This Transmission Electron Microscope (TEM) investigation uses the retrieved complex exit-wave function in combination with the geometric phase analysis (GPA) to analyze strain located at faceted Al-Pb hetero-interfaces. It was found that the outer regions of a Pb nanoparticle were strained (compressed) which explains, according to the Clapeyron equation, an elevated melting in terms of interface strain.
Research on confined systems as found in nano-materials has become a major topic in the last years since smaller dimensions cause a significant change of properties. One of the key parameters in this regard is the stress/strain located at internal interfaces or boundaries. A fundamental aspect where the role of interfaces plays a dominant role is melting. Binary monotectic Al-Pb composites composed of nanometer-sized Pb inclusions embedded in a polycrystalline Al matrix serve in this respect as a model system for melting studies at reduced sizes [1-3].
Although there is a large lattice mismatch of 22 % between the lattices, the metal/metal interfaces of Al-Pb were found to be semi-coherent having a cube-on-cube orientation relationship. However, the accommodation of this large lattice misfit was initially puzzling due to Moiré contrast that veiled a direct observation of the interface structure [4, 5].
Figure 1 summarizes the melting behavior of nanometer-sized Pb inclusions embedded in Al. Faceted Pb inclusions, as found for instance in melt-spun material, exhibit a melting point increase (see red curve of the DSC chart inserted in fig. 1) compared to bulk Pb (blue dotted line). However, Pb inclusions of the same size but differently shaped (round, non-faceted) exhibit an opposite melting characteristic; that is a melting point depression (see black curve of the DSC chart inserted in fig. 1). The 3-D morphologies of equi-sized Pb particles embedded in an Al matrix are depicted as a tomogram shown in figure 1 . Thus, the particle size alone cannot explain such melting behavior.
Focus Series Reconstruction and Geometric Phase Analysis (GPA)
As the spherical aberration of the objective lens leads to delocalization of the information , through focus series of the interface structure (typically 20 images) were acquired in an FEI Tecnai F20 G2.
The individual high-resolution micrographs were aligned and stacked. A flux-preserving iterative computer algorithm for the reconstruction of focal series was used , in which the defocus, the spherical aberration coefficient Cs, and the two-fold astigmatism were corrected. In this study 10 images with defocus values ranging from -154 to -203 nm were taken to retrieve the complex-valued exit-face wave function which was used as input for calculating the strain. The method used for measuring and mapping strain fields is the geometric phase analysis (GPA) developed by Hÿtch et al. . It allows analysis of quantitative atomic level strains and rigid-body rotations in plane of a TEM sample.
A modified version, which is based on the Hÿtch formalism  but allowing the geometric phase to be extracted from the complex-valued exit-face wave function, was implemented in Digital Micrograph (Version 1.71.38, Gatan) to calculate the components of the 2-D strain tensor with Pb as reference lattice. Using the complex exit-wave makes these measurements independent of variations in the mean inner potential. Circular masks with radii producing a lateral resolution of 1 nm in the geometric phase images were set around the centers of the [-1,-1,1] and [-1,0,0] reflections of the Al and Pb lattice . A cosine function which produced a gradual transition of the mask value from 1 to zero within the outer 20 % of the mask radius was used for smoothing the edges of the circular masks.
Results and Discussion
To exclude misinterpretations related to Moiré patterns or buried structures, an uncovered Pb inclusion exhibiting a hetero-structure with the Al matrix, was sought. The retrieved phase image (fig. 2a) shows such uncovered Al-Pb interface structures with a slight waviness, which appears to be more pronounced for the -interface. Strain maps (mean dilatation, shear and rigid-body rotation) were calculated as described in the preceding section.
Figure 2b shows the result of the strain analysis for the mean dilatation. Considerable strain (green color in fig. 2b) is present around the interface regions. Strain peaks (hot spots) originating from the localized cores of the misfit dislocations are observed, accommodating the misfit of 22 % between the two lattices. The general appearance of the interfaces in terms of strain (mean dilatation) is rough and wavy. The roughness and waviness would account for a non-equilibrium state of the particle-matrix interface. Figures 2c and 2d complete the strain picture displaying the results of the shear strain and rigid-body rotation. They show that the interface regions are neither sheared nor significantly rotated. This leads to the conclusion that the whole state of the particle-matrix interface is elastically strained but not plastically deformed. The experimental situation is compared with a model interface, which was constructed by two rigidly coupled lattices reflecting the lattice mismatch of Al and Pb. The resulting strain maps for the -interface are shown in figure 3a and 3b.
Strain profiles (fig. 3c and 3d) are performed across the interface (see indicated boxes) and compared. Compressive strain is found for the Pb side and tensile strain for the Al side. The strain profile across the model interface (fig. 3d) shows a strain gradient of 0.7 nm in total, which is mainly determined by the radius of the mask (resolution) to generate the geometric phase maps since the rigidly coupled interface excludes intrinsic strain. However, the situation at the real interface yields a value of 1.5 nm for the strain gradient (fig. 3c), which implies a net effect of at least 2-3 strained lattice planes. This indicates that the outer regions of the Pb particle are under compressive strain or, in terms of stresses, a hydrostatic pressure (applying Hooke‘s law). Thus, a higher melting temperature is expected according to the Clapeyron equation (volume change times pressure change divided by the entropy change equals a temperature change).
Focus series reconstruction is a necessary task, especially for interface studies using uncorrected microscopes, to exclude delocalization effects caused (mainly) by the spherical aberration of the objective lens. GPA has become a standard tool for precise strain measurements using high-resolution micrographs as input. The combination of both techniques, as applied in this study, allows extraction of precise strain information at interfaces. This analysis demonstrates that the outer regions of an embedded nanoparticle are compressed and thus provide an explanation for an elevated melting of faceted Pb inclusions embedded in Al matrix in terms of interface strain .
Support by the Deutsche Forschungsgemeinschaft (WI 1899/14-1) is gratefully acknowledged.
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