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Determination of dislocation density from hardness measurements in metals
Materials Letters 62 (2008) 3812–3814
Contents lists available at ScienceDirect
Materials Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m a t l e t
Determination of dislocation density from hardness measurements in metals S. Graça, R. Colaço ⁎, P.A. Carvalho, R. Vilar Dep. Engenharia Materiais and Instituto de Ciência e Engenharia de Materiais e Superfícies, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
A R T I C L E
I N F O
Article history: Received 31 March 2008 Accepted 24 April 2008 Available online 30 April 2008 Keywords: Hardness Metals and alloys Dislocation density TEM
A B S T R A C T For the first time the Nix–Gao model for indentation size effect (ISE) is used to estimate the dislocation density in a metal. The estimate of dislocation density obtained by this method, using Ni as a case study, is compared with the values obtained from direct observation by transmission electron microscopy. It is shown that the estimate of dislocation density from indentation hardness measurements, adjusted by the Nix–Gao model, gives values consistent with those obtained by TEM, provided that the proper procedures to minimize errors are adopted. Although the direct observation of dislocations by TEM gives additional structural information, the indirect method to estimate dislocation density based on hardness measurements is more efficient, since the sample preparation method, measurement procedure and analysis of results are easier and faster. © 2008 Elsevier B.V. All rights reserved.
1. Introduction The plastic properties of metals, in particular their yield stress, hardness, strain hardening coefficient and toughness, depend critically on the dislocation density (ρ). This density is usually determined by transmission electron microscopy (TEM) direct observation, but this method is rather complex and time consuming. Moreover, the small size of the regions observed, associated to the heterogeneity of the dislocation distribution in plastically deformed metals, may lead to ρ values that are not representative and are inconsistent with the plastic behaviour of the material. When a material is plastically deformed the dislocation density increases, leading to strain hardening. If no dislocations (or only sessile dislocations) are present in the deformed region, new dislocations must be nucleated for plastic deformation to occur. This phenomenon has been observed when performing extremely small indentations (nanoindentations) in low defect density materials [1–3]. Moreover, if strain gradients are present (e.g. in indentations created by pyramidal or conical indenters), geometrically necessary dislocations (GNDs) must be created to accommodate them [4,5]. Since the strain gradient increases when the deformation scale decreases, the density of GNDs and, consequently, the hardness of the material, increase when the size of the deformed region decreases: this is known as the indentation size effect (ISE) [5]. In the present work a method based on the application of an ISE model [5] is used to determine the dislocation density of a metal from
hardness measurements. The values of ρ obtained by this method are compared with those obtained by TEM direct observation of the samples. It is shown that the values obtained by the indirect method based on hardness measurements are rather consistent with the values obtained from TEM observations. The advantages and drawbacks of both methods are discussed. 2. Experimental methods A nickel layer about 1 mm thick was deposited on an AISI 304 stainless steel substrate by powder injection laser cladding [6] using a 3 kW fast axial flow CO2 laser with a TEM01⁎ mode. A detailed description of the preparation method and processing parameters was presented in a previous paper [7]. Foils for TEM specimens were cut from this layer using a low speed abrasive saw. Both faces of the foil were ground with SiC abrasive papers to reduce their thickness down to ~ 60 μm. 3 mm discs were cut from this foil with a Q-switched Nd/ YAG laser. In order to avoid changing the dislocation density, thinning was carefully performed on this foil by ion milling with a 4 kV Ar+ beam and a 12° incidence angle until electron transparency was obtained. The surfaces used for indentation testing were ground with SiC papers and polished with a sequence of diamond particles suspensions. Vickers indentation tests were performed using loads in the range 0.01 to 2 N. 3. Determination of ρ by TEM
⁎ Corresponding author. Tel.: +351 21 8418125; fax: +351 21 8418120. E-mail address: [email protected] (R. Colaço). 0167-577X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2008.04.072
The projected length of dislocation lines (lp) in the TEM images was measured using a commercial image analysis software. Assuming that
S. Graça et al. / Materials Letters 62 (2008) 3812–3814
3813
the dislocations are randomly oriented, lp can be related to the length of dislocation lines (l) by [8]: l¼
4 lp : p
ð1Þ
The dislocation density is given by: q¼
l l ¼ : V At
ð2Þ
The volume of observed material (V) was determined by multiplying the corresponding area (A) by the sample thickness (t), Fig. 2. Fitting of Eq. (3) to the micro and ultramicrohardness data of laser clad Ni.
estimated by application of the graphical method proposed by Allen [9] to convergent beam electron diffraction data [10]. The thin foil thickness measured, at the location “X” shown in Fig. 1a, was consistently about 157 nm. Taking into consideration the thin foil preparation method a variation in the range 100 to 200 nm can be expected. The dislocation density was then estimated in eighteen regions of the Ni thin foil. Fig. 1b shows a bright-field TEM micrograph of one of the observed regions, where dislocations are clearly visible, and Fig. 1c shows the dislocation path defined with the image analysis software. To minimize systematic errors in the dislocation length values the thin foil was tilted in order to put into evidence the largest possible number of dislocations present. The dislocation density calculated using the lower and upper limits of the thickness range are (8.3 ± 6.4) × 109 cm− 2 and (4.1 ± 3.2) × 109 cm− 2, respectively. 4. Determination of ρ from hardness measurements The hardness values, retrieved from the indentation tests, increase with decreasing indentation depth as shown in Fig. 2. This behaviour was previously observed by other authors [7,11–14] and is known as indentation size effect. According to Nix and Gao [5], the ISE results from an increase in the density of the geometrically necessary dislocations required to accommodate the plastic deformation gradient around the indentation. The authors developed a model and deduced a simple expression to relate hardness with indentation depth:
H H0
2
¼ 1 þ hT
1 ; h
ð3Þ
where H0 is the hardness in the limit of infinite depth (bulk hardness) and h⁎ a characteristic length. By fitting Eq. (3) to the experimental hardness values (see the regression curve in Fig. 2), a value of h⁎ = 2451 nm could be obtained. The Nix and Gao model enables relating this parameter to the dislocation density by [5,15]: qs ¼
31 2f3
tan 2 h ; bh4
ð4Þ
where ρs is the density of dislocations statistically stored in the lattice (SSDs), θ the angle between the surface of the material and the surface of the indenter, b the Burgers vector of the dislocations and f a correction factor for the size of the plastic zone. In the present work θ = 20°, f = 1.9 [15], and b = 0.25 nm [16]. Introducing these values in Eq. (4) a dislocation density of 4.7 × 109 cm− 2 is obtained. 5. Discussion Fig. 1. a) Bright-field TEM micrograph of a region in Ni where several estimations of the dislocation density were made. The letter “X” marks the location where the sample thickness was measured. b) Magnification of the area in a) delimited by a white dashed square. c) Same micrograph as in b), but after analysis with a commercial image analysis software, where the white lines define the path considered for the determination of lp.
The dislocation density estimated on the basis of indentation hardness measurements with different loads is in remarkably good agreement with that measured by TEM, showing that this method is
3814
S. Graça et al. / Materials Letters 62 (2008) 3812–3814
reliable to estimate the dislocation density in metals exhibiting ISE. The reliability of the method depends on the accuracy of hardness measurements at submicrometric scales. This imposes certain precautions, especially for submicrometric depth indentations. In this case (ultramicro and nanoindentations), deviations from the ideal geometry of the indenter and the compliance of the hardness tester should be corrected by using adequate procedures [17]. Although the TEM method has the advantage of providing substructural information (dislocation configuration), the volume of sampled material is much smaller than in the indentation hardness method proposed here (the indentation diagonal length varies between 3 and 53 μm, while the diagonal of the rectangular areas observed by TEM is in the range 1 to 3 μm). As a result, the latter method is statistically more representative. Another advantage of the hardness measurement method is that it leads to ρ values that are representative of the mechanical behaviour of the bulk material. It is also simpler and faster than the TEM method. 6. Conclusions A dislocation density of approximately 5 × 109 cm− 2 was determined for laser clad Ni by two independent methods: one based on the direct observation of dislocations by TEM and the other based on indentation hardness measurements. Although the TEM method enables retrieving information on the dislocation structure, it is laborious, time consuming and it only allows observing very small
volumes of material. On the contrary, the indentation hardness method is simpler and faster and leads to representative values of ρ. Acknowledgements The authors would like to thank FCT for the financial support of this research (Project POCTI/CTM/59376/2004). S. Graça also acknowledges FCT for the PhD grant SFRH/BD/17758/2004. References [1] Pethica JB, Oliver WC. Mat Res Soc Symp Proc 1989;130:13–23. [2] Lorenz D, Zeckzer A, Hilpert U, Grau P, Johansen H, Leipner HS. Phys Rev B 2003;67 (172101):1–4. [3] Asenjo A, Jaafar M, Carrasco E, Rojo JM. Phys Rev B 2006;73(075431):1–7. [4] Fleck NA, Muller GM, Ashby MF, Hutchinson JW. Acta Metall Mater 1994;42:475–87. [5] Nix WD, Gao H. J Mech Phys Solids 1998;46:411–25. [6] Vilar R. J Laser Appl 1999;11:64–79. [7] Graça R, Colaço R, Vilar R. Surf Coat Technol 2007;202:538–48. [8] Bailey JE, Hirsch PB. Philos Mag 1960;5:485–97. [9] Allen SM. Philos Mag A 1981;43:325–35. [10] Williams DB, Carter CB. Transmission Electron Microscopy: II Diffraction. New York: Plenum Press; 1996. [11] Ma Q, Clarke DR. J Mater Res 1995;10:853–63. [12] Liu Y, Ngan AHW. Scr Mater 2001;44:237–41. [13] Swadener JG, Misra A, Hoagland RG, Nastasi M. Scr Mater 2002;47:343–8. [14] Bull SJ. Z Metallkd 2003;94:787–92. [15] Durst K, Backes B, Göken M. Scr Mater 2005;52:1093–7. [16] Rodney D, Martin G, Bréchet Y. Mater Sci Eng A 2001;309:198–202. [17] Oliver WC, Pharr GM. J Mater Res 1992;7:1564–83.
Contents lists available at ScienceDirect
Materials Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m a t l e t
Determination of dislocation density from hardness measurements in metals S. Graça, R. Colaço ⁎, P.A. Carvalho, R. Vilar Dep. Engenharia Materiais and Instituto de Ciência e Engenharia de Materiais e Superfícies, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
A R T I C L E
I N F O
Article history: Received 31 March 2008 Accepted 24 April 2008 Available online 30 April 2008 Keywords: Hardness Metals and alloys Dislocation density TEM
A B S T R A C T For the first time the Nix–Gao model for indentation size effect (ISE) is used to estimate the dislocation density in a metal. The estimate of dislocation density obtained by this method, using Ni as a case study, is compared with the values obtained from direct observation by transmission electron microscopy. It is shown that the estimate of dislocation density from indentation hardness measurements, adjusted by the Nix–Gao model, gives values consistent with those obtained by TEM, provided that the proper procedures to minimize errors are adopted. Although the direct observation of dislocations by TEM gives additional structural information, the indirect method to estimate dislocation density based on hardness measurements is more efficient, since the sample preparation method, measurement procedure and analysis of results are easier and faster. © 2008 Elsevier B.V. All rights reserved.
1. Introduction The plastic properties of metals, in particular their yield stress, hardness, strain hardening coefficient and toughness, depend critically on the dislocation density (ρ). This density is usually determined by transmission electron microscopy (TEM) direct observation, but this method is rather complex and time consuming. Moreover, the small size of the regions observed, associated to the heterogeneity of the dislocation distribution in plastically deformed metals, may lead to ρ values that are not representative and are inconsistent with the plastic behaviour of the material. When a material is plastically deformed the dislocation density increases, leading to strain hardening. If no dislocations (or only sessile dislocations) are present in the deformed region, new dislocations must be nucleated for plastic deformation to occur. This phenomenon has been observed when performing extremely small indentations (nanoindentations) in low defect density materials [1–3]. Moreover, if strain gradients are present (e.g. in indentations created by pyramidal or conical indenters), geometrically necessary dislocations (GNDs) must be created to accommodate them [4,5]. Since the strain gradient increases when the deformation scale decreases, the density of GNDs and, consequently, the hardness of the material, increase when the size of the deformed region decreases: this is known as the indentation size effect (ISE) [5]. In the present work a method based on the application of an ISE model [5] is used to determine the dislocation density of a metal from
hardness measurements. The values of ρ obtained by this method are compared with those obtained by TEM direct observation of the samples. It is shown that the values obtained by the indirect method based on hardness measurements are rather consistent with the values obtained from TEM observations. The advantages and drawbacks of both methods are discussed. 2. Experimental methods A nickel layer about 1 mm thick was deposited on an AISI 304 stainless steel substrate by powder injection laser cladding [6] using a 3 kW fast axial flow CO2 laser with a TEM01⁎ mode. A detailed description of the preparation method and processing parameters was presented in a previous paper [7]. Foils for TEM specimens were cut from this layer using a low speed abrasive saw. Both faces of the foil were ground with SiC abrasive papers to reduce their thickness down to ~ 60 μm. 3 mm discs were cut from this foil with a Q-switched Nd/ YAG laser. In order to avoid changing the dislocation density, thinning was carefully performed on this foil by ion milling with a 4 kV Ar+ beam and a 12° incidence angle until electron transparency was obtained. The surfaces used for indentation testing were ground with SiC papers and polished with a sequence of diamond particles suspensions. Vickers indentation tests were performed using loads in the range 0.01 to 2 N. 3. Determination of ρ by TEM
⁎ Corresponding author. Tel.: +351 21 8418125; fax: +351 21 8418120. E-mail address: [email protected] (R. Colaço). 0167-577X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2008.04.072
The projected length of dislocation lines (lp) in the TEM images was measured using a commercial image analysis software. Assuming that
S. Graça et al. / Materials Letters 62 (2008) 3812–3814
3813
the dislocations are randomly oriented, lp can be related to the length of dislocation lines (l) by [8]: l¼
4 lp : p
ð1Þ
The dislocation density is given by: q¼
l l ¼ : V At
ð2Þ
The volume of observed material (V) was determined by multiplying the corresponding area (A) by the sample thickness (t), Fig. 2. Fitting of Eq. (3) to the micro and ultramicrohardness data of laser clad Ni.
estimated by application of the graphical method proposed by Allen [9] to convergent beam electron diffraction data [10]. The thin foil thickness measured, at the location “X” shown in Fig. 1a, was consistently about 157 nm. Taking into consideration the thin foil preparation method a variation in the range 100 to 200 nm can be expected. The dislocation density was then estimated in eighteen regions of the Ni thin foil. Fig. 1b shows a bright-field TEM micrograph of one of the observed regions, where dislocations are clearly visible, and Fig. 1c shows the dislocation path defined with the image analysis software. To minimize systematic errors in the dislocation length values the thin foil was tilted in order to put into evidence the largest possible number of dislocations present. The dislocation density calculated using the lower and upper limits of the thickness range are (8.3 ± 6.4) × 109 cm− 2 and (4.1 ± 3.2) × 109 cm− 2, respectively. 4. Determination of ρ from hardness measurements The hardness values, retrieved from the indentation tests, increase with decreasing indentation depth as shown in Fig. 2. This behaviour was previously observed by other authors [7,11–14] and is known as indentation size effect. According to Nix and Gao [5], the ISE results from an increase in the density of the geometrically necessary dislocations required to accommodate the plastic deformation gradient around the indentation. The authors developed a model and deduced a simple expression to relate hardness with indentation depth:
H H0
2
¼ 1 þ hT
1 ; h
ð3Þ
where H0 is the hardness in the limit of infinite depth (bulk hardness) and h⁎ a characteristic length. By fitting Eq. (3) to the experimental hardness values (see the regression curve in Fig. 2), a value of h⁎ = 2451 nm could be obtained. The Nix and Gao model enables relating this parameter to the dislocation density by [5,15]: qs ¼
31 2f3
tan 2 h ; bh4
ð4Þ
where ρs is the density of dislocations statistically stored in the lattice (SSDs), θ the angle between the surface of the material and the surface of the indenter, b the Burgers vector of the dislocations and f a correction factor for the size of the plastic zone. In the present work θ = 20°, f = 1.9 [15], and b = 0.25 nm [16]. Introducing these values in Eq. (4) a dislocation density of 4.7 × 109 cm− 2 is obtained. 5. Discussion Fig. 1. a) Bright-field TEM micrograph of a region in Ni where several estimations of the dislocation density were made. The letter “X” marks the location where the sample thickness was measured. b) Magnification of the area in a) delimited by a white dashed square. c) Same micrograph as in b), but after analysis with a commercial image analysis software, where the white lines define the path considered for the determination of lp.
The dislocation density estimated on the basis of indentation hardness measurements with different loads is in remarkably good agreement with that measured by TEM, showing that this method is
3814
S. Graça et al. / Materials Letters 62 (2008) 3812–3814
reliable to estimate the dislocation density in metals exhibiting ISE. The reliability of the method depends on the accuracy of hardness measurements at submicrometric scales. This imposes certain precautions, especially for submicrometric depth indentations. In this case (ultramicro and nanoindentations), deviations from the ideal geometry of the indenter and the compliance of the hardness tester should be corrected by using adequate procedures [17]. Although the TEM method has the advantage of providing substructural information (dislocation configuration), the volume of sampled material is much smaller than in the indentation hardness method proposed here (the indentation diagonal length varies between 3 and 53 μm, while the diagonal of the rectangular areas observed by TEM is in the range 1 to 3 μm). As a result, the latter method is statistically more representative. Another advantage of the hardness measurement method is that it leads to ρ values that are representative of the mechanical behaviour of the bulk material. It is also simpler and faster than the TEM method. 6. Conclusions A dislocation density of approximately 5 × 109 cm− 2 was determined for laser clad Ni by two independent methods: one based on the direct observation of dislocations by TEM and the other based on indentation hardness measurements. Although the TEM method enables retrieving information on the dislocation structure, it is laborious, time consuming and it only allows observing very small
volumes of material. On the contrary, the indentation hardness method is simpler and faster and leads to representative values of ρ. Acknowledgements The authors would like to thank FCT for the financial support of this research (Project POCTI/CTM/59376/2004). S. Graça also acknowledges FCT for the PhD grant SFRH/BD/17758/2004. References [1] Pethica JB, Oliver WC. Mat Res Soc Symp Proc 1989;130:13–23. [2] Lorenz D, Zeckzer A, Hilpert U, Grau P, Johansen H, Leipner HS. Phys Rev B 2003;67 (172101):1–4. [3] Asenjo A, Jaafar M, Carrasco E, Rojo JM. Phys Rev B 2006;73(075431):1–7. [4] Fleck NA, Muller GM, Ashby MF, Hutchinson JW. Acta Metall Mater 1994;42:475–87. [5] Nix WD, Gao H. J Mech Phys Solids 1998;46:411–25. [6] Vilar R. J Laser Appl 1999;11:64–79. [7] Graça R, Colaço R, Vilar R. Surf Coat Technol 2007;202:538–48. [8] Bailey JE, Hirsch PB. Philos Mag 1960;5:485–97. [9] Allen SM. Philos Mag A 1981;43:325–35. [10] Williams DB, Carter CB. Transmission Electron Microscopy: II Diffraction. New York: Plenum Press; 1996. [11] Ma Q, Clarke DR. J Mater Res 1995;10:853–63. [12] Liu Y, Ngan AHW. Scr Mater 2001;44:237–41. [13] Swadener JG, Misra A, Hoagland RG, Nastasi M. Scr Mater 2002;47:343–8. [14] Bull SJ. Z Metallkd 2003;94:787–92. [15] Durst K, Backes B, Göken M. Scr Mater 2005;52:1093–7. [16] Rodney D, Martin G, Bréchet Y. Mater Sci Eng A 2001;309:198–202. [17] Oliver WC, Pharr GM. J Mater Res 1992;7:1564–83.