Estimation of sphere‐size distributions in two‐phase polymeric materials from transmission electron microscopy data

Abstract

SYNOPSISThe blending of two immiscible polymer samples can lead to spherical inclusions of one component in a matrix of the other component. The mechanical solid-state properties as well as the flow behavior of the melt depend on the size of the spheres in the blend. For that reason, the sphere-size distribution is of major interest. Information about this distribution is often obtained by analyzing thin slices of the blend with transmission electron microscopy. In that way, however, the sphere-size distribution itself is not obtained. The reconstruction of the sphere-size distribution is introduced as a stereological problem, well known in fields as metallurgy, biology, geology, and medicine. It is shown that the spheresize distribution can be reconstructed using a regularization method as implemented in the program FTIKREG. 0 1994 John Wiley & Sons, Inc. I NTRO DU CTlO NMany polymeric materials consist of two or more immiscible polymers. In most of these materials, one component builds a matrix in which particles of the other components are embedded. In the case of twophase materials, the particles are often approximately spheres randomly dispersed in the matrix. Two well-known examples are rubber-toughened polymers like high-impact polystyrene lS2 and polymer blends of two immiscible thermoplastic^.^-^ Naturally, all the particles do not have the same size and a sphere-size distribution must be used to characterize them. This distribution depends on the conditions during the preparation of the material. It influences the mechanical solid-state properties 7-10 as well as the flow behavior of the melt."-'3 For that reason, the sphere-size distribution is of major interest and many experimental methods have been developed to characterize it.One widespread method to obtain this morphological information is miscroscopy, especially trans- CCC 0021-8995/94/0l0039-12 mission electron microscopy (TEM). To study a polymer blend with TEM, thin slices of 50-200 nm thickness have to be prepared, whereas typical particle radii range from about 50 nm up to several micrometers. Obviously, only a two-dimensional profile of the three-dimensional structure can be observed with TEM: Image-analysis of thin slices yields the radii of profiles of the particles rather than the radii of the spherical particles themselves. Similar problems arise in metallurgy, biology, geology, and medicine. Out of the common interest in such problems, the discipline of stereology was founded in the 1960s. In general, stereology deals with mathematical problems concerning the determination of parameters characterizing a three-dimensional structure from data obtained by studying two-dimensional profile^.'^^'^ The reconstruction of sphere-size distributions from so-called profile size distributions is a well-known problem in stereology and many scientists have dealt with the derivation of the relation between both distributions and with methods for reconstructing sphere-size distributions. '6-26 In this contribution, we show that stereological results ...

Rheology of Polymer Blends

“…The developed method of data treatment was found to be particularly useful for computation of the relaxation and retardation spectra [172,173]. It has also been used to compute the sphere-size distribution of the dispersed phase in binary blends [174], as well as the ratio of the dispersed drop diameter divided by the interfacial tension coefficient, n 12 [175]. Friedrich [176] modified Eq.…”

“…In short, the model systems often help in understanding the rheological behavior of concentrated and/or compatibilized polymer blends. Numerous examples can be found in the literature [174,175,308,[311][312][313][314][315][316][317][318][319][320]]. …”

Unfolding Sphere Size Distributions with a Density Estimator Based on Tikhonov Regularization

“…This method has been proposed in [26] and led to better results than other wellknown procedures for this purpose. In addition, the SC-method has successfully been applied to solve various linear as well as nonlinear ill-posed problems [4,20,[27][28][29][30][31][32]. Other possible methods to estimate a value of the regularization parameter are the cross-validation method and the discrepancy method [33].…”

“…There is a parameter controlling the smoothness of the result but in contrast to the work about kernel estimators, methods for the optimal estimation of this parameter have been applied. In addition, it was shown in [20] that positivity constraints can be taken into account in such a way that the result for the sphere size distribution does not have negative values. Nevertheless, there are some problems due to the histogram of the profile radii: the result for the sphere size distribution depends slightly on the number and the distribution of the histogram's bins.…”

“…In order to avoid the resulting instability, regularization methods [18] should be used. The determination of the sphere size distribution by calculation of a histogram of the profile radii and subsequent application of Tikhonov regularization has been proposed and discussed in [19,20]. There is a parameter controlling the smoothness of the result but in contrast to the work about kernel estimators, methods for the optimal estimation of this parameter have been applied.…”