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Temperature induced smoothing of initially fractal grain boundaries
ScriptaMaterialia,Vol. 34, No. 1, pp. ill-1 19. I996 ElsevierScience Ltd Copyright8 1995 Acta MetallurgicaInc. printedin the USA. All rightsreserved 1359-6462~96 $12.00 + .oO
Pergamon
0956-716X(95)00477-7
TEMPERATURE INDUCED SMOOTHING OF INITIALLY FRACTAL GRAIN BOUNDARIES P. Streitenberger, D. FGrster, G. Kolbe, P. Veit Otto-von-Guericke -Universitit Magdeburg, Institut fur Experimentelle Physik, Abteilung Materialphysik, PSF 4120, D-39016 Magdeburg, Germany (Received February 13,1995) (Revised July 9,199s)
Recently the effect of serrated or rugged grain boundaries on the mechanical properties of alloys and the numerical characterization of such a geometrically irregular microstructure by means of the concept of fiactal geometry [ 1,2] has attracted great attention. It has been reported (cf. [3 to 71) that the generation of serrated or rugged grain boundaries, e.g. by cold work or heat treatment, is one of the most effective methods to improve the high-temperature strength of alloys, especially the creep rupture properties. Rugged grain boundaries increase the grain boundary density without change in the average grain diameter [3] leading to an increased number of segregation sites for impurities and to different behaviour with respect to intergranular fracture [5 to 71. In many cases rugged grain boundaries exhibit an intrinsic hierarchical structure of length scales describable as a self-similar random pattern, which can quantitatively be characterized in terms of a fractal surface dimension D, with 2 < D s 3. On a micrograph of a plane section such a grain boundary acquires a l?actal dimension D, = D - 1 [ 1,2]. The grain boundary density is then related to the roughness of the boundary and depends on a microscopic characteristic length scale (resolution) and the fiactal dimension Ds of the boundary [3]. Several investigations have revealed that there is a numerical correlation between the value of Ds and strength properties such as fracture toughness and creep resistance [5 to 71. Almost no investigations are available with respect to the thermodynamic stability of fractal-like grain boundaries, especially with respect to the effect of the initially tiactal dimension of the boundaries on the coarsening kinetics during recovery. In [8] the change in the fiactal dimensionality of the grain structure in cold rolled pure Zn during recovery and recrystallization was investigated. In that investigation, however, the li-actal dimension was defined by application of the perimeter-area relation [l] without verification of the selfsimilarity of the grain structure. As shown in the previous paper [9], a fi-actal dimension determined in such a way indicates a correlation between grain shape and grain size rather than an intrinsic roughness of the grain boundaries. In the: present paper, for the first time, measurements of the change in the roughness of initially fiactal grain boundaries after annealing are presented. The experimental results are discussed on the basis of a coarsening model for self-similar interfaces, which predicts a dependency of the smoothing kinetics of the grain boundaries on their initially Iiactal dimension.
111
112
Vol. 34, No. 1
TEMPERATUREINDUCED SMOOTHING
Figure 1. SEM micrograph
of strongly serrated grain boundaries in pure Zn alk
deformation.
ExDerimentai Procedure and Results Cylindrical specimens of commercial 99.9% Zn were quasistatic deformed at room temperature by compression parallel to the cylinder axis to give a total reduction in thickness of 30 to 40%. Portions of the de.Sormed material were annealed at 423 K, 523 K and 623 K for a constant annealing time of 15 min. After the heat treatment the specimens were prepared for metallographic analysis by optical and scanning electron microscopy. The micrographs of the specimens were digitized to quantity the microstructure by means of image processing and analysis. Fig. 1 shows a scanning electron micrograph (at a magnification of 500x) of the typical microstructure of a sample atler &formation. The structure is characterized by rugged gram boundaries. The tractal dimension D, of the gram boundaries was determined by means of the yardstick method (Richardson plot) [9]. In order to verity the se~similarity of the gram boundary structure the measurements were carried out at different magnitications in the range of 100x to 1000x for the optical microscope and of 200x to 5000x for the scanning electron microscope. The measured fiactal dimension varies with magnification and position on the sample in the range D, = 1.04 to 1.2 1. The average value determined from nine randomly selected gram boundaries, where each of them was analyzed at different magnitications, results to D, = 1.14 f 0.04 (for more details see the previous paper [9]). Figs. 2, 3 and 4 show light micrographs of the typical grain boundary structure of the deformed and annealed specimens. After annealing a network of grams has been formed and the gram boundaries become smoother with higher temperature. In order to characterize the roughness of the grain boundaries quantitatively the roughness parameter
Vol. 34, No. 1
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Figure 2. Optical micrograph of a grain boundary after deformation and annealing at 423 K for 15 min.
has been defined, where L is the actual length of a grain boundary between two points and L, is the smooth or Euclidean distance between the same points (Fig. 5). More precisely, L, is defmed as the length of a polygon chosen in such a way that the global grain shape is approximately taken into account in the definition of the smooth part of the boundary length Using a light mi-pe, the roughness parameter of the grain boundaries
Figure 3. As for Fig. 2 but atIes annealing at 523 K.
114
TEMPERATURE INDUCED SMOOTHING
Vol. 34, No. 1
Figure 4. As for Fig. 2 but after annealing at 623 K.
as defined by equation (1) was measured for each of the deformed and annealed specimens at nine randomly selected positions and at four magnifications. Fig. 6 represents the isochronous plot of hr( p) versus l/T for the average roughness parameter (1) of each specimen determined at three different microscope magnikations. Each point represents an average value over the nine randomly selected gram boundaries at difkent places. The roughness parameter decreases with increasing temperature. Moreover, p decreases for each
Figure 5. Example of a digitized and processed grain boundary for the determination
of the roughness parameter (1).
TEMPERATURE
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115
INDUCED SMOOTHING
0.3
M8 0.2
0.1
0
0.002
I
CI.0104
0.003 l/T [l/K]
Figure 6. Isocbronc~us plot of In(p) versus l/T for the microscope magnification 200x (a), 500x (b) and 1000x (c). The till line represents the least+qare fit of eq. (8) (for both values of Q) to the last three points at magtScation b. The en-or bars indicate the scattering of the rougtums parameter in a sample.
specimen with decreasing resolution indicating a fractal-like character of the grain boundaries also during recovery. For all the three magnifications the initial state at room temperature and the two isochronous annealing states at 423 K and 523 K in Fig. 6 are close to straight lines with nearly the same slope. The deviation from the straight line at 523 K (i.e. at 0.0019 l/K) indicates the onset of the saturation of the smoothing process, i.e. atIer an annealing time of 15 min at 523 K the gram boundaries are almost smooth. For the three annealing states the grain diameters were measured by a commercial image processing and analysis system (SE) as the diameters of area equivalent circles. The average gram diameter R increases with increasing annealing temperature as shown by the isochronous plot of In(R) versus l/T in Fig. 7. From the slope of the straight line in Fig. 7 the activation energy for grain boundary migration may be estimated. As we will show in the next section, the slope of the straight lines for the roughness parameter in Fig. 6 is very sensitive to the initially fractal dimension of the grain boundaries. Model of the Smoothing
Kinetics
The rate of change of the global interface area S is described by the kinematic equation of stereology (e.g.
WJll>
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4.2
0.0016
0.0020
0.0024
l/T
[l/K]
Figure 7. Isocbronous plot of In(R) versus l/T. The fidl line represents the least-square fit of eq. (9) to the three annealing states. The diEere.nce due to the chosen n values is not visible iu the present graph.
where v, is the local velocity of an element of boundary along the normal direction of the surface element ~5’ and K the local mean curvature of the interface, the latter choosen to be positive if the normal points away from the nearest center of curvature and negative otherwise. The integral extends over the total grain boundary area S of the sample. Se is the contribution of the motion of edges to the rate of area change, which is neglected in the following analysis of the area smoothing kinetics. In the case of curvature driven coarsening v, is proportional to the gram boundary mobility I’ and the Iocal mean curvature K as the thaodynamic driving force: v,, = -rK [lo, 111. From (2), for the interface density S, = S/V, i.e. the grain boundary area per unit volume ( V is the constant volume of the sample), the equation of motion results db PC
dt
-
2rz*s,
(3)
where we have formally introduced the root-mean-square curvature R = [(l/S) J K * dS 1”. Normal grain growth in single-phase materials (cf. the review article [ 121) follows ti-om (3) as a special case of scale coarsening [ 10,l I], where the configuration of interfaces is characterized by a single tmiversal length scale I= l/S, In this case the mean curvature scales as R= l/l = S, and the equation of motion (3) takes the form dSJdt = -2llSG, with the solution S;Z - S$ a 4l7, or S, CCt-‘” (for t >> 1/(4X,*) with K0 as the initial root-meansquare curvature). The structural length scale I is proportional to the average grain size R and increases with time as I= R = t In (parabolic time law of growth [ 12]), while the number of grains decreases as N = M” 0~ t-*, the sbucture coarsens. The interface density can be obtained by the stereological relationship S, = (4/n)_& [ 131, where L,, = L/A is the boundary length per unit area, with L as the total length of the boundary lines measured on a plane section of area A. A grain boundary structure with rough boundaries has additionally to the average grain size or boundary distance R a structural length scale r
Vol. 34, No. 1
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117
-/ r . t2-
Figure 8. To illustratethe curvature driven smoothingkinetics of a grain boundary as a simpliiing random hctal.
with c as a constant. The meau curvature of the grain boundary on the scale of the boundary roughness is given by K= l/r. Since the boundary velocity is proportional to K, the rate of change of the boundary roughness at time t is controlled by the momentary characteristiclength r. The temporal smoothing of the boundary can then be understood as the time evolution of a simplifying random self-similar pattern, evolving step by step from a higher to the next lower generation of the random fiactal pattern characterized by the initially fractal dimension D, (Fig. 8). This corresponds to a model of interface dynamics first proposed by Toyoki and Honda [ 141far non-raudom liactals and further extended to random t?actal interfaces (and tested for disclinations in quenched liquid txystals) in [151.Here we adopt the model to describe the smoothing kinetics of tractal gram boundaries in deformed metallic polycrystals during recovery Express& by the roughness parameter p, eq. (l), the interface density S, = (4/x& and the mean curvature R= l/r= (UC>“R+) (from eq. (4)) take the form
kisaconstantdetiedbyk=(1/r,)p~ WA-’with r, and pOas the initial characteristic length and roughness pammeter, rapettively. For times where the curvature determined by the smooth part of the interface density is much smaherthan the curvature (5b) determined by the boundaty roughness, i.e. for l/R E LJA << l/r, the
Vol. 34, No. 1
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remain approximately constant. With this simplifying assumption, from termLJ4in(5a)canbeconsideredto (3), (5a) and (5b) we obtain the following equation of motion for the roughness parameter:
dp -= dt
- 2k21?p
(D,+lY(D,-1)
(6)
The solution of (6) is given by
P = PO
(Ds y.;
+
yDp=
PO[ (Ds
_:)r:)“”
(7)
[ leading to the time law of smoothing p = t’’ _%J”. From the definition of p and eq.(4) it follows that this time law for the roughness parameter corresponds to the parabolic time law r 0~t’” for the characteristic length scale or mean curvature radius r. We conclude the paper with a provisional first comparison of the above model with the experimental results, i.e. with the results of iscchronous anneahng presented in Fig.6. To this aim we have to consider eq.(7) for different temperatures T at the same time t. IfI& is independent of temperature, the temperature dependency of p is determined by the temperature dependency of the gram boundary mobility r, which is given by the well established Arrhenius-like relationship F = const x e Q’Rfl/T [ 161. Q is the activation energy for grain boundary migration and RG the molar gas constant. With this relationship for I?, from (7) we find for isochronous heat treatment
h(P)
In(T)
=
+ const.
(8)
A similar but empirical equation holds for the average grain diameter R, h(R)
= -
8 dG
1 T
-
1
h(T)
+ const,
n
where n is the inverse exponent in the empirical growth law R = (IV)“” for normal grain growth [ 12,161. For pure single-phase materials values between n= 2 (that is the theoretical value predicted by the above model) and n = 5 have been measured [12, 161. A least-square tit of (9) to the measured points in Fig. 7 yields the activation energies Q = 18.80 kJ/mol and Q = 40.75 kJ/mol for n = 2 and n = 5, respectively. With these limiting values for Q we have performed a least-square fit of eq.(8) to the measured roughening parameters represented in Fig. 6. Taking into account all the experimental points the least-square fit yields values for D, varying with increasing magnification from D, = 1.14 to 1.17 for Q = 18.80 kJ/mol and D, = 1.06 to 1.07 for Q = 40.75 kJ/mol. Since eq. (8) does not allow for the lower bound P = 1 (or equivalently In(p) = 0) at high temperatures, the points after onset of the saturation (i.e. the points at 0.0016 l/K in Fig. 6) should not be taken into account in the least-square fit. In this way we obtain as the average over the three magnitications in Fig. 6 D,= 1.17 and D, = 1.07, respectively, for the two bounds of the activation energy Q. Both of these values lie within the range of the initially fractal dimensions measured by geometric methods [9], i.e. Ds = 1.04 to 1.21. The results are not very difkrent horn those obtained by a linear least-square fit ignoring the term proportional to h-1(2’)in (8) and (9). Although the agreement between theory and experiment is plausible, it is not definitive because of the uncertainty in the assumption of the empirical grain growth exponent n and the resulting activation energy Q.
Vol. 34, No. 1
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Therefore, further experimental studies are in preparation with the aim to measure the time law of grain boundary smoothing of initially fractal grain boundaries directly by isothermal annealing. AcknowledPement
This work was supported by the Kultusministerium des Landes Sachsen-Anhalt. References
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York (1983). E. Hornbogen, Intern Mat. Rev. 3 277 (1989). E. Hombogen, 2. Metallkd a 622 (1987). Z. Huda and El. Ralph, Pratt. Met. 22,64 (1990). M. Tanaka and H. Iizuka, Z. Metallkd a 442 (1991). M. Tanaka, J. Mat. Sci. z 4717 (1992), a 5733 (1993). M. Tanaka., Z. Metallkd. @ 697 (1993). X. Wang, L. Dong and L Xiong, J. Phys.: Condens. Matter 2 3879 (1990). P. Streitenbeqer, D. F&ster, G. Kolbe and P. Veit, to be appear in Scripta metall. mater. R. T. De Ho@ Ata metall. z 43 (1984), Ata metall. mater. a 2349 (1991). W. W. Mull& and J. Vinals, Acta metall. z 991(1989). H. V. Atkinson, Acta metall. s 469 (1988). E. R Weibel, 5’tereological Methods, Vol. 2, p. 68, Academic Press, London, New York (1980). H. Toyoki and K Honda, Phyx. Lelters 111.367 (1987). H. Grihara and Y. Ishibasbi, J. Phys. Sot. Japan & 2340 (1987), and in Dynamics ofOrdering processeS in Condensed Matter (eds. S. Komura, H. Furukawa), p. 415, Plenum Press, New York (1988). F. Haessner and S. Hofinatm, in Recrystalliiation of Metallic Materials (ed. F. Haessner), p. 63, Dr. Riederer-Verlag GmbH, Stuttgart (1978).
Pergamon
0956-716X(95)00477-7
TEMPERATURE INDUCED SMOOTHING OF INITIALLY FRACTAL GRAIN BOUNDARIES P. Streitenberger, D. FGrster, G. Kolbe, P. Veit Otto-von-Guericke -Universitit Magdeburg, Institut fur Experimentelle Physik, Abteilung Materialphysik, PSF 4120, D-39016 Magdeburg, Germany (Received February 13,1995) (Revised July 9,199s)
Recently the effect of serrated or rugged grain boundaries on the mechanical properties of alloys and the numerical characterization of such a geometrically irregular microstructure by means of the concept of fiactal geometry [ 1,2] has attracted great attention. It has been reported (cf. [3 to 71) that the generation of serrated or rugged grain boundaries, e.g. by cold work or heat treatment, is one of the most effective methods to improve the high-temperature strength of alloys, especially the creep rupture properties. Rugged grain boundaries increase the grain boundary density without change in the average grain diameter [3] leading to an increased number of segregation sites for impurities and to different behaviour with respect to intergranular fracture [5 to 71. In many cases rugged grain boundaries exhibit an intrinsic hierarchical structure of length scales describable as a self-similar random pattern, which can quantitatively be characterized in terms of a fractal surface dimension D, with 2 < D s 3. On a micrograph of a plane section such a grain boundary acquires a l?actal dimension D, = D - 1 [ 1,2]. The grain boundary density is then related to the roughness of the boundary and depends on a microscopic characteristic length scale (resolution) and the fiactal dimension Ds of the boundary [3]. Several investigations have revealed that there is a numerical correlation between the value of Ds and strength properties such as fracture toughness and creep resistance [5 to 71. Almost no investigations are available with respect to the thermodynamic stability of fractal-like grain boundaries, especially with respect to the effect of the initially tiactal dimension of the boundaries on the coarsening kinetics during recovery. In [8] the change in the fiactal dimensionality of the grain structure in cold rolled pure Zn during recovery and recrystallization was investigated. In that investigation, however, the li-actal dimension was defined by application of the perimeter-area relation [l] without verification of the selfsimilarity of the grain structure. As shown in the previous paper [9], a fi-actal dimension determined in such a way indicates a correlation between grain shape and grain size rather than an intrinsic roughness of the grain boundaries. In the: present paper, for the first time, measurements of the change in the roughness of initially fiactal grain boundaries after annealing are presented. The experimental results are discussed on the basis of a coarsening model for self-similar interfaces, which predicts a dependency of the smoothing kinetics of the grain boundaries on their initially Iiactal dimension.
111
112
Vol. 34, No. 1
TEMPERATUREINDUCED SMOOTHING
Figure 1. SEM micrograph
of strongly serrated grain boundaries in pure Zn alk
deformation.
ExDerimentai Procedure and Results Cylindrical specimens of commercial 99.9% Zn were quasistatic deformed at room temperature by compression parallel to the cylinder axis to give a total reduction in thickness of 30 to 40%. Portions of the de.Sormed material were annealed at 423 K, 523 K and 623 K for a constant annealing time of 15 min. After the heat treatment the specimens were prepared for metallographic analysis by optical and scanning electron microscopy. The micrographs of the specimens were digitized to quantity the microstructure by means of image processing and analysis. Fig. 1 shows a scanning electron micrograph (at a magnification of 500x) of the typical microstructure of a sample atler &formation. The structure is characterized by rugged gram boundaries. The tractal dimension D, of the gram boundaries was determined by means of the yardstick method (Richardson plot) [9]. In order to verity the se~similarity of the gram boundary structure the measurements were carried out at different magnitications in the range of 100x to 1000x for the optical microscope and of 200x to 5000x for the scanning electron microscope. The measured fiactal dimension varies with magnification and position on the sample in the range D, = 1.04 to 1.2 1. The average value determined from nine randomly selected gram boundaries, where each of them was analyzed at different magnitications, results to D, = 1.14 f 0.04 (for more details see the previous paper [9]). Figs. 2, 3 and 4 show light micrographs of the typical grain boundary structure of the deformed and annealed specimens. After annealing a network of grams has been formed and the gram boundaries become smoother with higher temperature. In order to characterize the roughness of the grain boundaries quantitatively the roughness parameter
Vol. 34, No. 1
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INDUCED
SMOOTHING
113
Figure 2. Optical micrograph of a grain boundary after deformation and annealing at 423 K for 15 min.
has been defined, where L is the actual length of a grain boundary between two points and L, is the smooth or Euclidean distance between the same points (Fig. 5). More precisely, L, is defmed as the length of a polygon chosen in such a way that the global grain shape is approximately taken into account in the definition of the smooth part of the boundary length Using a light mi-pe, the roughness parameter of the grain boundaries
Figure 3. As for Fig. 2 but atIes annealing at 523 K.
114
TEMPERATURE INDUCED SMOOTHING
Vol. 34, No. 1
Figure 4. As for Fig. 2 but after annealing at 623 K.
as defined by equation (1) was measured for each of the deformed and annealed specimens at nine randomly selected positions and at four magnifications. Fig. 6 represents the isochronous plot of hr( p) versus l/T for the average roughness parameter (1) of each specimen determined at three different microscope magnikations. Each point represents an average value over the nine randomly selected gram boundaries at difkent places. The roughness parameter decreases with increasing temperature. Moreover, p decreases for each
Figure 5. Example of a digitized and processed grain boundary for the determination
of the roughness parameter (1).
TEMPERATURE
Vol. 34, No. 1
115
INDUCED SMOOTHING
0.3
M8 0.2
0.1
0
0.002
I
CI.0104
0.003 l/T [l/K]
Figure 6. Isocbronc~us plot of In(p) versus l/T for the microscope magnification 200x (a), 500x (b) and 1000x (c). The till line represents the least+qare fit of eq. (8) (for both values of Q) to the last three points at magtScation b. The en-or bars indicate the scattering of the rougtums parameter in a sample.
specimen with decreasing resolution indicating a fractal-like character of the grain boundaries also during recovery. For all the three magnifications the initial state at room temperature and the two isochronous annealing states at 423 K and 523 K in Fig. 6 are close to straight lines with nearly the same slope. The deviation from the straight line at 523 K (i.e. at 0.0019 l/K) indicates the onset of the saturation of the smoothing process, i.e. atIer an annealing time of 15 min at 523 K the gram boundaries are almost smooth. For the three annealing states the grain diameters were measured by a commercial image processing and analysis system (SE) as the diameters of area equivalent circles. The average gram diameter R increases with increasing annealing temperature as shown by the isochronous plot of In(R) versus l/T in Fig. 7. From the slope of the straight line in Fig. 7 the activation energy for grain boundary migration may be estimated. As we will show in the next section, the slope of the straight lines for the roughness parameter in Fig. 6 is very sensitive to the initially fractal dimension of the grain boundaries. Model of the Smoothing
Kinetics
The rate of change of the global interface area S is described by the kinematic equation of stereology (e.g.
WJll>
Vol. 34, No. 1
TEMPERATURE INDUCED SMOOTHING
116
4.2
0.0016
0.0020
0.0024
l/T
[l/K]
Figure 7. Isocbronous plot of In(R) versus l/T. The fidl line represents the least-square fit of eq. (9) to the three annealing states. The diEere.nce due to the chosen n values is not visible iu the present graph.
where v, is the local velocity of an element of boundary along the normal direction of the surface element ~5’ and K the local mean curvature of the interface, the latter choosen to be positive if the normal points away from the nearest center of curvature and negative otherwise. The integral extends over the total grain boundary area S of the sample. Se is the contribution of the motion of edges to the rate of area change, which is neglected in the following analysis of the area smoothing kinetics. In the case of curvature driven coarsening v, is proportional to the gram boundary mobility I’ and the Iocal mean curvature K as the thaodynamic driving force: v,, = -rK [lo, 111. From (2), for the interface density S, = S/V, i.e. the grain boundary area per unit volume ( V is the constant volume of the sample), the equation of motion results db PC
dt
-
2rz*s,
(3)
where we have formally introduced the root-mean-square curvature R = [(l/S) J K * dS 1”. Normal grain growth in single-phase materials (cf. the review article [ 121) follows ti-om (3) as a special case of scale coarsening [ 10,l I], where the configuration of interfaces is characterized by a single tmiversal length scale I= l/S, In this case the mean curvature scales as R= l/l = S, and the equation of motion (3) takes the form dSJdt = -2llSG, with the solution S;Z - S$ a 4l7, or S, CCt-‘” (for t >> 1/(4X,*) with K0 as the initial root-meansquare curvature). The structural length scale I is proportional to the average grain size R and increases with time as I= R = t In (parabolic time law of growth [ 12]), while the number of grains decreases as N = M” 0~ t-*, the sbucture coarsens. The interface density can be obtained by the stereological relationship S, = (4/n)_& [ 131, where L,, = L/A is the boundary length per unit area, with L as the total length of the boundary lines measured on a plane section of area A. A grain boundary structure with rough boundaries has additionally to the average grain size or boundary distance R a structural length scale r
Vol. 34, No. 1
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117
-/ r . t2-
Figure 8. To illustratethe curvature driven smoothingkinetics of a grain boundary as a simpliiing random hctal.
with c as a constant. The meau curvature of the grain boundary on the scale of the boundary roughness is given by K= l/r. Since the boundary velocity is proportional to K, the rate of change of the boundary roughness at time t is controlled by the momentary characteristiclength r. The temporal smoothing of the boundary can then be understood as the time evolution of a simplifying random self-similar pattern, evolving step by step from a higher to the next lower generation of the random fiactal pattern characterized by the initially fractal dimension D, (Fig. 8). This corresponds to a model of interface dynamics first proposed by Toyoki and Honda [ 141far non-raudom liactals and further extended to random t?actal interfaces (and tested for disclinations in quenched liquid txystals) in [151.Here we adopt the model to describe the smoothing kinetics of tractal gram boundaries in deformed metallic polycrystals during recovery Express& by the roughness parameter p, eq. (l), the interface density S, = (4/x& and the mean curvature R= l/r= (UC>“R+) (from eq. (4)) take the form
kisaconstantdetiedbyk=(1/r,)p~ WA-’with r, and pOas the initial characteristic length and roughness pammeter, rapettively. For times where the curvature determined by the smooth part of the interface density is much smaherthan the curvature (5b) determined by the boundaty roughness, i.e. for l/R E LJA << l/r, the
Vol. 34, No. 1
TEMPERATURE INDUCED SMOOTHING
118
remain approximately constant. With this simplifying assumption, from termLJ4in(5a)canbeconsideredto (3), (5a) and (5b) we obtain the following equation of motion for the roughness parameter:
dp -= dt
- 2k21?p
(D,+lY(D,-1)
(6)
The solution of (6) is given by
P = PO
(Ds y.;
+
yDp=
PO[ (Ds
_:)r:)“”
(7)
[ leading to the time law of smoothing p = t’’ _%J”. From the definition of p and eq.(4) it follows that this time law for the roughness parameter corresponds to the parabolic time law r 0~t’” for the characteristic length scale or mean curvature radius r. We conclude the paper with a provisional first comparison of the above model with the experimental results, i.e. with the results of iscchronous anneahng presented in Fig.6. To this aim we have to consider eq.(7) for different temperatures T at the same time t. IfI& is independent of temperature, the temperature dependency of p is determined by the temperature dependency of the gram boundary mobility r, which is given by the well established Arrhenius-like relationship F = const x e Q’Rfl/T [ 161. Q is the activation energy for grain boundary migration and RG the molar gas constant. With this relationship for I?, from (7) we find for isochronous heat treatment
h(P)
In(T)
=
+ const.
(8)
A similar but empirical equation holds for the average grain diameter R, h(R)
= -
8 dG
1 T
-
1
h(T)
+ const,
n
where n is the inverse exponent in the empirical growth law R = (IV)“” for normal grain growth [ 12,161. For pure single-phase materials values between n= 2 (that is the theoretical value predicted by the above model) and n = 5 have been measured [12, 161. A least-square tit of (9) to the measured points in Fig. 7 yields the activation energies Q = 18.80 kJ/mol and Q = 40.75 kJ/mol for n = 2 and n = 5, respectively. With these limiting values for Q we have performed a least-square fit of eq.(8) to the measured roughening parameters represented in Fig. 6. Taking into account all the experimental points the least-square fit yields values for D, varying with increasing magnification from D, = 1.14 to 1.17 for Q = 18.80 kJ/mol and D, = 1.06 to 1.07 for Q = 40.75 kJ/mol. Since eq. (8) does not allow for the lower bound P = 1 (or equivalently In(p) = 0) at high temperatures, the points after onset of the saturation (i.e. the points at 0.0016 l/K in Fig. 6) should not be taken into account in the least-square fit. In this way we obtain as the average over the three magnitications in Fig. 6 D,= 1.17 and D, = 1.07, respectively, for the two bounds of the activation energy Q. Both of these values lie within the range of the initially fractal dimensions measured by geometric methods [9], i.e. Ds = 1.04 to 1.21. The results are not very difkrent horn those obtained by a linear least-square fit ignoring the term proportional to h-1(2’)in (8) and (9). Although the agreement between theory and experiment is plausible, it is not definitive because of the uncertainty in the assumption of the empirical grain growth exponent n and the resulting activation energy Q.
Vol. 34, No. 1
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Therefore, further experimental studies are in preparation with the aim to measure the time law of grain boundary smoothing of initially fractal grain boundaries directly by isothermal annealing. AcknowledPement
This work was supported by the Kultusministerium des Landes Sachsen-Anhalt. References
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
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