Quantitative evaluation of orientation in cellulose fibres from the X‐ray fibre diagram

Abstract

The theory of fibre diagrams as given by P o 1 a n y i and first applied to cellulose fibres by S i s s o n and K r a t k y enables one to derive the orientation in a fibre from the intensity distribution in the diatropic interferences. Since these interferences are usually too weak in the X-ray diagrams of cellulose fibres, we must resort to the paratropic reflexions. The present paper shows how these paratropic reflexions must be evaluated to obtain the orientation factor fr as a rational measure of average orientation. It is shown on experimental grounds that the proximity of the A, interference does not invalidate the evaluation of the As sickle. A special procedure must be applied to fibres in which the orientation is not very pronounced, since here the interferences of the (021) planes partly overlap those of the paratropic A, planes.It is assumed throughout this article that in all experiments the B r a g g angle is negligible. The correction required when dealing with finite B r a g g angles is discussed in section 6.The relation between orientation and optical constants is set forth in section 5. Finally, section 8 gives an illustration along with a short summary of the method.In a previous study on the orientation of cellulose fibres, based upon optical and X-ray data, the results of the quantitative evaluation of X-ray diagrams of various fibre specimens were communicated and discussed *) . The theoretical basis of the evaluation and the experimental method applied were only briefly referred to in the publication quoted. In the present paper the underlying theory and the experiments performed in order to determine certain fundamental constants are given. W e may omit introductory remarks of a general nature, since these can be found in the publication referred to.It is assumed throughout this article that the incident beam of X-rays is perpendicular to the fibre axis. The film is a plane one at distance L from the fibre and perpendicular to the incident beam. W e have restricted the discussion to X-ray reflexions of the first order. Higher order reflexions are usually very weak in fibre diagrams: in the particular case of regenerated cellulose they play no part at all a). Besides, the extension to reflexions of higher order would be a very easy matter, if required.W e shall often make use of the fact that the B r a g g angle 6 is small. The error involved is discussed in section 6.The reader who is not interested in the mathematical details can find a summarizing description of the procedure, along with a few examples illustrating the method, in section 8. 1.1.1. Following P o 1 a n y i 4 ) we shall designate a lattice plane by its representative point on a sphere of unit radius: the tangent plane in this point Basic relations governing the fibre diagram.

Raman Anisotropy Measurements: An Effective Probe of Molecular Orientation in Conjugated Polymer Thin Films

“…The perfectly oriented fraction, f, corresponds to the Hermans' orientation function that can be derived from X-ray diffraction data. [27] Note also that if the Fraser distribution function is expanded in terms of spherical harmonics, f can be shown to be equal to the average of the second-order term in the expansion, namely áP 2 cos bñ. [28] We next assume that we are considering molecular vibrations with a highly uniaxial Raman tensor (as is the case for the~1600 cm ±1 in-plane stretching mode [25] ) and that their principal axis is coincident with the polymer-chain axis.…”

Effective Reduction of Volumetric Thermal Expansion of Aromatic Polyimide Films by Incorporating Interchain Crosslinking

“…Herman's orientation function of the polymer main chains along the in-plane direction (P 200 ) can be expressed as follows [14]:…”

Modeling Flow-Induced Crystallization

“…Herman's orientation factor [91,92] was applied to quantify the average level of molecular orientation in the sample thickness direction and at the location of the diamond window. The orientation factor is denoted by f H , which is defined as Reproduced with permission of the copyright owner [5] where ϕ is the angle between the crystallographic axis and a reference axis (in this case the flow direction) and ⟨cos 2 ϕ⟩ is the average value of the cosine squared of this angle given by…”