Mechanical Properties of Cementite

Abstract

This review focuses on the mechanical properties of single-phase cementite. The mechanical properties of interest are 1) sound velocity, 2) elastic constants, 3) hardness, 4) plastic deformation mechanism, 5) wear, 6) fracture toughness, and 7) crystal orientation anisotropy. The effects of temperature, magnetic transition, and alloying elements on the sound velocity, elastic constants, and hardness were reviewed. Furthermore, experimental values of the above mechanical properties as well as other parameters, such as the specimen shape, the quantity of alloying elements, and the measurement method, were collected. A large variation was observed in the reported experimental values. The main reason for this is that cementite is metastable, and it is difficult to prepare large single-phase samples. Other factors, such as sample shape, measurement method, alloying element, magnetic transformation, and crystal orientation anisotropy, also influenced the measured values. Studies using first-principles calculations of cementite were also reviewed. The crystal orientation anisotropy of each elastic constant of single-crystal cementite based on the first-principles calculations are summarized, and its comparison with the experimental results is discussed. By comparing the elastic constants obtained by the first-principles calculations with the measured values, the former values of Young’s modulus and shear modulus are several % and bulk modulus and Poisson’s ratio are several tens of % larger than those of the latter. This is thought to be because of the difference in temperature between 0 K (first-principles calculations) and room temperature (measured value), and theoretical and experimental studies in which the temperature is changed are expected.

1. Introduction

Cementite is the most prominent phase in steels, along with ferrite and austenite. Some 50 million tons of cementite is produced annually within about 1.6 billion tons of steel.¹⁾ Owing to its practical importance, a large number of investigations have been conducted on the mechanical properties of steels and cast irons containing cementite. However, the data on the characteristics of cementite itself are not well prepared. For example, even the Young’s modulus of a polycrystalline cementite varies by more than 10% depending on the researchers, and the single-crystal elastic constants of cementite have not yet been measured. There are no other materials whose characteristics are so unknown among the materials manufactured and used in such a large amount.²⁾ The main reason for this is that cementite is metastable, and it is difficult to prepare a large sample of single-phase cementite. Owing to the limited size and shape of the cementite samples used for experiments, the measurement methods are limited, which causes another variation in the measured values. Furthermore, the characteristics of cementite vary greatly depending on the alloying element, which also causes a difference in the measurement data. The concentration of alloying elements, such as Mn and Cr added to steels, in cementite is several times higher than that in ferrite. Thus, from the practical viewpoint, it is important to clarify the effect of alloying on cementite. The properties of cementite change significantly near the magnetic transformation temperature, which changes with the alloying and pressure. There is a large crystal orientation anisotropy in cementite, and its elastic constants and hardness depend on the crystallographic orientation. In this review, the influence of alloying on the mechanical properties of cementite was separated as much as possible, and the sample preparation and measurement methods were described together with the measured values of the mechanical properties of cementite.

Cementite, like Invar and Elinvar, is attracting the attention of not only steel scientists but also material physicists and geophysicists^3,4,5) as a promising material for the inner core of the Earth (the solid part at the center of the Earth). This review is not limited to research in steel science but also includes research from as many other fields as possible.

Newly developed experimental techniques and theories have been applied to cementite. By applying new techniques such as synchrotron radiation, nanoindentation, resonant ultrasonic spectroscopy, and nuclear resonant X-ray inelastic scattering, the understanding of the properties of cementite has deepened. A theoretical study using first-principles calculations revealed the single-crystal elastic constants of cementite. This review covers new experimental techniques and theories.

There are several reviews on the overall properties of cementite.^1,2,6) In this review, the focus is on the mechanical properties of cementite as a single-phase distinguishing pure cementite (without alloying elements) and alloyed cementite. Previous studies that determined the elastic constants of cementite were discussed, focusing on the sample preparation and measurement methods applied. The mechanical properties discussed in this review are 1) sound velocity (effect of temperature and alloying elements), 2) elastic constants (effect of temperature and alloying elements), 3) hardness (effect of temperature and alloying elements), 4) slip systems, 5) wear, 6) fracture toughness, and 7) crystal orientation anisotropy. Experiments using high pressure to determine the bulk modulus of cementite were also presented. In addition, we introduced the single-crystal elastic constants and crystal orientation anisotropy that were estimated using first-principles methods. Finally, based on literature search, the most reliable data on the mechanical properties of pure and alloyed cementite are summarized in Table 1. This was based on the specimen preparation and measurement methods used in the experiment and the crystal orientation anisotropy obtained using the first-principles methods.

Table 1. Mechanical properties of cementite at room temperature.^{8,10,13,23,26,57,58,59)} The numbers in the table are recommended values.

Reference	Characteristics	Values (range)
Fasiska and Jeffrey8)	Crystal structure	Orthorhombic, Space group: Pnma
Litasov et al.13)	Lattice parameters	a=5.084 (5.023–5.094) Å, b=6.747(6.701–6.753) Å, c=4.525(4.468–4.526) Å
	Density ρ	7.683(7.674–7.929)×103 kg/m3
Kagawa and Okamoto10)	Curie Temperature TC	493 K (453–513)
Umemoto et al.57)	Young’s Modulus E	189 GPa (140–298)
	Shear Modulus G	74 GPa (54–90)
	Bulk modulus B	159 GPa (105–240)
	Poisson’s ratio ν	0.301 (0.22–0.46)
	Sound velocity VL	5870 m/s (5140–6103)
	Sound velocity VS	3104 m/s (3012–3240)
Umemoto et al.26)	Increase in E	V > Cr > Mn ≈ Mo
Ghosh59)	Anisotropy in E	E110>E101>E100>E010>E001>E111>E011
Umemoto et al.26)	Hardness	1020 HV (1013–1340)
Inoue et al.58)	Slip system	(001)[100], (100)[010], and (010)[100]
Inoue et al.23)	Cleavage plane	{101}, {001}, and {102}
Prescher et al.46)	Magnetic transition, pressure	Ferro to para 8–10 GPa
		Para to nonmagnetic 22 GPa

( ) indicates the range of measured values reported by different researchers.

In this review, elastic constants such as E, G, B, and ν (refer to the list of nomenclature at the end of this paper) are discussed. Because there are four types of data on B depending on the experimental and calculation methods, the following symbols were used to distinguish them: 1) data obtained from the vibration or elastic wave experiments (B^E), 2) data obtained from pressure and volume experiments and equations of state (B^P), 3) data calculated from first-principles calculations and equations of state (B^F), and 4) data calculated from C_ijs obtained by first-principles calculations (B^C). When the bulk modulus is determined based on the equation of state, the bulk modulus at an arbitrary pressure is obtained using B₀^P or B₀^F, and the first pressure derivative, B’₀^P or B’₀^F. However, this review discusses the bulk modulus (B^E, B₀^P, B₀^F, and B^C) of cementite at ambient temperature and pressure or 0 K and 0 GPa as it is important for steels.

2. Cementite Samples Used for the Research of Physical Properties

Cementite has been studied for a long time and has been used in enormous amounts. However, compared to many other well-known materials, the physical properties of cementite are not as well understood as expected. The main reason for this is that cementite is metastable and it is still difficult to prepare a large single-phase cementite sample. The methods for measuring the physical properties are limited by the shape of the specimens. Because the cementite samples used in the experiments have various sizes and shapes depending on the preparation methods, various measurement methods have been adopted. Due to the diversity of sample preparation, shapes, and measurement methods applied, there is a wide difference in the reported data on the properties of cementite. The size and condition of the specimens used for the measurement of the physical properties of cementite are reviewed in the following sections.

2.1. Samples Used to Study the Physical Properties of Cementite

The following four types of samples were used to study the physical properties of cementite. One type is the sample in which cementite coexists with ferrite or other phases, such as carbon steel or cast iron. Experiments using such samples include the measurement of lattice parameters by X-ray diffraction,^{7,8,9,10,11,12)} synchrotron radiation X-ray diffraction,¹³⁾ neutron diffraction,^14,15) the measurement of thermal expansion coefficients,^{10,11,13,14,15)} and measurements of X-ray elastic constants.^16,17,18) In addition, hardness measurement^{19,20,21,22,23,24,25,26,27)} and the measurement of Young’s modulus by nanoindentation^27,28) were also performed using such samples. The second type involved the use of various samples with different volume fractions of cementite. Here, the physical properties of cementite were estimated by extrapolating the relationship between the properties and carbon content to the single-phase cementite (extrapolation method). In most cases, linear extrapolation was applied. The coefficient of thermal expansion^29,30) and Young’s modulus^{29,30,31,32,33)} were measured using this method. It must be noted that this method requires a large fraction of extrapolation because the maximum cementite volume fraction is 65%, which corresponds to the eutectic composition of 4.28 mass% C in the Fe–C system. The third type of samples is microcrystals or powder samples of cementite extracted from steel or cast iron by an electrolytic or chemical method (extraction method). Using these samples, characteristics such as mechanical properties,^34,35) thermal expansion,¹¹⁾ and magnetic properties³⁶⁾ were measured. Regarding the extracted cementite sample, caution must be taken to avoid the mixing other phases and surface contamination. The fourth type is a relatively large polycrystalline sample with a single-phase cementite. Two methods have been developed to synthesize a large sample of a single-phase cementite. One is the use of high temperature and high pressure^{13,37,38,39,40,41,42,43,44,45,46)} and the other is the sintering of mechanically alloyed powder^26,47,48,49) as will be described in detail later. In addition to these four types of samples, there are samples prepared by special techniques, including single-phase thin films^25,50) prepared by the vapor deposition method; samples in which iron and graphite are melted, crushed, and cementite is extracted and sintered;⁵¹⁾ cementite single-crystal foil electrolytically extracted from a pearlitic sample;³⁵⁾ and a sample in which the cementite volume fraction is locally increased by carburization.^29,52)

2.2. Preparation of Large Polycrystalline Samples with a Single-phase Cementite

To measure the mechanical properties of cementite with good accuracy, it is necessary to use a large cementite sample. The two preparation methods for a relatively large polycrystalline single-phase cementite sample are described in detail in the following paragraphs.

2.2.1. Sample Preparation Using High Temperature and High Pressure

One of the methods to prepare a large sample of single-phase cementite is to melt the raw mixed powder of iron and graphite with cementite composition at high temperature and high pressure and solidify as cementite. This method is popular among geophysicists.^{13,37,38,39,40,41,42,43,44,45,46)} In the Fe–C binary system, the cementite phase is metastable at all temperatures under one atm. Because the diffusion rate of carbon in liquid iron is very high, it is not possible to obtain a sample of single-phase cementite by solidification under normal pressure. However, at high pressures, the cementite phase can exist as a thermodynamically stable phase. The commonly used high-pressure process is as follows: Pressure is applied to the raw mixed powder of iron and graphite with cementite composition using a piston anvil and then heated. After melting the mixed powder at high temperature and high pressure, it is solidified as a single-phase cementite. The temperatures and pressures applied are in the range of 1273–1573 K and 1.5–10 GPa. The cementite phase obtained was cooled to room temperature and one atm without decomposition to iron and graphite. The size of the sample depends on the cell size of the piston anvil, which is usually a few millimeters. This method is popular among geophysicists.^{13,37,38,39,40,41,42,43,44,45,46)}

2.2.2. Sample Preparation Using Mechanical Alloying (MA) and Spark Plasma Sintering (SPS)

Another method to obtain a large sample of single-phase cementite is to use mechanically alloyed (MA) powder. From around 1990, many studies have reported the production of the cementite phase by ball milling a mixed powder of iron and graphite. Le Caer⁵³⁾ reported that a cementite phase was formed when a mixed powder of Fe₈₀C₂₀ was ball-milled, but an Fe₇C₃ phase was formed when a mixed powder of Fe₅₀C₅₀ composition was ball-milled. Matteazzi et al.⁵⁴⁾ reported that when a mixed powder of iron and graphite (Fe_0.75C_0.25) was ball-milled for a long time, three types of iron carbides, ε-carbide, χ-carbide, and cementite, were produced. They found that the volume fraction of the three carbides varied depending on the milling time and milling conditions (the weight ratio of balls to powder, the combination of milling time and interval time). Metastable cementite can be produced either by a long-time ball milling or by heating the short-time ball milled powder consisting of bcc iron with a supersaturated solid solution of carbon and nano-sized graphite particles. Based on thermodynamic considerations, Umemoto et al.⁵⁵⁾ suggested the following as an explanation of how ball milling produces metastable cementite. The enthalpy of the ball-milled powder is higher than that of cementite owing to the breaking of the carbon-carbon bond of graphite rather than the change in iron, and cementite is produced by the chemical reaction during ball milling or heating after ball milling. The final product obtained by ball milling depends not only on the thermodynamic conditions but also on the kinetics. The formation of cementite or decomposition to α-Fe and graphite from ball-milled powder by further ball milling or heating is a matter of nucleation and growth kinetics. Fortunately, the rate of cementite formation is faster than that of decomposition, and it is possible to obtain cementite from ball-milled powders. To sinter the MA powder into a high-density cementite sample, it is sintered at the lowest possible temperature to suppress the decomposition of cementite. In this respect, spark plasma sintering (SPS) is a suitable technique because sintering at low temperatures is possible. In addition, just before the formation of cementite, the ball-milled powder is softer and easier to make a green compact than the long-time ball-milled powder of 100% cementite. The size of the sintered cementite sample depends on the size of the SPS jig and the specimen size of several centimeters in diameter is attained. In recent years, the above methods for preparing large cementite single-phase samples using MA and SPS have also been implemented in China,^48,49) Germany,⁴⁷⁾ and the United States.⁵⁶⁾

3. Experimental Data of Mechanical Properties of Cementite

The selected mechanical properties of cementite at room temperature are summarized in Table 1.^{8,10,13,23,26,57,58,59)}*¹ Both experimentally measured (at room temperature and one atm) and theoretically estimated (at 0 K and 0 GPa) values are shown. The most reliable experimental data were chosen based on the sample condition (a large-size single-phase cementite, without alloying elements and a small void fraction) and measurement method (well established with small error).

3.1. Sound Velocity in Cementite

There are four elastic constants in isotropic polycrystalline materials: Young’s modulus, shear modulus, bulk modulus, and Poisson’s ratio, and two of the four elastic constants are independent. These elastic constants can be calculated from the sound velocities of the longitudinal and shear modes.^51,57) The sound velocity can be measured by the ultrasonic pulse-echo method, in particular, the laser ultrasonic method is advantageous for the measurement of the temperature and pressure dependence of sound velocity owing to its non-contact characteristics. The sound velocity itself is an important quantity for geophysicists because the sound velocity of the Earth’s inner core is one of a few measurable physical quantities.

Several experimental values have been reported for the sound velocity of cementite. These are summarized in Table 2.^{40,41,43,51,57)} A general measurement method is to use a vibrator or laser to apply ultrasonic vibration to one side of a sample with parallel surfaces, measure the time it takes for the ultrasonic waves to reflect on the front and back surfaces, and determine the speed of the sound waves. When the thickness of the sample is several millimeters, the time for the sound wave to move back and forth in the sample is approximately several microseconds.

Table 2. Compilation of longitudinal (V_L) and shear (V_S) sound velocity obtained from experimental measurements by different authors using different techniques in chronological order.^{40,41,43,51,57)}

Reference	VL (m/s)	VS (m/s)	Technique	Specimen
Umemoto et al.57)	5870	3104	Ultrasonic	MA+SPS 98%
Dodd et al.51)	5140 (method 1)*	3079 (method 1)*	Ultrasonic	Sintered 72%
	5333 (method 2)*	3012 (method 2)*		1.5 at% Cr
Fiquet et al.43)	6103		Inelastic X-ray scattering
Gao et al.40)	5890	3050	NRIXS	Few-crystal 57Fe-enriched
Gao et al.41)	5610	2630	NRIXS	Powder 57Fe-enriched
α-Fe	5960	3240

NRIXS: nuclear resonant inelastic X-ray scattering

*  The raw experimental data for the sample (porosity 28%) were corrected by 2 methods.

Umemoto et al.⁵⁷⁾ measured the sound velocity by the ultrasonic pulse method using a sample with 98% X-ray density prepared by MA and SPS and obtained V_L = 5770 ± 11 m/s and V_S = 3076 ± 10 m/s. Correcting the effect of porosity (2%) on the sound velocity, the sound velocity for the 100% density sample was estimated to be V_L0 = 5870 m/s and V_S0 = 3104 m/s. These values are 1.5% and 4.2% smaller than V_L (α-Fe) = 5960 m/s and V_S (α-Fe) = 3240 m/s, respectively, for α-Fe.

Dodd et al.⁵¹⁾ prepared a cementite sample by melting a mixture of iron and carbon powder, pulverizing the ingot, and chemically extracted the cementite. After sintering the cementite powder, they obtained almost single-phase cementite samples with an X-ray density of 72%. The sound velocity in the sample was measured using the pulse-echo overlap method. The measured sound velocities were V_L = 4057 ± 50 m/s and V_S = 2382 ± 30 m/s. The effect of porosity on the measured sound velocities was corrected using the self-consistent method (method 1) and the wave propagation theory method (method 2). They obtained the sound velocity at 100% density as V_L = 5140 m/s (method 1) and 5333 m/s (method 2), and V_S = 3079 m/s (method 1) and 3012 m/s (method 2). Because the sample used by Dodd et al. was highly porous, the sound velocities corrected for porosity were approximately 30% higher than the raw data. Fiquet et al.⁴³⁾ prepared a cementite sample by melting a mixture of iron and graphite powder at high temperatures and pressures. The longitudinal wave velocity was measured by the X-ray inelastic scattering method using a synchrotron radiation facility, and obtained V_L = 6103 ± 413 m/s, while Gao et al.^40,41) prepared a cementite sample using the same method and the X-ray diffraction measurement of the prepared sample revealed that the sample is composed of a small single-crystal part (termed ‘few-crystal’) and a powder state part (termed ‘powder’). For these two parts, the Debye sound velocity V_D (3/V_D³ = 1/V_L³ + 2/V_S³) was measured by nuclear resonant inelastic X-ray scattering (NRIXS). V_L and V_S were calculated using the coefficients of the equation of state (EOS) reported by Scott et al.⁶⁰⁾ and measured V_D. The resulting sound velocities at 300 K, 0 GPa were V_L = 5890 m/s, and V_S = 3050 m/s⁴⁰⁾ in the ‘few-crystal’ of the sample, and V_L = 5610 m/s and V_S = 2630 m/s⁴¹⁾ in the “powder”. The difference in sound velocities depending on the sample condition is probably due to the crystal orientation in the ’few-crystal’ sample.⁴¹⁾ In the case of cementite, the crystal orientation anisotropy is large, and the sound velocities are considered to vary depending on the crystal orientation.

The temperature dependence of the sound velocity was measured by Umemoto et al.⁵⁷⁾ (Fig. 1) and Dott et al.⁵¹⁾ The data shows that, the sound velocity increases as the temperature decreases and shows a drop at T_C. The decrease in V_L begins at approximately 100 K above T_C, and V_L shows a minimum at T_C. This minimum is approximately 5% smaller than that extrapolated from the paramagnetic state. However, the decrease in V_S occurs in the small temperature range near T_C, and the drop in V_S around T_C is 1% or less than that extrapolated from the paramagnetic state. In addition, as the temperature decreases below T_C, V_L increases, but V_S decreases. V_L shows a larger influence of the magnetic transition than V_S, suggesting that the magnetic transformation in cementite is accompanied by a volume change. At temperatures below T_C, both V_L and V_S in the ferromagnetic state are smaller than those extrapolated from the paramagnetic state. This indicates that the ferromagnetic state is elastically softer than the paramagnetic state. Dodd et al.⁵¹⁾ measured the change in sound velocities at temperature below room temperature (100–300 K) and found that V_L increased and V_S decreased as the temperature decreased below T_C. This is consistent with the results obtained by Umemoto et al.,⁵⁷⁾ but the absolute values of V_L and V_S cannot be compared because of the high porosity (28%) of the specimen used by Dodd et al.

Fig. 1.

Temperature dependence of longitudinal V_L and shear V_S wave velocities of Fe₃C (dark blue) and (Fe_0.95Mn_0.05)₃C (dark red). (a) longitudinal wave velocity V_L and (b) shear wave velocity V_S.⁵⁷⁾ (Online version in color.)

There are several reports on the effect of pressure on sound velocity. Generally, the higher the pressure, the higher the density of the material, thus, the higher the sound velocity in the material. In geophysics, the inner core of the Earth is considered to contain carbon, and may contain iron carbides such as Fe₃C or Fe₂₃C₆. To compare the measured values of the inner core density and the sound velocity with those of cementite, geophysicists are interested in the density and sound velocity of cementite under the conditions of the inner core of the Earth (330 GPa, 5300 K³⁾). Many studies carried out under high pressure have revealed that cementite changes from ferromagnetic to paramagnetic and then to non-magnetic with increasing pressure.^{39,40,43,46,61)} There is a linear relationship between the sound velocity and density of materials (ρ) (Birch rule⁶²⁾). In the case of non-magnetic cementite, the following relationships were reported: V_L (km/s) = −3.99 + 1.29ρ (g/cm³) and V_S (km/s) = 1.45 + 0.24 ρ (g/cm³).⁴⁰⁾ Because the density and pressure of a material are related to the bulk modulus, the effect of pressure on the sound velocity can be calculated using the bulk modulus. Therefore, studies^40,44,46,65) have been conducted to measure the pressure change in a unit cell volume of cementite, fitting the pressure–volume data using a third-order Birch–Murnaghan EOS^63,64) and the bulk modulus B^P was obtained. When compared at the same density of cementite, the sound velocity is lower in the ferromagnetic state than that extrapolated from the non-magnetic state.⁴⁰⁾

Mn alloying increases the sound velocity over the entire temperature range (Fig. 1); however, while the increase in V_L is greater in the paramagnetic state, the increase in V_S is greater in the ferromagnetic state. As can be seen from Fig. 1, there are two effects of Mn alloying: increasing the sound velocity and decreasing the T_C. In general, the addition of alloying elements affects the bonding strength of atoms (many elements strengthen the bond) and simultaneously changes T_C (many elements lower T_C).

3.2. Measurement of Elastic Constants of Polycrystalline Cementite

3.2.1. Measurement of Elastic Constants of Polycrystalline Cementite at Room Temperature

Measured values of the elastic constants of single-crystal cementite are rarely reported.³⁵⁾ In this subsection, the elastic constants of polycrystalline cementite are discussed. The crystal orientation anisotropy of the elastic constant of cementite will be described later in Subsection 3.4.4 and Section 4.4.

Various methods have been used to measure the elastic constants of cementite. For single-phase cementite samples, resonance methods, ultrasonic methods, and nuclear magnetic resonance methods have been applied. For samples containing phases other than cementite, such as steel or cast iron, the elastic constants of cementite were measured by the X-ray elastic constant measurement method or the nanoindentation method. In the X-ray elastic constant measurement method, various values of tensile elastic stress are applied to the sample, and the elastic constant is obtained from the lattice strain of the cementite phase. Using this method, the Young’s modulus and Poisson’s ratio were estimated. Instrumented indentation and nanoindentation have been used as tools for characterizing the mechanical properties of very small specimens or in very localized volumes. In addition to the above direct measurement method, the elastic constants are estimated indirectly by the extrapolation method, as described in Section 2.1. Geophysicists obtain the bulk modulus B^P by measuring the change in the unit cell volume by pressure and fitting the pressure–volume data to a third-order Birch–Murnaghan EOS.^63,64) This is described in detail in Subsection 3.2.2.

Several studies have been reported since the 1920s regarding the experimental elastic constants of cementite. In 1926, Honda and Tanaka⁶⁶⁾ obtained a Young’s modulus of E = 181 GPa by an extrapolation method using specimens with various carbon concentrations (maximum carbon content 1.4 mass%, cementite volume fraction 20%). In 1937, Forster⁶⁷⁾ reported E = 156 GPa, and the Metals Handbook⁶⁸⁾ published in 1948 states E = 170 GPa. In 1959, Laszlo and Nolle²⁹⁾ measured the Young’s modulus (E) and shear modulus (G) by the vibration method using a carburized cylindrical sample, and obtained E = 200 GPa and G = 74 GPa. Using these two values and assuming that the sample is isotropic, the bulk modulus B = 240 GPa and Poisson’s ratio ν = 0.361 were estimated using Eq. (10). In 1969, Hanabusa et al.¹⁶⁾ applied tensile stress to 1.4 mass% C tool steel (SK1) with a spherical cementite structure, and measured the strain on the (121) plane of cementite by X-ray. From this experiment, they obtained E = 212 GPa and ν = 0.46. It should be noted that the cementite phase contained various alloying elements.

In these early studies on the elastic constants of cementite, little attention was paid to the alloying conditions in cementite. Alloying elements such as Mn, Cr, V, and Mo are concentrated in cementite in steels (for example, Reference 69)), and these elements increase the elastic constant, as shown in Fig. 3. When discussing elastic constants, attention should be paid to the quantity of alloying elements in cementite. Table 3^{16,17,18,25,29,30,31,32,33,40,41,50,51,57,70,71)} is a summary of the experimental elastic constants of cementite reported after 1950. The table presents only data obtained using samples without alloying elements or samples with precise amounts of alloying elements. Glikman et al.³²⁾ measured the Young’s modulus by the vibration method using six samples with different carbon concentrations (maximum 4 mass% C, cementite volume fraction 60%), and estimated E in the range 176–186 GPa. Drapkin and Fokin³¹⁾ used seven samples with different carbon concentrations (maximum carbon concentration 2.7 mass% C, cementite volume fraction 42%) and measured the Young’s modulus using the resonance method. They linearly extrapolated the Young’s modulus to the composition of cementite and obtained E = 181 GPa. Kagawa et al.³⁰⁾ used five samples with different carbon concentrations (maximum carbon concentration 4.31 mass% C and cementite volume fraction of 65%) that were unidirectionally solidified after melting the mixture of iron and carbon. The Young’s modulus of the cementite was estimated by the extrapolation method, and the effect of temperature on Young’s modulus of cementite was determined. The crystallographic anisotropy in the Young’s modulus of the cementite was confirmed, and it was revealed that the Young’s modulus in the <100> direction was larger than those in the two directions perpendicular to it. They also found that at temperatures below T_C, the Young’s modulus decreased as the temperature decreased. Dodd et al.⁵¹⁾ measured the Young’s modulus of cementite by the ultrasonic technique using a sintered sample (72% X-ray density) containing 1.5 at% Cr. They corrected the effects of voids by two methods and obtained E = 175 GPa (method 1) and E = 176 GPa (method 2). Ledbetter³³⁾ measured the elastic constants of cementite by an extrapolation method using seven samples with different carbon contents of 0%–17.3 at%. By applying resonance-ultrasound spectroscopy, the author obtained E = 230 GPa. In the case of extrapolation, the constraints of the matrix and the influence of the interface between the matrix and cementite are unavoidable. Gao et al.^40,41) obtained E = 192 GPa (‘few-crystal’ sample)⁴⁰⁾ and E = 147 GPa (‘powder’ sample)⁴¹⁾ from the measured V_L and V_S values of the samples prepared by the high-temperature and high-pressure technique. They pointed out that crystallographic anisotropy is the reason the Young’s modulus of ‘few-crystal’ samples is higher than that of ‘powder’ samples.⁴¹⁾ Umemoto et al.^57,70) measured the elastic constants of cementite using 98% X-ray density cementite bulk samples prepared by MA and SPS. Both the vibration resonance method and laser ultrasonic technique were applied. They obtained E = 191 GPa⁷⁰⁾ and E = 185 GPa⁵⁷⁾ from the vibration resonance and the ultrasonic methods, respectively, which are close to each other. The reason for the slight difference is the difference in the measurement method and the measurement direction of the sample. The vibration frequency of the vibration resonance method is approximately 10–200 Hz, while that of the ultrasonic method is higher, approximately 1–20 MHz. Furthermore, in the vibration resonance method, the measurement is performed in the longitudinal direction of the sample, that is, in the direction perpendicular to the compression direction of sintering, while in the ultrasonic method, the measurement is performed along the direction parallel to the compression. Therefore, if a texture is formed during the sintering of the sample, a texture effect may appear. The bulk modulus was determined to be 155 GPa, and this value is close to the average value [B^E] = 145 GPa measured at 293 K a and 0.1 MPa, as described below, and the average value of B in the ferromagnetic state [B₀^P] = 173 GPa, measured by changing the pressure using cementite samples prepared at high- temperature and high pressure, as described in Subsection 3.2.2.

Fig. 3.

Young’s modulus of cementite as a function of the alloying element concentrations (x) in (Fe_1-xM_x)₃C, M = Cr, Mn, Mo, V.⁷⁰⁾ (Online version in color.)

Table 3. Compilation of elastic moduli (Young’s modulus E, shear modulus G, bulk modulus B (all in units of GPa)), and Poisson’s ratio ν obtained from experimental measurements by different authors using different techniques in chronological order.^{16,17,18,25,29,30,31,32,33,40,41,50,51,57,70,71)}

Reference	E	G	BE	ν	Experimental procedure
Laszlo and Nolle29)	200	74	224	0.35	Case carburized tube, E and G were measured with resonance method
Hanabusa et al.16)	212±44	–	–	0.46±0.15	Tool steel, X-ray elastic constants
Glikman et al.32)	176–186	–	–	–	Fe–C alloys, measured with resonance method, extrapolated from Fe–C alloys
Drapkin and Fokin31)	181	–	–	–	Fe–C alloys, measured with resonance method, extrapolated from Fe–C alloys, temperature dependence of E
Kagawa et al.30)	140	–	–	–	Oriented ledeburitic Fe–C cast irons, measured with resonance method, temperature dependence of E, evidences of anisotropy
Winholtz and Cohen17)	276–298	–	–	0.36–0.39	Eutectoid steel, X-ray elastic constant
Li et al.25)	160	63.5	111	0.26	Thin film (2.5 μm), polycrystalline (50 nm grain size), inverse analysis of surface waves
Mizubayashi et al.50)	177±1	–	–	–	Thin film (210–780 nm), polycrystalline (90 nm grain size), vibrating reed method
Umemoto et al.70)	191	–	–	–	Bulk-sintered polycrystalline sample, grain size 0.5 μm, resonance method, temperature dependence, alloying effect
Dodd et al.51)	175–176	69–72	105–125	0.22–0.27	Bulk-sintered polycrystalline sample including 1.5 at% Cr, grain size 5 μm, pulse-echo-overlap technique, values corrected for 28% porosity, temperature and pressure dependence
Che et al.18)	262–282	–	–	0.22–0.26	X-ray elastic constant, tool steels
Gao et al.40)	(192)	(73)	(175)	(0.32)	Bulk sample, Nuclear resonant inelastic X-ray scattering, sound velocity
Gao et al.41)	(147)	(54)	(175)	(0.36)	Powder sample, nuclear resonant inelastic X-ray scattering, sound velocity
Ledbetter33)	230±12	90±5	168±8	0.275±0.03	αFe–Fe3C alloys, extrapolated to Fe3C, resonance-ultrasound spectroscopy
Umemoto et al.57)	185	71	155	0.301	Bulk-sintered polycrystalline sample, grain size 0.5 μm, laser-ultrasonics technique, temperature dependence, Mn alloying effect
α-Fe Kim and Johnson71)	211	82	167	0.288	Summary of previous data

All the data are at RT. ( ) indicates indirect measurement. B is in ferromagnetic state or near ambient pressure.

The mean and standard deviation of the measured elastic constants shown in Table 3 were calculated excluding the data with alloying elements^{16,17,18,29,51)} and the data with crystal orientation anisotropy.³⁰⁾ The mean value and standard deviation obtained for E from seven data points^{25,31,32,33,50,57,70)} and for G, B, and ν from three data points^25,33,57) were [E] = 186 ± 20 GPa, [G] = 75 ± 11 GPa, [B^E] = 145 ± 24 GPa, and [ν] = 0.279 ± 0.017. The range of the experimental values is large for all elastic constants, and the coefficient of variation (standard deviation divided by the average value, and the larger this value, the greater the variability) is more than 10% for E, G, and B^E. Concerning the elastic constant of polycrystalline cementite, Umemoto et al.⁵⁷⁾ calculated elastic constants for 100% density specimens from the result of ultrasonic measurement (all four elastic constants were measured at the same time using this method). They obtained E (100%) = 189 GPa, G(100%) = 74 GPa, B(100%) = 159 GPa and ν (100%) = 0.301. Comparing these values with those of pure iron (E(α-Fe) = 211 GPa, G(α-Fe) = 82 GPa, B(α-Fe) = 167 GPa, ν (α-Fe) = 0.288), E and G were approximately 10% smaller, B was approximately 5% smaller, and ν was approximately 5% larger for cementite than for α-Fe. The bulk modulus and Poisson’s ratio of cementite change significantly with magnetic transformation, and the influence of magnetic transformation remains on these values at room temperature, which is 220 K below T_C. Comparing the Young’s modulus of metal elements and their carbides, all carbides other than Fe, such as Cr, Mo, Nb, Ti, V, W, and Zr, have much higher values than the base metal.³¹⁾ It is a unique phenomenon that the Young’s modulus of cementite is smaller than that of iron.

3.2.2. Bulk Modulus of Cementite Obtained by X-ray Diffraction in High-pressure Experiments

The bulk modulus of cementite has been studied by many geophysicists. The Earth’s inner core (the solid part at the center of the Earth) is under high pressure and high temperature (330 GPa, 5300 K³⁾), and iron is the main component. However, the measured density and seismic velocity were smaller than those of hcp iron. Therefore, it is expected that the inner core contains light elements such as C, S, or Si,⁷²⁾ and carbon is considered a promising element.³⁾ In the Fe–C binary system, Fe₃C³⁾ and Fe₇C₃^4,5) are stable phases under the condition of the Earth’s inner core. The density of the Earth’s inner core is estimated to be 13090 kg/m^{3 44)} in the Preliminary Reference Earth Model; therefore, the type of carbide and the ratio of iron to iron carbide that matches this density has been investigated mainly for Fe₃C.

Pressure-induced magnetic transformation should be considered when predicting the density of Fe₃C under the pressure and temperature conditions of the Earth’s inner core. Fe₃C is ferromagnetic (FM) at room temperature and 0.1 MPa, but when the pressure increases, it transforms into paramagnetic (PM) and then non-magnetic (NM). The bulk modulus of Fe₃C is increased nearly two times associated with this transition.

The method for experimentally determining the bulk modulus is to pressurize with diamond anvil, measure the lattice parameter by synchrotron X-ray diffraction, and calculate the bulk modulus from the pressure change of the unit cell volume using an EOS. The typical EOS used is the following 3rd order Birch–Murnaghan equation.^63,64)   

$$P=\frac{3}{2}{B}_0{}^P\left[{\left(\frac{V_0}{V}\right)}^{7/3}-{\left(\frac{V_0}{V}\right)}^{5/3}\right]\left\{1-\frac{3}{4}\left(4-{B_0^{\prime }}^P\right)\left[{\left(\frac{V_0}{V}\right)}^{2/3}-1\right]\right\}$$(1)

where P is the pressure, V₀, B₀^P, and B’₀^P are the volumes of the system at the equilibrium lattice parameter, the bulk modulus at room temperature and 0.1 MPa, and the first derivative of the bulk modulus at constant pressure B₀^P (=∂B₀^P/∂P). Table 4^{12,38,39,44,46,60,65,73,74)} shows B₀^P, B’₀^P, and V₀ of Fe₃C, the equation of state (EOS) used in the analysis, and the measured pressure range for each magnetic state. As can be seen from Table 4, the B₀^P of cementite varies greatly depending on the magnetic state. The pressure range applied in each magnetic state to measure B₀^P is FM (162–179 GPa), PM (190–199 GPa), and NM (288–290 GPa). An experimental example of the pressure-dependent change in the lattice parameter is presented in Fig. 2.⁴⁴⁾ As the pressure increased, the three-axis lattice parameters decreased, and at higher pressures, the rate of decrease in the lattice parameters decreased, resulting in a downward convex curve. The rate of decrease (coefficient of linear contraction) of the three-axis lattice parameter due to the increase in pressure differed depending on the axis, and decreased in the order c > b > a-axis.^38,44,65) This indicates that the bulk modulus exhibited crystallographic anisotropy. In addition, in Fig. 2, the lattice parameter of the b-axis decreases discontinuously when the pressure exceeds 55 GPa. Ono and Mibe⁴⁴⁾ carried out first-principles calculations and found that Fe₃C became NM at this pressure.

Table 4. Compilation of bulk modulus B₀ (in GPa), pressure derivative of the bulk modulus B’₀ (=∂B₀/∂P) and unit cell volume V₀ each at ambient pressure measured by different authors in chronological order.^{12,38,39,44,46,60,65,73,74)}

Reference	Magnetic state	Bulk modulus B0 (GPa)	B’0 (=∂B0/∂P)	Unit cell volume V0 (Å3)	EOS	Pressure P (GPa)
Jephcoat73)	–	162	6.4	–	V	<50
Scott et al.60)	FM	175.4±3.5	5.1±0.3	155.26±0.14	BM	<73
	FM	175.1±3.6	5.3±0.3	155.26±0.14	V	<73
Li et al.38)	FM	174±6	4.8±0.8	155.28	BM	<30.5
Lin et al.39)	FM	179.4±7.8	4.8±1.6	–	–	<45
	NM	288±42	4	–	–
Duman et al.12)	FM	174±8 RT	–	–	BM	<20
	PM	199±5550 K	–	–	BM	<20
Ono and Mibe44)	FM	167	6.7	–	BM	≤35
Sata et al.74)	NM	290 ± 13	3.76 ± 0.18	149.46	BM	25<P<187
Prescher et al.46)	–	161 ± 2	5.9 ± 0.2	–	BM	≤88
Litasov et al.65)	FM	175	5	155.2	VR	<7.2
	PM	190 ± 2	4.8 ± 0.1	154.56	VR	<31
	PM	194 ± 1	4.6 ± 0.1	154.42	VR	<31
	PM	191 ± 2	4.68 ± 0.08	154.56	VR	<31

FM: Ferromagnetic, PM: Paramagnetic, NM: Non-magnetic, BM: Birch-Murnaghan equation of state (EOS), VR: Vinet-Rydberg EOS, V: Vinet EOS.

Fig. 2.

Change in the lattice parameters of Fe₃C as a function of pressure at 300 K. The solid triangles, squares, and circles represent the experimental cell parameter ratios of the a-, b-, and c-axes, respectively, compared to the zero pressure values. The dashed lines represent the fitted curves of each cell parameter ratio. A significant reduction in the b-axis was observed at ~55 GPa.⁴⁴⁾ (Online version in color.)

Comparing the B^E values of Fe₃C (FM state) measured at room temperature and one atm (Table 3) with the B₀^P (Table 4) of the FM state obtained by X-ray diffraction under high-pressure experiments, there are differences in the values and deviations of the data. The B₀^P is in the range 162–179 GPa, which is a relatively narrow range (average value 173 GPa, standard deviation 5.2 GPa), but the B^E values in Table 3 are 111 GPa,²⁵⁾ 168 GPa,³³⁾ 155 GPa,⁵⁷⁾ showing a large difference among researchers. There are several causes of this difference between B₀^P and B^E. In the measurement of B^E, the sample preparation, amount of impurity, and measurement method are quite different among researchers. It is interesting to note that the B^E values of recent studies (168 GPa³³⁾ and 155 GPa⁵⁷⁾) are close to the value of B₀^P. It should be pointed out that B₀^P (measured from the volume-pressure relationship) is the isothermal bulk modulus and B^E (measured by ultrasonic or vibration method) is the adiabatic bulk modulus, but the difference between the two is as small as 3% or less.^3,65)

3.2.3. Alloying Effect on Young’s Modulus of Cementite at Room Temperature

Most metal elements can replace iron in cementite to form alloy cementite. Some elements dissolve in cementite in large quantities, but some other elements hardly dissolve in cementite. Ni and Si destabilize cementite, and Mo or V form its own carbide; thus, the amount that can be dissolved is limited.⁶⁾ The addition of alloying elements generally increases the Young’s modulus of cementite. Figure 3 shows the effect of the alloying element on the Young’s modulus (at room temperature) of cementite as measured by Umemoto et al.⁷⁰⁾ The alloyed samples were prepared using MA and SPS. The Young’s modulus increased as the concentration of alloying elements increased. The rate of increase in Young’s modulus is in the order Mo ≈ Mn < Cr < V. Regarding the Young’s modulus of alloyed cementite, Coronado et al.²⁷⁾ measured the Young’s modulus of cementite in mottled cast iron containing either V or Cr. The quantity of alloying elements in the cementite was measured by energy dispersive X-ray spectroscopy (EDS), and the Young’s modulus was measured by nanoindentation. They obtained E = 237 GPa for cementite containing 5.01 mass% Cr and E = 232 GPa for cementite containing 4.5 mass% V. Although the sample preparation and measurement methods used in their study differ from those used by Umemoto et al.,⁷⁰⁾ the values of the Young’s moduli obtained were similar.

3.2.4. Temperature Dependence of the Elastic Constant of Cementite

The temperature dependence of the Young’s modulus of cementite is shown in Fig. 4.^31,57,70) Umemoto et al. measured the temperature dependence of Young’s modulus by the resonance⁷⁰⁾ and ultrasonic⁵⁷⁾ methods, using samples prepared by MA and SPS. The Young’s moduli obtained by the two measurements were in excellent agreement. A clear minimum of Young’s modulus was observed at 471 K, just below T_C, when the ultrasonic method was applied but not seen with the resonance method. This is probably because the measured temperature interval is smaller for the ultrasonic method (several K) than for the resonance method (20–50 K). Figure 4 also shows the temperature dependence of Young’s modulus obtained by Drapkin and Fokin³¹⁾ who used the extrapolation method. It shows a more gradual temperature dependence than that obtained by Umemoto et al. who used the ultrasonic method.⁵⁷⁾ Figure 4 also shows the temperature dependence of the Young’s modulus of α-Fe for comparison. It should be noted that the temperature dependence of the Young’s modulus of cementite is minute compared to that of α-Fe. The small temperature dependence of the Young’s modulus indicates the Elinvar characteristics of cementite. When an external stress is applied to a magnetic material near its magnetic transformation temperature, its elastic constant decreases owing to the rotation of the magnetic moment or movement of the magnetic domain wall. This phenomenon is called the ΔE effect in the case of Young’s modulus (ΔG effect in the case of shear modulus). The ΔE effect was also observed in cementite. Based on a study on the temperature dependence of the Young’s modulus of the cementite, Drapkin and Forkin³¹⁾ estimated that the ΔE effect at room temperature is 17 GPa. This amount corresponds to an 8.5% reduction in Young’s modulus because the Young’s modulus of cementite at room temperature estimated from extrapolation in the PM state is 200 GPa. Similarly, Umemoto et al.⁵⁷⁾ reported that the ΔE effect is largest at temperatures just below T_C with a value of 8.2 GPa (corresponding to a 4.3% reduction).

Fig. 4.

Compilation of temperature dependence of Young’s modulus E of cementite obtained from experimental measurements by different authors.^31,57,70) Data of α-Fe⁷⁰⁾ is shown for comparison. (Online version in color.)

Figure 5 shows the temperature dependence of the four elastic constants of cementite measured by Umemoto et al.⁵⁷⁾ using the ultrasonic method. As can be seen in Fig. 5(a), E increased with decreasing temperature, showed a sharp drop at T_C, and became almost constant at temperatures below T_C. Figure 5(b) shows that G increased with decreasing temperature, showed a small drop at T_C, and decreased with decreasing temperature below T_C. B and ν show large minimums near T_C. Figure 5 also shows the result of the elastic constant of the (Fe_0.95Mn_0.05)₃C cementite. The addition of Mn increased E and G over the entire temperature range. The increases in E and G are larger in the FM state than in the PM state. With 0.05 Mn addition, E increases by 11.5 GPa in the PM state and by 12.0 GPa near room temperature in the FM state. Moreover, with 0.05 Mn addition, G increases by 4.2 GPa in the PM state and by 5.5 GPa near room temperature in the FM state. On the other hand, B (Fig. 5(c)) and ν (Fig. 5(d)) show a large decrease near T_C. With 0.05 Mn addition, B increases by 9.2 GPa in the PM state, but at room temperature, B decreases due to the Mn effect of lowering the T_C. As a result, these two factors cancel each other out at around room temperature, and the effect of 0.05 Mn addition on B was hardly observed. In addition, ν was not affected by the addition of Mn in the high-temperature PM state. When the ν values of Fe₃C and (Fe_0.95Mn_0.05)₃C were compared at room temperature, ν was seen to be smaller in the former due to the Mn effect of lowering the T_C.

Fig. 5.

Temperature dependence of elastic constants of Fe₃C (dark blue) and (Fe_0.95Mn_0.05)₃C (dark red). (a) Young’s modulus E, (b) shear modulus G, (c) bulk modulus B, and (d) Poisson’s ratio ν.⁵⁷⁾ (Online version in color.)

There are several reports on the temperature dependence of the cementite shear and bulk moduli. Dodd et al.⁵¹⁾ measured the shear modulus in the temperature range 75–295 K and reported that the shear modulus decreased by 1.3% with a temperature drop from 193 to 100 K. Regarding the temperature dependence of the bulk modulus, Duman et al.¹²⁾ calculated the bulk modulus by varying the pressure in the range 0–20 GPa at the temperatures 550 K and room temperature. As a result, B^P (550 K) = 199 GPa was obtained at 550 K (> T_C), and B₀^P (RT) = 174 GPa was obtained at room temperature (<T_C). It has been reported that the decrease in B^P due to the temperature decrease from 550 K to room temperature was 13%, which was similar to the change in bulk modulus due to the magnetic transformation of the Fe₆₅Ni₃₅ Invar alloy. On the other hand, the results of the ultrasonic measurement shown in Fig. 5(c) show B^E (550 K) = 157 GPa and B^E (RT) = 160 GPa, and there is no significant change due to magnetic transition. Regarding the observed large difference in the temperature dependence of B^P between the high-pressure and ultrasonic experiments, it is possible that the measurement by Duman et al. included pressure-induced magnetic transformation. Cementite is reported to undergo pressure-induced FM-PM transformation around 8–10 GPa at room temperature.⁴⁶⁾ Duman et al.¹²⁾ measured the bulk modulus by changing the pressure in the range 0–20 GPa, and it seems that the obtained B₀^P (RT) may include the effect of magnetic transition from a low-pressure FM state to a high-pressure PM state. Regarding the temperature change of the Poisson’s ratio, Dodd et al.⁵¹⁾ reported that the Poisson’s ratio increased slightly from 0.246 to 0.254 (without sample porosity correction) when the temperature decreased from 295 to 100 K.

3.3. Hardness of Cementite

Several studies have been conducted on the hardness of cementite at room temperature. The hardness values of pure cementite samples at room temperature reported after 1950 are summarized in Table 5.^{19,20,21,22,23,24,25,26)} Vickers hardness measurements were performed for the primary cementite in white cast iron and single-phase samples prepared using MA and SPS. Table 5 shows that the hardness of pure cementite at room temperature is in the range 1013–1340 HV. There are several possible causes for the difference in hardness values among researchers, in addition to the difference in the measurement weight (50 g–1 kg). The first is the crystallographic anisotropy of hardness. Inoue et al.²³⁾ measured the hardness of primary cementite in Fe-4.81C-5.80Cr (mass%) samples in various different orientations. As a result, it was reported that when the cementite surface was (010), the hardness was approximately 10% higher than that of the other orientations away from (010). Another reason for the difference in hardness is the thermal history of the sample. The carbon concentration of primary cementite is lower than that of the stoichiometric composition of Fe₃C.^10,75,76) Therefore, when annealed at a low temperature, ferrite precipitates,⁷⁶⁾ the carbon content in cementite increases, and the hardness increases.⁷⁷⁾ In the case of primary cementite, the thickness of the cementite plate and the structure of the surrounding ledeburite also affect the measured hardness of cementite. Thin films produced by physical vapor deposition and samples prepared using MA and SPS have a crystal grain size of 1 μm or less, which affects the hardness.

Table 5. Compilation of hardness of pure cementite (without alloying elements) at room temperature obtained from experimental measurements by different authors.^{19,21,22,24,25,26)}

Reference	Hardness HV	Specimen	Load and others
Sato et al.19)	1340	Primary cementite in Fe-4.8%C white cast iron	Load: 50 g
Gove and Charles21)	1220	Primary cementite in a white cast iron	Load: not noted, Temperature effect
Yakushiji et al.22)	1130	Primary cementite in the directionally solidified Fe-4.5%C alloy	Load: 50 g, Temperature effect, Hardness on (100)θ
Kagawa and Okamoto24)	1013	Primary cementite in the directionally solidified Fe-4.54% C alloy	Load: 100 g, Temperature effect, Hardness on (100)θ
Li et al.25)	1230	Plasma deposited Fe3C film (thickness 2.5 μm, grain size 50 nm)	Load: 2.5 g
Umemoto et al.26)	1020	MA+SPS* Grain size 0.5 μm	Load: 100 g

*  MA: Mechanical alloying, SPS: Spark plasma sintering

Several studies have been conducted on the effect of alloying on the hardness of cementite. These are summarized in Fig. 6.^{19,20,24,26,27,78,79)}*² The horizontal axis of Fig. 6 shows the amount of alloying elements in mass%. Metal solutes such as Cr, Mn, Mo, and V substitute iron at some sites and smaller atoms such as boron replace carbon at the interstitial sites of cementite. As shown in Fig. 6, there are many studies on Mn and Cr, which are often used in steels. Alloying elements such as Mn and Cr are concentrated in the cementite phase rather than in the ferrite phase. The ratio of the alloying element concentration (mass%) dissolved in cementite and ferrite phases is called the partition coefficient, which is a value specific to the alloying element and does not depend on the amount of the alloying element or the carbon content of the steel. Among the alloying elements, Cr and Mn have a large partition coefficient of 28.0 and 10.5, respectively, at 973 K.⁶⁹⁾ The partition coefficient increased as the temperature decreased, reaching 80 for Cr and 30 for Mn at 823 K. Therefore, it should be noted that in steels, the concentration of Cr or Mn in cementite is high even if the amount of Cr or Mn in the steel is small. As can be seen from Fig. 6, the addition of Cr, Mn, V, Mo, and B increases the hardness of cementite. Among Mn, Cr, V, and Mo, the rate of increase in hardness rises in the order Mo < Mn < Cr < V, which is very similar to the effect of alloying elements on Young’s modulus (Fig. 3).

Fig. 6.

Vickers hardness of cementite as a function of the alloying element concentrations (wt%).^{19,20,24,26,27,78,79)} Metallic solutes (Cr, Mn, Mo, V) substitute on to the iron sites and boron replace carbon at interstitial sites. (Online version in color.)

There are studies^{21,22,23,24,26,78)} on the high-temperature hardness of cementite, as it is important to understand the abrasive wear that occurs when the surface temperature of cast iron parts rises. According to Kagawa et al.,^24,78) there is a relationship between the Vickers hardness, HV, of cementite and the absolute temperature, T: HV = Aexp(−βT). Here, A and β are constants, and β is called the thermal softening coefficient, which is the magnitude of the temperature dependence of HV. The temperature dependence of cementite hardness can be expressed by this equation in three consecutive temperature ranges: lower than T_C (473 K), 473–710 K, and higher than 710 K. The thermal softening coefficient β increases as the temperature increases. In the temperature ranges lower than T_C (473 K) and between 473 and 710 K, where deformation by hardness creep (creep observed in hardness test) does not occur, β is proportional to the coefficient of thermal expansion.²⁴⁾ β is small in the FM state, where the coefficient of thermal expansion is small, whereas β is large in the PM state, where the coefficient of thermal expansion is large. The reason for the proportionality between β and the coefficient of thermal expansion is that both are related to the temperature dependence of the binding energy of atoms.²⁴⁾ Hardness creep deformation occurred in the temperature range above 710 K. Kagawa et al.⁷⁸⁾ obtained 254 kJmol⁻¹ as the activation energy for hardness creep deformation from the temperature dependence of the change in weight with time during high-temperature hardness measurements. This value corresponds to the activation energy of γ-Fe self-diffusion.

Most alloying elements for cementite increase the hardness creep start temperature.^24,78) Kagawa et al.⁷⁸⁾ measured the effect of Cr alloying (up to 14.2 mass%) on the temperature dependence of cementite hardness. They found that as the Cr concentration increased, the softening coefficient decreased, the hardness creep start temperature increased, and the activation energy for hardness creep increased. The effect of alloying elements on the high-temperature hardness of cementite has also been investigated. Figure 7⁷⁰⁾ shows the temperature dependence of the hardness of alloyed cementite with a composition in which 5 at% of Fe atoms are replaced with the substitutional element M (M = Cr, Mn, Mo, V), (Fe_0.95M_0.05)₃C. The hardness of all the alloyed cementite was higher than that of pure cementite and decreased with increasing temperature. Further, the high-temperature hardness depended on the alloying elements, and the hardness at 773 K was in the order Mn < Cr < V < Mo. This order corresponds well with that of the misfit parameter ε, which is the difference in atomic size between the iron and the alloying element, Mo (ε = 0.084), V (0.034), Mn (0.016), and Cr (0.014). This shows that the element having a larger misfit parameter has a greater influence on the high-temperature hardness of cementite. In the hardness measurement at 773 K (Fig. 7), a clear time-dependent hardness, that is, hardness creep, was observed.⁷⁰⁾ In the normal tensile creep of α-Fe, it is known that a solid solution element having a larger misfit parameter contributes more to strengthening, and it is considered that the same strengthening mechanism holds for the hardness creep in cementite.

Fig. 7.

Effect of alloying element on high-temperature Vickers hardness of cementite in (Fe_0.95M_0.05)₃C, M = Cr, Mn, Mo, V.⁷⁰⁾ (Online version in color.)

3.4. Other Mechanical Properties of Cementite

3.4.1. Deformation Mechanism of Cementite

It has been confirmed by electron microscopy that slip occurs in cementite during deformation.^58,80,81,82) Keh⁸⁰⁾ reported that the slip plane of cementite was the (001) plane based on the trace analysis of pile-up dislocations. Inoue et al.⁸²⁾ rolled eutectoid and hypereutectoid steels at room temperature and observed the slip system of cementite. The observed slip systems were (001)[100], (100)[010], and (010)[100] and the frequency of the slip system observed was in the order of {110} <(100) <(010), (001).⁸¹⁾ The magnitude of the Burgers vector of dislocations is considered to be the length of the unit cell edge from its simple crystal structure.⁵⁸⁾ This large Burgers vector is thought to be the reason for the high strength of cementite.

3.4.2. Wear Characteristics of Cementite

In carbon steel and white cast iron, the volume fraction of cementite significantly affects the abrasion resistance. However, because the volume fraction of cementite is less than 65% in ordinary steels and cast irons, the wear characteristics of the Fe–C samples containing more than 65% volume fraction of cementite are unknown. Zheng et al.⁴⁸⁾ studied abrasive wear using materials with different cementite volume fractions, that is, steels (cementite volume fractions of 0 and 7.83%), white cast irons (18.56 and 49.21%), and samples prepared using MA and SPS (81.75 and 100%). They observed that when the load pressure was less than 0.065 MPa, the amount of wear decreased as the volume fraction of cementite increased. When the load pressure reached 0.098 MPa, the amount of wear decreased as the volume fraction of cementite increased up to 50%, but when the volume fraction of cementite increased to more than 80%, the fracturing and peeling from the cementite increased, and the amount of wear increased. Sasaki et al.⁸³⁾ investigated wear characteristics using pure iron, S45C (cementite volume fraction 7%), and samples prepared using MA and SPS (50%, 75%, and 100%), at various load pressures. Figure 8 shows the relationship between wear resistance and Vickers hardness. In the figure, % denotes the volume fraction of cementite, and the pressure in the figure indicates the applied pressure during the wear experiment. The specific wear rate, which has the unit [m³/MPa·m], is the wear volume loss divided by the wear distance and applied pressure. The smaller the number, the less likely it is to wear. Wear resistance is the reciprocal of the specific wear rate, and the larger the number, the less likely it is to wear. As shown in Fig. 8, the wear resistance increases with increasing hardness of the sample up to a pressure of 0.15 MPa, that is, it becomes difficult to wear (dotted line). However, at pressures higher than 0.31 MPa, the wear resistance does not show a linear relationship with the hardness: when the volume fraction of cementite is larger than 75%, the wear resistance is lower than that expected from the linear relationship with the hardness of the sample. As described above, at pressures higher than 0.31 MPa, the wear resistance reaches a maximum at approximately 75% of the cementite volume fraction; moreover, the 100% cementite sample does not show a good wear resistance expected from its hardness.^48,83) This is attributed to the formation of large brittle chips of cementite, which occurs in pure cementite at high pressure. The maximum wear resistance at approximately 75% volume fraction of cementite suggests that the brittle cementite phase is wrapped in a ductile ferrite phase, and the formation of large brittle chips of cementite is suppressed.

Fig. 8.

Relationship between the abrasive wear resistance (reciprocal of specific wear rate) at applied pressures of 0.15 and 1.23 MPa and Vickers hardness for a variety of samples of ferrite containing different amounts of cementite (%). After Sasaki et al.⁸³⁾ (Online version in color.)

3.4.3. Fracture Toughness of Cementite

The fracture toughness of cementite has been measured using a nanoindenter.^84,85) Using this method, Coronado and Rodriguez⁸⁴⁾ measured the fracture toughness of the primary cementite of mottled cast iron (5 mass% C) to which Cr and V were added separately. The concentrations of the alloying elements in cementite were measured using EDS. The fracture toughness obtained for the cementite with different alloying elements were 2.24 MPa·m^1/2 (9.2 mass% Cr), 2.52 MPa·m^1/2 (5.1 mass% Cr), and 2.74 MPa·m^1/2 (4.5 mass% V). These values are in the range of measured fracture toughness (2.2–3.7 MPa·m^1/2) for other types of carbides (M₇C₃, M₆C, MC). Fernández and Vicente⁸⁵⁾ determined the fracture toughness of cementite using the nanoindentation method with spheroidal graphite cast iron. Two types of samples were used: SGP (spheroidized graphite pearlitic matrix i.e., spheroidized graphite + eutectic cementite + pearlite) and SGA (spheroidized graphite acicular matrix, i.e., spheroidized graphite + eutectic cementite + bainite + martensite). The results gave the fracture toughness of cementite (the content of alloying elements was not measured) as 4.09 MPa·m^1/2 for the SGP sample and 2.41 MPa·m^1/2 for the SGA sample. These values are similar to those of carbides such as ZrC, VC, Cr₇C₃, and NbC, but the hardness of cementite is reported to be the lowest among these carbides.

3.4.4. Anisotropy in the Mechanical Properties of Cementite

Although it is expected from the crystal structure that the physical properties of cementite are dependent on its crystal orientation, several studies show anisotropy in mechanical properties. Regarding the anisotropy of hardness, Inoue et al.²³⁾ measured the hardness of primary cementite in three orientations using a unidirectionally solidified Fe-4.81C-5.80Cr sample. It has been reported that the hardness of cementite is approximately 10% higher on the (010) cementite surface than on the other surfaces away from (010). Several experimental results have been reported regarding the anisotropy of the elastic constants. Kagawa et al.³⁰⁾ investigated the anisotropy in the Young’s modulus of cementite using specimens with three different longitudinal axes, that is, parallel to the [100] direction of eutectic cementite and normal to [100] ([010] and [001]) cut from iron-cementite castings and zone-melted bars. They found that the Young’s modulus of cementite in the [100] direction was larger than the Young’s modulus in the direction perpendicular to it. Koo et al.³⁵⁾ measured the Young’s modulus of a single crystalline cementite sheet using a bending test in a SEM using a 100–200 nm thick cementite sheet extracted from a pearlite structure. The cementite sheet was parallel to the (010) plane, and the Young’s moduli in the [100] and [001] directions were 262 GPa and 213 GPa, respectively. These values are respectively 9.8% and 14.0% smaller than the Young’s modulus calculated from the elastic stiffness constant C_ijs by Jiang et al.⁸⁶⁾ using first-principles calculation. However, the fact that the Young’s modulus is higher in the [100] direction than in the [001] direction agrees with the calculation result of Jiang et al.⁸⁶⁾ Alkorta and Sevillano²⁸⁾ used mottled cast iron to investigate the crystallographic anisotropy of the elastic constants of cementite. The orientation of the cementite was measured by electron backscatter diffraction, and the indentation modulus (composed of Young’s modulus of the indenter and the sample) was measured using the nanoindentation method. The measured indentation modulus showed the minimum value of 184 GPa in the [011] orientation, and showed the maximum value of 322 GPa in the [100] orientation, which almost doubled depending on the crystal orientation of cementite. Jiang et al.⁸⁶⁾ calculated the Young’s modulus using C_ijs obtained by first-principles calculation and showed that the maximum value of Young’s modulus was 317 GPa near the [101] orientation and the minimum value was 55.4 GPa at [011]. Alkorta and Sevillano²⁸⁾ further calculated the indentation modulus of each crystalline orientation by the finite element method using the value of C_ijs calculated by Jiang et al.⁸⁶⁾ and compared it with the measured value. They observed that the difference in elastic anisotropy in the measured indentation modulus was lower than that predicted by the first-principles calculation and suggested that the anisotropy of cementite is not as high as estimated by first-principles calculation.

*¹  In this review, the orthogonal axis of the unit cell is set to a = 0.5084 nm, b = 0.6747 nm, and c = 0.4525 nm to coincide with the space group Pnma. The crystallographic data expressed in other axial orientations were corrected to match the Pnma.

*²  In Fig. 10(a) of Reference 79), sample A4 is described as 9.76 mass% Cr, but when confirmed with the author, it was an error of 11.51 mass% Cr. In Fig. 6 of this review, it is plotted as 11.51 mass% Cr.

4. Estimation of the Elastic Constants of Cementite by First-principles Calculations

4.1. Cementite and First-principles Calculations

Theoretical studies on cementite using first-principles calculations have become popular since 2002. Owing to the difficulties in growing a large single-crystal cementite samples, the elastic constant of the single-crystal cementite has not been measured. Moreover, in the field of geophysics, it is difficult to conduct research at high pressures corresponding to the inner core of the Earth, and a theoretical approach using first-principles calculations is highly desired. Early studies of cementite using first-principles calculations included calculations of bonding state, formation energy, differences between η and χ carbides, and surface energy.¹⁾ The first-principles calculation of the elastic stiffness constant C_ijs of cementite was first carried out by Jiang et al.⁸⁶⁾ in 2008, and many studies have been conducted since then. Here is a summary of previous studies that applied first-principles calculations on the mechanical properties of cementite.

There are two main methods for obtaining the elastic constants by first-principles calculations. One is to find the bulk modulus B^F using the EOS as frequently seen in the studies by geophysicists. B^F is calculated from either the relationship between the total energy of the system E_tot and the unit cell volume V or from the relationship between the pressures P and V. The other method for estimating the elastic constants of cementite is using C_ijs. The first-principles calculation of C_ijs is described in detail in Section 4.3. Using C_ijs, the elastic constants of isotropic polycrystals can be estimated by the averaging method, and the elastic constants can be compared with the measured values. In the following sections, we first introduce these two methods for determining the elastic constants of cementite.

4.2. Bulk Modulus B^F of Cementite Estimated Using the Equation of State

Using the first-principles calculation, the total energy of system, E_tot, for a given unit cell volume V can be calculated. The bulk modulus B^F can be estimated using the EOS. The following third-order Birch–Murnaghan equation^63,64) is often used as the EOS.   

$$\begin{array}{l}{E}_{tot}=E_0\\ +\frac{3}{2}{V}_0{B}_0^F\left[\frac{3}{4}\left(1+2\eta \right){\chi }^{-4}-\frac{\eta }{2}{\chi }^{-6}-\frac{3}{2}\left(1+\eta \right){\chi }^{-2}+\frac{1}{2}\left(\eta +\frac{3}{2}\right)\right]\end{array}$$(2)

where, E₀ is the total energy of the system at the equilibrium lattice parameter, χ=(V/V₀)^1/3, η = (3−(3$B^{\prime }{}_{\mathit{0}}^F$/4), B₀^F is B^F at 0 K and 0 GPa and B’₀^F is the first pressure derivative of B₀^F (=∂B₀^F/∂P). The calculation process is as follows: calculate E_tot for various values of V under different pressure values, substitute E_tot and V in Eq. (2), and calculate B₀^F and B’₀^F, which minimizes the difference between the left and right sides of Eq. (2). The estimation of B₀^F and B’₀^F can also be done by using an equation similar to Eq. (1), which expresses the relationship between P obtained from ∂E_tot/∂V and V. This method of obtaining B₀^F has been attempted by many researchers as a method to obtain the bulk modulus of cementite using first-principles calculations. Table 6^{44,86,87,88,89,90,91,92,93,94)} summarizes the reported values of B₀^F, B’₀^F, and V₀, the EOS, and the range of pressure and volume values. Since cementite is FM at room temperature and becomes NM under high pressure, the values in Table 6 are shown separately according to the magnetic state, that is, either in the FM state or in the NM state. (The expression of magnetism is different from paper to paper, the expressions of FM and spin-polarized are classified as the expression of the FM state, and the expressions of NM and spin non-polarized are classified as NM states.) As can be seen from Table 6, B₀^F is approximately 200 GPa for the FM state and approximately 300 GPa for the NM state, which is approximately 100 GPa larger than the FM state. The B₀^P values at room temperature shown in Table 4 are different depending on the magnetic state and 162–179 GPa for the FM state, 190–199 GPa for the PM state, and 288–290 GPa for the NM state. Because B₀^F is the value at 0 K and B₀^P is at room temperature, it is not possible to compare the two directly, but the difference of approximately 100 MPa between FM and NM is in relatively good agreement.

Table 6. Compilation of bulk modulus B₀^F (GPa) and pressure derivative of the bulk modulus B’₀^F under zero pressure and at 0 K obtained from first-principles calculations by different authors in chronological order.^{44,86,87,88,89,90,91,92,93,94)}

Reference	Magnetic state	Bulk modulus B0 (GPa)	B0’ (=∂B0/∂P)	Unit cell volume V0 (Å3)	EOS	Data	Range of P or V
Vocadlo et al.87)	FM	173.02	5.79	153.25	BM	E-V	100<V<176.5 Å3
	FM	228.55	5.36	153.04	BM	E-V	144<V<176.5 Å3
Chiou and Carter88)	FM	142	4.60	154.54	BM	E-V
Huang et al.89)	FM	212	4.5	152	BM	E-V
Faraoun et al.90)	FM	235.13	–	–	BM	E-V	132<V<163 Å3
Jiang et al.86)	FM	204	–	–	BM	E-V	0<P<30 GPa,
	FM	223	–	–	BM	E-V	−3<P<6 GPa
Henriksson91)	FM	234	4.0	151.95	B	E-V
Jang et al.92)	FM	226.84	–	152.20	BM	E-V	140<V<168 Å3
Nisar and Shuja93)	FM	183.9	4.8	150.7	BM	E-V	P<73 GPa
							Pc=32–38 GPa
Ono and Mibe44)	FM	216.5	4.15	–	BM	P-V	P≤35 GPa
	FM	227.7	3.36	–	BM	E-V	P≤35 GPa
Mookherjee94)	FM	182.6	6.0	151.62	VR	P-V	100<V<160 Å3
	FM	184.0	5.6	151.75	BM	E-V
Vocadlo et al.87)	NM	316.62	4.30	143.49	BM	E-V	80<V<155.0 Å3
Huang et al.89)	NM	322.0	3.7	144.00	BM	E-V
Jang et al.92)	NM	319.7	–	143.27	BM	E-V	140<V<168 Å3
Nisar and Shuja93)	NM	297.3	4.9	142.9	BM	E-V
Ono and Mibe44)	NM	315.5	4.37	–	BM	P-V	P≤400 GPa
Mookherjee94)	NM	297.0	4.9	143.26	VR	P-V	90<V<160 Å3,
							0<P<400 GPa
	NM	303.6	4.5	143.35	BM	E-V	90<V<160 Å3

FM: Ferromagnetic, NM: Non-magnetic, BM: Birch-Murnaghan equation of state, B: Birch equation of state, VR: Vinet-Rydberg equation of state, E-V: from energy-volume data (energy-strain), P-V: from pressure-volume data (stress-strain), Pc: critical pressure to induce magnetic transition (high spin to low spin)

4.3. Elastic Constants of Single-crystal Cementite

When a small force is applied to a solid, the stress σ and the strain ε obey Hooke’s law:   

$$\sigma =\mathrm{M}\epsilon$$(3)

where M is an elastic constant. More generally, this relation is expressed with six stress components σ_i and six strain components ε_i:   

$${\sigma }_i={\sum }_i{C}_{ij}{\epsilon }_j$$(4)

C_ij is the elastic stiffness constant. In orthorhombic crystal structures like that of cementite, there are nine independent constants. Therefore, Voigt notation, the generalized Hooke’s law for an orthorhombic crystal, can be written as:   

$$\left(\begin{array}{l}{\sigma }_{\mathit{1}}\hfill \\ {\sigma }_{\mathit{2}}\hfill \\ {\sigma }_{\mathit{3}}\hfill \\ {\tau }_{\mathit{4}}\hfill \\ {\tau }_{\mathit{5}}\hfill \\ {\tau }_{\mathit{6}}\hfill \end{array}\right)=\left(\begin{array}{cccccc}{C}_{\mathit{1}\mathit{1}}& C_{\mathit{1}\mathit{2}}& C_{\mathit{1}\mathit{3}}& 0& 0& 0\\ C_{\mathit{2}\mathit{1}}& C_{\mathit{2}\mathit{2}}& C_{\mathit{2}\mathit{3}}& 0& 0& 0\\ C_{\mathit{3}\mathit{1}}& C_{\mathit{3}\mathit{2}}& C_{\mathit{3}\mathit{3}}& 0& 0& 0 \\ 0& 0& 0& C_{\mathit{4}\mathit{4}}& 0& 0\\ 0& 0& 0& 0& C_{\mathit{5}\mathit{5}}& 0\\ 0& 0& 0& 0& 0& C_{\mathit{6}\mathit{6}}\end{array}\right)\left(\begin{array}{l}{\epsilon }_{\mathit{1}}\hfill \\ {\epsilon }_{\mathit{2}}\hfill \\ {\epsilon }_{\mathit{3}}\hfill \\ {\gamma }_{\mathit{4}}\hfill \\ {\gamma }_{\mathit{5}}\hfill \\ {\gamma }_{\mathit{6}}\hfill \end{array}\right)$$(5)

where σ_i and τ_i are the normal stress and shear stress, respectively, and ε_i and γ_i are the normal strain and shear strain, respectively.

There are three methods for calculating single-crystal elastic constants by first-principles calculations: (1) energy-strain method, (2) stress–strain method, and (3) phonon method. In the energy-strain method, the elastic constants are derived by calculating the first derivatives of the energy density (defined as the total energy per unit volume) as a function of the appropriately selected strains. The total energy is expressed using C_ijs and the strain, and C_ijs can be calculated by applying simple tension or shear deformation to an appropriate cementite lattice and calculating the total energy change. In the stress–strain method, elastic constants are derived by calculating the first derivative of the stresses with respect to strain. In the phonon method, the elastic constants are derived by calculating the first derivative of the phonon branch with respect to an appropriate wave vector. Jiang et al.⁸⁶⁾ reported that the stress–strain method is more efficient than the energy-strain method, although the results obtained by these two methods are in good agreement. Ghosh⁵⁹⁾ stated that among the three methods, the stress–strain method has the shortest calculation time and lowest calculation cost.

Table 7^{59,86,91,94,95,96,97,98)} shows the nine independent C_ijs values of cementite obtained by the first-principles calculations, their calculation methods, and the elastic constants of the polycrystalline cementite calculated from C_ijs*³. The C_ijs value of Reference 94) was larger than those in other studies because Reference 94) is in the NM state while the others are in the FM state. Comparing the C_ijs values of the FM cementite in literature, the values other than C₄₄ are relatively close, and are in the range of ±17%. On the other hand, C₄₄ is a very small value compared to other C_ijs, and the value varies greatly depending on the literature.

Table 7. Compilation of elastic moduli (GPa) under zero pressure and at 0 K obtained from first-principles calculations by different authors in chronological order.^{59,86,91,94,95,96,97,98)}

Reference	C11	C22	C33	C12	C23	C13	C44	C55	C66	E	G	BC	v	Method
Jiang et al.86)	388	345	322	156	162	164	15	134	134	194	72	224	0.355	E-S, relaxed
	395	347	325	158	163	169	18	134	135	203	75	227	0.351	S-S, relaxed
	413	412	378	154	170	167	82	136	140	302*	117*	243*	0.293*	E-S, unrelaxed
	417	416	381	157	174	171	82	136	140	303*	117*	246*	0.295*	S-S, unrelaxed
	384	325	283	–	–	–	26	134	125	–	–	–	–	Phonon
Nikolussi et al.95)	385	341	316	157	167	162	13	131	131	184*	67.5	223.5	0.363*	S-S, relaxed
Henriksson et al.91)	394	412	360	157	166	146	83	133	136	296*	115*	233*	0.288*	E-S, unrelaxed
Lv et al.96)	393	340	319	144	149	141	−60	145	118	–	–	213	–	S-S, relaxed
Mookherjee94)	480	443	480	237	188	236	−6	149	153	177**	63	301	0.402**	S-S
														non-magnetic
Ghosh59)	375	339	298	161	172	144	13	132	130***	183	67	218	0.36	S-S
Huang97)	410	410	376	152	170	164	20	136	140	226*	84*	241*	0.344*	E-S
Mauger98)	315	321	299	136	175	131	24	142	138	221*	84*	202*	0.317*	self-consistent
	319	321	298	–	–	–	26	140	139	–	–	–	–	Phonon
	383	344	300	162	162	156	28	134	135	219*	82*	220*	0.334*	S-S
Fe2CrC59)	421	341	387	125	163	164	14	128	153	219	82	227	0.339
Fe2MnC97)	402	418	398	165	168	155	68	154	99	284*	109*	244*	0.306*

The listed values are in the ferromagnetic state unless otherwise noted.

‘E-S’ and ‘S-S’ denote that elastic constants were calculated from energy-strain and stress-strain relationships. ‘relaxed’: the unit cell volume and shape as well as all internal atomic positions are relaxed to their energetically more favorable lattice positions after the unit cell shape is deformed. ‘unrelaxed’: internal atomic positions were “frozen” at their zero-strain equilibrium values after the unit cell shape is deformed. Self-consistent method is a method which solves for the elastic constants of a system by direct comparison to calculated phonon frequencies at low wave vector generated from density functional theory force constant calculations.

*  Calculated by the present authors using the elastic stiffnesses C_ijs listed in this table. Bulk modulus B and shear modulus G for polycrystalline values are calculated by Hill average: B_H=(B_R+B_V)/2 and G_H=(G_R+G_V)/2, where the subscripts H, V and R represent the Hill, Voigt and Reuss approximations, respectively, as explained later in 4.5. Young’s modulus E and Poisson’s ratio v for polycrystalline values are calculated using the formula E=9GB/(G+3B) and v =(3B−2G)/(3B+G)/2.

**  Calculated by the present authors using the reported B and G.

***  We confirmed by contacting the author that the value 30 (GPa) of C₆₆ written in Table II of Ref.59 is an error of 130.

The necessary conditions for elastic stability in an orthorhombic crystal are presented by the following criteria:⁹⁹⁾   

$$\begin{array}{l}{C}_{\mathit{1}\mathit{1}}>0,\mathrm{\hspace{0.17em} } C_{\mathit{2}\mathit{2}}>0,\mathrm{\hspace{0.17em} } C_{\mathit{3}\mathit{3}}>0,\mathrm{\hspace{0.17em} } C_{\mathit{4}\mathit{4}}>0,\mathrm{\hspace{0.17em} } C_{\mathit{5}\mathit{5}}>0,\mathrm{\hspace{0.17em} } C_{\mathit{6}\mathit{6}}>0\\ \left(C_{\mathit{1}\mathit{1}}+C_{\mathit{2}\mathit{2}}-2C_{\mathit{1}\mathit{2}}\right)>0\\ \left(C_{\mathit{1}\mathit{1}}+C_{\mathit{3}\mathit{3}}-2C_{\mathit{1}\mathit{3}}\right)>0\\ \left(C_{\mathit{2}\mathit{2}}+C_{\mathit{3}\mathit{3}}-2C_{\mathit{2}\mathit{3}}\right)>0\\ \left(C_{\mathit{1}\mathit{1}}+C_{\mathit{2}\mathit{2}}+C_{\mathit{3}\mathit{3}}+2C_{\mathit{1}\mathit{2}}+2C_{\mathit{1}\mathit{3}}+2C_{\mathit{2}\mathit{3}}\right)>0 \end{array}$$(6)

According to the stability criteria, Reference 94) and Reference 96) do not satisfy the elastic stability criterion because C₄₄ is negative.

4.4. Crystal Orientation Anisotropy of Elastic Constant of Single-crystal Cementite

Once the single-crystal C_ijs values are obtained, it is possible to evaluate the crystal orientation anisotropy. In this section, the crystal orientation anisotropy for each elastic constant of cementite is discussed.

For an orthorhombic crystal, the Young’s modulus in any direction is given by:¹⁰⁴⁾   

$$\begin{array}{c}\frac{1}{E}=S_{\mathit{1}\mathit{1}}{l}_{\mathit{1}}{}^4+S_{\mathit{2}\mathit{2}}{l}_{\mathit{2}}{}^4+S_{\mathit{3}\mathit{3}}{l}_{\mathit{3}}{}^4+\left(2S_{\mathit{1}\mathit{2}}+S_{\mathit{6}\mathit{6}}\right)l_{\mathit{1}}{}^2{l}_{\mathit{2}}{}^2\\ +\left(2S_{\mathit{2}\mathit{3}}+S_{\mathit{4}\mathit{4}}\right)l_{\mathit{2}}{}^2{l}_{\mathit{3}}{}^2+\left(2S_{\mathit{1}\mathit{3}}+S_{\mathit{5}\mathit{5}}\right)l_{\mathit{1}}{}^2{l}_{\mathit{3}}{}^2\end{array}$$(7)

where S_ij is the elastic compliance constant, and l_i is the direction cosine. Ghosh⁵⁹⁾ reported the Young’s modulus of each crystal orientation calculated from C_ijs, as shown in Table 8. According to this, the magnitude of the Young’s modulus in the axial direction is in the order of E₁₀₀>E₀₁₀>E₀₀₁*⁴. Comparing this result with the Young’s moduli of 262 GPa and 213 GPa in the [100] and [001] directions, respectively, which were measured by Koo et al.³⁵⁾ using lamella cementite, the absolute values are different, however, the fact that E₁₀₀ is larger than E₀₀₁ is consistent. According to the stereographic projection of Young’s modulus drawn by Ghosh⁵⁹⁾ using C_ijs obtained by first-principles calculation,¹⁾ the maximum value of Young’s modulus is located in the direction between [100] and [101], and the minimum value is located near [011]. Similarly, according to the stereographic projection of Young’s modulus calculated by Alkorta et al.²⁸⁾ using C_ijs obtained by Jiang et al.⁸⁶⁾ the maximum value of Young’s modulus (317 GPa) is located in the direction between [100] and [101]. The minimum value (55.4 GPa) was located near [011]. Comparing these two stereographic projections, the absolute values of Young’s modulus are slightly different; however, the trends are similar.

Table 8. Calculated Young’s modulus (GPa) of cementite along different crystallographic directions (E_hkl) by the first principles calculation.⁵⁹⁾

E100	E010	E001	E110	E011	E101	E111
280.3	220.1	197.7	304.9	49.4	298.8	85.2

Regarding the anisotropy of the bulk modulus, Litasov et al.⁶⁵⁾ reported the value of B in each crystal axis direction using C_ijs obtained by first-principles calculation. They showed that B^C becomes smaller in the order of B^C (a axis) (= 195 GPa) > B^C (b axis) (= 186 GPa) > B^C (c axis) (= 163 GPa). Ghosh⁵⁹⁾ also calculated the directional bulk modulus along the orthorhombic crystallographic axes using C_ijs. The result showed that the order of the magnitude was B^C (b-axis) > B^C (a-axis) > B^C (c-axis). In both cases B^C (c-axis) is the smallest, but the order of B^C (a-axis) and B^C (b-axis) is different. The degree of magnitude of B^P in each crystal axis can be determined from the lattice parameter-pressure relationship; the larger the rate of decrease of the lattice parameter with increasing pressure, the smaller the axial bulk modulus B^P. All the experimental results of the lattice parameter-pressure relationship^38,44,65) show that the order of the magnitude was B^P (a-axis) > B^P (b-axis) > B^P (c-axis) (a typical example is shown in Fig. 2). This order agrees with that shown by Litasov et al.⁶⁵⁾ as B^C (a-axis) > B^C (b-axis) > B^C (c-axis). In the high-pressure experiments, the decrease of the lattice parameter in c-axis is larger than other axis. This agrees with the results in both References 65) and 59) where B^C (c-axis) is clearly smaller than in B^C (a-axis) and B^C (b-axis).

Regarding the anisotropy of shear modulus, several studies report that C₄₄ is smaller than C₅₅ and C₆₆. Nikolussi et al.⁹⁵⁾ found that the C₄₄ of cementite calculated using first-principles calculation was very small: 1/10 of C₅₅ and C₆₆. To confirm this, experiments were performed using a cementite layer grown on an α–Fe substrate. The cementite layer had a compressive residual stress due to the thermal misfit with the α–Fe substrate, caused by quenching the specimen from the nitrocarburizing temperature to room temperature. The contraction rate of the cementite atomic plane spacing owing to the compressive residual stress was measured by X-ray diffraction. A characteristic hkl-dependence of the stress-induced reflection shifts was observed, that is, the contraction rate is large on the (hkl) planes, where k and l are large, for example, (123), (122), and (031). The observed reflection shifts were found to be in good agreement with the contraction rate of each atomic plane calculated using C_ijs obtained by first-principles calculation. This is experimental proof of the extreme elastic anisotropy of Fe₃C.

On the other hand, when C₄₄ is smaller than C₅₅ and C₆₆, slip is expected to occur in the slip system (010) [001],²⁸⁾ where the ratio of the critical resolved shear stress to shear modulus is the largest. However, the observed slip systems were (001)[100], (100)[010], and (010)[100]⁸⁴⁾ and the observation of the (010) [001] slip system has not been reported. As mentioned above, it remains questionable whether C₄₄ is extremely small because cementite has a large shear strength. One interpretation for the abnormally small C₄₄ obtained by first-principles calculations is that C₄₄ is sensitive to the position of atoms, C₄₄ changes significantly with temperature, and C₄₄ at room temperature may not be as small as that calculated at 0 K.

The Poisson’s ratio ν_ij (where i indicates the tensile direction and j indicates the direction perpendicular to the tensile direction) of the single-crystal cementite in each axial direction was calculated by Ghosh.⁵⁹⁾ According to his calculation, ν₂₃ is the largest (0.4536); while ν₃₁ is the smallest (0.2087). It should be noted that ν_ij≠ν_ji.

Crystal orientation anisotropy also exists in pure iron single crystals. Pure iron has a cubic crystal structure, and there are three independent C_ijs: C₁₁ = 230, C₁₂ = 134, and C₄₄ = 116 GPa (at 300 K).¹⁰¹⁾ From these values, E, G, B, and ν for each crystal orientation of pure iron can be calculated. E is smallest in the [100] direction and largest in the [111] direction (E₁₀₀ = 141 GPa, E₁₁₁ = 293 GPa). G is largest in the [100] direction and smallest in the [111] direction (G₁₀₀ = 121 GPa, G₁₁₁ = 64 GPa). ν is largest in the [100] direction and smallest in the [111] direction (ν₁₀₀ = 0.367, ν₁₁₁ = 0.213). The Young’s modulus of cementite obtained by the first-principles calculations differs by more than five times depending on the crystal orientation;²⁸⁾ while that of pure iron measured is approximately twice. The anisotropy of the elastic constant of cementite is probably larger than that of pure iron.

4.5. Elastic Constants of Isotropic Polycrystalline Cementite Calculated from C_ijs and the Effect of Alloying

The elastic modulus of an isotropic polycrystal can be estimated from the single-crystal C_ijs. The methods used include the Voigt’s method,¹⁰²⁾ Reuss’s method,¹⁰³⁾ and Hill’s method.¹⁰⁴⁾ The Voigt model assumes that each crystal is in the same strain state, and the Reuss model assumes that each crystal is in the same stress state. Hill’s method is an average of Voigt’s and Reuss’s methods. In the Voigt and Reuss models, B and G are given by the following equations:¹⁰⁵⁾   

$$\begin{array}{l}{B}_V= 1/9\left(C_{\mathit{1}\mathit{1}}+C_{\mathit{2}\mathit{2}}+C_{\mathit{3}\mathit{3}}\right)+2/9\left(C_{\mathit{1}\mathit{2}}+C_{\mathit{1}\mathit{3}}+C_{\mathit{2}\mathit{3}}\right)\hfill \\ \begin{array}{l}{G}_V= 1/15\left(C_{\mathit{1}\mathit{1}}+C_{\mathit{2}\mathit{2}}+C_{\mathit{2}\mathit{3}}\right)-1/15\left(C_{\mathit{1}\mathit{2}}+C_{\mathit{1}\mathit{3}}+C_{\mathit{2}\mathit{3}}\right)\\ \hspace{1em}\hspace{1em}+1/5\left(C_{\mathit{4}\mathit{4}}+C_{\mathit{5}\mathit{5}}+C_{\mathit{6}\mathit{6}}\right)\end{array}\hfill \\ 1/B_R=\left(S_{\mathit{1}\mathit{1}}+S_{\mathit{2}\mathit{2}}+S_{\mathit{3}\mathit{3}}\right)+2\left(S_{\mathit{1}\mathit{2}}+S_{\mathit{1}\mathit{3}}+S_{\mathit{2}\mathit{3}}\right)\hfill \\ \begin{array}{l}1/G_R= 4/15\left(S_{\mathit{1}\mathit{1}}+S_{\mathit{2}\mathit{2}}+S_{\mathit{2}\mathit{3}}\right)-4/15\left(S_{\mathit{1}\mathit{2}}+S_{\mathit{1}\mathit{3}}+S_{\mathit{2}\mathit{3}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}+3/15\left(S_{\mathit{4}\mathit{4}}+S_{\mathit{5}\mathit{5}}+S_{\mathit{6}\mathit{6}}\right)\end{array}\hfill \end{array}$$(8)

where S_ij are elastic compliances, and their values can be obtained through an inversion of the elastic constant matrix S = C⁻¹. The subscripts ‘R and ‘V’ represent the Reuss and Voigt formulations, respectively. Hill¹⁰⁴⁾ showed that Reuss’s method gives the lower limit, the Voigt’s method gives the upper limit, and the average value of both is assumed to be the closest to the true value.   

$$\begin{array}{l}{B}_H=\left(B_V+B_R\right)/2\\ G_H=\left(G_V+G_R\right)/2\end{array}$$(9)

E and ν can be calculated by the following equation using B and G.   

$$\begin{array}{l}E=9GB/\left(G+3B\right)\\ \nu =\left(3B-2G\right)/\left(3B+G\right)/2\end{array}$$(10)

The right columns of Table 7 show the elastic constants (E, G, B, and ν) of the polycrystalline cementite calculated by the Hill formulation (The values with * are calculated by the present authors.) Comparing the values calculated by the Reuss and Voigt formulations, the difference between B_Vs and B_Rs is small (less than 0.5%), regardless of the researchers. However, the difference between G_Vs and G_Rs is large, sometimes close to 50%. The difference between the values calculated by the Reuss and Voigt formulations is larger for G than for B, and this is because normal stress is associated with B and shear stress is associated with G. The smaller the value of C₄₄, the larger the difference between G_V and G_R. The magnitude of variation among researchers were compared for the four elastic constants (E, G, B, and ν) of polycrystals calculated by the Hill formulation. For each elastic constant, the mean and standard deviation at 0 K of the FM state are calculated using seven data in Table 7 (excluding the data calculated under the unrelaxed condition, NM state, and negative C₄₄). The mean and standard deviation of the elastic constants are [E] = 204 ± 17 GPa, [G] = 76 ± 7 GPa, [B^C] = 222 ± 11 GPa, and [ν] = 0.346 ± 0.015. The coefficient of variation was smaller in B^C and ν than in E and G. This is thought to be due to the difference in shear stress, and especially the magnitude of C₄₄, among researchers. There are two bulk moduli in the FM state obtained by the first-principles calculations: [B₀^F] = 205 ± 27 GPa and [B^C] = 222 ± 11 GPa. Comparing these two, [B^C] is approximately 8% higher than [B₀^F], and the coefficient of variation of [B^C] was less than half that of [B₀^F]. The reason for [B^C]> [B₀^F] and the coefficient of variation in B^C being smaller than that of B₀^F is probably associated with the magnitude of the applied strain. The applied strain to evaluate C_ijs is usually smaller than that used to evaluate B₀^F, and the stress–strain relation was closer to linear in B^C than that in B₀^F.

The effect of the addition of alloying elements on the elastic constants of single-crystal cementite has been investigated using first-principles calculations.^59,97) Ghosh⁵⁹⁾ calculated C_ijs for Fe₂CrC (4c sites in Fe₃C are occupied by Cr), and Huang et al.⁹⁷⁾ calculated C_ijs for Fe₂MnC (8d sites in Fe₃C are occupied by Mn). These C_ijs and elastic constants are shown at the bottom of Table 7. There are few calculations on the effect of alloying elements on C_ijs. No clear tendencies for the addition of alloying elements to C_ijs are observed presently. Ghosh⁵⁹⁾ calculated the Young’s moduli of Fe₃C and M₃C (e.g., M = Cr, Mn, Mo, V) with a cementite structure using first-principles calculation. Assuming that the Young’s modulus of the alloyed cementite is given by the linear interpolation of the calculated Young’s modulus of Fe₃C and M₃C, the increase in Young’s modulus of Fe₃C with the addition of alloying elements should be in the order Cr > Mn > V > Mo. This is slightly different from the effect of alloying elements on E, which Umemoto et al.⁷⁰⁾ observed in experiments using polycrystalline alloyed cementite, V > Cr > Mn ≈ Mo.

4.6. Variations among Researchers in the Results of First-principles Calculations and Comparison between First-principles Calculations and Measured Values

As mentioned above, there are variations among researchers in the estimated elastic properties of cementite using first-principles calculations, although it is called ‘first-principles’. In the following paragraphs, we will discuss the causes of the variations in first-principles calculations in cementite and examine the differences from the measured values.

“First-principles calculation” is a method to calculate physical properties directly from basic physical quantities without adjusting to experimental data. However, in the actual calculation process, adjusting parameters or modeling are introduced to find agreement between theoretical calculations and experimental results, and the results differed from researcher to researcher. The main reason for this difference is the difference in the calculation method and condition or software. In general, first-principles calculations often refer to those based on density functional theory, in which the total energy of the ground-state electron system is described by the electron density function.¹⁰⁶⁾ Local-density approximation (LDA) and generalized gradient approximation (GGA) are used as exchange correlation terms that represent complex interactions between electrons.¹¹¹⁾ It is worth noting that GGA correctly predicts the FM bcc structure of Fe as its ground state, while the LDA incorrectly predicts its ground state to be a NM close-packed structure. Thus, the calculation by GGA has become the current mainstream method.⁸⁶⁾ In addition, to calculate the total energy of metals and compounds, there is a pseudopotential method that handles only valence electrons and replaces the inner shell electron part with an appropriate pseudopotential and a FLAPW method that handles all electrons.¹⁰⁶⁾ The calculation conditions include the number of atoms used in the calculation, the number of k points (k is the wave number, and k = 2π/λ (λ is the wavelength)), and the cut-off energy. For example, the data shown in Table 7 often use 12 Fe and 4 C as unit cells. In addition, Jiang et al.⁸⁶⁾ used a supercell (an integral multiple of the unit cell to obtain the energy of a bulk solid) containing 128 atoms when using the phonon method. The number of k points used in the calculation of C_ijs of the single crystal shown in Table 7 was in the range of 27 to 1331, and the cut-off energy was in the range of 330 to 1000 eV. This is a characteristic of cementite research—the adoption of a wide range of calculation conditions. In addition, the degree of calculation accuracy and the degree of convergence condition of the calculation have a balance with the calculation time and the calculation cost, and these influence the calculation result. As described above, because of the approximation of the electron density gradient and the selection of various parameters, the first-principles calculations results inevitably have some differences.

The variations in the predicted lattice parameters^{44,86,88,90,91,92,93,95,96)} and elastic constants (Tables 6 and 7) using first-principles calculations were studied. It was found that the variations of the predicted lattice parameters among researchers is less than 1% and the variations of the elastic constants among the researchers is several %. On the other hand, the difference between the measured and predicted values using the first-principles calculations is less than 1% in the lattice parameter and up to 53% in the elastic constant. (As mentioned below in detail, the comparisons of lattice parameters or elastic constants among predicted or measured values and between predicted and measured values are made using coefficient of variation and average value.) In general, the lattice parameters in pure elements can be evaluated with an error of about several percent from the measured value. However, because the elastic constant is a numerical differential of total energy by strain, the error may be about 10%.^107,108,109) It is noted that the difference in the values of lattice parameter of cementite between the predicted and the measured is smaller than those of pure elements, and the difference in the values of elastic constants of cementite between the predicted and the measured is larger than those of pure elements.

Regarding the four elastic constants, a comparison between the elastic constants of polycrystalline cementite calculated from C_ijs obtained by the first-principles calculations and the measured values was carried out using the respective mean values. The mean values of the elastic constants of polycrystalline cementite calculated from the seven data points listed in Table 7 are [E] = 204 GPa, [G] = 76 GPa, [B^C] = 222 GPa, and [ν] = 0.346. The mean values of the measured elastic constants calculated from the data listed in Table 3 (seven data points for E, three data points for G, B, and ν) are [E] = 186 GPa, [G] = 75 GPa, [B^E] = 145 GPa, and [ν] = 0.279. Comparing these data, the mean value estimated by the first-principles calculations is larger than the corresponding measured value; in particular, [B^C] is 53% larger than [B^E]. Compared with the elastic constants of 100% dense cementite estimated by Umemoto et al.⁵⁷⁾ (E (100%) = 189 GPa, G (100%) = 74 GPa, B (100%) = 159 GPa and ν (100%) = 0.301), the value predicted by the first-principles calculations is larger in every four elastic constants. When the [B^C] (calculated from C_ijs obtained by the first-principles calculations) is compared with the experimentally measured [B₀^P], [B^C] (=222 GPa) is 28% larger than [B₀^P] (173 GPa). Regarding the larger value of B predicted by the first-principles calculations than the measured one, one of the possible reasons for this difference in B is that the first-principles calculations is the value at 0 K while the measured value is at room temperature. However, there is an experiment⁹⁸⁾ by NRIXS showing the increase in B with increasing temperature from 0 K, which suggests that further detailed research is needed.

*³  C₆₆ in Table 7 is written as 30 (GPa) in the original paper (TABLE II⁵⁹⁾), but when we confirmed with the author, it was an error of 130 (GPa). If C₆₆ is 130 GPa, the values such as G in Reference 59) can be reproduced, so it is listed as 130 (GPa) in this review.

*⁴  The Young’s moduli of all crystal orientations are shown in Fig. 3 of Reference 59), but it should be noted that the vertical and horizontal axes in Fig. 3 are displayed in reverse.

5. Conclusion

Cementite is used in large quantities as the basic constituent phase of carbon steels and cast iron. Moreover, as a prominent candidate for the Earth’s inner core, it is of interest to geophysicists. However, since cementite is thermodynamically metastable, it is difficult to prepare a large sample with a single-phase cementite without defects. Therefore, many researchers have attempted various methods for preparing samples and measuring their properties. In the research of cementite, various types of cementite samples produced by various preparation methods have been used, and various measurement methods suitable for samples have been employed. In addition, the characteristics of cementite itself change significantly due to magnetic transition, crystal orientation anisotropy, and the influence of alloying elements, which causes a wide spread in the measured data. In this review, the most reliable experimental data were chosen based on the sample preparation method, without alloying elements, and the measurement method (well established with small error) and summarized in Table 1 together with theoretically estimated results. Future studies are expected to be conducted paying attention to the above factors. Moreover, a theoretical approach using first-principles calculations is strongly expected since the sample preparation of cementite single crystals has not been achieved. In recent years, the elastic constant of a cementite single crystal has been obtained by first-principles calculations, and the tendency of the obtained data is close to the measured value. However, the data obtained by the first-principles calculations is a single crystal with a temperature of absolute zero and zero atmospheric pressure, whereas the experimental data are obtained using polycrystalline sample and measured at room temperature under one atm; therefore, it is not possible to compare the experimental and theoretical data directly. To make a strict comparison between the measured and theoretical values, the sample to be used for measuring the elastic constant of cementite must be a single phase, it should be as large as possible, its impurities should be as small as possible, and in the case of sintered compacts, the density should be as high as possible. It is necessary to choose the measurement method with the smallest variation. In the future, a nonequilibrium process suitable for the preparation of cementite samples should be developed to produce larger single crystals, and parameters such as the elastic constants, crystal orientation anisotropy, effects of temperature and alloying elements, will be systematically elucidated by experiments.

Cementite is composed of elements such as iron and carbon, and is made of inexpensive and non-toxic substances; thus, its importance will not change in the future. It will continue to be used in our daily lives, and will continue to be of interest not only in the fields of steel science, but also in the fields of geophysics and astrophysics. It is expected that this review will help those who study cementite.

List of Symbols

B: bulk modulus (when used in a general sense without limiting the material or measurement method)

B^C: bulk modulus obtained from C_ijs calculated by first-principles calculations (0 K, 0 GPa).

B^E: bulk modulus measured from vibration or elastic wave experiments (ambient temperature and pressure).

B^F: bulk modulus calculated from first-principles calculations and equation of state

B₀^F: B^F at 0 K, 0 GPa

B^P: bulk modulus obtained from experiments on volume change by pressure and EOS

B₀^P: B^P at room temperature and 0.1 MPa

B’₀^F: first pressure derivative of B₀^F(=∂B₀^F/∂P)

B’₀^P: first pressure derivative of B₀^P (=∂B₀^P/∂P)

C_ij: single-crystal elastic stiffness constant

E: Young’s modulus

E_tot: total energy of the system

E₀: total energy of the system at equilibrium lattice parameter

G: shear modulus

P: pressure

S_ij: single-crystal elastic compliance constants

T_C: Curie temperature

V₀: volume of system at equilibrium lattice parameter

V_L: longitudinal sound velocity

V_S: transverse sound velocity

V_D: Debye sound velocity

β: thermal softening coefficient,

[E]: mean of E (± standard deviation) (same for G, B, and ν).

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