A. Density and microstructure

The densities, microstructure, and XRD patterns of B₄C samples sintered by SPS at temperatures varying from 1450 to 2000 °C are shown in Fig. 1. Figure 1(a) plots the densities of B₄C samples as a function of the processing temperature. It appears that with the sintering temperature increasing, the density of boron carbide increases. The density of B₄C achieves near theoretical density of 2.51 g/cm³ at 2000 °C. Three different sintering stages were observed within the temperature range of 1450–2000 °C. In the first stage of sintering, densification of B₄C is found to be slow when the temperature is <1600 °C. During the second stage of sintering, the density increases suddenly and it has steeper slope when the sintering temperature is 1600–1900 °C, which indicates that the B₄C powder has rapid densification during the temperature range and it plays a dominant role in the densification of B₄C. During the third stage or final stage of densification, the density curve of B₄C have slowest rate due to the elimination of residual porosities. Because of the different powder sizes of initial B₄C powder, the increase densification rate has some differences from the reported literature data. ^{Reference Sairam, Sonber, Murthy, Subramanian and Fotedar21}

FIG. 1.

(a) Density of SPS B₄C samples sintered as a function of sintering temperature for 20 min; (b) the optical microstructures of 99.6% dense B₄C compact sintered at 2000 °C for 20 min, (c) the XRD patterns of B₄C at 1450, 1700 and 2000 °C for 20 min prepared by SPS.

Figure 1(b) shows the typical optical microstructures of etched B₄C specimens sintered at 2000 °C. The grains are equiaxed and 2–10 μm in size. The average size of B₄C grains is 3.1 μm, which is close to the mean particle size (3.5 μm) of the initial powder. It is worthwhile pointing out that the grain shape and sizes of as-produced B₄C are different from hot-pressing B₄C samples, which appear to have straight grain boundaries and a wide range of particle sizes of 6–48 μm. ^{Reference Wu, Zeng and Yuan26} These results indicates that due to the short sintering time of SPS, the B₄C grains are not grown even at their full densification temperature, which reflects the advantage of SPS method.

XRD patterns of boron carbide samples sintered by SPS at 1450, 1700, 2000 °C are shown in Fig. 1(c). No additional oxidation phase was observed during sintering process. The peaks for B₄C can be clearly observed in the patterns and the main peak positions do not move with different sintering temperature, which indicates that B₄C is very stable in the SPS sintering process. A small peak in the XRD pattern is observed at about 2θ = 26.51°, which was assigned to graphite. ^{Reference Li, Jiang, Zhang, Lin, Chen and Huang18} There are two sources of graphite in the samples. The main source is from the raw powders of B₄C and other maybe diffused from the graphite foil during the sintering.

SEM micrographs of B₄C prepared at 1450, 1600, 1850, 2000 °C for 20 min are shown in Fig. 2. It is clearly observed that increasing sintering temperature, the porosity decreased and the density of B₄C was increased significantly. Grain growth was not observed at 1450 °C as shown in Fig. 2(a). At 1700 °C [Fig. 2(b)], sintering necks appeared and the samples still had interconnected porosity. When the sintering temperature was ≥1850 °C [Fig. 2(c)], there was isolated individual small pores displayed in the matrix and the relative density reached 87.7%. Nearly fully dense sample (relative density 99.6%) was obtained after sintering at 2000 °C [Fig. 2(d)]. In addition, some low-density crystal defects, such as twins, were observed in the sample [Fig. 2(d)]. It is well known that boron carbide has a α-rhombohedral crystal structure, which is hard to slip in plastic deformation. During applied pressure and high temperature; however, it can deformed by mechanical twinning or growth twinning mode, which need higher deformation stored energy and may shear to the matrix. The nucleation and growth mechanisms of these compression twins in boron carbide is need for further investigation.

FIG. 2.

SEM images of the B₄C samples sintered at (a) 1450 °C, (b) 1600 °C, (c) 1850 °C, and (d) 2000 °C under the heating rate of 100 °C/min and the holding time of 20 min.

B. Model and calculation

1. The Olevsky continuum model

The Olevsky continuum model is based on the traditional hot-pressing model and accounts for electric. The model framework includes three constitutive models which consider external load, surface tension, and a temperature gradient. ^{Reference Olevsky22}

a. Power-law creep mechanism of material transport

Considering the SPS pressing in a rigid die, the deformation of material occurs only in the x direction (the piston displacement direction). The equation was derived by Olevsky can be represented as:

(1) $${\dot \varepsilon _{{\rm{crx}}}} = - {\left\{ {{{{{\left( {{{3\theta } \over 2}} \right)}^{{{m + 1} \over 2}}}\left[ {{{3\alpha } \over {2G}}{{\left( {1 - \theta } \right)}^2} - {{\bar \sigma }_x}} \right]} \over {A{{\left( {1 - \theta } \right)}^{{{m + 3} \over 2}}}}}} \right\}^{{1 \over m}}}\quad ,$$

where ${\dot \varepsilon _{{\rm{crx}}}}$ is the macroscopic shrinkage rate, ${\bar \sigma _x}$ is effective (far-field) external stress in the x direction; α is the surface tension; A and m are the power-law creep frequency factor and the power-law creep exponent, respectively; G is the average particle radius, θ is the porosity.

Following Olevsky, ^{Reference Olevsky25} the power-law frequency factor can be written as:

(2) $$A = {A_0}\exp \left( { - {{{Q_{{\rm{cr}}}}} \over {RT}}} \right)\quad ,$$

where A ₀ is the power-law creep frequency factor, Q _cr is the creep activation energy; T is the absolute temperature, and R is the gas constant.

This equation mainly describes the macroscopic phenomena. For a specific material, the shrinkage rate due to the power-law mechanism was only influenced by G, T, ${\bar \sigma _x}$ , and θ.

b. Grain-boundary diffusion driven by an externally applied load and surface tension

Grain-boundary diffusion is another major mechanism of mass flux for SPS. Figure 3 shows the micro-mechanic model for a powder compact of simple-packed, rectangular grains with semi-axes a and c and elliptical pores located at the grain Quadra-junctions. This constitutive modeling was developed from early micro–mechanic model of the grain-boundary diffusion under applied pressure, and from the model of power-law creep under hot deformation of powders. The macroscopic shrinkage rate $\bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{st}}}$ of the grain-boundary diffusion mechanism can be represented as:

(3) $$ \bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{st}}} = - {{9\alpha {\delta _{{\rm{gb}}}}{D_{{\rm{gb}}}}\Omega } \over {4\pi kT{G^4}}}{{{{\left( {\sqrt \pi - 2\sqrt \theta } \right)}^2}} \over {1 - \theta }} \cr \times \left\{ {\left( {\sqrt \pi - 3\sqrt \theta } \right)\sqrt \theta - {{2\sqrt \pi \theta } \over {9{{\left( {1 - \theta } \right)}^2}\left( {\sqrt \pi - 2\sqrt \theta } \right)}}\left( {{{{{\bar \sigma }_x}G} \over \alpha }} \right)} \right\}\quad ,$$

where, D _gb and δ_gb are the diffusion coefficient and the thickness of the grain-boundary, respectively; k is the Boltzmann constant; Ω is the atomic volume.

FIG. 3.

Pore-grain structure unit cell. Rectangular grains with semiaxes a and c contain elliptical pores with semi-axis a _p and c _p located at the grains’ Quadra-junctions. Temperature and stress gradients are nonzero along grain boundaries. ^{Reference Olevsky22}

For a specific material, the shrinkage rate was only influenced by G, T, ${\bar \sigma _x}$ , and θ. This modeling only considered the effect of grain-boundary diffusion on the shrinkage rate or the density.

c. Thermal diffusion contributing to the intensification of densification during SPS

Based on the flux (along various paths) equation corresponding to thermal diffusion, the macroscopic contribution of thermal diffusion to the shrinkage rate $\bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{td}}}$ was derived:

(4) $$\bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{td}}} = - {{{\delta _{{\rm{gb}}}}{D_{{\rm{gb}}}}\Omega {C_{\rm{V}}}{H_{\rm{m}}}} \over {{\lambda _{\rm{T}}}k{{\left\langle T \right\rangle }^2}}}\left( {C{{\partial T} \over {\partial t}} - {\lambda _{\rm{e}}}{E^2}} \right) \times {{3{{\left( {\sqrt \pi - 2\sqrt \theta } \right)}^2}\theta } \over {2\pi G\left( {1 - \theta } \right)}}\quad ,$$

where, 〈T〉 is the spatial average of temperature; H _m is the enthalpy of vacancy migration; C _V is the vacancy concentration; λ_T is the thermal conductivity; C is the heat capacity; t is the time; λ_e is the specific electric conductivity; E is the electric field intensity.

In SPS method, the heating rate is more higher than hot-pressing methods. According to this model, when $\left| {C{{\partial T} \over {\partial t}} - {\lambda _{\rm{e}}}{E^2}} \right| \gg 0$ , the thermal diffusion has a great influence on SPS. Only the terms λ_e E ² and $C{{\partial T} \over {\partial t}}$ have the same order of magnitude, the heating rate ${{\partial T} \over {\partial t}}$ can be significantly influenced by the shrinkage rate $\bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{td}}}$ .

The combined contributions of the grain-boundary diffusion [Eq. (3)], power-law creep [Eq. (1)], and thermal diffusion [Eq. (4)] to the shrinkage rate under SPS are given by:

(5) $${\dot \varepsilon _x} = \bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{st}}} + {\dot \varepsilon _{{\rm{crx}}}} + \bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{td}}}\quad .$$

For the calculation of the porosity (relative density), the following continuity (mass conservation) equation is used:

(6) $${{\dot \theta } \over {1 - \theta }} = {\dot \varepsilon _x}\quad .$$

Substituting Eqs. (1)–(4) and (6) in Eq. (5), we obtain the Olevsky continuum model of SPS [Eq. (7)]:

(7) $$ {{\dot \theta } \over {1 - \theta }} = {{\dot \varepsilon }_x} = - {{9\alpha {\delta _{{\rm{gb}}}}{D_{{\rm{gb}}}}\Omega } \over {4\pi kT{G^4}}}{{{{\left( {\sqrt \pi - 2\sqrt \theta } \right)}^2}} \over {1 - \theta }} \cr \times \left\{ {\left( {\sqrt \pi - 3\sqrt \theta } \right)\sqrt \theta - {{2\sqrt \pi \theta } \over {9{{\left( {1 - \theta } \right)}^2}\left( {\sqrt \pi - 2\sqrt \theta } \right)}}\left( {{{{{\bar \sigma }_x}G} \over \alpha }} \right)} \right\} \cr {\rm{\ \ \ \ \ \ \ \ \ \ }} - {\left\{ {{{{{\left( {{{3\theta } \over 2}} \right)}^{{{m + 1} \over 2}}}\left[ {{{3\alpha } \over {2G}}{{\left( {1 - \theta } \right)}^2} - {{\bar \sigma }_x}} \right]} \over {{A_0}\exp \left( {{{{Q_{{\rm{cr}}}}} \over {RT}}} \right){{\left( {1 - \theta } \right)}^{{{m + 3} \over 2}}}}}} \right\}^{{1 \over m}}} \cr - {{{\delta _{{\rm{gb}}}}{D_{{\rm{gb}}}}\Omega {C_{\rm{v}}}{H_{\rm{m}}}} \over {{\lambda _T}k{{\left\langle T \right\rangle }^2}}}\left( {C{{\partial T} \over {\partial t}} - {\lambda _{\rm{e}}}{E^2}} \right) \times {{3{{\left( {\sqrt \pi - 2\sqrt \theta } \right)}^2}\theta } \over {2\pi G\left( {1 - \theta } \right)}}\quad .$$

2. Calculation of the SPS densification B₄C by the Olevsky continuum model

In the calculations, three parameters of C, λ_T and λ_e are functions of porosity θ ^{Reference Olevsky22,Reference Olevsky and Molinari27–Reference Maxwell, Parrott and Stuckes30} :

(8) $$\eqalign{C = {C_{{\rm{fd}}}}\left( {1 - \theta } \right)\quad {\rm{,}} \cr {\lambda _{\rm{T}}} = {\lambda _{{\rm{Tfd}}}}\left( {1 - {3 \over 2}\theta - {1 \over 2}{\theta ^2}} \right)\quad , \cr {\lambda _{\rm{e}}} = {\lambda _{{\rm{efd}}}}{{1 - \theta } \over {1 + 2\theta }}\quad , \cr}$$

where, C _fd, λ_Tfd, and λ_efd are the heat capacity, the heat conductivity and the electric conductivity of the fully dense material.

The contributions of different mass transport mechanisms ( $\bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{st}}}$ , ${\dot \varepsilon _{{\rm{crx}}}}$ , $\bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{td}}}$ ) to the overall shrinkage rate of B₄C under SPS conditions are calculated based on Eq. (7) and shown in Fig. 4. It shows that thermal diffusion contribution to the shrinkage rate is significant, especially for high temperatures and smaller particle sizes. For a boron carbide powder, the electric field is assumed to be 300 V/m, porosity θ is 0.3, the externally applied stress ${\bar \sigma _x}$ is 45 MPa, and the heating rate is 100 K/min. Two average particle sizes: G = 3.5 μm [Fig. 4(a)] and G = 7 μm [Fig. 4(b)] were analyzed. The material parameters for B₄C are given in Table I.

FIG. 4.

Contribution to shrinkage rate from different mechanisms of mass transport by different grain size of B₄C powder; (a) particle size G = 3.5 μm; (b) particle size G = 7 μm (electric field 300 V/m, applied stress 45 MPa, porosity 0.3, heating rate 100 K/min).

TABLE I.

Material parameters for boron carbide.

^a Refer to the parameters of the similar materials of silicon carbide.

During calculation, quantitative verification can be conducted by comparing the theoretically predicted densification kinetics with the experimental data. The temperature and pressure regimes used in the calculations replicate the conditions of the experiment. The pressure is expected to be constant and equal to 45 MPa. The boron carbide specimens with an initial porosity of 42% are assumed to be heated to 2000 °C at a rate of 100 °C/min. According to the previous SEM experimental results, we assumed that the particles did not grow during the SPS.

The shrinkage kinetics as well as the temperature is shown in Fig. 5. The modeling results for B₄C are obtained by the numerical solution of the ordinary differential Eqs. (7) and (8) by the Runge–Kutta method of the 4th order. The experimental data for the shrinkage kinetics is also shown in Fig. 5. The results show that with increasing sintering time, the relative density of B₄C decreases, and it is close to full density after 1000 s. The calculated results are agreement with the experimental points, except for a minor deviation in time at high temperature.

FIG. 5.

Relative density kinetics of B₄C powder versus sintering time during SPS. Comparison of the Olevsky continuum model (blue line) with experimental data (points).

The influence of different sintering parameters on sintering behavior of B₄C powder was calculated by continuum model. Figure 6(a) shows the shrinkage kinetics of B₄C powder with different average particle sizes. It showed that the full-density temperature was decreased from 2000 to 1900 °C with a decrease of the initial particle size from 4 to 3 μm. Figure 6(b) shows the shrinkage kinetics of B₄C powder with different heating rate. As the heating rate increased from 50 to 300 °C/min, the heating time required to achieve full density decreased from 1200 to 300 s. Figure 6(c) shows the shrinkage kinetics of B₄C powder with different applied stress. When the applied stress was increased, the sintering temperature decreased to achieve the same density. However, compared to the original particle size, the porosity is not so sensitive to applied stress in the range of 30–50 MPa during SPS.

FIG. 6.

Shrinkage kinetics of B₄C powder with: (a) different average particle size (applied stress 45 MPa, heating rate 100 K/min), (b) different heating rate (applied stress 45 MPa, particle size 3.5 μm), (c) different applied stress (particle size = 3.5 μm, heating rate 100 K/min).

C. Role of the current

Joule heating from the large current (800–6500 A) passing through the die (typically graphite) and sample (if the sample is conducting) is the major difference between convention of hot-pressing and the SPS method. Passing of high current may have three roles: providing direct Joule heating, creating plasma, and changing in defect or enhancing mobility of the defects, which can enhance the sintering process. ^{Reference Munir, Anselmi-Tamburini and Ohyanagi14} The recent results reported by Olevsky ^{Reference Aleksandrova, Ilyina, Grigoryev and Olevsky41} shows that the current passed through a copper specimen can achieve higher relative density than where no current was used at the same sintering condition. For the conducting sample, the current can pass through the sample and the die (if the die is conductive) directly, and then create a plasma between the contact point of two powder particles. This may lead to sintering enhancement, changing in defect, and enhancing mobility of the defects, and thus enhance mass transport. For the nonconducting sample, no current passed through the sample, and so the current only passes through the die. When the current crosses from the punch to the die (deflection current), a magnetic field was generated. ^{Reference Zhang42} Variable magnetic field can generate an electric field, so an electromagnetic field was generated when the nonconducting sample was sintered by SPS.

As a semiconductor, the electric conductivity of full density B₄C was about 33 s/m. This may allow a part of current through the sample and another part of current passing though the die. Therefore, there may be Joule heating, discharge, and electromagnetic field during sintering process of B₄C. In initial stage, the contact points between the particles offer high resistance to the current flow, the temperatures at those localized heating zones would exceed the melting temperature of the powder and electric discharge may happen in this zones. Sairam et al. ^{Reference Moshtaghioun, Gómez-García, Bernardi-Martín, Domínguez-Rodríguez and Monshi19} reported the partial melted microstructure of B₄C compact processed at 1800 °C by SPS, which is direct evidence for localized overheating mechanism. As the contacts between particles increase under pressure and heating, this phenomenon will becomes less and even disappear. The electromagnetic field only exists in heating-up period because there is no pulse current. During the late-stage of sintering, when the temperature has reached 2000 °C, it is found that thermal diffusion was the main diffusion mechanism (Fig. 4). At this stage, the effect of temperature on further densification was not obvious, where it mainly influence the grain shape and boundary. After holding for 20 min at 2000 °C, as shown in Fig. 2(d), the grain presented no-regular hexagonal shape and the partial melted structural may disappear. Particularly, because the current can enhance the sintering process, the creep activation energy used in the modeling was ${5 \over {33}}{Q_{{\rm{cr}}}}$ .

D. Initial grain size and heating rate

As shown in Fig. 4, the effect of initial particle size on shrinkage rate is significant because particle size can change the mass transport mechanisms of B₄C by SPS. For small particle size B₄C, thermal diffusion is the main mass transport mechanism. Figure 6(a) shows the shrinkage kinetics of B₄C powder with different average particle size. The full density temperature was decreased from 2000 to 1900 °C with decrease of initial particle size from 4 to 3 μm.

Shrinkage kinetics of boron carbide powder with different heating rate is shown in Fig. 6(b). As the heating rate increased from 50 to 300 °C/min, the time required to achieve full density was decreased from 1200 to 300 s, indicating that the higher heating rate can effectively reduce the sintering time. Johnson ^{Reference Shen, Johnsson, Zhao and Nygren43} qualitatively confirmed that high heating rates can minimize the surface diffusion processes of alumina particles during SPS, but the heating rate has no obvious influence on the sintering temperature and density of alumina. In the present work, the calculated curve of porosity versus temperature was unchanged with the different heating rate, which agrees very well with result of alumina.

In the Olevsky continuum model, the constitutive equation of thermal diffusion contains the parameters of the heating rate ${{\partial T} \over {\partial t}}$ . From the part of the modeling $\left( {C{{\partial T} \over {\partial t}} - {\lambda _{\rm{e}}}{E^2}} \right)$ , our calculation indicates that, even the heating rate range from 1 to 1000 °C/min, if assumed there is no particle coarsening during SPS, the shrinkage rate $\bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{td}}}$ only changes about 0.4‰. This change is relatively insignificant as compared with the total changes. Therefore, shrinkage kinetics of B₄C powder sintered by SPS was insensitive to heating rate. But in some cases, particle/grain growth and heating rate are necessary to consider. Olevsky ^{Reference Olevsky25} derived the grain growth kinetics of aluminum:

$$\dot G = \left\{ {\matrix{ {\alpha \left( {{1 \over K}} \right)\varpi \ln {{235{K \mathord{\left/ {\vphantom {K s}} \right. \kern-\nulldelimiterspace} s}} \over \varpi }{G_0}{{\left( {{{{G_0}} \over G}} \right)}^3}{\theta ^{ - 1.34}},} {{\rm{if\ }}T > 533{\rm{\ K}}} \cr {{\rm{0,}}} {{\rm{if\ }}T \le 533{\rm{\ K}}} \cr} } \right.\quad ,$$

where, α is a constant, and ϖ = dT/dt is the heating rate, K/s. From this expression, the grain growth kinetics was sensitive to particle size G and heating rate ϖ. The modeling indicates that small particle size and higher heating rate can lower the sintering temperature effectively. Thus for some materials such as aluminum, shrinkage rate may be sensitive to heating rate.

E. Applied stress

The constitutive modeling of power-law creep and sintering stress contain an important parameter of applied stress ${\bar \sigma _x}$ . Figure 6(c) shows the calculated results of four external applied stress: ${\bar \sigma _x}$ = 30 MPa, ${\bar \sigma _x}$ = 40 MPa, ${\bar \sigma _x}$ = 45 MPa and ${\bar \sigma _x}$ = 50 MPa. According to Eq. (1), it can be seen that the influence of stress on shrinkage rate ${\dot \varepsilon _{{\rm{crx}}}}$ depends on the balance between the term ${{3\alpha } \over {2G}}{\left( {1 - \theta } \right)^2}$ and the term ${\bar \sigma _x}$ . From our estimation, the value of ${\bar \sigma _x}$ is 2–10 times as large as the term ${{3\alpha } \over {2G}}{\left( {1 - \theta } \right)^2}$ . Combining Eq. (3) with our estimates, it shows that the influence of stress on shrinkage rate $\bar \dot \varepsilon _{{\rm{gbx}}}^{{\rm{st}}}$ is minor. As shown in Fig. 6(c), with the applied stress increased, the sintering temperature was decreased. However, compared with other parameter such as the original powder particle size, the influence of external applied pressure on the density of boron carbide is very small and can be ignored in the 30–50 MPa range.