Thermodynamics of Hard Phases

CLASSES OF MATERIALS

1.05 Cermets

1.05.1 Thermodynamics of Hard Phases

studies (Jung & Kang, 2000; Jung, Kang, Jhi, & Ihm, 1999; Kang, 1997) concluded that the mixing reaction of TiC and TiN or mixing of C and N in TiC (B1) structure does not happen ideally. Details to understand the characteristics of the Ti(CN) phase are as follows.

The standard state formation reaction of Ti(CxNy), at any temperature, from Ti, C andP_N2 ¼ 1 atm can be expressed as

TiþxCþy=2N₂/Ti CxNy

D_fG^o_TiðCNÞ (i)

wherexþy¼1 andxandyare the mole fractions of carbon and nitrogen, respectively, andD_fG^o_TiðCNÞis the free energy of Ti(CN) formation at standard state. This reaction is virtually the sum of the following three reactions:

TiþC/TiC D_fG^o_TiC (ii)

Tiþ1=2N₂/TiN D_fG^o_TiN (iii)

xTiCþyTiN/Ti CxNy

D_fG^M_TiðCNÞ (iv)

Thus, the free-energy change in the formation of a new solid solution at a particular temperature and any N₂ pressure, i.e. non-standard state,D_fG^M_TiðCNÞð ¼ G^M_TiðCNÞÞ, is the sum of the proportioned free-energy changes to form each component,xD_fG^o_TiCandyD_fG^o_TiNand the free-energy change in mixing to produce a complete solid solution. Therefore, in any N₂partial pressure, if ideal mixing is assumed,

G^M_TiðCNÞ ¼ D_fG^M_TiðCNÞ ¼ x

D_fG^o_TiCþRTlnx þy

D_fG^o_TiN1

2RTlnP_N2þRTlny

(1) The composition of the most stable carbonitride can be determined where the partial differential ofD_fG^M_TiðCNÞ with respect tox(ory) is zero. The formation energy of the most stable phase should be the lowest. From this calculation, a simple relationship can be derived for ideal mixing, betweenP_N2 and the C:N ratio,ðx=yÞ, in the equilibrium carbonitrides.

RTln x

y

P¹²_N

2 ¼

D_fG^o_TiND_fG^o_TiC

(2) Table 1lists the calculated N contents,y, in the Ti(CxNy) solid solution based on the ideal model. A sig- nificant loss of N from the carbonitride phase is predicted as the reaction temperature increases or as the N₂ pressure decreases. The values are compared with experimental results to examine the validity of the ideal solution model and the results are summarized inTable 2(Kang, 1997).

The equilibrium N content predicted byEqn (2)is 0.016 at 1500C and 10⁴atm. However, the amounts of C and N lost in the various Ti(CxNy) compositions during vacuum processing are insignificant compared with the starting compositions. Also, the C:N ratio remains, more or less, constant regardless of the exposure time

Table 1 Variation of N content,y, in Ti(CxNy) solid solutions with N2partial pressure and processing temperature. The values are calculated based on an ideal solution model: note decrease in cubic phase N content with decreasing N2pressure Processing temperature (C) N₂pressure (atm) N content

1500 0.01 0.14

0.1 0.33

1 0.61

10 0.83

1800 0.01 0.04

0.1 0.11

1 0.29

10 0.56

and processing temperature. This discrepancy limits the relevance of many estimates derived from an ideal solution approach to actual processing.

Another interesting observation made from this experiment is that the relative stability or loss of C and N depended on the composition, i.e. C:N ratio. As the processing time and temperature increases, the C content in the system continues to decrease gradually even though it looks like a minor quantity in terms of atomic percentage. This becomes obvious when the C:N ratio approaches zero. It implies that C is less stable in Ti(CxNy) than N in this temperature range. A measurable C and/or N loss of about 2 at% is noted in all cases except Ti(C_0.5N_0.5). Thus, the gas evolution during sintering with these solid solutions might be linked closely to the escape of C in the system. The highest stability is obtained from an equiatomic C:N ratio, as observed by other researchers (Nishigaki & Doi, 1980; Nishigaki, Ohnishi, Shiokawa, & Doi, 1974). Similar observations to those given inTable 2have also been made with Ti(CN)–WC–TaC–Ni–Co (Kang, 1996) and Ti(CN)–Mo₂C–Ni–Co (Park, 1994) systems. The level of O in all samples varied in the range 0.01–0.02 at%. A particularly high level of O (0.02–0.04 at%) was observed in TiN. The role of oxygen in the system has not yet been identified.

In a non-ideal solution case,Eqns (1) and (2)above can be changed as follows:

D_fG^M_TiðCNÞ ¼ x

D_fG^o_TiCþRTlng_TiCx þy

D_fG^o_TiN1

2RTlnP_N2þRTlng_TiNy

(3)

RTln x

y

P_N¹²

2

g_TiC g_TiN

¼

D_fG^o_TiND_fG^o_TiC

;

(4) whereg_TiCandg_TiNare the activity coefficients. These equations enable us to measure the deviation of this solid solution from the ideality. Thus, usingEqn (4)and the experimental data (Kieffer et al., 1971b; Schick, 1966) shown inFigure 1, the values of the term (g_TiC/g_TiN) are estimated as a function ofx(¼1y). An integration method (Belton & Fruehan, 1967) derived from the Gibbs–Duhem equation for a binary system is used to calculate the activities of TiC and TiN as a function ofxory. A variation of the Gibbs–Duhem equation is written as below:

xdðlna_AÞ þydðlna_BÞ dðlna_BÞ ¼ dðlna_BÞ (5) wherea_iis the activity ofith component. WhenEqn (5)is rearranged by replacing A and B with TiC and TiN, the following equations can be obtained:

lng_TiCjX_TiC ¼ Z^XTiC

X_TiC¼1

X_TiNd

ln g_TiN

g_TiC

; (6)

lng_TiNjX_TiN ¼ Z^XTiN

X_TiN¼1

X_TiCd

ln g_TiC

g_TiN

; (7)

whereX_TiCandX_TiNare the same asxandyinEqn (1) or (2), respectively.

Table 2 Variation of N content,y, in Ti(CxNy) as a function of maximum processing temperature and holding time

Maximum processing temperature (C) Holding time (h)

TiC Ti(C0.7N0.3) Ti(C0.5N0.5) Ti(C0.3N0.7) TiN

u v u v u v u v u v

As received – 0.97 0.01 0.68 0.28 0.51 0.46 0.31 0.64 0.03 0.95

800 0 – – 0.67 0.28 0.49 0.45 0.30 0.65 – –

1200 0 – – 0.66 0.27 0.49 0.46 0.30 0.65 – –

1200 1 0.96 0.02 0.66 0.28 0.49 0.46 0.30 0.65 0.02 0.94

1500 0 0.96 0.02 0.66 0.27 0.49 0.46 0.30 0.65 0.01 0.97

1500 1 0.96 0.02 0.66 0.27 0.50 0.45 0.30 0.65 0.02 0.98

1500 5 0.95 0.02 0.66 0.27 0.50 0.46 0.29 0.64 0.02 0.91

Cermets 141

Based on these equations, the activity coefficients for the TiC–TiN mixture are calculated; these are presented graphically inFigure 2, showing a strong negative deviation from ideality (non-ideal behavior) regardless of reaction temperature. This means that the formation of Ti(CN) occurs more readily than an ideal mixing of TiC and TiN. As expected, the tendency to form the Ti(CN) solid solution is much stronger at low temperatures (1400C) than at high temperatures (1800C). The activity coefficients of TiC vary from 0.04 to 0.3 at 1400–1500C as the N₂ pressure changes while high coefficients (0.2–0.7) are found at 1800C. The Figure 1 Variation of Ti(CuNv) composition with N2partial pressure.

Figure 2 Activitiesaof TiC and TiN in Ti(CN) solid solution at 1400 and 1800C: values obtained fromFigure 1.

coefficients of TiN remained in the same range (0.2–0.8) for both temperatures. This result suggests that TiC has a higher tendency to form a solid solution than TiN. This analysis confirms a higher thermal stability of C and N in Ti(CN) than expected by an ideal solid-solution model.

The diffusivity of C in TiC_0.89–0.97 is reported to be in the range 10¹¹–10¹²cm²s¹ at 1500C. The extrapolation of these data results in C diffusivity of 10¹²–10¹³cm²s¹at 1350C. Diffusivity increases as the non-stoichiometry of TiC increases (Matzke, 1989). Similarly, the diffusivity of N in TiN has been reported to be of the order 10¹⁴cm²s¹ at 0.5T_m (1350C). Nitrogen atoms move in the TiN matrix at a rate equivalent to or slower than C atoms in TiC at 1500C. Since the diffusivity of a species in a system is pro- portional to the gradient of the activity coefficient (affinity) with respect to concentration, the diffusivity data are in agreement with the activity data. It suggests further that the diffusivity of N remains low compared with that of C in the Ti(CN) solid solution. This accounts for the low dissociation rate of N observed. A similar diffusion behavior of C and N is expected to occur in complex carbonitride systems such as (Ti,M₁,M₂)(CN) solid so- lutions as long as the second and third metallic phases are minor (<20%) (Kang, 1996; Park, 1994).

Basically, the diffusivity and dissociation rate of nitrogen in Ti(CN)-based systems are largely controlled by the Ti–N bond strength in the structure. The strengths are normally quantified by measurements of enthalpy for the dissociation reaction. Some work has been done to estimate the Ti–C and Ti–N bond strengths in TiC and TiN (Gingerich, 1979, pp. 289–300; Kohl & Stearns, 1970), but this needs more investigation. However, the bond strengths in diatomic molecules have been measured spectroscopically for Ti–C and Ti–N bonds and the room temperature strengths are reported to be 423 and 476 kJ mol¹, respectively (Bilz, 1958; Gupta & Gin- gerich, 1980; Kohl & Stearns, 1974; Rundle, 1948). All this evidence indicates a stronger Ti–N bond than Ti–C bond in B1 (NaCl) structure. Overall results confirm that the reaction between TiC and TiN exhibits a strong negative deviation from the ideal solid-solution behavior in the formation of Ti(CN). The formation of Ti(CN) occurs more readily than an ideal mixing of TiC and TiN.

With the formation energy values of various Ti(CxNy) in the following section (Jung et al., 1999; Jung & Kang, 2000), a regular solution model, which is not a function of temperature, is examined for its applicability to Ti(CN) solid solutions. The variation of free energy with temperature can be obtained as long as the excess Gibbs free energy, G^XS, is known (Darken, 1967; Gaskell, 2003). In order to obtain G^Mð ¼ D_fG^M_TiðCNÞÞ as a function of temperature,G^XSis calculated by means of the results in the previous section. The heat of mixing for a regular solution is expressed by the equation below:

DH^M ¼ G^XS ¼ a⁰X_TiCX_TiN; (8) whereDG^M¼DH^MþTDS^Manda⁰must be a constant which is independent of temperature and whereX_TiC andX_TiNare the same asxandyinEqn (1) or (2), respectively. However, the data shown in previous work (Jung & Kang, 2000) did not show that a⁰ is a single value as required in the case of a regular solution, showing temperature dependence. Thus, it is concluded that the Ti(CN) solid solution fails to be a regular solution.

TheG^XSof the solution is expressed as a temperature function to see if a sub-regular solution model works. By the addition of a temperature function,G^XScan be represented as inEqn (5). The coefficient,a, is related to the extent of deviation from ideal behavior. The value ofsindicates the degree of sensitivity ofG^XSto temperature dependence. The larger the absolute value ofs, the weaker is its dependence with temperature. Assincreases, G^XSinEqn (5)becomes close to that of a regular solution. Infitting the data usingEqn (4), the values ofaands were calculated to be6.94 and 889.9 K, respectively (Jung & Kang, 2000).

G^XS ¼ aX_TiCX_TiN

1þT s

(9) A general function ofG^Mcan be expressed in the incorporation ofG^XSintoEqn (6)and the relative stability can be estimated as a function of temperature. TheG^Mcurves at various temperatures are shown inFigure 3.

G^M ¼ G^oþDG^M ¼ X_TiCD_fG^o_TiCþX_TiND_fG^o_TiNþ

DG^M;idþG^XS (10) As is seen inFigure 3, the free energy near the TiN composition is lower than that of TiC at low temperatures (below 1500 K), whereas TiC tends to have lower values at higher temperatures (above 2500 K). It means that the relative stability among Ti(CN) solid solutions is strongly dependent on temperature. Based on the above Cermets 143

analysis, the solution behavior is found to be beyond a regular solution range and it could show ordering tendency to some extent as reported.

1.05.1.2 Formation Energy of Ti(CN),G^M

Based on the TiC–TiN solution model, a value for the free energy of formation of Ti(CN) was obtained using the following equations and the data from JANAF tables (Jung & Kang, 2000).

G^M ¼ D_fG^M_TiðCNÞ ¼ X_TiCD_fG^o_TiCþX_TiND_fG^o_TiNþDG^M; (11) whereDG^M, the change in the free energy of mixing, is equal to entropy of mixing,RT{X_TiCln_TiCþX_TiNln_TiN}.

InFigure 4,G^M,idandG^M,realrepresent the absolute free energy of formation based on ideal mixing of TiC with TiN, and that obtained by using the derived activity coefficients in previous section, respectively. TheG^Mgraphs of 1700 and 2100 K were drawn as a solid and dashed curve, respectively. The data in the figure are those calculated by means of pathwise integration.

Each curve forG^M,realhas a unique minima at certain compositions. At 1700 K, this minimum is located in the nitrogen-rich composition (X_TiN¼0.6–0.7). However, the minimum point swings to carbon-rich compo- sition (X_TiN¼0.3–0.4) at 2100 K. This plot demonstrates that a stable composition exists for various temper- atures.G^Mis an indicator of the phase stability of a material. The solid solution with high nitrogen content would be expected to be more stable at 1700 K, while that with high carbon content to be more stable at 2100 K.

Figure 3 Curves of the free energy of a solid solution of Ti(CN) calculated from a subregular solution model.

Figure 4 The Gibbs free energy of formation of,G^M, Ti(CN)(kJ mol¹): the minimum points ofG^M,realare atXTiNw0.7 and 0.4 for the temperature of 1700 and 2100 K, respectively.

The behavior of the solid solution is in line with previousfindings, ruling out the previous understanding that nitrogen in Ti(CN)-based cermets has a destabilizing effect by dissociating in sintering at 1400–1500C.

However, the more negative is G^M, the less a substance evaporates or reacts. Thus, the composition with a minimumG^Mvalue would be expected to be the most stable, even if the stability region were to be extended further to the compositions in the vicinity of the minimum point.

The free energy of mixing for Ti(CN) solid solutions is calculated using theoretical models to obtain the free energies of the formation of various Ti(CN) at specific temperatures of interest (Jung et al., 1999). The stability regions of such solid solutions are then compared with empirical data obtained at 1700 and 2100 K (Jung &

Kang, 2000). Previous reports have discussed the relative strength of the chemical bonding in Ti(CN) solutions compared with TiC and TiN phases (Zhukov, Gubanov, Jepsen, Christensen, & Andersen, 1988; Zhukov, Medvedeva, & Gubanov, 1989). The maximum energy of formation of carbonitrides was calculated to be10 kJ mol¹at 0 K. These values deviate from linear behavior at varying nitrogen contents when the linear muffin-tin orbital method is used. Although the magnitude of the energy of formation itself shows some discrepancy relative to a recent pseudopotential method (Jhi & Ihm, 1997), the trend is consistent in both calculations.

Using thefirst-principles pseudopotential total energy method, the internal energy of various Ti(CN) solu- tions was calculated. Since the calculated equilibrium volume of a Ti(CN) solid solution nearly obeys Vegard’s law (Jhi & Ihm, 1997), the internal energy can be regarded as the enthalpies of formation. Hence the enthalpy change of mixing at 0 K is expressed as below:

DH^M_0k ¼ U_0k^MðTiC_1xNxÞ xU_0k^o ðTiNÞ ð1xÞU_0k^o ðTiCÞ (12) whereU^Mand U^orepresent the internal energies of the Ti(CN), TiC or TiN phases, respectively. As the tem- perature increases, the entropy contribution to phase stability becomes important. The lattice vibration energy and the electronic excitation energy as well as the configuration term all influence phase stability. A conven- tional approach was taken to estimate the thermal effect of lattice vibrations and electronic excitations. This was carried out within the framework of frozen-ion motion and the rigid band model (Ashcroft & Mermin, 1976;

Lebacq, Pasturel, Manh, et al., 1996). The free energy induced by lattice vibrations was calculated using the Debye model:

G_vibðTÞ ¼ H_vibðTÞ TS_vib ¼ 3RTDðq_DÞ 3RT 4

3Dðq_DÞ log

1exp

q_D T

(13) whereq_DandD(q_D) are the Debye temperature and its function, respectively. With the help of the rigid-band model and the density of states at a given temperature, the free energy of electrons at a given temperature,T, can be expressed as follows:

G_elðTÞ ¼ H_elðTÞ TS_el ¼ 1

2gT²þT Z^N

N

Nðε;TÞ½Fðε;TÞlogFðε;TÞ þ f1Fðε;TÞglogf1Fðε;TÞgdε (14) wheregrepresents the constant which is proportional to the density of states at the Fermi level (Guillermet, Haglund, & Grimvall, 1993), while N(ε,T) and F(ε,T) represent the density of states and the Fermi–Dirac distribution function, respectively. For the density of states,N(e, 0) was used in place ofN(e,T). The results indicate that the electronic part provides a significant contribution to the energy change of mixing although the absolute value itself is smaller than that of the lattice vibrations. The thermal excitation of electrons depends on the available states existing above the Fermi level as represented in the density of states. The electronic structure calculation shows that the density of states increases rapidly up to a nitrogen content of 0.5 in the Ti(CN) solution from the linearly interpolated values, thus resulting in a large electronic thermal energy. This behavior of the density of states is also found in other transition metal carbonitrides, e.g. Hf(CN) (Jhi, 1998).

Based on these models, the enthalpy of formation for various Ti(C₁xNx) solid solutions was calculated at 0, 1700 and 2100 K. It was found an exothermic reaction, which indicates that it is energetically favorable for the mixture of TiC and TiN to form a solid solution. The maximum reduction in the energy of formation was obtained at nitrogen concentrations in the range of 0.2–0.4 with a value of approximately4.5 kJ mol¹. With an increase in temperature,DH^M_T increases by 1 kJ mol¹at 2100 K, maintaining the original shape of the curve at 0 K.

Cermets 145

1.05.1.2.1 Modeling Real Solution

In a real solution, the atomic distributions will be gradually randomized with increasing temperature (Chris- tian, 1975). Hence, the configurational entropy change of mixing is different from that of ideal mixing. It is controlled by the heat of mixing in that a negative or positiveH^M_T value signifies a deviation from random mixing with a tendency toward ordering or segregation. Therefore, the entropy change in mixing requires modification by a parameter,b, which should be a function of temperature andDH_T^M. The parameter,b, may be expressed as an exponential function with some constants,εandn, as below. In a high-temperature range, a realisticDS^M_conf can be obtained by means of proper constants:

b ¼ exp

εDH^M_Tⁿ RT

ðε;n0Þ (15)

DS^M_conf ¼ bDS^M;ideal (16)

WithEqn (1)–(9), it is possible to obtain the enthalpy and entropy change for the mixing of TiC with TiN as well as the absolute enthalpy and entropy. Finally, the free-energy change of mixing,DG^M_T, at any temperature can be obtained using the equations below:

DH^M_T ¼ DH^M_0KþDH^M_vibþDH^M_el (17)

DS^M_T ¼ DS^M_conf þDS^M_thermal (18)

The value forbwas plotted with respect to temperature at nitrogen contents of 0.25, 0.50, and 0.75 whereε andnare given as unity using theDH^M_T values obtained as a function of temperature (Jung et al., 1999). The condition thatεandnare equal to unity is arbitrarily chosen as a case of ordering since no experimental data are available for the determination of the upper bounds for ordering.bis found to constantly increase from zero and the curve for Ti(C_0.25N_0.75) provides the highest values forbbecause of its small heat of mixing. That is, the tendency to have random mixing increases with temperature and nitrogen contents in Ti(CN).

The total free energy of a solid solution,G^M, then, is expressed as below:

G^M ¼ D_fG_TiðCNÞ ¼ X_TiCD_fG^o_TiCþX_TiND_fG^o_TiNþDG^M (19) D_fG^o_TiCandD_fG^o_TiN are the standard-state energies of the formation of TiC and TiN, respectively (Chase, Davies, Downey, et al., 1985). The free energy of mixing,G^M, for the formation of Ti(CN) at various temper- atures is plotted and found the same as inFigure 4.

The stability region at 1700 K, where the values ofG^Mare close to the minimum value, is predicted near 0.6 whenεandnare equal to unity in the study (Jung et al., 1999). In contrast, the experimental value is 0.7 while the ideal mixing (entropy) model (ε¼0) predicts 0.75. When the temperature reaches 2100 K, the equilibrium nitrogen content moves tow0.3 while the experimental results indicate a value of 0.4. Considering the limited number of data points, the predicted values do not differ greatly from the experimental value, 0.7. With changes in theεvalue for the model, the stability regions of Ti(CN) cover a wide area with respect to nitrogen content, resulting in less dramatic differences in practical processing than expected.

Based on the observation above, it can be concluded that the stability region of the Ti(C_1xNx) solid solution tends to shift toward Ti(C_0.5N_0.5) at temperatures higher than 1000C. The change in the com- positions of the highest stability is less rapid than predicted by the ideal mixing model. This appears to be due to the strong affinity of carbon and nitrogen in the octahedral sites of the B1 structure as nearest neighbors. Because of the interactions between interstitial atoms, the free energy for Ti(CN) becomes more negative and provides an extra measure of stability to its elements in Ti(CN). The affinity of carbon and nitrogen as neighbors should affect the atomic arrangement to some extent at 1700 and 2100 K. Thus, the value forεcannot be zero, i.e. a complete random mixing is not guaranteed. Even in such a case, however, the degree of discrepancy in the G^M values between theoretical and empirical results appears to be within acceptable limits. In other words, the sums of the inharmonic effect of lattice vibrations, deviations from the rigid-band model, the correction in the density of states at high temperatures, and experimental errors might mutually compensate one another.