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COMMON FACTORS CONTROLLING GRAIN AND PHASE BOUNDARY ENERGY
W. Lojkowski, H. Fecht
To cite this version:
W. Lojkowski, H. Fecht. COMMON FACTORS CONTROLLING GRAIN AND PHASE BOUNDARY ENERGY. Journal de Physique Colloques, 1988, 49 (C5), pp.C5-551-C5-556.
�10.1051/jphyscol:1988567�. �jpa-00228064�
JOURNAL DE PHYSIQUE
Colloque C5, suppl6ment au nO1O, Tome 49, octobre 1988
COMMON FACTORS CONTROLLING GRAIN AND PHASE BOUNDARY ENERGY
W. LOJKOWSKI and H.J. FECHT*
UNIPRESS, High Pressure Research Centre, Sokolowska 27, PL-01-142 Warsaw, Poland
"university of Wisconsin, Madison, U.S.A.
Abstract - The low energy interfaces identified recently for the case of silver-silver grain boundaries as well as gold-LiF and silver-nickel interfaces by the method of sintering of spheres to flat substrate, were analysed in a unified way. The special properties of some of those interfaces can be understood by taking into account that the following factors lead to a decrease of the interfacial energy:
i. a high fraction of atoms of one crystal locked in valleys on the surface of the other crystal,
ii. the locked atoms form close packed rows thus decreasing the elastic strain energy,
iii. in sectors separated by the locked atomic rows both crystals surfaces are parallel to low energy planes,
iv. the energy necessary to "unlock1' an atom is higher than the energy of thermal vibrations.
Introduction
Understanding the relation between the interface energy and structure is a basic problem of "interface engineering". Due to difficulties in atomistic calculations of the phase and grain boundary energy, geometric criteria for low energy relative orientations of the adj oining crystals have been proposed. However, recent review (1 ) and experimental work < 2 , 3 ) has shown that the above criteria, based on consideration of a single factor controlling the interface energy, do not match with experimental results.
The purpose of the present paper is to show that instead of a single geometrical criterium a set of physical factors characterise low energy interfaces. Taking those factors into account leads to an extension of the lock-in model for such interfaces (4). This will be demonstrated by analysing additional data obtained by investigating low energy interfaces formed by sintering single crystalline 801d spheres to a single crystalline LiF flat substrate and comparing them with results of previous studies by that method (2-4).
tal M e t h a d s l
The details of the specimens preparation are given elsewhere (3-5).
Briefly, the method consist of depositing about 18' single crystalline gold spheres about 5pm in diameter on the polished flat surface of a LiF single crystal and sintering. During sintering, the spheres rotate in such a way that after sintering the interface at their neck has a minimum energy. The surface of the LiF crystal was parallel to the (110) plane and the temperature of sintering was 833K. Further, the crystallographic parameters of the low energy interfaces were determined from X-ray texture plots of the sintered specimens. For
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988567
'25-552 JOURNAL DE PHYSIQUE
that purpose a pole figure displaying the space orientations of the poles of t111) planes of the spheres was plotted (fig. 1). Moreover, misorientation angles, axes and planes parallel in the single crystalline substrate and in the spheres were calculated.
Those data are listed in Tab. 1, where in a unified form given are as well the low energy interfaces for the case of grain boundaries in noble metals ( 2 ) and phase boundaries between silver spheres and a nickel substrate (3).
Fig.1 Pole figure representing the orientation in space of the poles of <Ill> planes of gold spheres sintered to a LiF substrate of surface macroscopically parallel to the (118) plane. Each pole is represented by a marker whose size is roughly proportional to the number of spheres with the given orientation. Note that some spheres are oriented in such a way that three t111) planes are distributed along circles of poles (818) or <I@@) while the fourth (111) plane is parallel to either the (100) or <@la) plane of the substrate.
Table 1. List of low energy interfaces identified by the rotating sphere method.
For each interface given is the number of crystallographically equivalent orien- tations, misorientation angle and axis, strength of the corresponding X-ray maxima on the texture plots and planes as well as directions parallel in both crystals. The ratio of the intensities of maxima is s/m/w/vw=4/3/2/1 for phase boundaries and 10/5/2/1 for grain boundaries. d is the deviation from exact parallelity of planes. After ref.(2-4).
I orientation I misorientation I I I
I number/ I angle I axis l intensity I parallel planes 0 I s [I
<>
IJ Au -de to a LiF m e I
I 1/1 1 0.0 1 all Is- I a l l w e s I
1 2/8 1 54.7 1 1 0 1 I strong I {Ill> 1 1 C100) C 1101 1 1 11101 I
I 1-54.7 1 1 0 - 1 1 I I
I 1 54.7 1 0 1 1 1 I I
I 1-54.7 1 0 1-1 I I I
I 1-54.7 1 1 0 1 I I I
I 1 54.7 1 1 0-1 1 I I
I 1 5 4 . 7 1 0 1 - 1 I I I
I 1-54.7 1 0 1 1 I I I
1 3/2 1 35.3 1 1-1 0 1 medium 1 1221) 1 1 1100) I I 1 -35.3 1 1-1 0 1 1 1111) 1 1 1110) C 1101 I I I1101 I 1 4/4 1 60.0 1 1-1-1 1 medium 1 1211) 1 I <I@@) I
I 1 60.0 1 1-1 1 I 1 1111) 1 1 1111) I
I 1 60.0 1 1 1-1 I I (115) 1 1 Clll) C 1101 I I 1 1101 I
I 1 60.0 1 1 1 1 1 I I
1 5/2 1 45.0 I 1 0 0 1 weak 1 {100) 1 1 1100) I 1101 1 1 1 1001 I
I 1 45.8 1 0 1 0 1 I I
1 6/4 1 114.6 1 a p I( I weak 1 1111) 1 1 {I001 I 1101 1 1 C 1001 I I 1114.6 1
- - -
B a y I 1 a=0.8724. B=0.4516. x=0.1870 1e I
I 1/1 1 0.0 1 all s-I -s I
1 2/2 1 54.7 1 1-1 0 1 medium I {lll)l I {I001 <I,-I,@>! l<l,-1,0>l
I 1 -54.7 1 1-1 0 1 I I
1 3/2 1 35.3 1 1-1 0 1 medium 1 1221) 1 I (100) I I 1 -35.3 1 1-1 0 1 1 <111)11 1110) <I.-1.0>11<1.-1.0>1 1 4/4 1 60.0 1 1-1-1 1 medium 1 {211) 1 1 {I001 I I 1 60.0 1 1-1 1 I 1 I1111 1 1 {111) <I,-1,0>11 <I,-1,0>1
I 1 60.0 1 1 1 - 1 I I 1115) 1 1 1111) I
I I 68.0 1 1 1 1 1 I I
1 5/4 1 114.6 1 a B ' X I weak I 1111) 1 1 {100) C110111 C 1001 I
I 1114.6 1 - - - B a y I I I
J As and Cu -ered to Ag-mui Cu su-tes. respecblvelp I
I l e s I
I 1 1 4 1 1 1 -50.5 1 1 0 1 1 strong I 1111)=l I {100),d=4.2' I
I 1 50.5 1 1 0 - 1 1 I t 1101 1 1 C 1101 I
I 1 50.5 1 0 1 1 1 I I
I 1-50.5 1 0 1 - 1 I I I
I IIb/2/33c 1 -59.0 1 -1 1 0 1 weak 1 1111)=1 1 1100>,d=4.2" I
I 1 59.0 1 - 1 1 0 I I C 1101 1 1 I 1101 I
I IIIa/4/9 1 38.9 1 1-1 0 1 weak 1 {221)=1 1 11001, d=4.6' I
I 1 -38.9 I 1-1 0 1 ! 1111)=1 1 1110),d=4.6' I
I 1-38.9 I 1 0 1 I 1 11101 1 11 1101 I
I L 1 38.9 1 1 0 - 1 1 I I
I IIIb/2/9 1 38.9 1 -1 1 0 1 medium I I
I 1 38.9 1 - 1 1 0 1 I I
I IV/4/3 1 60.0 1 1-1-1 1 medium 1 {211)1 1 {100) I I 1 60.0 1 1-1 1 I I {111)II~111> ~110111C1101 I I 1 60.0 1 1 1-1 I strong 1 {115>1 I {Ill) I
J 1 68.0 1 1 1 1 1 I I
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Qiscussion
The results quoted in tab.1 concern interfaces of w e e bonding across the boundary as well as grain boundaries. The bonding energy for ionic crystals
-
noble metals and silver-nickel interfaces is of the order of 10 mJ/mZ <7,8). For silver grain boundaries it is of the order of 160 mJ/m2 (9). The above difference is reflected in the low energy orientations for each case. For interfaces of week bonding energy, close packed planes and directions across the interface are exactly parallel. At the same time, for grain boundaries, low energy orientations were found to be of coincidence type and a deviation from exact parallelity of close packed planes across the boundary of up to 5' was found (2). However, in each case at least one close packed direction is parallel in both adjoining crystals.The above results can be understood in terms of the criteria for low energy grain boundaries described in ref (2) and the lock-in model
( 4 ) for phase boundaries. In fact, according to the lock-in model, low
energy interfaces are characterised by a high fraction of atoms of one of the adjoining crystals locked in valleys on the surface of the other crystal, thus increasing the bonding energy across the inter- face. Further, if the locked atoms form close packed rows, the lock-in type of structure avoids high elastic strain energy at points of intersection of ledges on the surface of both crystals (fig.2, ref .2).
However, besides the density of locked atoms, the interface energy depends on the structure of its sectors situated between the locked atomic rows (2). In fact, interfaces may facet in order to increase the fraction of atoms situated on low energy planes (1).
Taking into account the above physical factors controlling the interface energy, as well as the differences in interfacial bonding for the cases investigated, the following interpretation of the data presented in tab. 1 can be presented.
For the case of grain boundaries there is a strengthening of the bonding of the grains if atoms of one grain lock in valleys on the other grain surface, increasing the average coordination number. Local relaxation of the boundary structure may further increase the bonding energy. Threfore, coincidence boundaries may have a lower energy than general ones. Further, some C 1103 tilt coincidence boundaries are expected to be of even lower energy due to the parallelity of close packed atomic rows. However, of particularly low energy are those
C 1001 tilt boundaries where the locked rows of atoms are separated by sectors of close packed (111) or (100) planes (2).
On the other hand, for phase boundaries of weak bonding, no sub- stantial bt~undary energy decrease is expected by increasing the fraction of locked atoms. However, some energy decrease is expected when the locked atoms of one crystal form rows parallel to valleys on the surface of the other crystal. Therefore low energy interfaces could be obtained when two crystals are joined along low energy surfaces anti close packed rows of surface atoms are parallel.
This type of interactions across the interface i s illustrated by fig.1. The pole figure shows that a fraction of the spheres is oriented in such a way that a (111) plane of the sphere is parallel to either the (1 0 0) or (0 1 0) planes of the plate, while the angle of rotation around the above poles is random. However, the orientation where C 1101 directions are parallel in both materials collected more spheres than the random ones. The above observation can be understood in terms of faceting of the substrate parallel to (100) planes so that a I100) plane of the substrate is parallel to a (111) plane of the spheres and an additional small energy decrease is achieved when C1101 directions in both crystals are parallel to the boundary.
It follows, that if atoms of one crystal are not able to lock in valleys on the surface of the other grain, no energy cusps can be
observed (4). In fact, defining only one direction parallel in both crystals does not define their relative orientation.
It is known that no locking of atoms is possible if the atomic sizes in both crystals are significantly different ( 4 ) . However, it was observed that some low energy orientations of spheres can not be detected if the sintering temperature is increased <2,4,7,18). For the case of gold crystallites sintered to a A1,0, crystal, this temperature is above 1833K (7). The question arises whether that effect is not caused by "unlocking" atoms due to thermal vibrations.
If the energy necessary to "unlock'@ an atom is E,-,, then at temperatures higher than T,,,-E,,/k, where k-Boltzmann constant, the fraction of locked atoms would decrease.
Let us estimate T, for phase boundaries between a ionic crystal and gold as well as for a grain boundary. We assume the valid at high temperatures Einstein approximation, i.e, that atomic vibrations are not mutually coupled. If a locked atom occupies an area A, the fraction of locked atoms at low temperatures is f, as well as all the bonding energy B, i s concentrated on the locked atoms, one obtains the relationship :
A can be approximated as Taking the value %-75 mJ/m, (11) and f-1/18 one obtains T,-580K. In the case of grain boundaries in silver E, is twice higher (9) and T, is close to the melting point, unless
f
<.
81. Therefore, in accordance with experimental results(12>,boundaries of long periodicity are not expected to be associated with energy cusps at high temperatures
It can be concluded that according to the above rough calculations, the disappearance of some energy cusps at high temperatures can be explained in terms of unlocking of locked atoms due to thermal vibrations.
Conclusions
The energy of grain and phase boundaries seems to be controlled by four major factors:
1.Strain energy resulting from crossing of close packed rows of atoms.
2.Locking energy equal to the grain boundary energy decrease due to the locking of atoms of one crystal in valleys on the surface of the other grain.
3.Energy of the sectors of the boundaries which are separated by the locked rows of atoms. This energy is related to the energy of each grain surface in those sectors.
4. If, due to thermal vibrations the fraction of locked atoms for a given low energy boundary becomes small, the associated with that boundary energy cusp disappears.
The relative weight of the above factors for each individual boundary depends on type of atomic bonding across the boundary, its energy, on lattice mismatch and temperature, as well as possible structual relaxations in the boundary, for example interface faceting.
Acknowlednments
The authors are very grateful to Prof.H.Gleiter and dr.R.Maurer for stimulating discussions. L carried this work partly thanks to a Humboldt scholarship and in part thanks to the grant CPBR 2.4 of the Office for Research and Development, Poland.
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Figure.2. Schematic representation of a grain boundary where non parallelity of close packed ledges on the surfaces of the grains causes elastic strains in points where the ledges are crossing. The boundary plane is parallel to the plane of the paper. In order to make the picture more transparent, only one close packed row of atoms of the upper grain and only the edges of close packed planes emerging from the lower grain are shown.
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