Rheological Model for Viscous Flow Densification During Supersolidus Liquid Phase Sintering

A model is derived for the viscous flow densification of prealloyed powders heated to just over the solidus temperature, a process termed supersolidus liquid phase sintering. The model builds from the viscous flow concepts first introduced by Frenkel with a new porosity effect. Viscous flow densification starts with the formation liquid on the grain boundaries inside the particles, with subsequent spreading of liquid to form a capillary bond between contacting particles. Pores lower the initial semisolid viscosity, but as densification progresses the viscosity increases. On the other hand, viscosity decreases higher temperatures increase. Densification is induced by the capillary force acting against the semisolid system, but densification is delayed until the particles sufficiently softened from liquid spreading on the grain boundaries. Thus, both viscosity and strength vary with the liquid content and particle microstructure. Distortion in sintering traces to an excess of liquid that lowers the skeletal rigidity, mainly due to grain growth with a concomitant release of grain boundary liquid. This often occurs after full densification is achieved. This modification to the model includes the correction for porosity effects on viscosity. The model is compared with data on a 316L stainless steel doped with boron, subjected to in situ densification and slumping observations.


Introduction
As Frenkel realized many years ago, viscous particles will densify under the action of surface tension.Most liquid phase sintering begins by mixing two or more small powders of differing compositions [1].On heating, one powder melts or reacts to form a liquid between the particles that engulfs the more refractory phase.If the particle size is small, then capillary forces from the wetting liquid enhance densification [2].After sintering the product is a composite of refractory grains interlaced with a solidified liquid.To obtain significant advantage from the liquid requires a large capillary force from the wetting liquid, which in turn mandates a small initial particle size, often in the range of 1 µm or less.
A variant to traditional liquid phase sintering is to use alloy powders which are heated to a temperature between the liquidus and solidus, known as supersolidus liquid phase sintering (SLPS).The liquid forms inside the particles and spreads to the particle contacts, resulting in a capillary force acting on the semisolid particles.Densification is rapid once sufficient liquid forms to fragment the particles into individual grains [3].Most important, SLPS allows full-density sintering of larger particles.
Densification during SLPS is analogous to viscous flow sintering, since the semisolid particles flow once liquid spreads along the grain boundaries.The viscosity decreases as the liquid volume fraction increases, so higher temperatures give more liquid to induce faster sintering, often with a loss of dimensional precision.Consequently, temperature (which controls the solid-liquid ratio) is a main determinant of sintered density and dimensional precision.Densification occurs over a narrow temperature range, and densification without distortion often comes with shorter times at the peak temperature.A viscous flow model for SLPS has emerged from a variety of critical experiments with various alloy powders [3][4][5][6][7] and calculations on the rheology of semisolid structures [8,9].

Alloy composition effects
When an alloy particle is heated, first liquid forms at the locations of last solidification on cooling.During SLPS the liquid typically first nucleates inside the grains and along grain boundaries [10][11][12].A linear relation is assumed between the liquidus and solidus temperatures and composition to estimate the solid volume fraction as a function of temperature.The solidus and liquidus temperatures (T S and T L ) change linearly with alloy composition X A as follows: where T M is the base melting temperature, X A is the alloying content on a weight basis, and A and B are the slopes.Actual alloy melting behavior may be more complicated.With the assumed linear relation, the compositions at the liquidus and solidus lines (X L and X S , respectively) are functions of temperature are follows: The tie line between these two compositions allows calculation of the solid mass fraction M S in the particle (ignoring porosity) at a given sintering temperature T, In turn, ignoring pores, the solid volume fraction Φ depends on the solid mass fraction and the densities of the solid ρ S and liquid ρ L phases as follows: This allows calculation of the solid volume fraction for a given alloy and temperature.
Critical to SLPS is particle softening that comes from liquid penetration along the grain boundaries.Newly formed liquid can be located in three regions -at the interparticle neck, along grain boundaries inside the particles, and at pockets located inside the grains.The liquid located at the interparticle necks provides the capillary force for densification, but the liquid located on the grain boundaries lubricates grain sliding during densification.From quenched micrographs, it appears necks form early, but densification is delayed by the particle rigidity.Liquid that forms inside the grains plays no discernible role in densification.If a high fraction of internal liquid forms, then densification is delayed to a higher total liquid content.Because of significant differences in alloys, powders, and thermal cycles, the optimal liquid content for densification is highly variable between experiments and materials [3].

Microstructure evolution relations
A schematic of the sintering microstructure and its evolution is given in Fig. 1.Key features are associated with the particles, grains, necks, grain boundary liquid, and pores.The particle size is D, grain size inside the particles is G, neck size between particles is X, and width of grain boundary liquid film is δ.As densification progresses, spherical pores form with diameters of size d P .Initially the film thickness is constant until most of the grain boundaries are wetted, but it coarsens late in sintering [7].The particles are assumed to be spheres and the grains are assumed to be polygons, with a grain shape that varies with the relative liquid content [13,14], where the grain coordination number N G is estimated from the solid volume fraction Φ as, log 10 (N G )=1.15+1.25 log 10 ( Φ ) (7) The total liquid volume is represented by the sum, V L = V B + V N + V I , where V B , V N , and V I designate the liquid volumes (per particle) at the boundary, neck, and grain interior.With respect to a single spherical particle, the liquid volume fraction tied up in these three forms gives the solid volume fraction as, The liquid film on the grain boundaries inside the particles is assumed to be relatively small compared to the grain size G; thus, with S G being the surface area per grain, δ/2 being the width of the grain boundary film assigned to each grain, F C being the fractional grain boundary coverage by liquid (0 ≤ F C ≤ 1), and n G being the number of grains per particle.Most important is the fractional coverage of grain boundaries by liquid, since this determines particle rigidity.Initially, before liquid formation, the number of grains per particle varies with the cube of the grain size to particle size ratio, (10) assuming the approximate grain volume is V G = G 3 /2.At high liquid contents the grain boundaries are wetted by liquid and the film thickness on the grain boundaries contributes to the total volume.As liquid spreads on the grain boundaries, Equation 10 is modified to give an approximate solution for the volume of liquid and solid associated with each grain V G as, ( ) Accordingly, the number of grains becomes, as the volume of boundary liquid per particle.
The volume of liquid per particle located at the necks between particles depends on the neck size as measured by X; thus, where F C is the fractional grain boundary coverage by liquid and N P is the particle packing coordination.Quenched microstructures show the neck size approximates to the grain size as, The particle packing coordination number N P varies with the fractional density ρ for the system and can be empirically estimated for sintering structures as follows [14], Because the particles remain essentially spherical, but the grains are shape accommodated, there is a difference in the coordination numbers (Equations 7 versus 16).
The quantity of liquid at the grain interior is assumed to remain a constant fraction of the total liquid V L [5], where F I is a the fraction of liquid at the grain interior (0 ≤ F I ≤ 1).It depends on the details of the powder microstructure.
A combination of Equations 13,15, and 17 gives the liquid volume as follows: ( ) Equation 18 can be regrouped to give Equation 19 provides an important relation between liquid volume in a particle, as estimated from the phase diagram, and the microstructural features.

Rigidity of the semisolid structure
During SLPS the solid fraction is dictated by the sintering temperature.The rheological response is set by the fluidity of the solid-liquid system.Above the solidus temperature liquid films coat grain boundaries, resulting in a loss of strength [15].Then grain sliding occurs in response to the capillary forces.Accordingly, the fractional coverage of liquid on grain boundaries, F C , determines the densification kinetics.Combining Equations 8 and 19 with some numerical approximations gives, ( realizing the fractional coverage is also related to the number of grains per particle, n G .
During liquid formation and spreading there is grain growth.Additionally, as expressed by Equation 7, there is a change in the grain coordination number with solid content while grain size increases due to coarsening as follows [11]: where G o is the starting grain size and κ is the grain growth rate constant.Measurements of grain size just below the solidus versus in the two phase solid-liquid region often give a 20fold increase in the rate constant with liquid formation [11].Further, the grain growth rate constant is sensitive to the liquid content.To account for these factors, a hybrid grain growth rate constant is applied to the SLPS problem [14,16], resulting in the following form: where F C is the fractional coverage of grain boundaries, κ S is the solid-state grain growth rate constant, κ L is the liquid phase rate constant, and Φ is the solid volume fraction.
During melting and grain growth, initially there is too little liquid to fully coat grain boundaries, so the liquid films remain thin until most of the boundaries are penetrated.Subsequently, with an excess of liquid over that needed to coat the boundaries and necks, the grain boundary film coarsens with further melting and grain growth [7].In typical liquid phase sintering systems, the grain growth rate constant is in the range of a 0.1 to 100 µm 3 /s.Experiments with a boron-doped austenitic stainless steel during SLPS found κ in the range of 50 to 70 µm 3 /s [11], strength in the 100 kPa range, while viscosity during is near 250 Pa⋅s at the onset of densification [17].Thus, grain growth is relatively rapid during densification.Since the growing grains change the partition of liquid to the grain boundaries, grain growth lowers the rigidity and viscosity.The decreasing number of grains per particle leads to increased liquid coverage of the remaining grain boundaries.Thus, grain growth lowers the semisolid system viscosity.
Percolation concepts are important to determining the onset of viscous flow of the semisolid particles.If a high level of solid grain bonding exists, then the system is rigid and no densification occurs.Alternatively, if no bonding occurs, then the lack of rigidity results in compact slumping.The relation between the microstructural connectivity and viscous flow is given by the critical condition for loss of a percolated structure [5], ) where C N is the critical number of connections, N G is the grain coordination number, and P C is the probability of a connection between two contacting grains.The relation between F C the fractional grain boundary coverage and P C is given as follows: (24) For loss of a percolated structure, C N is approximately 1.5, which is the condition for the formation of a semisolid structure [5,18].The mushy condition important to SLPS occurs when C N is 2.4.Supersolidus liquid phase sintering occurs when the solid content is between 0.6 and 0.9 of the volume and 80 to 90 % of the grain boundaries are coated by liquid.
Prior reports show about 30 vol. % liquid is typical for SLPS densification, which gives N G = 9.0 (Equation 7) and F C = 0.83 for loss or rigidity.However, once the fractional coverage reaches unity there is an inability to retain compact shape during sintering.This explains the observed narrow sintering windows for SLPS, and further explains the eventual loss of shaped during SLPS, since grain growth leads to release of liquid to coat grain boundaries with a loss of rigidity.

Capillary Stress
The liquid at the particle contacts provides a capillary force [8] that induces neck growth and densification by viscous flow [14].For two spheres of diameter D, the contact stress Φ across the bond between particles is estimated as follows: where ( LV is the liquid-vapor surface energy, θ is the wetting angle (assumed to be zero), and ∆L/L o is the sintering shrinkage.The maximum shear stress is one-half the contact stress.It is the shear stress at the particle bonds that drives sintering densification in SLPS [9].

Densification by viscous flow
The rheological response of a semisolid particle depends on both the strength and viscosity.The initial structure is bonded to provide strength.As liquid spreads on the grain boundaries, the yield strength declines rapidly above the solidus temperature.Yield strengths of semisolid systems have been reported over a wide range from 0.01 kPa to 100 MPa [15,19,20].The capillary sintering stress is typically near 1 MPa.But the strength declines as liquid spreads in the microstructure.In the condition where densification occurs, the viscosity converges toward that encountered in highly in other net-shaping semisolid systems [20][21][22][23][24][25][26].As the pores are filled, the loss of porosity reduces the densification rate to zero.Note the porosity effect on viscosity is based on the liquid phase sintering observations by Gillia et al. [27], namely η = Ω exp[Bρ] where η is the viscosity, ρ is the fractional density, Ω is a temperature dependent preexponential, and B is reported as 23.
Besides the porosity effect, the SLPS system is sensitive to solid content and strain rate [19,23,28,29].At high liquid contents the behavior converges toward Newtonian viscous flow with a small or nonexistent yield strength.A fit to available data shows the yield strength τ Y variation with solid fraction Φ can be approximated as, ( ) where Φ Y is the solid content corresponding to loss of strength and τ o is the alloy strength just prior to liquid formation.For SLPS, a solid volume fraction near 0.6 is reasonable [10,11,30].Above this solid content the yield strength progressively increases, but the system strength is generally less than the sintering stress, meaning that τ o is probably near 1 MPa.Based on the observations by Wang and Raj [31], a Bingham response is assumed here where τ is the capillary induced shear stress, τ Y is the yield stress, η is the apparent viscosity, and dγ/dt is the shear strain rate.In the range where SLPS occurs, measured strengths are below 1 MPa.Calculations for various assumptions of Φ Y (from 0.3 to 0.6) and τ o (from 0.01 to 1 MPa) determined the densification temperature was not sensitive to these parameters.
The viscosity of a semisolid particle system depends on several factors.Rheological models were assessed for rheocasting, solder pastes, filled polymers, cements, superplastic glass-ceramics, and liquid phase sintering [5,[19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].Based on earlier models [5,9,22] as adapted to the conditions where the fractional grain boundary coverage by liquid allows viscous flow, the apparent viscosity is calculated as follows [35,36]: where η o is a combination of factors that includes numerical constants, the liquid viscosity preexponential, and a grain agglomeration factor; G o is the normalized grain size characteristic to the rheocasting measurements; Q is the activation energy for viscous flow; R is the gas constant; T is the absolute temperature; m is the strain rate sensitivity exponent; Φ C is the critical solids loading for viscous flow, Ω is a preexponential for the porosity effect on viscosity, ρ is the fractional density, and B = 23.When the solid content in the semisolid particles is above the critical level, the system strength resists viscous deformation.At low densities, the viscosity is low, similar to what is observed in foams.
From prior reports with loaded systems, rheocasting alloys, and SLPS alloys, it is evident Φ C depends on several microstructure parameters.From Equations 20, 22, and 23, the critical solids loading for the onset of viscous flow is calculated to range from 0.9 to 0.6 for most cases, with a mode of 0.7 [3].Thus, prior reports provide guides to the values for the model parameters; the viscosity of many pure metals is near 5 mPa⋅s, with preexponential viscosity terms near 0.3⋅10 -3 Pa⋅s and activation energies Q usually between 5 and 50 kJ/mol [37].The strain rate sensitivity m is usually between 0.7 and 1 [5,19,28,29,31,38] and is assumed to be 1 for these calculations.Accordingly, viscosity data give, This estimate arises since the sintering densification rate usually peaks at equivalent strain rates between 10 -2 and 10 -3 s -1 .Equation 29 allows calculations over a wide range of densification rates and grain sizes.
Equations 20, 24, and 25 provide a base for calculation of Φ C .For example, a Pb-Sn solder at 42 vol.% solid has an apparent viscosity below 1 Pa⋅s [19] at a shear rate of 230 s -1 , while a Cu-Al alloy at 50 vol.% solid has an apparent viscosity in the range from 0.5 to 10 Pa⋅s depending on shear rate [28].
Based on the SLPS analysis by Liu et al. [9], isothermal sintering shrinkage is expressed as follows: where γ LV is the liquid-vapor surface energy, η is the semisolid apparent viscosity, t is the sintering time, and D is the particle diameter.Equation 30 is valid for shrinkages up to 10 % [36].The sintered density can be calculated assuming constant mass and isotropic shrinkage.The sintered density for a compact starting at a fractional green density ρ G is given as, where ρ is the fractional sintered density.
In supersolidus liquid phase sintering, pore close into spheres and trap the process atmosphere at densities between 85 and 95 % of theoretical [47].Final stage densification depends on the elimination of these pores as covered by various models [14,48,49].When sintering occurs in vacuum, final densification is relatively fast because of the increasing capillary pressure as the pore size decreases.As densification occurs there is approximately one pore per particle, giving a relation between the pore size d P, particle size D, and sintered fractional density ρ, At a fractional density of 0.92 this gives a pore size to particle size ratio (d P /D) of 0.44, assuming no pore coarsening.
The driving stress for final stage viscous flow sintering is the surface tension of the spherical pore, reduced by the internal pressure from trapped gas, where γ LV is the liquid-vapor surface energy and P g is the gas pressure in the closed pore.As the pore shrinks the increasing stress accelerates densification, but simultaneous pore coarsening or gas pressurization retards densification [50].This gives a final stage densification model for densities over 0.9 (the point of pore closure) where gas pressurization during pore closure leads to progressively slower pore shrinkage [48,49], ( ) When the gas pressure is known during sintering, then the final density is estimated assuming no solubility of the gas in the sintering material.More complicated cases involving soluble gases are treated elsewhere [50].The pore size at the onset of the final stage sintering is given by Equation 32, designated as d PF .Subsequently, the pore size is limited by gas pressurization [14], recognizing that the gas pressure is set by the outgassing species, reaction species, or ambient atmosphere.For an insoluble gas such as argon, P P (36) where P gF is the gas pressure at pore closure.The maximum final density is then given by rearranging and solving Equation 32.Because of the densification retarding effect from a trapped insoluble gas, it is best to perform SLPS in vacuum.

Densification calculations
In SLPS, once the liquid forms, then densification is initiated by the capillary action of the wetting liquid at the particle contacts, but the initially high system viscosity limits sintering.The temperature for the onset of densification is dictated by several parameters, but the critical volume fraction of solid Φ C is a dominant factor.Low values of critical solid content delay densification to relatively higher temperatures.Accordingly, density depends on temperature and several other parameters, including hold times, heat rates, and grain growth rates.One characteristic aspect of SLPS is a narrow temperature or time range over which density changes rapidly, especially for coarse powders which exhibit negligible solid-state bonding during heating.
Computer simulations based on this model were conducted using parameters representative of SLPS.Calculation of the sintered density versus time was performed with an iterative, variable time step [51].As a new example, consider data published by Bollina and German [17] for the density versus temperature of a 49 µm water atomized 316L stainless steel powder doped with 0.3 wt.% boron.In situ video imaging observations and dilatometry were used to follow deflection for calculation of viscosity and density over temperatures from 1250 to 1400 o C using a 5 ο C/min heating rate.The SLPS model outlined above is ideally suited to simulation of such behavior.A comparison of the density predictions gives the following: The measured viscosity based on bending beam determinations are from 130 to 450 MPa⋅s, with densification being evident at about 250 MPa⋅s.Other reports provide similar ranges of semisolid viscosity during SLPS densification.Although the above density data are not a perfect fit, such findings are indicative of the value in modeling SLPS behavior.Several other systems have been treated in prior reports [4,5,10,12,42,[52][53][54][55][56][57][58][59][60].

Distortion during sintering
A common problem with SLPS is the loss of component shape at full density.The temperature-time combination required for densification is often close to the combination associated with distortion.Both densification and distortion are associated with the viscous flow attributes of the SLPS system [22], but the deviatoric stress from surface traction, substrate friction, and gravity induce distortion.If too much liquid is formed, then gravity causes distortion.From a practical standpoint it is necessary to form sufficient liquid to densify the compact, but not so much as to lose resistance to distortion [61,62].The model provides a mapping of the processing conditions associated with densification, showing long times at lower temperatures are one option to avoid distortion.Most important, distortion is minimized (besides that from density gradients in the green body, nonuniform heating, or friction with the substrate) if the material is slightly undersintered.On the other hand, model determination of the grain boundary coverage by liquid at densification provides a means to predict distortion.

Practical implications
Process control during SLPS is generally assisted by higher alloying levels to increase the separation between the liquidus and solidus temperatures.This minimizes the effects of slight compositional or temperature fluctuations.To maximize the SLPS benefit, the liquidus temperature should decrease with alloying.This ensures liquid nucleation on grain boundaries due to segregation on solidification.Alloying additions greatly improve the SLPS process if the addition segregates to grain boundaries and forms a low melting temperature phase.In many metallic systems, boron has an ideal impact on the melting behavior, so boron-doped alloys are well aligned with SLPS.Indeed, with boron the alloying addition is highly mobile during heating and can be co-mixed with the prealloyed powder to form a liquid [35,57,63,64].The melt from the admixed powder will then penetrate the grain boundaries of the major phase, giving full density at low temperatures.The role of boron caused one of the early misunderstandings on powder injection molding, since some of the early binders contained boron or boric acid.It was not injection molding that allowed full density, but the boron used in the binder system.
Applications for SLPS include carbon steels, tool steels, stainless steels, copper alloys, titanium alloys, precious metal alloys, nickel-base superalloys, and cobalt-base alloys.High sintered densities are possible with large grain sizes and 10 to 20 vol.% liquid.This model shows potent control of densification is through the rate of grain growth.A large grain size with a low grain growth rate constant minimizes the liquid content needed for densification and minimizes distortion.

Summary
Frenkel's model for viscous flow sintering densification is used to explain the rapid densification of coarse prealloyed powders as observed in supersolidus liquid phase sintering.The model uses percolation concepts to create a rheological response that matches well with observations of rapid densification over narrow temperature or time ranges.The model emphasizes the importance of controlling the fractional coverage of grain boundaries by liquid to ensure complete densification.

Fig. 1 .
Fig. 1.The conceptual outline of supersolidus liquid phase sintering densification for three particles: a) initial particle packing, b) formation of initial liquid with insufficient wetting of grain boundaries for densification, c) viscous flow densification of semisolid particles, and d) final stage densification with closed, spherical pores.

Fig. 2 .
Fig. 2.An image generated at 1260°C for a water atomized 316L stainless steel powder doped with boron, showing the midpoint deflection of a thin, flat sample as used to calculate the viscosity during SLPS.