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A Phenomenological Model (and Experiments) for Liquid Phase Sintering

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Two problems in liquid phase sintering (LPhS) are analyzed. In the first case, the densification obtained from particle rearrangement is related to the volume fraction of the liquid phase and the solubility of the solid in the liquid. The model is based upon the assumption that particle rearrangement is related to the dissolution of the solid into the liquid. Sintering experiments carried out with salt particles and water support the results from the model. These experiments provide further insights into the influence of particle size on densification. In the second problem, the kinetics of densification by particle rearrangement is related to the rate of dissolution of the solid into the liquid. These results are compared to the kinetics of densification by the solution-precipitation mechanism, which is shown to be much slower than particle rearrangement. The results from the two problems are then combined to delineate the regimes of particle rearrangement (stage I) and solution-precipitation (stage II) mechanisms. The implication of the contact angle on sintering behavior is discussed.

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Abbreviations

V A :

initial volume of the particles

V L :

initial volume of the liquid on a per particle basis

v L :

volume fraction of the liquid phase: v L  = V L /( V L  + V A ) ≈ V L /V A

Δv A :

volume fraction of particles, which are dissolved in the liquid at saturation

v * :

phenomenological parameter for relating densification to Δv A

Ω A :

molar volume of the solid phase, A, divided by the Avogadro’s number

x A :

mole fraction of solute A into the liquid

x sat :

mole fraction of the solute in the liquid at saturation

α A , a A :

activity coefficient and activity of the solute A in the liquid

Δμ A :

difference in the chemical potential of A in the solid and as a solute

k:

Boltzmann’s constant

k 1 :

dissolution rate constant at the solid liquid interface, expressed as interface velocity per one kT potential difference across the interface

τ D :

nondimensional time parameter to describe stage I sintering

τ SP :

nondimensional time parameter to describe stage II sintering

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Acknowledgments

This work was supported by a Faculty Fellowship from the Council of Research and Creative Work, the University of Colorado at Boulder, and by the University of Trento. This work was initiated under a Humboldt Senior Scientist award to RR under the auspices of the Institute for Metallforschung, Max Planck Institute (Stuttgart, Germany).

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Correspondence to Rishi Raj.

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Manuscript submitted March 29, 2006.

Appendices

Appendix A

Equations for relating shrinkage to relative density.

The volumetric shrinkage during sintering is related to the axial strain, ε z , and the transverse strains, ε x and ε y . If the corresponding sample dimensions are \( L^{o}_{x} ,\,L^{o}_{y} ,\,{\hbox{and}}\,L^{o}_{z} \) at time = 0 and L x , L y , and L z as a function of time, then

$$ \varepsilon _{z} = \ln {\left( {\frac{{L_{z} }} {{L_{z} ^{o} }}} \right)},\,\varepsilon _{x} = \ln {\left( {\frac{{L_{x} }} {{L_{x} ^{o} }}} \right)},\,{\text{and}}\,\varepsilon _{y} = \ln {\left( {\frac{{L_{y} }} {{L_{y} ^{o} }}} \right)} $$
(A1)

Writing the initial volume as V g , where the subscript stands for the volume of the green compact, and the time-dependent volume as V, the volumetric strain, ε a , is given by[25]

$$ \varepsilon _{a} = {\text{ln}}{\left( {\frac{V} {{V_{g} }}} \right)} = {\text{ln}}{\left( {\frac{{L_{x} L_{y} L_{z} }} {{L^{{\text{o}}}_{x} L^{{\text{o}}}_{y} L^{{\text{o}}}_{z} }}} \right)} = \varepsilon _{x} + \varepsilon _{y} + \varepsilon _{z} $$
(A2)

If shrinkage is isotropic, then

$$ \varepsilon _{a} = 3\varepsilon _{z} $$
(A3)

Our experience with the salt experiments suggests that gravity can influence the shape of the sample by creep during the sintering process; this made it necessary to use the full Eq. [A2] to calculate the sintering strain.

The relative density, ρ, the green density, ρ g , and the volumetric strain, ε a , are related by the following definitions:[25]

$$ \rho _{g} = \frac{{V_{{th}} }} {{V_{g} }},{\text{and }}\rho = \frac{{V_{{th}} }} {V} $$
(A4)

where V g is the volume of the green compact, and V th is the volume of the theoretically dense specimen. The volumetric strain can now be related to ρand ρ g :

$$ \varepsilon _{a} = {\text{ln}}\frac{V} {{V_{g} }} = {\text{ln}}{\left( {\frac{V} {{V_{{th}} }}.\frac{{V_{{th}} }} {{V_{g} }}} \right)} = {\text{ln}}\frac{{\rho _{g} }} {\rho } $$
(A5)

Equation [A5] can now be used to calculate the maximum shrinkage strain expected for a given green density because it shrinks to its theoretical value when ρ→1:

$$ {\left| {\varepsilon ^{{\max }}_{a} } \right|} = {\text{ln}}\frac{1} {{\rho _{g} }} $$
(A6)

Equations [A5] and [A6] were used to compare experiment with theory, the latter being given by Eq. [3].

Appendix B

Powder preparation for LPhS of silicon-nitride

The experiments were done with two liquid phase additives, LiAlSiO4 and LiYO2. The sintering additive LiYO2 was prepared by mixing as-received Y2O3 and Li2CO3 powders and calcining at 1400 °C for 4 hours. The additive LiAlSiO4 was used in the form of a Li-exchanged zeolite. Substitution of Li for Na in A-zeolite was accomplished by an ion exchange procedure that was repeated eight times to ensure complete displacement of ions. After the last exchange step, the product was heated at 800 °C in order to cause a collapse of the zeolite structure with simultaneous formation of a glassy phase of the composition LiAlSiO4. Phase analysis using X-ray diffraction confirmed the amorphous nature of the powder.

A commercially available Si3N4 powder was used as the starting powder (Silzot HQ, SKW, Trostberg, Germany; 80 pct α-phase content; average particle size of 1.7 μm; specific surface area of 3.2 m2/g; and impurities (in mass pct): Fe < 0.04, Al < 0.1, O < 0.5, SiC < 0.4, and free Si < 0.5). Mixtures of the Si3N4 powder with 5, 10, 15, and 20 wt pct of sintering additives were prepared by attrition milling for 4 hours in isopropanol, using polyamide containers and stirrers and Si3N4 milling media. The slurries were dried in a rotating vacuum evaporator, followed by heating for 16 hours at 65 °C in a drying oven. The powders were subsequently sieved to obtain granules with a maximum size of 160 μm. Green body compaction was carried out by cold isostatic pressing at 240 MPa. A relative density of about 57 pct was obtained.

The densification behavior was studied in constant heating rate experiments, carried out in a pushrod dilatometry system from Theta Industries, Port Washington, NY. This system gives data for the linear shrinkage in the specimen, without compensation for the thermal expansion. The thermal expansion strain was estimated to be less than 0.6 pct, which is more than an order of magnitude lower than the sintering strain. Nevertheless, the densification strain was measured not from the dilatometry data but by substituting the values for the final and green densities into Eq. [A5]. These density values were obtained from the Archimedes’ liquid displacement technique (in water). At least 0.5 mm of the material was removed from the sample surface before the density measurement in order to exclude effects of surface porosity from the sample surface by grinding.

The sintering experiments in the dilatometer were carried out under flowing nitrogen at a heating rate of 5 K/min. After a holding time of 5 minutes at the plateau temperature of 1500 °C, the furnace was switched off, which resulted in an initial cooling rate of about 50 K/min. The time derivative of the linear shrinkage data obtained from dilatometry (the derivative distinguishes sintering shrinkage from thermal expansion) showed the onset of sintering at 1080 °C for the LiAlSiO4 additive and 1200 °C for the LiYO2 additive. In all instances, the densification was completed before the end of the sintering cycle, as manifested by a zero value for the rate of change of displacement measured by the dilatometer. The samples were examined in a scanning electron microscope at different stages in the sintering cycle. There was no evidence of sintering before the onset of melting of the liquid phase. Full details of these sintering experiments are given in References 26 and 27.

The results for both sets of experiments are given in Figure A1. The plots in the left figure show the final densities achieved in the constant heating rate experiments as a function of the volume fraction, v L , of the liquid phase, as estimated from the starting compositions. The right-hand figure is a plot of the data according to Eq. [3] and is equivalent to the plot for the salt-water experiments that was given in Figure 3.

Fig. A1
figure 8

(Left) Experimental measurements of densification of silicon-nitride with the volume fraction of two different liquid phase additives. (Right) Plot of the densification data according to Eq. [3]

Appendix C

Characteristic time for stage II sintering by the solution-precipitation mechanism.

The characteristic time for sintering by the solution-precipitation mechanism is derived by using the equivalence between sintering and creep. As in the case of particle rearrangement, the kinetics of the process is assumed to be interface controlled. The equations for solution-precipitation creep have been derived in Reference 10. We draw an equivalence between creep and sintering equations to obtain the desired result, because the principal difference between them lies in the driving force. In creep experiments, it is the applied stress, while in sintering, it is the capillary force that arises from the surface tension and pore size. The pore size is assumed to scale with the particle size. The influence of the driving force may be integrated into the analysis by using equivalence, as follows:

$$ \frac{{\tau _{{SP}} }} {{\tau _{{SS}} }} = \frac{{\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }_{{{\text{Coble}}}} }} {{\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }_{{SP}} }} $$
(C1)

Here, τ SP is the sintering time by the solution-precipitation mechanism, τ SS is the time for solid-state sintering, while \( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }_{{SP}} \) and \( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }_{{{\text{Coble}}}} \) are the creep rates equations for solution-precipitation and solid-state diffusional creep. The inverse of the creep rates is used because we seek to calculate the time (not the rate) for the sintering process. The principle of equivalence used in Eq. [C1] is that the geometric and driving force factors embedded in Coble creep[9] and solid-state sintering carry over into solution-precipitation creep and solution-precipitation sintering. The equations for Coble creep and solution-precipitation creep are drawn from the literature:[9,10]

$$ \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }_{{{\text{Coble}}}} = 132\frac{{\sigma \Omega _{A} }} {{{\text{k}}T}}.\frac{{\delta _{B} D_{B} }} {{p^{3} }} $$
(C2)

and

$$ \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }_{{SP}} = \frac{{\sigma \Omega _{A} k_{1} }} {{{\text{k}}Tp}} $$
(C3)

In writing Eq. [C3], we have assumed that the load bearing area of the grain boundary containing the liquid phase, as discussed in Reference 10, is equal to 0.5.

We must now derive an expression for solid-state sintering by grain boundary diffusion. This expression is derived using the analysis for grain boundary diffusion controlled void growth at grain boundaries under the influence of a tensile applied tensile stress.[28,29] In the sintering process, the applied stress is replaced by the sintering stress;[30] this exercise leads to the following equation for solid-state sintering:

$$ \tau _{{SS}} = \frac{{{\text{k}}Tp^{4} }} {{18\Omega _{A} \delta _{B} D_{B} \gamma }}.\frac{{(1 - \rho _{g} )^{{3/2}} }} {{F_{B} (\theta )}} $$
(C4)

Note that the grain size (in the models for creep) has been replaced by the particle size, p. The term σ is the applied uniaxial stress in creep, δ B D B is the grain boundary width times the grain boundary diffusion coefficient, and γ is the surface energy. The quantity F B (θ) is the geometrical factor, which gives the projected area of the void in the boundary plane;[11] it is a function of the dihedral angle of the voids, θ. A reasonable value for F B (θ) is 2, which obtains for a dihedral angle of 75 deg. For simplification, we assume that ρ g  = 0.65.

Substituting Eqs. [C2] through [C4] into Eq. [C1] gives the desired result, which is quoted as Eq. [12] in the main text.

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Raj, R., Rixecker, G. & Valentinotti, M. A Phenomenological Model (and Experiments) for Liquid Phase Sintering. Metall Mater Trans A 38, 628–637 (2007). https://doi.org/10.1007/s11661-006-9072-7

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