Mathematical properties of polydisperse systems, 1. General relations

Greschner, Georg S.

doi:10.1002/macp.1973.021680122

Cited by 14 publications

(6 citation statements)

References 8 publications

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“…In contrast, as shown in Figure 2, in the case of the liquid-crystalline copolyester containing 60 mol % PET the results lead to a straight line at larger scattering angles only, while at small scattering angles the intensity is proportional to q~*, indicating that particle scattering predominates. 26 This may be due to some undissolved PHB clusters which may also cause the turbidity of the solution of this copolyester. This particle scattering, obviously, cannot be correctly subtracted when we perform the corrections described under Experimental Section.…”

Section: Resultsmentioning

confidence: 99%

Chain conformations in copolyesters of poly(ethylene terephthalate) and poly(p-hydroxybenzoic acid) as determined by small-angle neutron scattering

Olbrich¹,

Chen²,

Zachmann³

et al. 1991

Macromolecules

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Small-angle neutron scattering (SANS) experiments were performed on blends of deuterated and nondeuterated copolyesters of poly(ethy1ene terephthalate) (PET) and poly@-hydroxybenzoic acid) (PHB). The PET content varied between 80 and 60 mol % , the sample containing 60 mol ' ?6 being liquid crystalline. Depending on the time of melting during sample preparation, transesterification took place to various degrees resulting in samples in which the molecules contained deuterated and nondeuterated blocks of various lengths. The average degree of polymerization and radius of gyration of these blocks were measured and the length of the statistical element b was calculated as a function of composition. The transition from the amorphous to the liquid-crystalline state did not lead to an additional increase in b. From this it was concluded that, in the nematic state, the molecules are not elongated. From the length of the statistical element, the average degree of polymerization of the PHB sequences could be estimated.

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Section: Resultsmentioning

confidence: 99%

Chain conformations in copolyesters of poly(ethylene terephthalate) and poly(p-hydroxybenzoic acid) as determined by small-angle neutron scattering

Olbrich¹,

Chen²,

Zachmann³

et al. 1991

Macromolecules

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“…In a less ideal material, frequently many different but similar structural units can be found. This is the common notion of polydispersity (Higgins & Stein, 1978;Hosemann, 1950;Greschner, 1973;Glatter, 1980;Cohen & Thomas, 1987;Fö rster & Burger, 1998;Rieker et al, 1999;Pedersen et al, 2000;Keum et al, 2005;Triolo et al, 2005;. In the present work, polydispersity means that every structural unit in the sample is generated from a mean template by affine compression or expansion (dilatation).…”

Section: Introductionmentioning

confidence: 86%

“…At the same time, Hosemann (1939) addressed the effect of polydispersity on the particle scattering of spheres. Besides Glatter's method of direct inversion (Glatter, 1980), there have been several proposals to solve the problem analytically (Roess, 1946;Greschner, 1973;Schmidt & Brill, 1967). Before Stribeck (1993b) studied the mathematical properties of the related integral transform in the special case of a one-dimensional two-phase structure, Roess (1946) and Schmidt & Brill (1967) came rather close to a Mellin convolution approach.…”

Section: Introductionmentioning

confidence: 99%

Analysis of scattering from polydisperse structure using Mellin convolution

Stribeck

2006

J Appl Cryst

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This study extends a mathematical concept for the description of heterogeneity and polydispersity in the structure of materials to multiple dimensions. In one dimension, the description of heterogeneity by means of Mellin convolution is well known. In several papers by the author, the method has been applied to the analysis of data from materials with one-dimensional structure (layer stacks or fibrils along their principal axis). According to this concept, heterogeneous structures built from polydisperse ensembles of structural units are advantageously described by the Mellin convolution of a representative template structure with the size distribution of the templates. Hence, the polydisperse ensemble of similar structural units is generated by superposition of dilated templates. This approach is particularly attractive considering the advantageous mathematical properties enjoyed by the Mellin convolution. Thus, average particle size, and width and skewness of the particle size distribution can be determined from scattering data without the need to model the size distributions themselves. The present theoretical treatment demonstrates that the concept is generally extensible to dilation in multiple dimensions. Moreover, in an analogous manner, a representative cluster of correlated particles (e.g. layer stacks or microfibrils) can be considered as a template on a higher level. Polydispersity of such clusters is, again, described by subjecting the template structure to the generalized Mellin convolution. The proposed theory leads to a simple pathway for the quantitative determination of polydispersity and heterogeneity parameters. Consistency with the established theoretical approach of polydispersity in scattering theory is demonstrated. The method is applied to the best advantage in the field of soft condensed matter when anisotropic nanostructured materials are to be characterized by means of smallangle scattering (SAXS, USAXS, SANS).

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“…(1) design of a fully automatic PDC-chromatograph with a mean measuring error of some 0,1070, (2) formulation of a closed theory of the PDC-resolution effect and the PDC-calibration curves, (3) an analytical solution of the physico-mathematical problems of the spreading phenomena in a PDC-column, and (4) outline of a suitable mathematical method for the computation of the MWD by means of an effective numerical inversion of the corresponding integral operator') connecteed with problem (3).…”

Section: Introductionmentioning

confidence: 99%

“…Let mpBs and mp,, be the polymer masses of the considered P-mer in the sol and the gel, respectively, cp,s and cP,, the corresponding concentrations, u(P) = (dz/dt), the elution rate of the transported P-mer along the z-axis of the vertical column, and 6 the mean over-all linear rate of the column liquid; then a trivial integration of the thermodynamically and hydrodynamically defined retention coefficient of the P-mer in the column gives the equation of the PDC-column in a reversible thermodynamic equilibrium, if u(P) is constant to z and t: V ( P ) = V, 1 + -K ( P ) [ rL (3) In Eq. (3) V(P) is the elution volume of the P-mer measured at the elution time tE (P) and the position z = L of the thermostated column, V ( P ) = q li tE(P) (4) Vo the zero-volume (Eq.…”

Section: Introductionmentioning

confidence: 99%

Phase distribution chromatography (PDC) as a method for the investigation of polymers in solution 1. Thermodynamics and transport

Greschner

1980

Makromol. Chem.

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A phenomenological theory of the Phase Distribution Chromatography (PDC)-separation effect is outlined and a theoretical equation for the measured PDC-calibration curves is given. Assuming a reversible-thermodynamical equilibrium in the polystyrene-PDC-column, only a relatively small part of the measured PDC-calibration curves could be explained: namely those running below their tangents. In order to explain the whole sigmoidal shape of the experimental curves, a theory of steady state in the system sol/gel was developed assuming deformation of the polymer coil near the gel front due to the stress related to the velocity gradient. The resulting dynamical flow-equilibrium differs highly from the reversible-thermodynamic one at suitable low column temperatures. It leads to two typical calibration regions explaining the resolution power of the PDC-column as a function of temperature. Analysis of the first one (reversiblethermodynamical) starts with a three-parameter partition function and leads to two contributions of the heat of transfer of a P-mer from the gel into the sol of the column. These results are compared with those of Schulz, von Gunner and Gerrens, and of Schulz and Horbach. In this way a physical interpretation of the three parameters in the partition coefficient of the P-mer in the system gel/sol of the PDC-column can be given.

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Mathematical properties of polydisperse systems, 1. General relations

Cited by 14 publications

References 8 publications

Chain conformations in copolyesters of poly(ethylene terephthalate) and poly(p-hydroxybenzoic acid) as determined by small-angle neutron scattering

Chain conformations in copolyesters of poly(ethylene terephthalate) and poly(p-hydroxybenzoic acid) as determined by small-angle neutron scattering

Analysis of scattering from polydisperse structure using Mellin convolution

Phase distribution chromatography (PDC) as a method for the investigation of polymers in solution 1. Thermodynamics and transport

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