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Cr320 grain growth-lectureslides

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CERAMIC SINTERING

CERAMIC SINTERING

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  • 1. Grain Growth Shantanu K Behera Dept of Ceramic Engineering NIT Rourkela CR 320 CR 654Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 1 / 27
  • 2. Chapter Outline1 General Features Grain Growth and Coarsening NGG and AGG2 Ostwald Ripening3 Normal Grain Growth Burke and Turnbull Model Topology4 Abnormal Grain Growth5 Boundary Mobility Solute Drag Particle Inhibited Grain Growth Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 2 / 27
  • 3. General FeaturesIntroduction Engineering properties of materials are influenced by: Microstructure Shape and size of grains Porosity, Pore size, and their distribution Second phases, and their distribution The first step is to analyze grain growth in fully dense single phase ceramics/materials. This method lets you study only grain growth without other effects such as that of porosity, second phases, impurities, solutes, dopants etc. Subsequently, the influence of pores and second phases can be studied to design fabrication parameters. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 3 / 27
  • 4. General Features NGG and AGGGrain GrowthGrain Growth Coarsening is generally referred to the is generally referred to the increase in the average grain size simultaneous growth of grains as of a dense compact (either single well as pores (of course, in a phase, or containing a second porous solid). phase particle/precipitate. Both pores and grains increase in Grains grow at the expense of size, and decrease in number. other grains (Imagine a king Complex in nature. extending his empire by winning smaller states). Relatively simple analysis. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 4 / 27
  • 5. General Features NGG and AGGOccurrence of Grain Growth ˚ General width of a GB is 5A. Atoms from the convex side of the grain move to the concave side of the grain surface. Resulting atomic flux induces the boundary to move towards its center of curvature. Chemical potential difference across the two surfaces is responsible.Figure : Fig 3.1, Sintering of Ceramics,Rahaman, pg. 106 Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 5 / 27
  • 6. General Features NGG and AGGNormal and Abnormal Grain Growth Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 6 / 27
  • 7. General Features NGG and AGGNormal and Abnormal Grain Growth NGG: Self similar microstructural development. Only scaling dependence. AGG: Time invariant distribution is lost. Some grains grow at the expense of others, causing bimodal distribution. At a later stage, these large grains impinge to make a unimodal distribution again, but with much larger average grain size. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 7 / 27
  • 8. General Features NGG and AGGNormal and Abnormal Grain GrowthFigure : Normal and abnormal grain growth in alumina; Dillon,Behera,Harmer, LehighUniversity Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 8 / 27
  • 9. General Features NGG and AGGImportance of Grain Size 1 Strength varies with G as G− 2 Dielectric breakdown strength (ZnO) increases as G−1 . Dielectric constant increases with decreasing G (upto ∼ 1 µ. Densification decreases withFigure : Fig. 3.5, MN Rahaman, pg. 110; increase in grain size. [ ρ dρ = 1 dt K Gm ]Densificaiton mechanism for pores But, creep deformation increasesattached to a GB, and in the bulk, Arrows with decreasing grain size.indicate possible diffusion paths. Poreswhen detached from the GB can becomedifficult to be removed, thus limitingdensity. Therefore, keeping a low grain sizeis key to attainment of high density. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 9 / 27
  • 10. Ostwald RipeningOstwald RipeningOstwald ripening refers to the coarsening of particles or precipitates in a solidor liquid medium. Features of grain and pore growth are similar to O.R. Chemical potential on the surface of the particle with radius a is 2γΩ µ = µ0 + a The solute concentration dependence can be written asFigure : Fig. 3.6, MN Rahaman, C 2γΩ kT ln = µ − µ0 =pg. 111; Coarsening of particles C0 adue to materials transport from the C Csmaller particles to the larger Since ln C0 = C0 , thereforeones. C 2γΩ = C0 kTa Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 10 / 27
  • 11. Ostwald RipeningOstwald Ripening MechanismsOR controlled by Interface Reaction OR controlled by Diffusion αT C0 γΩ2 8DC0 γΩ2 [a ]2 − [a0 ]2 = t [a ]3 − [a0 ]3 = t kT 9kT αT is a transfer constant. a is the critical radius that neither Follows a parabolic growth law. grows nor shrinks. Interface reaction is rate Follows a cubic growth pattern. controlling. Diffusion of the solutes is rate Rate is independent of the volume controlling. fraction. Volume fraction of the media affects the rate. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 11 / 27
  • 12. Ostwald RipeningAn ExampleFigure : Fig. 15.8, Sintering, SJL Kang, pg. 219; Growth of a spinel crystal (MgAl2 O4 )from a glass melt. Diffusion controlled? Or..Interface controlled? Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 12 / 27
  • 13. NGG Burke and Turnbull ModelBurke-Turnbull Model Transfer of atoms across the grain boundary under the driving force of the pressure difference between the two internal interfaces. The grain boundary energy (γb ) is considered isotropic. The boundary width (δb is assumed to be constant.The grain boundary velocity, therefore, can be defined as: dG vb = dtwhere G is the average grain size.Additionally, the boundary velocity can also be defined in terms of the dragforce (Fb ), which is essentially the results of difference in curvature) and anadditional term called, boundary mobility (Mb , with units m.N−1 .s−1 ). vb = Fb MbSo, boundary mobility is the velocity of the boundary per unit drag force. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 13 / 27
  • 14. NGG Burke and Turnbull ModelBurke-Turnbull Model Contd.The pressure difference is 1 1 α P = γb + = γb r1 r2 Gwhere α is a geometrical constant.Force, as the gradient of chemical potential over the boundary width, can bewritten as dµ 1 1 Ωγb α Fb = = [Ω P] = dx dx δb GAtomic flux is Da dµ Da Ωγb α J= . = ΩkT dx ΩkT δb G dG Da Ω γ b α vb = = ΩJ = dt kT δb G Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 14 / 27
  • 15. NGG Burke and Turnbull ModelBurke-Turnbull Model Contd.Boundary mobility can be defined as Da Ω Mb = kT δbTherefore, dG γb α vb = = Mb dt GUpon integration, we have G2 − G2 = Kt 0where K = 2αγb MbThis is called the parabolic law for grain growth, quite similar to the interfacereaction-controlled Ostwald ripening. This expression generally describes thegrowth of grain in a pure material (metal or ceramic) that is not influenced byany solutes, segregants, pores, second phases etc. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 15 / 27
  • 16. NGG Burke and Turnbull ModelActivation Energy of Grain GrowthThe rate constant (or growth factor, as it is called sometime), K, has Arrheniusdependence, and can be written as Qa K = K0 e kTwhere, Qa is called the activation energy of grain growth. This can giveinformation on the type of diffusion. For example, in an ionic solid (generally aceramic) the rate controlling species (either the cation or the anion) will haveits diffusion activation energy, similar to that of the A.E of grain growth. Here, itis the slowest moving species.The boundary mobility (Mb ) in pure materials is called the intrinsic boundarymoblity, and the Da in the mobility expression represents the diffusioncoefficient of the rate limiting species. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 16 / 27
  • 17. NGG Burke and Turnbull ModelDeviation from Burke-Turnbull ModelIn practice, however, normal grain growth doesn’t follow the parabolic growthlaw in many ceramics, and in some metal systems. The growth law, therefore,is generalized as: Gm − Gm = Kt 0where m (called the grain growth exponent) can take any value from 2 to 4.The value of m = 3 is widely reported in ceramics. This is the cubic graingrowth law.The deviation from m = 2 is generally explained as the effect of solutes andimpurities. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 17 / 27
  • 18. Boundary MobilityIntrinsic Boundary MobilityFigure : Fig. courtesy: Shen Dillon. There is difference in the calculated andexperimental intrinsic boundary mobility of alumina by orders of magnitude. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 18 / 27
  • 19. Boundary Mobility Solute DragSolute DragDopants and/or impurities still can change the boundary mobility. Dopantsand impurities, if dissolved in the matrix, can cause solute drag.If the grain boundary has an interaction potential with the solute, which mayresult from elastic strain energy considerations (size mismatch between thehost and dopant cations) or from electrostatic potential energy (due to thecharge effect; eg. if the host and the dopant have different valencies).This could lead to a distribution of the solute across the grain boundary, whichcan become asymmetric once the boundary starts moving. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 19 / 27
  • 20. Boundary Mobility Solute DragSolute Drag: SchematicFigure : Distribution of the solutes across the boundary (a); Asymmetric distributiondue to a moving boundary (b); Left out solute cloud and boundary break away eventapproaching the mobility of a clean boundary (c). Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 20 / 27
  • 21. Boundary Mobility Solute DragSolute Drag: Cahn ModelSolute drag, Fs is analytically defined as αC∞ vb Fs = 1 + β 2 vbwhere, α can be defined as the solute drag per unit velocity per unit dopantconcentration (in the low velocity limit), and 1/β is the drift velocity with whichthe solute atom/ion moves across the grain boundary.The total drag force is αC∞ vb vb F = Fs + Fb = 2v + 1+β b MbIn the low velocity limit, we can neglect β 2 v2 . Therefore, b F vb = 1 Mb + αC∞ Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 21 / 27
  • 22. Boundary Mobility Solute DragSolute Drag: Boundary Velocity vs ForceFigure : The relationship between driving force and velocity for boundary migrationcontrolled by solute drag. Individual components of the intrinsic drag and the solutedrag, as well as combined drag on the boundary are indicated. Note that when the driftvelocity β −1 is is comparable to the boundary velocity, the dominance of solute dragdecreases (this refers to the boundary break-away event). Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 22 / 27
  • 23. Boundary Mobility Solute DragEffective MobilityThe effective boundary mobility can be defined in terms of the intrinsiccomponent Mb and the solute drag component Ms : eff 1 1 −1 Mb = + Mb Ms 1where Ms = αC∞ . For conditions where the solute segregates to the grainboundary core, the centre of the boundary contributes heavily to the drageffect.. Here, α can be approximated as: 4Nv kTδb Q α= Dbwhere Q is the partition coefficient for the dopant distribution between theboundary region and the bulk region (i.e the solute concentration in theboundary region is QC∞ ). Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 23 / 27
  • 24. Boundary Mobility Solute DragAdsorbate DragThe mobility due to solute drag is, therefore, Db Ms = 4Nv kTδb QC∞Some principles can be outlined for the selection of dopants that are mosteffective in reducing boundary mobility: When the diffusion coefficient of the rate limiting species (Db ) is low. which means that the oversized dopant ions (bigger than the host) can be effective since the bigger ions possess lower diffusivity in general. When the segregated solute concentration (QC∞ ) is high. which means that highly segregating dopants can be effective.Some examples in ceramics for grain growth control: Host(solute): Al2 O3 (Mg,Y, Zr), BaTiO3 (Nb, Co), ZnO(Al), Y2 O3 (Th), CeO2 (Y, Nd, Ca) Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 24 / 27
  • 25. Boundary Mobility Particle Inhibited Grain GrowthZener ModelAssumptions: Monosized, spherical, insoluble, immobile, and randomlydistributed particles in a polycrystalline matrix.The driving force (per unit area) of a grain boundary with principal radii ofcurvature a1 and a2 : 1 1 αγb Fb = γb + = a1 a2 Gwhere α is a geometrical factor (2 for spherical grains), γb is the boundaryenergy, G is the grain size. Figure : The zener model for particle inhibited grain growth. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 25 / 27
  • 26. Boundary Mobility Particle Inhibited Grain GrowthZener ModelWhen the boundary meets the particle, extra work is required for its motion.Therefore, the retarding force exerted is: Fr = γb Cosθ(2πrSinθ)Maximum retarding force is applied when θ = 45◦ . Thus Fmax = πrγb rFor NA inclusions, the total force is Fmax = NA πrγb dIf the volume fraction of the inclusions is f , the number of inclusions per unit 3fvolume is Nv = 4πr3 . Therefore, the total drag due to the particles is: 3f γb Fmax = d 2r Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 26 / 27
  • 27. Boundary Mobility Particle Inhibited Grain GrowthZener ModelTherefore, the net driving force per unit boundary area is: α 3f Fnet = Fb − Fmax = γb d − G 2rWhen Fnet = 0, boundary motion is ceased. The intrinsic drag is balanced bythe drag exerted by the particle. For this condition a limiting grain size can bedefined: 2αr GL = 3fThis is called the Zener Relationship. GL is proportional to the (second phase)particle radius, and inversely proportional to the fraction of the second phaseprecipitates/particles. Further grain growth could occur if the inclusion coarsens by Ostwald ripening, the inclusion dissolves and goes into solid solution, if AGG occurs. Shantanu Behera (NIT Rourkela) SINTERING CR 320 CR 654 27 / 27