It is known from [i] that appreciable graphitization of natural and synthetic diamonds
occurs at 1800-2300 K. The graphitization temperature of ultradispersed diamond powders, in
contact with graphite, is different. Material containing originally 74% of the ultradispersed
product, after heating for 30 min in vacuo at ~1300 K, contained only 66% of the ultradispersed material (for the separation of volatile impurities before calcining, the product was
dried for 2 h in air at T ~ 450 K). From this data and the Eqs. (4)-(6), the energy barrier
gex is calculated to be =5.6 eV. This should be compared with the bond energy between carbon
atoms of the diamond lattice calculated by [2] to be 7.3 eV.
Obviously, ultradispersed diamond powder is a complex material, and its properties are
significantly different from those of the diamond materials previously studied. Certain
qualitative conclusions concerning the behavior of ultradispersed diamond powders may be
drawn from the above data. The dimensions of the relatively stable diamond particles and
the energy barrier for the diamond to graphite transformation may be calculated.
LITERATURE CITED
i.
2.
3.
4.
5.
A. V. Kurdyumov and A. N. Pilyankevich, in: Phase Transformations in Carbon and Boron
Nitride [in Russian], Naukova Dumka, Kiev (1978).
C. Kittel, in: Introduction to Physics of Solid Bodies [Russian translation], Nauka,
Moscow (1978).
V. F. Anisichkin, Fiz. Goreniya Vzryva, 20, No. i, 77 (1984).
A. M. Stayer, N. V. Gubareva, et al., Fiz. Goreniya Vzryva, 20, No. 5, i00 (1984).
Tables of Physical Constants (Handbook) [in Russian], I. K. Kikoina (ed.), Atomizdat,
Moscow (1976).
THERMODYNAMICS OF GRAPHITE TO DIAMOND TRANSFORMATION
V. V. Danilenko
The thermodynamics of the graphite to diamond phase transformation are important in any
kinetic study of the synthesis of diamonds. For this, a knowledge of the thermodynamic potentials of graphite and diamond ( A ~ = ~
d -~g,
~=E--TS+pu
=
SevdT--T ST d T ~ p u )
over a
wide range of temperature and pressure is necessary to supplement inadequate data on the thermophysical properties. Only a few works [1-6], in which the graphite to diamond equilibrium line for comparatively low pressures has been roughly calculated, are known. However, for further studies under nonequilibrium conditions, and for determining the effects of pressure and temperature on the transformation, and the optimum conditions for the synthesis of diamonds, it is
essential to know the values of A#.
Thermodynamic calculations, based on the Debye and Gruneisen approximations, have been
made for pressures up to -200 GPa and temperatures up to -4000 K. It was assumed that the
Poisson and Gruneisen ratios for both graphite and diamond were independent of the pressure,
and that the compressibility and Debye temperature of diamond were constant. The accuracy
of the calculations was very dependent on the initial data, and therefore special attention
was paid to the selection of the most reliable experimental data, using only that for single
crystals or perfect graphite samples with a hexagonal lattice.
The thermophysical properties of graphite and diamond at p = 0 and T = 300 K, used in
the calculations, were as follows (data for diamond in brackets). Specific volume v 0 =
0.444(0.284) cm3/mole, Debye temperature @0= 1800(2200) K, volumetric coefficient of thermal
expansion = = 30.3.10 -6 (3.5.10 -~) I/K, Gruneisen coefficient ~ = 0.65 (1.3), compressibility
X0 = 28.9"10 -8 (1.6"10 -8 ) cm3/N. The graphite potential was determined from the KistyakovChelyabinsk. Translated from Fizika Goreniya i Vzryva, Vol. 24, No. 5, pp. 137-142,
September-October, 1988. Original article submitted February 4, 1987; revision submitted
June 5, 1987.
0010-5082/88/2405-0633512.50 9 1989 Plenum Publishing Corporation
633
cVD, Jlmole'K
@~F, ~- ~
--.
>
,\.
2000
,ooo
T=5OOOK
I
1ooo-.~ / . . 4
I
~" ~ ~ 0
25
p, GPa
40
Fig. i
he, kJlmole
? ~
P'
GPa[
IGraphite / xx
]
A 1 I
-~"
'~
i
0
~00
50
o!
i
p, GPa
~o0o
Fig. 2
Diamond
Graphite ~ xx
3o0o
Fig. 3
2000
f->
r~K
Fig. i. Debye temperature (solid line) and specific heat capacity dotted line) of graphite.
Fig. 2. Difference in thermodynamic potentials of graphite
and diamond.
Fig. 3. Theoretical phase diagram for carbon,
2) Av = 0; 3) max he; 4) 5r = 16 kJ/mole.
i, 5) he = 0;
Wilson Eq. [7], which was based on the impulse adiabatic curve for perfect Ceylon graphite,
plotted by Walsh. The equation was transformed by K. K. Krupnikov into the form
p = pz + p , = A ( 5 4 - i)+(al + a26)T + c Y~T2,
(1)
where A = 0 . 0 8 4 6 8 , a 1 = - 0 . 2 2 5 7 , a 2 = 0 . 2 7 1 2 , c = 0 . 0 3 5 8 , 6 = v 0 / v .
Each t y p e o f g r a p h i t e
( w i t h i t s own s p e c i f i c p a c k i n g o f t h e l a y e r s ) s h o u l d h a v e i t s own e q u a t i o n o f s t a t e , w h i c h
d i f f e r p r i m a r i l y i n t h e v a l u e o f t h e e x p o n e n t n i n t h e e x p r e s s i o n Px = h ( 6 n - 1 ) .
F o r example, according to [8], the equation for unannealed pyrographite
w i t h random p a c k i n g i s
p~=0,0746(66--1).
A s p e c i a l f e a t u r e , shown up by t h e c a l c u l a t i o n s ,
was t h e c h a n g e i n t h e t h e r m a l p r o p e r t i e s
of graphite with pressure.
The r e l a t i o n o f t h e h e a t c a p a c i t y CVD w i t h p ( o r w i t h t h e compressibility
5) was f o u n d from t h e e q u a t i o n r e l a t i n g t h e Debye t e m p e r a t u r e and p r e s s u r e [9,
10]
3n-i
In the general
t h e n f ( ~ ) = 1.
634
case,
Op/Oo = 6
6
f(~).
Since the Poisson ratio Y is assumed to be constant,
Analysis of Eq. (2) showed that 00 increased significantly, but CVD decreased with pressure (Fig. 1), leading to an increase in the theoretical temperature of impulse compression.
At T = 1300 K and p = 30 GPa, from Eqs. (i) and (2), graphite had the following values:
0p/
00 = 1.9, (CVD)p/(CVD) 0 = 0.68, Ep/E 0 = 0.6, and Sp/S 0 = 0.5. The theoretical increase in
the electron heat capacity of graphite was c e = cV - cVD, where c V = Cp - 12vgTag2/X0g.
Tak-
ing the experimental value for Cp from [ii] we find c e = 0.21-10-~(T - i000).
For diamond, the value of cV was derived from Debye and the value of cp = cV + 12v0T~2/
X0 was found with the help of the Gruneisen equation (~ = u
= 2.54"10 -~ cV [12]).
The specific volume of diamond was found from the equation
u;.r = v0(l + ~ T ) e x p ( - - % ~ ) = 0,284(t +
2,54.tO-6cvT)exp(--%~).
The a p p r o x i m a t e i n t e r n a l e n e r g y a n d e n t r o p y o f g r a p h i t e and d i a m o n d w e r e f o u n d by i n t e g r a t i o n
i n 100 ~ s t e p s .
F o r g r a p h i t e a t T = 100 K, t h e v a l u e s o f E0 = 71 J / m o l e [13] a n d So = 0 . 9 1 4
J/(mole'K)
[14] w e r e u s e d a n d a s s u m e d t o be i n d e p e n d e n t o f t h e p r e s s u r e .
For diamond a t T =
0 K, t h e v a l u e o f E0 = 2 . 4 k J / m o l e [1] a n d t h e v a l u e s o f t h e e n t r o p y up t o 1200 K, b a s e d on
t h e e x p e r i m e n t a l v a l u e s o f Cp [ 1 1 ] , w e r e u s e d .
The t h e o r e t i c a l
v a l u e s o f ~r a r e shown i n
F i g . 2.
The c a r b o n p h a s e d i a g r a m ( F i g . 3) was p l o t t e d f r o m t h e a b o v e t h e o r e t i c a l
results.
A
particular feature of the diagram is the existence of a second region where graphite is stable, i.e., when p > 65 GPa and T < 2000 K. The condition necessary for the existence of a
second equilibrium pressure is that the p(Vg - v d) = f(p) relation passes through a maximum,
which exists only if n < 5, i.e., only for perfect hexagonal graphite.
It should be noted
that it is impossible to obtain the temperature and pressure of the second region of stability
with a single impulse compression.
The theoretical equation for the low pressure (GPa) equilibrium line is
p :
1,3(i + i0-~T~.
(3)
This line lies below the line which has been established by other workers [2, 3, 15]. This
may be attributed to the use of a lower value for the thermal capacity of graphite at high
pressures.
The equilibrium line (3) intersects the experimental line for molten graphite
[16] at the triple point where p = 7 GPa and T = 4700 K (the temperature given in [16] is
increased by 100 ~ to account for the lower theoretical value of the heat capacity of graphite
under pressure).
The slope of the line for molten diamond is deduced on the assumption that
it is proportional to the initial compressibility.
In comparison with silicon,
GPa [18], we have:
for which X0 = 3.25 "10-6 cm3/N [17]
(oT/op)~ _ %~
(o~/ap)~i
-..-~,
hence
and (ST/Sp) s = -41 K/
(OT/Op)~ = m 2 0 K/GPa.
%o
According to [19], the slope of the line should be <30 K/GPa.
All materials with a diamond lattice which have been examined show a linear relation between the temperature of melting and the pressure, up to ~i0 GPa. Diamond also obeys this
relation.
The following features of the phase diagram are important for the synthesis of diamonds:
i) a line of equality of the specific volumes of graphite and diamond, ~v = Vg - v d = 0, above
which transformation does not take place (Fig. 2.2); 2) since A~ increases with temperature,
but has amaximumvalue depending on the pressure
(Fig. 2.3), there is a line of maximum difference between the thermodynamic potentials.
This line defines the optimum conditions for
the transformation of graphite into diamond.
For pressures p > 20 GPa
p ~
~ 20 + 0 , 0 1 7 ( T - - 1000).
As w i l l be s e e n f r o m F i g . 4, t h e i m p u l s e a d i a b a t i c c u r v e f o r n o n p o r o u s g r a p h i t e ( w i t h
p o r o s i t y k = 1) d e v i a t e s s i g n i f i c a n t l y
f r o m t h e l i n e o f maximum ~r
but is nearer to the
adiabatic
curve for porous graphite (k = 1.35).
Consequently, the latter
s h o u l d be t r a n s f o r m e d i n t o d i a m o n d t o a g r e a t e r e x t e n t and a t a l o w e r p r e s s u r e ( w h i c h h a s b e e n c o n f i r m e d e x perimentally).
635
LK
zIH<O nH>O .~8
7
Og- d" 'O ,rcrlO
I
2500-
"- L
/
,4
\-_-
D"
50g
20
, rm
40009
1500
-2
JO
d
\
40
50
,">.L>-
p, GPa
0
I0
Fig. 4
50
50
p~ G P a
Fig. 5
Fig. 4. Lines AH = 0 (i), maximum A~ (2), impulse adiabatic
line for graphite, porosity 1.35 (3) and 1.0 (6), temperature
of diamond formed by adiabatic impulse compression of graphite,
porosity 1.35 (4) and 1.0 (5). Figures at points -A~, kJ/mole.
Fig. 5.
Relation between Og_d, rcr , and p.
3) In the region of diamond stability, it is possible to find lines of constant potential difference (A# = const). A value of A@ = 16 kJ/mole (Fig. 3.4) corresponds to the dynamic synthesis of diamond at p = 30 GPa and T - 1300 K [20]. For the transformation of
graphite under static pressure, p = 13 GPa and T = 3500 K [15], A# was equal to 14.5 kJ/mole, but
in Bridgmen's experiments (diamond was not formed when p ~ 40 GPa and T = 300 K [21], A@ was
7.3 kJ/mole.
Heat of Transformation (AH = AE + pay = TAS + A@) and Temperature of Diamond. For the
transformation, when Av and AE < 0, AH is always <0, i.e., the reaction is exothermic (at 30
GPa and 1300 K, AH = -5 kJ/mole). At higher temperatures and pressures, AH falls. The heat
of transformation is zero under certain conditions (Fig. 4.1). The temperature of diamond,
T d, synthesized by the impulse compression of graphite, was determined from the temperature
of the compressed graphite and the increase in temperature due to the exothermic heat of
transformation. For nonporous graphite, T d ~ 1300 K and was virtually independent of the
pressure if p < 60 GPa (Fig. 4).
Surface Energy of Graphite to Diamond Interphase Boundary (O~_d). Compressed graphite
with a distorted crystal lattice (its layers become nonplanar, approximating to the diamond
lattice structure) can be regarded as a peculiar "graphitic semiliquid," for which a value
of ag_ d can be established from the formula for the boundaries of the crystal-melt [22]
~g_d ~ ~ - a d .
g
(4)
According to [23], the boundaries of the octahedron [with the (III) index] are the only
possible boundaries for diamond. From [24], a d (Ill) = (5.4-5).10 -~ J/cm 2 for temperatures
T = (300-3000) K, assuming that a d (III) is independent of pressure. From (4), it follows
that ag_ d = 05if) Av = 0, i.e., ag- d falls from ~1200 erg/cm 2 if p = 0, to -150 erg/cm 2 if p =
50 GPa (Fig.
.
Dimensions of Critical Seed (rcr = 2ag_d/A~). From the values of Og_ d and A@, it is
possible to derive the relation rcr = f(p, T). If p > 20 GPa, rcr is slightly reduced, so
that the seed consists of several atoms. On the other hand, with an increase in p, the
thickness of the surface layer (width of the interphase boundary) 6g_ d = ag_d/AP (where ap difference in the densities of the phases) should increase. Where 6g_ d = rcr a critical state
develops, in which it is impossible to form diamond crystals of stable volume.
On this basis, it is possible to explain the pronounced differences between the data for
gas dynamic measurements (based on the form of the impulse adiabatic curve) with the complete
transformation of the graphite phase into the diamond phase at p ~ 40 GPa [25] and the re-
636
sults for compressed graphite, containing 1-2% of the diamond phase after storage. This is
not in conflict with the role of the rapid crystallization of small diamond crystals, both
during rapid reductions in pressure (tensile and shear stresses reducing the metastability
of the diamond phase) and after the pressure reduction. The analysis can be used beneficially
in the study of the conditions and the results of the dynamic synthesis of diamonds.
LITERATURE CITED
1.
2.
3.
4.
.
6.
.
8.
9.
I0.
F. D. Rossini and R. S. Jessup, J. Res. Nat. Bur. Standards, 21, No. 491 (1938).
O. I. Leipunskii, Usp. Khim., 8, i0 (1939).
R. Berman and F. Simon, Z. Electrochem., 5_~9, No. 333 (1955).
R. Liljeblad, Arkiv. kemi, 8, No. 432 (1955).
A. Neuhans and H. J. Meyer, Angewandte Chem., 69, No. 17 (1957).
S. A. Gubin, V. V. Odintsov, and V. I. Pepekin, in: Thesis ist All-Union Symp. on Macroscopic Kinetics and Chem. Gas Dynamics [in Russian], Chernogolovka (1984).
R. D. Cowan and W. Fickett, J. Chem. Phys., 24, No. 5 (1956).
N. L. Coleburn, J. Chem. Phys., 40, No. 71 (1964).
Yu. N. Ryabinin, K. P. Rodionov, and E. S. Alekseev, Zh. Tekh. Fiz., 34, No. ii (1964).
Yu. N. Ryabinin, K. P. Rodionov, and E. S. Aiekseev, Fiz. Met. Metalloved., IO, No. 1
(1960).
iio
12.
13.
14.
15.
16
17
18
19
20
21
22
23
24
25
Thermodynamic Properties of Individual Substances (Handbook) [in Russian], Izd. Akad.
Nauk SSSR, Moscow (1962).
K. A. Crishan, Proc. Ind. Acad. Sci., 24A, No. 33 (1946).
W. Desorbo, and W. W. Tyler, J. Chem. Phys., 26, No. 2 (1957).
A. R. Ubbelohde and F. A. Lewis, in: Graphite and Its Crystal Compounds, Oxford (1960).
F. P. Bundy, J. Chem. Phys., 38, No. 3 (1963).
F. P. Bundy, J. Chem. Phys., 38, No. 3 (1963).
FES, Moscow, 2, 513 (1962).
F. P. Bundy, J. Chem. Phys., 41, No. 12 (1964).
A. Layaramon, W. Klement, and L. C. Kennedy, Phys. Rev. Ser. II, 130, No. 2 (1963).
P. J. DeCarli and A. C. Jamisson, Science, !33, 3467 (1961).
P. W. Bridgmen, J. Chem. Phys., 15, No. 2 (1947).
S. N. Zadumkin, Dokl. Akad. Nauk SSSR, 130, No. 4 (1960).
O. M. Ansheles, Uch. Zap. Leningr. State Univ., Ser. Geol., 215, No. 3 (1957).
S. N. Zadumkin, Fiz. Tverd. Tela, ii, No. 5 (1960).
B. A. Alder and R. H. Cristian, Phys. Rev. Lett., i, No. i0 (1961).
637