- Email: [email protected]
Forms of grain boundaries as may appear on primary recrystallization of deformed metals
Physica
IX,
no 7
Juli
FORMS OF GRAIN BOUNDARIES AS MAY ON PRIMARY RECRYSTALLIZATION OF DEFORMED METALS for Physical
APPEAR
J. SANDEE
IR. Laboratory
1942
Chemistry
of the Technical
University,
Delft
Zusammenfassung -4lle moglichen Formen von Korngrenzen, wie sie bei ,,isotroper” primlrer Rekristallisation von Metalen (konstante und gleiche lineare Wachstumsgeschwindi’gkeit in allen Richtungen innerhalb desselben Kristalliten) auftreten konnen, werden besprochen. Anschliessend wird ein kurzer cberblick gegeben iiber die anderen Bedingungen, welche gleichfalls die Formen der Korner beeinflussen konnen.
9 1. Primary recrystallization proceeds in such a way that relativelyhealthylattice areas, which are surrounded by a strong deformation-inhomogeneity act, on heating, as nuclei from where new crystallites start growing l). While developing, these meet each other and new grain boundaries are formed. Now it was noted by B u r g e r s 2), that in recrystallized aluminium plates some crystallites were entirely surrounded by larger ones, though the original material consisted of very fine grains (compared with the new crystallites) with randomorientations (fig. 1, LI)~). As was argued already in 2), this phenomenon can only be explained by assuming that the larger crystallites had a greater growing velocity and grew around the smaller ones. When a crystal develops in a plate of the above mentioned, finely grained, random oriented material it assumes a circular shape (fig. 2) “). This means that the growing velocity is equd in all directions. -
741 -
742
J. SANDEE
$ 2. In order to derive some forms suppositions must bc made :
of grain
boundaries
certain
1. A crystallite has an equal iinear growing velocity in all directiom. 2. This velocity is generally differelzt for each crystallite, but remains constant. 3. The crystallites generally start growing each at different moments. These conditions of recrystallization will be called ,,isotrofiic”.
Fiu 0’
1. Various
Fig.
Every
forms
2.
The
grain boundary
of isotropic aluminium
grain boundaries plates.
recrystallization in all directions.
is a boundary
velocity
between
in rccrystallizccl
is eclual
f~o grains.
To lx
FORMS
OF GRAIN
743
BOUNDARIES
able to analyse the shape of this boundary,,we must know about these crystallites : 1. the location of their nzlclei (conceived ns points), 2. the ratio of their growing velocities, 3. the dimensions of the ,,first” crystallite started to grow.
at the moment the ,,second”
The grain boundary surface shows revolution symmetry with the connecting line of the respective nucleus locations as its axis, so that is sufficient to study only the line of intersection of the boundary surface with a plane through this axis. S 3. ’ In the simplest case both crystallites start growing simultaneously with equal velocity. Their boundary becomes the locus of
Fig. 3. -4 and B started growing in the same moment with equal velocity.
those points that are equidistant from the nucleus points A and B (fig. 3), i.e. the bisecting perpendicular of AB. This is the only possibility for the occurrence of entirely straight grain boundaries. This case was treated before by B e 1 a i e w “) and B e n edicks5). If, however, one of the two, say A, started first and had developed into a circle of radius s by the time that the other one, B, started growing, one obtains another grain boundary (fig. 4a en b). This boundary is the locus of those points of which the distance to A is longer than their distance to B by a stretch s, i.e. one branch of a hyperbola with foci at A and B. The asymptotes pass through the middle of the line AB; they are perpendicular to the tangents through B to a circle of radius s around A.
744
J. SANDEE
This type was also discussed by B e n e d i c k s “). Now if the growing velocities of the crystallites A and B are no longer equal (but still in a constant ratio), one obtains the forms of grain boundaries sketched in fig. 5a and b. The analysis of these curves of 4th order is simple but will not be given here.
Fig. 4aand
b.
2 and
B
are growing with started first.
equal
velocity,
but A
.
Some attention may be paid, however, to the transition between the forms marked in fig. 5 by ,,v,., < vB” and ,,z)A
(1)
Let the first point of encounter Here ro = $$$
(2)
(+ = VA/% < 1). of the crystallites be 0 (fig. 7).
and
ro.= 6;
(3).
The grain boundary develops between the two circles and has, therefore, a vertical tangent in 0. The contact with this tangent is of a higher order if a point P, lying at an altitude Ah
FORMS
OF GRAIN
Fig.Sa and b. A and B are growing
745
BOUNDARIES
with different .
velocities.
A started
first.
-P
Fig. 6. Bipolar
coordinates.
above 0, is, for small values of A h, situated above 0. In this case in n APO: (r, + A+ = Ah2 + r; or 2r0 Av = Ah2 Likewise, in n BPO: 2p0Ap = Ah2
perpendicularly
(4) (5)
746
J. SANDEE
Hence : Differentiation
Y, Ar = ~~ Op. of (1) gives AY -= AP
Fig. 7. Contact
and from
of higher
-
(6)
P
order
with tangent
(6) it follows that
in 0.
,
F” = p z or, according to (2) and (3) : d-s -=
Pd+ s
fi,sothatp=
1-i.
The form ,,v4 v,& > 1 -s/d. An example of the latter remarkable curves is given in fig, 1, B. In fig. 5a and b only the beginnings of the grain boundaries were deliberately drawn. Merely the part between the first point of encounter of both crystallites on the line AB, and the contact R with the tangent from the nucleus point of the fastest growing crystallite has a physical meaning. For at P (fig. 8) the straight line BP intersects the crystallite A. But the crystallite B cannot have grown through A, but must have grown around it. For that reason the grain boundary follows after R another course. This has the essential property that: shortest connection A -+ P -
s = p x shortest connection B -+ P
(7)
where p = vA/vg -C 1. As the shortest connection B + P is composed of the constant straight intersection BR and the arc RP, differentiation of (7) gives: AAP
A arc RP =
P-
FORMS
OF GRAIN
747
BOUNDARIES
Then in fig. 9: AAP --=fi, ‘OS o: = A arc RP
Fig. 8. The grain boundary
has proceeded
past R.
Fig. 9. The radii vectores from A meet the new curve in every point at the same angle.
748
J. SANDEE
so that the grain boundary meets all radii vectores from A under the constant angle : a = arc cos p (8) This is a well-known property of the logarithmic spirals with A as their (asymptotic) centre. s of (7) is not used in the deduction, which therefore holds also if B started first. The spirals pass continuously into the curves of fig. 5a and b. For those curves at R: ++$-
= p (fig. 10)
Fig. 10. The grain boundary curves pass continuously into logarithmic spirals.
and, therefore,
L ARB = arc cos p = u. The spirals intersect the produced part of AB and meet their opposites at the same time (fig. 11). Beyond this point they lose their physical meaning. Examples of boundaries of this shape are given in fig. 1, A. As the produced part of AB may be regarded as a radius vector from B (or A respectively), the spirals meet the axis of symmetry at this terminal point at an angle u = arc cos P. $4. Fig. 1 shows that the above-discussed various types of isotropic boundaries may occur in reality, but it seemed impossible to
FORMS
OF GRAIN
749
BOUNDARIES
produce them at will. WC resorted therefore to an analogous phenomenon in the growth of micro-organisms in and on solid media.
-3
Fig.
11.
The point-angle difference in
2a growing
depends only velocities.
on
the
Part of microbiological technique consists in the ,,streaking” of a highly diluted suspension of micro-organisms on to the plane surface of a sterile jelly (e.g. a solidified solution of agar-agar or gelatin with nutrients). Wherever a single cell was deposited upon the ,,plate”, it
n
b
Fig. 12n and 0. Illustration of grain boundaries with moulcls. Figures a and b correspond to the actual grain bountlaries shown in figure 1 at A and B resp.
multiplies into a ,,colony”. Now a plate which was streaked with a suspension of the milk mould Oospora lactis Fres. showed conspicuously straight boundaries between those otherwise circular colo-
750
J. SANDEE
nies that had encountered each other. This could be explained by assuming that the colonies originated from ,,nuclei” (presumably single cells) that had started developing into colonies at the same moment (with a certain lag after the streaking) with the same constant, linear velocity. As two very similar types of this mould were available, of which the ,,white” type grew faster than the ,,brown” one, the various forms of isotropic grain boundaries could be illustrated in this way. For that purpose tiny groups of mould cells were transferred by means of an inoculation needle to separate spots of sterile malt extract agar-agar plates which were kept at 22°C. At this temperature the linear growing velocity was some mm per day, so that the development of the ,,grain boundaries” could be fixed photographically in various succeeding phases. In fig. 12A and B some examples of such photographs are given, corresponding to the grain boundaries of fig. 1, A and B. S;5. If the crystallites of a recrystallized metal plate are ,,isotroPit. ‘, i.e. if the linear growing velocity of each crystallite was conH
Fig.
13.
How
to find
s if A started
first.
stant and equal in all directions, an attempt may be made to determine the nucleus locations etc. of the crystallites present. The locations of the nuclei can usually be traced by taking into consideration that the connecting line of the nucleus locations of two
FORMS
OF
GRAIN
751
BOUNDARIES
adjacent crystallites is an axis of symmetry ofctheir boundary. A grain boundary which is straight over a fairly large length means that the incubation periods and the growing velocities of the adjacent crystallites were equal, so that the nuclei are located symmetrically with regard to the boundary.
Fig.
14.
How
to find
s if
B started
first.
In some cases where the boundaries were particularly smooth *) the following methods could be used. A large part of the far-cleveloped grain boundaries follows logarithmic spirals, in any case the parts between the sharp point and the broadest part HH’ (fig. 13 and 14). Some of these spirals with various o! = arc cos p were constructed (fig. 15), in such a way that:all
Fig.
15.
Logarithmic
spirals.
had the same maximal width. If the grain boundaries to be examined are magnified or reduced (e.g. optically )till they have the same maximal width as the given spirals, the best fitting spiral can be laid along them. In this way the nucleus location of *) This is the case if the grain size before recrystallization is very much smaller than aftcrwnrds. If the initial grain size is relatively coarse, the boundaries of the new crystallites arc often rathercrenated (cf. Hdb. d. Metallphys., lot. cit. fig. 80, p. 182).
752
J. SANDEE
the enclosed crystallite is found immediately. In order to detect the nucleus location of the surrounding crystallite, the point has to be traced, where the real grain boundary begins to deviate from the spiral. Through this point R the tangent is drawn; this intersects the axis in the required point. Now if R is found to the right of A (fig. 13), A started growing first and the radius of the circle which enveloped A the moment when B started too can be found by dropping from B a perpendicular BC onto AR. As cos L ARB = 9, RC = fi x BR; and hence AC = s. If, hawever, R is found to the left of A (fig. 14)) B started first. A perpendicular must now be erected in A to AR, which meets BR in D. In this case cos L ARB = #, so that AR = p x DR, and hence BD = s. If a crystallite is surrounded entirely by another one, the ratio of the growing velocities of the two may be determined simply by considering that the cosine of the half point-angle is equal to this ratio (cf. equation (8)). In this way velocity ratios were found in aluminium plates up to 0,75 : 1. f 6. From the differences in growing velocity, which were shown by adjacent crystallites, it follows once again, that the scattering in direct determinations of growing velocities “) need not be due only to experimental inaccuracies, but may bring out a real difference in the properties of crystals obtained by recrystallization. A preliminary investigation of the crystallographic orientation of those crystallites that differed much in growing velocity, did not show any special regularities, neither with regard to each other nor to the plate in which they developed ‘). Such regularities could not be expected very well as it can be inferred from the circular growth (see 5 1, fig. 2) that at least at the recrystallization temperature employed (600°C) no preference for any special crystallographic directions occurred. It seems to us, that the cause is to be found rather in a difference in ,,health” of the new crystallites, which makes itself felt also in the determinations of mechanical properties of crystals obtained by recrystallization “). Boundaries like C in fig. 1 originated because the nucleus of the small, pointed crystallite only started growing when the bigger one had approached very closely (cf. fig. 5a vs. b). In the aluminium plates mentioned before no gradual transition between these pointed
FORMS
OF
GRAIN
BOUNDARIES
753
crystallites and the very flat ones which generally occur was to be found. This gave the impression that the starting of the smaller crystallites was camed by the approach of the bigger ones. The relative crystallographic orientation of the two crystallites was determined in 10 cases and it was found that 6 pairs were spine1 tze+ns, at least within the experimental inaccuracy of about 2”. No relation could be foundbetween the twinning planes and the grain boundary symmetry axis of these pairs. A theoretical explanation (also of the problem why the ,,twin lamellae” that are so common with other metals, do not occur here), can not yet be given. Other causes influencing the ultimate shapes of grain boundaries are O): a. The surface tension, which has the tendency to straighten Erenated boundaries and to promote the formation of ,,Kapillarkijrner” “). b. coarse graiQ size of starting material compared with new crystal lites (cf. footnote *)). c. texture of starting material. d. preference for growth in certain crystallographic directions of the new crystallites. e. inhomogeneous deformation of starting material. f. unequal temperature of the recrystallizing metal. The author expresses his deep gratitude to Prof. Dr. MT. G. B u r: e r s, who drew his attention to the subject of this paper and gave invaluable help in many discussions. He is also much indebted to Prof. Dr. D. v a n D a n t z i g for his assistance in the mathematical preparation, to Prof. Dr. Ir. A. J. K 1 u y v e r, who provided the facilities for the microdiological part and to the ,,Delftsch Hoogeschoolfonds” for financial support. Received
Physica
June
IX
2tid,
1942.
754
FORMS
OF GRAIN
BOUNDARIES
REFERENCES 1) Cf. W. G. B u r g e r s, Hdb. d. Metallphys., Bd. III, 2 (1941), p. 152 etc. 2) Hdb. d. Metallphys., lot. tit , p. 189. 3) A. E. van Arkel, Polytechn. Weekbl. Z(i, 397, 1932; Hdb. d. Metallphys., lot. cit., fig. 77, p. 179. 4) N. T. B e 1 a i e w, Crystallisation of Metals, London 1922, p. 16. 5) C. Ben e d i c k s, Roll. Z. 01,217, 1940. 6) Hdb. d. Metallphys. lot. cit., p. 184-188. 7) Hdb. d. Metallphys lot. cit., p. 189. N.B. A misprint (2nd line from below: ,,eine” instead of ,,keine”) suggests the reverse. 8) E.g. R. I< a rn op and G. Sachs, cf. E. S c h m i d and W. Boas, Kristallplastizitlt, Berlin 1935, fig. 906, p. 127. 9) Also Hdb. d. Metallphys., lot. cit., 5 88 and 90.
IX,
no 7
Juli
FORMS OF GRAIN BOUNDARIES AS MAY ON PRIMARY RECRYSTALLIZATION OF DEFORMED METALS for Physical
APPEAR
J. SANDEE
IR. Laboratory
1942
Chemistry
of the Technical
University,
Delft
Zusammenfassung -4lle moglichen Formen von Korngrenzen, wie sie bei ,,isotroper” primlrer Rekristallisation von Metalen (konstante und gleiche lineare Wachstumsgeschwindi’gkeit in allen Richtungen innerhalb desselben Kristalliten) auftreten konnen, werden besprochen. Anschliessend wird ein kurzer cberblick gegeben iiber die anderen Bedingungen, welche gleichfalls die Formen der Korner beeinflussen konnen.
9 1. Primary recrystallization proceeds in such a way that relativelyhealthylattice areas, which are surrounded by a strong deformation-inhomogeneity act, on heating, as nuclei from where new crystallites start growing l). While developing, these meet each other and new grain boundaries are formed. Now it was noted by B u r g e r s 2), that in recrystallized aluminium plates some crystallites were entirely surrounded by larger ones, though the original material consisted of very fine grains (compared with the new crystallites) with randomorientations (fig. 1, LI)~). As was argued already in 2), this phenomenon can only be explained by assuming that the larger crystallites had a greater growing velocity and grew around the smaller ones. When a crystal develops in a plate of the above mentioned, finely grained, random oriented material it assumes a circular shape (fig. 2) “). This means that the growing velocity is equd in all directions. -
741 -
742
J. SANDEE
$ 2. In order to derive some forms suppositions must bc made :
of grain
boundaries
certain
1. A crystallite has an equal iinear growing velocity in all directiom. 2. This velocity is generally differelzt for each crystallite, but remains constant. 3. The crystallites generally start growing each at different moments. These conditions of recrystallization will be called ,,isotrofiic”.
Fiu 0’
1. Various
Fig.
Every
forms
2.
The
grain boundary
of isotropic aluminium
grain boundaries plates.
recrystallization in all directions.
is a boundary
velocity
between
in rccrystallizccl
is eclual
f~o grains.
To lx
FORMS
OF GRAIN
743
BOUNDARIES
able to analyse the shape of this boundary,,we must know about these crystallites : 1. the location of their nzlclei (conceived ns points), 2. the ratio of their growing velocities, 3. the dimensions of the ,,first” crystallite started to grow.
at the moment the ,,second”
The grain boundary surface shows revolution symmetry with the connecting line of the respective nucleus locations as its axis, so that is sufficient to study only the line of intersection of the boundary surface with a plane through this axis. S 3. ’ In the simplest case both crystallites start growing simultaneously with equal velocity. Their boundary becomes the locus of
Fig. 3. -4 and B started growing in the same moment with equal velocity.
those points that are equidistant from the nucleus points A and B (fig. 3), i.e. the bisecting perpendicular of AB. This is the only possibility for the occurrence of entirely straight grain boundaries. This case was treated before by B e 1 a i e w “) and B e n edicks5). If, however, one of the two, say A, started first and had developed into a circle of radius s by the time that the other one, B, started growing, one obtains another grain boundary (fig. 4a en b). This boundary is the locus of those points of which the distance to A is longer than their distance to B by a stretch s, i.e. one branch of a hyperbola with foci at A and B. The asymptotes pass through the middle of the line AB; they are perpendicular to the tangents through B to a circle of radius s around A.
744
J. SANDEE
This type was also discussed by B e n e d i c k s “). Now if the growing velocities of the crystallites A and B are no longer equal (but still in a constant ratio), one obtains the forms of grain boundaries sketched in fig. 5a and b. The analysis of these curves of 4th order is simple but will not be given here.
Fig. 4aand
b.
2 and
B
are growing with started first.
equal
velocity,
but A
.
Some attention may be paid, however, to the transition between the forms marked in fig. 5 by ,,v,., < vB” and ,,z)A
(1)
Let the first point of encounter Here ro = $$$
(2)
(+ = VA/% < 1). of the crystallites be 0 (fig. 7).
and
ro.= 6;
(3).
The grain boundary develops between the two circles and has, therefore, a vertical tangent in 0. The contact with this tangent is of a higher order if a point P, lying at an altitude Ah
FORMS
OF GRAIN
Fig.Sa and b. A and B are growing
745
BOUNDARIES
with different .
velocities.
A started
first.
-P
Fig. 6. Bipolar
coordinates.
above 0, is, for small values of A h, situated above 0. In this case in n APO: (r, + A+ = Ah2 + r; or 2r0 Av = Ah2 Likewise, in n BPO: 2p0Ap = Ah2
perpendicularly
(4) (5)
746
J. SANDEE
Hence : Differentiation
Y, Ar = ~~ Op. of (1) gives AY -= AP
Fig. 7. Contact
and from
of higher
-
(6)
P
order
with tangent
(6) it follows that
in 0.
,
F” = p z or, according to (2) and (3) : d-s -=
Pd+ s
fi,sothatp=
1-i.
The form ,,v4
s = p x shortest connection B -+ P
(7)
where p = vA/vg -C 1. As the shortest connection B + P is composed of the constant straight intersection BR and the arc RP, differentiation of (7) gives: AAP
A arc RP =
P-
FORMS
OF GRAIN
747
BOUNDARIES
Then in fig. 9: AAP --=fi, ‘OS o: = A arc RP
Fig. 8. The grain boundary
has proceeded
past R.
Fig. 9. The radii vectores from A meet the new curve in every point at the same angle.
748
J. SANDEE
so that the grain boundary meets all radii vectores from A under the constant angle : a = arc cos p (8) This is a well-known property of the logarithmic spirals with A as their (asymptotic) centre. s of (7) is not used in the deduction, which therefore holds also if B started first. The spirals pass continuously into the curves of fig. 5a and b. For those curves at R: ++$-
= p (fig. 10)
Fig. 10. The grain boundary curves pass continuously into logarithmic spirals.
and, therefore,
L ARB = arc cos p = u. The spirals intersect the produced part of AB and meet their opposites at the same time (fig. 11). Beyond this point they lose their physical meaning. Examples of boundaries of this shape are given in fig. 1, A. As the produced part of AB may be regarded as a radius vector from B (or A respectively), the spirals meet the axis of symmetry at this terminal point at an angle u = arc cos P. $4. Fig. 1 shows that the above-discussed various types of isotropic boundaries may occur in reality, but it seemed impossible to
FORMS
OF GRAIN
749
BOUNDARIES
produce them at will. WC resorted therefore to an analogous phenomenon in the growth of micro-organisms in and on solid media.
-3
Fig.
11.
The point-angle difference in
2a growing
depends only velocities.
on
the
Part of microbiological technique consists in the ,,streaking” of a highly diluted suspension of micro-organisms on to the plane surface of a sterile jelly (e.g. a solidified solution of agar-agar or gelatin with nutrients). Wherever a single cell was deposited upon the ,,plate”, it
n
b
Fig. 12n and 0. Illustration of grain boundaries with moulcls. Figures a and b correspond to the actual grain bountlaries shown in figure 1 at A and B resp.
multiplies into a ,,colony”. Now a plate which was streaked with a suspension of the milk mould Oospora lactis Fres. showed conspicuously straight boundaries between those otherwise circular colo-
750
J. SANDEE
nies that had encountered each other. This could be explained by assuming that the colonies originated from ,,nuclei” (presumably single cells) that had started developing into colonies at the same moment (with a certain lag after the streaking) with the same constant, linear velocity. As two very similar types of this mould were available, of which the ,,white” type grew faster than the ,,brown” one, the various forms of isotropic grain boundaries could be illustrated in this way. For that purpose tiny groups of mould cells were transferred by means of an inoculation needle to separate spots of sterile malt extract agar-agar plates which were kept at 22°C. At this temperature the linear growing velocity was some mm per day, so that the development of the ,,grain boundaries” could be fixed photographically in various succeeding phases. In fig. 12A and B some examples of such photographs are given, corresponding to the grain boundaries of fig. 1, A and B. S;5. If the crystallites of a recrystallized metal plate are ,,isotroPit. ‘, i.e. if the linear growing velocity of each crystallite was conH
Fig.
13.
How
to find
s if A started
first.
stant and equal in all directions, an attempt may be made to determine the nucleus locations etc. of the crystallites present. The locations of the nuclei can usually be traced by taking into consideration that the connecting line of the nucleus locations of two
FORMS
OF
GRAIN
751
BOUNDARIES
adjacent crystallites is an axis of symmetry ofctheir boundary. A grain boundary which is straight over a fairly large length means that the incubation periods and the growing velocities of the adjacent crystallites were equal, so that the nuclei are located symmetrically with regard to the boundary.
Fig.
14.
How
to find
s if
B started
first.
In some cases where the boundaries were particularly smooth *) the following methods could be used. A large part of the far-cleveloped grain boundaries follows logarithmic spirals, in any case the parts between the sharp point and the broadest part HH’ (fig. 13 and 14). Some of these spirals with various o! = arc cos p were constructed (fig. 15), in such a way that:all
Fig.
15.
Logarithmic
spirals.
had the same maximal width. If the grain boundaries to be examined are magnified or reduced (e.g. optically )till they have the same maximal width as the given spirals, the best fitting spiral can be laid along them. In this way the nucleus location of *) This is the case if the grain size before recrystallization is very much smaller than aftcrwnrds. If the initial grain size is relatively coarse, the boundaries of the new crystallites arc often rathercrenated (cf. Hdb. d. Metallphys., lot. cit. fig. 80, p. 182).
752
J. SANDEE
the enclosed crystallite is found immediately. In order to detect the nucleus location of the surrounding crystallite, the point has to be traced, where the real grain boundary begins to deviate from the spiral. Through this point R the tangent is drawn; this intersects the axis in the required point. Now if R is found to the right of A (fig. 13), A started growing first and the radius of the circle which enveloped A the moment when B started too can be found by dropping from B a perpendicular BC onto AR. As cos L ARB = 9, RC = fi x BR; and hence AC = s. If, hawever, R is found to the left of A (fig. 14)) B started first. A perpendicular must now be erected in A to AR, which meets BR in D. In this case cos L ARB = #, so that AR = p x DR, and hence BD = s. If a crystallite is surrounded entirely by another one, the ratio of the growing velocities of the two may be determined simply by considering that the cosine of the half point-angle is equal to this ratio (cf. equation (8)). In this way velocity ratios were found in aluminium plates up to 0,75 : 1. f 6. From the differences in growing velocity, which were shown by adjacent crystallites, it follows once again, that the scattering in direct determinations of growing velocities “) need not be due only to experimental inaccuracies, but may bring out a real difference in the properties of crystals obtained by recrystallization. A preliminary investigation of the crystallographic orientation of those crystallites that differed much in growing velocity, did not show any special regularities, neither with regard to each other nor to the plate in which they developed ‘). Such regularities could not be expected very well as it can be inferred from the circular growth (see 5 1, fig. 2) that at least at the recrystallization temperature employed (600°C) no preference for any special crystallographic directions occurred. It seems to us, that the cause is to be found rather in a difference in ,,health” of the new crystallites, which makes itself felt also in the determinations of mechanical properties of crystals obtained by recrystallization “). Boundaries like C in fig. 1 originated because the nucleus of the small, pointed crystallite only started growing when the bigger one had approached very closely (cf. fig. 5a vs. b). In the aluminium plates mentioned before no gradual transition between these pointed
FORMS
OF
GRAIN
BOUNDARIES
753
crystallites and the very flat ones which generally occur was to be found. This gave the impression that the starting of the smaller crystallites was camed by the approach of the bigger ones. The relative crystallographic orientation of the two crystallites was determined in 10 cases and it was found that 6 pairs were spine1 tze+ns, at least within the experimental inaccuracy of about 2”. No relation could be foundbetween the twinning planes and the grain boundary symmetry axis of these pairs. A theoretical explanation (also of the problem why the ,,twin lamellae” that are so common with other metals, do not occur here), can not yet be given. Other causes influencing the ultimate shapes of grain boundaries are O): a. The surface tension, which has the tendency to straighten Erenated boundaries and to promote the formation of ,,Kapillarkijrner” “). b. coarse graiQ size of starting material compared with new crystal lites (cf. footnote *)). c. texture of starting material. d. preference for growth in certain crystallographic directions of the new crystallites. e. inhomogeneous deformation of starting material. f. unequal temperature of the recrystallizing metal. The author expresses his deep gratitude to Prof. Dr. MT. G. B u r: e r s, who drew his attention to the subject of this paper and gave invaluable help in many discussions. He is also much indebted to Prof. Dr. D. v a n D a n t z i g for his assistance in the mathematical preparation, to Prof. Dr. Ir. A. J. K 1 u y v e r, who provided the facilities for the microdiological part and to the ,,Delftsch Hoogeschoolfonds” for financial support. Received
Physica
June
IX
2tid,
1942.
754
FORMS
OF GRAIN
BOUNDARIES
REFERENCES 1) Cf. W. G. B u r g e r s, Hdb. d. Metallphys., Bd. III, 2 (1941), p. 152 etc. 2) Hdb. d. Metallphys., lot. tit , p. 189. 3) A. E. van Arkel, Polytechn. Weekbl. Z(i, 397, 1932; Hdb. d. Metallphys., lot. cit., fig. 77, p. 179. 4) N. T. B e 1 a i e w, Crystallisation of Metals, London 1922, p. 16. 5) C. Ben e d i c k s, Roll. Z. 01,217, 1940. 6) Hdb. d. Metallphys. lot. cit., p. 184-188. 7) Hdb. d. Metallphys lot. cit., p. 189. N.B. A misprint (2nd line from below: ,,eine” instead of ,,keine”) suggests the reverse. 8) E.g. R. I< a rn op and G. Sachs, cf. E. S c h m i d and W. Boas, Kristallplastizitlt, Berlin 1935, fig. 906, p. 127. 9) Also Hdb. d. Metallphys., lot. cit., 5 88 and 90.