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Effect of faceting on twin grain boundary motion in zinc
Materials Letters 64 (2010) 105–107
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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
Effect of faceting on twin grain boundary motion in zinc Vera G. Sursaeva Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Russia
a r t i c l e
i n f o
Article history: Received 10 July 2009 Accepted 11 September 2009 Available online 6 October 2009 Keywords: Twin grain boundaries Faceting Migration
a b s t r a c t ̅ flat single crystals has been Faceting and migration of incoherent twin grain boundary in Zn [112 0] investigated. The stationary shape of the slowly migrating incoherent twin grain boundary of the twin plate tip was studied and migration velocity was measured in situ in the range from 473 K to 692 K using polarized light. Below 623 K the incoherent twin grain boundary represents the facet which forms at a 43° angle to the coherent twin grain boundary. Above 623 K the incoherent twin grain boundary represents the facet, whose position changes from the initial to 75° to the coherent twin grain boundary as temperature increases. Below 623 K the incoherent twin, grain boundary moves at very low experimentally determined activation enthalpy 19.3 kJ/mol of facet migration. Above 623 K the experimentally determined activation enthalpy for the facet migration is 154.4 kJ/mol, which is higher than the activation enthalpy of grain boundary diffusion in Zn. These results clearly indicate that there is a strong effect of the grain boundary shape on the migration velocity of the twin tip. We suppose that there is a grain boundary structural phase transition in this system: facet with high coincidence site lattice transforms to facet with low coincidence site lattice with disordered structure at 623 K. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Boundaries which have markedly different values of properties than an average boundary have come to be known as ‘special’. For many years the coincidence site lattice (CSL) model, which describes the grain boundary (GB) in terms of misorientation between neighboring grains, was a cornerstone of GB research [1]. It was originally thought that any CSL boundary with a low Σ value (where Σ is the reciprocal density of coinciding sites) had special properties. However Wolf [2] indicated that a low-Σ CSL was a necessary but not sufficient criterion for specialness. For instance, so-called ‘coherent twin grain boundary (CTGB)’ which is a Σ3 boundary on {111} always possesses special properties, whereas so-called ‘incoherent twin grain boundary (ICTGB)’, which is a Σ3 boundary assumed to be on {211}, may be characterized by ‘less special’ behaviour. CTGBs are immobile, whereas ICTGBs migrate readily, thus indicating a very different response between the two boundary types. Importance of knowing the grain boundary plane indices in addition to the misorentation is emphasized in [3]. In our previous work [4] the compensation effect, namely, the linear relation between activation enthalpy and pre-exponential factor was observed. The compensation temperature Tc was 623 K. As a rule, the compensation temperature Tc is the temperature of first order phase transition. Phase transition should be reflected on a kinetic property of ICTGB—grain boundary mobility.
E-mail address: [email protected]. 0167-577X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2009.09.024
The purpose of presented work is in investigation of ICTGB motion for clarification of a question: what phase transition takes place at 623 K.
2. Experimental ̅ flat single crystals were grown from Zn of 99.999 wt.% Zn [1120] using the modified Bridgman technique [1,5,6]. Stress was applied to introduce deformation twins into the single crystal, and we could see ̅ a bicrystal shaped as a halfloop with two on the surface (112 0) parallel CTGBs and one ICTGB. The produced twin plates were ̅ axes in both perpendicular to the surface of the sample. The [112 0] grains were also normal to the surface of the sample. In this case twin GB can be described in terms of axis and angle as GB formed by 86° ̅ axis. Twin GB is a special [112 0] ̅ tilt GB with rotation around [112 0] Σ = 15. Due to its optical anisotropy, zinc allows one to study the shape of a GB with help of polarized light. The stationary shape of slowly migrating tip of the twin plate was studied in situ in a hot stage of an optical microscope in the temperature interval 473 K ÷ 692 K. In the same temperature interval the GB migration velocity was measured and, subsequently, GB mobility was calculated. This method of studying the GB migration was originally developed in Refs. [1,6]. It has been shown previously that driving forces originated from the GB phase transitions (such as faceting) are usually rather low to cause a shape change of stationary migrating GB in a reasonable time [7–9], so that the steady-state shape characteristic for a given temperature can be readily established.
106
V.G. Sursaeva / Materials Letters 64 (2010) 105–107
Fig. 1. a) The GB shape of the twin plate tip at temperature below 623 K. b) The GB shape of the twin plate tip at temperature above 623 K.
3. Results and discussion The micrographs in Fig. 1 show the shape of the GB at the tip of the twin plates in Zn at temperatures below and above 623 K. The shape of the twin tip differs drastically from the rounded shape of the GB halfloops in Zn bicrystals containing non-special GB [1,6,10,11]. Below 623 K the twin tip contains flat facet lying at an angle 43° to CTGB boundary (Fig. 1a). Above 623 K the angle between CTGB and the facets depends on temperature (Fig. 2) and the twin tip contains flat facet lying at angle 55 ÷ 75° to CTGB (Fig. 1b). Close to the Zn melting temperature Tm the edges of the facet at the intersection with CTGBs become rounded and the GB roughening transition proceeds [12]. Since the ratio between c and a lattice spacing is irrational in Zn, the exact coincidence site lattice (CSL) exists for Zn lattice only for rotations around c axis. In all other cases the so-called constrained CSL (CCSL) is used for description of coincidence in Zn GB. Schemes in Fig. 3 demonstrate the CCSL for angle misorientation 86° around axes ̅ CTGB coincides with (11 02 ̅ )̅ planes in both lattices. Typical [112 0]. time dependence of ICTGB displacement is linear. ICTGB mobility has been calculated from least square linear fit of this dependence under conditions of steady-state GB migration with the constant shape. CTGB is stable at all temperatures. For isotropic GB the velocity of its normal motion, V, is given by [1] V =M
γb Rc
ð1Þ
where M, γb and Rc are the GB mobility, energy and radius of curvature, respectively. For a flat GB facet the velocity of normal motion is determined by the weighted mean curvature, (WMC): V = M⋅WMC:
ð2Þ
Fig. 2. The temperature dependence of the facet position relatively to CTGB position.
– Fig. 3. So-called coincidence site lattice for 86° misorientation around axis [112 0]. Facet forms at 43° to CTGB below 623 K.
The latter is defined as a negative of total interfacial energy change of the system after infinitesimal displacement of the given facet while all other interfaces are immobile, divided by the volume swept by the facet [12]. Facet mobility is considered as phenomenological length-independent quantity that depends on temperature according to Arrhenius law. For the twin considered in this work (Fig. 1) the facet velocity V in the framework of WMC concept for the facets can be easily calculated: aV =
2γ0 M sinα
ð3Þ
where γ0 is the interfacial energy of the CTGB, M is mobility of facet, respectively, a—facet length, α—angle between facet and CTGB. In Fig. 4 temperature dependence of the grain boundary mobility of the twin plate tip in the Arrhenius coordinates is shown. The solid lines represent Arrhenius-type fit to the low- and high-temperature data and characterize the intrinsic mobility of the facets. The temperature dependence is strongly non-linear indicating that the twin tip migration cannot be described by a unique value of migration activation enthalpy in the case in which the steady-state GB shape changes at temperature 623 K. Below 623 K ICTGBs manifest motion with very low value (19.3 kJ/mol) of the experimentally determined activation enthalpy for facet migration. Above 623 K the experimentally determined activation enthalpy for facet migration is 154.4 kJ/mol, and it is above the activation enthalpy of grain boundary diffusion in Zn. These results clearly indicate that there is a strong effect of the GB shape on the
Fig. 4. Temperature dependence of the facet mobility.
V.G. Sursaeva / Materials Letters 64 (2010) 105–107
migration velocity of the twin tip. We suppose that there is GB structural phase transition in this system: facet with high coincidence site lattice transforms to facet with low coincidence site lattice with disordered structure at 623 K. The material selected for the study was hexagonal zinc, which has differential thermal expansion between the a- and c-axes, and while thermoelastic stresses are different across grain boundaries, the force appears and facet rotates. The differential grain boundary stress is simply proportional to the difference in the expansion coefficients, Δa , and the temperature difference (ΔТ). The thermal expansion coefficient for one grain can be represented as an ellipse. The thermal expansion coefficient for two grains can be represented as two imposed ellipses. For GB in our sample there is a situation when the caxes of the two grains (the acute angle between c-axes is θ = 86°) lie at angles ϕ and γ to the facet, two ellipses representing the thermal expansion in the boundary plane are of different sizes. The difference in expansion coefficients in an arbitrary direction ω is [13] 2
2
2
2
Δa = ðac −aa Þ−½cos ϕ cos ω−aa cos γ cos ðω + θÞ:
ð4Þ
107
We considered the hypothesis: the existence of facet below 623 K is governed by interfacial energy, while the position of facet above 623 K is determined by the influence of thermoelastic stress due to anisotropy of thermal expansion. 4. Conclusions The temperature dependence of incoherent twin grain boundary (ICTGB) mobility and incoherent twin grain boundary shape at the tip of the twin in Zn were studied simultaneously. Two branches are seen in the Arrhenius plot of ICTGB mobility with different activation enthalpies above and below Tc = 623 K. This result is explained in terms of GB structural phase transition from the facet with a high density of coincident lattice sites to the facet with the disordered structure. Acknowledgements Presented investigations were supported by the Russian Foundation for Basic Research (contract 09-02-91339). References
The condition |a| = 0 corresponds to the minimum separation of the ellipses. If the facet lies in direction with |a| = 0 the thermoelastic stress is balanced. The torque will be zero and grain boundary will occupy stable position in bicrystal. In Fig. 2 the temperature dependence of the facet position relatively to position of CTGB is shown. In the experimental runs facet rotates because the condition |a| = 0 depends on temperature. The angle between facet position and CTGB increases gradually as temperature arises. By cooling of the sample, the facet rotates backwards and the angle between the facet and the coherent twin grain boundary decreases. This temperature dependence is reversible because crystal anisotropy is temperature reversible. The facet position change stimulates the structural change, and, consequently, mobility changes.
[1] Gottstein G, Shvindlerman LS. Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications. Boca Raton: CRC Press; 1999. [2] Wolf D, Merkle KL. Materials Interfaces: Atomic-Level Structure and Properties. London: Chapman&Hall; 1992. [3] Randle V. Acta Mater 2004;52:4067. [4] Tuflin AU, Sursaeva VG, Czubayko U. Defect Diffus Forum 2001;194–199:1253. [5] Straumal BB, Sursaeva VG, Shvindlerman LS. Phys Met Metallogr 1980;49:102. [6] Czubayko U, Sursaeva VG, Gottstein G, Shvindlerman LS. Acta Mater 1998;46:5863. [7] Straumal BB, Sursaeva VG, Polyakov SA. Interface Sci 2001;9:275. [8] Maksimova EL, Shvindlerman LS, Straumal BB. Acta Metal 1988;36:1573. [9] Maksimova EL, Shvindlerman LS, Straumal BB. Acta Metal 1989;37:2855. [10] Sursaeva VG, Czubayko U, Gottstein G, Shvindlerman LS. Mater Sci Forum 1999;294–296: 517. [11] Tuflin AU, Sursaeva VG, Czubayko U. Defect Diffus Forum 2001;194–199:1253. [12] Straumal BB, Rabkin E, Sursaeva VG, Gornakova AS. Zt Metallkd 2005;96:161–6. [13] Davidge RW. Acta Metall 1981;29:1695.
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
Effect of faceting on twin grain boundary motion in zinc Vera G. Sursaeva Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Russia
a r t i c l e
i n f o
Article history: Received 10 July 2009 Accepted 11 September 2009 Available online 6 October 2009 Keywords: Twin grain boundaries Faceting Migration
a b s t r a c t ̅ flat single crystals has been Faceting and migration of incoherent twin grain boundary in Zn [112 0] investigated. The stationary shape of the slowly migrating incoherent twin grain boundary of the twin plate tip was studied and migration velocity was measured in situ in the range from 473 K to 692 K using polarized light. Below 623 K the incoherent twin grain boundary represents the facet which forms at a 43° angle to the coherent twin grain boundary. Above 623 K the incoherent twin grain boundary represents the facet, whose position changes from the initial to 75° to the coherent twin grain boundary as temperature increases. Below 623 K the incoherent twin, grain boundary moves at very low experimentally determined activation enthalpy 19.3 kJ/mol of facet migration. Above 623 K the experimentally determined activation enthalpy for the facet migration is 154.4 kJ/mol, which is higher than the activation enthalpy of grain boundary diffusion in Zn. These results clearly indicate that there is a strong effect of the grain boundary shape on the migration velocity of the twin tip. We suppose that there is a grain boundary structural phase transition in this system: facet with high coincidence site lattice transforms to facet with low coincidence site lattice with disordered structure at 623 K. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Boundaries which have markedly different values of properties than an average boundary have come to be known as ‘special’. For many years the coincidence site lattice (CSL) model, which describes the grain boundary (GB) in terms of misorientation between neighboring grains, was a cornerstone of GB research [1]. It was originally thought that any CSL boundary with a low Σ value (where Σ is the reciprocal density of coinciding sites) had special properties. However Wolf [2] indicated that a low-Σ CSL was a necessary but not sufficient criterion for specialness. For instance, so-called ‘coherent twin grain boundary (CTGB)’ which is a Σ3 boundary on {111} always possesses special properties, whereas so-called ‘incoherent twin grain boundary (ICTGB)’, which is a Σ3 boundary assumed to be on {211}, may be characterized by ‘less special’ behaviour. CTGBs are immobile, whereas ICTGBs migrate readily, thus indicating a very different response between the two boundary types. Importance of knowing the grain boundary plane indices in addition to the misorentation is emphasized in [3]. In our previous work [4] the compensation effect, namely, the linear relation between activation enthalpy and pre-exponential factor was observed. The compensation temperature Tc was 623 K. As a rule, the compensation temperature Tc is the temperature of first order phase transition. Phase transition should be reflected on a kinetic property of ICTGB—grain boundary mobility.
E-mail address: [email protected]. 0167-577X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.matlet.2009.09.024
The purpose of presented work is in investigation of ICTGB motion for clarification of a question: what phase transition takes place at 623 K.
2. Experimental ̅ flat single crystals were grown from Zn of 99.999 wt.% Zn [1120] using the modified Bridgman technique [1,5,6]. Stress was applied to introduce deformation twins into the single crystal, and we could see ̅ a bicrystal shaped as a halfloop with two on the surface (112 0) parallel CTGBs and one ICTGB. The produced twin plates were ̅ axes in both perpendicular to the surface of the sample. The [112 0] grains were also normal to the surface of the sample. In this case twin GB can be described in terms of axis and angle as GB formed by 86° ̅ axis. Twin GB is a special [112 0] ̅ tilt GB with rotation around [112 0] Σ = 15. Due to its optical anisotropy, zinc allows one to study the shape of a GB with help of polarized light. The stationary shape of slowly migrating tip of the twin plate was studied in situ in a hot stage of an optical microscope in the temperature interval 473 K ÷ 692 K. In the same temperature interval the GB migration velocity was measured and, subsequently, GB mobility was calculated. This method of studying the GB migration was originally developed in Refs. [1,6]. It has been shown previously that driving forces originated from the GB phase transitions (such as faceting) are usually rather low to cause a shape change of stationary migrating GB in a reasonable time [7–9], so that the steady-state shape characteristic for a given temperature can be readily established.
106
V.G. Sursaeva / Materials Letters 64 (2010) 105–107
Fig. 1. a) The GB shape of the twin plate tip at temperature below 623 K. b) The GB shape of the twin plate tip at temperature above 623 K.
3. Results and discussion The micrographs in Fig. 1 show the shape of the GB at the tip of the twin plates in Zn at temperatures below and above 623 K. The shape of the twin tip differs drastically from the rounded shape of the GB halfloops in Zn bicrystals containing non-special GB [1,6,10,11]. Below 623 K the twin tip contains flat facet lying at an angle 43° to CTGB boundary (Fig. 1a). Above 623 K the angle between CTGB and the facets depends on temperature (Fig. 2) and the twin tip contains flat facet lying at angle 55 ÷ 75° to CTGB (Fig. 1b). Close to the Zn melting temperature Tm the edges of the facet at the intersection with CTGBs become rounded and the GB roughening transition proceeds [12]. Since the ratio between c and a lattice spacing is irrational in Zn, the exact coincidence site lattice (CSL) exists for Zn lattice only for rotations around c axis. In all other cases the so-called constrained CSL (CCSL) is used for description of coincidence in Zn GB. Schemes in Fig. 3 demonstrate the CCSL for angle misorientation 86° around axes ̅ CTGB coincides with (11 02 ̅ )̅ planes in both lattices. Typical [112 0]. time dependence of ICTGB displacement is linear. ICTGB mobility has been calculated from least square linear fit of this dependence under conditions of steady-state GB migration with the constant shape. CTGB is stable at all temperatures. For isotropic GB the velocity of its normal motion, V, is given by [1] V =M
γb Rc
ð1Þ
where M, γb and Rc are the GB mobility, energy and radius of curvature, respectively. For a flat GB facet the velocity of normal motion is determined by the weighted mean curvature, (WMC): V = M⋅WMC:
ð2Þ
Fig. 2. The temperature dependence of the facet position relatively to CTGB position.
– Fig. 3. So-called coincidence site lattice for 86° misorientation around axis [112 0]. Facet forms at 43° to CTGB below 623 K.
The latter is defined as a negative of total interfacial energy change of the system after infinitesimal displacement of the given facet while all other interfaces are immobile, divided by the volume swept by the facet [12]. Facet mobility is considered as phenomenological length-independent quantity that depends on temperature according to Arrhenius law. For the twin considered in this work (Fig. 1) the facet velocity V in the framework of WMC concept for the facets can be easily calculated: aV =
2γ0 M sinα
ð3Þ
where γ0 is the interfacial energy of the CTGB, M is mobility of facet, respectively, a—facet length, α—angle between facet and CTGB. In Fig. 4 temperature dependence of the grain boundary mobility of the twin plate tip in the Arrhenius coordinates is shown. The solid lines represent Arrhenius-type fit to the low- and high-temperature data and characterize the intrinsic mobility of the facets. The temperature dependence is strongly non-linear indicating that the twin tip migration cannot be described by a unique value of migration activation enthalpy in the case in which the steady-state GB shape changes at temperature 623 K. Below 623 K ICTGBs manifest motion with very low value (19.3 kJ/mol) of the experimentally determined activation enthalpy for facet migration. Above 623 K the experimentally determined activation enthalpy for facet migration is 154.4 kJ/mol, and it is above the activation enthalpy of grain boundary diffusion in Zn. These results clearly indicate that there is a strong effect of the GB shape on the
Fig. 4. Temperature dependence of the facet mobility.
V.G. Sursaeva / Materials Letters 64 (2010) 105–107
migration velocity of the twin tip. We suppose that there is GB structural phase transition in this system: facet with high coincidence site lattice transforms to facet with low coincidence site lattice with disordered structure at 623 K. The material selected for the study was hexagonal zinc, which has differential thermal expansion between the a- and c-axes, and while thermoelastic stresses are different across grain boundaries, the force appears and facet rotates. The differential grain boundary stress is simply proportional to the difference in the expansion coefficients, Δa , and the temperature difference (ΔТ). The thermal expansion coefficient for one grain can be represented as an ellipse. The thermal expansion coefficient for two grains can be represented as two imposed ellipses. For GB in our sample there is a situation when the caxes of the two grains (the acute angle between c-axes is θ = 86°) lie at angles ϕ and γ to the facet, two ellipses representing the thermal expansion in the boundary plane are of different sizes. The difference in expansion coefficients in an arbitrary direction ω is [13] 2
2
2
2
Δa = ðac −aa Þ−½cos ϕ cos ω−aa cos γ cos ðω + θÞ:
ð4Þ
107
We considered the hypothesis: the existence of facet below 623 K is governed by interfacial energy, while the position of facet above 623 K is determined by the influence of thermoelastic stress due to anisotropy of thermal expansion. 4. Conclusions The temperature dependence of incoherent twin grain boundary (ICTGB) mobility and incoherent twin grain boundary shape at the tip of the twin in Zn were studied simultaneously. Two branches are seen in the Arrhenius plot of ICTGB mobility with different activation enthalpies above and below Tc = 623 K. This result is explained in terms of GB structural phase transition from the facet with a high density of coincident lattice sites to the facet with the disordered structure. Acknowledgements Presented investigations were supported by the Russian Foundation for Basic Research (contract 09-02-91339). References
The condition |a| = 0 corresponds to the minimum separation of the ellipses. If the facet lies in direction with |a| = 0 the thermoelastic stress is balanced. The torque will be zero and grain boundary will occupy stable position in bicrystal. In Fig. 2 the temperature dependence of the facet position relatively to position of CTGB is shown. In the experimental runs facet rotates because the condition |a| = 0 depends on temperature. The angle between facet position and CTGB increases gradually as temperature arises. By cooling of the sample, the facet rotates backwards and the angle between the facet and the coherent twin grain boundary decreases. This temperature dependence is reversible because crystal anisotropy is temperature reversible. The facet position change stimulates the structural change, and, consequently, mobility changes.
[1] Gottstein G, Shvindlerman LS. Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications. Boca Raton: CRC Press; 1999. [2] Wolf D, Merkle KL. Materials Interfaces: Atomic-Level Structure and Properties. London: Chapman&Hall; 1992. [3] Randle V. Acta Mater 2004;52:4067. [4] Tuflin AU, Sursaeva VG, Czubayko U. Defect Diffus Forum 2001;194–199:1253. [5] Straumal BB, Sursaeva VG, Shvindlerman LS. Phys Met Metallogr 1980;49:102. [6] Czubayko U, Sursaeva VG, Gottstein G, Shvindlerman LS. Acta Mater 1998;46:5863. [7] Straumal BB, Sursaeva VG, Polyakov SA. Interface Sci 2001;9:275. [8] Maksimova EL, Shvindlerman LS, Straumal BB. Acta Metal 1988;36:1573. [9] Maksimova EL, Shvindlerman LS, Straumal BB. Acta Metal 1989;37:2855. [10] Sursaeva VG, Czubayko U, Gottstein G, Shvindlerman LS. Mater Sci Forum 1999;294–296: 517. [11] Tuflin AU, Sursaeva VG, Czubayko U. Defect Diffus Forum 2001;194–199:1253. [12] Straumal BB, Rabkin E, Sursaeva VG, Gornakova AS. Zt Metallkd 2005;96:161–6. [13] Davidge RW. Acta Metall 1981;29:1695.