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The thermal expansion of grain boundaries
Actu mm//. Vol. 35, No. 8, pp. 2101-2104, 1987 Printed in Great Britain. All rights reserved
THE THERMAL
COOI-6160/87 $3.00 + 0.00
Copyright 0 1987Pergamon Journals Ltd
EXPANSION
OF GRAIN BOUNDARIES
H. J. KLAM, H. HAHN and H. GLEITER Universitat des Saarlandes, D-6600 Saarbriicken, F.R.G. (Received
25 November
1986)
Ahstraet-The thermal expansion of grain boundaries in copper was measured by a newly developed method comparing the thermal expansion of copper polycrystals (same chemical composition) with small (17 pm) and large (19 mm) grain sizes. From these m~su~ments, the thermal expansion coefficient of a grain boundary in copper was deduced to be 40 to 80 *10e6K-’ which is about 2.5 to 5 times the expansion coefficient of a copper crystal suggesting large anharmonic atomic vibrations in grain boundaries. The thermal expansion of grain boundaries (which was not measurable so far) may be used to test structural models of interfaces. R&sun&--Nous avons mesure la dilatation thermique des joints de grains dans le cuivre ri l’aide dune methode recente basQ sur la comparaison de la dilatation thermique de polycristaux de cuivre de mdme composition chimique, mais de tailles de grains petite (17 pm) ou grande (19 mm). Nous dtduisons de ces mesures que le coefficient de dilatation thermique dun joint de grains dans le cuivre varie entre 40 et 80. 10e6 K-‘, ce qui est 2,5 a 5 fois le coefficient de dilatation dun cristal de cuivre et qui suppose d’importantes vibrations atomiques non harmoniques dans les joints de grains. On peut utiiiser la dilatation thermique des joints de grains (qui n&it pas jusqu’a present mesurable) pour tester les modeles structuraux des interfaces. Z~~~f~~Die thermische A~dehn~g von Komgrenzen in Kupfer wurde mit einer neu entwickelten Methode gemessen; die Methode vergleicht die therm&he Ausdehnung von Kupferpoly kristallen (dieselbe chemische Zusammensetzung) mit kleinen (17 pm) und grol3en (19 pm) Komem. Aus diesen Messungen ergibt sich der therm&he Ausdehnungskoefficient einer Komgrenze in Kupfer zu 40 bis 80. lO-6 I(-‘, welches etwa 23 bis 5 ma1 gr%er als der Ausdehnungsk~ffizient eines Kupfer-kristalles ist. Dieser Refund deutet auf grol3e a~~oni~he Atom~h~n~ngen in der Komgrenze hin. Die thermische Ausdehnung von Komgrenzen (die bisher nicht gemessen werden konte) kann genutzt werden, urn Strukturmodelle von Grenzfliichen zu prilfen.
~RODU~ON Two types of approaches have been utilized to study the atomic structure. of grain or interphase boundaries. Direct methods (such as X-ray diffraction 111or high resolution electron microscopy [2]) and measurements of boundary properties (such as the free energy, the mobility or the diffusivity) which may be used to test existing structural models. However, the testing of structural models by means of property measurements is limited by the fact that many interfacial properties which would provide direct information about the structure of interfaces cannot yet be measured; for example, the boundary free volume, the elastic properties of interfaces or the thermal expansion of the inter-facial region. This paper reports a first attempt to measure the thermal expansion of grain boundaries in Cu. METHOD If the thermal expansion of a grain boundary in Cu differs from the thermal expansion of a Cu single crystal, the thermal expansion of a coarsegrained and fine-grained polycrystal is expected to be different. Hence, from measurements of the thermal expansion of polycrystals as a function of crystal size, the
thermal expansion of the grain ~unda~es (averaged over many boundaries) may be deduced. In order to investigate the thermal expansion of a coarse and a fine-grained polycrystal of the same material, the following method was applied (Fig. 1). A thin (thickness t = 25 .fim) polycrystalline foil (average crystal size 17 pm, foil length 1 = 235.8 mm) of 99.99% Cu was clamped at both ends to a coarse-grained crystal (crystal size 19 mm) block (thickness b = 5 mm) of C u with the same chemical compostion as the foil. Initially (at the temperature To = 20°C) the foil lies flat on the polished panar surface of the Cu block. When the temperature is raised from T, to T,, the Cu block expands and thus changes its length by Al, = cq .I(?“, - To) where u, is the thermal expansion of the Cu block. If the thermal expansion coefficient of the fine-grained Cu foil is az, the corresponding expansion of the foil amounts to Al* = a&T, - To). The differential thermal expansion A/,,, = Ai* - Al, can be accomodated by a compressive strain in the foil (and a small tensile strain in the block) or, by buckling of the foil, or by both. According to the linear theory of elasticity, buckling is the dominant deformation mode of a foil, if
2101
n2t2 tb 2 312
(1)
2102
KLAM er al.:
THERMAL EXPANSION OF GRAIN BOUNDARIES
Fig. 2. Shape of the specimens used. The meaning of the symbols A, B and C is the same as in Fig. 1. The points 1, 2,3,4 and 5 indicate the positions at which the buckling was measured interferometrically.
Fig. 1. Experimental arrangement (schematically) used to measure the thermal expansion of grain boundaries by the buckling of a thin foil. The foil is shown in the (initially) flat and in the buckied configuration. A = clamps at the end of the foil (B); C = coarse-grained copper block.
where t and 1 are explained in Fig. 1. tt, is the compression strain due to the differential thermal expansion of the foil and the block. Under the conditions used here (Fig. 1) the shape [w(x)] of the buckled foil follows from the linear theory of elasticity w(x) = B(cos /3x - 1) where B = -h/2 and @= 21[/1. The length of the bucked foil is obtained integration over w(x) L=
r ‘&V)z
Jo .
smZ ’ @x + 1 dx.
The differential thermal expansion relative to the block follows from AI=L-1.
(2)
respectively. A heating coil was attached to the outer surface of the furnace. The temperature of the specimen was controlled by 6 thermocouples. The entire system was placed on a mechanically damped heavy table to reduce vibrational motions of the foil. The buckling of the foil was studied by measuring w(x) (Fig. 1) during heating and cooling at five equally spaced points (1, 2, 3, 4, 5 in Fig. 2) along the foil. The measurements of w(x) were carried out by means of speckle interferometry [3,4], using a He-Ne-Laser of 5 mW. The interference fringes (Fig. 3) were generated by a two beam standard optical system (Fa. Spindler and Hoyer, Gottingen, F.R.G.). One of the laser beams was reflected at the polished surface of the block. The corresponding beam was directed towards the foil surface (points 1 . . .5). The motion of the interference fringes was recoreded by a video system as the typical time for heating and for cooling the speicmen was about 8 h. Slow heating and cooling was chosen to guarantee uniform heating and cooling of the block and the foil.
by -RESULTS (3)
(AZ) of the foil
(4)
The measurements were divided in two groups. In the first set of experiments, the reliability and resolution limit of the method was tested. Subsequently, the method was applied to study the differential thermal expansion of a fine-grained Cu foil and a coarsegrained Cu block.
Hence, if the buckling distance, h, is measured, Al can be computed from equations (3) and (4). The advantage of the method is that for an initially (T = To) flat foil, the buckling distance, h, is much larger than the differential expansion Al. In other words, the buckling is utilized to magnify the differential expansion effect. For example, if I = 235.8 mm, the foil buckles h = 10.1 pm (Fig. 1) if the thermal expansion is AI = 1.07 nm. In other words, the buckling magnifies the thermal expansion about lo4 times and therefore permits the measurement of differential thermal expansions less than 1 nm. EXPERIMENTAL The shape of the specimens used is shown in Fig. 2. The specimens were heated in a furnace consisting of two concentric metal cyclinders of Al and stainless steel with a wall thickness of about 80 and 20mm
Fig. 3. Speckle interference fringes originating by the interference of the light reflected from the surface of the copper foil and the copper block.
KLAM et a!.:
THERMAL
EXPANSION
OF GRAIN
BOUNDARIES
30 51
2103
I1200
29.5 28.5 -
1000 In k? z 800 ,”
27 5
$ 255-
Fi Q 24
//
Q SPECIMEN
A INTERFERENCE
5
- 600 E
TEMPERATURE FRINGES
z f: 23 5 p
::
22.5 21.5 J t
5 ii - LOO -
iz - 200 !?j
r=
i
2o5w 0
180
240 TIME
300
/
360
L20
/ ‘0 L80
(min)
Fig. 4. Variation of the speicmen temperature and the simultaneously recorded motion of the speckle interference fringes at point 3 (Fig. 2) of the thin, fine-grained foil clamped to the coarse-grained stainless steel block at both ends: 0 = specimen tem~rature; a = numbers of interference fringes.
The basic assumption of the method is that the shape of the buckled foil can be described by equation (2). In order to test this assumption, a Cu foil (t = 25 pm, average grain size 17 pm) was mounted on a stainless steel block (X5 Cr Ni 18.9). Due to the different thermal expansion coefficients of both materials (xc” = 16.6~10-6K-‘, ass= 15.6~10-bK~‘) an initially straight Cu foil is expected to buckle during heating. In order to check the shape of the buckled foil, w(x) was measured at the points 1, 2, 3, 4, and 5 after increasing the temperature of the system. w(x) was less than 0.25 of the width of the interference fringes at the points 1 and 5 at both ends of the foil. w(x) at 2 and 4 was equal. The maximum value w(x) was observed at point 3 (center of the foil). In order to test the reliability and the resolution limit of the method, the computed values of w(x) at point 3 were compared with the measured values w(x), when the temperature of the system was varied. An example is shown in Fig. 4. In this case the observed coupling between w(x) and T can be described in the entire temperature range (21.5-29.5”C) within a resolution of about 25% of the width of an interference fringe by the equations (2), (3) and (4), if it is assumed that the foil deviated initially (at 21.5”C by Ah, = 100.3 pm from an ideally straight line. The fit between the computed and measured w(x) is unique for each experiment. Due to the non-linear character of equation (4), there is only one Ahho,which allows to fit the measured w(x) by the equations (2), (3) and (4) in the entire temperature regime. The resolution limit of the method depends primarily on the width (IV) of the speckle interference fringes (Fig. 3). A movement of the fringes by 0.25 W can be measured directly on the screen of the t.v. monitor. 0.25 W correspond to a differential thermal expansion of the foil and the block of 0.15 nm
(resolution limit of the method). The reversibility of the thermal expansion was tested by cooling the specimens back to the initial temperature. The number of fringes counted during the “forward” and “backward” cycle was the same in all cases. In a second set of experiments, the differential thermal expansion of a Cu foil (thickness 25 pm, average crystal size 17 pm) and a Cu block (average crystal size 19 mm, identical chemical composition) was studied. The total solute concentration of the Cu used was found to be less than 0.01%. The heating (cooling) rate was 0.1 K/min. During each cycle, the specimen was held for 300 min at the highest and lowest temperature. The temperature intervals covered in the various cycles and the measured maximum buckling {point 3) of the Cu foil are listed in Table 1. If the buckling is converted [equations (2), (3), and (4) into a differential thermal expansion coefficient (AE) between the foil and the block, obtains Acr = ufol,- abloclt= 1.86. 10-9K-‘.
DISCUSSION The idea of this study was to investigate the thermal expansion of grain boundaries by measuring the difference of the thermal expansion coefficient between a fine and a coarse-grained polycrystal of Cu. If such a difference exists, it is expected to result in the buckling of a thin, fine-grained Cu foil attached at both ends to a coarse-grained Cu block, if the tem~rature is increased simultaneousIy. The advantage of the buckling is that it magnifies the differential thermal expansion of both materials by about four orders of magnitude or more. In order to test the suitability of this method, a thin Cu foil was attached at both ends to a stainless steel block. The
KLAM et al.: THERMAL EXPANSION OF GRAIN BOUNDARIES
2104
Table 1. Experimental results Temperature interval (“C)
Maximum buckling (h. lo- 3mm)
21-26 2631 31-36 3&41 41-36 3651 21-26&
0.8701 0.8701 0.9492 0.9492 0.8701 2.6894 0.8701
*Results obtained by using a second specimen with the same (99.99%) nominal purity, but not identical chemical composition.
different thermal expansion of both materials causes the Cu foil to buckle. The observed shape of the buckled foil and the amount of buckling was found to agree well with the predictions of the classical linear theory of elasticity. This conclusion agrees with equation (1) predicting that the strain above which the fine-grained Cu foil will deform by buckling is L = 3.7~10~*. This strain is achieved in a Cu-stainless steel specimen if AT > 3.7.10-* K. In summa~, these experiments suggest that the buckling of a thin foil is suitable to reveal a small difference ( 3 0.15 nm) between the thermal expansion of a fine-grained and a coarse-grained Cu crystal. According to the results listed in Table 1, the thermal expansion coefficient of a Cu foil (grain size 17~10-~ mm) deviates by Aa = 1.86. 10-9K-’ from the coe~cient of a Cu polycrystal with a grain size of 19 mm. The different thermal expansion of the foil and the block may be understood if it is assumed that Aa is primarily due to the thermal expansion of the grain boundaries in the fine-grained Cu foil. If this is so, a further reduction of the grain size increase Aa further. This was indeed observed. Recent measurements of the thermal expansion coefficient of Cu with an average grain size of 8 nm revealed (5) a thermal expansion coefficient of a = 31 ~10-6K-’ i.e. the deviation of a from the value of single crystalline Cu was Aa = 15 * 10m6K-l. The observed difference between the thermal expansion of the fine-grained and the coarse-grained copper is unlikely to result from different chemical compositions of the foil and the block. The foil and the block were produced from
the same material. Chemical analysis revealed a total solute content of < 0.01%. This conclusion also agrees with the observation that the same Aa was found for two Cu specimens of the same nominal purity (Table l), but with probably different types of solute atoms. If the enhanced thermal expansion of the poly crystal is interpreted in terms of the contribution from the grain boundaries, one is led to conclude. (i) The coefficient of the thermal expansion of a grain boundary (ab) is larger than the expansion of the crystalline state. Depending on the assumptions made about the thickness and the elastic properties of a boundary, one obtains (6) for the thermal expansion coefficient of a grain boundary a,, = 40 . . .80.10-6 K-l which is about 2.5.. .5 times the thermal expansion of the Cu lattice. (ii) An enhanced thermal expansion of the grain boundary region agrees with the recently observed enhanced Debye Wailer factor of grain boundaries deduced by Miissbauer spectroscopy 171, with the enhance specific heat of Cu polycrystals with a crystal size of 8 nm [S] and the theoretically predicted difference between the phonon spectrum of the lattice and grain boundaries [9]. Ackn~~le~gem~nts-~~~ financial support of the Alcoa Foundation, and the Deutsche Forschungsgemeinschaft are
gratefully acknowledged. REFER~C~ P. Lamarre, F. Schmiickle, M. D. Vandin and S. L. Sass. J. Phvs. CON.C4 46. 71 (1985).
1. K. R. Milkove,
2. Y. Ishida, Trans. J&m. i&t. Metals, &pi. i7, j3 (1986). 3. C. S. Viktram and K. Veolam, Optik 64, 171 (1983).
4. H. Goslowsky and I. Menzel, Optik 60, 149 (1982). 5. Xing Zhu, Ph. D. thesis, Univ. of Saarbriicken (1986). 6. A. A. Berlin, Principles ~~Polyrney ~~rnp~sites, p. 56. Springer, Berlin (1986). 7. U. Herr, J. Jing, R. Birringer, H. Gleiter and U. Gonser, Appl. Phys. Lett. In press. 8. R. Rupp and R. Birringer. To be published. 9. R. Yamamoto, M. Imafuku, Y. Sasajima and M. Doyama, Trans. Japan Inst. Metals, Suppl. 27, 329 (1986).
THE THERMAL
COOI-6160/87 $3.00 + 0.00
Copyright 0 1987Pergamon Journals Ltd
EXPANSION
OF GRAIN BOUNDARIES
H. J. KLAM, H. HAHN and H. GLEITER Universitat des Saarlandes, D-6600 Saarbriicken, F.R.G. (Received
25 November
1986)
Ahstraet-The thermal expansion of grain boundaries in copper was measured by a newly developed method comparing the thermal expansion of copper polycrystals (same chemical composition) with small (17 pm) and large (19 mm) grain sizes. From these m~su~ments, the thermal expansion coefficient of a grain boundary in copper was deduced to be 40 to 80 *10e6K-’ which is about 2.5 to 5 times the expansion coefficient of a copper crystal suggesting large anharmonic atomic vibrations in grain boundaries. The thermal expansion of grain boundaries (which was not measurable so far) may be used to test structural models of interfaces. R&sun&--Nous avons mesure la dilatation thermique des joints de grains dans le cuivre ri l’aide dune methode recente basQ sur la comparaison de la dilatation thermique de polycristaux de cuivre de mdme composition chimique, mais de tailles de grains petite (17 pm) ou grande (19 mm). Nous dtduisons de ces mesures que le coefficient de dilatation thermique dun joint de grains dans le cuivre varie entre 40 et 80. 10e6 K-‘, ce qui est 2,5 a 5 fois le coefficient de dilatation dun cristal de cuivre et qui suppose d’importantes vibrations atomiques non harmoniques dans les joints de grains. On peut utiiiser la dilatation thermique des joints de grains (qui n&it pas jusqu’a present mesurable) pour tester les modeles structuraux des interfaces. Z~~~f~~Die thermische A~dehn~g von Komgrenzen in Kupfer wurde mit einer neu entwickelten Methode gemessen; die Methode vergleicht die therm&he Ausdehnung von Kupferpoly kristallen (dieselbe chemische Zusammensetzung) mit kleinen (17 pm) und grol3en (19 pm) Komem. Aus diesen Messungen ergibt sich der therm&he Ausdehnungskoefficient einer Komgrenze in Kupfer zu 40 bis 80. lO-6 I(-‘, welches etwa 23 bis 5 ma1 gr%er als der Ausdehnungsk~ffizient eines Kupfer-kristalles ist. Dieser Refund deutet auf grol3e a~~oni~he Atom~h~n~ngen in der Komgrenze hin. Die thermische Ausdehnung von Komgrenzen (die bisher nicht gemessen werden konte) kann genutzt werden, urn Strukturmodelle von Grenzfliichen zu prilfen.
~RODU~ON Two types of approaches have been utilized to study the atomic structure. of grain or interphase boundaries. Direct methods (such as X-ray diffraction 111or high resolution electron microscopy [2]) and measurements of boundary properties (such as the free energy, the mobility or the diffusivity) which may be used to test existing structural models. However, the testing of structural models by means of property measurements is limited by the fact that many interfacial properties which would provide direct information about the structure of interfaces cannot yet be measured; for example, the boundary free volume, the elastic properties of interfaces or the thermal expansion of the inter-facial region. This paper reports a first attempt to measure the thermal expansion of grain boundaries in Cu. METHOD If the thermal expansion of a grain boundary in Cu differs from the thermal expansion of a Cu single crystal, the thermal expansion of a coarsegrained and fine-grained polycrystal is expected to be different. Hence, from measurements of the thermal expansion of polycrystals as a function of crystal size, the
thermal expansion of the grain ~unda~es (averaged over many boundaries) may be deduced. In order to investigate the thermal expansion of a coarse and a fine-grained polycrystal of the same material, the following method was applied (Fig. 1). A thin (thickness t = 25 .fim) polycrystalline foil (average crystal size 17 pm, foil length 1 = 235.8 mm) of 99.99% Cu was clamped at both ends to a coarse-grained crystal (crystal size 19 mm) block (thickness b = 5 mm) of C u with the same chemical compostion as the foil. Initially (at the temperature To = 20°C) the foil lies flat on the polished panar surface of the Cu block. When the temperature is raised from T, to T,, the Cu block expands and thus changes its length by Al, = cq .I(?“, - To) where u, is the thermal expansion of the Cu block. If the thermal expansion coefficient of the fine-grained Cu foil is az, the corresponding expansion of the foil amounts to Al* = a&T, - To). The differential thermal expansion A/,,, = Ai* - Al, can be accomodated by a compressive strain in the foil (and a small tensile strain in the block) or, by buckling of the foil, or by both. According to the linear theory of elasticity, buckling is the dominant deformation mode of a foil, if
2101
n2t2 tb 2 312
(1)
2102
KLAM er al.:
THERMAL EXPANSION OF GRAIN BOUNDARIES
Fig. 2. Shape of the specimens used. The meaning of the symbols A, B and C is the same as in Fig. 1. The points 1, 2,3,4 and 5 indicate the positions at which the buckling was measured interferometrically.
Fig. 1. Experimental arrangement (schematically) used to measure the thermal expansion of grain boundaries by the buckling of a thin foil. The foil is shown in the (initially) flat and in the buckied configuration. A = clamps at the end of the foil (B); C = coarse-grained copper block.
where t and 1 are explained in Fig. 1. tt, is the compression strain due to the differential thermal expansion of the foil and the block. Under the conditions used here (Fig. 1) the shape [w(x)] of the buckled foil follows from the linear theory of elasticity w(x) = B(cos /3x - 1) where B = -h/2 and @= 21[/1. The length of the bucked foil is obtained integration over w(x) L=
r ‘&V)z
Jo .
smZ ’ @x + 1 dx.
The differential thermal expansion relative to the block follows from AI=L-1.
(2)
respectively. A heating coil was attached to the outer surface of the furnace. The temperature of the specimen was controlled by 6 thermocouples. The entire system was placed on a mechanically damped heavy table to reduce vibrational motions of the foil. The buckling of the foil was studied by measuring w(x) (Fig. 1) during heating and cooling at five equally spaced points (1, 2, 3, 4, 5 in Fig. 2) along the foil. The measurements of w(x) were carried out by means of speckle interferometry [3,4], using a He-Ne-Laser of 5 mW. The interference fringes (Fig. 3) were generated by a two beam standard optical system (Fa. Spindler and Hoyer, Gottingen, F.R.G.). One of the laser beams was reflected at the polished surface of the block. The corresponding beam was directed towards the foil surface (points 1 . . .5). The motion of the interference fringes was recoreded by a video system as the typical time for heating and for cooling the speicmen was about 8 h. Slow heating and cooling was chosen to guarantee uniform heating and cooling of the block and the foil.
by -RESULTS (3)
(AZ) of the foil
(4)
The measurements were divided in two groups. In the first set of experiments, the reliability and resolution limit of the method was tested. Subsequently, the method was applied to study the differential thermal expansion of a fine-grained Cu foil and a coarsegrained Cu block.
Hence, if the buckling distance, h, is measured, Al can be computed from equations (3) and (4). The advantage of the method is that for an initially (T = To) flat foil, the buckling distance, h, is much larger than the differential expansion Al. In other words, the buckling is utilized to magnify the differential expansion effect. For example, if I = 235.8 mm, the foil buckles h = 10.1 pm (Fig. 1) if the thermal expansion is AI = 1.07 nm. In other words, the buckling magnifies the thermal expansion about lo4 times and therefore permits the measurement of differential thermal expansions less than 1 nm. EXPERIMENTAL The shape of the specimens used is shown in Fig. 2. The specimens were heated in a furnace consisting of two concentric metal cyclinders of Al and stainless steel with a wall thickness of about 80 and 20mm
Fig. 3. Speckle interference fringes originating by the interference of the light reflected from the surface of the copper foil and the copper block.
KLAM et a!.:
THERMAL
EXPANSION
OF GRAIN
BOUNDARIES
30 51
2103
I1200
29.5 28.5 -
1000 In k? z 800 ,”
27 5
$ 255-
Fi Q 24
//
Q SPECIMEN
A INTERFERENCE
5
- 600 E
TEMPERATURE FRINGES
z f: 23 5 p
::
22.5 21.5 J t
5 ii - LOO -
iz - 200 !?j
r=
i
2o5w 0
180
240 TIME
300
/
360
L20
/ ‘0 L80
(min)
Fig. 4. Variation of the speicmen temperature and the simultaneously recorded motion of the speckle interference fringes at point 3 (Fig. 2) of the thin, fine-grained foil clamped to the coarse-grained stainless steel block at both ends: 0 = specimen tem~rature; a = numbers of interference fringes.
The basic assumption of the method is that the shape of the buckled foil can be described by equation (2). In order to test this assumption, a Cu foil (t = 25 pm, average grain size 17 pm) was mounted on a stainless steel block (X5 Cr Ni 18.9). Due to the different thermal expansion coefficients of both materials (xc” = 16.6~10-6K-‘, ass= 15.6~10-bK~‘) an initially straight Cu foil is expected to buckle during heating. In order to check the shape of the buckled foil, w(x) was measured at the points 1, 2, 3, 4, and 5 after increasing the temperature of the system. w(x) was less than 0.25 of the width of the interference fringes at the points 1 and 5 at both ends of the foil. w(x) at 2 and 4 was equal. The maximum value w(x) was observed at point 3 (center of the foil). In order to test the reliability and the resolution limit of the method, the computed values of w(x) at point 3 were compared with the measured values w(x), when the temperature of the system was varied. An example is shown in Fig. 4. In this case the observed coupling between w(x) and T can be described in the entire temperature range (21.5-29.5”C) within a resolution of about 25% of the width of an interference fringe by the equations (2), (3) and (4), if it is assumed that the foil deviated initially (at 21.5”C by Ah, = 100.3 pm from an ideally straight line. The fit between the computed and measured w(x) is unique for each experiment. Due to the non-linear character of equation (4), there is only one Ahho,which allows to fit the measured w(x) by the equations (2), (3) and (4) in the entire temperature regime. The resolution limit of the method depends primarily on the width (IV) of the speckle interference fringes (Fig. 3). A movement of the fringes by 0.25 W can be measured directly on the screen of the t.v. monitor. 0.25 W correspond to a differential thermal expansion of the foil and the block of 0.15 nm
(resolution limit of the method). The reversibility of the thermal expansion was tested by cooling the specimens back to the initial temperature. The number of fringes counted during the “forward” and “backward” cycle was the same in all cases. In a second set of experiments, the differential thermal expansion of a Cu foil (thickness 25 pm, average crystal size 17 pm) and a Cu block (average crystal size 19 mm, identical chemical composition) was studied. The total solute concentration of the Cu used was found to be less than 0.01%. The heating (cooling) rate was 0.1 K/min. During each cycle, the specimen was held for 300 min at the highest and lowest temperature. The temperature intervals covered in the various cycles and the measured maximum buckling {point 3) of the Cu foil are listed in Table 1. If the buckling is converted [equations (2), (3), and (4) into a differential thermal expansion coefficient (AE) between the foil and the block, obtains Acr = ufol,- abloclt= 1.86. 10-9K-‘.
DISCUSSION The idea of this study was to investigate the thermal expansion of grain boundaries by measuring the difference of the thermal expansion coefficient between a fine and a coarse-grained polycrystal of Cu. If such a difference exists, it is expected to result in the buckling of a thin, fine-grained Cu foil attached at both ends to a coarse-grained Cu block, if the tem~rature is increased simultaneousIy. The advantage of the buckling is that it magnifies the differential thermal expansion of both materials by about four orders of magnitude or more. In order to test the suitability of this method, a thin Cu foil was attached at both ends to a stainless steel block. The
KLAM et al.: THERMAL EXPANSION OF GRAIN BOUNDARIES
2104
Table 1. Experimental results Temperature interval (“C)
Maximum buckling (h. lo- 3mm)
21-26 2631 31-36 3&41 41-36 3651 21-26&
0.8701 0.8701 0.9492 0.9492 0.8701 2.6894 0.8701
*Results obtained by using a second specimen with the same (99.99%) nominal purity, but not identical chemical composition.
different thermal expansion of both materials causes the Cu foil to buckle. The observed shape of the buckled foil and the amount of buckling was found to agree well with the predictions of the classical linear theory of elasticity. This conclusion agrees with equation (1) predicting that the strain above which the fine-grained Cu foil will deform by buckling is L = 3.7~10~*. This strain is achieved in a Cu-stainless steel specimen if AT > 3.7.10-* K. In summa~, these experiments suggest that the buckling of a thin foil is suitable to reveal a small difference ( 3 0.15 nm) between the thermal expansion of a fine-grained and a coarse-grained Cu crystal. According to the results listed in Table 1, the thermal expansion coefficient of a Cu foil (grain size 17~10-~ mm) deviates by Aa = 1.86. 10-9K-’ from the coe~cient of a Cu polycrystal with a grain size of 19 mm. The different thermal expansion of the foil and the block may be understood if it is assumed that Aa is primarily due to the thermal expansion of the grain boundaries in the fine-grained Cu foil. If this is so, a further reduction of the grain size increase Aa further. This was indeed observed. Recent measurements of the thermal expansion coefficient of Cu with an average grain size of 8 nm revealed (5) a thermal expansion coefficient of a = 31 ~10-6K-’ i.e. the deviation of a from the value of single crystalline Cu was Aa = 15 * 10m6K-l. The observed difference between the thermal expansion of the fine-grained and the coarse-grained copper is unlikely to result from different chemical compositions of the foil and the block. The foil and the block were produced from
the same material. Chemical analysis revealed a total solute content of < 0.01%. This conclusion also agrees with the observation that the same Aa was found for two Cu specimens of the same nominal purity (Table l), but with probably different types of solute atoms. If the enhanced thermal expansion of the poly crystal is interpreted in terms of the contribution from the grain boundaries, one is led to conclude. (i) The coefficient of the thermal expansion of a grain boundary (ab) is larger than the expansion of the crystalline state. Depending on the assumptions made about the thickness and the elastic properties of a boundary, one obtains (6) for the thermal expansion coefficient of a grain boundary a,, = 40 . . .80.10-6 K-l which is about 2.5.. .5 times the thermal expansion of the Cu lattice. (ii) An enhanced thermal expansion of the grain boundary region agrees with the recently observed enhanced Debye Wailer factor of grain boundaries deduced by Miissbauer spectroscopy 171, with the enhance specific heat of Cu polycrystals with a crystal size of 8 nm [S] and the theoretically predicted difference between the phonon spectrum of the lattice and grain boundaries [9]. Ackn~~le~gem~nts-~~~ financial support of the Alcoa Foundation, and the Deutsche Forschungsgemeinschaft are
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