Microstructural temperature gradient-driven diffusion: Possible densification mechanism for flash sintering of zirconia?

Author’s Accepted Manuscript Microstructural temperature gradient - driven diffusion: possible densification mechanism for flash sintering of zirconia? Mattia Biesuz, Vincenzo M. Sglavo www.elsevier.com/locate/ceri

PII: DOI: Reference:

S0272-8842(18)32791-3 https://doi.org/10.1016/j.ceramint.2018.09.311 CERI19715

To appear in: Ceramics International Received date: 1 September 2018 Revised date: 25 September 2018 Accepted date: 30 September 2018 Cite this article as: Mattia Biesuz and Vincenzo M. Sglavo, Microstructural temperature gradient - driven diffusion: possible densification mechanism for flash sintering of zirconia?, Ceramics International, https://doi.org/10.1016/j.ceramint.2018.09.311 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

temperature gradient - driven diffusion: possible densification mechanism for flash sintering of zirconia?

Mattia Biesuza,b,#,

[email protected]

Vincenzo M. Sglavoa,b,

[email protected]

a

University of Trento, Department of Industrial Engineering

Via Sommarive 9, 38123 Trento –ITALY–

b

INSTM, Research Unit of Trento

Via G. Giusti 9, 50121 Firenze –ITALY–

#

Corresponding author:

[email protected]

Keyword: flash sintering; field-assisted sintering; sintering; thermodiffusion; yttria stabilized zirconia (YSZ); electric current-activated sintering

Abstract Flash sintering represents an innovative electric field-assisted sintering technology which allows a drastic reduction of the consolidation time and temperature of many ceramics. It is still under debate whether and how the electric field and current boost the densification process. 1

In this work, we investigate the impact of microstructural temperature gradients, associated to an overheating of the interparticle contact points, on mass transport in the different stages of flash sintering of yttria-stabilized zirconia (YSZ). Although the temperature difference between grain bulk and boundary is limited, it is associated to extremely high gradients as a result of the small particle size. In such conditions, a temperature gradient-driven atom flux can be activated. The calculations point out that thermodiffusion can significantly accelerate densification of micrometric YSZ powder upon flash sintering if the thermal gradients exceed 106 K m-1.

1 Introduction Flash Sintering (FS) represents an innovative, energy-saving consolidation technology for ceramics, it allowing significant reduction of processing time and temperature [1–4]. So far, it has been successfully applied to many different ceramics and glasses [1–17]. Flash sintering belongs to the family of the Field Assisted Sintering Techniques (FAST) like Spark Plasma Sintering (SPS) or microwave sintering. Nevertheless, FS possesses some peculiar characteristics mainly associated with the presence of the so called “flash event” where the material undergoes to a transition from electrical insulator to semiconductor or conductor [1,18], a strong photoemission is produced [19–22] and the green compact densifies very quickly. The process is usually divided into three stages. The first one is the incubation: the system works in voltage control and power dissipationelectrical current slowly increase. The second stage is a “far from the equilibrium stage”, where the material undergoes to the flash event. During such period, most of the densification takes place, the electrical resistivity consistently drops and heating rates in the order of 10 4 K/min are achieved; 2

the electrical power reaches a peak and the system is typically switched from voltage to current control [23]. When the system reaches a new equilibrium condition (i.e., the electrical parameters and power dissipation are stabilized), the third stage, also known as steady stage, begins. Here, residual densification occurs, sometimes accompanied also by grain growth. Although the trigger for the flash event was demonstrated to reside in the Joule heating thermal runaway [24–26], the mechanisms responsible for the electrical resistivity drop and the ultra-fast densification have not been clarified yet. Raj et al., working on yttria stabilized zirconia (YSZ), argued that the sample temperature reached during the steady stage of flash sintering is not high enough for explaining the densification observed upon flash [27]. In particular, they calculated that the observed densification rates during the flash event could be achieved only at 1900°C, the real sample temperature being only approximately 1250°C [27]. Other authors working on YSZ and ZnO concluded that the densification upon the flash event is basically accelerated by the extremely high heating rates [28–30], these changing the densification-to-coarsening rate ratio in a sort of fast firing-like process [31– 33]. It has been also proposed that, during rapid heating process, the grain boundary structure differs from that at the equilibrium, thus providing a preferential fast diffusion path [28]. Nevertheless, this matter is still very controversial and different explanations have been proposed: some authors pointed out that the high heating rate is crucial for enhancing densification upon flash sintering [34] while others concluded that the heating process by itself can not explain the densification achieved during the flash event [35]. In any case, at present, the field/current induced effects on densification upon flash sintering are still a matter of discussion. Different mechanisms have been proposed to explain how the flash process can accelerate densification. One is based on the assumption that Frenkel pairs are nucleated during flash sintering: this could increase the defects population and atomic diffusion [1,36,37] although very limited experimental confirmations have been collected. A second effect observed during flash 3

sintering is associated to the partial oxide reduction due to electrical current flow [9,38,39]. The cations oxidation state modification could decrease the activation energy for diffusion [40,41] and accelerate the densification processes. Such phenomenon have been clearly observed in DC flash sintering experiments where different grain coarsening kinetics occurred in the anodic and cathodic areas [41–45]. Nevertheless, coarsening mechanisms are mainly activated during the third stage of the process in the electrodes region and, therefore, it is not clear whether partial reduction could also interact with densification upon the second stage of FS. An additional mechanism is based on the assumption that the sample temperature is not homogeneous. In particular, the grain boundaries are thought to be hotter than the grains bulk and this could be the result of higher electrical resistance associated to the presence of the space charge region [37] and to the lower cross section available in the necks [46–48]. Therefore, local diffusion and densification at the grain boundaries are much faster than corresponding phenomena occurring at an “average” sample temperature. Some authors have also hypothesized a partial melting of the particle surface/grain boundaries, this leading to fast densification via liquid phase sintering [46–48]. The formation of microstructural temperature gradients was also proposed to interact with coarsening mechanisms. Ghosh et al. have shown that the application of small DC field reduces grain coarsening in YSZ [49]. They assumed that the current flow causes a sort of grain boundary pinning due to the development of temperature gradients. In fact, the grain boundary energy (

) can be expressed as: (1)

where

and

are the grain boundary enthalpy and entropy,

respectively, and T the absolute temperature. Since for

> 0, a local minimum

can be identified where the temperature reaches a maximum;

4

therefore, microstructural temperature gradients and grain boundary hotter than the bulk cause a pinning effect due to the localized minimum for

.

The real existence of microstructural temperature gradients during the flash event is still under debate. Some experimental evidences suggest localized melting/overheating, these having been associated in different works with the development of a core/shell structure in Na0.5K0.5NbO3 [50], secondary phase formation at the grain boundary in BaTiO3 [51] and phase segregation in magnesium silicide stannide [52]. In another work, Holland et al. simulated the temperature differences between the grain boundary in the neck region and the particle core during field-assisted sintering of YSZ [53]. They determined that the temperature differences are generally lower than 1 K in the micrometric or sub-micrometric scale; therefore, no significant effects could be associated with grain boundary overheating. Nevertheless, one can observe that, although the grain boundary overheating is very limited, the temperature gradient is quite large because of the small particle size. In particular, according to the results reported by Holland et al. (Fig. 2 in [53]), a temperature gradient of about 102 – 107 K m-1 is generated for 0.1 – 10 μm particles. They also pointed out that the temperature difference between the neck region and the particle core partially decays with time; however, this effect does not change the order of magnitude of the gradient in the time scale of 1 s. Holland et al. simulation refers to 10 mA mm-2 current density in the central part of the particle; such value represents an extreme condition for SPS but it is quite low in flash sintering. Baraki et al. used 65 mA mm-2 current density [16], while Cologna et al., in the very first works on flash sintering, used currents in the order of 100 mA mm-2 or higher. In addition, Holland et al. interrupted the simulation when the temperature at the particle core increased by 50 K because of Joule effect. This value is much lower than that usually achieved in flash sintering experiments where the temperature increases by hundreds of degrees. Therefore, it seems reasonable that the gradients predicted by Holland et al. can be substantially amplified during 5

flash sintering experiments, according to previous experimental findings reported for different materials [50–52]. Other factors like plasma formation at the contact point between particles or in the neck region [19], local field intensification at the neck surface [54], rapid release of surface energy due to the fast sintering process can also increase the temperature gradient. Under said conditions, thermodiffusion can be activated during flash sintering and can interact with the sintering mass transport. Thermodiffusion is also known as thermophoresis or Ludwig-Soret effect[55] and it accounts for vacancy/atom flux driven by thermal gradients. For example, a vacancies flux from the hottest region toward the coldest one can be considered [56]: in other words, vacancies move from the neck (hotter) toward the bulk of the grain (colder). The atoms flow in the opposite direction (from the bulk to the neck), thus promoting densification. Therefore, under thermal diffusion regime in materials where diffusion is controlled by vacancies motion, densification is not driven only by the surface curvature and an additional contribution depending on temperature gradients must be considered. On these bases, Young and McPherson in 1989 [57] and, more recently, Olevsky and Froyen [58] proposed a thermal diffusion contribution during SPS of alumina. The aim of the present work was to identify the thermal gradients needed for thermodiffusion in YSZ and to understand whether they can contribute to densification during flash sintering. The fundamental idea was therefore to decouple the different phenomena which are involved in flash sintering and investigate the role of thermodiffusion as possible pivotal step in the very quick consolidation process.

2 Theory and calculations 2.1 Thermodiffusion

6

The Ludwig-Soret effect [55] is based on the well-known equation [57–59]: ( where

)

,

and

(2)

are the thermal-induced flux, the concentration and the

diffusing species transport heat, respectively, the absolute temperature,

the diffusion coefficient,

the perfect gas constant.

Under thermal diffusion conditions, the defects concentration is determined by the Schottky or Frenkel equilibrium, depending on the specific dominant lattice defects. According to previous works [60], one can confidently assume that the fundamental lattice defects in YSZ are determined by Schottky disorder and, by using the Kroeger-Vink notation, .

(3)

Therefore, the defects concentration can be determined as [61]: [

][

where

]

⁄

(

,

and

)

( )

(

⁄

)

(4)

are the Gibbs free energy, the entropy and the enthalpy

for Shottky disorder formation, respectively. Since cation vacancies diffusion is much slower than oxygen vacancies diffusion in YSZ, one can assume that densification is controlled by cations self-diffusion. Moreover, the oxygen vacancy concentration is extrinsically determined by the doping element concentration which is, in general, orders of magnitude larger than the intrinsic defect concentration. Therefore, at a first approximation, one can assume that [ [

] is constant and equal to

]/2.

The thermal concentration gradient of [

(

]

[

) ]

(

can be determined from Eq. 4 as:

)

(5)

By substituting Eq. 5 into Eq. 1, one can obtain[57,58]: (

( [

) ]

(

)

)

7

(

) .

(6)

According to Wirtz [62]: (7) where

is the enthalpy of vacancies migration. Therefore, Eq. 6 becomes: (

).

(8)

During conventional sintering processes, atomic diffusion is driven by curvature differences which lead to different vacancies concentration beneath concave and convex surfaces [31]. The difference of defects concentration with respect to a flat surface can be determined by the Kelvin’s equation[31,63]: (

)

(9)

where C0 is the vacancy concentration as determined from the thermodynamic equilibrium under a flat surface (with infinite curvature radius),

the surface energy,

the molar volume, ⁄

curvature radii of the surface and

⁄

and

the principal

the curvature.

Therefore, an average vacancies concentration gradient between two sites “A” and “B” at distance

can be calculated as: (

)

.

(10)

The defects flux induced by the curvature is: (

)

.

(11)

By combining Eqs 8 and 11, the ratio between thermal and curvature-driven diffusion can be calculated as: (12) If

>1, thermal diffusion is the dominant densification mechanisms;

conversely, if

< 1, surface curvature controls densification.

values can be

calculated at different sintering stages, assuming the presence of thermal

8

gradients in the range 105 – 107 K m-1, i.e. within the temperature gradient range previously reported by Holland et al. [53].

2.2 Early stage of sintering: surface diffusion In the early stage of sintering, due to the limited temperature, the dominant transport mechanism (from the particle surface to the neck) is surface diffusion [31]. This is responsible for inter-particles neck formation and growth while densification is very limited. Two different diffusion paths shown in Figure 1 are considered here: i.

From site “2” (zero curvature) to site “1” (negative radius of

curvature); ii.

From site “3” (particle surface, positive radius of curvature) to

site “1”. Initially, the neck geometry must be identified. The radius of curvature of the neck surface can be calculated as [64]: (13) where

and

are the particle and neck region radius, respectively. One can

estimate the angle

in Figure 1 by using the assumption proposed by Burke

[64]: if the distance between the neck center (dashed line in Figure 1) and the particle center is equal to ( (

)

), then:

.

(14)

Therefore, the diffusion distance for atoms moving from “2” to “1” is approximately: (

)

(15)

From “3” to “1” it is: (

)(

) .

(16)

9

The curvature in “1” is: ;

(17)

in “3” it is: .

(18)

In site “2” one of the two principal curvature radii is infinite, this point being an inflection point in Figure 1. The other radius of curvature is equal to (

(

). Since, from Eq. 14,

)⁄(

),

can be easily

calculated and we can estimate the curvature in site “2” as: .

√

(19)

The ratio between thermal and curvature driven densification (Eq. 12) can be therefore calculated for the two diffusion paths as: (

(

)

)(

( )

)

√

(

)

(20)

(21)

2.3 Intermediate stage: diffusion from the neck center to the neck surface At higher temperature, bulk diffusion is activated and it accounts for shrinkage and densification. The transport mechanisms correspond to lattice and/or grain boundary diffusion and atoms move from the region between the particles centers to the neck surface. Therefore, the centers of the particles get closer, the ceramic component shrinks and densifies. In the present analysis, we assume that atoms move from the grain boundary region located at the center of the neck toward the neck surface, following a path similar to that depicted in Figure 2. In this case, the diffusion path length is one half of the inter-pore distance (2ξ). . Particles are assumed to be tetrakaidecahedra (with body centered cubic packing) with circular pores

10

located at the grain corners (pore radius =

) [31]. One can estimate the

pore curvature in “1” (Figure 2) assuming that [57]: ( where

)

(22)

is the relative density. In addition, can be calculated as [57]: [

√

(

)]

.

(23)

In order to consider particle coarsening as well, the radius can be corrected as [65]: ( where

) and

(24) are the particle size and the relative density of the green

body, respectively. The geometrical model developed by Coble [66] for the intermediate sintering stage assumes that the pores reach their equilibrium shape. Pores are approximated with perfect cylinders (“spaghetti-like”) located at the grain corners. In such condition the curvature on the pore surface is

.

If the vacancies concentration at the center of neck region is assumed to be similar to the thermodynamic equilibrium value [31], it is possible to obtain the ratio between thermal and curvature diffusion as: (25) One could dispute whether this model is actually applicable to FS. As said, it is usually assumed that pores attain their equilibrium shape in the intermediate sintering stage, thus forming flat grain boundaries [31]. This might be not the case of FS; as a matter of fact, the rapid sintering and grain boundary formation might not give “enough time” to reach this condition as recently proposed by Ji et al. [29]. Such statement seems to be partially supported by previous findings by Zhang et al. [67] on alumina densified under extremely rapid heating conditions (as in FS experiments); they showed that the rapidlysintered ceramic presents singular features like curved grain boundaries, 11

non-equilibrium angles at the triple points and absence of the dihedral angle at the grain/pore interface (even in an advanced sintering stage). Young and McPherson [57] developed a geometrical model for sintering which assumes that the pores maintain also a curvature radius ξ. Albeit this may not be the case of conventional sintering, where the equilibrium pores and grain boundary shape are reached, it could be applicable to flash processes. In this case (

becomes: )

.

(26)

It is important to observe that Eqs. 25 and 26 are simply derived from the defects concentration gradient between the neck center and pore surface. They are, therefore, valid in the case of grain boundary or lattice diffusioncontrolled sintering. 2.4 Final stage: isolated pores, lattice diffusion In the final stage of sintering, pores are isolated at the grain boundary or within the grains. The pores volume can be reduced by lattice or grain boundary diffusion depending on the pore location and the fastest diffusion path. In this simplified model, one can assume that the pore is perfectly spherical and isolated within a grain. In this case, the curvature on the pore surface is given by: (27) where

is the pore radius. The vacancies distribution close to the pore can

be calculated by solving the second Fick’s law, in steady state condition: .

(28)

In a spherical geometry, a possible solution is [68]: (29)

( )

where

and

are integration constants and r the distance from the center

of curvature of the pore. The vacancies concentration gradient is therefore: 12

.

(30)

This solution ensures that the net vacancies/atoms flux in any spherical shell around the pore is zero, leading to a steady-state condition. In fact, the net atomic flux in the region indicated by dashed lines in Figure 3 is: ( ( )

( ) )

)(

(

)

( )

)

(

(

)

(

.

) (31)

The boundary conditions can be identified as follows: (i) at virtual infinite distance from the pore curvature the vacancies concentration is equal to the equilibrium one (C0); (ii) the concentration on the pore surface is: (

.

)

(32)

Therefore, the analytical relationship for vacancies distribution is: ( )

( )

.

(33)

The ratio between lattice curvature and thermal diffusion can be now determined as a function of the distance from the pore center: .

(34)

2.5 Thermal gradient effect on grain boundary mobility At this point, it is useful to estimate how grain boundary overheating can affect grain coarsening. To this purpose, one can analyze the variation of Gibbs free energy associated with the motion of some atoms (

moles)

across the grain boundary. Basically, the free energy variation is associated to the pressure difference across the grain boundary [31]: (

) .

By assuming a spherical geometry (

(35) =

= ), Eq. 34 reduces to: (36)

13

Therefore, the chemical potential variation, correlated with atoms motion across the grain boundary can be calculated as: (37) and .

(38)

If one considers the grain boundary energy as a function of temperature, Eq. 1 can be differentiated as: .

(39)

Therefore, the Gibbs free energy variation associated with the grain boundary motion in a thermal gradient becomes: .

(40)

Since the grain volume variation is (41) one can obtain: .

(42)

This contribution to the Gibbs free energy causes grain boundary pinning, in the hypothesis that the grain boundary is hotter than the grain bulk. Finally, the ratio between the driving forces promoting coarsening and grain boundary pinning can be calculated as: .

(43)

3 Results and Discussion 3.1 Factors generating microstructural temperature gradient

14

Different factors contribute to the generation of microstructural temperature gradients; among them, the limited cross-section available for the current flow in the neck region and the formation of the space charge (responsible for an extra grain boundary resistance) were considered in the simulations in ref. [53]. Nevertheless, some additional factors can participate to the development of such gradients and are discussed here below. The very rapid sintering process observed during flash sintering is associated with a large surface enthalpy release rate. Surface ( ) and grain boundary (

) energy for YSZ can determined

according to Ref. [69]: (44) .

(45)

Therefore, we can estimate the enthalpy excess correlated with surface ( ) and grain boundary (

) as 1.927 J m-2 and 1.215 J m-2, respectively. Any

effect of the applied field on the surface and grain boundary energy is therefore neglected here; as a matter of fact, the field involved during FS of YSZ are quite low, typically in the order 50 – 200 V cm-1, and, according to previous findings, only much higher fields (10 6 – 107 V cm-1) can induce variations on the surface chemistry of zirconia [70]. The heat release associated with surface reduction upon sintering can be calculated under the assumption that the grain boundary surface created during the process is one half of the reduced external surface. Therefore, the power released per unit volume is: ( where and

)

(46)

is the specific surface area,

the time needed for densification

the ceramic body density.

At this point, in order to quantify the phenomenon, one can assume that the densification is completed in 2 s during flash sintering, the relative density of 15

the green body is 0.55, the theoretical density is 6.1 g/cm3 [71] and the SSA is 16 m2/g [72]. The power release corresponding to the surface reduction is 35.4 mW mm-3 or 64.4 mW mm-3 for the green (initial sintering stages) and dense body (final sintering stages), respectively. Such power evolution is similar to that needed for triggering FS, but one-two orders of magnitude lower than the power peak reached during the second stage of the process and about one order of magnitude lower than the power dissipation typically generated during the third stage (steady stage). Nevertheless, it is important to point out that the surface energy evolution is concentrated within the neck/grain boundary region. Therefore, the local power dissipation can be calculated as: (47) where

and

are the grain boundary and bulk volume, respectively, and: (48)

Therefore: .

(49)

The combination of Eqs 46 and 49 yields: .

(50)

Here, it is important to point out that

does not depend on the

of the

used powder. In previous works, grain boundary thickness in the range 1 - 5 nm was reported for YSZ [73,74]. Therefore, in the present case, the power released at the grain boundary ranges between 264 mW mm-3 and 1319 mW mm-3. Such values are comparable if not larger than that measured during the third stage of flash sintering. Very significantly, the considered power dissipation can account for conspicuous thermal gradients between the neck region and the bulk and for the consequent activation of thermodiffusion processes. Interestingly, said thermal effect can influence not only flash

16

sintering, but all processes characterized by very quick heating/densification, like SPS, microwave sintering and fast firing. In addition to the power dissipation associated with surface reduction, additional phenomena can contribute to the generation of the thermal gradients. In particular, the presence of an interface between different dielectrics (i.e., the ceramic and the air) significantly intensifies the electric field [75]. Holland et al. calculated the electric field intensification in proximity of the neck surface region for a ceramic with dielectric permittivity equal to 30, very similar to YSZ [54]. According to the diagram reported in Ref. [54], the field intensification factor is ~ 50, 20 and 8 for neck/particle size radius equal to 0.05, 0.10 and 0.20, respectively. If one considers, as a limit condition, a perfect spherical pore, the field is intensified 1.95 times close to the pore surface (dielectric permittivity = 30)[76]. The specific power dissipation is: (51)

( )

where

( )

and

are the electrical conductivity and field, respectively.

One can assume that the electrical conductivity is substantially constant within the neck region, the temperature being roughly the same. This hypothesis does not disagree with the assumption that strong thermal gradients are present since the neck radius is typically very small, in the micrometric/submicrometric range. Therefore, the local power dissipation close to the neck surface is quadratic with respect to the electric field and the specific power increases ~2500, 400, 64 times with respect to the nominal value in the case of neck/particle size radius equal to 0.05, 0.10 and 0.20, respectively. As a limit condition, in the case of a perfectly spherical pore, the power dissipation intensification is 3.8 times. Such field intensification reasonably amplifies the estimated temperature gradients between the grain core and the neck surface. Moreover, very importantly, it generates gradients within the neck region: the higher power dissipation close to the neck surface, with respect to the neck center, causes a localized overheating of the 17

surface itself. Therefore, it activates mass transport from the neck center to the surface by thermal diffusion. These gradients, localized within the neck, are also increased by the surface enthalpy reduction and, therefore, with reference to Figure 1, the surface is depleted on site “1”, this causing an overheating of the same with respect to the neck center. Finally, a further contribution to neck overheating can be pointed out. Many authors suggested the formation of a plasma during field-assisted sintering processes [19,75,77,78]. The field intensification in the neck region calculated by Holland et al. may be consistent with this hypothesis [54]. The possible formation of air glow or arc discharge can increase locally the temperature gradients and the neck surface temperature and promote the formation of surface liquid phase. In conclusion, it is possible to state that the microstructural temperature gradient can also be in excess of 105 – 107 K m-1 (the values estimated by Holland et al. [53]) during flash sintering of YSZ, thus becoming essential in the activation of thermodiffusion phenomena.

3.2 Thermodiffusion The impact of microstructural temperature gradients on thermal diffusion and, hence, on sintering can be evaluated by Eqs 20 and 21, using the data reported in Table I. The material considered here is 8 mol% yttria-stabilized zirconia (8YSZ). The sample temperature is 1250°C, identical to that estimated by Raj et al. during flash sintering experiment [27]. Different particle size and temperature gradients are considered, in the range 0.1 – 10 μm and 105 – 107 K m-1, respectively. The surface energy is estimated at 1250°C by Eq. 44; the molar volume is calculated from the molecular weight of 8YZS and the theoretical density (equal to 6.1 g/cm3) [71]. The activation energy for cation vacancies migration in 8YSZ is in the range 4.5 – 6.1 eV [60], the average being 5.3 eV (= 511 kJ/mol).

18

The ratio ( ) between thermal diffusion and curvature-driven diffusion is represented in Figure 4 for the earlier sintering stages as a function of the neck-to-particle size ratio (

). Both diffusion paths “2→1” and “3→1” (as

in Figure 1) are considered. The ratio

is lower than 1 in most cases, this

pointing out the very limited thermodiffusion contribution associated with the temperature gradient-driven atomic flux, which is generally orders of magnitude lower than the curvature-driven diffusion. Only for the largest particles ( = 10 μm) and the longest diffusion path (“3→1” in Figure 1), thermal diffusion is dominant (

≥ 106 K m-1), the curvature driving force for

diffusion drastically decreasing. Some possible thermal diffusion effects ( ≥107 K m-1) can also be observed in “3→1” path for 1 μm particles and in “2→1” path for 10 μm particles. These conditions are very extreme for the incubation of typical flash sintering process (power dissipation lower than 10 – 30 mW mm-3) but could be reached during the flash transition. It is therefore crucial to determine whether the flash takes place before or after that the initial sintering stages are completed. In any case, the assumed thermal gradients have an effect on diffusion only in the case of coarse particles.

It is clear that, as the sintering process advances, the neck

curvature decreases and this reduces the driving force for curvature-driven diffusion, leading to larger

values (Figure 4). In any case, even for neck-to-

particle size radius ratio equal to 0.2, a dominant thermal diffusion contribution can not be substantially pointed out. The overall results suggest the very limited thermal gradient contribution during flash sintering incubation, when necks are formed by surface diffusion mechanisms. This is consistent with the results reported by Cologna et al. [79] who showed that the application of DC field/current does not interact with neck growth rate and surface diffusion kinetics. The situation is substantially different in the intermediate stage of sintering, as shown in Figure 5. Here, B slowly decreases as a function of the relative density (

), since the model assumes that the pore radius progressively

decreases during the densification and this accelerates the curvature-driven

19

diffusion. In any case, thermodiffusion plays a fundamental role on mass transport both considering Young and McPherson (Figure 5 (a,b,c)) and Coble (Figure 5 (d,e,f)) sintering geometry, although the first model points out a more important thermodiffusion effect on sintering. An intense thermal gradient,

~ 107 K m-1, is needed to induce a thermal flux of atoms

comparable to that activated by the curvature for 0.1 μm particle size. In the case of 1 μm particles, thermodiffusion is always curvature-driven diffusion for

more intense than

~ 106 K m-1 - 107 K m-1 (about one - two

orders of magnitude larger, depending on the sintering model and relative density). Finally, in the case of 10 μm particles, thermal diffusion is always dominant in the entire thermal gradient range. Such temperature gradientdriven diffusion can be therefore fundamental for the densification of YSZ upon flash sintering. The ratio between thermal and curvature-driven diffusion around a spherical and isolated pore is shown in Figure 6 as a function of the distance from the center of curvature of the pore itself. This geometry well approximates the final stage of sintering. One can observe that for

= 105, 106 and 107 K m-1,

thermal diffusion is dominant only at 1.2, 0.4 and 0.12 μm from the pore center, respectively. Therefore, also in this case, a partial contribution of temperature gradient-driven diffusion to mass transport and densification must be considered. One may ask whether there is any experimental evidence to support the thermodiffusion mechanism for flash sintering. Francis et al. investigated the particle size effect on flash and conventional sintering of YSZ [80] and, interestingly, they observed that in the latter case the fired specimens relative density was 96%, 84%, 72% and 68% using powders with D50 = 0.3, 0.6, 1.2, and 1.7 μm, respectively (firing at 1450°C, heating rate 10°C min-1); in other words, the final density strongly decreases, as expected, with the particle size. Conversely, the relative density after flash sintering was 96%, 84%, 82% and 82% for particles with D50 = 0.3, 0.6, 1.2, and 1.7 μm, respectively, the densification during flash sintering being much less 20

influenced by the particle size; in addition, when powders larger than 0.6 μm are used, an asymptotic behavior is observed, the fired body density becoming substantially constant. Such result appears to be not consistent with the curvature-driven diffusion, whose driving force rapidly decreases with the powder size. Therefore, thermodiffusion could represent a possible explanation of the behavior reported by Francis et al. [80]; as a matter of fact, the temperature gradient (ΔT/Δx) between particle core and neck region is not strongly varying with the particle size [53]. When small powders are used (between 0.3 and 0.6 μm), the curvature plays an important role on the densification behavior (the material being denser when the particle size decreases); nevertheless, when larger particles are used, the driving force for curvature-driven diffusion decreases and an alternative driving force (less dependent on the particle size) becomes dominant, this accounting for the asymptotic behavior observed for particles larger than 0.6 μm. Although the present work points out that thermal diffusion could accelerate the densification phenomena during FS of YSZ, other concomitant mechanisms can not be excluded. Among them, the partial reduction of the oxide in DC polarization or the enhanced densification associated with nonconventional heating rates seem to be the most convincing ones. A more accurate determination of the thermal gradients via numerical simulation could help to provide a definitive clarification of the densification mechanisms during flash sintering.

3.3 Thermal gradients effect on grain boundary motion The microstructural thermal gradient effect on the driving force for grain boundary motion can be calculated for grains with different size (0.1 – 10 μm). According to Eq. 45, the grain boundary energy is 0.67 J m-2 at 1250°C; from literature data, the grain boundary entropy is 0.358 mJ m-2 K-1 [69]. The plot in Figure 7 represents the ratio (Q) between the driving force for grain coarsening (based on curvature) and the driving force for grain 21

boundary pinning (associated to the thermal gradient effect on grain boundary stabilization). One can basically observe that, for thermal gradients up to 107 K m-1, there are no significant pinning effects on the grain boundary associated with grain boundary overheating, the driving force for coarsening being two to four orders of magnitude larger than that for pinning. The grain boundaries are effectively stabilized only at

= 3.7·108, 3.7·109 and 3.7·1010

K m-1 for 10, 1 and 0.1 μm grains, respectively. Such thermal gradients are unreasonably high and would induce a complete grain boundary melting, this determining a temperature difference between the grain core and boundary in excess to 3000 K. Therefore, grain boundary stabilization due to the entropic contribution to the grain boundary energy is very unlikely to take place during field-assisted sintering of YSZ. Nevertheless, since thermal gradients can help densification, they cause an increase of the densification-to-coarsening rate ratio, thus facilitating the consolidation of fine-grained microstructures.

4 Conclusions Microstructural temperature gradients in the order of 106 – 107 K m-1 stimulate the densification of micrometric YSZ powder and could have a pivotal role in flash sintering. Such gradients, are sufficient to accelerate sintering in the intermediate/advanced stages of the process, but are not high enough to increase the surface diffusion processes in the initial stages of sintering, during the incubation of the flash process. In addition, the development of microstructural temperature gradients seems unlikely to be at the base of the grain boundary pinning observed in YSZ under electric field/current application. Therefore, during the flash sintering process, thermal gradients account for larger densification-to-coarsening ratio, this facilitating the consolidation of fine-grained microstructures. 22

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Table I: Data used for the calculation of thermal-to-curvature driven diffusion ratio. T T a γs Ω Hm

1250°C (1523 K) 105 – 107 K m-1 0.1 – 10 · 10-6 m 1.275 J m-2 2.15 · 10-5 m3 mol-1 5.11 · 105 J mol-1

Figure 1: Particles and neck geometry considered for surface diffusion from sites “2” and “3” to site “1”.

32

Figure 2: Particles and pore geometry considered for diffusion from the grain boundary center (site “2”) to the pore surface (site “1”). Diffusion can take place both thought the grain boundary or through the lattice, depending on the relative diffusivities.

Figure 3: Atomic flux toward a closed and isolated pore (radius = Rp).

33

Figure 4: Ratio between thermal and curvature-driven diffusion (Bsurf) as a function of the neck-to-particle size ratio (x/a) for different particle size corresponding to the paths shown in Figure 1: “2→1” (a,b,c), “3→1” (d,e,f).

Figure 5: Ratio between thermal and curvature-driven diffusion (Bint21) from the center of the grain boundary as a function of relative density of the ceramic body relative density ( rel) for “2→1” path in Figure 2 .Three different particle size, 0.1 (a,d), 1 (b,e) and 10 μm (c,f) are considered. Two different geometrical models of particels/pores arrangement are considered corresponding to Eq. 25 (d,e,f), Coble model [66], and Eq. 26 (a,b,c), Young and McPherson model [57].

34

Figure 6: Ratio between thermal and curvature driven diffusion (Blatt) as a function of the distance (r) from the center of curvature of an isolated pore.

Figure 7: Ratio between pinning and grain coarsening (Q) thermodynamic driving force as a function of microstructural thermal gradients and for three different grain size (Π).

35