SCIENCE CHINA Structure and property of metal melt IV—Evolution

SCIENCE CHINA
Physics, Mechanics & Astronomy
• Article •
August 2012 Vol.55 No.8: 1371–1375
doi: 10.1007/s11433-012-4804-8
Structure and property of metal melt IV—Evolution of titanium
melt residual bond structure and its effect on dynamic viscosity
MI GuangBao1,2*, CAO JingXia1, HUANG Xu1, CAO ChunXiao1, LI PeiJie2 & HE LiangJu2
1
Aviation Key Laboratory of Science and Technology on Advanced Titanium Alloys, Beijing Institute of Aeronautical Materials,
Beijing 100095, China;
2
National Center of Novel Materials for International Research, Tsinghua University, Beijing 100084, China
Received October 31, 2011; accepted December 13, 2011; published online June 20, 2012
Based on the concept of melt residual bonds, a calculating model quantitatively describing the evolution of the residual bond
structure of titanium melt at the melting point or in a certain range above the melting point was established; i.e., both the size
dS and the bond number n of the residual bond structure decrease monotonously with the increase of temperature. By mathematical deduction, a linear relationship between the residual bond structure size dS and the dynamic viscosity  of Titanium
melt was revealed, i.e., = 0.876 + 0.471·dS, which is of great significance to the investigation of the relationship between the
melt microstructure and the macroscopic properties of metals with high melting temperature.
titanium melt, residual bond structure, dynamic viscosity, calculating model
PACS number(s): 61.25.Mv, 61.20.Gy, 36.40.-C, 66.20.+d
Citation:
Mi G B, Cao J X, Huang X, et al. Structure and property of metal melt IV—Evolution of titanium melt residual bond structure and its effect on dynamic viscosity. Sci China-Phys Mech Astron, 2012, 55: 13711375, doi: 10.1007/s11433-012-4804-8
Titanium has obtained increasing applications due to many
excellent properties such as high specific strength and good
corrosion resistance. Currently, people have acquired a clear
understanding of the crystal structure of solid titanium [1].
Titanium is a transition metal with atomic number 22 and
has two allotropic forms at room condition: -Ti below
1155.5 K, in hcp structure with the lattice constants a=
2.9504 Å and c/a=1.587; -Ti above 1155.5 K, in bcc
structure with the lattice constant a=3.282 Å; the phase
transformation from  to  results in a 0.17% change in
volume. In addition, titanium is the metal with the highest
melting point among light metals, whose melting point is
1943 K and the density is 4.50 g/cm3. However, due to the
unstability and uncertainty of melt structure and the limitations of experimental conditions, experiments on the struc*Corresponding author (email: [email protected]; [email protected].
cn)
© Science China Press and Springer-Verlag Berlin Heidelberg 2012
ture and properties of titanium melt are difficult to conduct,
corresponding to a few research reports. Therefore, on the
basis of earlier relevant researches [2–4], this paper theoretically studied the evolution of the titanium melt structure
and its effect on viscosity at the melting point or in a certain
range above the melting point, which laid a foundation for
deeper investigations on the liquid structure of metals with
high melting temperature and the microscopic nature of
viscosity.
1 Evolution of the residual bond structure of
titanium melt
When the metal melts, the local atomic distribution still
shows certain regularity; i.e., in the range of dozens, hundreds even thousands of atomic distance, the atomic arrangement is similar to that of solid and the change of orphys.scichina.com
www.springerlink.com
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Mi G B, et al.
Sci China-Phys Mech Astron
dering is small. In melt, the region with such arrangement
pattern is called local ordered structure or atomic cluster.
Such facts have been confirmed by X-ray diffraction, neutron diffraction and other experiments [5–11]. Besides, with
the development of cluster physics, it has been proved that
the atomic cluster constructed by a certain quantity of atoms
possesses a relatively stable structure [12–17]; e.g., the
atomic cluster Al13 shows a stable structure, which provides
additional evidence for the existence of local ordered structure in melt.
When the liquid metal transforms into vapor, the mass
spectrometry analysis on aluminium and copper showed
that atomic clusters with a certain size exist in the vapor
(Figure 1) [18]. As Figure 1 shows, the size of atomic clusters changes with inert gas environment; i.e., the cluster size
increases gradually with the atomic size of inert gas and the
cluster size increases obviously with the pressure of inert
gas. The results above indicated that the bonds retained
from the liquid still preserved in the metal vapor. When the
atomic number of inert gas increases or the atomic size of
inert gas enlarges, the chance of the interaction among the
greatly weakened bonds in the vapor may grow from the
point of probability. The bonds bind together and the atomic
clusters form, while the inert gas atoms still move freely.
To sum up, the local ordered structure of melt can be described from the view of chemical bond: assume that the
atoms in free state with chemical bonds totally destructed
are active atoms, and the atoms linked by the retained
chemical bonds after phase transformation are defined as
local ordered structure; i.e., the active atoms yield due to the
destruction of a certain quantity of chemical bonds in the
metal crystal, while large numbers of undestroyed chemical
bonds retain in the local ordered region. From the point of
statistical average, the retained chemical bonds bind the
atoms other than the active atoms together in a certain arrangement pattern, and a time average of the space positions
of all atoms composes local ordered structure. The retained
chemical bonds in the local ordered structure are called melt
residual bonds. In this sense, the local ordered structure
constructed by melt residual bonds is called melt residual-
Figure 1 Relationship between atomic cluster (particle) and inert gas
pressure after the liquid to vapor transformation of liquid metals [18].
August (2012) Vol. 55 No. 8
Figure 2
Structure model of the metal crystal after melting [3].
bond structure, as shown in Figure 2. With atoms aggregating and dispersing randomly, the melt residual bond structure fluctuates under the statistical law. The main parameters describing the melt residual bond structure are the statistical average size ds and the residual bond number n in the
structure.
Based on the concept of melt residual bonds, the evolution of residual bond structure can be described quantitatively by the melt structure information calculating model
[3,4]; i.e., ds and n are:

 Z  1  C
 Q  1 1   
0
d (T )  2r   1 
exp       1   1 ,
 k T T   

 2  C0
m 
 
 


3
(1)

 Q  1 1  
Z 3 1  C0
exp       1 ,
n(T )  1 
k T T 
8  C0
m 
 


(Tm  T  TC ),
where k is the Boltzmann constant; α, the geometrical morphology factor (0<1); Z1, the coordination number of
metal before melting; r, the half atomic distance in the residual bond structure, which is assumed to be equal to the
single-bond radius of atom; Q, the activation energy; C0, the
relative concentration of active atoms at the melting point;
Tm, the melting point; TC, the temperature when the first
transformation of melt structure happens during the process
from liquid to vapor.
The thermophysical and structure parameters of titanium
are [19,20]: Tm =1943 K, Hb=425.0 kJ/mol, Hm=14.15
kJ/mol, r=1.467 Å. Substitute the parameters into eq. (1)
and obtain the formulas for dS and n:


1 
1
d (T )  79.218exp  5615.025   
   16.137,
 T 1943  



3
(2)


1  
1






n
(
T
)
216
6
exp
5615.025
1


 T 1943    ,


 



(Tm  T  TC ).
Mi G B, et al.
Sci China-Phys Mech Astron
Using eq. (2), the residual bond structure parameters of
titanium melt between the melting point and 200 K above
the melting point were calculated, as shown in Figure 3. As
shown in Figure 3, the average size of the titanium melt
residual bond structure and the bond number in the structure
decrease monotonously with the increase of temperature.
2 Relationship between the dynamic viscosity
and the residual bond structure parameters of
titanium melt
Using function transformation on the model to calculate the
residual bond structure parameters of melt, the relationship
between the size of the residual bond structure and melt
temperature is obtained:
T (ds ) 
1
, (Tm  T  TC ). (3)

C0  ds  2r  
1 k
 ln 
 1 

Tm Q 1  C0   z1r

Substituting eq. (3) into the Arrhenius equation [21] we
obtain the melt kinematic viscosity:
 H
 C  d  2r  
H
v(ds )  A exp  v  v ln  0  s
 1 
 RTm
R 1  C0   z1r
 

(Tm  T  TC ),
k
Q

,


(4)
v(ds ) 

 H 
C0  2
 1 A exp  v 

1  C0   z1 
 RTm 

 H 
C0 1 1
A exp  v   ds , (Tm  T  TC ), (5)
1  C0  z1 r
 RTm 
Substituting the relation between kinematic viscosity and
dynamic viscosity v 
Figure 3

 H 
 C0  2
 1 A exp  v 

1  C0   z1 
 RTm 
 H 
C0  1

A exp  v   ds , (Tm  T  TC ). (6)
1  C0  z1 r
 RTm 
 (d s ) 
According to the dynamic viscosity at the melting point
 H 
 m  A  exp  v  , the dynamic viscosity above the
 RTm 
melting point is:
 (ds ) 
C0
1  C0
 2

C0  m 1
 1  m 
 ds ,

1  C0  z1 r
  z1 
(Tm  T  TC ).

into eq. (5) we obtain:

(7)
Define
0 

C0  2
C0  m 1
.
 1m , K1 

1  C0   z1 
1  C0  z1 r
Thus, eq. (7) can be expressed as:
  0  K1  ds ,
(8)
where ds is the average size of the residual bond structure;
0 and K1 are the constants.
Based on the semi-empirical formula of viscosity derived
by Andrade, which considers Lindemann melting law [21],
the dynamic viscosity at the melting point is obtained:
 m  1.8  107
After simplification:
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August (2012) Vol. 55 No. 8
( MTm )
2
Vm
3
1
2
,
(9)
where M is the atomic mass (kg), Tm is the absolute temperature of the melting point (K) and Vm is the atomic volume
at the melting point (m3).
Combining eq. (7), eqs. (8) and (9), substituting the basic
parameters of titanium (M=0.04788 kg, Tm=1943 K, Vm=
1.165×105 m3) into the equations, we obtain the relationship between the dynamic viscosity and the residual bond
structure size of titanium melt:
Evolution of the residual bond structure size dS of Titanium melt and the bond number n in the structure. (a) dS; (b) n.
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Mi G B, et al.
  0.876  0.471  ds .
Sci China-Phys Mech Astron
(10)
According to eq. (10), the dependence of the dynamic
viscosity of titanium melt on the residual bond structure size
in the temperature range from 1943 K to 2143 K was calculated, as shown in Figure 4. At 1943 K, the dynamic viscosity  = 3.850×103 Pa·s, corresponding to the earlier work 
= 4.42×103 Pa·s with deviation smaller than 15% [22].
Thus, on the one hand, the relationship between the
macroscopic viscosity and the microstructure parameters of
titanium melt is revealed by eq. (10); i.e., the viscosity increases linearly with the residual bond structure size of the
melt, which provides a new way to calculate the viscosity of
metal melt. On the other hand, the microscopic mechanism
of the dependence of the titanium melt dynamic viscosity on
temperature is reflected by eq. (10); i.e., the microscopic
nature of viscosity can be described as the size variation of
the viscous flow unit (residual bond structure), which provides direct theoretic evidence for views such as the concentration of atoms in the crystal-like atomic clusters of tin
melt is only a function of temperature [23], aluminium
atomic cluster is an independent unit of viscous flow [24],
the kinematic viscosity of magnesium and aluminium has a
one-to-one correspondence with the microscopic structure
parameters [25–27] and the increase of viscosity reflects the
improvement of ordering [28].
+0.471·ds, which provides a new way to calculate the viscosity of titanium melt.
(3) The microscopic mechanism of the dependence of
viscosity on temperature is attributed to the evolution of the
residual bond structure size, which provides direct theoretic
evidence for experiment results about the viscosity and the
atomic cluster size of tin and aluminium melts, etc.
This work was supported by the National Basic Research Program of China (Grant Nos. 2007CB613803 and 2007CB613702).
1
2
3
4
5
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3 Conclusions
9
(1) Based on the concept of melt residual bonds, a model
quantitatively describing the evolution of the residual bond
structure of titanium melt at the melting point or in a certain
range above the melting point was established; i.e., the residual bond structure parameters of titanium melt (ds, n)
decrease monotonously with the increase of temperature.
(2) By mathematical deduction, a relationship between
the average size of the residual bond structure and the dynamic viscosity of titanium melt was established: =0.876
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Figure 4 Relationship (ds) between the viscosity and the residual bond
structure size of titanium melt.
August (2012) Vol. 55 No. 8
19
Moiseyev V V. Titanium Alloy Russian Aircraft and Aerospace Applications. New York: CRC Press, 2006
Mi G B, Li P J, He L J, et al. EET research on melt structural information of magnesium Alloy. Rare Metal Mater Eng, 2010, 39(11):
1881–1887
Mi G B, Li P J, He L J. Structure and property of metal melt I. The
number of residual bonds after solid-liquid phase changes. Sci China-Phys Mech Astron, 2010, 53(9): 1571–1577
Mi G B, Li P J, He L J. Structure and property of metal melt II. Evolution of atomic clusters in the not high temperature range above
liquidus. Sci China-Phys Mech Astron, 2010, 53(10): 1823–1830
Khrushchev B I. Nature of the bond and change in the structure of
metals during melting. Strukturnoi Khimii, 1971, 12(6): 958–963
Данилов В И. Строение и кристаллизация жидкости: Избранные
статьи. Киев: АН УССР, 1956: 2–4
Бухаренко В В, Чень С Ш, Ильинский А Г, и др. Рентгеновское
исследование структуры жидких сплавов системы индий-галлий.
Металлофизика, 1991, 19(10): 92
Ляшко А С, Полтавцев Ю Г. Рентгенографическое исследование
жидкого галлия в широком интервале температуре. Украинский
Физический журнал, 1968, 13(9): 1579–1583
Кпименков Е А, Гельб П В, Баум Б А, и др. О структуре
ближнего порядка в жидком железе, кобальте и никеле. Докл. АН
СССР, 1976, 230(1): 71–73
Ершов Г С, Бычнов Ю Б. Высокопрочные алюминиевые сплавы
на основе вторичного сырья. Москва: Металлургия, 1979: 5–60
Казимиров В П, Роик А С, Самсонников А В, и др. Характер
упорядочения атомов в расплаве и поверхностные свойства
систем с интерметаллическими соединениями. Сверхтвердые
материалы, 2009, 4: 40–54
Rao B K, Jena P. Evolution of the electronic structure and properties
of neutral and charged aluminum clusters: A comprehensive analysis.
J Chem Phys, 1999, 111(5): 1890–1904
Arnold G L, Anbar A D, Barling J, et al. Formation of Al13I: Evidence for the superhalogen character of Al13. Science, 2004, 304:
84–87
Chacko S, Deshpande M, Kanhere D G. Structural and electronic
properties of aluminium-based binary clusters. Phys Rev B, 2001, 64:
155409
Medel V M, Reveles J U, Khanna S N, et al. Hund’s rule in superatoms with transition metal impurities. PNAS Early Ed, 2011: 1–5
Woodward W H, Eyet N, Shuman N S, et al. Aluminum Cluster anion reactivity with singlet oxygen: Evidence of Al9-stability. J Phys
Chem C, 2011, 115: 9903–9908
Milligan J, Heard D W, Brochu M. Formation of nanostructured
weldments in the Al-Si system using electrospark welding. Appl Surf
Sci, 2010, 256: 4009–4016
Granqvist C G, Buhrman R A. Ultrafine metal particles. J Appl Phys,
1976, 47(5): 2200–2219
Zhang R L. Empirical Electron Theory of Solids and Molecules (in
Chinese). Changchun: Jilin Science and Technology Press, 1993.
66–67
Mi G B, et al.
20
21
22
23
24
Sci China-Phys Mech Astron
Dean J A. Lange’s Handbook of Chemistry. 15th ed. New York:
McGraw-Hill, 1999. 6.124–6.142
Iida T, Rodarick I L. The Properties of Liquid Metal. Oxford: Clavendon Press, 1993. 148–153
Paradis P F, Ishikawa T, Yoda S. Non-contact measurements of surface tension and viscosity of niobium, zirconium, and titanium using
an electrostatic levitation furnace. Int J Thermophys, 2002, 23(3):
825–842
Швидковский Е Г, Горяга Г И. Метод измерения вязкости
металлических расплавов: влияние примеси на вязкость. Вестн.
МГУ, 1953, 9: 63
Sklyarchuk V, Plevachuk Y, Yakymovych A, et al. Structure sensitive properties of liquid Al-Si alloys. Int J Thermophys, 2009, 30:
1400–1410
25
26
27
28
August (2012) Vol. 55 No. 8
1375
Mi G B, Li P J, Okhapkin A V, et al. Relationship between liquid
structure and property (i). Kinematic viscosity of Mg melt and its relationship with the microstructure (in Chinese). Acta Phys Sin, 2011,
60(4): 046601
Mi G B, Li P J, Okhapkin A V, et al. Relationship between liquid
structure and property (ii). Kinematic viscosity of Mg-9Al melt and
its relationship with the microstructure (in Chinese). Acta Phys Sin,
2011, 60(5): 056601
Mi G B, Li P J, Popel P S, et al. Structure and property of metal melt
III. Relationship between kinematic viscosity and size of atomic
clusters. Sci China-Phys Mech Astron, 2010, 53(11): 2054–2058
Qi J G. Research on Electric Pulse Treatment and Liquid Structure of
Aluminum Melt (in Chinese). Beijing: University of Science and
Technology Beijing, 2006. 39–55