both of these parameters highly sensitive to the properties of the film. However, whereas a knowledge of A and 4 allowed the direct calculations of the optical properties of an uncoated metal via equations (2.80) and (2.81), these equations no longer apply in the case of a coated substrate. At this point, the central problem in ellipsometry is encountered. To characterise the film completely, three parameters are required n, k and L, However, the above discussion includes only the determination of two parameters, a. and c, giving 4 and A. One of two approaches is commonly employed to solve the equations to give /i, k and L. 

Whereas the XSW technique takes advantage of the standing wave established on the total reflection of X-rays from a mirror surface, a conceptually more straightforward approach is that of simply specularly reflecting an X-ray beam from an electrode coated with the film of interest, measuring the ratio of the intensities of the incident and reflected rays, and fitting the data, using the Fresnel equations, to a suitable model an approach similar to optical ellipsometry. 

Takahashi et al.67) prepared ionene-tetrahydrofuran-ionene (ITI) triblock copolymers and investigated their surface activities. Surface tension-concentration curves for salt-free aqueous solutions of ITI showed that the critical micelle concentration (CMC) decreased with increasing mole fraction of tetrahydrofuran units in the copolymer. This behavior is due to an increase in hydrophobicity. The adsorbance and the thickness of the adsorbed layer for various ITI at the air-water interface were measured by ellipsometry. The adsorbance was also estimated from the Gibbs adsorption equation extended to aqueous polyelectrolyte solutions. The measured and calculated adsorbances were of the same order of magnitude. The thickness of the adsorbed layer was almost equal to the contour length of the ionene blocks. The intramolecular electrostatic repulsion between charged groups in the ionene blocks is probably responsible for the full extension of the... 

McCrackin, F. L., Colson, J. P. Computational techniques for the use of the exact drude equations in reflection problems, in Ellipsometry in the Measurement of Surfaces and Thin Films (eds.) Passaglia, E., Stromberg, R. R., Kurger, J., p. 61, NBS Miscellaneous Publication 256, Washington, D. C., Superintendent of Documents, U. S. Government Printing Office 1964... 

The technique can also be used for multilayered structures. The corresponding equations are then more complicated and are usually applied to inorganic semiconductors [36-38] due to their better defined interfaces and geometry compared with organic semiconductors. In the case of transparent media (k = 0), the ellipsometric equations can be used to determine both n and the thickness of the film with sensitivity below 1 A. This is much better than can be achieved by methods based on 7Z and T, thus reducing the uncertainty in the n determination. Several examples of ellipsometry applied to CPs are reported in the literature [32,43,44],... 

We next seek the fundamental equation of ellipsometry. In the isotropic case (where /("and n are both scalar) the traceless 2x2 reflection matrix R is given by... 

Both parameters are combined in the basic equation of ellipsometry... 

Ellipsometry measures the relative attenuation and phase shift of polarized light reflected from a polymer-coated surface. The Drude equations (Drude, 1889a,b, 1890 Stromberg et ai, 1963 McCrackin and Colson, 1964) relate the attenuation and phase shift to the average refractive index and thickness tel of an equivalent homogeneous film. Interpretation of fel in terms of the actual refractive index distribution or the polymer distribution [Pg.189]

Besides ellipsometry, reflectometry has proven its value. By this technique adsorbed masses can conveniently be obtained and. if the measurements are carried out with polarized light, also the orientation of the adsorbed molecules. Experiments are usually done at near-normal Incidence, when // Another variant, pertaining to adsorption from solution and sketched in fig. 2.15, can be made fast enough for the kinetics of adsorption to be followed. In the mode shown, fluid is admitted to the surface from bottom to top ("impinging jet") equations are available for the rate of supply in the stagnation point (the "core" of the fluid flow, which hits the surface perpendicularly). The quotient of the reflected Intensities = S is obtained by electronic division, it is... 

Returning to the basic equations for the surface tension 12.5.27 and 28 or 28a], we automatically ask how useful these are for predicting y for a given liquid. The answer is that for practical use these equations are not very suitable because the problem is translated into two esoteric sub-problems, viz. the measurement of the profile p (z) and the determination of van der Waals constant C. Determination of the profile is not simple. To that end optlcail methods such as ellipsometry and reflectivity measurements are needed, whose interpretation is not unambiguous because the profile cannot be obtained without making certain assumptions. We return to this in sec. 2.8. 

The principles of ellipsometry have been set forth in sec. 1.7.10b. For the ellipsometric thickness of an interlayer over which the index of refraction n changes with position we gave the following Drude equation... 

Gas adsorption (N2, Kr) can be used to estimate the relative quantity of zeolite deposited on the support (BET- or Langmuir equation). When a dense substrate is used, ellipsometry gives the film thickness and void volume fraction. Absorption spectroscopies such as FTIR, are adapted to study the membrane material short range structure. 

Ellipsometry is based fundamentally on the Fresnel equations for the reflection and refraction of light at a sharp two-phase boundary. They can... 

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