Pressure sintering kinetics of tungsten and titanium carbides

Int. Journal of Refractory Metals and Hard Materials 39 (2013) 32–37

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Pressure sintering kinetics of tungsten and titanium carbides M.S. Kovalchenko ⁎ Frantsevich Institute for Problems of Materials Science, National Academy of Sciences of Ukraine, Ukraine

a r t i c l e

i n f o

Article history: Received 31 August 2012 Accepted 10 March 2013 Keywords: Pressure sintering Kinetics Rheology Power-law and diffusional creep Tungsten and titanium carbides

a b s t r a c t The rheological model of deformable, irreversibly compressible, porous body based on mechanics of continua, and creep theory of crystalline materials, is used to describe quantitatively the compaction of the tungsten and titanium carbides powders under pressure sintering in isothermal conditions. Densification of the porous body occurs under action of Laplace's pressure, generated by surface tension, and applied pressure. The estimated mean value of Laplace's pressure was determined to be 5.8 MPa for tungsten carbide and 7.2 MPa for titanium carbide. The densification kinetics of tungsten carbide in the sintering range of 2100–2500 °C and titanium carbides in the sintering range of 2100–2700 °C are controlled by the mechanism of nonlinear steady-state creep, which occurs at a rate proportional to the fourth power of stress in carbide matrix forming porous material. The estimated values of activation energy for the powder particle power-law creep rate are 591 kJ/mol for tungsten carbide and 573 kJ/mol for titanium carbide during the pressure sintering in initial and intermediate stages. These values indicate that a climb dislocation mechanism controlled the creep, and the values are consistent with the activation energies of bulk diffusion in metal sublattice of carbides. A diffusional creep controls the pressure sintering kinetics in a later stage. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Refractory carbides are used as oxygen-free ceramics in hightemperature engineering applications due to high melting temperature, hardness, elastic moduli, wear resistance, electric conductivity and high-temperature strength. Refractory carbides are also widely applied as a base of hardmetals. Applications of refractory carbides include structural, heating and reflecting functions as well as tool materials in composition with other refractory compounds [1–3]. The compacted products of these compounds are made by powder metallurgy technology where sintering methods are of decisive importance. The sintering process is driven by surface tension that induces Laplace's pressure (pL ~ α/r) in pores of a powder body, where α is surface tension, r is the mean pore radius [4–8]. In initial stage of sintering, the porous body structure is characterized by fluctuations in density and coordination (contact) number of powder particles due to their random packing [9]. The fluctuations lead to the formation of mesostructure, which is intermediate between the macrostructure of the body as a whole and its microstructure [10]. The compressive sintering force in 3-dimension space is determined as F ðdim ¼ 3Þ ¼ −2παnhλia:

ð1Þ

⁎ 3 Krzhizhanovskogo Str., Kyiv 03142, Ukraine. Tel.: +380 44 424 2101; fax: +380 44 424 2131. E-mail address: [email protected]. 0263-4368/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijrmhm.2013.03.001

Here n is the number of particles in a porous body; 〈λ〉 is the average coordination number of particles; a is the radius of necks between the particles [11]. Since the average coordination number of particles in the cores of mesoelements is higher than that in their boundaries, the sintering force in the cores is greater than in the boundaries. This difference promotes the compaction of the cores into dense particle clusters while loosening the mesoelement boundaries. The residual porosity of mesoelement boundaries may be eliminated only under pressure sintering. Applied pressure higher than Laplace's pressure, along with the elimination of zonal isolation, promotes densification, structure formation, and attainment of high mechanical and functional properties of sintered materials. Such sintered materials are needful for high-temperature technique, structural and tool application. The experimental data analysis of the relative density change during the compaction of tungsten and titanium carbides powders in isothermal conditions of pressure sintering is carried out in the framework of volume viscous flow theory for irreversibly compressible body.

2. Theoretical background The theory of bulk viscous flow of porous bodies is based on the continuum mechanics approach [12–16]. This approach considers the viscous behavior of a composite consisting of a matrix or solid phase that forms the porous body.

M.S. Kovalchenko / Int. Journal of Refractory Metals and Hard Materials 39 (2013) 32–37

The porous body strain under the bulk action of external forces is determined by the shear strains of matrix material particles. The matrix viscous flow under low shear stress corresponds to the Newton's fluid flow at which hτi ¼ 2ηð0Þ hε_ i

ð2Þ

where 〈τ〉 and hε_ i ¼ ðd=dtÞhεi are the mean-square-root stress and the mean-square-root strain rate respectively [14], η(0) is the matrix dynamic viscosity, and t is time. The values 〈τ〉 and hε_ i depend on relative density, ρ. Under pressure sintering of the porous body in a die 〈τ〉 and hε_ i are expressed [17,18] as follows P hτi ¼ pffiffiffiffiffiffiffi ; hε_ i ¼ ρψ

sffiffiffiffiffi ψ dρ : ρ3 dt

ð3Þ

Here P is pressure and value ψ is expressed by the formula 2−ρ2=ρ ð2:5−ρÞ=ρ ρ : ψ¼  2 1−ρ2=ρ

ð4Þ

Eq. (4) also used to determine the dependence of porous body bulk viscosity, Z, upon relative density: Ζ = 2η(0)ψ. The increase in bulk viscosity with porous body relative density causes volume flow deceleration, until the flow stops at ρ = 1. Such a body does not deform under bulk compression, but remains deformable under pure shear. Such properties are similar to incompressible fluids [19] and are successfully used in shaping of compact materials under pressure processing. As seen from Eqs. (3) and (4), the mean-square stress decreases with densification that leads a reduction in the mean-square-root strain rate. Approaching the limit of relative density, i.e. ρ → 1, the mean-square-root stress approaches zero, 〈τ〉 → 0. In the initial stage of pressure sintering, high mean-square-root pffiffiffiffiffiffi stress takes place due to small values of the denominator, ρψ, in Eq. (3). For high 〈τ〉, the linear law given in Eq. (2) cannot be applied for crystalline materials. In this case, the flow in shear is described by power-law creep equation: n ε_ ¼ Aðτ=τ0 Þ =2

ð5Þ

where τ0 is the elastic limit; n ≥ 1 is the non-linearity exponent; A is given as follows

A ¼ const 

D0 μ ð0Þ b kB T

  U exp − : kB T

D0 is the pre-exponential factor for the bulk self-diffusion coefficient; U is activation energy; μ(0) is shear modulus; b is the Burgers vector, and kB is the Boltzmann constant [20]. After the substitution of Eq. (3) into Eq. (5) and some algebra an equation for pressure sintering kinetics of porous polycrystalline material can be obtained:  n n−3 nþ1 dρ 1 P ρ2 ψ2 ¼ A : dt 2 τ0

ρ

n−3 2

ρ0

nþ1 2

ψ

dρ ¼

ρ

2:5ðnþ1Þ−4ρ 2ρ

Χd ðn; ρÞ ¼ ∫ ρ ρ0

2−ρ2=ρ   2 1−ρ2=ρ

!nþ1 2

dρ:

ð9Þ

The Eq. (8) shows that integral function is linear with respect to time. The function increases considerably, approaching infinity as the relative density goes to unity, ρ → 1. However, the mean-square-root stress decreases during the late stage of pressure sintering causing a transition from power-law creep to linear diffusional matrix creep in accordance with Eq. (2). In this case, the matrix fluidity is determined as follows   1 DΩ Db b 2 ¼ ¼ 2 : ηð0Þ L kB T kB T L

ð10Þ

Here D is the bulk diffusion coefficient; L is the characteristic size of microstructure; Ω = b3 is the atomic volume. According to the theory proposed by Nabarro [21] L equal to subgrain size. However, in the Pines' theory [5] value L represents the characteristic pore size, and in the Herring's theory of diffusional viscous creep [6] L is the grain size. In the intermediate stage of sintering, grain growth occurs as the contact area between particles increases. A large body of data shows that the grain growth kinetics is described as follows m

m

L −L0 ¼ Kt

ð11Þ

where L0 is the mean grain size at t = 0; K = K0 exp(− Q/(kBT)), Q is the activation energy; m is the exponent which depends on the mechanism of mass transfer. According to the theory developed by Geguzin (Geguzin YE - www.scholar.google.com) and Krivoglaz [22] the exponent can be defined as, m = c + 2. The value c = 2 corresponds to the diffusion mechanism of mass transfer in solid phase if gas pressure in the pores is equal to Laplace's pressure; c = 3 corresponds to the volume diffusion mechanism and diffusional mass transfer in gas phase at constant gas pressure in pores; and c = 4 in cases of surface diffusion. The analysis of the grain growth kinetics during the sintering of chromium carbide [23], as well as our data for grain growth during the annealing of pressure sintered titanium and molybdenum carbides, has shown that m = 4. Taking this value, the grain growth kinetics during the pressure sintering may be described as 4

4

L ¼ L0 ð1 þ βt Þ

ð12Þ

K/L04.

Then Eq. (10) for matrix fluidity takes the form

1 DΩ : ¼ ηð0Þ L20 ð1 þ βt Þ1=2 kB T

ð13Þ

Combining Eqs. (2), (3) and (13), taking exponent n = 1 into account, with following integration results in the dependence of integral relative density function Χd(n = 1,ρ) upon time for matrix diffusional viscous flow during the late stage of the porous body pressure sintering: ρ

1 2PDΩ 1=2 ð1 þ βt Þ : Χd ðn ¼ 1; ρÞ ¼ ∫ ψdρ ¼ 2 ρ βL 0 kB T ρ

ð14Þ

0

ð7Þ

After integration, this equation takes the form:

Χd ðn; ρÞ ¼ ∫ ρ

Here the integral is represented as an integral function of the relative density Χd(n, ρ), with respect to the die (index d). This integral expression may be represented in detailed form as follows

where β = ð6Þ

33

 n 1 P A t: 2 τ0

ð8Þ

At βt > > 1 this can be taken approximately as Χd(n = 1,ρ) ~ t 1/2. It enables the estimation of the availability of the diffusional viscous flow of solid phase (matrix), which forms a porous body. The dependence of grain growth upon time during the sintering of alumina [24,25] may be expressed as 3

3

L ¼ L0 ð1 þ βt Þ

ð15Þ

34

M.S. Kovalchenko / Int. Journal of Refractory Metals and Hard Materials 39 (2013) 32–37

where β = K/L03. In this case, the dependence of Χd(n = 1,ρ) upon pressure sintering time takes the following form: ρ

1 3PDΩ 1=3 ð1 þ βt Þ : Χd ðn ¼ 1; ρÞ ¼ ∫ ψdρ ¼ 2 ρ k T βL B 0 ρ

ð16Þ

0

Analysis of the densification kinetics of glass powder during the pressure sintering in isothermal conditions [26] has shown that the dependence of the integral function, Χd(n = 1, ρ), on time is linear [17]. Such observation agrees with theory if the matrix is an amorphous material, similar to fluid. It has been established that the densification of porous glass differs from the densification of metal powders in terms of slower initial flow, and the weaker deceleration of densification in the subsequent stage of sintering. In case of crystalline copper electrolytic powder [26] the pressure sintering kinetics [16] are consistent with the power-law creep of copper matrix controlled by dislocation climb mechanism with n = 4.5 in accordance to theory [27,28]. 3. Initial materials and pressure sintering procedure The tungsten carbide powder used in this study contained (in weight) 93.4% W, 6.2% bonded C, and 0.1% Fe. As the free carbon was absent, the starting powder composition consisted of stoichiometric WC, impurities of Fe, and other carbides formed by extraneous materials. The starting titanium carbide powder contained (in weight) 79.9% Тi, 19.4% total C, 0.4% free C, and 0.3% Fe. After purification (by washing in hydrochloric acid solution) the content of elements slightly changed to 80.3% Ti, 19.3% total C, 0.3% free C, and trace of Fe. The free carbon was not found in sintered samples, and chemical composition of which was fitted to the formula TiC0.98. The mean powder particles size was approximately 5 μm for WC and 8 μm for TiC. The experimental study of the carbide powder pressure sintering was carried out in laboratory apparatus with fixed loads, applied to porous samples in graphite moulds with internal diameters of 8 and 14 mm (Fig. 1). The larger moulds served as an alternative electrical

current heater. The smaller mould was placed inside of the larger one when it was used for sintering. Axial pressure was applied to both sides in order to obtain more homogeneous densification of samples. The final height of the samples was not to exceed 10 mm. A graphite heat shield surrounded the heater. Temperature measurements were taken with the help of optical pyrometer through the small window, the diameter of 3–4 mm, in the shield. Temperature in the centre of the mould was higher than on the heater surface because of losses due to radiation. Because the graphite emissivity factor was ε = 0.7, the measured temperature was lower than true temperature, i.e. temperature of the ideal black body. Therefore, a correction determined experimentally and by calculation was applied to the measured temperature. Plotted corrections for the 30 mm diameter heater are shown in Fig. 2. Most experiments were carried out in an atmosphere CO + N2 generated from the interaction of air and graphite. Verification experiments were carried out in both a vacuum and an argon atmosphere. Comparison of results revealed that the choice of atmosphere did not influence essentially densification of carbide powders in initial and intermediate stages of pressure sintering. External pressure was applied after reaching the preset temperature. Five experiments were done for each temperature and pressure condition in order to obtain reliable results. In order to determine the dependences of density on time, the displacement of the upper punch was recorded with the help of a rheochord which carried electric current was connected to electron potentiometer. The rheochord was rigidly attached to the support of press, and its brush arm was connected with the press lever. The device ensured a measuring precision of ±0.01 mm. The final density of samples was determined by Archimedes' method of hydrostatic weighing. The relative density of samples was calculated using the principle of mass conservation (γV = γSh = const), where γ is the density, V is the volume, S is the cross-section area, and h is the height of sample. At constant S = S0 the principle above transforms to: γ0h0 = γh = γfhf, where γ0, γ and γf, are the initial, current, and final density, respectively; h0, h, and hf are the initial, current, and final height of the sample,

Fig. 1. A sketch of the die-assembly for pressure sintering.

M.S. Kovalchenko / Int. Journal of Refractory Metals and Hard Materials 39 (2013) 32–37

0.8

3000

4

2500

→

1500

3

0.4

1000

0.2

1600

2000

2400

3

1

2

2 1

0

2100 oC

0.08

1

0

2400 oC

1.6

4

0.06 1200

2

3

Ts

800

4

4

Xd (n = 4, ρ)

Tc

2000

2500 oC

2300 oC

0.6

→

Temperature in the center of a mould, Tc [°C]

35

1.2

3

0.04

4

0.8

3

Surface temperature of a heater, Ts [°C] 0.02 Fig. 2. Plotted corrections to determine the true temperature in the centre of graphite mould for the 30 mm diameter heater.

0

0.4

2 0

5

10

15

1 25

20

0

2 0

5

10

15

20

1 25

Sintering time, t [minutes]

ρ¼

γf hf : ðh0 −xi Þγth

Here xi is the recorded current linear shrinkage; γth is theoretical density. This formula shows that the relative density is in inversely proportion to sample height during densification. 4. Results and discussion The initial relative density as a function of sintering time was averaged and smoothed (by piecewise polynomial approximation using least-squares criterion). However, data processing by this method is time-consuming. Since integration smoothes data, the Χd(n, ρ) values were computed with the author's custom Fortran code for nonlinearity exponent values from n = 1 up to n = 5, including n = 3.5 and n = 4.5, were plotted for each relative density curve, in addition to the exponents corresponding to the sintering condition. Based on the calculations described above, it was determined that n = 4 for both of carbides powders densification during the pressure sintering. Obtained time dependence linear functions of Χd(n = 4,ρ) for each sintering condition enabled averaging easily from the slopes of the lines. Plots of Χd(n = 4,ρ) as functions of time t, under different external pressures, at chosen temperatures in the range from 2100 °C to 2500 °C during the WC powder pressure sintering are shown in Fig. 3, and for TiC powder in the sintering range from 2100 °C to 2700 °C in Fig. 4. Inverse transformation of Χd(n = 4,ρ) into ρ, taking into account linear dependence of integral function on time, enabled the time dependent plotting of relative density, ρ, for WC and TiC samples during the pressure sintering (Figs. 5 and 6 respectively). The linearity of the integral relative density function Χd(n = 4,ρ) on time enables easy determination of rate and dependence on pressure. As the forth power of pressure in accordance with Eq. (8) results in large numerical values, it is preferable to plot the values of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 4 ðd=dt ÞΧd ðn ¼ 4; ρÞ as functions of applied pressure P (Fig. 7). It can be seen in the figure that the function values are defined and positive for negative values of pressure, suggesting the presence of Laplace's pressure that is unknown in advance. The estimated mean value of Laplace's pressure was determined to be 5.8 MPa for WC and 7.2 MPa for TiC, even though the initial mean size of WC particles was less than that for TiC particles. This contradiction may be explained by angularity,

Fig. 3. The integral function, Xd(n = 4,ρ), vs. time, t, for the sintering of WC powder under applied pressure of 3.49 (1), 6.86 (2), 12.76 (3) and 16.21 MPa (4).

characteristic of WC particles, which could cause the decrease of effective Laplace's pressure. The plots in Fig. 7 may be used for selection of temperature and pressure conditions, ensuring higher relative density and mechanical properties of material in experimenting. For this purpose, it is necessary to extrapolate Fig. 7. In such an approach the WC sample relative density of 0.992 can be reached at approximately 2500 °C under pressure of 30 MPa during 25 minutes or under pressure of 50 MPa during 5.5 minutes. When pressure of 50 MPa is taken for sintering time of 25 minutes the reachable relative density will be 0.99754, which is near the non-porous state. However, it has to be taking into account the transition to the stage of viscous flow controlled by diffusion creep. Temperature sensitivity of pressure sintering is determined by the activation energy of the carbide matrix creep mechanism. The activation energy can be estimated with the slopes of the lines in the plots of Fig. 7. The plot of ln(T(d/dP)(d/dt)Χd(n = 4,ρ)) in dependence on

2.0 2300 oC

0.08

2700 oC 5

1.5

5

4

4

1.0

3

0.04

Xd (n = 4, ρ)

respectively. The initial height, h0, was calculated as a sum of final height of sample, hf, and linear shrinkage, xf, i.e. h0 = hf + xf. The current relative density ρ was calculated as follows

2

3 2

0.5 1

1

0

0 2500 oC

2100 oC

0.012

0.4

5

5

4

0.008

4 3

3

0.2

2

2

0.004

1

1

0

0

5

10

15

20

25

0 0

5

10

15

20

25

Sintering time, t [minutes] Fig. 4. The integral function, Xd(n = 4,ρ), vs. time, t, for the sintering of TiC powder under applied pressure of 5.68 (1), 6.86 (2), 9.31 (3), 11.18 (4), and 12.76 MPa (5).

M.S. Kovalchenko / Int. Journal of Refractory Metals and Hard Materials 39 (2013) 32–37

Temperature: 1 − 2100 C; 2 − 2300 C; 3 − 2400 C; 4 − 2500 C 1

4 3 2

4 3 2

0.9

1

0.8

1

Relative density, ρ

0.7

P=6.86 MPa

0.6

P=16.21 MPa

((d/dt)Xd (n= 4, ρ))1/4 [s−1/4]

36

0.25

4 WC

0.2

3

0.16

0.15

2

0.12

0.1

1

0.08

4 TiC

3 2 1

0.04

0.05 0

−5

0

5

10

15

0

−5

0

5

10

15

Applied pressure, P [MPa] 0.5 0.9

0.7

1

1

0.6 0.5

4 3 2

4 3 2

0.8

P=12.76 MPa

P=3.49 MPa 0

5

10

15

20

0

5

10

15

20

25

Sintering time, t [minutes] Fig. 5. The mean relative density, ρ, vs. time, t, during the sintering of WC powder.

reciprocal thermodynamic temperature, T−1, is shown in Fig. 8. Evaluating the activation energy during the pressure sintering of metal powders allows one to neglect the decrease in the elastic limit, τ0, of the matrix, which occurs at higher temperatures. The estimated activation energy are 591 kJ/mol for WC and 573 kJ/mol for TiC powder particle creep during the pressure sintering. The estimated value for WC is consistent, within the limits of experimental precision, with the activation energy of W in WC which is 138 kcal/mol = 577.8 kJ/mol [29]. The activation energy of Ti in TiC, 148 kcal/mol = 620.9 kJ/mol [30], is higher than our estimation. However, compared with the data for other carbides

Pressure: 1 − 5.68; 2 − 6.86; 3 − 9.31; 4 − 11.18; 5 − 12.76 MPa

1

[3], the value 620.9 kJ/mol seems to be somewhat overestimated. As it was shown [3], there is a derived relationship between the activation energy Q for self-diffusion of the metal atoms in carbides, as well as in pure metals, with cubic crystal structure and melting point, Tm, in the form: Q/Tm ≡ 36 cal/mol. If the melting point of TiC is assumed to be 3530 K [30], then the activation energy for self-diffusion, as well as chemical diffusion, of Ti in TiC is 127.1 kcal/mol = 532 kJ/mol, which is less than the activation energy for TiC creep during the pressure sintering. Our value is intermediate between the data above. The measured creep rate value affirms that the power law creep of carbides in study is determined by the dislocation climb mechanism [27,28] with diffusion in metal sublattice of the carbide crystalline grains. In order to elucidate the transition from power law creep to diffusional creep during pressure sintering of WC powder at 2500 °C the dependences of integral functions Xd(n = 1,ρ) upon t 1/2 are plotted (Fig. 9) in accordance with estimation of Eq. (14). As it is seen in this figure, all the curves are approximately linear in the late stage of pressure sintering as mean-square root stress, 〈τ〉, is decreasing. The linear approximation is more apparent at lower applied pressures. The results suggest that the transition to the late stage of pressure sintering is controlled by kinetics of diffusional creep.

Pressure sintering eliminates mesostructural defects, which characterize pressureless sintering of powder materials. Defects elimination promotes densification, structure formation, and attainment of high

5 4 3 2 1 1 2 3

0.8

Fig. 7. The fourth-root rate of integral function ((d/dt)Χd(n = 4,ρ))1/4 vs. applied pressure, P, for the pressure sintering of WC powder at temperature of 2100 (1), 2300 (2), 2400 (3), and 2500 °C (4) as well as TiC powder at temperature of 2100 (1), 2300 (2), 2500 (3), and 2700 °C.

5. Conclusions

5 4 0.9

3.6

0.7

2300 oC

o

2700 C

ln(T(d/dP)(d/dt)Xd(n = 4, ρ))

Relative density ρ

0.2

0.6 0.5

5

0.9 0.8

5 4 3 2

4

1 1

0.7 0.6 0.5

10

15

20

2.4

TiC

2

2500 C

2100 C 5

2.8

o

o

0

3

2

WC

3.2

0

5

10

15

20

25

1.6 3.2

3.6

4

4.4

104/T [K-1]

Sintering time, t [minutes] Fig. 6. The mean relative density, ρ, vs. time, t, during the sintering of TiC powder.

Fig. 8. The logarithm, ln(T(d/dP)(d/dt)Xd(n = 4,ρ)), vs. reciprocal thermodynamic temperature, T 1, for the pressure sintering kinetics of WC and TiC powders.

M.S. Kovalchenko / Int. Journal of Refractory Metals and Hard Materials 39 (2013) 32–37

t1/2 [minutes]1/2 0

2500 oC

0.6

Xd (n = 1, ρ)

1

0.4

2

References

4

5

16

25

Pa M 1 .2 16 .76 12

6.86 3.49

0.2

0

3

0

1

4

9

37

Sintering time, t [minutes] Fig. 9. The integral function, Xd(n = 1,ρ), vs. square-root time, t1/2, for the sintering of WC powder at 2500 °C.

mechanical and functional properties of sintered materials for structural and tool applications. The rheological models of deformable, irreversibly compressible, porous bodies that are based on continuum mechanics and creep theory of crystalline materials are used to quantitatively describe the compaction of the tungsten and titanium carbides powders under pressure sintering in isothermal conditions. Densification of the porous body occurs under action of Laplace's pressure, generated by surface tension, and applied pressure. The estimated mean value of Laplace's pressure was determined to be 5.8 MPa for WC and 7.2 MPa for TiC. The densification kinetics of tungsten and titanium carbides is controlled by nonlinear steady-state creep, which occurs at a rate proportional to the fourth power of stress in carbide matrix forming porous material. Based on the results in Fig. 8, the estimated average values of activation energy of powder particle creep are 591 kJ/mol for WC and 573 kJ/mol for TiC during the pressure sintering, which are consistent with diffusion of the metal in its respective carbide. It is shown that the kinetics are controlled by power law creep in an initial and intermediate stages of pressure sintering. However, in the later stage of sintering the kinetics are controlled by linear diffusional creep.

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