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Effects of local Joule heating during the field assisted sintering of ionic ceramics
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Journal of the European Ceramic Society 32 (2012) 3667–3674
Effects of local Joule heating during the field assisted sintering of ionic ceramics Troy B. Holland a,∗ , Umberto Anselmi-Tamburini b , Dat V. Quach a , Tien B. Tran a , Amiya K. Mukherjee a a
Department of Chemical Engineering and Materials Science, University of California, Davis, CA 95616, United States b Department of Physical Chemistry, University of Pavia, V.le Taramelli 16, 27100 Pavia, Italy Received 13 December 2011; received in revised form 15 February 2012; accepted 21 February 2012 Available online 17 March 2012
Abstract Recent investigations regarding the role of applied fields on the grain growth and densification behavior of ionic ceramics are providing strong insights into the efficacy of Field Assisted Sintering Technique (FAST), aka Spark Plasma Sintering (SPS). Explanations of the observed behaviors, such as grain growth suppression and densification enhancement, are based upon the conjectured presence of a Joule heating driven temperature differential between grain interfaces and grain cores. These differentials were thought to be responsible for providing increased densification rates and lower densification temperatures through grain growth suppression and/or increased local kinetics at the forming necks. In this paper, we analyze the energetic, thermal, and practical details of this process in the context of the commonly accepted stages of sintering. © 2012 Elsevier Ltd. All rights reserved. Keywords: Sintering; Thermal conductivity; Electrical conductivity; ZrO2 ; SPS
1. Introduction Recent investigations1–6 have resulted in a great deal of progress toward the understanding of the role applied fields may have on the sintering of ionic materials using either FAST/SPStype or4–7 “flash sintering” configurations.2,3 In these processes, the effects are currently often ascribed to the effects produced by localized Joule heating, from the flux of current during sintering and/or annealing. It has been proposed that such localized heating might affect grain growth and atomic mobility at the grain boundary. The increase in mobility at grain boundaries, due to this higher grain boundary temperature, has previously been proposed to describe the efficacy of microwave sintering ionic ceramic powders in the 1980s and 1990s.8,9 However, it was shown, by analytically modeling heat flow in sapphire slabs, that the heat flow from the boundaries rapidly ameliorated any severe temperature differences existing between bulk grain regions and grain boundaries.10 These approaches, however, did not account for the diminished interface areas consistent with the
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neck geometries during the early stage of sintering. On the other hand, Ghosh et al.1 have shown that an applied field, with a resulting flow of current, can greatly suppress grain growth in fully dense 3 mol% yttria-stabilized tetragonal zirconia (3YSZ). In a rather novel experimental approach, Ghosh et al.1 conducted a series of grain growth studies on 3YSZ samples with varied field strengths (<400 V/m), and an associated increase in current, across the cross section of a single sample. The grain sizes were then analyzed in accordance with the applied fields and a clear correlation between lessened grain growth and the increased electric field strength was shown. Applying this idea to sintering experiments, Yang and Conrad have demonstrated dramatically increased densification rates as well as reduced final grain sizes in the presence of modest applied DC (2000 V/m),67 and AC fields (1390 V/m, 60 Hz)4 in 3YSZ. They have also demonstrated an interrelationship between heating rates and the applied DC fields at similar field strengths.5 The explanation for grain growth suppression in the investigation requires that an applied field be in direct electrical contact with the samples, producing Joule heating from the flow of current across the grains and their interfaces. In all cases the experiments were conducted in air, assuring that the primary mechanism of conduction is ionic, and that no electrochemical reduction would be expected. Also in
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this case localized Joule heating are again suggested to be the operative mechanism by which increased densification rates and final grain sizes are achieved. Three aspects of the applicability of this mechanism to FAST/SPS experiments need addressing: (1) the thermodynamics and kinetics of the mechanism, (2) grain boundary versus grain core conductivities in the FAST/SPS environment, and (3) the time dependent nature of the temperature difference between grain cores and grain boundary regions. Analysis of the particular details of the process is critical to the determination of their applicability to traditional FAST/SPS sintering processes as well as to extend them into other manufacturing processes. In this paper, we analyze the thermodynamics and kinetic role that temperature gradients localized at the grain boundaries might play on the sintering behavior of ionic ceramics, focusing on the initial and final stages of sintering by FAST/SPS. We then numerically model the thermal development through Joule heating at the particle interfaces for various neck sizes and resistivities. 1.1. Grain growth suppression by localized Joule heating Grain growth suppression has been shown to provide easier densification as described by the sintering equation developed by Wang and Raj11 shown in here in Eq. (1), ρ˙ =
Af (ρ) (−Qb /kb T ) e Td n
(1)
where A is a material constant, f(ρ) is a function of density, T is temperature, kb is Boltzmann’s constant, d is grain size, and n is the grain size exponent and Qb is the activation energy for diffusion at grain boundaries. The grain size exponent, n, has been found to be 4 or 3 for grain boundary or volume diffusion-driven sintering, respectively. As a result, any process producing grain growth suppression during sintering has a potential beneficial effect on process itself. A localized heating of the grain boundaries can be shown to have such an effect. When grain boundary energy is taken into account, the differential of the total Gibbs free energy for a polycrystalline solid can be represented as in Eq. (2), dG = VdP − SdT + γgb dA
(2)
where A is total grain boundary area, γ gb is grain boundary energy (per unit area) and other parameters have their usual thermodynamic meanings. The pinning effect due to Joule heating at grain boundaries originally proposed by Ghosh et al.1 can be explained using the classical grain growth analysis introduced by Burke and Turnbull.12 Here the velocity of grain boundary migration νb is a product of grain boundary mobility Mb and the driving force for grain growth as shown in Eq. (3), 1 1 , + (3) ν = Mb γgb r1 r2 where r1 and r2 are principle radii of curvature of grain boundary surface. At constant pressure, the Gibbs free energy can be rewritten as dG = −SdT + γgb dA.
(4)
Cross-differentiation gives δS δγgb =− ≡ Sg , δA T,P δT A,P
(5)
where Sg is defined as grain boundary entropy (or entropy per unit grain boundary area) that is always a positive number.13 Therefore, the temperature coefficient of grain boundary energy γ is always negative. According to the model proposed by Ghosh et al.,1 when an electric current flows through polycrystalline sample the grain boundaries incur more Joule heating because the intrinsic resistivity at the grain boundary in ionic materials is higher than in the bulk of the grain and as a result of this increase in temperature, the driving force for grain growth is reduced through the −SdT contribution. This reduction gives rise to a pinning effect for grain growth even though the mobility, Mb , may be increased by the higher temperature at grain boundaries. Effectively, the source of grain growth is kinetically limited since the reduction in grain boundary energy lowers the driving force for grain growth. It is assumed here, consistent with all of the discussions on this subject to date, that no Soret Effect,14,15 which would provide a variation in concentration of the diffusing species, is present. If the Soret Effect were active, the mobility of the grain boundaries would be further lowered due to the slower diffusion of the rate limiting species from the grain cores to the interface region. Therefore, the hypothesis of Ghosh et al., necessarily presumes that the cores of the grains remain at a significantly lower temperature than the grain boundaries. Otherwise no reduction in grain boundary mobility would be expected or found by their argument. 1.2. Local temperature distributions in FAST/SPS Most ionic ceramic materials present some level of electrical conductivity at high temperature. The mechanism responsible for this conduction varies with the chemical nature of the material and with the experimental conditions. Ceramic materials are usually ionic, electronic, or mixed conductors. In the first case the charge carriers are ions only, while in the second case conduction by electrons or holes is observed. The details of the conduction mechanism could be quite complex, particularly in the case of mixed conductors. In these materials the relative concentration of the charge carriers is often controlled by complex chemical equilibria, while the carrier mobility might vary largely, even for the same type of carrier, from one compound to the other. This carrier mobility can vary according to grain size as well. For example, coarse polycrystals of pure ceria exhibit both electronic and ionic conduction while nanocrystalline ceria (∼10 nm) shows an increase of four orders of magnitude in electronic conductivity.16 In this work we focuses on purely ionic materials in which it has been noted that a significant fraction of the overall resistivity is due to the poor conductivity at the grain boundaries. In ionic conductors such as zirconia and doped ceria this “blocking effect” of the grain boundaries has been well-characterized. In early studies, it was considered to be associated with the segregation of impurities at the grain boundaries; later on it was recognized as a natural characteristic of
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the materials, caused by the depletion of charge carriers in the small region, often called the space charge layer (SCL), adjacent to the grain boundary core. This depletion derives from the electrostatic interaction between the charged point defects and the intrinsic charge always associated with the grain boundary “core” of the ionic materials.14 The presence of this grain boundary resistivity spurred the idea that when high current densities flow through an ionic conductor a local increase in temperature can be achieved at the grain boundaries. Cologna et al.2 describe the overall heat distribution in the “flash sintering” of nanograin zirconia by assuming uniform sample heating through Joule heating, and with black body radiation as the sole source of power dissipation. The analysis of the temperature increase would then follow as shown in Eqs. (6)–(8). In the absence of significant convection, Joule heating from the electrical current through the sample would be the source of temperature differences between the sample and furnace. In this case, the heat balance equation is represented by Eq. (6). W = Aσ(T14 − T04 )
(6)
where W is the electrical Joule heating from the grain boundaries, T0 is the furnace temperature, T1 is the temperature at the surface of the sample, A is the sample area, and σ is the black body radiation constant. Assuming that the furnace temperature is roughly equal to the surface temperature, the increase in the overall sample temperature is readily determined from Eqs. (7) and (8). T14 − T04 = (T12 + T02 )(T1 + T0 )T
(7)
W T = T0 4AσT04
(8)
Eq. (8) would be used to describe the overall temperature increase of the sample, versus the Joule heated temperatures of the grain boundaries, described by the growing neck areas and the current flow in the samples. Using this reasoning it has been suggested that the bulk of the power dissipation is within the grain boundaries alone due to the resistive differences between the grain boundaries and the grain cores in YSZ. Due to the small volume of the grain boundaries in comparison to the sample it is therefore assumed that the temperature of the grain boundaries is increased greatly (up to 200 ◦ C) while the average sample temperature may only demonstrate a 10 ◦ C difference. The resulting difference in temperature between boundary and core would therefore provide the driving force difference responsible for the extremely rapid increases in sintering rate. Presumably this analysis would be appropriate in describing the temperature distribution described qualitatively by Ghosh et al.1 In the case of nanometric grain sizes, nearly ubiquitous in use for FAST/SPS experiments, it may be helpful to consider the thermal conductivity controlled heat flow which occurs from the developed temperature gradient. The assumption of only grain boundary thermal generation may lead to significant errors in the microstructural thermal gradients, which could hamper the proper analysis of the operating mechanisms.
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2. Analysis In order to clarify the physical soundness of a model based on a localized Joule heating it is clearly important to evaluate the actual extent and duration of the resulting local increase in temperatures when a samples undergo a high density current flux. As we noted earlier an instantaneous T, between the grain boundaries and grain cores of over 200 ◦ C has been proposed, but the role of thermal conductivity in dissipating such a temperature difference is never been investigated. In the case of nanosized grains, such as those typically used in FAST/SPS processing, the mass of the grain cores is not particularly high relative to the grain boundaries, and as such, the thermal volume to be heated by the resulting heat flow into the nanograin can be expected to equilibrate quickly. Besides, it is important to note that any significant temperature difference between a nanometric grain’s boundary and its core produces extremely steep thermal gradients. In order to maintain a temperature difference of 10 K in a grain that is 100 nm in diameter a minimum temperature gradient of 2 × 107 K/m must be sustained despite thermal conductivity driven dissipation. Classical heat transfer analysis17 demonstrates, that in a slab of the same thickness, any thermal gradient would be disrupted in approximately 10−7 s using thermal conductivity values consistent with nanometric zirconia. A more quantitative understanding of the thermo-electrical details associated with the current flowing in a polycrystalline, ionically conductive ceramic can be developed using a few simple models. It is important to recognize that the models we developed account for, and represent, each stage of the sintering process. In the initial stage of sintering, particles are connected only by narrow necks producing a complex three-dimensional path for the flowing current. The reduced cross-sectional area of the necks produces a localized resistance increase that must be discussed and compared with the contribution from the known conductivity changes in the developing grain boundaries in the necks. Using numerical modeling we characterize the separate contributions of the reduced cross-section and of the intrinsic grain boundary resistivity to the temperature distribution within the particles. Fig. 1a represents our model using a linear array of spherical ceramic particles of radius r0 connected by rounded necks characterized by a radius rn . This is a simplified representation of a powder bed during the early stages of a sintering process, but it contains all the essential elements necessary to demonstrate the interrelationships of Joule heating and thermal conductivity. In the model without grain boundaries (Fig. 1a) the local heating is produced only by the restriction to the current flux introduced by the neck geometry. The electro-thermal problem can be modeled using Eqs. (9) and (10), = ∇ · |(−σ∇V ) = 0 ∇ · J = ∇ · (σ E) ρcp
δT = ∇ · (k∇T ) + q˙ j δt
(9) (10)
represents the current density, E the electric field, where J = σ E σ the electrical conductivity, V is the potential gradient, ρ the density, Cp the heat capacity, and q˙ j the heat generated by Joule
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Fig. 1. Schematic of the linear array of spherical particles used in modeling the temperature differentials between the grain boundaries and grain cores in the (a) absence and (presence) of a differently conductive grain boundary.
heating per unit volume per unit time. Here our assumed electric fields differ slightly from those we have reported elsewhere on the modelled18 effects of a field upon the densification of ionic ceramics. In that paper, the model assumes that no conduction occurs, while here we assume that the opposite is true in regards to free charges: no charge accumulation occurs to provide an increase in field strength. In principle this simply means that the accumulation of free charges does not occur, consistent with the homogenous solution of Gauss’ law in that work,18 and that at each point in the model the field strength is assumed to be a single value, in contrast with our findings of how a field interacts with dielectric particles and that at each point in the model the field strength is assumed to be a single value, in contrast with our findings of how a field interacts with dielectric particles. As discussed later, this simplification is largely inconsequential to the outcomes of the model. If grain boundaries are introduced into the neck, the situation becomes similar to one depicted in Fig. 1b. The SCL thickness is exaggerated for grains above 30 nm, but is consistent with many recent FAST/SPS experiments on nanometric powders. To accurately represent the contribution of a grain boundary to the local temperature we must use the electrical resistance associated with a single grain boundary. In YSZ, as in other polycrystalline ionic conductors, the contribution of the grain boundaries to the overall resistivity can be identified very accurately with impedance spectroscopy measurements.
This analysis relies on the fact that the grain boundary and the bulk regions are characterized not only by a specific contribution to the overall resistance (Rb and Rgb ) but also by a different capacitance. Determination of the impedance at different frequencies allows the individual contributions of Rb and Rgb to be calculated as long as the time constants of the RC elements are sufficiently different. This technique has been extensively used on ionic conductors.14 The experimental data are generally analyzed using a model for the material microstructure termed the “brick layer model”. In this model, the sample consists of a 3D network of conducting “bricks” (grain cores) surrounded by a resistive “mortar” (grain boundaries).19 The components of the overall resistance measured experimentally Rex = Rb + Rgb are related using Eq. (11), σgb D Rb = Rgb σb w
(11)
where σ b is the bulk specific conductivity, σ gb the specific grain boundary conductivity, D the grain size and w the grain boundary thickness. The value of w can be estimated on the basis of difference in capacitance between the bulk and the grain boundary components, and in YSZ it is generally found to be between 2 and 5 nm. The specific grain boundary conductivity is often found to be roughly two orders of magnitude lower than the bulk specific conductivity in ionically conductive ceramics. The
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resistance per unit surface of a single grain boundary may be estimated from the value of the specific grain boundary conductivity and the grain boundary thickness (corresponding to the SCL). In this paper we model heat transfer within multiple powder particles, and the boundary condition is conservatively considered to be adiabatic since these particles are surrounded by other particles in the pellet having very much the same temperatures. Furthermore, we consider a very short time scale for our simulations and this removes the need to account for heat conduction on a macroscopic scale. The models always included at least five spheres. The values for the electrical resistivity, heat capacity, and thermal conductivity of bulk zirconia have been taken from20 and represented by the relation log (σT). The simulations have been performed using a neck radius to particle ratios of 5% and 15%, corresponding to roughly the very early and middle of stage 1. A current flux of 104 A/m2 , measured at the largest cross-section of the particles, was allowed to flow through the system modeled to be at 1000 ◦ C at the beginning of the simulation. This current flow is unrealistically high for any FAST/SPS processing, but we used it in order to produce measurable temperature gradients. The field strengths necessary to provide this current flow are between 6.3 × 104 and 3.57 × 106 V/m, with smaller grain sizes and/or the presence of grain boundaries obviously requiring higher fields. These fields are higher than even the local field strengths modeled to occur by dielectric polarization,18 and in the case of FAST/SPS processing with a conductive graphite die as an alternative current pathway they are almost impossible to achieve in practice. Therefore, the results reported here can be considered the most extreme case possible for the achievement of high thermal gradients in typical FAST/SPS experiments. They are not unreasonably high in comparison to the external field strengths used in the “flash sintering” experiments of Cologna et al.,23 if one considers the possible contributions of polarization. It is important to note, however, that only the rate of heating is calculated from Joule heating, and if the mechanisms of Flash sintering are such that faster heating occurs, the models might underestimate the developed gradient. However, the dissipation of the gradient through the thermal conductivity and mass of the particles will be shown to be exceedingly rapid. The simulations have been interrupted when the temperature of the center of grains was increased by more than 50 K which is well above the overall temperature increase calculated in the experimental reports on current effects. The specific assumptions are: (1) the external surfaces of the particle are adiabatic, (2) the region in which Joule heating occurs in the grain boundary is significantly wider than the physical boundary, (3) complex impedance resistivity values reflect accurate values of conductivity, (4) the divergence of the electric field is zero, and (5) Joule heating alone provides the increase in local temperatures. As stated previously (1) is a conservative estimation, and would tend to overestimate the thermal gradients. Assumption (2) deals with the thermal boundary resistance, commonly referred to as the “Kapitza” resistance.21,22 The physical boundaries are contained within the “electrical grain boundaries” that generate the local Joule heating, for which there is no structural discontinuity. This implies
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Fig. 2. Peak temperature difference attained between the grain boundary (Tgb ) and grain cores (Tb ) with and without the presence of a differently conductive grain boundary. Model utilizes a fixed current flow and the relevant thermophysical and thermoelectrical properties of nanocrystalline zirconia.
that the area in which the Joule heating occurs has no real discontinuity in phonon distribution occurring in the presence of the regions at differing temperatures. This is not much different from rapid contact heating in various techniques such as FAST/SPS or microwave sintering with susceptors. Through assumption (3), we have accounted for increased resistances in the grain boundary of up to two orders of magnitude, which is, again, conservative in favor of the development of a thermal gradient. Assumptions (4) and (5) are intertwined in that the nature of charge development and accumulation are currently unknown, and the possibility of dielectric breakdown, plasma formation, or other unknown sources of heating are unaccounted. Two neck sizes consistent with early and middle density values for Stage 1 were modeled with and without the contributions of differently conductive grain boundaries with the results shown in Fig. 2. This graph helps to clarify a crucial point in the discussion of localized heating induced by the current flow. It is clear that the local temperature gradient is strongly dependent on the size of the grains. While spheres with a dimension in the millimeter range might experience a significant temperature gradient between the center and the boundary, when the size is reduced to the tens of microns, the temperature difference becomes insignificant. In the case of grains smaller than the micron scale, the thermal conductivity equilibrates any significant temperature gradient inside the grain, providing an evenly heated grain. Although the introduction of the grain boundaries provides a significant increase in the local temperature gradient, particularly when the neck size is small, the absolute values remain very low particularly as grain size decreases to sub-micron levels. Even accounting for possible differences in developing grain boundary conductivities, versus the equilibrium grain boundary conductivities often measured using impedance spectroscopy, it is clear that even across two orders of magnitude the thermal gradients are minimal at small grain sizes.
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Fig. 4. Thermal differential of the grain boundaries and cores of several starting grain sizes as a function of time. The changes in scale demonstrate the sensitivity with respect to grain size.
While there may occur temperature differentials by other means, such as dielectric breakdown, exceedingly small necks, or massively difference local conductivities, it is important to understand the rapidity by which the intrinsic thermal conductivity of the grains ameliorates the differential. To model this we use the geometry shown in Fig. 3a with a temperature delta between the boundary and core of 200 ◦ C and the thermal conductivity of nanometric zirconia. Fig. 3b shows the time it takes for the microstructure to remove the differential as a function of grain size (r0 ) and relative neck size (rn /r0 ). It is clear from the model that, as the starting grain size of the material approaches the nanometric dimension, as so often happens in FAST/SPS processing, the rate at which temperature gradients are removed is exceedingly fast. This helps to explain why the models shown in Fig. 2 demonstrate minimal gradients even with high current densities. The time frames with which the combined Joule heating and thermal equilibration mechanisms achieve their maximums and steady states at fixed current values are shown in Fig. 4 for various grain sizes. This clearly demonstrates that while the maximums gradients shown in Fig. 2
Fig. 5. Generalized grain boundary and core resistance model for Stage 3 sintering.
are somewhat small, as grain size is reduced the temperature gradient after a single second is much reduced. This would further limit the kinetic contribution of local Joule heating as it is extremely transient in nature as well. The sensitivity with respect to grain size is evident in the orders of magnitude changes of scale for each grain size, as well as an increase in the rate of equilibration. The 1 m grain size is the smallest grain size reasonably calculated due to the exceedingly small temperature differentials that develop and the short equilibration times. In the late stages of sintering pores have closed and ultimately are removed, and so a different model such as that shown in Fig. 5 must be considered. Again the process is found to be a function of the rates of heat generation and redistribution. Fig. 5 describes the classic slab heat transfer problem discussed previously, and as such need not be solved with numerical methods. The equilibration times are much faster due to the larger interface cross section providing better contacts and heat flow. As the more intense gradients in
Fig. 3. (a) Temperature equilibration model schematic and the (b) time required for thermal equilibration between the neck and grain cores as a function of grain size with Tn starting at 1200 ◦ C and the grain cores starting at 1000 ◦ C.
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the case of narrow necks are already found to be quite low, the thermal gradients in the fully dense specimens are a moot point.
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straight DC sintering experiments and SPS experiments which typically operate at <300 Hz DC pulsing). 4. Conclusions
3. Discussion It is clear from the analysis that the thermal conductivity, even in ceramics, reduces the effectiveness of localized Joule heating tremendously as grain sizes are reduced to those typically used in FAST/SPS operations. This is true across the spectrum of sintering, from the early neck formations in Stage 1 to geometries to the fully densified samples in Stage 3. The local temperature gradients would therefore seem to provide little benefit in practice to the thermodynamics (discussed in Eqs. (2)–(5)) or to the local kinetics. In describing the effects observed in very recent works on FAST/SPS, the new and novel works on “flash sintering”,2,3 and our own non-contact field experiments reported in elsewhere,23 it may be necessary to look for explanations outside of local Joule heating. This does not mean, however, that the local current densities are not contributing to the increased kinetics. Nor does it mean that the Joule heating at the grain boundaries is not contributing to the overall temperature and perhaps heating rate of the bulk sample. The role of fields and/or currents on the diffusion mechanisms dominant in each stage of sintering are still quite unclear. In terms of diffusivity, it has been shown that grain boundary diffusion is quite influenced by the presence of a field as has volume diffusion. In the case of surface diffusion, this is not the case. Presuming that surface diffusivities are insensitive to an applied field, then the balance of diffusivities24 would have a profound effect on the nature and temperature of the transitions from Stage 1 to Stage 2 sintering.25 This would be particularly true if the diffusivities of grain boundaries and volumes are themselves differently affected by the field. It has been shown,24,25 that the balance of the ratios of the grain boundary to surface diffusivity and bulk to surface diffusivity have strong influences on the chemical potential distributions at necks. Another explanation for the role of applied fields is perhaps necessary due to the lack of achievable temperature gradients in nanometric grains. In another work,18 we discuss the role of fields in the initial stages of sintering. The local field strengths are found to be high enough to (potentially) increase the likelihood of dielectric breakdown, alter the thermodynamic driving forces for cationic motion, or influence SCL development. However, in describing the late stages of sintering, the role of polarization is minimal at best. In determining the local field strengths in dense parts, Vollman et al.26 have shown that the local field strengths from conductivity differences scale, with some simplifications, as E = Eox (dg /dgb ) where dg is the thickness of the grain, Eox is the applied field strength, and dgb is the width of the grain boundary SCL. The criterion for making this simplification are that the Eox is comparatively small with respect to the SCL field strength, the resistance of the bulk is very much less than that of the grain boundary, and the time of Eox application is larger than the Maxwell–Wagner relaxation time,27,28 (a safe assumption in
The thermodynamic basis for the suppression of grain growth in the presence of a thermal gradient was developed. Using numerical models the thermal gradients attainable in nanometric powder specimens were calculated with the contribution of Joule heating. It is shown that no significant contribution of temperature gradient can be expected, in mixed or ionically conducting ceramics, due to the thermal conductivity and nanometric grain sizes used in most FAST/SPS processing. This is the case even at the unreasonably high current densities utilized in our numerical models. The observations of the contributory effects of an applied field or current to the densification rates of ionic ceramics still defy well-developed explanations. The role of the applied fields themselves may be a useful avenue to explore, since they are shown to provide contributions to the initial and final stages of sintering in many ionic ceramics. Acknowledgement This investigation was supported by a Grant from the Office of Naval Research (ONR Grant # N00014-10-1-0632) with Dr. Lawrence Kabacoff as the Program Manager. References 1. Ghosh S, Chokshi AH, Lee P, Raj R. A huge effect of weak dc electrical fields on grain growth in zirconia. Journal of the American Ceramic Society Aug. 2009;92(8):1856–9. 2. Cologna M, Rashkova B, Raj R. Flash Sintering of Nanograin Zirconia in <5 s at 850 ◦ C. Journal of the American Ceramic Society 2010;(September):3556–9. 3. Cologna M, Prette ALG, Raj R. Flash-sintering of cubic yttria-stabilized zirconia at 750 ◦ C for possible use in SOFC manufacturing. Journal of the American Ceramic Society 2010;(December). Online only currently. 4. Yang D, Conrad H. Enhanced sintering rate of zirconia (3Y-TZP) by application of a small AC electric field. Scripta Materialia 2010;63(3):328–31. 5. Yang D, Conrad H. Enhanced sintering rate and finer grain size in yttriastablized zirconia (3Y-TZP) with combined dc electric field and increased heating rate. Materials Science & Engineering A 2010;528(3):1221–5. 6. Yang D, Raj R, Conrad H. Enhanced sintering rate of zirconia (3Y-TZP) through the effect of a weak dc electric field on grain growth. Journal of the American Ceramic Society 2010;93(10):2935–7. 7. Conrad H, Yang D. Influence of an applied dc electric field on the plastic deformation kinetics of oxide ceramics. Philosophical Magazine 2010;90(March (9)):1141–57. 8. Janney MA, Kimrey HD. Microwave Sintering of Alumina at 28 GHz. In: Messing GL, Fuller Jr ER, Hausner H, editors. Ceramics powder science II, vol. 1. American Ceramics Society: Westerville, OH; 1988. p. 638–40. 9. Meek TT. Proposed model for the sintering of a dielectric in a microwave field. Journal of Materials Science Letters 1987;6(6):638–40. 10. Johnson DL. Microwave heating of grain boundaries in ceramics. Journal of the American Ceramic Society 1991;74(4):849–50. 11. Wang J, Raj R. Estimate of the activation energies for boundary diffusion from rate-controlled sintering of pure alumina, and alumina doped with zirconia or titania. Journal of the American Ceramic Society 1990;73(5):1172–5. 12. Burke J, Turnbull D. Recrystallization and grain growth. Progress in Metal Physics 1952;3:220–92.
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13. Ragone D. Thermodynamics of materials, vol. 2; 1995. 14. Mehrer H. Diffusion in solids: fundamentals, methods, materials, diffusioncontrolled processes. Berlin: Springer Verlag; 2007. 15. Olevsky EA, Froyen L. Impact of Thermal Diffusion on Densification During SPS. Journal of the American Ceramic Society 2009;92(s1): S122–32. 16. Chiang Y-M, Lavik EB, Kosacki I, Tuller HL, Ying JY. Defect and transport properties of nanocrystalline CeO[sub 2 − x]. Applied Physics Letters 1996;69(2):185–7. 17. Carslaw HS, Jaeger JC. Conduction of heat in solids. 2nd ed. Oxford University Press: New York; 1959. 18. Holland TB, Anselmi-Tamburini U, Quach DV, Tran TB, Mukherjee AK. Mukherjee., Local field strengths during early stage field assisted sintering (FAST) of dielectric materials. Journal of the European Ceramic Society, in press. 19. Kidner NJ, Perry NH, Mason TO, Garboczi EJ. The brick layer model revisited: introducing the nano-grain composite model. Journal of the American Ceramic Society 2008;91(6):1733–46. 20. Leclercq B, Mévrel R, Liedtke V, Hohenauer W. Thermal conductivity of zirconia-based ceramics for thermal barrier coating. Materialwissenschaft und Werkstofftechnik 2003;34(4):406–9.
21. Swartz E, Pohl R. Thermal boundary resistance. Reviews of Modern Physics 1989;61(3):605. 22. Every A, Tzou Y, Hasselman D, Raj R. The effect of particle size on the thermal conductivity of ZnS/diamond composites. Acta Metallurgica et Materialia 1992;40(1):123–9. 23. Holland TB, Tran TB, Quach DV, Anselmi-Tamburini U, Groza JR, Mukherjee AK. Athermal and thermal mechanisms of sintering at high heating rates with and without an externally applied field. Journal of the European Ceramic Society, in preparation. 24. Bouvard D, McMeeking RM. Deformation of interparticle necks by diffusion-controlled creep. Journal of the American Ceramic Society 1996;79(3):666–72. 25. Maximenko AL, Olevsky EA. Effective diffusion coefficients in solid-state sintering. Acta Materialia 2004;52(June (10)):2953–63. 26. Vollmann M, Hagenbeck R, Waser R. Grain-boundary defect chemistry of acceptor-doped titanates: Inversion layer and low-field conduction. Journal of the American Ceramic Society 1997;80(9):2301–14. 27. Kremer F, Schonhals A. Broadband dielectric spectroscopy. Berlin: Spring; 2003. 28. Yager WA. The distribution of relaxation times in typical dielectrics. Physics 1936;7(12):434–50.
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Effects of local Joule heating during the field assisted sintering of ionic ceramics Troy B. Holland a,∗ , Umberto Anselmi-Tamburini b , Dat V. Quach a , Tien B. Tran a , Amiya K. Mukherjee a a
Department of Chemical Engineering and Materials Science, University of California, Davis, CA 95616, United States b Department of Physical Chemistry, University of Pavia, V.le Taramelli 16, 27100 Pavia, Italy Received 13 December 2011; received in revised form 15 February 2012; accepted 21 February 2012 Available online 17 March 2012
Abstract Recent investigations regarding the role of applied fields on the grain growth and densification behavior of ionic ceramics are providing strong insights into the efficacy of Field Assisted Sintering Technique (FAST), aka Spark Plasma Sintering (SPS). Explanations of the observed behaviors, such as grain growth suppression and densification enhancement, are based upon the conjectured presence of a Joule heating driven temperature differential between grain interfaces and grain cores. These differentials were thought to be responsible for providing increased densification rates and lower densification temperatures through grain growth suppression and/or increased local kinetics at the forming necks. In this paper, we analyze the energetic, thermal, and practical details of this process in the context of the commonly accepted stages of sintering. © 2012 Elsevier Ltd. All rights reserved. Keywords: Sintering; Thermal conductivity; Electrical conductivity; ZrO2 ; SPS
1. Introduction Recent investigations1–6 have resulted in a great deal of progress toward the understanding of the role applied fields may have on the sintering of ionic materials using either FAST/SPStype or4–7 “flash sintering” configurations.2,3 In these processes, the effects are currently often ascribed to the effects produced by localized Joule heating, from the flux of current during sintering and/or annealing. It has been proposed that such localized heating might affect grain growth and atomic mobility at the grain boundary. The increase in mobility at grain boundaries, due to this higher grain boundary temperature, has previously been proposed to describe the efficacy of microwave sintering ionic ceramic powders in the 1980s and 1990s.8,9 However, it was shown, by analytically modeling heat flow in sapphire slabs, that the heat flow from the boundaries rapidly ameliorated any severe temperature differences existing between bulk grain regions and grain boundaries.10 These approaches, however, did not account for the diminished interface areas consistent with the
∗
Corresponding author. Tel.: +1 530 752 6290; fax: +1 530 752 9554. E-mail address: [email protected] (T.B. Holland).
0955-2219/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jeurceramsoc.2012.02.033
neck geometries during the early stage of sintering. On the other hand, Ghosh et al.1 have shown that an applied field, with a resulting flow of current, can greatly suppress grain growth in fully dense 3 mol% yttria-stabilized tetragonal zirconia (3YSZ). In a rather novel experimental approach, Ghosh et al.1 conducted a series of grain growth studies on 3YSZ samples with varied field strengths (<400 V/m), and an associated increase in current, across the cross section of a single sample. The grain sizes were then analyzed in accordance with the applied fields and a clear correlation between lessened grain growth and the increased electric field strength was shown. Applying this idea to sintering experiments, Yang and Conrad have demonstrated dramatically increased densification rates as well as reduced final grain sizes in the presence of modest applied DC (2000 V/m),67 and AC fields (1390 V/m, 60 Hz)4 in 3YSZ. They have also demonstrated an interrelationship between heating rates and the applied DC fields at similar field strengths.5 The explanation for grain growth suppression in the investigation requires that an applied field be in direct electrical contact with the samples, producing Joule heating from the flow of current across the grains and their interfaces. In all cases the experiments were conducted in air, assuring that the primary mechanism of conduction is ionic, and that no electrochemical reduction would be expected. Also in
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this case localized Joule heating are again suggested to be the operative mechanism by which increased densification rates and final grain sizes are achieved. Three aspects of the applicability of this mechanism to FAST/SPS experiments need addressing: (1) the thermodynamics and kinetics of the mechanism, (2) grain boundary versus grain core conductivities in the FAST/SPS environment, and (3) the time dependent nature of the temperature difference between grain cores and grain boundary regions. Analysis of the particular details of the process is critical to the determination of their applicability to traditional FAST/SPS sintering processes as well as to extend them into other manufacturing processes. In this paper, we analyze the thermodynamics and kinetic role that temperature gradients localized at the grain boundaries might play on the sintering behavior of ionic ceramics, focusing on the initial and final stages of sintering by FAST/SPS. We then numerically model the thermal development through Joule heating at the particle interfaces for various neck sizes and resistivities. 1.1. Grain growth suppression by localized Joule heating Grain growth suppression has been shown to provide easier densification as described by the sintering equation developed by Wang and Raj11 shown in here in Eq. (1), ρ˙ =
Af (ρ) (−Qb /kb T ) e Td n
(1)
where A is a material constant, f(ρ) is a function of density, T is temperature, kb is Boltzmann’s constant, d is grain size, and n is the grain size exponent and Qb is the activation energy for diffusion at grain boundaries. The grain size exponent, n, has been found to be 4 or 3 for grain boundary or volume diffusion-driven sintering, respectively. As a result, any process producing grain growth suppression during sintering has a potential beneficial effect on process itself. A localized heating of the grain boundaries can be shown to have such an effect. When grain boundary energy is taken into account, the differential of the total Gibbs free energy for a polycrystalline solid can be represented as in Eq. (2), dG = VdP − SdT + γgb dA
(2)
where A is total grain boundary area, γ gb is grain boundary energy (per unit area) and other parameters have their usual thermodynamic meanings. The pinning effect due to Joule heating at grain boundaries originally proposed by Ghosh et al.1 can be explained using the classical grain growth analysis introduced by Burke and Turnbull.12 Here the velocity of grain boundary migration νb is a product of grain boundary mobility Mb and the driving force for grain growth as shown in Eq. (3), 1 1 , + (3) ν = Mb γgb r1 r2 where r1 and r2 are principle radii of curvature of grain boundary surface. At constant pressure, the Gibbs free energy can be rewritten as dG = −SdT + γgb dA.
(4)
Cross-differentiation gives δS δγgb =− ≡ Sg , δA T,P δT A,P
(5)
where Sg is defined as grain boundary entropy (or entropy per unit grain boundary area) that is always a positive number.13 Therefore, the temperature coefficient of grain boundary energy γ is always negative. According to the model proposed by Ghosh et al.,1 when an electric current flows through polycrystalline sample the grain boundaries incur more Joule heating because the intrinsic resistivity at the grain boundary in ionic materials is higher than in the bulk of the grain and as a result of this increase in temperature, the driving force for grain growth is reduced through the −SdT contribution. This reduction gives rise to a pinning effect for grain growth even though the mobility, Mb , may be increased by the higher temperature at grain boundaries. Effectively, the source of grain growth is kinetically limited since the reduction in grain boundary energy lowers the driving force for grain growth. It is assumed here, consistent with all of the discussions on this subject to date, that no Soret Effect,14,15 which would provide a variation in concentration of the diffusing species, is present. If the Soret Effect were active, the mobility of the grain boundaries would be further lowered due to the slower diffusion of the rate limiting species from the grain cores to the interface region. Therefore, the hypothesis of Ghosh et al., necessarily presumes that the cores of the grains remain at a significantly lower temperature than the grain boundaries. Otherwise no reduction in grain boundary mobility would be expected or found by their argument. 1.2. Local temperature distributions in FAST/SPS Most ionic ceramic materials present some level of electrical conductivity at high temperature. The mechanism responsible for this conduction varies with the chemical nature of the material and with the experimental conditions. Ceramic materials are usually ionic, electronic, or mixed conductors. In the first case the charge carriers are ions only, while in the second case conduction by electrons or holes is observed. The details of the conduction mechanism could be quite complex, particularly in the case of mixed conductors. In these materials the relative concentration of the charge carriers is often controlled by complex chemical equilibria, while the carrier mobility might vary largely, even for the same type of carrier, from one compound to the other. This carrier mobility can vary according to grain size as well. For example, coarse polycrystals of pure ceria exhibit both electronic and ionic conduction while nanocrystalline ceria (∼10 nm) shows an increase of four orders of magnitude in electronic conductivity.16 In this work we focuses on purely ionic materials in which it has been noted that a significant fraction of the overall resistivity is due to the poor conductivity at the grain boundaries. In ionic conductors such as zirconia and doped ceria this “blocking effect” of the grain boundaries has been well-characterized. In early studies, it was considered to be associated with the segregation of impurities at the grain boundaries; later on it was recognized as a natural characteristic of
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the materials, caused by the depletion of charge carriers in the small region, often called the space charge layer (SCL), adjacent to the grain boundary core. This depletion derives from the electrostatic interaction between the charged point defects and the intrinsic charge always associated with the grain boundary “core” of the ionic materials.14 The presence of this grain boundary resistivity spurred the idea that when high current densities flow through an ionic conductor a local increase in temperature can be achieved at the grain boundaries. Cologna et al.2 describe the overall heat distribution in the “flash sintering” of nanograin zirconia by assuming uniform sample heating through Joule heating, and with black body radiation as the sole source of power dissipation. The analysis of the temperature increase would then follow as shown in Eqs. (6)–(8). In the absence of significant convection, Joule heating from the electrical current through the sample would be the source of temperature differences between the sample and furnace. In this case, the heat balance equation is represented by Eq. (6). W = Aσ(T14 − T04 )
(6)
where W is the electrical Joule heating from the grain boundaries, T0 is the furnace temperature, T1 is the temperature at the surface of the sample, A is the sample area, and σ is the black body radiation constant. Assuming that the furnace temperature is roughly equal to the surface temperature, the increase in the overall sample temperature is readily determined from Eqs. (7) and (8). T14 − T04 = (T12 + T02 )(T1 + T0 )T
(7)
W T = T0 4AσT04
(8)
Eq. (8) would be used to describe the overall temperature increase of the sample, versus the Joule heated temperatures of the grain boundaries, described by the growing neck areas and the current flow in the samples. Using this reasoning it has been suggested that the bulk of the power dissipation is within the grain boundaries alone due to the resistive differences between the grain boundaries and the grain cores in YSZ. Due to the small volume of the grain boundaries in comparison to the sample it is therefore assumed that the temperature of the grain boundaries is increased greatly (up to 200 ◦ C) while the average sample temperature may only demonstrate a 10 ◦ C difference. The resulting difference in temperature between boundary and core would therefore provide the driving force difference responsible for the extremely rapid increases in sintering rate. Presumably this analysis would be appropriate in describing the temperature distribution described qualitatively by Ghosh et al.1 In the case of nanometric grain sizes, nearly ubiquitous in use for FAST/SPS experiments, it may be helpful to consider the thermal conductivity controlled heat flow which occurs from the developed temperature gradient. The assumption of only grain boundary thermal generation may lead to significant errors in the microstructural thermal gradients, which could hamper the proper analysis of the operating mechanisms.
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2. Analysis In order to clarify the physical soundness of a model based on a localized Joule heating it is clearly important to evaluate the actual extent and duration of the resulting local increase in temperatures when a samples undergo a high density current flux. As we noted earlier an instantaneous T, between the grain boundaries and grain cores of over 200 ◦ C has been proposed, but the role of thermal conductivity in dissipating such a temperature difference is never been investigated. In the case of nanosized grains, such as those typically used in FAST/SPS processing, the mass of the grain cores is not particularly high relative to the grain boundaries, and as such, the thermal volume to be heated by the resulting heat flow into the nanograin can be expected to equilibrate quickly. Besides, it is important to note that any significant temperature difference between a nanometric grain’s boundary and its core produces extremely steep thermal gradients. In order to maintain a temperature difference of 10 K in a grain that is 100 nm in diameter a minimum temperature gradient of 2 × 107 K/m must be sustained despite thermal conductivity driven dissipation. Classical heat transfer analysis17 demonstrates, that in a slab of the same thickness, any thermal gradient would be disrupted in approximately 10−7 s using thermal conductivity values consistent with nanometric zirconia. A more quantitative understanding of the thermo-electrical details associated with the current flowing in a polycrystalline, ionically conductive ceramic can be developed using a few simple models. It is important to recognize that the models we developed account for, and represent, each stage of the sintering process. In the initial stage of sintering, particles are connected only by narrow necks producing a complex three-dimensional path for the flowing current. The reduced cross-sectional area of the necks produces a localized resistance increase that must be discussed and compared with the contribution from the known conductivity changes in the developing grain boundaries in the necks. Using numerical modeling we characterize the separate contributions of the reduced cross-section and of the intrinsic grain boundary resistivity to the temperature distribution within the particles. Fig. 1a represents our model using a linear array of spherical ceramic particles of radius r0 connected by rounded necks characterized by a radius rn . This is a simplified representation of a powder bed during the early stages of a sintering process, but it contains all the essential elements necessary to demonstrate the interrelationships of Joule heating and thermal conductivity. In the model without grain boundaries (Fig. 1a) the local heating is produced only by the restriction to the current flux introduced by the neck geometry. The electro-thermal problem can be modeled using Eqs. (9) and (10), = ∇ · |(−σ∇V ) = 0 ∇ · J = ∇ · (σ E) ρcp
δT = ∇ · (k∇T ) + q˙ j δt
(9) (10)
represents the current density, E the electric field, where J = σ E σ the electrical conductivity, V is the potential gradient, ρ the density, Cp the heat capacity, and q˙ j the heat generated by Joule
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Fig. 1. Schematic of the linear array of spherical particles used in modeling the temperature differentials between the grain boundaries and grain cores in the (a) absence and (presence) of a differently conductive grain boundary.
heating per unit volume per unit time. Here our assumed electric fields differ slightly from those we have reported elsewhere on the modelled18 effects of a field upon the densification of ionic ceramics. In that paper, the model assumes that no conduction occurs, while here we assume that the opposite is true in regards to free charges: no charge accumulation occurs to provide an increase in field strength. In principle this simply means that the accumulation of free charges does not occur, consistent with the homogenous solution of Gauss’ law in that work,18 and that at each point in the model the field strength is assumed to be a single value, in contrast with our findings of how a field interacts with dielectric particles and that at each point in the model the field strength is assumed to be a single value, in contrast with our findings of how a field interacts with dielectric particles. As discussed later, this simplification is largely inconsequential to the outcomes of the model. If grain boundaries are introduced into the neck, the situation becomes similar to one depicted in Fig. 1b. The SCL thickness is exaggerated for grains above 30 nm, but is consistent with many recent FAST/SPS experiments on nanometric powders. To accurately represent the contribution of a grain boundary to the local temperature we must use the electrical resistance associated with a single grain boundary. In YSZ, as in other polycrystalline ionic conductors, the contribution of the grain boundaries to the overall resistivity can be identified very accurately with impedance spectroscopy measurements.
This analysis relies on the fact that the grain boundary and the bulk regions are characterized not only by a specific contribution to the overall resistance (Rb and Rgb ) but also by a different capacitance. Determination of the impedance at different frequencies allows the individual contributions of Rb and Rgb to be calculated as long as the time constants of the RC elements are sufficiently different. This technique has been extensively used on ionic conductors.14 The experimental data are generally analyzed using a model for the material microstructure termed the “brick layer model”. In this model, the sample consists of a 3D network of conducting “bricks” (grain cores) surrounded by a resistive “mortar” (grain boundaries).19 The components of the overall resistance measured experimentally Rex = Rb + Rgb are related using Eq. (11), σgb D Rb = Rgb σb w
(11)
where σ b is the bulk specific conductivity, σ gb the specific grain boundary conductivity, D the grain size and w the grain boundary thickness. The value of w can be estimated on the basis of difference in capacitance between the bulk and the grain boundary components, and in YSZ it is generally found to be between 2 and 5 nm. The specific grain boundary conductivity is often found to be roughly two orders of magnitude lower than the bulk specific conductivity in ionically conductive ceramics. The
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resistance per unit surface of a single grain boundary may be estimated from the value of the specific grain boundary conductivity and the grain boundary thickness (corresponding to the SCL). In this paper we model heat transfer within multiple powder particles, and the boundary condition is conservatively considered to be adiabatic since these particles are surrounded by other particles in the pellet having very much the same temperatures. Furthermore, we consider a very short time scale for our simulations and this removes the need to account for heat conduction on a macroscopic scale. The models always included at least five spheres. The values for the electrical resistivity, heat capacity, and thermal conductivity of bulk zirconia have been taken from20 and represented by the relation log (σT). The simulations have been performed using a neck radius to particle ratios of 5% and 15%, corresponding to roughly the very early and middle of stage 1. A current flux of 104 A/m2 , measured at the largest cross-section of the particles, was allowed to flow through the system modeled to be at 1000 ◦ C at the beginning of the simulation. This current flow is unrealistically high for any FAST/SPS processing, but we used it in order to produce measurable temperature gradients. The field strengths necessary to provide this current flow are between 6.3 × 104 and 3.57 × 106 V/m, with smaller grain sizes and/or the presence of grain boundaries obviously requiring higher fields. These fields are higher than even the local field strengths modeled to occur by dielectric polarization,18 and in the case of FAST/SPS processing with a conductive graphite die as an alternative current pathway they are almost impossible to achieve in practice. Therefore, the results reported here can be considered the most extreme case possible for the achievement of high thermal gradients in typical FAST/SPS experiments. They are not unreasonably high in comparison to the external field strengths used in the “flash sintering” experiments of Cologna et al.,23 if one considers the possible contributions of polarization. It is important to note, however, that only the rate of heating is calculated from Joule heating, and if the mechanisms of Flash sintering are such that faster heating occurs, the models might underestimate the developed gradient. However, the dissipation of the gradient through the thermal conductivity and mass of the particles will be shown to be exceedingly rapid. The simulations have been interrupted when the temperature of the center of grains was increased by more than 50 K which is well above the overall temperature increase calculated in the experimental reports on current effects. The specific assumptions are: (1) the external surfaces of the particle are adiabatic, (2) the region in which Joule heating occurs in the grain boundary is significantly wider than the physical boundary, (3) complex impedance resistivity values reflect accurate values of conductivity, (4) the divergence of the electric field is zero, and (5) Joule heating alone provides the increase in local temperatures. As stated previously (1) is a conservative estimation, and would tend to overestimate the thermal gradients. Assumption (2) deals with the thermal boundary resistance, commonly referred to as the “Kapitza” resistance.21,22 The physical boundaries are contained within the “electrical grain boundaries” that generate the local Joule heating, for which there is no structural discontinuity. This implies
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Fig. 2. Peak temperature difference attained between the grain boundary (Tgb ) and grain cores (Tb ) with and without the presence of a differently conductive grain boundary. Model utilizes a fixed current flow and the relevant thermophysical and thermoelectrical properties of nanocrystalline zirconia.
that the area in which the Joule heating occurs has no real discontinuity in phonon distribution occurring in the presence of the regions at differing temperatures. This is not much different from rapid contact heating in various techniques such as FAST/SPS or microwave sintering with susceptors. Through assumption (3), we have accounted for increased resistances in the grain boundary of up to two orders of magnitude, which is, again, conservative in favor of the development of a thermal gradient. Assumptions (4) and (5) are intertwined in that the nature of charge development and accumulation are currently unknown, and the possibility of dielectric breakdown, plasma formation, or other unknown sources of heating are unaccounted. Two neck sizes consistent with early and middle density values for Stage 1 were modeled with and without the contributions of differently conductive grain boundaries with the results shown in Fig. 2. This graph helps to clarify a crucial point in the discussion of localized heating induced by the current flow. It is clear that the local temperature gradient is strongly dependent on the size of the grains. While spheres with a dimension in the millimeter range might experience a significant temperature gradient between the center and the boundary, when the size is reduced to the tens of microns, the temperature difference becomes insignificant. In the case of grains smaller than the micron scale, the thermal conductivity equilibrates any significant temperature gradient inside the grain, providing an evenly heated grain. Although the introduction of the grain boundaries provides a significant increase in the local temperature gradient, particularly when the neck size is small, the absolute values remain very low particularly as grain size decreases to sub-micron levels. Even accounting for possible differences in developing grain boundary conductivities, versus the equilibrium grain boundary conductivities often measured using impedance spectroscopy, it is clear that even across two orders of magnitude the thermal gradients are minimal at small grain sizes.
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Fig. 4. Thermal differential of the grain boundaries and cores of several starting grain sizes as a function of time. The changes in scale demonstrate the sensitivity with respect to grain size.
While there may occur temperature differentials by other means, such as dielectric breakdown, exceedingly small necks, or massively difference local conductivities, it is important to understand the rapidity by which the intrinsic thermal conductivity of the grains ameliorates the differential. To model this we use the geometry shown in Fig. 3a with a temperature delta between the boundary and core of 200 ◦ C and the thermal conductivity of nanometric zirconia. Fig. 3b shows the time it takes for the microstructure to remove the differential as a function of grain size (r0 ) and relative neck size (rn /r0 ). It is clear from the model that, as the starting grain size of the material approaches the nanometric dimension, as so often happens in FAST/SPS processing, the rate at which temperature gradients are removed is exceedingly fast. This helps to explain why the models shown in Fig. 2 demonstrate minimal gradients even with high current densities. The time frames with which the combined Joule heating and thermal equilibration mechanisms achieve their maximums and steady states at fixed current values are shown in Fig. 4 for various grain sizes. This clearly demonstrates that while the maximums gradients shown in Fig. 2
Fig. 5. Generalized grain boundary and core resistance model for Stage 3 sintering.
are somewhat small, as grain size is reduced the temperature gradient after a single second is much reduced. This would further limit the kinetic contribution of local Joule heating as it is extremely transient in nature as well. The sensitivity with respect to grain size is evident in the orders of magnitude changes of scale for each grain size, as well as an increase in the rate of equilibration. The 1 m grain size is the smallest grain size reasonably calculated due to the exceedingly small temperature differentials that develop and the short equilibration times. In the late stages of sintering pores have closed and ultimately are removed, and so a different model such as that shown in Fig. 5 must be considered. Again the process is found to be a function of the rates of heat generation and redistribution. Fig. 5 describes the classic slab heat transfer problem discussed previously, and as such need not be solved with numerical methods. The equilibration times are much faster due to the larger interface cross section providing better contacts and heat flow. As the more intense gradients in
Fig. 3. (a) Temperature equilibration model schematic and the (b) time required for thermal equilibration between the neck and grain cores as a function of grain size with Tn starting at 1200 ◦ C and the grain cores starting at 1000 ◦ C.
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the case of narrow necks are already found to be quite low, the thermal gradients in the fully dense specimens are a moot point.
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straight DC sintering experiments and SPS experiments which typically operate at <300 Hz DC pulsing). 4. Conclusions
3. Discussion It is clear from the analysis that the thermal conductivity, even in ceramics, reduces the effectiveness of localized Joule heating tremendously as grain sizes are reduced to those typically used in FAST/SPS operations. This is true across the spectrum of sintering, from the early neck formations in Stage 1 to geometries to the fully densified samples in Stage 3. The local temperature gradients would therefore seem to provide little benefit in practice to the thermodynamics (discussed in Eqs. (2)–(5)) or to the local kinetics. In describing the effects observed in very recent works on FAST/SPS, the new and novel works on “flash sintering”,2,3 and our own non-contact field experiments reported in elsewhere,23 it may be necessary to look for explanations outside of local Joule heating. This does not mean, however, that the local current densities are not contributing to the increased kinetics. Nor does it mean that the Joule heating at the grain boundaries is not contributing to the overall temperature and perhaps heating rate of the bulk sample. The role of fields and/or currents on the diffusion mechanisms dominant in each stage of sintering are still quite unclear. In terms of diffusivity, it has been shown that grain boundary diffusion is quite influenced by the presence of a field as has volume diffusion. In the case of surface diffusion, this is not the case. Presuming that surface diffusivities are insensitive to an applied field, then the balance of diffusivities24 would have a profound effect on the nature and temperature of the transitions from Stage 1 to Stage 2 sintering.25 This would be particularly true if the diffusivities of grain boundaries and volumes are themselves differently affected by the field. It has been shown,24,25 that the balance of the ratios of the grain boundary to surface diffusivity and bulk to surface diffusivity have strong influences on the chemical potential distributions at necks. Another explanation for the role of applied fields is perhaps necessary due to the lack of achievable temperature gradients in nanometric grains. In another work,18 we discuss the role of fields in the initial stages of sintering. The local field strengths are found to be high enough to (potentially) increase the likelihood of dielectric breakdown, alter the thermodynamic driving forces for cationic motion, or influence SCL development. However, in describing the late stages of sintering, the role of polarization is minimal at best. In determining the local field strengths in dense parts, Vollman et al.26 have shown that the local field strengths from conductivity differences scale, with some simplifications, as E = Eox (dg /dgb ) where dg is the thickness of the grain, Eox is the applied field strength, and dgb is the width of the grain boundary SCL. The criterion for making this simplification are that the Eox is comparatively small with respect to the SCL field strength, the resistance of the bulk is very much less than that of the grain boundary, and the time of Eox application is larger than the Maxwell–Wagner relaxation time,27,28 (a safe assumption in
The thermodynamic basis for the suppression of grain growth in the presence of a thermal gradient was developed. Using numerical models the thermal gradients attainable in nanometric powder specimens were calculated with the contribution of Joule heating. It is shown that no significant contribution of temperature gradient can be expected, in mixed or ionically conducting ceramics, due to the thermal conductivity and nanometric grain sizes used in most FAST/SPS processing. This is the case even at the unreasonably high current densities utilized in our numerical models. The observations of the contributory effects of an applied field or current to the densification rates of ionic ceramics still defy well-developed explanations. The role of the applied fields themselves may be a useful avenue to explore, since they are shown to provide contributions to the initial and final stages of sintering in many ionic ceramics. Acknowledgement This investigation was supported by a Grant from the Office of Naval Research (ONR Grant # N00014-10-1-0632) with Dr. Lawrence Kabacoff as the Program Manager. References 1. Ghosh S, Chokshi AH, Lee P, Raj R. A huge effect of weak dc electrical fields on grain growth in zirconia. Journal of the American Ceramic Society Aug. 2009;92(8):1856–9. 2. Cologna M, Rashkova B, Raj R. Flash Sintering of Nanograin Zirconia in <5 s at 850 ◦ C. Journal of the American Ceramic Society 2010;(September):3556–9. 3. Cologna M, Prette ALG, Raj R. Flash-sintering of cubic yttria-stabilized zirconia at 750 ◦ C for possible use in SOFC manufacturing. Journal of the American Ceramic Society 2010;(December). Online only currently. 4. Yang D, Conrad H. Enhanced sintering rate of zirconia (3Y-TZP) by application of a small AC electric field. Scripta Materialia 2010;63(3):328–31. 5. Yang D, Conrad H. Enhanced sintering rate and finer grain size in yttriastablized zirconia (3Y-TZP) with combined dc electric field and increased heating rate. Materials Science & Engineering A 2010;528(3):1221–5. 6. Yang D, Raj R, Conrad H. Enhanced sintering rate of zirconia (3Y-TZP) through the effect of a weak dc electric field on grain growth. Journal of the American Ceramic Society 2010;93(10):2935–7. 7. Conrad H, Yang D. Influence of an applied dc electric field on the plastic deformation kinetics of oxide ceramics. Philosophical Magazine 2010;90(March (9)):1141–57. 8. Janney MA, Kimrey HD. Microwave Sintering of Alumina at 28 GHz. In: Messing GL, Fuller Jr ER, Hausner H, editors. Ceramics powder science II, vol. 1. American Ceramics Society: Westerville, OH; 1988. p. 638–40. 9. Meek TT. Proposed model for the sintering of a dielectric in a microwave field. Journal of Materials Science Letters 1987;6(6):638–40. 10. Johnson DL. Microwave heating of grain boundaries in ceramics. Journal of the American Ceramic Society 1991;74(4):849–50. 11. Wang J, Raj R. Estimate of the activation energies for boundary diffusion from rate-controlled sintering of pure alumina, and alumina doped with zirconia or titania. Journal of the American Ceramic Society 1990;73(5):1172–5. 12. Burke J, Turnbull D. Recrystallization and grain growth. Progress in Metal Physics 1952;3:220–92.
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