Singularities of phase boundaries and values of the second-order derivative of the correlation function at the origin

Abstract

The connection between the zero-distance value of the second-order derivative of correlation functions used in light or small-angle x-ray-scattering theory and the singularities of phase boundaries is more thoroughly exploited. It is shown that, besides sharp edges, only smooth contact points contribute. The explicit expression of the latter contribution is obtained.Small-angle x-ray-scattering experiments allow us to determine j'(0+) and j'(0+), i.e, the zero-distance values of the firstand second-order derivatives of the correlation function y(r) relevant to the sample that one is interested in. j(0+} is determined through the relation' -8n.y(0+)= lim k i(k), k~Oo where k and i (k} are the momentum transfer and the standard scattered intensity, respectively. j'(0+ ) is obtained from Porod's relation, ' y(0+) = -f [k i(k)+ 8my(0+)]dk/12m. So far the latter relation is not as commonly used as the former one, for two reasons. On the one hand, unavoidable experimental uncertainties on the large k values of i (k) make the determination of j(0+) by Eq. (1) much less accurate than that of y(0+).On the other hand, the relation between y'(0+) and some geometrical features of the phase boundaries has not been as fully analyzed as in the j (0+) case. Until a few years ago, in fact, one knew that j'(0+ } was null when the surfaces separating the different phases of the sample were smooth functions (i.e. , their parametric equations were continuous doubly differentiable functions, more briefiy, C2 functions). Moreover, it has also been argued that y'(0+ ) should be different from zero and positive whenever boundaries have sharp edges. 'Only recently the explicit relation between this )&cot[apj(y)]I .In Eq. (2), the integral is performed along the edge, and p,j(y) is the value of the dihedral angle at the edge point characterized by the curvilinear abscissa y. Equation (2) allows one to evaluate the contribution of a sharp edge to j'(0+), since y(r) is related to SPF's by y(r)=1g(n; nj) PJ(r)/(ri -) .(3) We recall that n; and (g ) in Eq. (3) are known quantities. They denote, respectively, the electron density of the ith phase and the average of the squared electron density fiuctuation.We stress, however, that we do not know whether y'(0+) is different from zero only if sharp edges are present. To the best of our knowledge the kinds of boundary singularities responsible for a j'(0+) value different from zero have not been classified yet. In this paper we shall carry through such an analysis.We shall show, indeed, that in addition to sharp edges only "smooth contact" points contribute to y(0+). Moreover, the explicit expression of these kind of surface singularity and the j'(0+) value has been obtained.In this paper it was shown that the contribution of an edge on the surface separating the ith phase from the jth one to the zero-length limit of the second-order derivative of the relevant "stick probability function" (SPF) is~1 P; (0+)=-6 V dy[1 -[n. -P; (y)] lJ 26 6384 Q~1982 The American Physical Society

The vertex contribution to the Kirste–Porod term

“…When interfaces have a very arbitrary shape or are related to a very polydisperse sample, it appears reasonable to assume that the sum of the oscillatory contributions, present on the r.h.s, of (1.5), averages to zero owing to their large number.I" In these cases, the above statement implies that the intensities scattered by samples whose interfaces consist of planar * It should also be noted that the results obtained by Ciccariello, Cocco, Benedetti & Enzo (1981) and by Ciccariello & Benedetti (1982) for the y~2)(0÷) value indicate that the smoothinterface value y~2)(0+)= 0 must be corrected for possible singularities of the interface and that the correct value is simply obtained by summing up the 'corrections' owed to the interface singularities. Moreover, the non-null contributions to y~2)(0÷) come from edges and contact points only.…”

Preparation of Porous Silica Materials via Sol−Gel Nanocasting of Nonionic Surfactants: A Mechanistic Study on the Self-Aggregation of Amphiphiles for the Precise Prediction of the Mesopore Size

“…According to Ciccariello et al, a value of g(0) > 0 is indicative of angular structures (edges, vertexes). [43][44][45] The behavior of the CLD at larger r is related to the size and shape of the mesopores and their spatial distribution. Since the mesopore sizes were determined by nitrogen sorption, the evaluation of the CLD was primarily used to investigate the microporous character of these silicas.…”

Edge contributions to the Kirste–Porod formula: the truncated circular right cone case

“…The explicit relations between the 3,"(0+) * value and the singularities of the interphase surface were obtained by Ciccariello, Cocco, Benedetti & Enzo (1981) for the case of sharp edges and by Ciccariello & Benedetti (1982) for the case of contact points and vertices were shown not to contribute. The derivation of these results was made possible by the use of the two general integral expressions of y'(r) and y"(r) obtained by Ciccariello et al (1981).…”