IN
DEGREE PROJECT MATERIALS SCIENCE AND ENGINEERING, SECOND CYCLE, 30 CREDITS
STOCKHOLM SWEDEN 2016,
Simulation of Gradient Formation in Cemented Carbides
ARMIN SALMASI
KTH ROYAL INSTITUTE OF TECHNOLOGY
Simulation of Gradient Formation in Cemented Carbides
Armin Salmasi
salmasi@kth.se
Master Thesis in Engineering Martials Science School of Industrial Engineering and Management
KTH Royal Institute of Technology
Supervisors: Henrik Larsson, Andreas Blomqvist Examiner: Joakim Odqvist
Principal: Sandvik Coromant
Stockholm - June 2016
Abstract
The aim of the present work is to study the formation of the cubic carbide phase ( phase)free gradient zone and the cone structure at the edges of gradient sintered cemented carbides. Four types of cemented carbides; WC- Ti(C,N)-Co, WC-Ti(C,N)-Ni, WC-Ti(C,N)-Fe, WC-(Ti,Ta,NB)(C,N)-Co were gradient sintered and the thicknesses of the gradients were measured. Formation of the gradients is simulated and the simulations results are compared with experimental data.
For all of the one-dimensional simulations, the DICTRA [1] software is used.
The two-dimensional simulations are carried out by using a new simulation tool which is called “YAPFI”. The YAPFI software is a tool for simulation of di↵usion in multiphase systems along one, two, or three spatial coordinates.
Various numerical parameters have been studied by running less computa- tionally demanding one-dimensional simulations. The optimized parameters are used to setup the two-dimensional simulations.
Two di↵erent kinetic databases were used in the simulations. The e↵ect of di↵erent so-called labyrinth factors were studied systematically. The simulation results are in close agreement with the experimental observations, although some anomalies are present in the results. Results of the two-dimensional simulations show the formation of the cone at the edges of the insert.
Sammanfattning
M˚alet med det aktuella arbetet ¨ar att studera bildandet av den gamma-fas fria gradient zonen samt strukturen hos den kon av gamma-fas som bildas vid kanterna av gradient sintrad h˚ardmetall.
Fyra olika h˚ardmetaller unders¨oktes; WC-Ti (C, N) -Co, WC-Ti (C, N) -Ni, WC-Ti (C, N) -Fe, och WC- (Ti, Ta, Nb) (C, N) -Co d¨ar samtliga gradient sintra- des och tjockleken hos gradient zonen m¨attes. En datasimulering f¨or bildandet av gradienten gjordes och resultaten j¨amf¨ordes med data fr˚an experimentella f¨ors¨ok. Programvaran DICTRA anv¨andes vid samtliga en-dimensionella simu- leringar.
Vid tv˚a-dimensionella simuleringar anv¨andes en ny programvara kallad “YAP- FI”vilken ¨ar ett verktyg f¨or just simulering av di↵usion av flera faser i en-, tv˚a- eller tre dimensioner. Tv˚a olika kinetiska databaser har anv¨ants och e↵ekten av s˚a kallade labyrint faktorer studerades systematiskt.
Olika parametrar har studerats vid de simplare och snabbare en-dimensionella simuleringarna och efter optimering anv¨andes dessa vid de mer kr¨avande tv˚a- dimensionella simuleringarna.
Contents
1 Introduction 6
1.1 Background . . . 6
1.2 Aim and Contribution . . . 7
1.3 Outline . . . 8
2 Cemented Carbides 9 2.1 Structure and Composition . . . 9
2.2 Application . . . 10
2.3 Development History . . . 11
2.3.1 Alloy Development . . . 11
2.3.2 Alternative Binders . . . 12
2.4 Economy and Market . . . 12
2.5 Production Methods . . . 13
2.5.1 Powder Metallurgy . . . 13
2.5.2 Sintering and Coating . . . 14
3 Gradient Sintering 16 3.1 Gradient Sintering, History and Development . . . 16
3.1.1 Liquid Phase Migration . . . 17
3.1.2 Di↵usion Mechanism . . . 17
3.1.3 Inverse Gradient . . . 19
3.1.4 Related Work . . . 19
3.2 Gradient Sintering, Modeling and Simulation . . . 22
3.2.1 Model Description-Multiphase Di↵usion and Homogeniza- tion Model . . . 23
3.2.2 Related work . . . 28
3.2.3 Two and Three Dimensional Simulation Of Di↵usion . . . 30
4 Methodology 32 4.1 Experiments . . . 32
4.1.1 Sample Preparation and Measurements . . . 33
4.2 Model Setup . . . 35
4.2.1 Thermodynamics . . . 35
4.2.2 Kinetics . . . 36
4.2.3 Global Conditions . . . 37
4.2.4 Grid Setup . . . 37
4.2.5 System Setup . . . 39
4.2.6 Simulation Conditions . . . 39
4.2.7 Boundary Conditions . . . 41
4.2.8 Homogenization Function . . . 41
4.2.9 Simulation Time Setup . . . 44
5 Results and Discussion 45 5.1 The Gradient Shape . . . 45
5.2 Adjustment Parameters . . . 47
5.2.1 Sintering Time-Temperature Profile . . . 47
5.2.2 Grid Setup . . . 50
5.2.3 Ideal Flux Contribution . . . 52
5.3 Thermodynamics . . . 52
5.4 Kinetic Database . . . 53
5.4.1 Labyrinth Factor . . . 56
5.5 Chemical Composition . . . 57
5.5.1 Carbon Window . . . 59
5.5.2 Boundary Conditions . . . 60
5.6 Two-Dimensional Simulations . . . 62
6 Summary 70
7 Future Work 73
A Appendix: Main Configurations 75
List of Figures
1.1 Di↵erent shapes of turning inserts. . . 6
3.1 Di↵usion mechanism at corners . . . 18
3.2 Examples of sharp and round corners . . . 24
4.1 Geometry of SNUN samples. . . 34
4.2 Sintering cycle QT1450 1, 2 and 3 hours. . . 34
4.3 samples after preperation for microscopy. . . 35
4.4 Isopleth of grade C. . . 36
5.1 The phase cone at the edge of the gradient sintered sample. . . 45
5.2 The phase cone at di↵erent grades sintered in 1450 C for 2 hours. 46 5.3 The cubic carbonitride free layer at the top of the gradient sin- tered sample. . . 47
5.4 Thickness of the gradient in di↵erent samples sintered at 1450 C for one, two, and three hours. . . 48
5.5 Simulated sintering cycles. . . 48
5.6 Simulation results of sintering cycles for grade A. . . 49
5.7 Thickness profile as a function of the modified time. . . 49
5.8 Grid points profile close to the active boundary of the cell. . . 50
5.9 Minimum distance between the first two points in di↵erent setups. 50 5.10 The phase fraction profiles obtained from simulations of grade A by using various grid point densities. . . 51
5.11 Simulation results of grade A without using ideal flux contribution. 52 5.12 E↵ect of di↵erent ideal flux contributions on the simulation re- sults of grade A. . . 53
5.13 E↵ect of the miscibility gap on the gradient sintering simulation. 54 5.14 The growth rate of the gradient in grade A simulated by using the MOB2 databse and the AIMD database. . . 54
5.15 Distribution profile of phases after one hour gradient sintering at 1450 C of grade A. . . 55
5.16 Distribution profile of elements in grade A after one hour gradient sintering at 1450 C. . . 56
5.17 Simulation of gradient zone thickness of di↵erent grades with (f ) = f2 and (f ) = f3. . . 58
5.18 E↵ect of chemical composition on the growth of the gradient and distribution profiles of phases. . . 60
5.19 E↵ect of chemical composition on the growth kinetics and distri- bution profiles of elements. . . 61
5.20 The e↵ect of carbon content on the simulated distribution profiles in grades A and B. . . 62 5.21 Results of using di↵erent nitrogen activities at the boundaries in
simulation of sinteirng of grade A . . . 63 5.22 Results of the 2D simulation of gradient formation in grade A –
carbon. . . 64 5.23 Results of the 2D simulation of gradient formation in grade A –
cobalt. . . 65 5.24 Results of the 2D simulation of gradient formation in grade A –
nitrogen. . . 66 5.25 Results of the 2D simulation of gradient formation in grade A –
titanium. . . 67 5.26 Results of the 2D simulation of gradient formation in grade A
–tungsten. . . 68 5.27 Results of the 2D simulation of gradient formation in grade A;
overlay of all elements. . . 69
List of Tables
4.1 Chemical composition of grades (wt%). . . 33 4.2 Choices for grid point distribution in DICTRA and YAPFI. . . . 38 4.3 Major simulation conditions and selected values. . . 40 4.4 Choices of the Homogenisation function in the Homogenization
model. . . 43 5.1 Thickness of the gradient at the middle top of the samples . . . . 47 5.2 Di↵erent nitrogen activities and corresponding nitrogen pressure
at 1450 C and the predicted thickness of the gradient for each activity. . . 62 A.1 Main setup configurations . . . 76
Chapter 1
Introduction
“During the past few years, industrial inventions have resulted less and less from inspiration, a sudden fortunate idea, or even chance.
New industrial methods are usually discovered only after the expen- diture of great e↵ort and the application of every investigative faculty.
Hard metal carbide is no child of chance.” K. Schr¨oter 1934 [2]
1.1 Background
Di↵erent types of turning tools are produced for various applications. Figure 1.1 shows a collection of turning tools and inserts which are produced by SAND- VIK COROMANT. In designing a tool insert, four parameters are of utmost importance: the substrate, i.e., the core material, the coating, the cutting edge, and the geometry of the insert [3]. The most important geometrical elements
Figure 1.1: Di↵erent shapes of turning inserts [4].
of a turning insert are its edges. The edges are exposed to a high concentrated
stress during the turning operation. The extent of the stress is a function of the edge radius. The higher the rounding (equivalent radius), the higher the stress.
The lower the edge radius, the higher the risk of shattering of the edge. Flank wear, crater wear, and plastic deformation are examples of the di↵erent types of wear at the edges of a turning insert.
Di↵usion and phase transformation during the gradient sintering change the microstructure of cemented carbides. When a cemented carbide which contains cubic titanium carbonitride ( phase) is sintered in a nitrogen-free atmosphere, a layer free of the phase forms at the surface of the material. This layer is called Cubic Carbonitride Free Surface Layer (CFL). At the edges of the sintered material, di↵usion along two spatial directions forms the cone microstructure.
The cone increases the hardness and the wear resistance at the edges. For better performance, the sintered inserts are then coated with a layer of wear resistant hard material.
1.2 Aim and Contribution
The cubic carbonitride free surface layer forms when a cemented carbide which contains titanium carbonitride is sintered in vacuum or in an atmosphere with a very low nitrogen partial pressure. Adversely, a phase rich surface layer forms when the same material is sintered in a high-pressure nitrogen atmo- sphere. Thermodynamics and di↵usion control the formation of the gradient.
To explain the gradient sinteing mechanism, a model based on thermodynamics and kinetics was developed during the 1980’s and 1990’s [5–7]. Since the for- mation of the gradient zone is controlled by di↵usion, it is possible to simulate the the phenomena by using the DICTRA software.
The DICTRA software is capable of simulating the di↵usion along one spatial direction. Di↵usion along only one direction is a valid assumption at the regions which are relatively far from the edges. Recently, a new simulation tool for modeling of the two- and three-dimensional di↵usion in multiphase systems, the so-called YAPFI [8](H.Larsson, unpublished research), has been developed.
Like DICTRA, the new simulation tool uses the homogenization model [9, 10].
The aim of the present work is to study the formation mechanism of the cubic carbonitride free layer and the cone structure in cemented carbides by using DICTRA and YAPFI respectively. The simulation results are then validated by comparing them with the gradient thicknesses of the real samples.
There are many assumptions and adjustments that can be made which de- crease the necessary computer power for running a two/three-dimensional sim- ulation. Running a one-dimensional simulation by using the DICTRA software is relatively fast and requires much less computer power. Such 1D simulations were used to investigate the e↵ects of multiple parameters on the simulation time and the computer power demands. These parameters are the labyrinth factor, the mobilities and solubilities of di↵using elements in the liquid binder phase, the chemical composition, atmosphere and temperature of the sintering furnace, the grid point setup, and some numerical parameters which a↵ect the response time and the accuracy of the solver.
1.3 Outline
The outline of the report is as follows:
1. The first chapter - Introduction:
(a) Background, aim, and outline of the presented work.
2. The second chapter - Cemented Carbides:
(a) A short introduction on the structure, and composition of cemented carbides.
(b) The areas of application of cemented carbides.
(c) The history of the development of cemented carbides..
(d) An overview of the production methods of cemented carbides.
3. The third chapter - Gradient Sintering:
(a) An introduction on the gradient sintering of cemented carbides.
(b) The history of the development of the gradient sintering.
(c) An overview of the formation mechanism of the gradient zone.
(d) A summary of the state of the art in the gradient sintering of ce- mented carbides.
(e) An overview of the state of the art in the simulation and modeling of the gradient sintering:
i. A summary of the computational thermodynamics, Thermo-Calc software, and DICTRA.
ii. A short introduction to the modeling of the multiphase di↵usion and phase transformation.
iii. A brief introduction to the homogenization model.
iv. A review on the previous works in the field of di↵usion modeling in the gradient sintering of cemented carbides.
v. A discussion on the necessity of the spatial di↵usion modeling in the gradient sintering of cemented carbides.
vi. A discussion on the shortcomings of the phase field method in the simulation of the gradient layer formation in cemented carbides.
4. The fourth chapter - Methodology:
(a) A description of the simulation setup and parameters such as grid- point setup, Boundary conditions, numerical parameters, choice of homogenization function, the labyrinth factor, thermodynamic cal- culations.
5. The fifth chapter - Result and discussion:
(a) Results of the simulations and experiments and an extensive discus- sion.
Chapter 2
Cemented Carbides
2.1 Structure and Composition
Cemented carbide is a composite material which consists of grains of one or several hard phases in a tough binder matrix. The main hard phase is tung- sten carbide(WC). Usually, a second hard phase is added to cemented carbides for high temperature hardness. The second hard phase is a refractory metal carbonitride. An example of such a carbonitride is (Ti,Ta,Nb)(C,N). Metal car- bides such as Mo2C, Cr3C2, and VC are also added to some cemented carbide grades. The binder metal is usually cobalt [11–29]. The predominant cemented carbide grade has been di↵erent variations of the plain WC-Co grade since the first days of the production.
The raw materials of cemented carbides are di↵erent compounds such as, titanium carbide, tantalum carbide, niobium carbide, tungsten carbide, cobalt, and recycled materials. Tungsten carbide (the ↵ phase) is the primary phase in most of the cemented carbides. Tungsten carbide has a hexagonal crystal structure and because of the minimization of the surface energy, its grains are prismatic. These grains are partially connected and form a skeleton like mi- crostructure. In the turning cemented carbide inserts, the size of the tungsten carbide grains are usually smaller than a micrometer and cobalt is the dominant binder phase.
The second hard phase which is added to cemented carbides consists of the cubic metal carbonitrides (the phase), which has a Cubic-rock-salt (NaCl) crystal structure. Also, the metal constituents of the cubic carbonitride phase acts as a grain growth inhibitor in cemented carbides. High hardness and fine grain size improve the plastic deformation resistance, fracture toughness, and wear resistance. [11, 14, 22, 30–32]. The cubic carbonitride phase is a hard but brittle phase [19]. There are some reports on the increases of the thermal sta- bility and the oxidation resistance of cemented carbides at higher temperatures by addition of the cubic carbonitrides [27].
During the liquid sintering, tungsten carbide and cubic carbonitride grains partially dissolve in the liquid binder phase. The level of the dissolution depends on the solubility limit and the amount of the cubic phase in the cemented carbide. In the grades with a high amount of the cubic carbonitride phase, some of the primary grains remain undissolved during the sintering process.
During the cooling of the sintered material, a rim of near equilibrium cubic phase reprecipitates around the remaining undissolved grains and the complex core-rim structure forms. The composition of the rim depends on the nitrogen and carbon content of the material. [18, 33].
A high carbon concentration in cemented carbides results in the precipita- tion of the graphite phase (C-porosity) and a low carbon content results in the precipitation of the double carbide Eta phase. Depending on the composition and the temperature the Eta phase can be M6C or M12C. For these reasons, controlling of the narrow carbon concentration window is very important for the performance of the cemented carbide products.
Near full density is achieved by sintering of cemented carbides and the small amount of the remaining porosity is usually negligible. Any remaining poros- ity is measured in a light optic microscope and compared with a standardized scale. Classification of porosity in cemented carbides is based on the size of the pores [27]:
1. A-porosity: Pores no larger than 10 µm.
2. B-porosity: Pores with sizes between 10 and 25 µm.
3. Coarser pores are designated with their size and the number of pores per unit of area
If the main constituents of the hard metal are the cubic metal carbonitride phases, then the material is called a cermet. The cubic hard phase in cer- mets forms spherical particles [34]. In the cermets, nickel is a common binder metal in addition to Cobalt [11]. Cermets usually have a higher hot hardness, wear resistance and lower toughness in comparison to conventional cemented carbides. These properties makes cermets suitable for certain specific cutting applications [21, 26].
2.2 Application
Cutting, drilling, milling, grinding and polishing are the essential parts of any industrial manufacturing process. Even the last step of the near-net-shape pro- duction method is a finishing operation. The drilling and exploitation activities are at the heart of the mining and petroleum industries.
Turning inserts experience a high temperature and an intense stress situation during the operation. To tolerate the harsh working conditions, a material with good thermal and mechanical properties is required. Nevertheless, cutting tools have a short lifetime. The inserts must have a reliable and stable performance during the operation to ensure the productivity [3].
Due to the high hardness, wear resistance, excellent mechanical properties and the thermal stability, sintered cemented carbides are used in the production of the cutting tools, drill bits, road headers, sharp cutting teeth, wear parts and rock drilling. Major application areas of cemented carbides are in the mining and petroleum, car manufacturing, and in the aerospace industries. Even microelectronic industries use cemented carbide inserts to drill the very fine holes in the printed circuit boards [2, 3, 11, 13, 16–18, 21–26, 26, 29–33, 35, 36, 36–40].
Some of the most important properties to adjust the performance of ce- mented carbides are the composition, the carbide grain size, the volume fraction
of the binder and the production procedure. These parameters determine the application area of the material [3, 27, 33].
2.3 Development History
The most important minerals of tungsten are tungstates such as wolframite ((Fe,Mn)WO4) and scheelite (CaWO4) [29]. During the 18th-century, Swedish researcher Georg Brandt refined the metallic cobalt from the ore. Another Swedish researcher Axel Cronstedt discovered the scheelite ore and called it tungsten. In 1783 based on works of Von Scheele, The Spanish D’Elhuyar brothers produced metallic tungsten. The French chemist Henri Moissan has first synthesized tungsten carbide with an electric arc furnace in 1896 [27].
Cemented carbide first produced in Germany during the first world war in 1914 to replace the diamond in the drawing dies which had been used in the production of the tungsten filaments for light bulbs. During the 1920’s, the production resumed for the light bulb dies and rock drilling purposes [2, 27, 29].
In 1923 Karl Schr¨oter -OSRAM Studiengesellschaft- produced the first hard- metal by combining tungsten carbide with a small amount of iron binder. Fol- lowing a proposal by Dr. Franz Skaupy from the German Incandescent Gaslight Joint Stock Company, Iron binder was replaced by cobalt, which has an excellent wetting with tungsten carbide. Cobalt is still the dominating binder phase in ce- mented carbides today. In 1925-26 Fried Krupp AG secured the production right of the invention and called the material “Widia”. The word is a combination of German words “Wie” and “Diamond”, meaning “like diamond” [2, 21, 29, 34].
Commercial cemented carbide cutting tools were presented in the Leipzig spring fair in 1927. Although many significant discoveries and breakthroughs were made in Germany, further developments took place in USA, Austria, Swe- den, Japan, and other countries [2].
Because of cutting performance of cemented carbides, Sweden lost its market in the high-speed steel cutting tools. In 1932 production of the hardmetals started in Fagersta Jernverks AB. The product was called SECO (“I cut”), for it was meant for cutting processes, but initially the dominant application was rock drilling. During the second world war, Sandviken Jernverks AB acquired AB Lumalampan in Stockholm to investigate the possibility of production of cemented carbides. In 1942 Sandvik started the production of cemented carbides for rock drilling and called the material “COROMANT” which is a combination of the words “CORONA” and “DIAMANT”. Today, Sandvik and its subside companies are the leading manufacturers of hardmetals in the world [2, 27].
2.3.1 Alloy Development
Many e↵orts have been made for improving the production techniques, design- ing new grades, and increasing thermal and mechanical properties of cemented carbides and cermets [34].
A big mark in the history of the development of cemented carbides was the addition of cubic carbides to the material. In 1942 in Germany, Schwarzkopf dis- covered that the addition of cubic carbides such as TiC improves the properties of cemented carbides. This innovation was the starting point for the develop- ment of multi-carbide cutting tools for the high-speed machining of steel. This
development and the growing concern over the supplies of raw materials resulted in the next big step in the history of cemented carbides; production of cermets.
The industrial production of the cermets started around 1960 after research by Humenik and Moskowitz at Ford’s Dearborn Research Laboratories. It was only after the addition of titanium, niobium, and tantalum, the application of cemented carbides and cermets in metal cutting processing grew [2,21,27,27,34].
The next historical achievement was the discovery of the e↵ect of nitrogen addition in the form of titanium nitride on the properties of (TiC)-(Ni/Mo) cermets by Kie↵er in 1971 [41, 42]. Titanium nitride inhibits the grain growth during the sintering [12, 20, 33, 43, 44]. Japanese companies first scaled up the production of nitrogen containing cermets of the type (Ti,Mo)(C,N)-Ni [33, 34].
Although the cermet materials are developing rapidly, they lack the univer- sality of application and despite all of the e↵orts to replace WC-based cemented carbides, still, tungsten carbides are superior among all carbides. [2].
2.3.2 Alternative Binders
Cobalt is the dominating binder phase of the cemented carbide composites. The volume fraction of the tough binder is an important property of the material.
The binder provides a continuous matrix for di↵usion of the elements.
Because of the high wetting of the tungsten carbide grains with cobalt and the good mechanical properties, application of alternative binders had been very limited for many years. Recently economic considerations, health issues related to the carcinogenicity of the mixture of cobalt and tungsten carbide, in addition to the instability of the virgin sources of the raw minerals, sparked the attempts for replacing cobalt with other metals.
European Union regulation “REACH”, and the U.S. National Toxicology Program “NTP”, both states that the WC-Co dust particles are more carcino- genic than either pure cobalt or tungsten carbide individually. These regula- tions increased the concern over the replacement of cobalt with the alternative binders.
Iron, nickel, a combination of iron-nickel-cobalt with metals such as chromium and copper, some specific grades of stainless steel, and complex binders such as nickel aluminides and iron aluminides have been suggested as alternatives to pure cobalt. Today only the iron-based alternative binders are commercially produced for soft woodworking applications. Nevertheless moving toward the replacement of cobalt with alternative binders is inevitable [2, 29, 35, 45–47].
2.4 Economy and Market
From the 1920’s when the first cemented carbide had been produced, plain WC- Co grades were the main produced cemented carbides. Although other metal carbides, such as TiC, are also used in cutting and turning tools, around 95%
of all of the hardmetals are tungsten carbides.
Until the middle of the 1980’s, steel was the primary material for cutting purposes. The addition of titanium, niobium, and tantalum enhanced the ap- plication of the cemented carbide inserts in steel works. This development increased the share cemented carbide inserts of the market in the mid of 1990’s
when the high-speed steel had 45% of the market, ceramics about 4%, polycrys- talline diamond (PCD) and Cubic Boron Nitride (CBN) 1%, and the rest of the market was dominated by cemented carbide inserts [2, 21, 27, 29].
In the 1930’s, annual world consumption of cemented carbides was only ten tons per year. The consumption increased to 10000 tons in the 1960’s.
Considering the share of recycling hard metal industry consumed 61,750-66,500 tons of the total production of 103,500-111,500 tons of tungsten worldwide in the year 2011. In the same year 67% of the total production of cemented carbides went into metal working applications, 13% to the mining section, 11% to the oil drilling and tunneling industries and 9% to the woodworking and construction industries [2, 26, 29, 35].
2.5 Production Methods
2.5.1 Powder Metallurgy
In the production of cemented carbides, the raw materials with a relatively high melting point and high hardness must be alloyed together to form a complex microstructure. Also for many of the application areas of cemented carbides a fine grain size is necessary. Besides the final product must have a very high dimensional precision, but machining of cemented carbides is very hard. For all of these reasons, conventional metal production method, i.e. melting; alloying;
casting, machining is not applicable in the manufacturing of cemented carbides.
The only manufacturing process which suits is the Powder Metallurgy method.
The production process starts with the mixing and milling of raw material powders. All the raw materials alongside a compaction aid; e.g. the Poly Ethylene Glycol (PEG) powder or liquid, ethanol and/or water are charged to a ball mill. The inner surface of the mill chamber and the milling bodies are made of the same final material to prevent any unwanted contamination.
After proper milling, the slurry is dried in a spray dryer. The spray dryer produces a fine size spherical agglomerate which is called Ready to Press Powder (RTP). The RTP has a high flowability compared to the raw material powders.
The flowability is essential for the repeatability of the pressing. The grain size of the cemented carbide phases in the RTP pellets can be in a sub-micron region if it is necessary for the given application area e.g. the turning inserts.
The RTP is consolidated to form the green body which is larger than the final product to compensate for the densification shrinkage during the sintering.
Inserts are sintered in a controlled atmosphere in a sintering furnace. The interiors of the furnace and the heating elements are made of graphite. The reason is to maintain a high carbon activity in the furnace and compensate the loss of carbon during the sintering.
The sintering procedure consists of several steps; debinding, deoxidation, densification are the main steps, but additional steps for specific processes such as gradient sintering can also be the part a part of the procedure.
Sintered inserts are machined. Depending on the application, they might be CVD or PVD coated by a wear resistance material. [27, 48].
2.5.2 Sintering and Coating
Sintering is forming a continuous solid mass from a consolidated powder by heat treatment. Minimization of the total energy of the system by decreasing the surface area provides the initial driving force for the densification process.
When a fine grain solid powder is consolidated to form a green body, the system reduces its total energy by decreasing the surface area by forming of di↵use joints at the contact point of the particles.
Heat treatment provides the driving force for di↵usion of atoms in the sys- tem. The curvature of the surface of the powder particles adds potential to the system. Atoms di↵use from the high-pressure regions toward the low-pressure area around the contact point. The di↵usion takes place through the bulk and the surface of the particle. The process results in the necking and growth.
Growing of the neck at the intersection between the particles led to densifi- cation and pore closer. New grain boundaries form at the former contact points between the particles. The di↵usion takes place through the grain boundaries in this step. Giving enough time and thermal activation energy the system reaches a fully dense continuous structure.
Higher surface energy enhances the solid sintering process, but, in general, the decrease in the surface energy is smaller than the activation energy which is needed to reach a fully dense structure. Naturally the additional thermal and mechanical energy solve the problem, but high pressure-temperature sintering is not always practical.
One way of enhancing the di↵usion process is to increase the movement of the di↵using elements in the solid. The addition of some materials can change the mobility of elements by altering the crystalline structure. For instance, in the case of iron based alloys the volume di↵usivity in ferrite at 910 C is 300 times higher compared to austenite. Therefore, any chemical ingredient which stabilize the ferrite enhances the sintering process of iron.
When a liquid phase forms during the heat treatment, the sintering is much faster. A liquid phase provides a full density di↵usion bridge between the solid particles. The di↵usion is much faster in the pore-free liquid compared to the fastest di↵usion mechanism in the solid state. The liquid must have a good wetting with the solid particles to spread out between the particles and form a continuous bound. In addition the di↵using elements must have an acceptable solubility in the liquid. The cycle of dissolution and re-precipitation forms the final microstructure.
Sintering of cemented carbides consists of two main stages, the first stage is the debinding of the compaction aid at a low temperature between 300 and 400 C. The furnace atmosphere in this stage is suitable for burning out the polymeric compaction aid material e.g. hydrogen gas. Cemented carbides have a high carbon activity. Removing the compaction aid may result in the decrease of carbon activity and formation of the undesirable Eta phase. For this reason, the sintering furnace has a carbon saturated environment. Inserts are placed on the graphite boats in a furnace. The interior parts of the furnace and the heating elements are all made of Graphite. A small extra amount of carbon is added to the green body, to compensate for any possible carbon loss. High carbon content also causes graphite formation.
Brittle oxides must be reduced before the densification. Grades which con- tain titanium, niobium and tantalum have a high amount of oxide. The low
temperature Hydrogen has a limited reduction capability. After the debinding, the temperature is increased gradually toward the liquidus line. Based on the amount of the cubic carbonitride phases, at a temperature between 700 to 1200 C the carbon monoxide starts forming. Formation of carbon monoxide at high temperature and the low pressure in the furnace, together, result in the com- plete reduction of the oxides. During the solid sintering stage, titanium nitride partially decomposes, and nitrogen loss to the atmosphere is inevitable due to the high amount of the open pores. Slow heating rate results in the loss of ni- trogen. Introducing a counter pressure of nitrogen during the heating prevents the nitrogen loss.
Solid state compaction and densification starts at the higher temperatures after the debinding stage. Under the right conditions, before the binder phase melts, a near full density structure can be achieved.
By the time the first melt forms a rapid densification takes place. The liquid binder wets the solid and the capillary force results in the rearrangement of particles. The driving force for the rearrangement comes from the di↵erences in the spacing between the particles. The spacing between the larger particles is higher, and consequently, the volume fraction of the binder phase is larger.
Any local di↵erence in the size distribution of the particles results in the liquid phase migration.
At temperatures higher than 1100 C the dissolution of smaller tungsten carbide and cubic carbonitride grains in the binder starts.
In industrial scale sintering, liquid state sintering takes place between 1350 and 1500 C. The furnace is evacuated, and a stream of carbon monoxide and argon is introduced into the furnace. In this stage if the material contains titanium nitride, the formation of the gradient starts. The gradient formation depends on the partial pressure of nitrogen in the atmosphere.
The gradient formation is a function of the nitrogen pressure, the activity of nitrogen and carbon in the bulk, sintering temperature, volume fraction of the binder, concentration, the solubility limit of di↵using elements in the liquid and the sintering time.
During the cooling, the reprecipitation of dissolved elemnts results in the growth of tungsten carbide and cubic carbonitride grains [22, 23, 27, 46, 48].
After sintering, inserts are polished and machined to the desirable shape and geometry. The next step is the coating which highly depends on the application of the insert.
Coating increases the performance of the tool. The coating layers are consist of di↵erent inert, high wear, and temperature resistance materials, e.g., Al2O3, TiC, TiN, and Diamond. These layers are applied on the surface by CVD or PVD methods.
CVD method is the most common coating technique. CVD is a relativly high temperature process and takes place around 1000 C. The substrate and the coating materials have di↵erent thermal expansion coefficients which leads to crack formation. These cracks propagate to the core of the material and cause failure.
To prevent crack propagation one way is to form a tougher bu↵er layer under the coating. A cubic carbonitride free layer with a thickness of approximately 20 µm surves the purpose [21, 29].
Chapter 3
Gradient Sintering
3.1 Gradient Sintering, History and Develop- ment
In the early days of the cemented carbide industry, the aim of the sintering process was to produce a homogeneous insert and coat it with a hard anti- abrasive layer. The material which is produced this way has a tougher core under a very hard coated surface [49]. Chemical vapor deposition (CVD) is a high- temperature process (1000 C). The di↵erence between the thermal expansion coefficient of the coating and the substrate results in crack initiation and failure.
The substrate must be tough enough to dump the thermal stresses [2].
Much work has been devoted to the development of the gradient multilayer coatings. One of the advantages of multilayer coatings is that the overall load is dissipated over a larger interface area and volume of the material [34, 36].
This approach is still in use today, but multilayer coating is expensive and complicated.
The coating process accounts for more than 15% of the total manufacturing costs compared to the 5% share of the sintering [42]. Many researchers tried to modify the local microstructure and composition of the bulk. These e↵orts led to the production of the Functionally Graded Sintered Materials (FGM). A gradual change of microstructure, phases, and element over the bulk volume are the characteristics of these materials. Functionally graded cemented carbides are tailored for di↵erent applications but the main type is the functionally graded near surface hard metals [29, 34, 50].
The methods for production of a functionally graded hard metal can be classified to the following:
1. Bulk processing.
2. Preform processing.
3. Layer processing.
4. Melt processing.
3.1.1 Liquid Phase Migration
Liquid binder migration is one of the earliest mechanisms which has been used for the production of functionally graded cemented carbides. Liquid phase mi- gration (LPM) takes place in a two-phase liquid-solid system.
Cemented carbides are produced by liquid phase sintering. Surface energy- related phenomena are one of the controlling parameters in liquid sintering.
The liquid binder dissolves the smaller tungsten carbide grains. Capillary force pushes the liquid toward the regions with smaller grain size distribution. Dur- ing the cooling, dissolved elements reprecipitate in these areas and form new particles [51].
Engineering of the size distribution of the tungsten carbide particle, the carbon content in the green body, and the cooling rate; gives the possibility of modifying the microstructure. Building the gradient in the green body requires many extra steps which make the production complex and time consuming [50].
Nomura et al. [52] in 1999 reported the production of a graded cemented carbide by sintering of a green body made of multi-layers of di↵erent grades of hard metals. Rosso et al. [24] studied the formation of the gradient in a WC- (Ta,Nb)C-Co system by sintering of cylindrical di↵usion couples and observed a link between the composition of the coupled grades and the appearance of a di↵usion zone. Fan et al. [37] in 2013 reviewed the principles of the liquid phase migration in the sintering of cemented carbides and described the di↵erent gradient sintering phenomena. They reported that a post-sintering carburizing process could produce a cobalt depleted zone. Guo et al. [13,53,54] in 2011 and 2013 reported a new method for formation of the gradient by controlling the carbon di↵usion during the post-carburization of the sintered plain cemented carbide. Cheng et al. [44] in 2014 investigated the liquid phase migration by interrupted sintering of di↵usion couples. Based on the CALPHAD calculations of the relation between the volume fraction of the liquid and the mass fraction of cobalt they developed a numerical model to describe the formation of the gradient by liquid phase migration. Recently Konyashin and Lengauer [50]
reviewed the previous researches on the liquid phase migration mechanism in the sintering of cemented carbides.
3.1.2 Di↵usion Mechanism
Despite all of the e↵ort which researchers put to study the liquid binder mi- gration, the procedure of forming a gradient in a green body layer by layer is complicated and unpractical. Forming a functionally graded cemented carbide from a uniform homogeneous green body by a thermochemical process is another option which is less complex and feasible[25]. In a multiphase cemented carbide system, manipulation of the chemical potential and activity of constituents of the phases can provide the necessary driving force for di↵usion of elements. A new class of cemented carbides with a near surface functionally graded layer have been designed and produced by using thermodynamics and kinetics principle.
The addition of nitrogen in the form of the cubic carbonitride phase started a new era in the production of cemented carbides [41]. Suzuki et al. [6] in 1981 for the first time reported the formation of layer free of cubic phase near the surface of a vacuum sintered WC-Ti(C,N)-Co cemented carbide. They interpreted that the outward di↵usion of nitrogen forms the tough gradient and showed that
the thickness of the layer varies linearly with the square root of the sintering time. In 1988, Shwarzkwopf et al. [5] investigated the kinetics of formation of the gradient. They argued that if the outward di↵usion of nitrogen is the controlling parameter, then the graded layer must be thicker at the edges of the inserts due to the overlaying of the di↵usion fields. Besides, depletion of nitrogen in the gradient must transform the titanium carbonitride grains to titanium carbide particles, and these particles must remain in the gradient. The presence of titanium carbide particles in the gradient zone contradicts the observations by Suzuki et al. [6] in 1981. Furthermore, from the thermodynamics point of view titanium nitride is a very stable phase and the di↵erence between the activity of nitrogen in the bulk and in the atmosphere is not sufficient to provide the driving force for the formation of the gradient. Shwarzkwopf et al. suggested that the rate controlling parameter is the inward di↵usion of titanium and the outward di↵usion of nitrogen only activates the process. They formulated a simple kinetic model to describe the zone formation. The model predicts a parabolic growth rate and fits the experimental observations. Figure 3.1 illustrates the di↵usion mechanism according to the Shwarzkwopf’s model.
Figure 3.1: from ref. [5] the figure illustrate the phase distribution when (I) titanium and (II) nitrogen di↵usion is rate determining (The arrows indicate directions of nitrogen and titanium fluxes .
In 1994, Gustafson and ¨Ostlund [7] improved Schwarzkopf’s model. They made a kinetic model based on thermodynamic calculations. By using the Thermo-Calc software, they evaluated the thermodynamic properties of the in- dividual phases and applied a simple kinetic model for di↵usion. They concluded that the growth rate of the gradient is proportional to the di↵erence in nitrogen activity within the sintered body and the furnace atmosphere. Equation 3.1 shows the relation between aN: the di↵erence in N activity, f : the volume fraction of the cubic nitride phase, f : the volume fraction of the liquid phase, and t: the sintering time spent at the liquid stage. In addition, increasing the carbon content and sintering temperature increases the bulk nitrogen activity which must be considered when using Gustafson and ¨Ostlund model. The model
is successful in prediction of the formation of a thinner gradient at the corners.
Gradient zone thickness / f f
p aN.t (3.1)
Gustafson and ¨Ostlund’s model was based on very simple thermodynamic and kinetic data and it was necessary to adjust several experimental parameters to model the gradient formation in a given composition, but it provided a founda- tion for future works on the modeling of gradient formation [11, 17, 22, 46, 50].
The gradient sintering method based on the counter di↵usion of titanium and nitrogen soon became a standard industrial approach. The process was patented in the early 1980’s [55, 56] and several other modifications to the method have been patented since.
3.1.3 Inverse Gradient
Gradient sintering of cemented carbides started with the standard procedure of cubic carbonitride free layer formation by sintering of the nitrogen containing grades in a vacuum or a low-pressure atmosphere free of nitrogen. In addition to the cubic carbonitride free layer, many researchers worked on the formation of a gradient layer enriched with the cubic carbonitride phase particles. Such a gradient forms when a titanium-containing cemented carbide is sintered in a high-pressure nitrogen atmosphere. Nitrogen di↵uses in and causes the counter di↵usion of titanium toward the surface. Cubic carbonitride phase accumulate near the surface and forms a Functionally Graded Cubic Carbonitride rich layer (FGCC) or the inverse gradient.
An example of the formation of the enriched cubic carbonitride layer is a two- step process. In the first step, regular cubic carbonitride free gradient forms by a liquid sintering in a vacuum. In the second step, nitrogen pressure is applied at a temperature lower than the solidus line. Interstitial nitrogen atoms have a high di↵usivity in the solid binder whereas substitutional titanium atoms di↵use much slower, as a result, the cubic carbonitride free layer acts a membrane in the process. The phase enriched layer is a hard anti-abrasive layer which is placed on top of a tougher core.
Inverse gradient sintering started in the mid 1990’s. Despite many e↵orts to replace the coating layers with the phase enriched layers, the standard gradient sintering procedure has dominated the cemented carbide industry [25, 33, 34, 57–59].
3.1.4 Related Work
Chen et al. [33,42] in 2000 studied the formation of cubic carbonitride free and phase enriched gradients in WC-TiC-MoC-(Ni,Co), WC-Ti(C,N)-Co, and WC- (Ti,Ta)(C,N)-Co systems, by using the thermal analysis scanning calorimetry, dilatometry,and microscopy.
Ekroth et al. 2000 [60] and Frisk et al. 2001 [61] developed the first ther- modynamic database for cemented carbides with a full description of the alloy system C-Co-N-Nb-Ta-Ti-W and new assessments of quaternary Ti-W-C-N, Ta- W-C-N, and Nb-W-C-N systems. Following the development of the thermody- namic database, Ekroth et al. [11] in 2000 studied the formation of the cubic
carbonitride free layer in WC-Ti(C,N)-Co cemented carbides by using DICTRA software for the first time.
Andr´en et al. [26] in 1994 studied the microstructure of the WC-Co, Ti(CN)- (Mo2C)-(Ni,Co), and (Ti,W,Ta)(C,N)-(Co,NI) grades by using the Atom Probe Field Ion Microscopy method (APFIM). They reported that from the kinetic and thermodynamic point of view tantalum behavior during the sintering is same as titanium and tungsten behavior is like molybdenum.
Zackrisson and Andr´en [59,62] in 1999 and 2000 heat treated a WC-Ti(C,N)- Co cermet in a high pressure nitrogen atmosphere. They reported that increas- ing the carbon content decreases the concentrations of tungsten and titanium in the binder phase, the amount of the undissolved titanium carbonitride cores, but changes in the carbon content does not a↵ect other phases.
From 1999 to 2003, Frykholm and Andr´en studied the formation mechanism and structure of graded cemented carbides [21, 63–66]. They reported that at the end of the solid state sintering the microstructure of cemented carbides is already formed. Also, open porosities during the solid state gradient sintering decrease the thickness of the graded layer. By analyzing the core-rim structure, they found out that the liquid binder is in a near equilibrium state during the liquid sintering. Frykholm and Andr´en [22, 23] in 2001 studied the formation of the gradient zone in grades containing niobium and tantalum and used DIC- TRA and Thermo-calc to simulate the gradient formation in these grades. They reported that, despite the strong coupling between nitrogen and titanium, such a bond does not exist between nitrogen and tantalum or niobium. Therefore, outward di↵usion of nitrogen during the gradient sintering does not drive the inward di↵usion of tantalum or niobium. They suggested that inward di↵usion of titanium drags tantalum and niobium with it. They also found a gradient in the number density of the undissolved cubic cores which they related to the pen- etration depth of the nitrogen during the sintering. Frykholm et al. [18] studied the e↵ect of carbon content and activity of nitrogen on the thickness of the gradient in 2002. They found out changing the content of other elements rather than nitrogen can change the gradient thickness. For instance, higher activity of carbon increases the nitrogen activity and the gradient depth in the simpler grades. On the other hand tantalum and niobium neutralize the e↵ect of car- bon. To understand the mechanism of di↵usion which leads to the formation of the gradient layer. Frykholm et al. [19], analyzed the rim structure at di↵erent depths from the surface of the gradient sintered cemented carbides in 2002. In 2003 [17] they investigated the e↵ect of the volume fraction of the binder on the formation of the gradient. They used DICTRA to simulate the formation of the gradient. They reported that the gradient zone thickness growth has a linear relation to the volume fraction of the binder and the earlier assumption that the gradient width is proportional to the square root of the binder content is wrong. This new relation has been used extensively in simulation and modeling of the gradient formation.
Between 2009 and 2011 Weidow and Andr´en [67,68] performed a systematic quantitative measurement of the impact of titanium, niobium and zirconium carbides on the microstructure development of cemented carbides.
Bellosi et al. [69] in 2001 studied the sintering of the WC-Ti(C,N)-Co and WC-Ti(C,N),(Co-Ni) grades in argon and nitrogen atmosphere and the hot pressing of the material in the vacuum.
Andersson et al. [20] studied the solubility of the cubic carbide former ele-
ments in the liquid cobalt. By using the interrupted sintering procedure they studied di↵erent parameters such as the di↵usion, surface energy, and viscous flow during the sintering.
Lengauer and Dreyer [34,49,58] in 2002, 2005, and 2006, and Eder et al. [57]
in 2005 summarized the previous works on the di↵usional near-surface micro- structure modification of hard-metals. They studied the formation mechanism of the regular and inverse gradient in cemented carbides. They measured the release of nitrogen during the gradient sintering by analyzing the exhaust gas samples from the di↵erent stages of sintering.
In 2013 Gl¨uhmann et al. [70] described the thermodynamics and kinetics of the formation of the gradient. They reported that the graded cemented carbides with a phase enriched layer outperform the regular grades because of the good adhesion between the substrate and the coating.
Garcia [47] in 2011 studied the formation of cubic carbonitride free layer in the WC-(Ti,Ta,Nb)(C,N)-(Fe/Ni/Co) cemented carbides. By using the Thermo- Calc software, they calculated the solubilities of elements in the binder. They reported that the addition of iron increases the solubility of nitrogen in the alloyed binder phase. In 2013 Garcia and Pitonak [3] investigated the influence of the gradient sintering of the cemented carbides on the wear performance of the coated inserts. They reported an increase in the wear performance of the cemented carbides with a phase enriched surface layer.
Shi et al. [32] in 2012 presented a physical model and kinetic equation for the gradient sintering mechanism. They used analytical thermodynamics and kinetics rather than numerical methods for modeling of cubic carbonitride free gradient formation. Chen et al. [71] in 2013, and Zhang et al. [14, 31, 43] in 2013 and 2016 used the developed Arrhenius formula of the modified Sutherland equation to calculate the di↵usivities of the di↵using elements in the liquid binder consist of cobalt, tantalum, titanium, and tungsten. To validate the calculated mobilities they compared the thickness of the cubic carbonitride free surface layers in the experimentally sintered samples with the results of the DICTRA simulations.
Sun et al. in 2014 [12] studied the vacuum sintering of cemented carbides containing the nanoparticles of titanium nitride. They indicated that the gradi- ent sintered cemented carbide has an excellent hardness and transverse rupture strength compared to the conventional grades.
Zhou et al. [72] in 2016 predicted that the ultra fine grain cemented carbides are the major development trend in the future because of their excellent me- chanical properties. They reported that the gradient thickness increases with the decrease in the grain size of the tungsten carbide and titanium carbonitride powder particles. They introduced a one step combined cycles of vacuum and pressure sintering and claimed that this procedure is more e↵ective for produc- tion of fine-grained cemented carbides.
Walbr¨uhl [30] in 2014 calculated the mobilities of elements in the liquid binder by using the Ab initio molecular dynamic calculations. He reported that due to the low solubility of nitrogen in the liquid binder, an unreal high nitrogen supersaturation occurs in the matrix phase during the di↵usion simulations us- ing the DICTRA software which leads to the numerical di↵usion. His simulation results showed that the previous assumptions that all of the mobilities in the liquid phase are equal is not correct. He reported that the calculated mobilities are higher than the values which had been reported in previous publications
and are not equal.
Larsson [46] investigated the formation of the gradient in the grades with nickel-iron alloy binder. He investigated the best sintering process conditions for forming a 26 µM thick gradient in cemented carbides with 85Ni:15Fe binder by thermodynamic calculations using the Thermo-Calc software and the exper- imental study of the sintered samples.
Chen et al. [73] in 2016 studied the e↵ect of titanium carbide on the for- mation of the cubic rich gradient. They reported that titanium carbide is the most critical component for the modification of the surface microstructure mod- ification where the higher titanium carbide content results in a thicker surface layer and larger Ti(C,N)/TiN ratio. They reported the observation of the three distinct zones in the gradient structure;
1. An outer layer which is composed of cubic phase and a small amount of tungsten carbide and cobalt binder.
2. The intermediate layer with a coarser tungsten carbide grains and a higher Cobalt content.
3. The bulk composition.
Norgren et al. [35] in 2015 reviewed the R&D trends in the field of cemented carbide. Their report contains a survey of the studies on the simulation and modeling of the gradient layer formation. Konyashin and Lengauer [50] in 2016 described di↵erent types of the functionally graded cemented carbides as follows:
1. The gradient in the plain WC-Co system comprising gradients of tungsten carbide grain sizes and/or cobalt contents
2. The gradient in the titanium nitride containing cemented carbides com- prising gradients of nitrogen, cobalt, and titanium-based cubic carbides.
They reviewed the studies on the functionally of the graded cemented carbides of both types.
3.2 Gradient Sintering, Modeling and Simula- tion
Cemented carbides have a wide area of application, and each application de- mands a grade with an optimized composition and microstructure which re- quires the tailoring of raw material and the production procedures. Developing the new high-performance grades is a complicated, time-consuming, and costly process. The classic trial-and-error approach of material design does not fit the modern needs of the cemented carbide industry. On the other hand the modern approach, hierarchical material design, consists of developing novel materials with specific properties for a particular application which requires contempo- rary physical theory, advanced computational methods and models, materials properties databases and complex calculations in an integrated process involv- ing all appropriate scales of length and time[74, 75]. This procedure helps to obtain desired properties at minimum cost and time by a multi-length scale engineering approach as well as the application of computer modeling and sim- ulation to predict material behavior. The validated materials data is essential
for successful prediction of materials properties and design of production pro- cedure. Industrial alloys and materials are complex systems with many phases and constituents. Data of these phases and the interactions between them in a given system are rarely available. The CALPHAD approach (calculation of phase diagrams method), as implemented in some software -e.g., Thermo-Calc and DICTRA, provides a method of modeling of thermodynamic and di↵usion in complex systems. [76, 77].
For coupling of the phase diagrams with thermochemistry by computational techniques, computational thermodynamics, CALPHAD, started in 1950’s. The method evolved toward kinetic simulations integrated with thermodynamic cal- culations, e.g., computational kinetics, including di↵usion-controlled phase trans- formation, precipitation simulation, and phase field simulation. Thermody- namic, mobility, and physical property databases are evaluated by the CAL- PHAD approach combining key experiments, first-principles calculations, statis- tic methods, and empirical theories. The combination of the computational techniques with the databases makes it possible to simulate phase transforma- tions and predict the microstructure evolution [77].
DICTRA and Thermo-Calc [1] are established computational tools for ma- terials science applications in design and process of materials. The software Thermo-Calc was developed at the KTH Royal Institute of Technology in Stock- holm. In further cooperation between KTH, Thermo-Calc software and the Max Planck Institute f¨ur Eisenforschung in D¨usseldorf, the software DICTRA (DI↵usion-Controlled TRAnsformations) is developed for the simulations of the di↵usion in the multicomponent alloys. Thermo-Calc equilibrium calculations have been successfully used in di↵erent alloy systems. DICTRA uses Thermo- Calc for calculating of the thermodynamic quantities needed for solving the di↵usion equations. In DICTRA the system geometries are limited to spheres, planes, and cylindrical shapes. Thermodynamic and kinetic databases and the choice of a model are important in solving any given problem. Examples of applications of DICTRA are the simulation of carburization heat treatment of steel and the formation of the gradient zones in cemented carbides [30, 78]. The latter case is of interest in the present work where we use the homogenization model in DICTRA to simulate gradient sintering of cemented carbides.
3.2.1 Model Description-Multiphase Di↵usion and Homog- enization Model
Larsson and H¨oglund [10] defined multiphase di↵usion as di↵usion through a multiphase mixture where the changes in the phase constitution are the phenom- ena of interest. Formation of the gradient in cemented carbides is a typical exam- ple of such a di↵usion process. For 1D simulation of multiphase di↵usion prob- lems, a model was first presented by Larsson and Engstr¨om in 2006 [9]. They used a homogenization approach to simulate the di↵usion of iron, chromium, and nickel along one spatial dimension inside and between one- and two-phase regions in a duplex stainless steel. Larsson and H¨oglund developed the latest description of the homogenization model which is incorporated in the DICTRA software which we used in this work [10].
The homogenization model has replaced the old dispersed system model by Engstr¨om et al. [79] in DICTRA software. The model by Engstr¨om et al. had been used in DICTRA for many years and described as a general
model to treat multicomponent di↵usion in multiphase dispersions based on the multicomponent di↵usion data and basic thermodynamic data. The dispersed system model placed no restriction on the number of components or phases in the calculations when the necessary thermodynamic and kinetic data were available.
The dispersed system model has been applied to many cases successfully, but it has shortcomings:
1. Di↵usion is only considered in a single continuous matrix phase.
2. The model uses an inherently explicit method, which limits the size of time steps required that the size of time-steps be kept down[9].
3. The model does not conserve matter, although the error is quite small if the volume fraction of dispersed phases is small [80].
The homogenization model is developed to overcome the shortcomings of the old dispersed system model and is based on certain assumptions:
1. The composition is only allowed to vary along one spatial coordinate which means that it must be possible to simplify the problem to a planar, cylin- drical, or spherical symmetry. In the case of gradient sintering of cemented carbides, in regions far from the sharp edges it is reasonable to consider the 1D di↵usion of elements in the continuous liquid binder phase. However, this assumption is not valid at the edges where the di↵usion is a↵ected by multiple di↵usion fields. To simulate the di↵usion at the edges, it is necessary to consider the edge rounding. If the edge rounding is relatively large (Figure 3.2 (a)), it is possible to reduce the geometry of the prob- lem to one dimension. However, at the edges with low rounding radius, geometry e↵ects must be considered (Figure 3.2 (b))
(a) (b)
Figure 3.2: Examples of sharp and round corners (a) A round corner with large nose radius reducible to one dimension di↵usion model. (b) A sharp corner a↵ected by two di↵usion fields must be treated with a 2D di↵usion model.
2. Equilibrium holds locally and at any given volume element, phase frac- tions, compositions, chemical potentials, etc. are equilibrated correspond- ing to the local composition, pressure, and temperature. Therefore, any
di↵erence in equilibrium, except along a tie-line, between neighboring el- ements provides a driving force for the di↵usion.
3. From the computational point of view, the method resembles a single- phase problem, and it does not rely on a continuous matrix phase.
4. Based on the suggestion by Andersson and ˚Agren’s [81], the model is devel- oped on the assumption that the volume is only made up of substitutional elements and that the molar volume of Vsof all substitutional elements is constant and equal.
Under this assumption, it is convenient to use the u-fraction which is defined as:
uk= Xk
P
j2SXj, (3.2)
Where Xk is the mole fraction of component k. By j 2 S should be inferred that the summation must be taken only over the substitutional elements. The flux Jk of component k in the z direction in the lattice is then given by:
Jk= MkCk
@µk
@z = 1 VSMkuk
@µk
@z , (3.3)
where Ck, Mk and µk are the concentration, mobility, and the chemical potential of component k, respectively. The following transformation maps the fluxes to a more practical volume-fixed frame of reference:
Jk0 = Jk uk
X
j2S
Jj. (3.4)
To simulate the multiphase di↵usion in a one dimension problem, the problem is transferred to a single phase problem by using a homogenization function. In the first version of the homogenization model in 2006, Larsson and Engstr¨om [9] considered the mobility of each phase for calculations of di↵usion. In the latest version in 2009, the model relies on the averaging function to find local kinetic properties. All averaging functions consider the product k = MkuK of each phase and the volume fraction f of phases. Introducing the product to Equation 3.3 we get:
Jk= 1 VS
⇤k
@µI.eq.k
@z . (3.5)
The asterisk indicates that a locally averaged kinetic coefficient is used and ”l.eq.” stands for the local equilibrium. An example of using the product is the simulation of nitrogen di↵usion in the gradient sintering of cemented carbides where the nitrogen content is only one or two atomic percent, and the solubility of nitrogen in the liquid phase is in order of 10 2. Ekroth et al. [11] used the old dispersed system model [79] and reported a good agreement with the experimental observations.
The solver divides the system into a number of slices. For each time step, the fluxes are calculated by solving a coupled system of equations between all of the neighboring slices by an implicit procedure. The nature of the model and the implementation ensures that the mass is conserved. Independently
assessed thermodynamics and kinetics data in the simulations are calculated by the Thermo-calc software [9].
The driving force for long range di↵usion, i.e., di↵usion between the slices, comes from the di↵erence in the equilibrium states of grid points. Because of the local equilibrium assumption, at a given point, same components of di↵er- ent phases has the same chemical potential, and this means that the driving force between two points, which hold the equilibrium locally, is the same for all components in the di↵erent phases. Helander in 1999 [82] pointed out that the assumption of local equilibria in each of the volume elements at each timestep, neglects the e↵ect of a finite rate of the growth and dissolution of individual pre- cipitates. This might be important e.g. when considering the non-isothermal conditions. Then the fraction of the precipitates in a volume element might change more rapidly in the calculation than the rate of growth, or dissolution of precipitates might allow for in reality.
The transport capability of a phase depends on the mobility of elements and concentration (the di↵usivity product ). To consider the transport capability of a multiphase mixture, the averaging of kinetics properties is made by a range of homogenization functions. Three groups of homogenization functions are implemented in the homogenization model [9, 10]:
1. Wiener bounds:
The upper Wiener bound, i.e., the rule of mixture, corresponds geomet- rically to continuous layers parallel with the direction of di↵usion and is given by:
⇤k=X
f k (3.6)
Where f is the volume fraction of the phase . The Lower Wiener bound, i.e., the inverse rule of mixture, which corresponds to continuous layers orthogonal to the direction of di↵usion and is given by:
⇤k= [X f
k
] 1 (3.7)
Summation in both cases are taken over all phases 2. Hashin-shtrikman bounds:
The bounds are given by:
⇤k= ↵k+ A↵K 1 3A↵K↵
k
(3.8)
Where:
A↵K=X
6=↵
f
1
k ↵
k
+31↵ k
(3.9) A lower bound is obtained by taking:
↵
k= min[ k, k, ✏k, ...] (3.10) the upper bound is obtained by taking:
↵k= max[ k, k, ✏k, ...] (3.11)
The geometrical interpretation of the Hashin-shtrikman bounds is a dis- tribution of spherical volume that fills up all of the space. The spherical volume is made of concentric spherical shells of each of the phases. The outermost shell corresponds to the ↵ phase. Besides the upper and lower bound the following varieties have been implemented:
(a) Predefined matrix where the phase ↵ is specified by the user.
(b) Majority phase as a matrix where the ↵ is the phase with the highest local volume fraction
(c) Upper and lower bounds with excluded phase where specified phase are excluded from Equations 3.10 or 3.11, respectively.
3. Labyrinth factor:
The labyrinth factor corresponds to the elongation of the long range dif- fusion paths due to the presence of dispersed non-di↵usion phases in a continuous matrix phase ↵ and is given by:
⇤k= (f↵)n ↵k (3.12)
Where f is the volume fraction of the continuous matrix phase. Larsson and H¨oglund [10] suggested that n can be 1 or 2. Recently, the model has been modified, and it is possible to set n to any arbitrary positive number. This new option makes it possible to investigate the mobilities of di↵using elements in the continuous matrix. It is possible to use the real values of mobilities in the di↵usion matrix instead of fitting the simulation conditions to the experimental data. Powering the volume fraction by a number larger than one, the product of the two terms decreases, which leads to the reduction of the di↵usivity matrix. Increasing the n to values larger than one corresponds to dispersed phase obstacles which hinder the di↵usion path. Based on the number, size, and the shape of the dispersed phase, the value of n might change. n = 1 reproduces the upper Wiener bound where the dispersed phase are arranged parallel to the di↵usion direction. It must be considered that for a given fixed microstructure the labyrinth factor must be independent of the chemical composition.
In the case of contradictory results with di↵erent composition sets, the mobility and solubility of elements in the di↵usion matrix must be wrong.
The solubility of elements in the di↵usion matrix provides the di↵using elements and restricts the di↵usion. A correct thermodynamic description for calculation of the solubilities is necessary.
Prior knowledge of the system is useful when selecting the homogenization function [10]. Sandwiching the results between lower and upper bounds is an option when the knowledge about the system is limited. The Wiener bound is based on the assumption of the presence of a continuous di↵usion path whereas using the Hashin-shtrikman bounds are tighter. In the cases where the di↵usiv- ities are available using the Hashin-shtrikman bounds is a convenient choice.
In the modeling of the gradient sintering of cemented carbides, the labyrinth factor is the proper choice. This is because the transport capabilities depend on the liquid binder phase volume fraction. The irregular prismatic tungsten carbide grains and the undissolved spherical gamma phase particles hinder the long range di↵usion path.
