Uniaxial sample

Only two birefringence indices are independent and necessary to describe the anisotropy of a biaxial system, while a single birefringence measurement is required for uniaxial samples because nx — ny. 

The parameters K1/ K2/ and K3 are defined by the refractive indices of the crystal and sample and by the incidence angle [32]. If the sample has uniaxial symmetry, only two polarized spectra are necessary to characterize the orientation. If the optical axis is along the plane of the sample, such as for stretched polymer films, only the two s-polarized spectra are needed to determine kz and kx. These are then used to calculate a dichroic ratio or a P2) value with Equation (25) (replacing absorbance with absorption index). In contrast, a uniaxial sample with its optical axis perpendicular to the crystal surface requires the acquisition of spectra with both p- and s-polarizations, but the Z- and X-axes are now equivalent. This approach was used, through dichroic ratio measurements, to monitor the orientation of polymer chains at various depths during the drying of latex [33]. This type of symmetry is often encountered in non-polymeric samples, for instance, in ultrathin films of lipids or self-assembled monolayers. 

Specular reflection IR spectroscopy has been used by Cole and coworkers to study the orientation and structure in PET films [36,37]. It has allowed characterizing directly very highly absorbing bands in thick samples, in particular the carbonyl band that can show saturation in transmission spectra for thickness as low as 2 pm. The orientation of different conformers could be determined independently. Specular reflection is normally limited to uniaxial samples because the near-normal incident light does not allow measuring Ay. However, it was shown that the orientation parameter along the ND can be indirectly determined for PET by using the ratio of specifically selected bands [38]. This approach was applied to the study of biaxially oriented PET bottles [39]. 

For a uniaxial sample, there are five independent quantities (a,yap( ) or (a ) given... 

For oriented systems, the determination of molecular conformation is a complex problem because Raman spectra contain signals inherently due to both molecular conformation and orientation. To extract only the information relative to the conformation, one has to calculate a spectrum that is independent of orientation, in a similar way to the A0 structural absorbance of IR spectroscopy (Section 4). Frisk et al. [57] have shown that for a uniaxial sample aligned along the Z-axis, a spectrum independent of orientation (so-called isotropic spectrum), fso, can be calculated from the following linear combination of four polarized spectra [57]... 

The subscript i refers to the two polarization states parallel (i = ) and perpendicular [i = L) to the c-axis, which have to be distinguished for optically uniaxial samples, for instance wurtzite-structure ZnO or sapphire. Cubic crystals, for instance rocksalt-structure Mg Zni- O, are optically isotropic and have only one DF, because the dielectric tensor is reduced to a scalar. 

When neither s-polarized light (light polarized perpendicular to the plane of incidence) is converted into p-polarized light (light polarized parallel to the plane of incidence) nor vice versa, standard SE is applied. This is the case for isotropic samples and for uniaxial samples in the special case, where the optical axis is parallel to the sample normal, for example (0001) ZnO [119]. 

Molecular order is descnhedhy the orientational distribution function P(0) [Mcbl]. This is the probability density of finding a preferential direction n in the sample under an angle 0 in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). For simplicity, macroscopically uniaxial samples with cylindrically symmetric molecules are considered. Then, one angle is sufficient to characterize the orientational distribution function. In practice, not the angle 0 itself but its cosine is used as the variable and for weak order the distribution function is expanded into Legendre polynomials P/(cos 0),... 

The validity of (3.2.9) is restricted to the symmetries mentioned above, that is to cylindrical molecules, macroscopically uniaxial samples, and r] = 0. For many samples, these conditions are fulfilled when using NMR, because the quadrupole coupling tensor of aliphatic deuterons is often found to be axially symmetric. In wideline NMR, the anisotropy of the magnetic shielding is used. Here the angular resolution is lower, and the calculation has to be extended to include p > 0 [Henl]. In combination with MAS (cf. Section 3.3), the Legendre subspectral analysis has been used successfully for the determination of molecular order in partially ordered polymers [Harl]. 

As an example, it can be shown that the Reuss average compliance 53333 for a uniaxial sample consisting of an aggregate of transversely isotropic units is given in terms of the compliances Sy/y of the unit by... 

The second type of explanation for finding values of R less than 3 involves the assumption that the emission and absorption axes of the fluorescent molecule are not coincident. Kimura et al. have considered a model in which the absorption and emission axes each have, independently, a cylindrically symmetric distribution of orientations around a third unique axis in the molecule, and Nobbs et al. have considered a model which includes both this and the possibility that there is a fixed angle between the emission and absorption axes which are otherwise uniformly distributed around a third unique axis. In the more general model at least three parameters are required to specify the relationships between the directions of the emission and absorption axes and that of the unique axis of a fluorescent molecule and these are not generally known. For orthotropic symmetry, five v, are required to characterise the distribution of orientations of the unique axes and if the constant NqIo is included, there is a total of at least nine unknown quantities. No attempt has so far been made to evaluate these from intensity measurements on an orthotropic sample. For a uniaxial sample only two parameters, cos O and cos O, are required for characterising the distribution of orientations and by making various approximations the total number of unknown quantities can be reduced to six. Their evaluation then becomes a practical possibility. 

The same conclusion could be drawn from Table 11.2, The randomly oriented reinforcement has a deleterious effect on both the modulus of elasticity (decrease of 10%), and the ultimate tensile strength, a, (decreases of 65%) when compared to the neat PP film. The crossply samples show a significant improvement in stiffness over neat PP of 44% on average, with an insignificant improvement in a. The uniaxial samples show a very good reinforcement effect, with nearly twice the modulus, E, at 3.43 GPa, and a 50% improvement in a at 42,5 MPa. 

In summary, it should be noted that the uniaxial and crossply samples perform well, with 77% and 44% increase in tensile modulus, respectively. These are approximately in line with what would be expected for the crossply sample from simply halving the improvement of the uniaxial sample. Compared to the recorded tensile modulus for the uniaxial MFC as tested by Fuchs et al. and discussed above [35], the samples tested here exhibited a significantly lower a value with 65 MPa being recorded by Fuchs et al. [35] versus 42.5 MPa in this study. This may be accounted for by the fact that previous samples were 10 mm in width, decreasing the edge effects experienced. 

NJ Everall, JM Chalmers, R Ferwerda, JH van der Maas, PJ Hendra. Measurements of poly (aryl ether ether ketone) crystallinity in isotropic and uniaxial samples using Fourier transform-Raman spectroscopy a comparison of univariate and partial least-squares calibrations. J Raman Spectrosc 35 43-51, 1994. 

Assume a small uniaxial sample is loaded with a strain input,... 

DRS provides a particularly usdul means of monitoring the nature and extent of macroscopic alignment in SCLC samples that have been subjected to E fields, B fields, surface forces or are aligning/disaligning after electrical and/or thermal treatments. As we have shown [5Q, the complex permittivity of a uniaxial sample of intermediate alignment is given, to a good approximation, by the linear-addition relationship... 

In uniaxial samples, f(Q ) has the same value for all angles a and depends only on p and y. In the special case of molecules for which all angles of rotation about their own z axis are equally likely, f(Q ) also has the same value for all angles Y-A redundant but convenient set of quantities that specify the orientation of the molecular x y z axes relative to the laboratory X, Y, Z axes are the nine angles between the two sets. In uniaxial samples the angles and q. between x y z respectively,... 

Using the orientation factors Equation [22], Equations [16]-[21] provide expressions for polarized absorption and emission by molecules photoselected from isotropic samples and represent a special case of the treatment given below, which handles both one-photon events simultaneously and describes photoinduced absorption and luminescence from all uniaxial samples. 

For uniaxial samples and molecules whose symmetry dictates the orientation axes x, y, z, there are only three independent orientation factors L and Equation [15] simplifies ... 

In uniaxial samples, seven distinct combinations of photon polarization are important. The observed intensities are given by = 7 7 = 7, ... 

Here, Sj, S2 nd 2 e the molar decadic absorption coefficients and the refractive indices for a linearly polarized light beam with a polarization parallel and perpendicular to the optical axis of a uniaxial sample, and are the absorption coef-... 

For partially oriented uniaxial samples, two linearly independent spectra can be obtained from an experiment, irrespective of the mode of measurement and the polarization state of the absorbed light. The obvious procedure is to record two absorption spectra, one with the electric vector of light parallel to U, the unique axis of the sample Ei), the other with the electric vector perpendicular to the unique axis (Ey). The spectra must be baseline-corrected by subtracting the curves recorded on pure solvents. This yields the largest difference between the tow linearly independent spectra. Other alternatives are also possible. A combination of Ey or Ey with the spectrum of an isotropic sample, E =[E + 2Ey]/3) is easily obtained in liquid crystal. In cases where the sample geometry precludes the measurement of both Ey and the isotropic spectrum (membranes), the two spectra are Ey and the latter recorded with the incidence angle nlZ-co, and with the plane of polarization containing the U axis (the tilted plate method ). 

The use of a polarizer with the ATR technique will allow one to study the molecular orientation of surfaces. Any solid sample will have three optical constants kx, ky, and k. For isotropic samples, all three optical constants will be identical. For uniaxial samples, two of the constants will be the same and differ from the third one. For biaxial samples, all three optical constants will be distinct. Flournoy and Schaffers (6) have developed the theoretical basis for the ATR from anisotropic media. Sung (7) has described a modified ATR dichrolc technique whereby most of the experimental uncertainties could be minimized. Suppose, the sample under study is a polymer with its draw axis the x-axis (Figure 9). 

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