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Dependence of lattice parameters of small particles on the size of the nuclei
Surface Science 106(1981)64-69 North-Holland Publishing Company
DEP~~ENCE OF LATTICE P~~~ ON THE SIZE OF THE NUCLEI
OF SMALL P~~CLES
J. WOLTERSDORF, AS. NEPIJKO* and E. PIPPEL Ins/i&t fiir Festk6rperphysik und Elektronenmikroskopie der Akademie der
Wissenschaften der DDR, Halle (S.), DDR
Received 8 September 1980;accepted for pubIication 22 October 1980
1. Introduction The influence of surface on the crystal properties can be studied in a convenient manner when going to smaller and smallest particles where the proportion of the surface with respect to the bulk increases considerably. There are some theoretical and experimental investigations (e.g. Bore1 (11, De Planta et al. [2], Mays et al. [3]) suggesting that the lattice spacings will decrease with decreasing particle size. Thermodynamically this effect is commonly explained using the Gauss-Laplace formula (see e.g. Defay and Prigogine [4]). The present paper is devoted to the experimental measurement of the size dependence of lattice parameters of small metallic particles; a method is suggested which is not an integrale one -in contrast to the commonly used methods of electron diffraction or neutron scattering- and which is utmost sensitive to very small changes in lattice parameters. Besides that, the change of lattice constant within the small nuclei depending on the distance from the centre is investigated.
2. Experimental For measuring the lattice parameters of individual small metallic particles we employed the moire fringe method. To produce the moire fringes in the electron microscope it is necessary that the mono~~st~line particles to be examined are grown epitaxially in a parallel alignment on a different monocrystalline substrate material having a slightly different lattice parameter. Then, in a transmission electron microscope the electron beams are diffracted also at slightly different angles. Thus, in the bright-field as well as in the dark-field mode moire fringes arise due the interference of neighbouring beams. Moire fringes are always situated in directions perpendicular to the difference between correspondent reciprocal lattice vectors of substrate and deposit which in case of parallel moire are again situated perpendicular to the adequate lattice planes. So the moire *Permanent address: Institute of Physics of the Ukrainian Academy of Sciences, Kiev, USSR.
~3~28/81/~~/$02.50
@ North-Holland ~blishing Company
J. Woltersdorfet al. / Dependence of latticeparameters
65
technique can be regarded as an indirect method of lattice plane imaging. Moreover, the moire technique is very sensitive to small values of lattice parameter difference and an additional change in one of the lattice parameters leads to a strong enlargement of the moire magnification, So this method is a very sensitive one with respect to our problem and it allows to work with relatively low electron microscopic magnifications. Because of its excellent electron microscopic transmissibility even at 100 keV monocrystalline MgO was used as substrate material. Moreover, MgO is relatively hard, and elastic or plastic relaxations of the bicrystal are not expected during specimen preparation. The investigations were carried out using aluminium particles. The lattice parameters of compact aluminium (aN = 0.405 nm) and compact MgO (aMgo= 0.42 nm) lead to a moire fringe spacing of 6 = 5.7 nm and 6 = 3.9 nm in the case of the (200) reflexion and (220) reflexion, respectively (employed in our experiments). Another advantage of the substance chosen is that increasing lattice parameters of aluminium cause an increase in 8. For specimen preparation a method was used developed by two of the authors [5]. Due to the small interactions in the interface (cf. refs. [6,7]), the growth of aluminium on MgO starts with hemispherical nuclei coalescing finally and yielding continuous layers in the further film growth process.
3. Results Fig. 1 shows an example of a transmission micrograph of the sandwich system Al/MgO. The aluminium lattice parameters of individual particles were determined by photometric measurements of the moire fringe spacing directly from photographic plates. As the moire fringes have a distance of about 3 to 5 nm in our case, the lattice parameter estimated by this method is, of course, only an average value valid for an area between two neighboured moire fringes. Therefore, the measured values illustrated in the subsequent graphical representations in form of discrete points are to be understood as representative for the centres of the area between two moire fringes. To attain a high precision the magnification of each photographic plate must be calibrated. For this purpose we treated only photographic plates which contain small particles as well as continuous film areas and supposed a continuous aluminium layer to have the lattice parameter of compact aluminium. A calibration in the described manner allows a relative lattice parameter determination with the accuracy of ?0.0005 nm in the present case. Our main results may be divided into two groups: Firstly, the lattice parameter of small particles was related to the linear dimensions of those particles, and secondly, the change of lattice parameter within a single particle (from centre to border) was observed. For illustrating the dependence of the lattice parameter on the particle size the lattice parameter in the centre of the particle was used. In fig. 2 the corresponding results are shown. The graphic representation gives the results of about 50 particles ranging from 6 to 30 nm in diameter. Below a diameter of 20 nm a significant decrease in lattice parameter is found, which amounts to 1.5% for 6nm diameters of the particles. The insert in fig. 3 shows that the moire fringes do not propagate in straight lines over the particle, but with
J. Woltersdorfet al. 1 Dependence of latticeparameters
Fig. 1. Transmission electron micrograph of the sandwich system Al on MgO. The Al particles of different size are characterized by moirt patterns.
distances decreasing from the center to the edges. To estimate the distribution of the lattice parameters within a single particle the moire fringes were measured along the lines as indicated in the insert of fig. 3 by starting from the central line of the particle to its edge. From fig. 3 it can be deduced that the lattice spacings remain nearly constant within the measuring accuracy over most parts of the particle along the three lines in the central part.
l-m
A bulk
moteriol
4396
20
10 particle
30 nm
diameter
Fig. 2. Dependence of the measured lattice parameter on particle size.
J. Woltersdorfet al. I Dependence of lam’ceparameters
Fig. 3. Lattice parameter at different distances from the central line as indicated in the figure.
A significant decrease is only found at the edges as well as along the outer lines. Fig. 4 contains a summary of the typical results: for three different nuclei the lattice parameters are given as a function of the distance from the particle centre. On the one hand, smaller aluminium particles are characterized by smaller overall lattice parameters. On the other hand, a clear tendency was found for the lattice parameter to decrease from the centre to the edges by nearly 0.5%.
A
A
5 distance
Fig. 4. Variation of the lattice parameter particles.
from
the particle
10
nm
- centre
with the distance from the particle centre, measured at three different
J. Wohsdmf et al. i Dependence of latticeparameters
68
4. Discussion
In contrast to the integral methods like electron di~raction or neutron scattering the advantage of the electron microscopic moire method is the possibility of measuring the lattice parameters of individual particles as well as the parameter changes within one particle with a high accuracy as mentioned above. Fig. 2 represents our first result concerning the decrease of lattice parameter a at decreasing particle size d. This behaviour may be expressed on principle by the help of the Gauss-Laplace formula. For justifying the application of the drop model as a first approximation fig, 3 can be used, indicating that the deformation is homogeneous over large parts of the particle volume. It follows from simple thermodynamical considerations that the equilibrium of a liquid drop is given by ApdV=ydF,
(1)
where y dF is the additional surface energy connected with a virtual enlargement of the drop (y is surface stress, dF surface element), and the left-hand side is the ~o~esponding work (Ap the pressure inside the drop). For a sphere with diameter d therefore one has: Ap=4yd-’
.
If the small nuclei are crystalline with cubic structure and lattice constant a, then with the compressibility # = AV(A~V~)-’and V. = a3, from (I) one obtains Aaia =+/d
.
(2)
However, this formula (2) should only allow qualitative predictions, because it does not take into account the change of surface stress with the particle size, which should possibly be important for very small nuclei. However, recently Kern et al. IS] believe eq. (2) to be valid also in such cases. Moreover, Stoneham [9] has shown that an equation equivalent to (2) can be used to explain contraction phenomena in cubic crystals having (111) and (100) facets. Using the values resulting from our experiments given in fig. 2 it follows from formula (2) that the surface stress increases with decreasing particle diameter: If K is approximated by the reciprocal of Young’s modulus, then, for example, the surface stress of particles with diameter of 20 nm is y = 2000 dyn cm-’ and for particles with diameters of 10 nm it is y %5500dyncm-‘. Regarding that the value of the Al bulk material is given by y = 860 dyn cm-‘, these results indicate that a relatively high surface stress is connected with the lattice contractions observed in our experiments (if the drop model is valid). Besides the restrictions concerning formula (2) as mentioned above, some difficulties in quantitative inte~retations of expe~mental results are due to the following facts: (1) Large particles can be produced by incomplete coalescence of smaller ones - a process leading to uncontrolled stresses in the particle to be examined. (2) The kind of surface stress and of interaction between substrate and deposit may be influenced by a thin oxide layer forming on the aluminium particles. For understanding the lattice parameter change from the centre to the edges, as shown
J. Woltersdorfet al. / Dependence of lam’ceparameters
69
in figs. 3 and 4 we assume the aluminium particles to be flattened on the top. Such a mode of growth is connected with a stronger curvature near the particle edges, and according to the Gauss-Laplace formula a decrease of the lattice parameter near the border may be deduced. Moreover, the stress state at the particle edges should be decisively influenced by a complicated transition region with respect to the MgO substrate. Concerning the structure and the effective interface stresses of this region no assured informations are available. For more detailed information cf. ref. [lo].
Acknowledgements The authors would like to express their appreciation to Professor H. Bethge for his encouragement in this study. Special thanks are due to Professor R. Kern and Dr. M. Krohn for helpful discussions.
References [l] [2] [3] [4] [S] [6] [7] [8] [9] [lo]
J.-P. Borel, MCm. Sot. Vaudoise Sci. Nat. 11 (1955) 33. T. de Planta, R. Ghez and F. Pinz, Helv. Phys. Acta 37 (1964) 74. C.W. Mays, J.S. Vermaak and D. Kuhlmann-Wilsdorf, Surface Sci. 12 (1968) 124. R. Defay, J. Prigogine, A. Bellemans and D.H. Everett, Surface Tension and Adsorption (London, 1966). H. Bethge, E. Pippel and J. Woltersdorf, Phys. Status Solidi (a) 37 (1976) 457. E. Pippel, in: Strukturen Kristalliner Phasengrenzen - Elektronenmikroskopischer Bildkontrast, Eds. H.G. Schneider and J. Woltersdorf (Verlag fiir Grundstoffindustrie, Leipzig, 1977) pp. 58-78. J. Woltersdorf, in: Phasengrenzflachen, Sitzungsberichte Akad. Wiss. 17 N (Akademieverlag, Berlin, 1978) pp. 9-43. R. Kern, G. Le Lay and J.J. Metois, in: Current Topics in Materials Science, Vol. 3, Ed. E. Kaldis (North-Holland, Amsterdam, 1979) pp. 1311119. A.M. Stoneham, J. Phys. Cl0 (1977) 1175. A.S. Nepijko, E. Pippel and J. Woltersdorf, Phys. Status Solidi (a) 61 (1980) 469.
DEP~~ENCE OF LATTICE P~~~ ON THE SIZE OF THE NUCLEI
OF SMALL P~~CLES
J. WOLTERSDORF, AS. NEPIJKO* and E. PIPPEL Ins/i&t fiir Festk6rperphysik und Elektronenmikroskopie der Akademie der
Wissenschaften der DDR, Halle (S.), DDR
Received 8 September 1980;accepted for pubIication 22 October 1980
1. Introduction The influence of surface on the crystal properties can be studied in a convenient manner when going to smaller and smallest particles where the proportion of the surface with respect to the bulk increases considerably. There are some theoretical and experimental investigations (e.g. Bore1 (11, De Planta et al. [2], Mays et al. [3]) suggesting that the lattice spacings will decrease with decreasing particle size. Thermodynamically this effect is commonly explained using the Gauss-Laplace formula (see e.g. Defay and Prigogine [4]). The present paper is devoted to the experimental measurement of the size dependence of lattice parameters of small metallic particles; a method is suggested which is not an integrale one -in contrast to the commonly used methods of electron diffraction or neutron scattering- and which is utmost sensitive to very small changes in lattice parameters. Besides that, the change of lattice constant within the small nuclei depending on the distance from the centre is investigated.
2. Experimental For measuring the lattice parameters of individual small metallic particles we employed the moire fringe method. To produce the moire fringes in the electron microscope it is necessary that the mono~~st~line particles to be examined are grown epitaxially in a parallel alignment on a different monocrystalline substrate material having a slightly different lattice parameter. Then, in a transmission electron microscope the electron beams are diffracted also at slightly different angles. Thus, in the bright-field as well as in the dark-field mode moire fringes arise due the interference of neighbouring beams. Moire fringes are always situated in directions perpendicular to the difference between correspondent reciprocal lattice vectors of substrate and deposit which in case of parallel moire are again situated perpendicular to the adequate lattice planes. So the moire *Permanent address: Institute of Physics of the Ukrainian Academy of Sciences, Kiev, USSR.
~3~28/81/~~/$02.50
@ North-Holland ~blishing Company
J. Woltersdorfet al. / Dependence of latticeparameters
65
technique can be regarded as an indirect method of lattice plane imaging. Moreover, the moire technique is very sensitive to small values of lattice parameter difference and an additional change in one of the lattice parameters leads to a strong enlargement of the moire magnification, So this method is a very sensitive one with respect to our problem and it allows to work with relatively low electron microscopic magnifications. Because of its excellent electron microscopic transmissibility even at 100 keV monocrystalline MgO was used as substrate material. Moreover, MgO is relatively hard, and elastic or plastic relaxations of the bicrystal are not expected during specimen preparation. The investigations were carried out using aluminium particles. The lattice parameters of compact aluminium (aN = 0.405 nm) and compact MgO (aMgo= 0.42 nm) lead to a moire fringe spacing of 6 = 5.7 nm and 6 = 3.9 nm in the case of the (200) reflexion and (220) reflexion, respectively (employed in our experiments). Another advantage of the substance chosen is that increasing lattice parameters of aluminium cause an increase in 8. For specimen preparation a method was used developed by two of the authors [5]. Due to the small interactions in the interface (cf. refs. [6,7]), the growth of aluminium on MgO starts with hemispherical nuclei coalescing finally and yielding continuous layers in the further film growth process.
3. Results Fig. 1 shows an example of a transmission micrograph of the sandwich system Al/MgO. The aluminium lattice parameters of individual particles were determined by photometric measurements of the moire fringe spacing directly from photographic plates. As the moire fringes have a distance of about 3 to 5 nm in our case, the lattice parameter estimated by this method is, of course, only an average value valid for an area between two neighboured moire fringes. Therefore, the measured values illustrated in the subsequent graphical representations in form of discrete points are to be understood as representative for the centres of the area between two moire fringes. To attain a high precision the magnification of each photographic plate must be calibrated. For this purpose we treated only photographic plates which contain small particles as well as continuous film areas and supposed a continuous aluminium layer to have the lattice parameter of compact aluminium. A calibration in the described manner allows a relative lattice parameter determination with the accuracy of ?0.0005 nm in the present case. Our main results may be divided into two groups: Firstly, the lattice parameter of small particles was related to the linear dimensions of those particles, and secondly, the change of lattice parameter within a single particle (from centre to border) was observed. For illustrating the dependence of the lattice parameter on the particle size the lattice parameter in the centre of the particle was used. In fig. 2 the corresponding results are shown. The graphic representation gives the results of about 50 particles ranging from 6 to 30 nm in diameter. Below a diameter of 20 nm a significant decrease in lattice parameter is found, which amounts to 1.5% for 6nm diameters of the particles. The insert in fig. 3 shows that the moire fringes do not propagate in straight lines over the particle, but with
J. Woltersdorfet al. 1 Dependence of latticeparameters
Fig. 1. Transmission electron micrograph of the sandwich system Al on MgO. The Al particles of different size are characterized by moirt patterns.
distances decreasing from the center to the edges. To estimate the distribution of the lattice parameters within a single particle the moire fringes were measured along the lines as indicated in the insert of fig. 3 by starting from the central line of the particle to its edge. From fig. 3 it can be deduced that the lattice spacings remain nearly constant within the measuring accuracy over most parts of the particle along the three lines in the central part.
l-m
A bulk
moteriol
4396
20
10 particle
30 nm
diameter
Fig. 2. Dependence of the measured lattice parameter on particle size.
J. Woltersdorfet al. I Dependence of lam’ceparameters
Fig. 3. Lattice parameter at different distances from the central line as indicated in the figure.
A significant decrease is only found at the edges as well as along the outer lines. Fig. 4 contains a summary of the typical results: for three different nuclei the lattice parameters are given as a function of the distance from the particle centre. On the one hand, smaller aluminium particles are characterized by smaller overall lattice parameters. On the other hand, a clear tendency was found for the lattice parameter to decrease from the centre to the edges by nearly 0.5%.
A
A
5 distance
Fig. 4. Variation of the lattice parameter particles.
from
the particle
10
nm
- centre
with the distance from the particle centre, measured at three different
J. Wohsdmf et al. i Dependence of latticeparameters
68
4. Discussion
In contrast to the integral methods like electron di~raction or neutron scattering the advantage of the electron microscopic moire method is the possibility of measuring the lattice parameters of individual particles as well as the parameter changes within one particle with a high accuracy as mentioned above. Fig. 2 represents our first result concerning the decrease of lattice parameter a at decreasing particle size d. This behaviour may be expressed on principle by the help of the Gauss-Laplace formula. For justifying the application of the drop model as a first approximation fig, 3 can be used, indicating that the deformation is homogeneous over large parts of the particle volume. It follows from simple thermodynamical considerations that the equilibrium of a liquid drop is given by ApdV=ydF,
(1)
where y dF is the additional surface energy connected with a virtual enlargement of the drop (y is surface stress, dF surface element), and the left-hand side is the ~o~esponding work (Ap the pressure inside the drop). For a sphere with diameter d therefore one has: Ap=4yd-’
.
If the small nuclei are crystalline with cubic structure and lattice constant a, then with the compressibility # = AV(A~V~)-’and V. = a3, from (I) one obtains Aaia =+/d
.
(2)
However, this formula (2) should only allow qualitative predictions, because it does not take into account the change of surface stress with the particle size, which should possibly be important for very small nuclei. However, recently Kern et al. IS] believe eq. (2) to be valid also in such cases. Moreover, Stoneham [9] has shown that an equation equivalent to (2) can be used to explain contraction phenomena in cubic crystals having (111) and (100) facets. Using the values resulting from our experiments given in fig. 2 it follows from formula (2) that the surface stress increases with decreasing particle diameter: If K is approximated by the reciprocal of Young’s modulus, then, for example, the surface stress of particles with diameter of 20 nm is y = 2000 dyn cm-’ and for particles with diameters of 10 nm it is y %5500dyncm-‘. Regarding that the value of the Al bulk material is given by y = 860 dyn cm-‘, these results indicate that a relatively high surface stress is connected with the lattice contractions observed in our experiments (if the drop model is valid). Besides the restrictions concerning formula (2) as mentioned above, some difficulties in quantitative inte~retations of expe~mental results are due to the following facts: (1) Large particles can be produced by incomplete coalescence of smaller ones - a process leading to uncontrolled stresses in the particle to be examined. (2) The kind of surface stress and of interaction between substrate and deposit may be influenced by a thin oxide layer forming on the aluminium particles. For understanding the lattice parameter change from the centre to the edges, as shown
J. Woltersdorfet al. / Dependence of lam’ceparameters
69
in figs. 3 and 4 we assume the aluminium particles to be flattened on the top. Such a mode of growth is connected with a stronger curvature near the particle edges, and according to the Gauss-Laplace formula a decrease of the lattice parameter near the border may be deduced. Moreover, the stress state at the particle edges should be decisively influenced by a complicated transition region with respect to the MgO substrate. Concerning the structure and the effective interface stresses of this region no assured informations are available. For more detailed information cf. ref. [lo].
Acknowledgements The authors would like to express their appreciation to Professor H. Bethge for his encouragement in this study. Special thanks are due to Professor R. Kern and Dr. M. Krohn for helpful discussions.
References [l] [2] [3] [4] [S] [6] [7] [8] [9] [lo]
J.-P. Borel, MCm. Sot. Vaudoise Sci. Nat. 11 (1955) 33. T. de Planta, R. Ghez and F. Pinz, Helv. Phys. Acta 37 (1964) 74. C.W. Mays, J.S. Vermaak and D. Kuhlmann-Wilsdorf, Surface Sci. 12 (1968) 124. R. Defay, J. Prigogine, A. Bellemans and D.H. Everett, Surface Tension and Adsorption (London, 1966). H. Bethge, E. Pippel and J. Woltersdorf, Phys. Status Solidi (a) 37 (1976) 457. E. Pippel, in: Strukturen Kristalliner Phasengrenzen - Elektronenmikroskopischer Bildkontrast, Eds. H.G. Schneider and J. Woltersdorf (Verlag fiir Grundstoffindustrie, Leipzig, 1977) pp. 58-78. J. Woltersdorf, in: Phasengrenzflachen, Sitzungsberichte Akad. Wiss. 17 N (Akademieverlag, Berlin, 1978) pp. 9-43. R. Kern, G. Le Lay and J.J. Metois, in: Current Topics in Materials Science, Vol. 3, Ed. E. Kaldis (North-Holland, Amsterdam, 1979) pp. 1311119. A.M. Stoneham, J. Phys. Cl0 (1977) 1175. A.S. Nepijko, E. Pippel and J. Woltersdorf, Phys. Status Solidi (a) 61 (1980) 469.