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Recovery of strain amplitude independent dislocation damping in single crystal lif
RECOVERY
OF
STRAIN
AMPLITUDE INDEPENDENT DISLOCATION IN SINGLE CRYSTAL LiF*
DAMPING
S. H. CARPENTER? The recovery of strain amplitude independent damping in single crystal LiF, after driving at high st,rain amplitudes, has been investigated. Results show that the amplitude independent damping is increased by driving at high strain amplit.udes, but upon removal of the high strains the amplitude independent damping recovers with time to very near its original value. The rate of recovery was found to be temperature dependent. with an activation energy of -0.3 eV. The kinetics of the recovery were found to obey a tlf3 a@ng law rather than the more famlhar t*‘3 aging law. Measurements of t,he amplitude independent dampmg while simultaneously applying a st,at,ic stress, below the yield stress, gave similar results.
RESTAURATION
DE L’AMORTISSEMENT IKDEPENDANT DEFORMATION DAR‘S UN MONOCRISTAL
DE L’AMPLITUDE DE LiF
DE
On Btudie la restauration de l’amort~issement indbpendant. de l’amplitude de deformation dans un monocristal de LiF port& B de fortes amplitudes de d&formation. Les r&ultats montrent que l’amortissement indbpendant de l’amplitude est accru quand on Porte le cristal B de fortes amplitudes de deformation mais quand on supprime les fortes d&formations, l’amortissement indkpendant de l’amplitude se restaure avec le temps pour atteindre une valeur t&s proche de sa valeur init,iale. La vitesse de restauration dbpend de la temperature avec une Bnergie d’activation de -0,3 eV. La cinQtique de la restauration ob&t B une loi de vieillissement en tl/a plut8t qu’it la loi plus familibre en 1*/a. On obtient des r&ultats semblables par mesure de l’amortissement indbpendant de l’amplit,ude quand on applique simultan&ment une contrainte statique infbrieure B la limite Qlastique. DIE
ERHOLUNG DER VON DER VERSETZUNGSDdMPFUNG
DEHNUNGSAMPLITUDE IS EINKRISTALLINEM
UNABHbNGIGEN LiF
Die Erholung der von der Dehnungsamplitude unabhiingigen DImpfung in LiF-Einkristallen wurde nach Anlegen hoher Dehnungsamplituden untersucht. Die amplitudenunabh(ingige Dilmpfung nimmt bei Versuchen mit hoher Dehnungsamplitude zu. Bei Ent,fernung der hohen Dehnungen erholt sich die amplitudenunabhiingige Diimpfung jedoch bis fast auf ihren urspriinglichen Wert. Die Erholungsgeschwindigkeit erwies sich als bemperaturabh&ngig mit einer Aktivierungsenergie von etwa 0,3 eV. Die Kinetik folgte einem tl/g-Alt.erungsgesetz und nicht dem iiblichen tl/3-Gesetz. lihnliche Messungen, bei denen simultan einest,atische Spannung angelegt wurde (unterhalb der Fliellspannung), ergaben keine wesentlich anderen Resultate. INTRODUCTION
location
damping
It is now a well accepted fact that dislocations in ionic crystals are electrically charged. The electric
studying
both the dislocation
charge
amplitude independent
associated
the fact that and
negative
unequal.
with
the dislocation
the energies required ion
Hence,
vacancies in thermal
expect the dislocations
results
to form positive
in ionic equilibrium
crystals
are
Accordingly
of impurities
both the charge on the dislocation
This
somewhat
will be
the
may modify
where
charge
of
is particularly
of
true
of
damping, where the magnit,ude
the
on
the
line.
damping charge
measurements
at
cloud
be
would
mobile can be used as a method to evaluate
validity
dislocation
and also the charge
method
motion and pinning
depends in a sensitive manner dislocation
temperatures
by a charge cloud of vacancies
The presence
line.
an excellent
a,verage length of free dislocation
one would
of the opposite sign so that electric neutrality
dislocation
of the damping
in the crystal to be electrically
charged and surrounded maintained.
from
the
provides
of the motion
cloud,
charge
cloud
is restricted
the
amplitude
model.
by
If
the
a surrounding
independent
damping
cloud around the dislocations.
will be maintained
Eshelby et uZ.(~) have suggested that the interaction between the dislocation and the charge cloud might
st,ress sufficient to pull the dislocation from the charge cloud should produce
be very important
independent
in determining
the strength of the
ionic crystals. The effectiveness of the dislocationcharge cloud interaction will depend on the mobility
damping.
original value.
* Received April 28, 1967; revised May 31, 1967.
ACTA
METALLURGICA,
University
VOL.
1968
However,
of
in the amplitude if the charge
cloud
by the
diffusional
of the damping energy
of the
defects which make up the charge cloud.
of Denver, Denver,
16. JANUARY
an increase
The rate of recovery
will be governed of Physics,
The application
is mobile it will try to follow the dislocation by diffusion and hence the amplitude independent damping should recover with time back to near its
of the charge cloud and how effective it is in pinning or restricting the motion of the dislocations. Dis-
t Department Colo. 80210.
at a low value.
Bakerf2) 73
observed
that
driving
single crystal
LiF
_kCTA
54
METALLURGICA,
VOL.
16,
196s
at high vibratory stresses above the breakaway stress, (stress at which amplitude dependent. damping occurs), at; room tem~rature, caused an increase in the amplitude independent damping. This increase was followed by a time dependent recovery of the damping to near its value before application of the stress. Although Baker@) observed this effect he did not investigat#eit thoroughly. This paper presents the results of an investigation on the recovery of amplitude independent damping in single crystal LiF. Samples were subjected to both static compressive stresses and prolonged vibratory stresses. The results are interpreted in terms of a dislocation charge cloud model. An activation energy for the recovery process has been determined and its relationship to the model is discussed. EXPERIMENTAL
PROCEDURE
SampIes used in this investigation were high purity single crystals of LiF purchased from the Harshaw Chemical Co. An impurity analysis of the samples is given in Table 1. Samples used were approximately & of an inch square and were hand ground to one half wavelength for longitudinal resonance at 50 kc/s. All samples were of the same orientation, with the (100) direction parallel to the length of the sample TABLE
1. Impurity
analysis of the single crystal LiF used in this investigation
Impurity
Quantity
Aluminum 1-2 ppm Calcium 3 ppm Copper less than 1 ppm Iron l-2 ppm Lithium in excess of 5% Magnesium l-2 ppm Silicon 3 ppm Only elements which were det,ected
Damping measurements were made using a Marx@) composite piezoelectric oscillator. All measurements were made at a frequency of approximately 50 kc/s, over a tem~rature range of OYXO”C. A dummy rod technique,fa) using a dummy rod of single crystal A&O,, was used to measure the damping while the sample was maintained at temperature. The test Oemperature was monitored by a chrome1 alumel thermocouple and was controIled to within X*C. A series of room tem~rature damping measurements were made while the sample was simultaneously loaded with a static compressive stress. The apparatus used is shown in Fig. 1 and is thoroughly discussed elsewhere.(5)
FIG. 1. Schematic representation of the apparatus used to me&sure damping while simultaneously applying a static compressive stress.
Figure 2 shows a curve of the damping, expressed in terms of the log decrement, versus strain ampltude for a typical LiF single crystal. The curve clearly shows a strain amplitude independent and a strain amplitude dependent region. Measurements were made in the following manner. The damping was first measured at a low strain amplitude (sr in Fig. 2) in the amplitude independent region. The strain amplitude was then increased well above the breakaway strain into the amplitude dependent region (es in Fig. 2) and held there for an extended time. The strain a.mplitude was then immediately returned to its original value (sl in Fig. 2) and the damping at that st,rain amplitude was then measured as a function of time. The entire process was carried out while maintaining the sample at a constant temperature. Another series of meaaurements was carried out in the following manner. The damping was first measured at a low value of strain amplitude in the amplit.ude independent region. Then while maintaining the same low strain amplitude a static compressive load was simultaneously applied to the sample. The damping was then measured as a function of time immediately after loading, while sjmultaneously maintai~ng the load. Great care was taken to make certain the application of the static load did not shift the damping into the amplitude dependent region. In all cases the applied loads gave stresses well below the yield
CARPEKTER
: DISLOCAiTION
IO
I
STRAIN
DAMPING
AMPLITUDE
IS
LiF
100
X IO’
FIG. 2. Damping of single crystal LiF aa a fun&ion of strain amplitude. Ed- The strain amplitude at which the recovery measurements were made. Ed-- The strain amplitude used to drive the sample.
stress of the material. These measurements were only performed at room temperature. RESULTS
Data taken show that the amplitude independent damping of single crystal LiF was increased by driving the sample at a high strain amplitude. Upon removal of the high strain amplitude driving force the damping was found to decrease with time. After a sufficiently long time the damping approaches very nearly its original value before application of the high strain amplitude. Typical data obtained at different temperatures are shown in Figs. 3 and 4. These figures show curves of damping, in terms of the log decrement vs. the logarithm of time. Zero time is at the instant the high strain is removed. The dashed line in the lower corner gives the original value of the damping. Changes in resonant frequency were also observed to occur during the recovery process. Resonant frequency data are also shown in Figs. 3 and 4. In a like manner, t,he application of a static stress (always below the yield stress) was found to cause an immediate increase in the amplitude independent damping. Following the immediate increase, the damping was found to recover with time. Figure 5 shows data taken for two different static loads. In Fig. 5 the log decrement is plotted against the logarithm of time, where zero time is the instant the load was applied. The recovery process wit811a
static stress is similar in all details except one to the recovery process for driving at high strain amplitudes. The one exception is that, even for very long times, the damping of the sample while statically loaded is always higher than its original value before loading. If the static load is maintained until the damping reaches an equilibrium value and then removed, results similar to loading are obtained. The damping increases immediately upon removal of the load and recovers back to near its original value with time. DISCUSSION
The results obtained by this investigation are consistent with the dislocation-charge cloud model. The results can be explained clearly with the aid of the illustrations shown in Figs. 6 and 8. These figures show graphically the interaction between the dislocation and the surrounding charge cloud. At low strain amplitudes the dislocation is constrained to move within the surrounding charge cloud, Fig. 6A. As the strain amplitude is increased above the breakaway strain the dislocation is able to move outside of the charge cloud for a considerable amount of time during each cycle, Fig. 6B. Maintaining the high strain amplitude for an extended time will allow the charge cloud to readjust itself. The charge cloud will move by diffusion to a new configuration which will enclose the new envelope of vibrations of the dislocation, Fig. 6C. Upon removal of the high strain amplitude the dislocation can move over a
ACT-4
5
+
METALLURGICA,
ORlOlNAL I
VOL.
“ILUE
I
‘I
OF
16,
DAYPING-
1968
-
-
IO TIME,
-
-
-
-
-
-
100
MINUTES FIG. 3 HELD
AT
..-
7X
Id’p0.Q
MEASUREMENTS
‘\.,
l,=
AT
TEMPERATURE
35
MINUTES
7X
IO.’
1.c
+
‘A.
‘-,
i..... -.\.
+
‘Y
11
‘\,
4
‘\ +
+
‘-q
++
+
‘L.
-...
‘.\
+
4
+++
*
ORIGINAL
VALUE
OF
DAMPING-
‘1.
---‘_’
+.b
+ + +
I
4.
I
I
I
I
100
TIME,
I 1000
I
kN”TES FIG. 4
FIGS.
3 & 4. Amplitude independent damping and resonant frequency of single crystal LiF, as a function of time after driving at a high skain amplitude.
greater distance since the spread out charge cloud is less effective in restricting the dislocation motion. The ability of the dislocation to move over larger distances will in turn result in a higher damping as was observed. However, with time the charge cloud will diffuse in toward the dislocation, Fig. 6D. The closing of the charge cloud around the dislocation will impede the dislocation motion giving the observed recovery of the damping. After sufficient time the charge cloud will diffuse back to very near its original configuration, Fig. 6E, giving a value of the damping near the original value.
The rate at which the damping recovers is directly related to how fast the defects composing the charge cloud can diffuse at the temperatures in question. Results show the charge cloud to be quite mobile at the temperatures investigated. If one treats the recovery of the damping as a thermally activated process it is possible to obtain an activation energy for the recovery. By normalizing the recovery curves of damping vs time one can measure the relaxation time for the recovery process as a function of temperature. The relaxation time is deflned as the time required for the damping to recover one half of the
CARPESTER:
DISLOC_4TION MEASUREMENTS
AT
DAMPING
LiF
B,= 6X10-’
*-STATIC
LOAD
= 3kq
e,= 16XIO“
c-
LOAD
= 2kq
e,= 5.3X10.’
STATIC
IN
“Y, *,
.\..
=\..
IS-
I-* ORIGINALLY------
*-DAMPING IO_
____
‘~__________
I I
100
IO
1000
TIME,MINUTES FIG. 5. Amplitude independent damping of single crystal as a LiF function of time aft,er the application of a static compressive load.
total amount recovered during a complete recovery run. Mea,surements of the relaxation t,imes show they
energy
obey an equation
of the predominant
of the form 7 = @IRT
where 7 is the relaxation act,ivation
energy
Thus a plot
Q/R. Figure
this investigation line
time, 70 a constant,
RT
of log 7 vs. l/T
line with slope straight
and
plotted
(1)
has its usual should
give a straight
7 shows the results of
in such a manner.
fit is obtained
Q t,he
meaning.
giving
an
of ~0.3
eV.
The
energy should correspond SproulP
value
to be positive. by Whitworth
the
activation energy
defects in the charge cloud.
has investigated
on the dislocations
of
with the diffusional the sign of the
charge
in single crystal LiF and found it
However, a more recent investigation on NaCl indicates that the disloca-
tions are negatively
charged.
If the dislocat,ions
are
positively charged the most likely defect for the charge
A good
activation
b-
I-
E
FIG. 6. Schematic representation of the dislocation charge cloud interaction produced by the application and removal of a high strain amplitude vibratory stress. A.Before application of the high strain amplitude vibratory stress. B.-Immediately after application of the driving stress. C.-Long time after application of the driving stress. D.-Immediately after removal of the driving stress. E.-Long time after removal of the driving stress.
I
I
I
I
I
I
3. IO
3.20
3.30
3.40
3.50
3.60
(I/T)X
IO3
FIG. 7. Arrhenius plot of the relaxation times for the recovery of amplitude independent damping in single LiF.
ACTA
METALLUR,GICd,
STATIC STRESS
1 STATIC STRESS
FIG. 8. Schmatic representation of the dislocation charge cloud interaction produced by application of a static stress. A.-Before application of the static stress. B.Immediately following application of the static stress. C.-Long t,ime after application of the static stress.
cloud would be anion vacancies. Haven@) has shown that anion vacancies in LiF do have appreciable mobility at room temperature, but they are characterized by a bulk diffusional energy of 0.65, over twice that measured in this investigation. This leads one to believe that more likely the dislocations are negatively charged and the defects in the charge cloud are cation vacancies and/or positively ionized impurities. An exact determination of the defects composing the charge cloud has not been made. A comment should be made that the diffusion of the defects making up the charge cloud all takes place in close proximity of the dislocation core. It might well be that the diffusion in this region would take
0.5
1.0
I.5
VOL.
16,
1968
place at, a eignificant,ly lower energy than observed for bulk diffusion. Results obtained in the static et,ressexperiments can also be explained in a manner consistent with the charge cloud model. Before application of t.he st’at,ic load the dislocation oscillates at low amplitude within the charge cloud as described earlier: Fig. 8A. Application of a static load of sufficient magnitude will force the dislocation to a new equilibrium position, where a majority of the dislocation line is outside the charge cloud, Fig. 8B. Damping from the dislocation in the new position will be increased due to the fact that the equilibrium length of the dislocation is greater and its motion is not inhibited by the charge cloud. The charge cloud will immediately start to readjust by diffusion and will eventually surround the dislocation in its new configuration, Fig. SC. The fact that the equilibrium length of dislocation is longer, when under the applied stress, accounts for the fact that the damping never reaches its original value without load. Granato et a1.(g)have shown that if the amplitude independent damping follows the Granto-Liicke(l*Jl) theory of damping and the defects in the charge cloud diffuse to the dislocation according to the t2J3 aging law of Cottrell and Bilby(12) the time dependence of the damping can be expressed as
where 6, is the amplitude independent damping, t is the time and C and /3 are constants for constant temperature. This expression has shown to be valid in many instances in describing the recovery of the
2.0
2.5
3.0
3.5
l/3
(TIME,
MINUTES)
FIG. 9. Plot of the data given in Figs. 3 and 4 according to equation (3).
:
CARPENTER
damping
following
plast,ic deformat’ion.
data, however, taken in this investigation (2).
If t,he 3 in equa,tion (2) is replaced
good fit is obt,ained. following
DISLOCATION
D.1MPISG
None of the
planar
fit equation
cloud
by Q a fairly
Rewrit,ing t#he equation
in the
fashion (3)
/w/3
t5ype as pictured using
Bilbyd2)
the
would
same
in Fig. S. argument’s
Such
The recovery
give a’
of amplitude
charged dislocation cloud
plotted
diffusion
independent
that
fair
straight
is
mobile
charge cloud model. at
room
energy of -0.3
the charge cloud were not determined. of the recovery
These resultIs are consistent
point
on AgCl.
One should
out that the PI3 time dependence
has been observed
in metal
for ageing
systemso4)
and is not
unique to the charge cloud model presented
here.
theory
of damping
charge cloud diffuse according rather than the more familar
to a t1J3 aging law t2j3 aging law. The
the National
good
with
of
the
data
is somewhat
the
of free dislocation charge
since the
points,
but
“smeared two
as such
has no
number
along the dislocation
should
pinning
of defects
line.
That the
give the same results
is by no
Robinson
published
and
a theory
using a charge cloud
however, present
has a large
obvious.
recently crystals
model
rather
out”
models
means
line between minor pinning points.
cloud
are difficult
Birnbaumd5)
of
damping
model.
to interpret
have
in ionic
Their results,
in terms
of the
experiment.
The fact that a t1j3 aging law is obeyed is also of interest. One possible explanation of this result is that the charge cylindrical
as usually
pictured
cloud
not
the t213 aging
Fair agreement
law of
is obtained
if
is indebted
and assistance
measurements.
to
0.
This investigation Aeronautics
Quist
in making
for
sample
the necessary
was supported
by
and Space Administration.
Granato-
surprising
theory is based on a pinning point model with loops The
but
ACKNOWLEDGMENT
The author preparation
theory
The kinetics
a t1f3 aging law is used.
theory of damping is probably quite accurate in describing the damping, but that the defects in the
agreement
a
composing
agree with the Granato-Liicke(lopll)
Cottrell and Bilby.(12)
These results indicate that the Granato-Liicke(lO*ll)
Liicked”*ll)
having
The defects
of the recovery
with the
The charge
temperature
eV.
damping
consistent
lines are obtained for the times well past the midpoint with recent work by Kirn’ls)
and
dependence.04)
More work will be needed to answer t’hese quest,ions adequa.tely.
straight, line. Figure 9 shows the data of Figs. 3 and 4,
curves.
a. defect
as Cott,rell
lead to a t113 time
in single crystal LiF is basically
one can see a plot of (l/61)1/4 vs. P3 should Notice
i9
LiF
CONCLUSIONS
!$)“” =(i)1’4 + (;) in this manner.
IN
may
not be
but is more of a flat
REFERENCES 1. J. D. ESHELBY, C. W. A. NEU’EY and P. L. PRATT, Phil. Mag. 3, 75 (1958). 2. G. S. BAKER, J.appZ. Phys. 33, 1730 (1962). 3. J. MARX, Rev. scient. In&rum. 22, 503 (1951). 4. J. MARX and J. M. SIVERSTEN, J. appl. Phys. 24,Sl (1953). 5. G. S. BAKER and S. H. CARPENTER, Rev. 8cient. In&rum.
36. 29 119651. --I
6. R. SPRbULL,‘PhiZ. Msg. 5, 815 (1960). 7. R. W. WHITWORTH, Phil. Mag. 15, 305 (196;). 8. Y. HAVEN, Fran. China. Pays-&8 69, 1471 (1950). 9. A. GRANATO. A. HIKATA and K. L%xE, Acta Met. 6. 470 (19581. ’ 10. A. G~ANA~O and K. L~~cKE, J. appl. Phys. 27,583 (1956). 11. A. GRANATO and K. LOCKE, J. appl. Phys. 27, 789 (1956). 12. A. H. COTTRELL and B. A. BILBY, Proc. phys. Sm. A62, 49 (1949). 13. J. S. KIM, BUZZ.Am. phys.Soc. 12, 338 (1967). 14. B. S. LEMENT and M. COHEN, Acta Met. 4,496 (1956). 15. W. K. ROBINSON and H. K. BIRNBAUM, J. appl. Phys. 37, 3754 (1966).
OF
STRAIN
AMPLITUDE INDEPENDENT DISLOCATION IN SINGLE CRYSTAL LiF*
DAMPING
S. H. CARPENTER? The recovery of strain amplitude independent damping in single crystal LiF, after driving at high st,rain amplitudes, has been investigated. Results show that the amplitude independent damping is increased by driving at high strain amplit.udes, but upon removal of the high strains the amplitude independent damping recovers with time to very near its original value. The rate of recovery was found to be temperature dependent. with an activation energy of -0.3 eV. The kinetics of the recovery were found to obey a tlf3 a@ng law rather than the more famlhar t*‘3 aging law. Measurements of t,he amplitude independent dampmg while simultaneously applying a st,at,ic stress, below the yield stress, gave similar results.
RESTAURATION
DE L’AMORTISSEMENT IKDEPENDANT DEFORMATION DAR‘S UN MONOCRISTAL
DE L’AMPLITUDE DE LiF
DE
On Btudie la restauration de l’amort~issement indbpendant. de l’amplitude de deformation dans un monocristal de LiF port& B de fortes amplitudes de d&formation. Les r&ultats montrent que l’amortissement indbpendant de l’amplitude est accru quand on Porte le cristal B de fortes amplitudes de deformation mais quand on supprime les fortes d&formations, l’amortissement indkpendant de l’amplitude se restaure avec le temps pour atteindre une valeur t&s proche de sa valeur init,iale. La vitesse de restauration dbpend de la temperature avec une Bnergie d’activation de -0,3 eV. La cinQtique de la restauration ob&t B une loi de vieillissement en tl/a plut8t qu’it la loi plus familibre en 1*/a. On obtient des r&ultats semblables par mesure de l’amortissement indbpendant de l’amplit,ude quand on applique simultan&ment une contrainte statique infbrieure B la limite Qlastique. DIE
ERHOLUNG DER VON DER VERSETZUNGSDdMPFUNG
DEHNUNGSAMPLITUDE IS EINKRISTALLINEM
UNABHbNGIGEN LiF
Die Erholung der von der Dehnungsamplitude unabhiingigen DImpfung in LiF-Einkristallen wurde nach Anlegen hoher Dehnungsamplituden untersucht. Die amplitudenunabh(ingige Dilmpfung nimmt bei Versuchen mit hoher Dehnungsamplitude zu. Bei Ent,fernung der hohen Dehnungen erholt sich die amplitudenunabhiingige Diimpfung jedoch bis fast auf ihren urspriinglichen Wert. Die Erholungsgeschwindigkeit erwies sich als bemperaturabh&ngig mit einer Aktivierungsenergie von etwa 0,3 eV. Die Kinetik folgte einem tl/g-Alt.erungsgesetz und nicht dem iiblichen tl/3-Gesetz. lihnliche Messungen, bei denen simultan einest,atische Spannung angelegt wurde (unterhalb der Fliellspannung), ergaben keine wesentlich anderen Resultate. INTRODUCTION
location
damping
It is now a well accepted fact that dislocations in ionic crystals are electrically charged. The electric
studying
both the dislocation
charge
amplitude independent
associated
the fact that and
negative
unequal.
with
the dislocation
the energies required ion
Hence,
vacancies in thermal
expect the dislocations
results
to form positive
in ionic equilibrium
crystals
are
Accordingly
of impurities
both the charge on the dislocation
This
somewhat
will be
the
may modify
where
charge
of
is particularly
of
true
of
damping, where the magnit,ude
the
on
the
line.
damping charge
measurements
at
cloud
be
would
mobile can be used as a method to evaluate
validity
dislocation
and also the charge
method
motion and pinning
depends in a sensitive manner dislocation
temperatures
by a charge cloud of vacancies
The presence
line.
an excellent
a,verage length of free dislocation
one would
of the opposite sign so that electric neutrality
dislocation
of the damping
in the crystal to be electrically
charged and surrounded maintained.
from
the
provides
of the motion
cloud,
charge
cloud
is restricted
the
amplitude
model.
by
If
the
a surrounding
independent
damping
cloud around the dislocations.
will be maintained
Eshelby et uZ.(~) have suggested that the interaction between the dislocation and the charge cloud might
st,ress sufficient to pull the dislocation from the charge cloud should produce
be very important
independent
in determining
the strength of the
ionic crystals. The effectiveness of the dislocationcharge cloud interaction will depend on the mobility
damping.
original value.
* Received April 28, 1967; revised May 31, 1967.
ACTA
METALLURGICA,
University
VOL.
1968
However,
of
in the amplitude if the charge
cloud
by the
diffusional
of the damping energy
of the
defects which make up the charge cloud.
of Denver, Denver,
16. JANUARY
an increase
The rate of recovery
will be governed of Physics,
The application
is mobile it will try to follow the dislocation by diffusion and hence the amplitude independent damping should recover with time back to near its
of the charge cloud and how effective it is in pinning or restricting the motion of the dislocations. Dis-
t Department Colo. 80210.
at a low value.
Bakerf2) 73
observed
that
driving
single crystal
LiF
_kCTA
54
METALLURGICA,
VOL.
16,
196s
at high vibratory stresses above the breakaway stress, (stress at which amplitude dependent. damping occurs), at; room tem~rature, caused an increase in the amplitude independent damping. This increase was followed by a time dependent recovery of the damping to near its value before application of the stress. Although Baker@) observed this effect he did not investigat#eit thoroughly. This paper presents the results of an investigation on the recovery of amplitude independent damping in single crystal LiF. Samples were subjected to both static compressive stresses and prolonged vibratory stresses. The results are interpreted in terms of a dislocation charge cloud model. An activation energy for the recovery process has been determined and its relationship to the model is discussed. EXPERIMENTAL
PROCEDURE
SampIes used in this investigation were high purity single crystals of LiF purchased from the Harshaw Chemical Co. An impurity analysis of the samples is given in Table 1. Samples used were approximately & of an inch square and were hand ground to one half wavelength for longitudinal resonance at 50 kc/s. All samples were of the same orientation, with the (100) direction parallel to the length of the sample TABLE
1. Impurity
analysis of the single crystal LiF used in this investigation
Impurity
Quantity
Aluminum 1-2 ppm Calcium 3 ppm Copper less than 1 ppm Iron l-2 ppm Lithium in excess of 5% Magnesium l-2 ppm Silicon 3 ppm Only elements which were det,ected
Damping measurements were made using a Marx@) composite piezoelectric oscillator. All measurements were made at a frequency of approximately 50 kc/s, over a tem~rature range of OYXO”C. A dummy rod technique,fa) using a dummy rod of single crystal A&O,, was used to measure the damping while the sample was maintained at temperature. The test Oemperature was monitored by a chrome1 alumel thermocouple and was controIled to within X*C. A series of room tem~rature damping measurements were made while the sample was simultaneously loaded with a static compressive stress. The apparatus used is shown in Fig. 1 and is thoroughly discussed elsewhere.(5)
FIG. 1. Schematic representation of the apparatus used to me&sure damping while simultaneously applying a static compressive stress.
Figure 2 shows a curve of the damping, expressed in terms of the log decrement, versus strain ampltude for a typical LiF single crystal. The curve clearly shows a strain amplitude independent and a strain amplitude dependent region. Measurements were made in the following manner. The damping was first measured at a low strain amplitude (sr in Fig. 2) in the amplitude independent region. The strain amplitude was then increased well above the breakaway strain into the amplitude dependent region (es in Fig. 2) and held there for an extended time. The strain a.mplitude was then immediately returned to its original value (sl in Fig. 2) and the damping at that st,rain amplitude was then measured as a function of time. The entire process was carried out while maintaining the sample at a constant temperature. Another series of meaaurements was carried out in the following manner. The damping was first measured at a low value of strain amplitude in the amplit.ude independent region. Then while maintaining the same low strain amplitude a static compressive load was simultaneously applied to the sample. The damping was then measured as a function of time immediately after loading, while sjmultaneously maintai~ng the load. Great care was taken to make certain the application of the static load did not shift the damping into the amplitude dependent region. In all cases the applied loads gave stresses well below the yield
CARPEKTER
: DISLOCAiTION
IO
I
STRAIN
DAMPING
AMPLITUDE
IS
LiF
100
X IO’
FIG. 2. Damping of single crystal LiF aa a fun&ion of strain amplitude. Ed- The strain amplitude at which the recovery measurements were made. Ed-- The strain amplitude used to drive the sample.
stress of the material. These measurements were only performed at room temperature. RESULTS
Data taken show that the amplitude independent damping of single crystal LiF was increased by driving the sample at a high strain amplitude. Upon removal of the high strain amplitude driving force the damping was found to decrease with time. After a sufficiently long time the damping approaches very nearly its original value before application of the high strain amplitude. Typical data obtained at different temperatures are shown in Figs. 3 and 4. These figures show curves of damping, in terms of the log decrement vs. the logarithm of time. Zero time is at the instant the high strain is removed. The dashed line in the lower corner gives the original value of the damping. Changes in resonant frequency were also observed to occur during the recovery process. Resonant frequency data are also shown in Figs. 3 and 4. In a like manner, t,he application of a static stress (always below the yield stress) was found to cause an immediate increase in the amplitude independent damping. Following the immediate increase, the damping was found to recover with time. Figure 5 shows data taken for two different static loads. In Fig. 5 the log decrement is plotted against the logarithm of time, where zero time is the instant the load was applied. The recovery process wit811a
static stress is similar in all details except one to the recovery process for driving at high strain amplitudes. The one exception is that, even for very long times, the damping of the sample while statically loaded is always higher than its original value before loading. If the static load is maintained until the damping reaches an equilibrium value and then removed, results similar to loading are obtained. The damping increases immediately upon removal of the load and recovers back to near its original value with time. DISCUSSION
The results obtained by this investigation are consistent with the dislocation-charge cloud model. The results can be explained clearly with the aid of the illustrations shown in Figs. 6 and 8. These figures show graphically the interaction between the dislocation and the surrounding charge cloud. At low strain amplitudes the dislocation is constrained to move within the surrounding charge cloud, Fig. 6A. As the strain amplitude is increased above the breakaway strain the dislocation is able to move outside of the charge cloud for a considerable amount of time during each cycle, Fig. 6B. Maintaining the high strain amplitude for an extended time will allow the charge cloud to readjust itself. The charge cloud will move by diffusion to a new configuration which will enclose the new envelope of vibrations of the dislocation, Fig. 6C. Upon removal of the high strain amplitude the dislocation can move over a
ACT-4
5
+
METALLURGICA,
ORlOlNAL I
VOL.
“ILUE
I
‘I
OF
16,
DAYPING-
1968
-
-
IO TIME,
-
-
-
-
-
-
100
MINUTES FIG. 3 HELD
AT
..-
7X
Id’p0.Q
MEASUREMENTS
‘\.,
l,=
AT
TEMPERATURE
35
MINUTES
7X
IO.’
1.c
+
‘A.
‘-,
i..... -.\.
+
‘Y
11
‘\,
4
‘\ +
+
‘-q
++
+
‘L.
-...
‘.\
+
4
+++
*
ORIGINAL
VALUE
OF
DAMPING-
‘1.
---‘_’
+.b
+ + +
I
4.
I
I
I
I
100
TIME,
I 1000
I
kN”TES FIG. 4
FIGS.
3 & 4. Amplitude independent damping and resonant frequency of single crystal LiF, as a function of time after driving at a high skain amplitude.
greater distance since the spread out charge cloud is less effective in restricting the dislocation motion. The ability of the dislocation to move over larger distances will in turn result in a higher damping as was observed. However, with time the charge cloud will diffuse in toward the dislocation, Fig. 6D. The closing of the charge cloud around the dislocation will impede the dislocation motion giving the observed recovery of the damping. After sufficient time the charge cloud will diffuse back to very near its original configuration, Fig. 6E, giving a value of the damping near the original value.
The rate at which the damping recovers is directly related to how fast the defects composing the charge cloud can diffuse at the temperatures in question. Results show the charge cloud to be quite mobile at the temperatures investigated. If one treats the recovery of the damping as a thermally activated process it is possible to obtain an activation energy for the recovery. By normalizing the recovery curves of damping vs time one can measure the relaxation time for the recovery process as a function of temperature. The relaxation time is deflned as the time required for the damping to recover one half of the
CARPESTER:
DISLOC_4TION MEASUREMENTS
AT
DAMPING
LiF
B,= 6X10-’
*-STATIC
LOAD
= 3kq
e,= 16XIO“
c-
LOAD
= 2kq
e,= 5.3X10.’
STATIC
IN
“Y, *,
.\..
=\..
IS-
I-* ORIGINALLY------
*-DAMPING IO_
____
‘~__________
I I
100
IO
1000
TIME,MINUTES FIG. 5. Amplitude independent damping of single crystal as a LiF function of time aft,er the application of a static compressive load.
total amount recovered during a complete recovery run. Mea,surements of the relaxation t,imes show they
energy
obey an equation
of the predominant
of the form 7 = @IRT
where 7 is the relaxation act,ivation
energy
Thus a plot
Q/R. Figure
this investigation line
time, 70 a constant,
RT
of log 7 vs. l/T
line with slope straight
and
plotted
(1)
has its usual should
give a straight
7 shows the results of
in such a manner.
fit is obtained
Q t,he
meaning.
giving
an
of ~0.3
eV.
The
energy should correspond SproulP
value
to be positive. by Whitworth
the
activation energy
defects in the charge cloud.
has investigated
on the dislocations
of
with the diffusional the sign of the
charge
in single crystal LiF and found it
However, a more recent investigation on NaCl indicates that the disloca-
tions are negatively
charged.
If the dislocat,ions
are
positively charged the most likely defect for the charge
A good
activation
b-
I-
E
FIG. 6. Schematic representation of the dislocation charge cloud interaction produced by the application and removal of a high strain amplitude vibratory stress. A.Before application of the high strain amplitude vibratory stress. B.-Immediately after application of the driving stress. C.-Long time after application of the driving stress. D.-Immediately after removal of the driving stress. E.-Long time after removal of the driving stress.
I
I
I
I
I
I
3. IO
3.20
3.30
3.40
3.50
3.60
(I/T)X
IO3
FIG. 7. Arrhenius plot of the relaxation times for the recovery of amplitude independent damping in single LiF.
ACTA
METALLUR,GICd,
STATIC STRESS
1 STATIC STRESS
FIG. 8. Schmatic representation of the dislocation charge cloud interaction produced by application of a static stress. A.-Before application of the static stress. B.Immediately following application of the static stress. C.-Long t,ime after application of the static stress.
cloud would be anion vacancies. Haven@) has shown that anion vacancies in LiF do have appreciable mobility at room temperature, but they are characterized by a bulk diffusional energy of 0.65, over twice that measured in this investigation. This leads one to believe that more likely the dislocations are negatively charged and the defects in the charge cloud are cation vacancies and/or positively ionized impurities. An exact determination of the defects composing the charge cloud has not been made. A comment should be made that the diffusion of the defects making up the charge cloud all takes place in close proximity of the dislocation core. It might well be that the diffusion in this region would take
0.5
1.0
I.5
VOL.
16,
1968
place at, a eignificant,ly lower energy than observed for bulk diffusion. Results obtained in the static et,ressexperiments can also be explained in a manner consistent with the charge cloud model. Before application of t.he st’at,ic load the dislocation oscillates at low amplitude within the charge cloud as described earlier: Fig. 8A. Application of a static load of sufficient magnitude will force the dislocation to a new equilibrium position, where a majority of the dislocation line is outside the charge cloud, Fig. 8B. Damping from the dislocation in the new position will be increased due to the fact that the equilibrium length of the dislocation is greater and its motion is not inhibited by the charge cloud. The charge cloud will immediately start to readjust by diffusion and will eventually surround the dislocation in its new configuration, Fig. SC. The fact that the equilibrium length of dislocation is longer, when under the applied stress, accounts for the fact that the damping never reaches its original value without load. Granato et a1.(g)have shown that if the amplitude independent damping follows the Granto-Liicke(l*Jl) theory of damping and the defects in the charge cloud diffuse to the dislocation according to the t2J3 aging law of Cottrell and Bilby(12) the time dependence of the damping can be expressed as
where 6, is the amplitude independent damping, t is the time and C and /3 are constants for constant temperature. This expression has shown to be valid in many instances in describing the recovery of the
2.0
2.5
3.0
3.5
l/3
(TIME,
MINUTES)
FIG. 9. Plot of the data given in Figs. 3 and 4 according to equation (3).
:
CARPENTER
damping
following
plast,ic deformat’ion.
data, however, taken in this investigation (2).
If t,he 3 in equa,tion (2) is replaced
good fit is obt,ained. following
DISLOCATION
D.1MPISG
None of the
planar
fit equation
cloud
by Q a fairly
Rewrit,ing t#he equation
in the
fashion (3)
/w/3
t5ype as pictured using
Bilbyd2)
the
would
same
in Fig. S. argument’s
Such
The recovery
give a’
of amplitude
charged dislocation cloud
plotted
diffusion
independent
that
fair
straight
is
mobile
charge cloud model. at
room
energy of -0.3
the charge cloud were not determined. of the recovery
These resultIs are consistent
point
on AgCl.
One should
out that the PI3 time dependence
has been observed
in metal
for ageing
systemso4)
and is not
unique to the charge cloud model presented
here.
theory
of damping
charge cloud diffuse according rather than the more familar
to a t1J3 aging law t2j3 aging law. The
the National
good
with
of
the
data
is somewhat
the
of free dislocation charge
since the
points,
but
“smeared two
as such
has no
number
along the dislocation
should
pinning
of defects
line.
That the
give the same results
is by no
Robinson
published
and
a theory
using a charge cloud
however, present
has a large
obvious.
recently crystals
model
rather
out”
models
means
line between minor pinning points.
cloud
are difficult
Birnbaumd5)
of
damping
model.
to interpret
have
in ionic
Their results,
in terms
of the
experiment.
The fact that a t1j3 aging law is obeyed is also of interest. One possible explanation of this result is that the charge cylindrical
as usually
pictured
cloud
not
the t213 aging
Fair agreement
law of
is obtained
if
is indebted
and assistance
measurements.
to
0.
This investigation Aeronautics
Quist
in making
for
sample
the necessary
was supported
by
and Space Administration.
Granato-
surprising
theory is based on a pinning point model with loops The
but
ACKNOWLEDGMENT
The author preparation
theory
The kinetics
a t1f3 aging law is used.
theory of damping is probably quite accurate in describing the damping, but that the defects in the
agreement
a
composing
agree with the Granato-Liicke(lopll)
Cottrell and Bilby.(12)
These results indicate that the Granato-Liicke(lO*ll)
Liicked”*ll)
having
The defects
of the recovery
with the
The charge
temperature
eV.
damping
consistent
lines are obtained for the times well past the midpoint with recent work by Kirn’ls)
and
dependence.04)
More work will be needed to answer t’hese quest,ions adequa.tely.
straight, line. Figure 9 shows the data of Figs. 3 and 4,
curves.
a. defect
as Cott,rell
lead to a t113 time
in single crystal LiF is basically
one can see a plot of (l/61)1/4 vs. P3 should Notice
i9
LiF
CONCLUSIONS
!$)“” =(i)1’4 + (;) in this manner.
IN
may
not be
but is more of a flat
REFERENCES 1. J. D. ESHELBY, C. W. A. NEU’EY and P. L. PRATT, Phil. Mag. 3, 75 (1958). 2. G. S. BAKER, J.appZ. Phys. 33, 1730 (1962). 3. J. MARX, Rev. scient. In&rum. 22, 503 (1951). 4. J. MARX and J. M. SIVERSTEN, J. appl. Phys. 24,Sl (1953). 5. G. S. BAKER and S. H. CARPENTER, Rev. 8cient. In&rum.
36. 29 119651. --I
6. R. SPRbULL,‘PhiZ. Msg. 5, 815 (1960). 7. R. W. WHITWORTH, Phil. Mag. 15, 305 (196;). 8. Y. HAVEN, Fran. China. Pays-&8 69, 1471 (1950). 9. A. GRANATO. A. HIKATA and K. L%xE, Acta Met. 6. 470 (19581. ’ 10. A. G~ANA~O and K. L~~cKE, J. appl. Phys. 27,583 (1956). 11. A. GRANATO and K. LOCKE, J. appl. Phys. 27, 789 (1956). 12. A. H. COTTRELL and B. A. BILBY, Proc. phys. Sm. A62, 49 (1949). 13. J. S. KIM, BUZZ.Am. phys.Soc. 12, 338 (1967). 14. B. S. LEMENT and M. COHEN, Acta Met. 4,496 (1956). 15. W. K. ROBINSON and H. K. BIRNBAUM, J. appl. Phys. 37, 3754 (1966).