Handbook of Mathematical Relations in Particulate Materials Processing: Ceramics, Powder Metals, Cermets, Carbides, Hard Materials, and Minerals: 3 ... on Processing of Engineering Materials) - Hardcover

German, Randall M.; Park, Seong Jin

Hardcover

ISBN 10: 0470173645 ISBN 13: 9780470173640

Verlag: Wiley-Interscience, 2008

Inhaltsangabe

The only handbook of mathematical relations with a focus on particulate materials processing

The National Science Foundation estimates that over 35% of materials-related funding is now directed toward modeling. In part, this reflects the increased knowledge and the high cost of experimental work. However, currently there is no organized reference book to help the particulate materials community with sorting out various relations. This book fills that important need, providing readers with a quick-reference handbook for easy consultation.

This one-of-a-kind handbook gives readers the relevant mathematical relations needed to model behavior, generate computer simulations, analyze experiment data, and quantify physical and chemical phenomena commonly found in particulate materials processing. It goes beyond the traditional barriers of only one material class by covering the major areas in ceramics, cemented carbides, powder metallurgy, and particulate materials. In many cases, the governing equations are the same but the terms are material-specific. To rise above these differences, the authors have assembled the basic mathematics around the following topical structure:

• Powder technology relations, such as those encountered in atomization, milling, powder production, powder characterization, mixing, particle packing, and powder testing

• Powder processing, such as uniaxial compaction, injection molding, slurry and paste shaping techniques, polymer pyrolysis, sintering, hot isostatic pressing, and forging, with accompanying relations associated with microstructure development and microstructure coarsening

• Finishing operations, such as surface treatments, heat treatments, microstructure analysis, material testing, data analysis, and structure-property relations

Handbook of Mathematical Relations in Particulate Materials Processing is suited for quick reference with stand-alone definitions, making it the perfect complement to existing resources used by academic researchers, corporate product and process developers, and various scientists, engineers, and technicians working in materials processing.

Über diesen Titel

Über die Autorin bzw. den Autor

Randall M. German, PhD, is the CAVS Chair Professor of Mechanical Engineering and Director of the Center for Advanced Vehicular Systems at Mississippi State University. He holds an Honorary Doctorate from the Universidad Carlos III de Madrid in Spain, is a Fellow of APMI and ASM, holds the Tesla Medal, and is listed in various issues of Who's Who. His accomplishments comprise 850 published articles, twenty-three issued patents, nineteen edited proceedings, and fourteen books, including Sintering Theory and Practice (Wiley).

Seong Jin Park, PhD, is Associate Research Professor in the Center for Advanced Vehicular Systems at Mississippi State University. He is the recipient of numerous awards and honors, including Leading Scientists of the World and Outstanding Scientists Worldwide, both awarded by the International Biographical Centre in 2007. Dr. Park is the author of over 190 published articles and three books, holds four patents, and created four commercialized software programs. His areas of specialization and interest include materials processing technology, numerical technology, and physics.

The National Science Foundation estimates that over 35% of materials-related funding is now directed toward modeling. In part, this reflects the increased knowledge and the high cost of experimental work. However, currently there is no organized reference book to help the particulate materials community with sorting out various relations. This book fills that important need, providing readers with a quick-reference handbook for easy consultation.

This one-of-a-kind handbook gives readers the relevant mathematical relations needed to model behavior, generate computer simulations, analyze experiment data, and quantify physical and chemical phenomena commonly found in particulate materials processing. It goes beyond the traditional barriers of only one material class by covering the major areas in ceramics, cemented carbides, powder metallurgy, and particulate materials. In many cases, the governing equations are the same but the terms are material-specific. To rise above these differences, the authors have assembled the basic mathematics around the following topical structure:

• Powder technology relations, such as those encountered in atomization, milling, powder production, powder characterization, mixing, particle packing, and powder testing

• Powder processing, such as uniaxial compaction, injection molding, slurry and paste shaping techniques, polymer pyrolysis, sintering, hot isostatic pressing, and forging, with accompanying relations associated with microstructure development and microstructure coarsening

• Finishing operations, such as surface treatments, heat treatments, microstructure analysis, material testing, data analysis, and structure-property relations

Handbook of Mathematical Relations in Particulate Materials Processing is suited for quick reference with stand-alone definitions, making it the perfect complement to existing resources used by academic researchers, corporate product and process developers, and various scientists, engineers, and technicians working in materials processing.

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Handbook of Mathematical Relations in Particulate Materials Processing

By Randall M. German Seong Jin Park

John Wiley & Sons

Copyright © 2008 John Wiley & Sons, Inc.
All right reserved.
ISBN: 978-0-470-17364-0

Chapter One

A

ABNORMAL GRAIN GROWTH (Worner et al. 1991; Kang 2005)

Abnormal grain growth involves the excessively rapid growth of a few grains in an otherwise uniform microstructure. It is a particular problem in the later stages of sintering. It is characterized by certain grains or crystallographic planes exhibiting faster growth than average. Figure A1 is a sketch of a microstructure formed as a consequence of abnormal grain growth where one large grain at the top is growing at the expense of the surrounding smaller grains. Abnormal grain growth is favored when segregation changes the grain-boundary mobility or grain-boundary energy. When grain growth occurs, there is an interfacial velocity [V.sub.ij] for the grain boundary between the i-j grain pair given by the product of the mobility [M.sub.ij] and the force per unit area on the grain boundary [F.sub.ij],

[V.sub.ij] = [M.sub.ij][F.sub.ij]

where the grain-boundary velocity varies between individual grain boundaries, as indicated by the subscript. The force [F.sub.ij] is given by the product of the interfacial energy and the curvature,

[F.sub.ij] = - [[gamma].sub.ij] (1/[g.sub.i] - 1/[g.sub.i])

where [G.sub.i] and [G.sub.j] are the grain size for contacting grains, and [[gamma].sub.ij] is the corresponding interfacial energy for the i-j interface. Although not routinely recorded, the interfacial energy depends on the misorientation between grains. Effectively, the energy per unit volume scales with the inverse grain size, so if [G.sub.i] > [G.sub.j], then the force is pushing the grain boundary toward the smaller grain center. A critical condition occurs when the mobility of an individual grain boundary, [M.sub.ij], greatly exceeds the average or when the individual grain-boundary energy is excessively low. This critical condition is expressed by the following inequality:

[M.sub.ij]/[M.sub.m] > 16/9 ([[gamma].sub.ij] / [[gamma].sub.m]) where [M.sub.m] is the mean grain-boundary mobility, [[gamma].sub.ij] is the individual grain-boundary energy, and [[gamma].sub.m] is the mean grain-boundary energy. With respect to abnormal grain growth, the two situations of concern are a twofold higher individual grain-boundary mobility, for example, because of a segregated liquid, or a twofold lower individual grain-boundary energy, for example, due to segregation or near coincidence in grain orientation. In sintering practice, most examples of abnormal grain growth are caused by impurities that segregate on the grain boundaries even at the sintering temperature. For example, in sintering alumina ([Al.sub.2][O.sub.3]), abnormal grain growth is favored by a high combined calcia (CaO) and silica (Si[O.sub.2]) impurity level.

[F.sub.ij] = grain-boundary force per unit area between the i-j grain pair, N/[m.sup.2]

[G.sub.i], [G.sub.j] = grain size for corresponding grain, m (convenient units: [micro]m)

[M.sub.ij] = grain-boundary mobility between the i-j grain pair, [m.sup.3] / (s x N)

[M.sub.m] = mean grain-boundary mobility averaged over the body, [m.sup.3] /(s x N)

[V.sub.ij] = interfacial velocity for the grain boundary between the i-j grain pair, m/s

[[gamma].sub.ij] = individual grain-boundary energy between the i-j grain pair, J/[m.sup.2]

[[gamma].sub.m] = mean grain-boundary energy averaged over the body, J/[m.sup.2].

ABRASIVE WEAR

See Friction and Wear Testing.

ACCELERATION OF FREE-SETTLING PARTICLES (Han 2003)

An assumption in Stokes' law, as applied to both particle-size classification and particle-size distribution analysis, is that the particles instantaneously reach terminal velocity. However, this is not the case in practice, and the acceleration of the particle to the free-settling terminal velocity adds an error in a particle-size analysis. The approach to the Stokes' law terminal velocity [v.sub.T] is described by the following equation for spherical particles initially at rest:

v = [v.sub. T] [1 - exp (- 18 t[eta]/[rho][D.sup.2])]

where v is the velocity after time t when the particle starts from rest, [eta] is the fluid viscosity, [rho] is the theoretical density of the particle, and D is the particle diameter. A plot of this equation is given in Figure A2, where the actual velocity is normalized to the terminal velocity for the case of a 1-mm stainless steel particle settling in water.

D = particle diameter, m (convenient units: [mu]m)

t = time, s

v = velocity (starting with v = 0 at t = 0), m/s

[v.sub.T] = Stokes' law terminal velocity, m/s

[eta] = fluid viscosity, Pa x s

[rho] = theoretical density of the particle, kg/[m.sup.3](convenient units: g/[cm.sup.3]).

ACTIVATED SINTERING, EARLY-STAGE SHRINKAGE (German and Munir 1977)

Activated sintering is associated with a treatment, usually by an additive, that greatly increases sintering densification at lower temperatures than typically required. In activated sintering the initial sintering shrinkage depends on the rate of diffusion in the activator, which is segregated to the interparticle grain boundary. Figure A3 provides a schematic of the sintering geometry used to model first-stage activated sintering. The growth of the interparticle bond results in attraction of the particle centers, which gives compact shrinkage [DELTA]L/[L.sub.0] as follows:

[DELTA]L/[L.sub.0] = L - [L.sub.0]/[L.sub.0] = g]OMEGA][deltaITL [[gamma].sub.SV] [D.sub.A]t / [D.sup.4] RT

where [DELTA]L is the change in length, [L.sub.0] is the initial length, L is the instantaneous length during sintering, g is a collection of geometric terms, [OMEGA] is the atomic volume, [delta] is the width of the second-phase activator layer coating the grain boundary, ITLITL is the solubility of the materials being sintered in the second-phase activator, [[gamma].sub.SV] is the solid-vapor surface energy, [D.sub.A] is the diffusivity of the material being sintered in the activator (note this changes dramatically with temperature), t is the sintering time, D is the particle size, R is the gas constant, and T is the absolute temperature. Faster diffusion in the activator induces early sintering gains, but this mandates that the solid be soluble in the activator. The controlling step is the diffusivity in the activator layer. The difference in effectiveness between various activators is explained by their differing diffusivities and solubilities.

ITLITL = volumetric solubility in the activator, [m.sup.3]/[m.sup.3](dimensionless)

D = median particle size, m (convenient units: [micro]m)

[D.sub.A] = diffusivity of the base material in the activator layer, [m.sup.2]/s

L = instantaneous length, m (convenient units: mm)

[L.sub.0] = initial length, m (convenient units: mm)

R = universal gas constant, 8.31 J/(mol x K)

T = absolute temperature, K

g = collection of geometric terms, dimensionless

t = isothermal sintering time, s

DL = change in length, m (convenient units: mm)

[DELTA]L/[L.sub.0] = sintering shrinkage, dimensionless (convenient units: %)

[OMEGA] = atomic volume, [m.sup.3]/mol

[delta] = activator phase width on the grain boundary, m (convenient units: nm or [micro]m)

[gamma].sub.SV] = solid-vapor surface energy, J/[m.sup.2].

ACTIVATION ENERGY

See Arrhenius Relation.

ADSORPTION

See BET Specific Surface Area.

AGGLOMERATE STRENGTH (Pietsch 1984)

Powder that is wetted by a relatively small quantity of liquid or polymer will agglomerate. If the fluid phase is not solidified or hardened, then the crush strength [sigma] for an agglomerated mass of powder depends on the fractional porosity [epsilon] and the degree of pore saturation S,

[sigma] = 7S][gamma].sub.LV] 1 - [epsilon] / D]epsilon]

where the saturation S is the fraction of pore volume that is filled with liquid (often as small as 0.01), [[gamma].sub.LV] is the liquid-vapor surface energy, and D is the particle size. Unless the agglomerate is wetted by a high-strength polymer, the strength of a typical agglomerated powder is dominated by capillarity effects.

D = particle diameter, m (convenient units: [micro]m)

S = degree of pore saturation, dimensionless fraction [0, 1]

[epsilon] = fractional porosity, dimensionless [0, 1]

[[gamma].sub.LV] = liquid-vapor surface energy, J/[m.sup.2]

[sigma] = strength of the agglomerate, Pa.

[Also see Capillarity.]

AGGLOMERATION FORCE

When a small powder is exposed to water or other condensable vapor, a liquid bridge can form at the contact points between particles. Initially the liquid bridges are small and do not merge, giving a structure termed the pendular state. As long as the liquid is wetting, then at low concentrations the resulting capillary bonds provide an attractive force. As an approximation, the attractive force F between contacting particles varies with the liquid-vapor surface energy [[gamma].sub.LV], and particle size D, as follows:

F = 3D][gamma].sub.LV]

D = particle diameter, m (convenient units: [micro]m)

F = attractive force between contacting particles, N

[[gamma].sub.LV] = liquid-vapor surface energy, J/[m.sup.2].

AGGLOMERATION OF NANOSCALE PARTICLES

See Nanoparticle Agglomeration.

ANDREASEN SIZE DISTRIBUTION (Andreasen 1930)

Originally isolated in colloidal particle-packing studies, the Andreasen particle size distribution is applicable to all powders where a high packing density is desired. The cumulative particle-size distribution is expressed in terms of the weight fraction of particles F(D) given as the fractional weight of powders with a size less than particle size D. The Andreasen size distribution is described as follows:

F(D) = A [(D/[D.sub.L]).sup.q]

Where A is a fitting parameter, [D.sub.L] is the largest particle size in the distribution, and q is the distribution exponent. For the highest packing densities, it is observed that the exponent q tends to range near 0.6. As an alternative, the cumulative particle-size distribution can be expressed with respect to a limiting size by defining a distribution constant B = A/[D.sup.q.sub.L], giving

F(D) = B[D.sup.q]

A = fitting parameter, dimensionless

B = distribution constant, 1/[m.sup.q]

D = particle size, m (convenient units: [micro]m)

[D.sub.L] = size of the largest particle, m (convenient units: [micro]m)

F(D) = cumulative weight-based particle-size distribution, dimensionless [0, 1]

q = distribution exponent, dimensionless.

APPARENT DIFFUSIVITY (Porter and Easterling 1981)

In cases where both volume diffusion [D.sub.V] and grain-boundary diffusion [D.sub.B] are acting to induce sintering shrinkage, the data from shrinkage experiments only provide a means to extract an apparent diffusivity, not an absolute diffusivity. For a constant temperature, the combined or apparent diffusivity [D.sub.A] depends on the two contributions as follows:

[D.sub.A] = [D.sub.V] + [beta][delta] / G [D.sub.B]

where G is the grain size of the microstructure, [delta] is the grain-boundary width (usually assumed to be 5 to 10 times the atomic size), and [beta] is an adjustable parameter near unity (typically ranges from 0.5 to 1.5). Both diffusivities are functions of temperature.

[D.sub.A] = apparent diffusivity, [m.sup.2]/s

[D.sub.B] = grain boundary diffusivity, [m.sup.2]/s

[D.sub.V] = volume diffusivity, [m.sup.2]/s

G = grain size, m (convenient units: [micro]m)

[beta] = adjustable parameter, dimensionless

[delta] = grain-boundary width, m (convenient units: [micro]m or nm).

ARCHARD EQUATION (Archard 1957)

Sliding wear is commonly treated in terms of the loss of material as a function of the hardness, sliding distance, and normal load. The coefficient of friction between the substrate and sliding component is a factor that can greatly change wear rates. The Archard equation calculates the wear behavior by assuming asperity removal, where a single circular cross section is acted upon by an intense wear event. Fragments form and contribute to the mass loss based on the assumption that the hardness and yield strength of the material are proportional. The resulting wear equation is given as follows:

Q = kWL / H

where Q is the volume of material removed from the test or wear material, k is a wear constant that provides a measure of the wear resistance, W is the normal (perpendicular to the surface) load causing wear, L is the total sliding length for the wear event, and H is the material hardness (assuming units of Pa or N/[m.sup.2], where it is assumed the opposing material is much harder). The first derivative of this equation with respect to time then says the wear rate (volume per unit time) is proportional to the sliding velocity.

H = hardness, Pa (convenient units: MPa)

L = sliding length, m (convenient units: mm)

Q = wear volume, [m.sup.3](convenient units: m[m.sup.3])

W = normal load, N (convenient units: kN or MN)

k = wear constant, dimensionless.

[Also see Friction and Wear Testing.]

ARCHIMEDES DENSITY

A standard means to determine the volume of an irregular shape is based on fluid displacement when the component is immersed in a fluid such as water. The measurement must prevent fluid intrusion into surface-connected pores to extract an accurate volume. Combined with the dry mass determined prior to the test, a density calculation follows. First, the sample is weighed dry ([W.sub.1]), then again after oil impregnation of the evacuated pores ([W.sub.2]), and finally the oil-impregnated sample is immersed in water for the final weight ([W.sub.3]). Usually a wire is used to suspend the sample in the water and its weight [W.sub.W] must be measured in water too. Then the actual or Archimedes density [rho] can be calculated from the weight determinations as follows:

[rho] = [W.sub.1] [[phi].sub.W] / [W.sub.2] - [W.sub.3] - [W.sub.W]

where [[phi].sub.W] is the density of water in kg/[m.sup.3], which is temperature dependent as given here,

[[phi].sub.W] = 1001.7 - 0.2315T

with T being the water temperature in C. Dividing the measured density by the theoretical density gives the fractional density. One variant uses water impregnation instead of oil to fill the pores, which still involves two immersion events, but there is no oil trapped in the pores.

T = water temperature, [degrees]C

[W.sub.1] = dry mass of the sample prior to testing, kg (convenient units: g)

[W.sub.2] = wet mass of the sample after filling pores with fluid, kg (convenient units: g)

[W.sub.3] = mass of the component immersed in water, kg (convenient units: g)

[W.sub.W] = mass of the suspension wire, kg (convenient units: g)

[[phi].sub.W] = temperature-corrected density of water, kg/[m.sup.3] (convenient units: g/[cm.sup.3])

[rho] = component density, kg/[m.sup.3](convenient units: g/[cm.sup.3]).

[Also see Fractional Density.]

(Continues...)

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Details zur Ausgabe

• Verlag: Wiley-Interscience
• Erscheinungsdatum: 2008
• Sprache: Englisch
• ISBN 10: 0470173645
• ISBN 13: 9780470173640
• Einband: Tapa dura
• Anzahl der Seiten: 464
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar