Study on elastic deformation of interstitial alloy FeC with BCC structure under pressure

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Study on elastic deformation of interstitial alloy FeC with BCC structure under pressure. The elastic deformations of main metal A is special case of elastic deformation for interstitial alloy AB. The theoretical results are applied to alloy FeC under pressure. The numerical results for this alloy are compared with the numerical results for main metal Fe and experiments.
VNU Journal of Science: Mathematics Physics, Vol. 35, No. 1 (2019) 1-12
Review article
Study on Elastic Deformation of Interstitial Alloy FeC
with BCC Structure under Pressure
Nguyen Quang Hoc1, Tran Dinh Cuong1, Nguyen Duc Hien2,*
1Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
2Mac Dinh Chi High School, Chu Pah District, Gia Lai, Vietnam
Received 03 December 2018
Revised 16 January 2019; Accepted 04 March 2019
Abstract: The analytic expressions of the free energy, the mean nearest neighbor distance between
two atoms, the elastic moduli such as the Young modulus E, the bulk modulus K, the rigidity
modulus G and the elastic constants C11, C12, C44 for interstitial alloy AB with BCC structure under
pressure are derived from the statistical moment method. The elastic deformations of main metal A
is special case of elastic deformation for interstitial alloy AB. The theoretical results are applied to
alloy FeC under pressure. The numerical results for this alloy are compared with the numerical
results for main metal Fe and experiments.
Keywords: interstitial alloy, elastic deformation, Young modulus, bulk modulus, rigidity modulus,
elastic constant, Poisson ratio.
1. Introduction
Elastic
properties
of
interstitial
alloys
are
specially
interested
by
many
theoretical
and
experimental researchers [1-4, 7-12]. For example, in [3] the strengthening effects interstitial carbon
solute atoms in (i.e., ferritic or bcc) Fe-C alloys are understood, owning chiefly to the interaction of C
with crystalline defects (e.g., dislocations and grain boundaries) to resist plastic deformation via
dislocation glide. High-strength steels developed in current energy and infrastructure applications
include alloys where in the bcc Fe matrix is thermodynamically supersaturated in carbon. In [4],
structural, elastic and thermal properties of cementite (Fe3C) were studied using a Modified Embedded
Atom Method (MEAM) potential for iron-carbon (Fe-C) alloys. The predictions of this potential are in
good
agreement
with
first-principles
calculations
and
experiments.
In
[7],
the
thermodynamic
________
Corresponding author.
E-mail address: n.duchien@gmail.com
https//doi.org/ 10.25073/2588-1124/vnumap.4293
1
r
r
1
n
i
r
r
r
r
r
i
1
0
1
r
1
2
1
(2)
1
2
(r
r
1
(1)
r
1
(
(r )+
= =
r
i
1
i
(
r
1
)
2 +
(r
4 5
(3)
r
1
(r
(r )+
(r )
(r )+
=
=
r
2 2
1 1 1
(r
1
)+
(r
2
1
(r
)
+
2
3
1
(r
(r
+
1
3
1
(r
2
N.Q. Hoc et al. / VNU Journal of Science: Mathematics Physics, Vol. 35, No. 1 (2019) 1-12
properties of binary interstitial alloy with bcc structure are considered by the statistical moment
method (SMM). The analytic expressions of the elastic moduli for anharmonic fcc and bcc crystals are
also obtained by the SMM and the numerical calculation results are carried out for metals Al, Ag, Fe,
W and Nb in [12]
In this paper, we build the theory of elastic deformation for interstitial AB with body-centered
cubic (BCC) structure under pressure by the SMM [5-7]. The theoretical results are applied to alloy
FeCunder pressure.
2. Content of research
2.1. Analytic results
In interstitial alloy AB with BCC structure, the cohesive energy of the atom B (in face centers of
cubic unit cell) with the atoms A (in body center and peaks of cubic unit cell) in the approximation of
three coordination spheres with the center B and the radii 1, 1
2,r
5 is determined by [5-7]
u0B = AB ( i ) = 2AB ( 1) + 4AB ( 1
i=1
2) +8AB ( 1
5),
(2.1)
where AB
is the interaction potential between the atom A and the atom B, ni
is the number of
atoms on the ith coordination sphere with the radius r(i =1,2,3), 1 r B =r 1B +y0A (T) is the nearest
neighbor distance between the interstitial atom B and the metallic atom A at temperature T, 01B is the
nearest neighbor distance between the interstitial atom C and the metallic atom A at 0K and is
determined from the minimum condition of the cohesive energyu0B ,
y0A (T) is the displacement of
the atom A1 (the atom A stays in the body center of cubic unit cell) from equilibrium position at
temperature T. The alloy’s parameters for the atom B in the approximation of three coordination
spheres have the form [5-7]
kB = 2 i uiAB eq =AB (r)+
2 (1)
AB 1
1
2)+ 5r 65AB ( 1
5),
B = 4(1B +2B ),
1 4AB 1 (4) 1 (2)
1B 48 u4 24 AB 1 8r2 AB 1
eq
2)
2 (1)
16r3 AB 1
1 (4)
150 AB 1
2)+125r AB ( 1
5),
6 4AB 1 (3) 1 (2) 5 (1) 2 (3)
2B 48 i ui ui eq 4r AB 1 4r2 AB 1 8r3 AB 1 8 1 AB 1
2)
1 (2)
8r2 AB 1
1 (1)
8r3 AB 1
2 (4)
25 AB 1
5)+25r
(3)
5 AB 1
5)+
2 (2)
25r2 AB 1
5)25r3
(1)
5 AB 1
5),
(2.2)
(m
i i
r
1 1
( ) ( )
k +
r
r
+
k =k + =
,
1 1
1
i
1A
i
eq
1
1
1
1 1
1
+
(r )
(r )+
+ =
=
(r ),
u
24
8r 8r
48
1 1 1
1
1 1
1
1
+
+
(r ).
(r )
(r )
=
+ =
(2.3)
2 3
2 2
1
1 1 1
1 1 1
1 1 1
1
1
r
2
2
( ) (
)
4
1
k +2
r +
k = k +
=
r
,
2 2 2
i
1A
i
e
q
2
1
A
2 2
2
1 1
1
(r )
+
(r )+
+ =
=
2
2
2
1A
i
i
eq
2
1
2
)+
(r
(r ),
8r 8r
2 2
2 2
4
1
6
(4)
AB
1
2
2
i
i
i
eq
1
2
(r )+
(r )
+
(r ),
8r
4r 8r
2 2 2
A
2 2
N.Q. Hoc et al. / VNU Journal of Science: Mathematics Physics, Vol. 35, No. 1 (2019) 1-12
3
where AB) mAB (r ) / rm (m =1,2,3,4,, = x, y.z,
and ui
is the displacement of the
ith atom in the direction.
The cohesive energy of the atom A1 (which contains the interstitial atom B on the first
coordination sphere) with the atoms in crystalline lattice and the corresponding alloy’s parameters in
the approximation of three coordination spheres with the center A1 is determined by [5-7]
u0A =u0A +AB (1A ),
1 2AB (2) 5 (1)
A A 2 u2 A AB 1A 2r AB 1A
r=r A
A = 4(1A +2A ),
1 4AB 1 (4) 1 (2) 1 (1)
1A 1A 4 1A AB 1A 2 AB 1A 3 AB 1A
i i eq r=r A 1A 1A
6 4AB 1 (3) 3 (2) 3 (1)
2A 2A 48 i uiui eq r=r A 2A 2r A AB 1A 4r A AB 1A 4r A AB 1A
where .. is the nearest neighbor distance between the atom A1 and atoms in crystalline lattice.
The cohesive energy of the atom A2 (which contains the interstitial atom B on the first
coordination sphere) with the atoms in crystalline lattice and the corresponding alloy’s parameters
in the approximation of three coordination spheres with the center A2 is determined by [5-7]
u0A =u0A +AB (1A ),
2
AB (2) (1)
A A 2 u2 A AB 1A r AB 1A
r=r 2
A = 4(1A +2A ),
4
AB (4) (3)
1A 1A 48 u4 1A 24 AB 1A 4r AB 1A
r=r A
1 (2) 1 (1)
2 AB 1A 3 AB 1A
1A 1A
2A =2A + 48u2 u2 =2A +8AB (rA )+
r=r A
1 (3) 3 (2) 3 (1)
AB 1A 2 AB 1A 3 AB 1A
1A 1A 1 2
(2.4)
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Study on elastic deformation of interstitial alloy FeC with BCC structure under pressure. The elastic deformations of main metal A is special case of elastic deformation for interstitial alloy AB. The theoretical results are applied to alloy FeC under pressure. The numerical results for this alloy are compared with the numerical results for main metal Fe and experiments..

Nội dung

VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 1-12 Review article Study on Elastic Deformation of Interstitial Alloy FeC with BCC Structure under Pressure Nguyen Quang Hoc1, Tran Dinh Cuong1, Nguyen Duc Hien2,* 1Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam 2Mac Dinh Chi High School, Chu Pah District, Gia Lai, Vietnam Received 03 December 2018 Revised 16 January 2019; Accepted 04 March 2019 Abstract: The analytic expressions of the free energy, the mean nearest neighbor distance between two atoms, the elastic moduli such as the Young modulus E, the bulk modulus K, the rigidity modulus G and the elastic constants C11, C12, C44 for interstitial alloy AB with BCC structure under pressure are derived from the statistical moment method. The elastic deformations of main metal A is special case of elastic deformation for interstitial alloy AB. The theoretical results are applied to alloy FeC under pressure. The numerical results for this alloy are compared with the numerical results for main metal Fe and experiments. Keywords: interstitial alloy, elastic deformation, Young modulus, bulk modulus, rigidity modulus, elastic constant, Poisson ratio. 1. Introduction∗ Elastic properties of interstitial alloys are specially interested by many theoretical and experimental researchers [1-4, 7-12]. For example, in [3] the strengthening effects interstitial carbon solute atoms in (i.e., ferritic or bcc) Fe-C alloys are understood, owning chiefly to the interaction of C with crystalline defects (e.g., dislocations and grain boundaries) to resist plastic deformation via dislocation glide. High-strength steels developed in current energy and infrastructure applications include alloys where in the bcc Fe matrix is thermodynamically supersaturated in carbon. In [4], structural, elastic and thermal properties of cementite (Fe3C) were studied using a Modified Embedded Atom Method (MEAM) potential for iron-carbon (Fe-C) alloys. The predictions of this potential are in good agreement with first-principles calculations and experiments. In [7], the thermodynamic ________ ∗Corresponding author. E-mail address: n.duchien@gmail.com https//doi.org/ 10.25073/2588-1124/vnumap.4293 1 2 N.Q. Hoc et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 1-12 properties of binary interstitial alloy with bcc structure are considered by the statistical moment method (SMM). The analytic expressions of the elastic moduli for anharmonic fcc and bcc crystals are also obtained by the SMM and the numerical calculation results are carried out for metals Al, Ag, Fe, W and Nb in [12] In this paper, we build the theory of elastic deformation for interstitial AB with body-centered cubic (BCC) structure under pressure by the SMM [5-7]. The theoretical results are applied to alloy FeCunder pressure. 2. Content of research 2.1. Analytic results In interstitial alloy AB with BCC structure, the cohesive energy of the atom B (in face centers of cubic unit cell) with the atoms A (in body center and peaks of cubic unit cell) in the approximation of three coordination spheres with the center B and the radii 1, 1 2,r 5 is determined by [5-7] u0B = AB ( i ) = 2AB ( 1) + 4AB ( 1 2) +8AB ( 1 5), (2.1) i=1 where AB is the interaction potential between the atom A and the atom B, ni is the number of atoms on the ith coordination sphere with the radius r(i =1,2,3), 1 r B =r 1B +y0A (T) is the nearest neighbor distance between the interstitial atom B and the metallic atom A at temperature T, 01B is the nearest neighbor distance between the interstitial atom C and the metallic atom A at 0K and is determined from the minimum condition of the cohesive energyu0B , y0A (T) is the displacement of the atom A1 (the atom A stays in the body center of cubic unit cell) from equilibrium position at temperature T. The alloy’s parameters for the atom B in the approximation of three coordination spheres have the form [5-7] 1  2AB  (2) 2 (1) B 2 i  ui eq AB 1 r AB 1 2)+ 5r 65AB ( 1 5), B = 4(1B +2B ), 1  4AB  1 (4) 1 (2) 1B 48 u4 24 AB 1 8r2 AB 1 eq 2)− 2 (1) 1 (4) 16r3 AB 1 150 AB 1 2)+125r AB ( 1 5), 6  4AB  1 (3) 1 (2) 5 (1) 2 (3) 2B 48 i ui ui eq 4r AB 1 4r2 AB 1 8r3 AB 1 8 1 AB 1 2)− 1 (2) 1 (1) 2 (4) 8r2 AB 1 8r3 AB 1 25 AB 1 3 (3) 25r 5 AB 1 5)+ 2 (2) 25r2 AB 1 5)−25r3 (1) 5 AB 1 5), (2.2) N.Q. Hoc et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 1-12 3 where AB)  mAB (r ) / rm (m =1,2,3,4,, = x, y.z,   and ui is the displacement of the ith atom in the direction. The cohesive energy of the atom A1 (which contains the interstitial atom B on the first coordination sphere) with the atoms in crystalline lattice and the corresponding alloy’s parameters in the approximation of three coordination spheres with the center A1 is determined by [5-7] u0A =u0A +AB (1A ), 1 2AB   (2) 5 (1) A A 2  u2  A AB 1A 2r AB 1A r=r A A = 4(1A +2A ), 1 4AB   1 (4) 1 (2) 1 (1) 1A 1A 4 1A AB 1A 2 AB 1A 3 AB 1A i  i eq r=r A 1A 1A 6  4AB   1 (3) 3 (2) 3 (1) 2A 2A 48 i uiui eq r=r A 2A 2r A AB 1A 4r A AB 1A 4r A AB 1A where .. is the nearest neighbor distance between the atom A1 and atoms in crystalline lattice. The cohesive energy of the atom A2 (which contains the interstitial atom B on the first coordination sphere) with the atoms in crystalline lattice and the corresponding alloy’s parameters in the approximation of three coordination spheres with the center A2 is determined by [5-7] u0A =u0A +AB (1A ),  2  AB (2) (1) A A 2  u2  A AB 1A r AB 1A r=r 2 A = 4(1A +2A ),  4  AB (4) (3) 1A 1A 48  u4  1A 24 AB 1A 4r AB 1A r=r A 1 (2) 1 (1) 2 AB 1A 3 AB 1A 1A 1A 2A =2A + 48u2 u2   =2A +8AB (rA )+ r=r A 1 (3) 3 (2) 3 (1) AB 1A 2 AB 1A 3 AB 1A 1A 1A 1 2 (2.4) 4 N.Q. Hoc et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 1-12 where..is the nearest neighbor distance between the atom A2 and atoms in crystalline lattice at 0K and is determined from the minimum condition of the cohesive energy u0A ,y0B(T) is the displacement of the atom C at temperature T. In Eqs. (2.3) and (2.4),u0A,kA,1A,2A are the coressponding quantities in clean metal A in the approximation of two coordination sphere [5-7] The equation of state for interstitial alloy AB with BCC structure at temperature T and pressure P is written in the form Pv = −r 6 r0 +xcth x 2k r . (2.5) where 4r3 is the unit cell volume per atom, r1 is the nearest neighbor distance, θ = k T , 3 3 k is the Boltzmann constant, k ω , m is the atomic mass and ω is the vibrational 2θ m 2θ frequencies of atoms. At temperature T =0 K, Eq. (2.5) will be simply reduced to Pv = −r  6 r0 + 4k r . (2.6) Note that Eq.(2.5) permits us to find r1 at temperature T under the condition that the quantities k, x, u0 at temperature T0 (for example T0 = 0K) are known. If the temperature T0 is not far from T, then one can see that the vibration of an atom around a new equilibrium position (corresponding to T0) is harmonic. Eq.(2.5) only is a good equation of state in that condition. Eq. (2.6) also is the equation of state in the case of T0 = 0K. In Eq. (2.6), the first term is the change of energy potential of atoms in euilibrium position and the second term is the change of energy of zeroth vibration. If knowing the form of interaction potential i0,eq. (2.6) permits us to determine the nearest neighbor distance rX (P,0)(X = B,A,A,A ) at 0 K and pressure P. After knowing , we can determine alloy parametrs kX (P,0),1X (P,0),2X (P,0),X (P,0),ωX (P,0) at 0K and pressure P. After that, we can calculate the displacements [5-7] y0X (P,T) = 23k((P,0)2 AX (P,T), AX = a X + X i aiX ,kX = mωX ,xX = i=2 X 2 ,a1X =1 + 2 , 13 47 23 2 1 3  25 121 50 2 16 3 1 4  2X 3 6 X 6 X 2 X 3X  3 6 X 3 X 3 X 2 X  43 93 169 2 83 3 22 4 1 5 4X 3 2 X 3 X 3 X 4 X 2 X 103 749 363 2 733 3 148 4 53 5 1 6  5X  3 6 X 3 X 3 X 3 X 6 X 2 X  N.Q. Hoc et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 1-12 5 a6X =65+ 561YX +1489Y2 + 927Y3 + 733YX +145YX + 31YX + 1YX ,YX  xX cothxX . (2.7) From that, we derive the nearest neighbor distance rX (P,T)at temperature T and pressure P rB (P,T) = rB (P,0) + yA (P,T),rA (P,T) =rA (P,0) + yA (P,T), rA (P,T)  rB (P,T),rA (P,T) = rA (P,0) + yB (P,T). (2.8) Then, we calculate the mean nearest neighbor distance in interstitial alloy AB by the expressions as follows [5-7] 1A (P,T)= rA (P,0)+ y(P,T), 1A (P,0) = (1−cB )rA (P,0) + cBrA (P,0),rA (P,0) = 3rB (P,0), (2.9) where 1A(P,T) is the mean nearest neighbor distance between atoms A in interstitial alloy AB at pressure P and temperature T, rA(P,0) is the mean nearest neighbor distance between atoms A in interstitial alloy AB at pressure P and 0K, rA(P,0) is the nearest neighbor distance between atoms A in clean metal A at pressure P and 0K, rA(P,0) is the nearest neighbor distance between atoms A in the zone containing the interstitial atom B at pressure P and 0K and cB is the concentration of interstitial atoms B. The free energy of alloy AB with BCC structure and the condition cB << cA has the form AB =(1−7cB )A +cB B +2cB A +4cB A −TSc, X  N u0X +0X +3N 2 2X XX − 21X 1+ XX  + X + 23 32X XX 1+ 2 −2(1X + 21X 2X )1+ 2 (1+ XX ), 0X = 3N xX + ln(1− e−2xX ), XX  xX coth xX , (2.10) where X is the free energy of atom X, AB is the free energy of interstitial alloy AB, Sc is the configuration entropy of interstitial alloy AB. The Young modulus of alloy AB with BCC structure at temperature T and pressure P is determined by   B + 22 A + 42 A  EAB cB ,P,T = EA 1− 7cB + cB 2 , EA = , A 1A 1A  2  A A = kA 1+2A2 1+ XA (1+XA ), 6 N.Q. Hoc et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 1-12 2 X 1 2u0X 3 ωX 2kX 1  kX 2  2 2 2 rX 4 kX  rX 2kX rX   01X + 2 rX + 2 ωX cthxX 2kX rX 2r1X , (2.11) where  is the relative deformation. The bulk modulus of BCC alloy AB with BCC structure at temperature T and pressure P has the form KAB (cB ,P,T)= EA(1 cB ,P,T). (2.12) The rigidity modulus of alloy AB with BCC structure at temperature T and pressure P has the form GAB (cB ,P,T)= EAB (cB ,P,T). (2.13) AB The elastic constants of alloy AB with BCC structure at temperature T and pressure P has the form C 1AB (cB ,P,T)= EAB (cB ,P,T)(1−AB ), (2.14) AB AB C 2AB (cB ,P,T)= EAB (cB ,P,T)AB , (2.15) AB AB C44AB (cB,P,T)= E2((cB,P,T). (2.16) The Poisson ratio of alloy AB with BCC structure has the form AB = cA A +cBB A, (2.17) where A and B respectively are the Poisson ratioes of materials A and B and are determined from the experimental data. When the concentration of interstitial atom B is equal to zero, the obtained results for alloy AB become the coresponding results for main metal A. 2.2. Numerical results for alloy FeC For pure metal Fe, we use the m – n potential as follows n m (r) = n−m m r  −n r  , where the m – n potential parameters between atoms Fe-Fe are shown in Table 1. For alloy FeC, we use the Finnis-Sinclair potential as follows N.Q. Hoc et al. / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 1 (2019) 1-12 7 U =−A ( ij )+ 1( ij ), ij ij (r) =t1 (r −R )2 +t2 (r −R )3 (r) =(r − R2 )2 (k1 +k2r +k3r2 ) (r < R ), (r < R2 ). (2.19) where the Finnis-Sinclair potential parameters between atoms Fe-C are shown in Table 2. Our numerical results are summarized in tables and illustrated in figures. Our calculated results for Young modulus E of alloy FeC in Table 3, Table 4, Fig.5 and Fig.6 are in good agreement with experiments [10]. Table 1. The m-n potential parameters between atoms Fe-Fe [8] Interaction m Fe – Fe 7.0 n D(eV) 11.5 0.4 o r A 2.4775 Table 2. The Finnis-Sinclair potential parameters between atoms Fe-C [9] (eV) 2.958787 R1 o A 2.545937 t1 o −2 A 10.024001 t2 o −3 A 1.638980 R2 A 2.468801 k1   o −2      8.972488 k2 −3 eVA    -4.086410 k3   o −4      1.483233 Table 3. The dependence of Young modulus E(1010Pa) for alloy FeC with cC = 0.2% from the SMM and alloy FeC with cC  0.3% from EXPT[10] at zero pressure T(K) 73 144 200 294 422 SMM 22.59 22.03 21.58 20.75 19.49 EXPT 21.65 21.24 20,82 20.34 19.51 533 589 644 18.28 17.65 16.96 18.82 18.41 17.58 700 811 866 16.26 14.81 14.06 16.69 14.07 12.41 Table 4. The dependence of Young modulus E(1010Pa) for alloy FeC with cC = 0.4% from the SMM and alloy FeC with cC  0.3% from EXPT[10] at zero pressure T(K) 73 144 SMM 22.46 21.90 EXPT 21.51 21.10 200 294 21.45 20.62 20.68 20.20 422 533 19.38 18.18 19.37 18.62 589 644 700 17.53 16.87 16.17 18.27 17.44 16.55 811 866 922 14.72 13.98 13.21 13.93 12.34 10.62

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