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The interface stress is brought on through the interaction of solid-solid interfaces and their (elastic) deformations necessary to maintain coherency of the grains; in this case, it is assumed that no atomic bonds are broken in the interface plane [Gurtin and Murdoch (1975), Cahn (1978)]. As a result, it is likely that grain boundary atoms are not in their perfect lattice

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position, nor in their lowest energy state. Considering that the grain boundary atoms are not in their perfect lattice position, a qualitative illustration of this effect can be given on an atomistic level. In order to do this, the Lennard-Jones pair potential model is considered [Lennard-Jones (1924)], which according to Wolf (1989a,b) yields results that are similar to the multi-body potential model. The general form of the Lennard-Jones pair potential U, for two physical bodies, as a function of the two-body separation x [e.g., Atkins (1997)] is,

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where, ε is the depth potential well and  is the pair separation where U = 0. The bond energy, U, and bond force, F  U / x, curves are shown in Fig. 6.8.

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Figure 6.8.

Lennard-Jones pair potential model: (a) bond-energy curve where a0 is the equilibrium interatomic spacing value, U a is the attractive bond-energy, U r is the repulsive bond-energy, and U t is the total energy, (b) bond-force curve where a0 is the equilibrium interatomic spacing value, Fa is the attractive bond-force, Fr is the repulsive bond-force, and Ft is the total force [Lennard-Jones (1924)].

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In the case of the bond-energy curve, the total energy is at its lowest when the equilibrium interatomic spacing, a0, is achieved. Consequently, the force-energy curve at equilibrium has a net zero force. If the interatomic spacing is increased or decreased, as in the case of the grain boundary atoms, there will be a net attractive or repulsive force respectively, and an overall increase in energy which contribute to the generation of an interface stress in a network of grains leading to the observed macrostresses. Ultimately, it is a net elastic stress field that is generated over the network of grains in the system giving rise to the macrostresses observed in these materials. As the grain size decreases, or the intercrystal volume fraction (i.e., grain boundaries and triple junctions) increases, a corresponding increase in elastic field intensity is generated resulting in an increase in macrostress.

6.4. Elastic Response Due to Internal Stress The effect of the localized strain at grain boundaries on XRD lines in nanocrystalline materials was investigated by Qin et al. (2008). In addition to XRD line broadening, they proposed that there is a superimposed XRD line shift. In order to illustrate this, Qin et al.

(2008) modeled the variation of the lattice parameter in nanocrystalline materials with distance from grain boundaries and determined that conventional XRD lines resulting from lattice planes near grain boundaries will also shift to lower 2 angles or take on an increased lattice parameter. This is consistent with the presence of internal stresses. Note that conventional XRD produced lines from crystallographic planes which are parallel to the sample plane. Therefore, such XRD line shifts correspond to changes in interplanar spacing, d, normal to the sample plane.

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If the interplanar spacing increases there is a net tensile strain and if it decreases there is a net compressive strain. The lattice parameter change or resulting change in interplanar spacing is given as follows,

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where,  is the strain, d is the measured interplanar spacing, and d 0 is the interplanar spacing of the unstressed condition.

The macrostresses measured in this study are bi-axial and result only in the plane of the samples. Since the macrostresses at the smallest grain sizes are predominantly compressive in nature, the net interplanar spacing is bi-axially contracted and thus smaller than that of an unstressed material. On the contrary, there are numerous reports on the effect of decreasing grain size on the lattice parameter or interplanar spacing determined using conventional XRD, e.g., Ni [Liu et al. (1994)] and Fe [Zhao et al. (2001)], whereby it tends to increase, especially at grain sizes less than 10 nm. As mentioned earlier, the modeling of Qin et al.

(2008) determined that conventional XRD lines resulting from lattice planes near grain boundaries will contribute to a shifting to lower 2 angles or take on an increased interplanar spacing. Note that in Chapter 4, local strain at grain boundaries was evident in the HR-TEM images (Section 4.4.2) and the lattice parameters for the current nanocrystalline Ni and Ni-Fe samples were observed to be slightly increased when compared to their large grain polycrystalline counterparts (Section 4.5.1). This implies that while the material is undergoing a bi-axial contraction in the plane it is also likely that the material is undergoing an expansion in the normal direction to the plane.

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Figure 6.9.

Schematic diagram of the sample planar cross-section showing conventional XRD lines from an unstressed lattice with equally spaced atoms and a Bragg angle,  0, corresponding to the interplanar spacing, d 0.

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Figure 6.10.

Schematic diagram of the sample planar cross-section showing conventional XRD lines from a lattice in a state of (bi-axial) compressive stress. A contraction of the lattice in the planar direction is accompanied by an expansion that is normal to the plane.

The net result is a lower Bragg angle, , and increased interplanar spacing value, d.

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In order to clarify this, we consider the irradiated area near the surface of the sample in the planar cross-section on an atomic scale shown in Fig. 6.9 for the case where there is no apparent stress in the material and the lattice parameter is equivalent to d 0 at a corresponding Bragg angle of  0. In the presence of interface stresses, mechanical equilibrium requires that macrostresses are present in order to compensate [Birringer et al. (2002)] or analogously to equilibrate within the whole body [Macherauch and Kloos (1987)].

If an elastic response can be assumed, then the elastic deformation which takes place will obey the Poisson ratio,

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Considering this, a negative value for strain in the planar directions,  11 and  22, should result in a positive strain in the normal direction,  33. If there is a net interplanar spacing contraction in the plane, there must be a net interplanar spacing expansion normal to it. Fig.

6.10 illustrates the result of a bi-axial compressive stress where the lattice contracts in the planar directions but expands in the normal direction causing a diffraction line shift to a lower Bragg angle and thus, an increased interplanar spacing, d, in the normal direction to the sample plane. As a result, there is an inherent relationship between the microscopic and macroscopic strain measurements. Such a relationship can also be seen in the microstrain and macrostress relationship with grain size, i.e., a similar inverse relationship with grain size.

If the sources of these stresses are considered to be the same, i.e., the consequences of interface stresses [Cahn and Larche (1982)] or long-range elastic fields since local elastic

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deformation is necessary for joining of the crystals [Valiev et al. (1986)], then they are distinguished only by way of the method used to detect them. That is, if microstrain induced XRD line broadening is accompanied by a superimposed XRD line shift [Qin et al. (2008)] then by conventional XRD it is detected as interplanar spacing changes in the direction normal to the plane. Since the macrostresses are resolved bi-axially, they are detected as interplanar spacing changes in the planar directions.

6.5. Summary The microstrain for the series of nanocrystalline Ni and Ni-Fe alloys produced by electrodeposition was determined from XRD pattern analysis based on line broadening. A general increasing trend with decreasing grain size was observed. The microstrain was found increase dramatically at grain sizes less than 20 nm. This increasing microstrain trend is approximately proportional to the inverse of the grain size and consistent with the increase of intercrystal defects (i.e., grain boundaries and triple junctions) with decreasing grain size.

This microstrain is associated with the local strains observed at grain boundaries in the HRTEM image analysis. The microstrain values for the nanocrystalline Ni-Fe samples were noticeably higher than the Ni samples and showed a distinct increasing trend with both decreasing grain size and increasing Fe concentration. In Chapter 4, both HR-TEM image analysis (Section 4.4.2) and growth fault probabilities determinations (Section 4.5.4) showed an increase of growth faults with increasing Fe concentration. As a result, for the nanocrystalline Ni-Fe alloys, a contributing effect of microstrain induced XRD line broadening is considered to be due to the presence of growth faults in addition to the effect of grain size reduction.

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Macrostresses were systematically determined for the series of electrodeposited nanocrystalline Ni and Ni-Fe samples over a broad grain size range based on the detection of strain from the (311) ring using 2D-XRD. At grain sizes below 20 nm, there was a dramatic increase in the compressive macrostress with decreasing grain size. Similar to the microstrain, macrostress was found to be approximately proportional to the inverse of the grain size. The origin of the compressive macrostress is proposed to be an equilibrating response to the presence of interface stresses. These interface stresses arise from elastic deformations at grain boundaries which generate a long-range elastic stress field. The intensity of this elastic stress field is related an increasing intercrystal volume fraction that results in a corresponding increase in the observed macrostresses.

Furthermore, microstrain in these materials was identified to be a result of localized strain near grain boundaries. These localized strains are proposed to be caused by the interface stresses that result in the observed macrostresses. The sources of microstrain and macrostresses are considered to be the same, but detected in different manners. That is, microstrain is detected by XRD line broadening and simultaneously a superimposed XRD line shift. The XRD line shift in this case results in an increased interplanar spacing. On the other hand, macrostresses are detected as XRD line shifts or changes in the interplanar spacing corresponding to the planar directions. In the present study, the macrostresses were found to be increasingly compressive at grain sizes less than 20 nm and thus, there is a decreased interplanar spacing. It follows that the material is undergoing an elastic response that obeys the principle behind the Poisson ratio.

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6.7. References Atkins, P., “The Elements of Physical Chemistry”, 2nd Ed., W.H. Freeman and Company, New York, 1997.

Biju, V., N. Sugathan, V. Vrinda, and S.L. Salini, J. Mater. Sci. 43 (2008) 1175.

Birringer, R., M. Hoffmann and P. Zimmer, Phys. Rev. Lett. 88 (2002) 206104.

Brooks, H., “Metal Interfaces”, American Society for Metals, Metals Park, OH, 1952.

Cahn, J.W., in: “Interfacial Segregation”, Ed.: W.C. Johnson and J.M. Blakely, Metals Park Ohio, American Society for Metals, 1978.

Cahn, J.W. and F. Larche, Acta Metall. 30 (1982) 51.

Callister, W.D., “Fundamentals of Materials Science and Engineering: An Integrated Approach – 2nd Edition”, John Wiley & Sons, 2005.

Cammarata, R.C., Prog. Surf. Sci. 46 (1994) 1.

Cammarata, R.C., Mater. Sci. Eng. A237 (1997) 180.

Cammarata, R.C. and R.K. Eby, J. Mater. Res. 6 (1991) 888.

Cullity, B.D. and S.R. Stock, “Elements of X-Ray Diffraction, Third Edition”, Prentice Hall, New Jersey, 2001.

Czerwinski, F., J. Electrochem. Soc. 143 (1996) 3327.

Derlet, P.M., S. Van Petegem and H. Van Swygenhoven, Phys. Rev. B 71 (2005) 024114.

Eastman, J.A., M.A. Beno, G.S. Knapp and L.J. Thompson, Nantostruct. Mater. 6 (1995) 543.

Eckert, J., J.C. Holzer, C.E. Krill, III and W.L. Johnson, J. Mater. Res. 7 (1992) 1751.

El-Sherik, A.M., J. Shirokoff and U. Erb, J. All. Comp. 389 (2005) 140.

Gurtin, M.E. and A.I. Murdoch, Arch. Rat. Mech. Anal. 57 (1975) 291.

Haasz, T.R., G. Palumbo, K.T. Aust, A.M. El-Sherik and U. Erb, Scripta Metall. Mater., 32 (1995) 423.

He, B.B., “Two-Dimensional X-Ray Diffraction”, John Wiley & Sons, Hoboken, New Jersey, 2009.

Jiang, Q., D.S. Zhao and M. Zhao, Acta Mater. 49 (2001) 3143.

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Krstic, V.D., U. Erb and G. Palumbo, Scripta Metall. Mater. 29 (1993) 1501.

Lehwald, S., F. Wolf, H. Ibach, B.M. Hall and D.L. Mills, Surf. Sci. 192 (1987) 131.

Lennard-Jones, J.E., Proc. R. Soc. Lond. A 106 (1924) 463.

Li, H.Q. and F. Ebrahimi, Mater. Sci. Eng., A347 (2003) 93.

Li, D.X., D.H. Ping, J.Y. Huang, Y.D. Yu and H.Q. Ye, Micron 31 (2000) 581.

Liu, X.D., H.Y. Zhang, K. Lu and Z.Q. Hu, J. Phys.: Condes. Matter 6 (1994) L497.

Lu, J., “Handbook of Measurement of Residual Stress”, The Fairmont Press, Lilburn, GA, 1996.

Macherauch, E. and K. H. Kloos, in: Residual Stresses in Science and Technology (E.

Macherauch and V. Hauk, ed.), Informationsgesellschaft, Oberursel, 1987.

Malow, T.R., C.C. Koch, Acta Mater. 45 (1997) 2177.

Mishra, R., B. Basu and R. Balasubramaniam, Mater. Sci. Eng. A373 (2004) 370.

Nieman, G.W. and J.R. Weertman, J. Mater. Res. 6 (1991) 1012.

Noyan, I.C. and J.B. Cohen, Residual Stress: Measurement by Diffraction and Interpretation, Springer-Verlag, New York, 1987.

Palumbo, G., S.J. Thorpe and K.T. Aust, Scripta Metall. Mater., 24 (1990) 1347.

Paterson, M.S., J. Appl. Phys. 23 (1952) 805.

Ping, D.H., D.X. Li and H.Q. Ye, J. Mater. Sci. Lett. 14 (1995) 1536.

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