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Small-angle X-ray scattering on spinodal, nucleotic or coarsening processes in two-phase systems containing spherical, fibrillar or lamellar inhomogeneities

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Zeitschrift für Physik B Condensed Matter

Abstract

We study the effects of some of the most important and typical structural changes in two-phase systems on selected structural parameters obtained from small-angle x-ray scattering (SAXS) measurements. To limit the present study, it was assumed that the Phase, 1, embedded in the matrix

  1. i)

    is monodispersed and homogeneous,

  2. ii)

    possesses one of the three most extreme shapes (spherical, fibrillar or lamellar) and

  3. iii)

    changes its behaviour

  1. a)

    through type change (spinodal or nucleotic or coarsening), without changing the shape,

  2. b)

    through a change of the shape only, or

  3. c)

    through a) (type change) and b) (shape change) simultaneously.

To find the type of change for three basically different shapes of Phase 1 and to calculate its intensity (amount of the change) the following three SAXS parameters must be compared before and after the treatment of the system:

  1. i)

    chord lengthl 1 (and/or radius of gyrationR),

  2. ii)

    volume partw 1 of the Phase 1, and

  3. iii)

    relative inner surfaceS v of the system.

It is shown by this comparison that by the pure type change in the case of

  1. i)

    spinodal change, all three SAXRS parameters are increasing or decreasing simultaneously and proportional to a power of the intensity of the change,

  2. ii)

    nucleotic change,l 1 (and/orR) is unchanged, the other two (w 1 andS v ) are increasing or decreasing simultaneously and directly proportional to the intensity,

  3. iii)

    coarsening change,w 1 is unchanged and anincreasing ofl 1 is always accompanied by adecreasing ofS v and vice versa.

In addition to this type change, the cases of mere changes of the shape (“shape change”) and finally of simultaneous type and/or shape change are studied. For the case of pure shape change the alteration of the dimensions (chord lengthl 1 and/or radius of gyrationR) must be followed. The limitations of the analyses of the simultaneous type and/or shape change are discussed in detail.

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Abbreviations

D :

dimension of one inhomogeneity (e.g.R, l r ,r h etc.)

F :

(=F 1+F2) cross-section area of a fibre system

F 1, F2 :

cross-section area of the Phase 1 and 2 in a fiber system

f c :

cross-section area of a cylinder

k :

number of the inhomogeneities

l 1, l2 :

chord (or intersection) length (Porod) of the Phase 1 (inhomogeneities) and the Phase 2 (matrix)

l c :

length of the coherence (Porod)

m :

(=w′ 1/w1) intensity, of the spinodal change

n :

(=w′ 1/w1) intensity of the nucleotic change

O :

inner surface common to the Phase 1 and 2

O r1 :

(=O/V 1) relative surface of one inhomogeneity

p :

(=k′/k, w 1=const.) intensity of the coarsening change

R :

radius of gyration of one inhomogeneity

r :

radius of the sphere

r H ,r h ,r t :

cross-section radius of a cylinder

S v :

(=O/V) relative inner surface of the system

T :

(=T 1+T2) thickness of a lamellar system;t, thickness of a lamellar inhomogeneity

T 1, T2 :

thickness of the Phase 1 and 2 in a lamellar system

V :

(=V 1+V2) volume of the system

v :

volume of one inhomogeneity

V 1 V2 :

volume of the Phase 1 and 2

w 1, w2 :

volume part of the Phase 1 and 2 (w 1+w2=1)

References

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Authors and Affiliations

  1. Institute of Physical Chemistry, University of Graz, Heinrichstrasse 28, A-8010, Graz, Austria

    A. Jánosi

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  1. A. Jánosi
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Dedicated to Prof. Dr. J. Schurz on the occasion of his 65th birthday

This work was sponsored by the Jubiläumsfonds of the Österreichische Nationalbank grant no. 2664

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Jánosi, A. Small-angle X-ray scattering on spinodal, nucleotic or coarsening processes in two-phase systems containing spherical, fibrillar or lamellar inhomogeneities. Z. Physik B - Condensed Matter 80, 393–400 (1990). https://doi.org/10.1007/BF01323522

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  • DOI: https://doi.org/10.1007/BF01323522

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