Modeling and Simulation for Material Selection and Mechanical ...

2 

Design Simulation of Kinetics 

of Multicomponent Grain 

Boundary Segregations in the 

Engineering Steels Under 

Quenching and Tempering 

Anatoli Kovalev and Dmitry L. Wainstein 

Physical Metallurgy Institute, Moscow, Russia 

I. INTRODUCTION 

The basic factors controlling grain boundary segregations (GBS) in engineering 

steels are discussed. In contrast to single-phase alloys, in engineering 

steels, the multicomponent segregation is developed simultaneously 

with undercooled austenite transformations and martensite decomposition. 

Based on these reasons, the influence of steel phase composition and 

kinetics on concurrent segregations is discussed. It is established that grain 

boundary enrichment by harmful impurities (S and P) is possible after carbon 

and nitrogen segregation dissolution. Two models of GBS are 

described. The dynamic model of segregation during quenching is based 

on the solution of independent diffusion and adsorption–desorption equations 

for various impurities in steel. The model of multicomponent segregation 

under tempering considers the influence of alloying and tempering 

parameters on concentration and thermodynamic activity of carbon in 

the a-solid solution. 

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

II. GRAIN BOUNDARY SEGREGATION AND PROPERTIES 

OF ENGINEERING STEELS 

Chemical composition and structure of the grain boundary influences various 

properties of engineering steels. The following are some of these properties: 

inclination to temper and heat brittleness, resistance to hydrogen 

embrittlement, corrosion, delayed fracture, and creep. The intergranular 

fracture is the main reason for decrease of many steel exploitation properties. 

Application of modern physical experimental or calculation methods 

has successfully helped in solving the old metallurgical problem of intergranular 

fracture. 

The affinity of various kinds of intergranular brittleness is associated 

with two main unfavorable factors that decrease intergranular bonds. The 

impurity segregation to grain boundaries (GBS) and localization of internal 

microstresses are necessary and sufficient conditions that could initiate 

embrittlement [1]. 

Despite the common features, certain kinds of steel brittleness are distinguishable 

from each other and are stipulated by complex interaction of 

these factors. The concentration of internal stresses on grain boundaries 

could be an effect of martensite transformation, hydrogen accumulation, 

or carbide precipitation; and grain boundary segregations could appear during 

the equilibrium or non-equilibrium processes of element redistribution in 

steel. 

The concentration of microstresses on grain boundaries cause the 

initiation of cracking and acts as the primary reason for brittleness. The 

enrichment of grain boundaries by harmful impurities is a major and common 

condition for development of various intercrystalline brittleness phenomena 

and it specifies crack propagation entirely along grain boundaries 

at low stresses. 

The concept of intercrystalline internal adsorption [2] that was confirmed 

by theoretical [3], and experimental work [4], the thermodynamic analysis 

of chemical element interaction during equilibrium grain boundary 

segregation [5], and investigations of quenched and tempered steel [6,7]. This 

made it possible to interpret the tempering embrittlement phenomenon. 

Elemental impurities enrich grain boundaries in thin layers up to 

several atoms and change the type and value of interatomic bonds that 

lead to intercrystalline fracture. Embrittlement power is commonly 

attributed to the elements of the 3rd to 5th periods of groups IV to 

VI in the periodic system [1]. Sulfur, phosphorus, arsenic, selenium, tellurium, 

antimony, bismuth, and oxygen are the most harmful impurities 

that segregate in grain boundaries. The concentration in grain boundaries 

could reach several atomic percentages exceeding the volume one 


y several hundreds. Intercrystalline brittleness, as caused by GBS, due 

to harmful impurities is observed, as a rule, in BCC metals and 

alloys. The austenite alloys are significantly more resistant to this kind 

of fracture. 

The motonic increase of plasticity that is expected after martensite 

decomposition due to tempering of engineering steels is disturbed by two 

anomalies resulting in a relative decrease of impact strength. These anomalies 

are accompanied by intergranular fracture. Steels may be susceptible to 

embrittlement when they are heated for prolonged period in the temperature 

range 350–5508C, or when slowly cooled through it. Depending on the heattreatment 

cycle, the phenomenon is called either temper embrittlement (3508 

embrittlement) or reversible temper embrittlement (5508 embrittlement). 

Common indications of embrittlement are a loss of toughness, segregation 

of harmful impurities to grain boundaries and the fracture path usually 

along prior austenite grain boundaries, and the impact transition temperature 

(FATT—the fracture appearance transition temperature) which is displaced 

towards higher values. 

The irreversible temper embrittlement of low-alloyed steels (

Figure 1 Impact strength at room temperature of samples with V-shape cut for 

several melts of industrial steel 4340 after quenching (8508C, 1 hr) in oil and 1 hr 

tempering at various temperatures. (From Ref. 8.) 

Figure 2 (a) Lamellar parts of cementite Fe 3C on primary austenite grains and (b) 

martensite packs boundaries. Steel 0.35C–1.5Mn–0.1P. Tempering at 3508C, 1 hr 

(TEM, replicas). 


Figure 3 (a) Brittle fracture through primary austenite grains and (b) martensite 

pack boundaries in steel 0.35C–1.5Mn–0.1P. Tempering at 3508C, 1 hr (SEM). 

deformation and, consequently, formation of grain boundary cracks. The 

brittle crack develops, in this case, within boundaries of primary austenite 

grains and martensite packs (Fig. 3a, b). The significant enrichment of grain 

boundaries by phosphorus is confirmed by Auger spectroscopy. The decisive 

role of austenitization when compared with tempering in achieving equilibrium 

GBS is shown by low diffusion mobility of harmful impurities’ atoms 

(P, As, S, and Sb) for medium tempered steel. Calculations show that only 

nitrogen has significant diffusion mobility in this temperature range which is 

sufficient for diffusion at a distance of 10 mm for 1 hr at 3508C [1]. The 

temperature of steel for quenching is sufficiently high for intensive diffusion 

of P in austenite with the formation of equilibrium GBS [9], and quenching 

fixes this enrichment. The level of P segregation depends on the austenitization 

temperature and increases when the temperature decreases below 

10508C. This is related to the decrease of P solubility in austenite. It is confirmed 

by a significant decrease of the quenched steel delay fracture resistance 

with respect to temperature in the austenite region (Fig. 4). 

Phosphorous content in steel influences its embrittlement at 3508C tempering. 

Its concentration in GBS is several hundred times higher than in the 

volume, and this harmful element segregates in austenite even at low concentrations, 

about 0.01% mass, leading to a significant decrease in the 

impact strength (Fig. 5). 

Change of the relative energy of grain boundaries results in segregation 

enrichment by impurities during austenitization. The interferometry is 

one such direct technique for grain boundary energy determination. For this 


Figure 4 Delayed fracture diagram of quenched steel 0.35C; 1.5Mn; 0.1P after 

austenitization at 1423K, 3 hr and 30 min interim cooling to temperatures: 1, 1123K; 

2, 1223K; 3, 1273K and cooling in water. (From Ref. 10.) 

purpose, the opening angle of GB slot (y) is determined on metallographic 

grinds after vacuum etching at various temperatures. The grinds are austenitized 

in vacuum at high temperature, then subjected to interim cooling to 

the desired temperature and then quenched at a high cooling rate. The relative 

energy of GB is calculated by the equation 

gb ¼ 2cos 

gs y 

ð1Þ 

2 

where symbols b and s correspond to boundary and bulk, respectively. 

Figure 6 shows the change of relative surface energy with respect to the 

interim cooling temperature during austenitization and P content in the 

0.35% C, 1.5% Mn. For the steel with low phosphorus content, the GB 

energy regularly decreases slowly as the temperature increases within the 

austenite region. Significant segregation enrichment of grain boundaries 

by phosphorus in the steel with 0.1% P does not show this dependence 

and decreases the surface energy of GB with decreasing of the temperature. 

The decrease of P solubility in austenite with decreasing temperature assists 

its adsorption on the GB and decreases significantly its surface tension 

energy. At the same time, one can observe another process: P enriches 


Figure 5 Temper embrittlement of steel Fe–0.3C–3.5N–1.7Cr: 1, high purity steel; 

2, 0.01 mass% P; 3, 0.03 mass% P; 4, 0.06 mass% P. (From Ref. 8.) 

non-metallic inclusions. The dependence of P content on Mn sulfides in steel 

0.35% C, 1.5% Mn, 0.1% P from austenitization temperature is shown in 

Fig. 7. When annealing temperature decreases, P redistributes between the 

bulk and grain boundaries enriching them and the non-metallic inclusions. 

Steel grain size reduction can decrease significantly its tendency to temper 

embrittlement. Increase of the specific surface of the GB at dispersion of 

the steel structure decreases the concentration of the impurity in the grain 

boundary that leads to growth of steel brittle fracture resistance (Fig. 8) [3]. 

Phosphorous segregations decrease the surface energy of intergranular cohesion. 

Using the approach proposed in Ref. [11], one can estimate the role of 

phosphorus in change of surface energy of intergranular cohesion for development 

of this kind of embrittlement. 


Figure 6 Change of relative surface energy of grain boundaries in dependence of 

temperature of austenitization. Fe–0.35C–1.5Mn steel: 1, 0.03% P; 2, 0.1% P. 

Figure 7 Phosphorus concentration in manganese sulfides in 0.35C–1.5Mn–0.1P 

steel vs. austenitization temperature. 


Figure 8 FATT (T50) of steel vs. grain size d 1=2 and specific grain boundary surface 

P S. Phosphorus content: 1, 0.03%; 2, 0.1%. (DT50 ¼ T50(0.1% P) T50(0.03% P)). 

Influence of grain size on ductile–brittle transition temperature FATT 

is determined from the well known Petch–Hall equation 

sf ¼ so þ Ky d 1=2 

where Ky ffi aðGgÞ 1=2 ; a is a constant (1–3); g is the surface energy of intergranular 

cohesion for generation of cracks on grain boundaries; G is the 

shear modulus. 

It is described by the equation 

dT50 

d d 1=2 

1 

¼ 

ð Þ b 

where b is determined as a tangent of inclination angle of straight lines in 

Fig. 8. Taking into account that 

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved. 

ð2Þ 

ð3Þ

dT50 

d d 1=2 

ð Þ 

dT 

¼ ð K 

dsf 

0 yÞ; ð4Þ 

it is possible to use the change of the inclination angle of lines 1 and 2 in 

Fig. 8 to determine the influence of P on GBS by the coefficient b and 

the effective intergranular cohesion surface energy g that is proportional 

to b 2 . The values of b for steel samples containing 0.03% and 0.1% P at 

the temper embrittlement state are equal to 0.54 and 0.22 sec, respectively. 

Taking into account dependencies (2) and (3), one can conclude that effec- 

tive intergranular cohesion surface energy in steel with higher P concentra- 

¼ 6:04 times. 

tion decreases in g 0:03=g 0:1 ¼ b 2 

0:03 =b2 

0=1 

B. Ductile Intergranular Fracture of Overheated Steel 

The ductile fracture of steel as well as brittle fracture could be characterized 

by the lowest energy capacity. Such a fracture occurs when the inclusions are 

located along grain boundaries occupying a very large volume near the 

boundary. The intergranular microvoid fracture is observed in this case 

due to overheating of the steel. The samples are exposed to high temperature 

heating to dissolve inclusions. As a rule, the large and lamellar oxysulfides 

that did not embrittle steel are dissolved. After their dissolution, segregation 

of O, S, P, and precipitation, disperse particles during steel cooling occurs. 

In low-alloyed steels, precipitates could be sulfides of chromium and manganese 

(MnS, CrS), and aluminum nitride, AlN. These particles build a 

dense network on grain boundaries. The fracture occurs at higher or room 

temperatures by intergranular microvoid coalescence at low stress intensity. 

The micrometer scale cavities nucleate on the intergranular fine dispersion 

of sulfide or nitride particles (Fig. 9). The segregation of harmful impurities 

is observed on grain boundaries in this case. 

C. Reversible Temper Embrittlement 

The reversible temper embrittlement (RTE) is observed in engineering steel 

alloyed by carbide-forming elements after quenching and high tempering 

(500–6008C). This phenomenon is developed in steels of industrial purity. 

It consists of a large decrease in the steel impact strength after slow cooling, 

but after rapid cooling at 6508C, the steel has a standard impact strength. 

The RTE phenomenon was identified for the first time in 1883 [12], when 

blacksmiths observed that some steels had to be water quenched after tempering, 

to avoid embrittlement. This decrease of impact strength is not 

accompanied by a change of physical or other mechanical properties of 


Figure 9 Ductile fracture of steel through grain boundaries (SEM). 

steel. The duration of the exposure time within a definite range of tempering 

temperatures (brittleness zone) plays a decisive role in the development of 

RTE. The brittle fracture goes through the primary austenite grains 

(Fig. 10). The duration of the exposure time of the normalized or annealed 

steel within the dangerous temperature range also leads to this kind of 

embrittlement. Steels sensitive to RTE are subjected to rapid cooling from 

6508C during different heat treatments. 

The GBS of phosphorus is the main reason for this kind of brittleness. 

The RTE of alloyed steels is mainly sensitive to two factors: the chemical 

composition of the grain boundaries, and the mechanical and microstructural 

parameters of the alloy. The direct correspondence of embrittlement 

kinetic features and GBS of P at steel tempering has been established. Figure 11 

shows the iso-FATT curves for temper embrittlement of Ni–Cr steel [13]. 


Figure 10 SEM image of intergranular fracture of Fe–0.35C–1.5Mn–0.1P steel 

after quenching with temperature of 9508C, 1 hr and tempering at 5508C, 2 hr. 

At a constant temperature of embrittlement tempering, the FATT increases 

with time. The embrittlement is reversible. It can be rejected by short heating 

above the nose of the C-curve in the ferrite range. Renewed aging and slow 

cooling of a de-embrittled steel in the critical temperature region gives reembrittlement. 

These processes are accompanied by a redistribution of 

impurities on the grain boundaries. 

III. FACTORS DETERMINING MULTICOMPONENT 

INTERFACE ADSORPTION IN ENGINEERING STEELS, 

AND THE METHODS OF ITS CALCULATION 

The intercrystalline internal adsorption, or grain boundary segregation 

phenomenon, means the increased concentration of small impurities on 


Figure 11 Diagram of time–temperature FATT, 8C: (1) 60; (2) 55; (3) 50; (4) 

45; (5) 40; (6) 35; (7) 30; (8) 25; (9) 20; (10) 15; (11) 10; (12) 5; (13) 0; 

(14) þ5; (15) þ10; (16) þ15; (17) þ20; (18) þ25; (19) þ30. 

the grain boundary (compared to bulk) is caused by decreasing boundary 

energy. The energy of impurity segregation corresponds to energy gain in 

the ‘‘bulk-boundary’’ system that accompanies the transition of one impurity 

atom from bulk to boundary. Thermodynamic description of this phenomenon 

is based on the Gibbs theory of the equilibrium segregation on 

free surface. The decrease of redundant energy of GB is the thermodynamic 

stimulus to change its chemical composition compared to bulk. The impurities 

that decrease the energy of interfaces are surface-active. These elements 

could form equilibrium segregations under favorable conditions. 

The impurities that increase surface tension escape from surface. Gibbs’ 

adsorption isotherm for grain boundary segregation in a solid binary system, 

where the matrix obeys Raoult’s law and Henry’s law of diluted solutions, 

may be expressed 

Gb ¼ Xc dgb kT dXc 

where Gb is the surplus concentration on the GB, mol=m 2 ; Xc is bulk mole 

part of impurity; dg b=dX c is the tendency to adsorb. 


ð5Þ

McLean [14] has developed a statistical model of intercrystalline internal 

adsorption. The main points of this theory are the following: 

1. A definite number of adsorption centers (one monolayer) are 

present on the GB. They have equal adsorption potential. 

2. The impurity atoms are adsorbed independently on each of the 

centers. 

3. The adsorption decreases at temperature growth. 

This model adequately describes the GBS process for two-component 

system: 

X 0 b 

Xb 

Xb 

¼ Xc 

exp 

1 Xc 

DG 

RT 

where Xb and Xc are the impurity concentrations on boundary and in bulk, 

respectively; X 0 b is the ultimate equilibrium concentration of the impurity 

on boundary; DG is the segregation energy. 

Hondros and Seah [15] have established the interrelation of GB enrichment 

and solubility limit X 0 c 

X 0 b 

Xb 

Xb 

¼ Xc 

X0 exp 

c 

DG 

RT 

The enrichment factor, determined as the GB content=bulk concentration 

ratio, is of the order of magnitude 10 4 and 10 1 for impurity and alloying elements, 

respectively (see Fig. 12). 

A. Binding Energy of Impurities with Grain Boundaries 

One can determine the adsorption energy as an alteration of the system free 

energy during transition of the dissolved atom from the grain bulk to boundary 

E ¼ðEb E 0 b Þ ðEc E 0 c Þ ð8Þ 

where Eb and Ec are the free energy of the system with impurity atoms on the 

grain boundary or in the bulk, respectively; Eb 0 and Ec 0 are the free energies 

of boundary and bulk in the pure solvent. 

The value of energy E is tied with the elastic (dimensional) and chemical 

interaction of impurity with boundary. 

The influence of dimensional discordance of atoms is accounted by the 

Eshelby equation [16] 

E ¼ 16 

3 pGr3 ðs 1Þ 2 

where G is the shear modulus of the solvent; r is the solvent atom radius; s is 

the ratio of impurity and solvent atoms radii. 


ð6Þ 

ð7Þ 

ð9Þ

Figure 12 Calculated and experimental enrichment coefficients for surface 

segregations. (From Ref. 15.) 

The decrease of elastic distortion energy during the transition of the 

impurity atom from ideal bulk lattice to distorted boundary lattice is the 

driving force adsorption. 

According to Ref. [17], one can describe the chemical part of adsorption 

Ech ¼ DZðe12 e11Þ ð10Þ 

where DZ is the difference of the atom coordination numbers in bulk and 

on boundary; eij is the energy of interaction between the nearest neighbors 

(1 for the solvent, 2 for impurity). 


The Fowler–Guggenheim theory [18] accounts for the interaction of 

atoms adsorbed on GB: 

X 0 b 

Xb 

Xb 

¼ Xc 

X0 exp 

c 

E1 ZoXb=X0 b 

KT 

ð11Þ 

where Z is the coordination number for impurity atoms; o is the interaction 

energy of nearest atoms. 

The portion of segregation energy ZoXb=X 0 b corresponds to interaction 

of the same and different atoms on boundary, and depends on the 

degree of neighbor centers filled. At o > 0, the atoms repulse mutually, 

and they attract at o < 0. The attraction increases the adsorption energy 

substantially. Interaction of atoms on interface influences segregation 

enrichment and promotes formation of 2D phases with ordered atomic 

structure. 

B. Impurities’ Concurrence During Adsorption 

None of the existing adsorption theories adequately describe the micromechanism 

of impurities’ concurrence on the adsorption centers. This is related 

to the peculiarities of adsorption from gaseous phase to the free surface to 

describe the grain boundary segregation mechanism [19–21]. According to 

this point of view, all segregating elements (for example N, C, S, and P) 

occupy equal positions on GB, described by the ‘‘site competition’’ term. 

The peculiarity of GBS formation consists of diffusion of alloying elements 

and impurities from bulk to interface. The migration mechanisms for substitial 

and interstitial impurities are different. The reason for this is that the 

adsorption centers on interface are different for these two kinds of impurities. 

Therefore, the interstitial impurities (C, N) are located in interstices, but 

S, P, Sb, Bi, etc. occupy substitution position on the GB. Adsorption of any 

surface-active impurity on GB decreases its free energy and lowers the thermodynamic 

stimulus for adsorption of other impurity in a similar way. This 

causes a site competition between atoms on the GB [22]. 

The concurrence of impurities A and B segregating at the same temperature 

and occupying the same positions in crystalline lattice (lattice 

points or interstices) with different binding energies to GB was investigated 

in Ref. [23]. 

The equilibrium concentration of competing impurities A and B could 

be calculated using equations: 

X A b ¼ 

XA C expðEAseg =kTÞ 

1 þ XA C expðEAseg =kTÞþXB C expðEBseg =kTÞ 


ð12Þ

X B b ¼ 

XB C expðEBseg =kTÞ 

1 þ XA C expðEAseg =kTÞþXB C expðEBseg =kTÞ 

ð13Þ 

where Eseg is the segregation energy; Xb I is the GB concentration of element I 

(atomic fraction). As one can see from these equations, at EA seg > EB seg , the 

concentration of element A decreases with increasing temperature. 

In this case, the adsorption level of the element B reaches its maximum 

at critical temperature: 

E A seg 

Tcr ¼ 

k lnðEA seg =ðEBseg EA seg ÞXA b Þ 

ð14Þ 

The grain boundaries are enriched in this case by element B at low 

temperatures, and by element A at high temperatures. 

C. Thermodynamic Calculations of the 

Segregation Energy 

The segregation energy calculations are based on various models of solid 

solution electronic structure or quasi-liquid model of grain boundary. 

Thermodynamic properties of the solid solution determine firstly the 

surface activity of small impurities in the formation of equilibrium GBS. 

The components of the alloy influence the energy of interaction with grain 

boundaries significantly. The parameters of such interaction are determined 

by thermodynamic calculations, phase equilibrium diagram analysis, computer 

modeling of GBS. The heat of solution of different atoms in solid 

solution is the thermodynamic measure of their interaction. 

The model establishing interrelationship segregation energy and heat 

of solutions is proposed in Ref. [24] 

Eseg ¼ FHsol Pðg A g BÞV 2=3 

A 

ð15Þ 

where F and P are empirical coefficients; H sol is the heat of solution of A in 

B [25–30]; g A, g B is the surface enthalpy of elements A and B [31,32]; V A is 

the molar volume of A. 

Using the liquid grain boundary model approximation, the segregation 

energy of impurities (Eseg) could be determined from an analysis of 

the solidus and liquidus curves on phase equilibrium diagrams: 

Eseg ¼ KT LnðK0Þ ¼ KT LnðCL CSÞ ð16Þ 


where K0 is the coefficient of equilibrium distribution of the element 

between solid and liquid phases [33]; T is the melting temperature of the 

pure solvent; CL and CS are concentrations of impurity in the liquid and 

solid solutions, respectively. 

An experimental method for determination of binding energy of 

impurity atoms to grain boundary is used. 

The analysis of large number of phase equilibrium diagrams has led 

to the establishment of the basic property of two-component solid solutions 

consisting of periodic variation of the segregation formation energy 

of an element as a function of its location in the periodic table (atomic 

number). As seen in Fig. 13, the impurities could have positive or negative 

surface activity, or be neutral. The elements So, Mo, Ni, and Co are 

neutral in two-component alloys with Fe. At the same time, it is well 

known that molybdenum is the surface-active element in steels and it 

reduces the tendency of steel to the reversible temper embrittlement. This 

change of surface activity is observed only in multicomponent alloys, and 

it is due to the mutual influence of elements on its thermodynamic activity. 

The binding energy of an impurity to the GB depends significantly on 

boundary structure. The wide spectrum of Eseg exists for the given substance 

analogous to the spectrum of the GB energy. This circumstance 

explains the wide dispersion of the segregation energy for various impurities 

that are listed in literature sources. Based on this reasoning, it is 

useful, for segregation modeling, to apply the unified approach for determination 

of the generalized characteristic of definite impurity segregation 

in definite solvent. The thermodynamic calculations of segregation energy 

are the most suitable way for its estimation. Auger electron spectroscopy 

(AES) for investigation of segregation kinetics on the free surface of polycrystalline 

foils is a reliable experimental technique for the averaged E seg 

determination. 

The part of elastic and chemical interaction in GBS process could be 

estimated experimentally based on concentration dependencies of segregation 

energy. These dependencies were determined for alloys whose compositions 

are listed in Table 1. 

The segregation energy was determined based on AES of equilibrium 

free surface segregations of phosphorus. The polycrystalline foil samples 

were tempered at 823K for 4 hr in a work chamber of electron spectrometer 

ESCALAB MK2 after quenching from austenitization temperature of 

1323K. The segregation energy of P was determined using Eq. (6) based 

on surface and bulk impurity concentration. Figure 14 shows the dependence 

of the P segregation energy and its bulk content in alloy. For the 

diluted solid solutions, Eseg is independent of concentration or temperature. 

It is caused only by elastic distortions that are formed by impurity atoms in 


Figure 13 Change of calculated E seg of impurities in Fe-base alloys in accordance 

to its number in periodic system. Calculations were based on Hsol in the following 

publications: (a) Refs. 25 and 26; (b) Refs. 27 and 28; (c) Ref. 33. 


Table 1 Composition (at.%) of the Fe–P and Fe–P–Mo Alloys 

Chemical composition, at.% 

C S P Mo 

0.01 0.002 0.017 0 

0.01 0.003 0.10 0 

0.01 0.002 0.15 0 

0.01 0.003 0.093 3.1 

0.01 0.002 0.033 3.1 

0.01 0.002 0.14 3.1 

0.01 0.005 0.09 0.3 

0.01 0.014 0.074 0.02 

bulk and on the interface. As seen in Fig. 14, the elastic interaction energy of 

the P atoms with grain boundaries in iron is equal to 0.53 eV=at and 

decreases significantly at molybdenum alloying to 0.24 eV=at in the alloy 

Fe–3.1at.% Mo. Decrease of segregation energy of the impurity at its 

Figure 14 Change of E seg of phosphorus with its volume concentration in Fe (1) 

and Fe–3.1 at.% Mo–P alloys (2). Auger electron spectroscopy of free surface 

segregations at 823K. 


volume concentration growth is caused by chemical pair interaction of the 

atoms in alloy. 

Using the example of the Fe–P system, we could determine chemical 

interaction of elements by applying the approach proposed in Ref. [34]. 

Analyzing the solidus and liquidus equilibrium (volume and GB) on the 

equilibrium phase diagram at three temperatures permits the construction 

of a system of three equations that describe this equilibrium 

100 Xs 

kT qa ln 

100 Xl 

¼ X 2 s W0 X 2 l W00 þ kqaTa ð17Þ 

where k is the Boltzmann constant; T a is the melting temperature of 

Fe; qa is melting entropy per atom divided by Boltzmann constant; 

W 0 and W 00 are the mixing energies in solid and liquid states; Xs and 

Xl are the impurity concentration in solid and liquid phases at the temperature 

T. 

Solving these equations for the phase diagram of Fe–P binary system 

[35], the sign and value of mixing energy in liquid phase equal 

0.425 eV=at were determined. The positive value (in accordance with physical 

sense) means that binding force of P–P and Fe–Fe atoms is higher 

than for Fe–P atoms: 

W ¼ WFe P 

1 

2 ðWFe Fe þ WP PÞ ð18Þ 

emphasizing the tendency for solid solution tendency for stratification or 

intercrystalline internal adsorption. 

D. Effect of Solute Interaction in Multicomponent System 

on the Grain Boundary Segregation 

Guttman has expanded the concept for synergistic co-segregation of alloying 

elements and harmful impurities at the grain boundaries. His theory is 

very important for analysis of steels and alloys that contain many impurities 

and alloying elements. In accordance with the theory, the interaction 

between alloying elements and the impurity atoms could be estimated from 

enthalpy of formation of the intermetallic compounds (NiSb, Mn2Sb, Cr3P, 

etc.). The alloying elements could influence on the solubility of impurities in 

the solid solution. Only the dissolved fraction of the impurity takes part in 

the segregation [36]. When preferential chemical interaction exists between 

M (metal) and I (impurity) atoms with respect to solvent, the energy of 


segregation becomes functions of the intergranular concentrations of I 

and M: 

DGI ¼ DG 0 I 

DGM ¼ DG 0 M 

þ bb 

MI 

C b Yb M 

þ bb 

MI 

a b Yb I 

b a 

MI 

C a Xa M 

ð19Þ 

b a 

MI 

a a Xa I ð20Þ 

where C b and a b are the fractions of sites available in the interface for I and 

M atoms, respectively ðab þ Cb ¼ 1Þ; Y b is the partial coverage in the interface; 

X a is the concentration in the solid solution a; bMI is the interaction 

coefficient of M and I atoms in a-solid solution (a) or on the grain boundary 

(b). For a preferentially attractive M–I interaction, the bMIare positive and 

the segregation of each element enhances that of the other. If the interaction 

is repulsive, the bMI are negative and the segregations of both elements will 

be reduced. For a high attractive M–I interaction in the a-solid solution, the 

impurity can be partially precipitated in the matrix into a carbide, or intermetallic 

compound. The interface is then in equilibrium with an a-solution 

where the amount of dissolved I, XI a , may become considerably smaller than 

its nominal content. 

In the ternary solid solutions, the segregation of impurity (I) could be 

lowered or neglected at several critical concentrations of the alloying 

element (M) whose value (C M a) depends on surface activity of each compo- 

I,M 

nent (ESeg ) and interaction features of the dissolved atoms (bMI): 

C M a ¼ 

E I Seg 

bMIðexpðEM Seg =RTÞ 1Þ 

ð21Þ 

The critical concentration of alloying element is accessible for segregation of 

impurity and alloying element E I;M 

Seg > 0 and repulsion of different atoms 

bMI > 0; or without segregation of alloying element EM Seg < 0 and with 

attraction of different atoms bMI < 0. 

I,M 

In this case, the dependence of ESeg on the dissolved element concentration 

is not taken into account. Indeed, for systems with limited solubility, the 

alteration of value and sign of segregation energy is possible at a definite content 

of alloying element. The phase equilibrium diagram analysis allows the 

determination of mutual influence of components on their surface activity. 

The equilibrium distribution of solute elements between solid and 

liquid phases in iron-base ternary system (distribution interaction coefficient 

K0) is known to be an important factor in relation to microsegregation during 

the solidification of steels. As it was shown above, these analogies 

are useful for the prediction of GBS and for impurity segregation energy 


Figure 15 Change of the equilibrium distribution coefficient of some elements with 

carbon concentration in Fe–C-based ternary systems. (From Ref. 37.) 

determination in the given solvent. The K0 of some elements, especially in 

multicomponent systems, is considered to be different from those in binary 

systems because of the possible existence of solute interactions, but the 

mechanisms are so complicated that detailed information has not yet been 

obtained. Therefore, it would be very useful if the effect of an addition of 

an alloying element on the distribution could be determined by the use of 

a simple parameter. 

Equilibrium distribution coefficient K0 1 of various elements in Fe–C 

base ternary system is calculated from equilibrium distribution coefficient in 

iron-base binary systems [40–43]. In Fig. 15, the calculated results are compared 

with the measured values by various investigators. The changes of the 

K 0 1 of P and S with various alloying elements are shown in Fig. 16(a, b) in 

Fe–P and Fe–S base ternary system, respectively. 

These data could be applied for calculation of phosphorus segregation 

energy change under the alloying element influence in Fe–Me–0.1at.% P 

alloys (Fig. 17) or for calculation of the segregation energy change of 

alloying elements with concentration of carbon in Fe–0.1Me–C alloys 

(Fig. 18). For the growth of carbon volume content, the segregation energy 

of C and P decreases which means lowering of the segregation stimulus for 

these elements. 


Figure 16 (a) Change of the equilibrium distribution coefficient of phosphorus 

with the concentration of alloying elements; and (b) change of the equilibrium 

distribution coefficient of sulfur with the concentration of alloying elements. Solid 

line: a-phase. Chain line: g-phase. (From Ref. 38.) 

E. Kinetics of Segregation 

The existing models of multicomponent adsorption do not analyze in detail 

the kinetics of the process. But in reality, GBS forms only during a limited 

time of the heat treatment process. The difference of segregation level from 

the equilibrium one depends on temperature and time. At low temperatures 

and limited time of heat treatment, segregation is controlled by diffusion. As 

the temperature increases, segregations with lower equilibrium concentra- 


Figure 16 (Continued) 

tion are developed, but rich segregations dissolve. Distinguishing diffusion 

mobility and mutual influence of elements on their diffusion coefficients 

determines much of their segregation ability. Amplification or suppression 

of adsorption could be due to a kinetic factor. This peculiarity determines 

the fundamental factor of distinguishing adsorption from gas phase to free 

surface when comparing it to intercrystalline internal adsorption: GBS is 

controlled by diffusion during heat treatment of steels and alloys. 

Many GBS features in multicomponent systems cannot be predicted 

adequately using the equilibrium segregation thermodynamic accounting 

basis. Particularly, the thermodynamic concept of the cooperative (synergis- 


Figure 17 Change of E seg of phosphorus with the concentration of alloying 

elements in Fe–Me–0.1% P alloys. 

tic) adsorption of elements is disturbed when they do not segregate at the 

same temperature. The concurrence of impurities at GBS could be tied not 

only due to their attractive or repulsive interaction, but also with higher diffusion 

mobility of some impurities. In many cases, the determination interatomic 

interaction on grain boundary that is proposed in Guttmann’s 

theory has a significantly lower effect for segregation prediction than accounting 

of mutual influence of elements on their thermodynamic activity in the 

grain bulk. 

Mutual influence of the alloy components on their surface activity is 

caused by their interaction in solid solution in the bulk. The interaction 

on grain boundaries could be analyzed only for those elements that 

segregate in near temperature ranges. Many postulates of the thermodynamic 

theory of equilibrium grain boundary segregation could not be 

applied simply for heat treatment of multicomponent alloys. This is especially 

important for steels, which have complex phase transformations during 

treatment that accompany change of the solid solution composition. 

Auger electron spectroscopy permits the investigation of multicomponent 

adsorption kinetics. The composition of grain boundaries on the intercrystalline 

fracture surface made under high vacuum is analyzed for this purpose. 

In these cases, the experimental modeling of GBS is widely used. 

The chemical composition of free surface of thin poly-crystalline foils that 

are heated in situ is investigated using Auger electron spectrometers. The 

P grain boundary adsorption isotherms for samples of three Fe–Cr–Mn 


Figure 18 Change of Eseg of alloying elements (Me) with the concentration of 

carbon in Fe–0.1% Me–C alloys. 

steels after quenching from 1273K and tempering at 923K for 25 min, 1 and 

2 hr with air cooling are presented in Fig. 19. Dissolution of Ti and V carbonitrides 

after steel quenching promotes enrichment of the solid solution 

by these elements. They have high values of Gibbs energy for phosphide formation, 

decrease the thermodynamic activity of phosphorus in solid solution 

and reduce its GBS. 

Most models of kinetics are classically analyzed in terms of the law 

derived by McLean [14] for binary alloys 

XbðtÞ Xbð0Þ 

Xb Xbð0Þ 

4Dit 

¼ 1 exp 

ðXb=Xa i Þ2d2 " # pffiffiffiffiffiffiffi 

2 Dit 

erfc 

ðXb=Xa i Þ 


ð22Þ

Figure 19 Kinetics of P GBS in steel 0.3C–1.6Mn–0.8Cr–008P (1) with adds of 

0.047Ti (2) or (0.07Ti and 0.026V) (3), quenched from 1273K and tempered at 923K. 

where Xb(t) is the interfacial coverage of element, at time t; Xb(0) — is its 

initial value and Xb its equilibrium value as defined by Eq. (7); Xi a —is 

its volume concentration; Di is the bulk diffusivity of i and d is the interface 

thickness. 

a 

Assuming Xb=Xi ¼ const, using Laplace transformation for (22), one 

can obtain the approximate expression 

XbðtÞ Xbð0Þ 

Xb Xbð0Þ ¼ 2Xa i 

Xbd 

rffiffiffiffiffiffiffiffiffi 

FDti 

p 

ð23Þ 

where F ¼ 4 for grain boundaries and F ¼ 1 for free surface. 

The kinetics of segregation dissolution could be described by these 

equations (22) and (23). But, in this case, the variables Xb(0) and Xb 

exchange places. The influence of Mo, Cr, and Ni additions on kinetics of 

P segregation has been studied in six Fe–Me–P alloys, whose base compositions 

are listed in Table 1. These materials were austenitized for 1 hr at 

1323K and quenched in water. The tempering of foils at 773K was carried 

out in a work chamber of an electron spectrometer ESCALAB MK2 (VG). 

The kinetics of P segregation studied for Fe–Me–P alloys (Figs. 20–22) show 

that equilibrium is reached within several hours. Based on the starting position 

of adsorption isotherms, the phosphorus diffusion coefficients in these 

alloys were calculated using Eq. (22). The data are presented in Table 2. 

Molybdenum reduces significantly P surface activity and decelerates its 

diffusion. Nickel is not a surface-active element in carbonless alloys, Fe– 

P–Ni. It increases sharply P thermodynamic activity and equilibrium GB 

concentration, and accelerates its diffusion. Chromium segregates poorly 


Figure 20 Kinetics of P segregation on free surface in Fe–P–Mo alloys with 

different relative concentration of Mo=P at 773K. 

in these alloys. It also, as Ni, increases diffusion mobility of P and its grain 

boundary adsorption. 

The adsorption isotherms of various elements have non-monotonous 

shape in multicomponent alloys. The isodose thermokinetic diagrams 

present the averaged information on segregation of all components. Such 

Figure 21 Kinetics of P and Cr segregation on free surface in Fe–0.04P–2.3 Cr 

alloy at 773K. 


Figure 22 Kinetics of P segregation on free surface in Fe–P–Ni alloys with 

different relative concentration of Ni=P at 773K. 

diagrams for steel with 0.2–0.3% C alloyed by Cr, Mo, Ni, Mn, V, and Nb 

are presented in Figs. 23–27. 

Chemical composition of steel is listed in Table 3. The T–t diagrams 

are obtained based on adsorption isotherms on free surface of foils that were 

heat treated in Auger spectrometer ESCALAB MK2 at vacuum about 

10 10 Torr. The isodose curves characterizing time of definite segregation 

level access depending on temperature are shown in these diagrams. At elevation 

of an isothermal exposition temperature, the mobility of impurities 

increases, and time for reaching of definite segregation level decreases. 

The lower branch of isodose curve means decrease of segregation formation 

Table 2 Composition (at.%) of the Fe–Me–P Alloys and Kinetics Characteristics 

of P Free Surface Segregation, Deduced from the Segregation Kinetics 

Chemical Composition, at.% 

P Mo Ni Cr 

Surface 

activity X b=X i a 

Bulk diffusivity of 

P DP 10 18 (m 2 =sec) 

0.07 0 0.9 0 285 5.34 

0.03 0 3.1 0 2530 1048 

0.04 0 0 2.3 1450 160 

0.16 1.0 0 0 380 27 

0.21 2.1 0 0 250 2.28 

0.15 3.1 0 0 80 0.3 


Figure 23 The isodose C-curves of multicomponent interface segregation in 0.3C– 

Cr–Mo steel (see Table 3). Auger electron spectroscopy of free surface segregations. 

time at increasing temperature. With temperature increase, the solubility of 

impurity in solid solution increases, and its GB concentration reduces. It follows 

that the probability to form the segregation with high impurity content 

reduces, and time for such segregation increases extensively. The upper 

branch of isodose curves corresponds to dissolution of rich segregations 

and access to new equilibrium with lower impurity concentration. The 


Cr–Mn–Ni–Si steel (see Table 3) under its tempering. Auger electron spectroscopy of 

free surface segregations. 



Cr–Mn–Nb steel (see Table 3) under its tempering. Auger electron spectroscopy of 


adsorption patterns for engineering steels have common as well as individual 

features. As a rule, carbon segregates at temperatures lower than 

523K, nitrogen—in 523–623K range, phosphorus—in 523–823K range, sulfur 

segregates at temperatures higher than 723K. 

The substitual and interstitial element concurrence promotes blocking 

of adsorption centers by mobile impurities and impedes P segregation at 


Cr–Mn–V steel (see Table 3) under its tempering. Auger electron spectroscopy of free 

surface segregations. 



Cr–Mn–Si–Ti steel (see Table 3) under its tempering. Auger electron spectroscopy of 


temperatures lower than 523–673K. The preferential enrichment of GB by P 

and S becomes possible after dissolution of C and N segregations. The alloying 

elements change significantly the segregation stability regions for various 

elements. Fig. 28(a,b) shows the P adsorption isotherms in the 

investigated steels at 723K. Molybdenum sharply slows down the P 

segregation formation. The differences in diffusion mobility of elements 

and temperature intervals of segregation stability are the reasons for nonequilibrium 

enrichment of grain boundaries. The rich segregations are 

formed at the initial stage of isothermal exposition, and they are dissolved 

after longer exposition. Comparing the behavior of 0.3C–Cr–Mn–Nb (1) 

and 0.22C–Cr–Mn–Si–Ni (3) steels at 673K tempering, one can see that 

small (lower than 20 min) expositions 0.3C–Cr–Mn–Nb, and longer ones 

(about 1 hr 20 min) are dangerous for 0.22C–Cr–Mn–Si–Ni steel. Analyzing 

Table 3 Chemical Composition of Steels 

No. Steel 

Concentration of elements, wt.% 

C Si Mn Cr V Al Ti Nb Ni Mo S P 

1 0.3C–Cr–Mn–V 0.32 0.25 0.88 0.92 0.088 0.014 0.024 0 0 0 0.016 0.027 

2 0.3C–Cr–Mn–Nb 0.29 0.33 1.04 1.07 0 0.007 0.036 0.025 0 0 0.014 0.027 

3 0.3C–Cr–Mo 0.33 0.23 0.56 0.96 0.003 0.014 0.025 0 0 0.25 0.005 0.004 

4 0.3C–Cr–Mn–Si–Ti 0.28 0.61 1.15 0.75 0 0.029 0.016 0 0.32 0 0.013 0.022 

5 0.2C–Cr–Mn–Ni–Si 0.22 0.43 0.92 0.89 0 0.030 0.015 0 0.91 0 0.015 0.025 


Figure 28 Influence of alloying on the kinetics isotherms of P free surface 

segregation at 723K. The following steels were investigated (see Table 3): 1, 3C–Cr– 

Mn–Nb; 2, 3C–Cr–Mn–Si–Ti; 3, 2C–Cr–Mn–Ni–Si; 4, 3C–Cr–Mo; 5, 3C–Cr– 

Mn–V. 

the thermokinetic diagrams for ternary Fe–Me–P alloys based on Eqs. (23) 

and (6), the mutual influence of elements on their binding energy to GB was 

determined [36] 

E P seg ¼ 20:6 þ 183CP a 

4:8C Al 

a 

7:2C Mo 

a 

3:4C Ni 

a 

7141C B a 

þ 4:9C Cr 

a 444C S a 183E Mo 

seg 87E N seg ð24Þ 

E S seg ¼ 6:9 151CS a 1:5C Al 

a þ 14:5CP a 39E Sn 

seg 

E N seg 

¼ 16 2:6CAl 

a 

þ 3CMo 

a 

þ 4:2CCr a 


2625C Ti 

a 

þ 175EMo 

seg 

ð25Þ 

ð26Þ

E C seg ¼ 7:9 1:4CAl a þ 5CMo a þ 676CB a þ 1:2CCr a 

130E N seg þ 116EP seg 

ð27Þ 

E Mo 

seg ¼ 0:7 þ 32EN seg 28E P seg ð28Þ 

E Ti 

seg ¼ 17 þ 3CC a E P seg ð29Þ 

E Al 

seg ¼ 1:4CAl a 

E Sn 

seg ¼ 21; ENi seg ¼ 14; EBseg ¼ 54; ECu seg ¼ 20 kJ/mol 

ð30Þ 

I 

where Esegis 

segregation energy of the I element, Ca j is bulk concentration of 

j impurity. 

F. Stability of the Segregation 

The equilibrium GBS dissolves as temperature increases. Analysis of the 

kinetic development of the equilibrium segregation level of P shown in 

Fig. 29 gives the T–t plot of segregation directly. Obviously that segregation 

level close to the maximum exists only within a specific temperature range. 

This range is characterized by a maximum temperature stability T max, over 

which the intensive dissolution of the segregates is observed. This temperature 

can be calculated by computer analysis of Eq. (7) at dCb max =dT ¼ 0. 

The temperature Tmax depends on Eseg and temperature dependencies of 

solubility limits, which can be determined from analysis of phase equilibrium 

diagrams [43]. 

Using these dependencies as a generalizing criterion, it is possible to 

simplify the analysis of data on element segregation kinetics in iron alloys. 

The interrelationship of maximum temperature of stability (T max) of rich 

equilibrium segregations and segregation energies of different elements is 

presented in Fig. 30. 

The common features of kinetics show the following groups: 

1. enriching grain boundaries at low- and medium-tempering 

temperatures—B, C, N, and Cu; 

2. co-segregating with P at high tempering—P, Sn, Ti, and Mo; 

3. segregating at high temperatures—S and Al. 

Phosphorus in Fe alloys has abnormally weak dependence of Tmax 

on Eseg in reversible temper embrittlement temperature range. In other 


Figure 29 The calculated segregation level of P as a function of temperature 

according to Eq. (7). Tmax is the maximal temperature of stability of rich segregation 

level. (From Ref. 42.) 

words, this means that the temperature of P segregation stability in the 

RTE development interval weakly depends on segregation energy or alloy 

composition. This circumstance is associated with the specific shape of the 

temperature dependence of P solubility in Fe. The established regularity 

allows to explain the difficulties with rational alloying of engineering steels 

for RTE suppression. 

G. Nature of Reversibility of Temper Embrittlement 

The reversibility of temper embrittlement is usually associated with precipitation 

or dissolution of carbide phase at various modes of quenched steel 

heat treatment below Ac1 [44–46]. The complex character of multicomponent 

GB adsorption—namely interrelation of two opposite processes: concurrence 

between impurities, and their cooperative segregation—is not 

taken into account using this approach. 


Figure 30 The interconnection of Tmax-segregation stability temperature and 

Eseg-energy of impurities segregation in Fe-base alloys. 


Figure 31 Thermo-kinetics diagrams of multicomponent segregation on free 

surface in steel 0.35C–1.58Mn–0.1P–0.6Al. 

Figure 31 presents the thermokinetic diagram of element segregation 

in 0.35C–1.5Mn–0.1P–0.6Al steel. The chemical composition of free surface 

segregations was determined by AES for a set of isothermal conditions in 

the spectrometer ESCALAB MK2 (VG). The temperature–time interval 

of preferential segregation of chemical elements is the result of different diffusion 

mobility and binding energy of elements with GB. The temperature 

interval of P preferential segregation is caused by concurrence of this impurity 

with mobile interstitial elements C and N. This process determines 

temperature and exposition necessary for RTE development. Direct investigation 

of grain boundary composition by AES confirms the conclusion 

about the prevailing role of concurrent segregation in RTE. The composition 

of several grain boundaries on brittle intercrystalline fracture of 

0.35C–Mn–Al steel after heat treatment: quenching from 1223K, tempering 

at 923K for 1 hr with rapid (a) and slow (b) cooling is presented in Fig. 32 

[47]. These data are in good correspondence with those in Fig. 31. Accelerated 

cooling of steel, does not provide enough time for the development of 

segregations with high P content, and GB are enriched by C. During slow 

cooling, phosphorus has enough time to enrich the grain boundaries. In this 

case, the carbon concentration on GB is sufficiently lower than at rapid 

cooling of steel. Carbon segregations are unstable at temperatures higher 

than 500–673K, and they are dissolved. At slow cooling, P segregates to 

grain boundaries, decreasing the GB redundant energy. This circumstance 

lessens the thermodynamic stimulus for carbon segregation as the temperature 

decreases. Carbon and phosphorus in steels are responsible for RTE 

development. They have high surface activity and diffusion mobility that 


Figure 32 Chemical composition of GB in steel 0.35C–1.58Mn–0.1P–0.6Al (AES); 

(a) tempering at 923K, water cooling; (b) tempering at 923K, cooling with furnace. 

(From Ref. 47.) 

predetermines their segregation on GB at heat treatment. Difference of diffusion 

mobility as well as difference of maximum temperature of segregation 

stability is the reason for preferential segregation of an impurity. This is the 

reason for the characteristic temperature region of RTE development. 

Reversibility (i.e. disappearance) of temper embrittlement is associated 

with full dissolution of rich GBS of phosphorus at high temperatures and 


enrichment of GB by carbon at rapid cooling [48]. Undoubtedly, carbide 

transformation, internal stresses, substructure transformations are very 

important for RTE. One should take into account such circumstances 

where kinetics of C and P segregation are dependent significantly on steel 

alloying. 

IV. DYNAMIC SIMULATION OF GRAIN BOUNDARY 

SEGREGATION 

A. Interface Adsorption During Tempering of Steel 

1. Decomposition of Martensite 

The common laws of multicomponent GBS and analysis of experimental 

diagrams on elements segregation kinetics in iron alloys are used to develop 

the computer models of these processes. The exact solution of McLean’s 

diffusion Eq. (21) accounting for temperature dependant of diffusion and 

element solubility is a complex problem. In low-alloyed steels, the concentration 

of surface-active impurities (S, P, and N) is rather small, and based 

on this reason, it is possible to analyze the diffusion of each element separately. 

The model takes into account mutual influence of bulk and surface 

concentration of elements with respect to segregation energies. 

Carbon in solid solution has maximum influence on phosphorus GBS 

kinetics. Concentration of C in martensite changes significantly during 

quenched steel tempering and mainly depends on alloying element content. 

Based on this reason, one should take into account the solid solution composition 

altering segregation processes modeling during tempering. 

Investigations of martensite tetragonality at alloyed steel tempering 

[6,7] are the basis for calculations of mutual influence of alloying elements 

on martensite decomposition kinetics and carbon content in solid solution. 

The carbon content change in solid solution during tempering of engineering 

steels is well described by equation 

where 

DX C a 

X C a ð0Þ ¼ 1 exp KDot exp 

Q 

RT 

n 

ð31Þ 

DX C a ¼ XCa ð0Þ XCa ðtÞ ð32Þ 

X a C (0) and Xa C (t) are the carbon content in quenched steel and after a time t; 

Do is the carbon diffusion coefficient; Q is the activation energy associated 

with the interstitial diffusion of carbon atoms; K is the constant associated 


Table 4 Coefficient A in Eq. (28) for Low-Alloying Engineering Steels 

Coefficient A 

with the nucleation; n is the constant independent of both temperature and 

X a C (0); R is the gas constant and T is the temperature. 

Influence of C and alloying elements on parameters Q, K, and n in 

Eq. (31) is determined for various steels. The activation energy Q in lowalloyed 

steel depends on the concentration of carbon and alloying elements 

in solid solution: 

Qðcal=molÞ ¼8571:5X C a 

þ A XMe 

a 

Alloying element 

Ni Si Mn Cr Mo 

433.56 1,432.54 726.35 2,898.91 971.51 

þ 18; 000 ð33Þ 

where Xa C and Xa Me are concentrations of C and alloying elements, mass%; 

A is a constant depending on alloying element. The values of coefficients in 

Eq. (31) are presented in Tables 4 and 5. The diffusion activation energy of 

Table 5 Influence of Carbon and Alloying Elements on Parameters Q, K, 

and n in Eq. (31) 

Steel, wt.% Q, cal=mol Ln K n 

0.4C–0.24Ni 21,532 15.364 0.26 

0.39C–3.0Ni 22,643 17.481 0.22 

0.37C–5.6Ni 23,599 18,575 0.24 

0.4C–0.32Mn 21,196 15.737 0.24 

0.4C–1.32Mn 20,298 14.241 0.22 

0.4C–2.43Mn 19,406 13.713 0.24 

0.4C–0.2Cr 20,848 15.366 0.21 

0.4C–2.1Cr 15,348 10.076 0.24 

0.4C–3.6Cr 10,992 5.481 0.42 

0.4C–6.7Cr 2,005 1.698 2.32 

0.4C–0.37Si 21,929 16.351 0.19 

0.38C–1.75Si 23,764 15.234 0.22 

0.4C–2.75Si 25,368 10.050 0.15 

0.4C 21,429 16.72 0.24 

1.4C 30,000 40.881 0.07 

1.2C–2.0Mo 26,343 29.768 0.08 


Figure 33 Change of carbon concentration in solid solution with temperature and 

time of tempering. Steel 0.43C–2.43 Mn (mass%). Isodose curves for: 1, 1 at.% C; 2, 

0.5 at.% C; 3, 0.1 at.% C; 4, 0.05 at.% C; 5, 0.03 at.% C. 

carbon decreases on the growth of carbide-forming element (Mn, Cr, and 

Mo) concentration. The contrary effect is observed for Ni and Si. 

Obviously, it is associated with the different influence of these elements 

on thermodynamic activity of carbon in ferrite. These dependencies are 

basic for calculations of segregation kinetics of C since carbon is the element 

that influences on P segregation highly. The kinetics of carbon content in 

solid solution change during tempering of quenched steel 0.43C–2.43Mn 

(mass%) are shown in Fig. 33. These data are obtained by computer modeling 

using Eqs. (31–33) and those from Tables 4 and 5. 

This model provides the possibility of calculating the influence of 

alloying on cementite formation temperature interval, growth rate of its 

particles, and many other parameters of martensite decomposition at tempering 

[49]. 

Fig. 34 presents the calculation results of effective growth rate of Fe 3C 

nucleus at tempering of engineering alloyed steels. The calculations were 

carried out using expression [49]: 

VmaxR ¼ð27D=256pÞN ð34Þ 

where R is the cementite particle radius; N is the right part of Eq. (30). Manganese 

decreases martensite stability significantly promoting its decomposition 

at low temperatures. Silicon, at a concentration greater than 1%, 

activates martensite decomposition at 700–800K and inhibits it at lower 


Figure 34 Change of effective growth rate of Fe 3C nucleus with alloying of 0.4% C 

steel: 1, unalloyed steel with 0.4C(mass%); 2, alloyed with 0.35Si; 3, alloyed with 

2.1Cr; 4, alloyed with 1.75Si; 5, alloyed with 2.43Mn. 

content. Chromium does not change the temperature of intensive cementite 

growth. 

2. Calculation of Thermokinetic Diagrams of Impurities’ 

Segregation During Tempering of Steel 

Modeling of multicomponent adsorption kinetics is carried out using a 

sequence of computer calculations. 

At the initial stage, thermodynamic characteristics of surface activity 

in Fe-base binary and ternary alloys are determined. 

Analysis of phase equilibrium diagrams permits the determination of 

the impurity segregation energy Eseg 

i 

and temperature dependence of ulti- 

mate solubility X8c. These two parameters are very important for determination 

of equilibrium GB concentration of impurity X i b using Eq. (17). 

Examples of such calculations for binary and ternary alloys have been 

presented. Mutual influence of alloy components on their surface activity 

could be refined experimentally. The equations for binding impurity segregation 

energy with solid solution composition could be obtained by regression 

analysis of multicomponent adsorption diagrams. These experiments 

allow the determination of the effective diffusion coefficient of elements. 

The diagram of the equilibrium impurity concentration calculation on grain 

boundary in engineering steel is presented in Fig. 35. 

Carbon concentration in martensite changes drastically during tempering, 

as it depends on chemical composition of steel, temperature, and duration 


Figure 35 Calculation scheme of equilibrium impurity GBS. X a i (0) is the initial 

concentration of ith element in the steel; Xa C (T,t) is the running carbon concentration 

in martensite during its tempering; Xb i (T) is the maximal equilibrium GBS of ith 

element; E i seg Fe–i is the segregation energy of ith element in two-component Fe–I 

alloy; E i seg Fe–i–j is the segregation energy of ith element in multicomponent alloy; 

Di(T) is the diffusion coefficient of ith element in austenite, martensite, and ferrite. 

of treatment. This factor influences on thermodynamic activity of all steel 

components and on their energy of GB segregation. 

The second important stage of GBS modeling includes calculation 

of C volume concentration in martensite Xa C (T), depending on steel 

chemical composition Xa i (0) and parameters of tempering. New segregation 

energy values of each element at changing of treatment temperature or 

time and new equilibrium GBS level have been calculated in this way (see 

Fig. 35). 

The final stage of modeling includes a set of independent calculations 

of various element diffusion to GB zone, and their desorption. The limited 

capacity of boundary and its effective width (about 0.5 nm) are shown. It is 

assumed that interstitial and substitial impurities occupy different positions 

on GB. Time t of reaching the definite concentration of impurity in segregation 

X b(t) at given temper temperature T is calculated by (22), and it is controlled 

by diffusion D i(T). 

Adsorption in multicomponent system is accompanied by concurrence: 

arrival of some surface-active impurity decreases GB energy and, in 

this way, the thermodynamic stimulus for segregation of other impurities. 

Dissolution of segregations is observed at increasing temperature. Impurity 

desorption to grain bulk is analogous to adsorption, however it is tied not 

with concentration Xi(0) but with Xb(t), and it is also controlled by diffusion 

D i(t). 


The model is restricted to initially homogeneous bulk concentrations 

X i b ð0Þ ¼Xi a 

ð35Þ 

The kinetics of segregation to surfaces or grain boundaries from the 

bulk are determined by volume diffusion of impurities with bulk concentrations 

X i a(t) which can be treated as a one-dimensional problem. Since both 

bulk concentrations are very small, Arrhenius type diffusion coefficients: 

Di ¼ D i 0 exp 

Qi 

RT 

ð36Þ 

can be used which are independent of X i a(t). In the case of site competition, 

the GB impurities concentration is 

qi ¼ 

Xi 

1 P 

J Xj 

exp 

Ei 

KT 

ð37Þ 

The equations describing the time evolution of segregation for homogeneous 

initial condition [60] are 

pffiffiffi pd 

XiðtÞ ¼Xið0Þþ2 X0 i 

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

Z t 

0 

Diðt 0 Þ dt 0 

1 

pffiffiffi pd 

Z t 

0 

qiðt0ÞDiðt0 Þ 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

R t0 t Diðt00Þ dt00 q ð38Þ 

In the case of constant temperature (i.e. Di ¼ const), Eq. (38) can be 

simplified: 

pffiffiffiffi 

2 D 

XiðtÞ ¼Xið0Þþpffiffiffi ½X 

pd 

0 i 

pffiffi 

qiðtÞŠ t 

ð39Þ 

Diffusion coefficient for impurities in Fe and Fe-base alloys in ferrite 

interval is present in Table 6. 

The calculated diagrams of multicomponent adsorption in steels 0.3C– 

Cr–Mo, 0.3C–Cr–Mn–V, 0.3C–Cr–Mn–Si–Ti (see Table 3) are presented in 

Figs. 36–38. Comparing these diagrams with the experimental ones (Figs. 

24, 26, and 27), a good correlation of segregation kinetic features 

for various elements is observed, that confirms the basic principles of the 

proposed model of GBS in steels. According to this model, the main 

role of carbide precipitation in GBS consists of changing solid solution 


Table 6 Coefficients of Diffusion for Impurities in a-Fe and Steels 

Solute System Temperature, K D, m 2 Sec 1 

D 0 (m 2 Sec 1 ) 

Q, 

kcal=mol Reference 

C 0.3C–10Ni 

Martensite 

723–873 5.26 10 5 

15.2 [50] 

C a-Fe 623 5.3 10 14 

[51] 

P a-Fe 723 2.8 10 19 

[52] 

P a-Fe 748 7.7 10 19 

[52] 

P a-Fe 773 2.0 10 18 

[52] 

P a-Fe 798 4.8 10 18 

[52] 

P a-Fe 9.55 10 6 

50.6 [53] 

P Fe–2.1Mn 1.43 10 4 

54.2 [53] 

P Fe–Ni–P 0.51 10 4 

55 [54] 

P a-Fe 0.108 exp( 288=RT) [55] 

P 0.1P–0.15Cr a 0.336 exp( 296=RT) [55] 

P 0.1P–0.13Si a 18 exp( 329=RT) [55] 

P 0.1P–0.17Mn a 0.235 exp( 292=RT) [55] 

P 0.1P–0.14Mo a 3.23 10 6 exp( 434=RT) [55] 

P 0.1P–0.14Ni a 43.9 exp( 336=RT) [55] 

S Fe–3Si 973 1.7 10 2 

61.2 [56] 

Sn a-Fe 973–1303 5.4 55.5 [57] 

Cr Fe–Cr 1048 2.33 10 4 

57.1 [58] 

Co Fe–6.8Co 903–1073 4.69 10 5 

44.7 [59] 

C Fe–0.79Si 803 3.6 10 12 

[60] 

C Fe–0.79Si 873 1.4 10 11 

[60] 

C Fe–0.79Si 973 1.9 10 10 

[60] 

C Fe–0.6Ni 803 4 10 4 

[60] 

C Fe–0.6Ni 873 16 10 4 

[60] 

C Fe–0.6Ni 973 6.9 10 4 

[60] 

C Fe–0.56Mo 873 7.2 10 4 

[60] 

C Fe–0.56Mo 923 21 10 4 

[60] 

C Fe–0.56Mo 973 55 10 4 

[60] 

composition, as it is exactly this factor that controls mutual influence of elements 

on their surface activity. Only such elements that segregate in near 

temperature ranges mutually influence GBS. The computer calculations of 

segregation kinetic diagrams predict these effects with small changes of steel 

chemical composition. Figures 39 and 40 present the modeling data on influence 

of sulfur content in 0.3C–Cr–Mn–Si–Ti steel on phosphorus segregation 

kinetics. Sulfur and Phosphorus are strong surface-active elements, 

and they can compete at grain boundaries. Desulfurization of steel significantly 

slows down GBS of S. Indeed, P adsorption increases with a decrease 

of S content. According to calculations (see Fig. 40), the time of 6% P GBS 

formation exceeds 4000 sec at a sulfur content more than 0.02 at.%. This 

time it is significantly longer than the usual duration of quenched steel 



Cr–Mn–Ni–Si steel (see Table 3) under its tempering. Computer simulation. 

tempering. Deeper cleaning of steel by S activates GB adsorption of P, drops 

down time of segregation formation, and increases the maximum temperature 

of segregation stability (see Figs. 39 and 40). Such calculations 

are very useful for the design of optimal alloying and purification degree on 

harmful impurities, since they permit the determination of the influence of 

alloying on ultimate concentration of harmful impurities. 


Cr–Mn–V steel (see Table 3) under its tempering. Computer simulation. 



Cr–Mn–Si–Ti steel (see Table 3) under its tempering. Computer simulation. 

B. Interface Adsorption During Quenching of 

Engineering Steels 

Mathematical models of GBS [61] and phase transformations permit the 

analysis of heat treatment with respect to the accompanying phenomena 

in a greater detail than that of a simple summary of the experimental 

knowledge. 

Figure 39 Dependence of Tmax of P GBS as a function of sulfur containing in 

0.3C–Cr–Mn–Si–Ti steel under its tempering. Computer simulation. 


Figure 40 Dependence of time of 6 at.% GBS of phosphorus and sulfur as a 

function of sulfur concentration in 0.3C–Cr–Mn–Si–Ti steel during its tempering at 

700K. Computer simulation. 

The results of mathematical modeling provide backgrounds for reasonable 

planning of full-scale experiments when seeking for the optimum 

technological procedures and steel composition and they enable the 

extrapolation of the consequences of variations in the technological conditions 

even outside the boundary of the empirical experience we have available. 

Interaction of GB segregation enrichment and phase transformations 

during heat treatment of steels in the austenitic region is hard to imagine. 

Nb and V carbonitride precipitation in microalloyed austenite, precipitation 

of free ferrite, change chemical composition of austenite, and influence 

on GBS kinetics to a large extent. The experiments show that nonequilibrium 

grain boundary phenomena occur for a rather short time up 

to 100 sec. The minimum time of 5% volume fraction of Nb and V carbonitride 

precipitation is about 1000 sec [62,63]. Precipitation of free ferrite 

needs from several seconds to several minutes depending on steel chemical 

composition. Therefore, the non-equilibrium GBS in steels with a wide 

region of undercooled austenite stability independently from phase transformations. 

This computer model has some limitations but redistribution 

of harmful impurities between grain bulk and boundaries permits the analysis 

of steel quenching. 

The modeling of non-equilibrium GB phenomena allows during investigation 

of such short-time changes of chemical composition that could not 

be measured experimentally and that has an extreme importance for modern 

heat-treatment processes with high heating and cooling velocities in 

controlled media. 


Figure 41 The presentation of C-curve on simulating TTT diagrams. (Scheme.) 

Parameters U and S correspond to Table 7. 

1. Phase Transformations of Undercooling Austenite 

At present, many computer models of evolution of structure and phase composition 

of steels during quenching have been developed. Most of them are 

based on physical models of phase transformations [64–66]. But physical 

models cannot describe adequately all kinetic features of undercooled austenite 

transformations. The computer models based on regression analysis of 

experimental data can best predict steel phase composition changes during 

steel cooling. 

It was introduced directly by Davenport and Bain [67] and the time– 

temperature-transformation (TTT) diagram was the predominant tool to 

describe the isothermal decomposition kinetics of supercooled austenite. 

In most TTT diagrams, general S- or C-curves are used to represent the 

kinetics of a number of isothermal transformation products: ferrite, pearlite, 

upper bainite, lower bainite, and martensite. Conversely, many experimental 

results demonstrate that each type of transformation product has a separate 

C-curve. 

To build a mathematical model, all TTT diagrams published in Refs. 

[68–71] were analyzed. The rationalization of the kinetics of isothermal 

decomposition of austenite permitted the establishment of a metastable 

product (phase) diagram of a number of steels of different compositions 

with 6% of total content of all alloying elements. 


Most isothermal transformations take place by nucleation at the austenite 

grain boundaries, so the original austenite grain size will affect the isothermal 

decomposition kinetics of austenite. 

From the total number of factors characterizing austenite matrix, the 

present day experimental knowledge allows only an approximate examination 

of the statistically recrystallized proportion and estimation of the size 

of deformed austenite grains. 

The grain growth kinetics satisfy the law [73] 

dðtÞ ¼d0 þ kt exp 

Q 

RT 

ð40Þ 

where d(t) is averaged grain size at moment t; d0 is initial grain size; Q is activation 

energy; k is a constant. 

The algorithm of calculating the size of austenite grains is described in 

Ref. [73]. 

The procedure for calculation of the structural proportions of anisothermal 

decomposition of austenite at engineering steel cooling is given 

in Tables 7 and 8 and shown in Fig. 41. 

The cooling curve is approximated partially by a constant function 

and at the individual time intervals Dt and the rate of decomposition is calculated 

as isothermal transformation corresponding to the mean temperature 

of that interval. The required kinetic data are available from the TTT 

diagrams [68–71] that can be digitized (see Table 8) by procedures shown 

in Fig. 41, using equation 

S S0 

SN S0 

1=2 U U0 

¼ e 

UN U0 

1=2 

exp 

1 

2 

U U0 

UN U0 

ð41Þ 

where S ¼ Int-time interval, s; U ¼ 1000=(T þ 273). 

Since it is necessary to distinguish between the parts of the C-curves 

representing the formation of ferrite, pearlite, and bainite, only those 

diagrams having readily distinguishable component curves were used in 

the analysis. 

The calculation method includes the effect of the size of austenite 

grains on the kinetics of phase transformations. The main precondition is 

knowledge of this effect on the course of C-curves showing the start and 

end of transformations in the graph of isothermal decomposition of austenite 

for the relevant steel. 


Table 7 The Algorithm of Calculation of the Structural Proportions 

1. Temperature of start of transformations yAc3, yAc1, yBa, yMs 

2. t 0; y(0) y0; V1(0) 1; Vi(0) 0, i ¼ 2, 3, 4, 5; i ¼ 1-austenite, 

2-ferrite, 3-pearlite, 4-bainite, 5-martensite 

2.1 Mean temperature at the interval of ht, t þ Dti 

y ¼ (y(t) þ y(t þ Dt))=2; 

if y < ¼ yMs pass to 3; 

if y < ¼ min (yjs) then n j; 

2.2 for i ¼ 2, ..., n carried out as follows: 

– calculation of the transformable proportion of austenite 

Vmi(t) for i ¼ 2: 

Vm2(t)¼0; for y ¼ > yA3, 

Vm2ðtÞ ¼V 0 yA3 y 

m2 yA3 yA1 ; for yA1 < y < yA3; 

; for y 2 and Vmi ¼ 1 

– calculation of the start and end of transformation tsi, tfi and exponent 

ki for y 

ki ¼ 6.127=ln (tsi=tfi); for ferrite k2 ¼ 1; 

– the fictive volume fraction of the transformed proportion 

Xi ¼ Vi(t)=[Vi(t) þ Vi(t) Vmi(t)]; – the fictive time of isothermal transformation required for reaching 

the proportion Xi 

t 0 i ¼ 

tki lnð1 XiÞ 

si 

bS 1=ki; 

– the fictive volume proportion of the structural component at time 

t þ Dt 

Xiðti þ DtÞ ¼1 exp b2 tiþDt 

tsi 

; 

– the volume proportion of the structural component at time t þ Dt 

Viðt þ DtÞ ¼Xiðt0 i þ DtÞ½ViðtÞþViðtÞVmiðtÞŠ; 

2.3 the new value of residual content of austenite 

V1ðt þ DtÞ ¼1 Pn i¼2 Viðt þ DtÞ; if V1ðt þ DtÞ

Table 8 The Constants Blk for Calculations of the YL Parameters of C-Curves 

Transformation YL 

k 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 

B lk 

Const C Mn Si Ni Cr Mo GS Const C Mn Si Ni Cr Mo GS 

Ferrite-start T8C T OFS 727 13 9 17 22 3.5 0 229 30 66 22 70 5 0 

T NFS 572 33 44 9 25 50 1.01 48 57 25 15 15 25 3.90 

Time IntOFS 10.13 2.5 2.84 0.46 4.8 5.9 0.095 1.92 2.80 3.55 1.50 3.00 3.50 0.0096 

IntNFS 1.04 0.6 0.19 0.07 4.8 5.9 0.038 2.58 0.20 1.10 0.04 2.55 3.50 0.021 

Pearlite-start T8C TOPS 727 13 9 17 22 3.5 0 0 15 5 5 22 0 

T NPS 572 33 44 9 25 50 1.01 48 32 20 5 27 23 3.20 

Time Int OPS 10.13 2.5 2.84 0.46 4.8 5.9 0.095 1.92 10.12 3.55 3.30 3.00 4.20 0.096 

Int NPS 1.04 0.6 0.19 0.07 4.8 5.9 0.038 2.58 1.31 0.20 0.20 2.55 2.50 0.041 

Pearlite-finish T8C TOPF 727 13 9 17 22 3.5 0 0 0 15 5 5 22 0 

TNPF 577 2 35 7 53 58 0.52 6 4 5 5 36 25 1.31 

Time IntOPF 10.55 6.86 3 1.81 4.8 12.3 0.095 0.03 14.08 0.76 1.99 3.00 4.35 0.0096 

IntNPF 0.15 2.0 0.19 0.07 2.65 10.6 0.038 0.03 1.31 1.56 0.65 0.98 4.35 0.021 

Bainite-start T8C T OBS 570 12 16 12 20 22 0 107 12 –5 –5 –5 –40 0 

T NBS 485 12 2 7 40 32 0.50 107 30 35 5 36 2 0.37 

Time Int OBS 3.52 2.14 1.07 0.05 2.3 3.1 0.070 5.44 1.46 1.64 0.40 3.49 3.00 0.0626 

IntNBS 0.96 1.30 0.30 0.02 2.2 2.2 0.024 1.40 1.29 1.64 0.20 2.04 6.60 0.0248 

Bainite-finish T8C TOBF 570 12 16 7 22 22 0 76 12 5 7 5 51 0 

TNBF 488 48 25 7 47 61 0.52 76 30 28 7 40 96 0.37 

Time Int OBF 6.90 5.23 2.8 0.7 6.2 9.5 0.070 7.73 0.75 5.44 0.40 0.90 6.88 0.0813 

Int NBF 0.15 3.00 2.8 0.7 3 5.9 0.024 1.98 1.17 5.84 0.02 2.47 4.00 0.0340 

M S T MS 539 423 30.4 0 17.7 12.1 7.5 0 0 0 0 0 0 0 0 

Ck 

C1 C2 C3 C4 C5 C6 C7 C8 

1 %C %Mn %Si %Ni %Cr %Mo 2 

YL ¼ P8 k¼1 Blk ½ þ Dlk þ 8ð0:8 C2ÞŠCk; 

GS—grain size (ASTM). 


D lk

The program involves the calculation of temperatures of transformations 

of bainite and twinned, athermal and lamellar martensite [74] 

BS ¼ 720 585:63ðCÞþ126:6ðCÞ 2 

31:66ðCrÞþ2:17ðCrÞ 3 

42:37ðMoÞþ9:16ðCoÞ 0:125ðCoÞ 2 

66:34ðNiÞþ6:06ðNiÞ 2 

91:68ðMnÞþ7:82ðMnÞ 2 

M TM 

S ¼420 208:33ðCÞ 72:65ðNÞ 43:36ðNÞ2 16:08ðNiÞ 

M LM 

S 

M A S 

þ0:78ðNiÞ 2 

36:02ðCuÞ ð42Þ 

0:025ðNiÞ 3 2:47ðCrÞ 33:428ðMnÞþ1:296ðMnÞ 2 

þ30:0ðMoÞþ12:86ðCoÞ 0:2665ðCoÞ 2 7:18ðCuÞ ð43Þ 

¼ 540 356:25ðCÞ 260:64ðN 24:65ðNiÞþ1:36ðNiÞ2 


47:59ðMnÞþ2:25ðMnÞ 2 

þ 17:5ðMoÞþ21:87ðCoÞ 16:52ðCuÞ ð44Þ 

¼ 820 603:76ðCÞþ247:13ðCÞ2 


0:196ðCoÞþ0:165ðCoÞ 2 

55:72ðNiÞþ3:97ðNiÞ 2 

66:24ðMnÞ 24:29ðMoÞ 

The size of ferritic grain is expressed as follows [75] 

da ¼ 11:7 þ 0:14dg þ 37:7V 0:5 

C 

31:88ðCuÞ ð45Þ 

ð46Þ 

where dg is the size of austenitic grain, (mm); VC is the cooling speed, 

(8Cmin 1 ). 

The interlamellar distance of pearlite can be estimated as follows [75]: 

S ¼ X 

" 

18:0DVPðyiÞ=ð996 

# 

yiÞ =VP 

ð47Þ 

i 

where DVP(yi) is the volume proportion of pearlite transformed at yi temperature. 

The thickness of the ferritic and carbide lamellae of pearlite is approximately 

lf ¼ 0.885S; lc ¼ 0.115S 

The size of martensitic and bainitic particles is identical with the 

original size of the austenite grains. The course of the anisothermal 

decomposition of austenite in several steels has been calculated by the 


just-described method and by applying the digitized TTT diagrams of 

Table 7. 

2. Determination of the Kinetics of Carbonitride 

Precipitation in Austenite 

Microalloying of steels with Ti, V, Nb, and Zr affects decomposition of 

supercooled austenite, its recrystallization and grain boundary segregations 

of harmful impurities. These changes of material properties are associated 

with carbonitride precipitation and changes of austenite chemical composition. 

Based on these reasons, the modeling of the kinetics of carbonitride 

precipitation is important. 

The nucleation time t of carbonitrides per unit volume N at any temperature 

T, can be expressed as [76] 

t ¼ C exp Q 

RT exp 

B 

T3ðLnKSÞ 2 

! 

t ¼ C exp Q 

RT exp 

B 

T3ðLnKSÞ 2 

! 

ð48Þ 

where C ¼ 6 10 13 for homogeneous nucleation; activation energy of Nb 

diffusion Q ¼ 270 kJ=mol; 

B ¼ 16pg 3 V 2 3 

mN0=3R ð49Þ 

Vm ¼ 1.28 10 5 m 3 =mol; g ¼ 0.5 J m 2 ; N0 are numbers of nucleus by 

radius R per molar volume Vm; KS is supersaturation [77] 

LgðKSÞ ¼ A 

þ B ð50Þ 

T 

where thermodynamics parameters A and B for various carbides and 

nitrides are calculated in Ref. [78] and presented in Table 9. Thus, the calculation 

of carbonitride nucleation time necessary to reproduce the C-curves 

Table 9 Thermodynamics Parameters in Eq. 50 (From Ref. 77) 

Chemical compound 

Parameter AlN VC VN TiC TiN NbC NbN ZrC ZrN 

A 7,130 9,500 7,985 8,872 15,573 7,714 10,440 8,464 13,968 

B 1.463 6.72 3.09 4.04 3.82 3.27 3.87 4.96 3.08 


corresponds to start or finish of this transformation in austenite under heat 

treatment of steel. 

3. Calculation of Thermokinetic Diagrams of Impurity 

Segregation During Quenching of Steel 

Computation of grain boundary multicomponent adsorption kinetics could 

be simplified for steels with high undercooled austenite stability. The GBS 

develops in this case in austenite in short time and has no dependence on 

phase and structure transformations at steel quenching. Enrichment of grain 

boundaries by various impurities as well as their desorption is treated as a 

result of multicomponent diffusion of impurities from near-boundary 

volume to the boundary. Impurity binding energy with GB includes 

mutual influence of elements in grain bulk and on the boundary in accordance 

with Guttmann’s theory [Eqs. (18) and (19)]. Auger electron spectroscopy 

is the technique for experimental investigation of GBS kinetics. These 

experiments are basic for analysis of correlation of impurity segregation 

energy with the content of other elements in the bulk and on boundaries 

(see Section 2.5, Eqs. (23)–(29). 

Adsorption and desorption of impurities on GB (qi) at steel quenching 

is modeled well using the equation 

qi ¼ qið0Þþ 2q0 i 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

Z t 

pffiffiffiffiffi 

Diðt 

pd 

0Þ dt0 s 

1 

pffiffiffiffiffi 

pd 

0 

Z t 

0 

C i a ðt0 ÞDiðt 0 Þ 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

R t 

t0 Diðt00Þ dt00 q dt 0 

ð51Þ 

where d is the grain boundary thickness; Di(t 0 ) is the diffusion coefficient of 

impurity which depends on the temperature and phase composition (austenite, 

martensite, and ferrite); in the case of adsorption 

C i a ¼ 

i 

where CGB 

Ci GB 

1 P 

j Cj 

expð Gi=kTÞ ð52Þ 

GB 

is the element i concentration on grain boundary; Ca i is the con- 

centration of ith element in the adjacent bulk layer; Gi is segregation energy. 

Desorption is determined by GB concentrations of impurities, and in this 

case, the parameter C a i in Eq. (51) is equivalent to GB concentration Xb I in 

Eqs. (12) and (13). 

The change of temperature at cooling or isothermal exposition is 

described by equation 

TðtÞ ¼½Tð0Þ Tð/ÞŠ expð rtÞþTð/Þ ð53Þ 


Table 10 Cooling Rates for Some Metallurgical Technologies 

Name of the treatment Cooling rates r (K sec 1 ) 

Quenching 100–10 

Controlled cooling 10 

Air cooling of hot-rolled metal 10–0.1 

Cooling with furnace 0.01 

Controlled cooling of large-size forging 0.001 

Table 10 presents cooling rates r for heat-treatment processes. The block 

diagram of multicomponent intercrystalline adsorption model is shown in 

Fig. 42. Adsorption of P, C, and S is determined by parameters K1, K2, 

K3, and their desorption by parameters K2, K4, K6. The parameters Ci 

are equivalent to GB concentration of element i. This model allows the computation 

of the condition when there is change of GB composition in steels 

and alloys at preselected arbitrary mode of cooling including isothermal 

exposition. 

Given below are the examples of investigation of phosphorus and sulfur 

grain boundary adsorption in Cr–Ni–Mo steel (see Table 11). 

The components of steel mutually influence their diffusion mobility 

and GBS activation energy. Based on this reason, one should take into 

account the stochastic fluctuations of diffusion flows of various impurities 

on GBS kinetics. For this purpose, the random fluctuation of diffusion 

coefficients up to 30% of its mean value was used in the model. Figures 43 

and 44 present the GBS kinetics calculation results at cooling of various purity 

steels cooling that were carried out using the stochastic model. As one 

can see, the self-regulation of adsorption is observed which is developing 

despite significant short-time oscillations of impurity concentration on grain 

boundaries. The significant non-equilibrium enrichment of GB by impurities 

is observed at initial stage of the heat treatment. This effect is determined 

by cooling velocity as well as impurities content. Increasing cooling 

velocity from 0.001 to 1000K s 1 decreases the non-equilibrium GBS of P 

and S. Formation of non-equilibrium rich GBS of harmful impurities at 

small cooling times could be established only by using computer modeling 

methods. The experimental verification of such phenomena needs special 

techniques which allow to open grain boundaries: hydrogenation of 

quenched samples or delayed fracture tests. Since these techniques are 

conducted in air and could not be applied in the vacuum chamber of 

electron spectrometer; for most of engineering steels, the regularities of 

non-equilibrium GBS formation at quenching could only be estimated by a 

computer experiment. 


Figure 42 Calculation scheme of three-component GBS (phosphorus, sulfur, and 

carbon). 

Table 11 Chemical Composition of Cr–Ni–Mo Steels 

Smelting 

number 

Chemical composition, mass% 

C S P Ni Cr Mo 

Time of austenite 

stability at 600 hr 

82 0.38 0.027 0.054 3.95 3.0 0.51 2.0 

83 0.38 0.01 0.006 4.02 3.0 0.50 2.0 


Figure 43 The change of GBS during quenching of Cr–Ni–Mo steel containing 

0.027S and 0.054P (mass%). Computer simulation of fast (a) and slow (b) cooling. 


Figure 44 The change of GBS during quenching of Cr–Ni–Mo steel containing 

0.01S and 0.006P (mass%). Computer simulation of fast (a) and slow (b) cooling. 


Figure 45 Chemical composition of GB in Cr–Ni–Mo steel containing 0.027S and 

0.054P (mass%) after austenitization at 1373K (30 min), interim cooling up to 873K 

and quenching in water (a) and in furnace (b). Auger electron spectroscopy of 

intergranular fracture. 


Figure 46 Chemical composition of GB in Cr–Ni–Mo steel containing 0.01S and 

0.006P (mass%) after austenitization at 1373K (30 min), interim cooling up to 873K 

and quenching in water (a) and in furnace (b). Auger electron spectroscopy of 

intergranular fracture. 


The validation of calculation reliability was done for steel composition 

82 and 83 (see Table 10) by Auger spectroscopy. The samples after austenitization 

at 1373K (30 min) were in the interim cooled to 873K with further 

cooling in water or with furnace cooling. The undercooled austenite in this 

steel has high stability and does not transform in ferrite region for 2 hr. 

After cooling the samples had martensite–baintite structure. To investigate 

the chemical composition of grain boundaries by Auger spectroscopy, special 

samples were crushed in the electron spectrometer ESCALAB MK2 at 

vacuum at about 10 8 Pa at temperature 83K. The fields with intercrystalline 

fracture type were investigated on the fracture surface. The variation 

of phosphorus and sulfur content in GBS in Cr–Ni–Mo steel of several melts 

after heat treatment is shown in Figs. 45 and 46. At accelerated cooling the 

GB are significantly enriched by carbon. The P concentration in GB 

increases only at slow cooling of samples, and P segregation is strongly suppressed 

in pure steel. A good correspondence of calculated and experimental 

results is observed for all cases to be analyzed. 

The results of numerical modeling give information about the equilibrium 

and non-equilibrium character of a GB adsorption processes, which 

are frequently unavailable from experiments. Moreover, these simulation 

methods explain the phenomenon of reverse temper embrittlement as the 

result of non-equilibrium concurrent GBS of carbon and phosphorus. These 

results explain many questions in the multicomponent GB adsorption 

kinetics in engineering steels that were dynamically developed in the last 

10 years. Further investigations in this direction are required especially 

for competitive internal adsorption in engineering steels treated by using 

newest schemes of heat treatment. 

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