Hansen coefficients

C. M. Hansen, The Three-Dimensional Solubility Parameter and Solvent Diffusion Coefficient, Danish Technical Press, Copenhagen, Denmark, 1967. 

Ostergaard, J., Hansen, S. H., Larsen, C., Schou, C., Heegaard, N. H. Determination of octanol-water partition coefficients for carbonate esters and other small organic molecules by microemulsion electrokinetic chromatography. Electrophoresis 2003, 24, 1038-1045. 

The least-squares Molly program based on the Hansen-Coppens model [10] was used to determine atomic coordinates, thermal parameters and multipolar density coefficients in scolecite. In the Hansen-Coppens model, the electron density of unit cell is considered as the superposition of the pseudo-atomic densities. The pseudoatom electron density is given by... 

Helweg, C., Nielson, T., Hansen, RE. (1997) Determination of octanol-water partition coefficients of polar polycyclic aromatic (NPAC) by high performance liquid chromatography. Chemosphere 34, 1673-1684. 

For completeness it should be mentioned that the passivation of gold, presumably via the same AuH complex, has also been studied in p-type silicon, where it is the donor rather than the acceptor level of gold that is active (Hansen et al., 1984). Though no profiles were reported in this work, apparent hydrogen diffusion coefficients inferred by these authors are of the same order as the Pearton (1985) points of Fig. 16 at temperatures 110°C and below. 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

The parameters Pim , Pcore, and k can be refined within a least square procedure, together with positional and thermal parameters of a normal refinement to obtain a crystal structure. In the Hansen and Coppens model, the valence shell is allowed to contract or expand and to assume an aspherical form [last term in (11)], as it is conceivable when the atomic density is deformed by the chemical bonding. This is possible by refining the k and k radial scaling parameters and population coefficients Pim of the multipolar expansion. Spherical harmonics functions yim are used to describe the deformation part. Several software packages [68-71] are available for multipolar refinement of the electron density and some of them [68, 70, 72] also compute properties from the refined multipolar coefficients. 

Hansen ADA, Rosen H, Novakov T (1982) Real-time measurement of the absorption coefficient of aerosol particles. Appl Opt 21 3060-3062... 

One approach to a solution of this problem was put forward by Hansen (14), who derived general equations which express the overall experimental film absorbance in terms of the external reflectance of the substrate. These relations contain within them expressions for the individual anisotropic extinction coefficients in each geometric orientation. Solution of these general equations for the anisotropic extinction coefficients allows for an unambiguous description of the dipole orientation distribution when combined with a defined orientation model. 

Ugelstad and Hansen (1976) also derived a rate coefficient for radical desoiption similar to those given by Eqs, (92) and (95). 

Allahyarov, E., Lowen, H., Hansen, J.P., and Louis, A.A. Nonmonotonic variation with salt concentration of the second virial coefficient in protein solutions. Phys. Rev. E, 2003, 67, No. 5, p.051404. 

A number of methods based on regular solution theory also are available. Only pure-component parameters are needed to make estimates, so they may be applied when UNIFAC group-interaction parameters are not available. The Hansen solubility parameter model divides the Hildebrand solubility parameter into three parts to obtain parameters 8d, 5p, and 5 accounting for nonpolar (dispersion), polar, and hydrogenbonding effects [Hansen,/. Paint Technot, 39, pp. 104-117 (1967)]) An activity coefficient may be estimated by using an equation of the form... 

Once a decision of the chemical functionality or host structure is made and a sensing film is included in a sensor device, the next goal would be to model the sensor response of the film in the device. Sensor response to an analyte is a complex function of the partitioning of the target analytes based on the interactions within the film as well as the transport properties of the analyte in the sensor. The sensor responses for polymer-based sensors have been modeled by various approaches using (1) first principles techniques such as Hansen solubilities, (2) multivariate techniques such as QSAR to correlate sensor response with molecular descriptors, and (3) simulations and empirical formulations used to calculate the partition coefficient, such as linear solvation energy relationships, to provide a measure of selectivity and sensitivity of the material under consideration. 

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