INTROD UCTIO N It is known th a t if the free energy of a system is known, we can find the thermo dynamic properties.. So it is vory important to dpterrnine the free energy V' although
Trang 1VNU J O U R N A L O F S C I E N C E Nat Sci t.xv n^2 - 1999
IN V E S T IG A T IO N OF T H E R M O D Y N A M IC PR O PE R T IE S
OF B IN A R Y A - B ALLOYS B Y T H E M O M E N T M E T H O D
Vu Van Hung, Hoang Vail Tich, Nguyen Van Quang
F a cu k y o f Physics, Teacher's Training College - V N U
A b s t r a c t B y the m o m e n t method the thermodynamic quantities of binary A - B
a l l o y s w ith f c c s t r u c t u r e are c o n s id e r e d T h e a n a l y t i c e x p r e s s i o n o f t h e r m o d y n a m i c
quavfiUes for the binary A - B alloys as the isothermal compressibility X t ^ linear thermal expansion coefficient a, the specific heat at constant volum C y are obtained The obtained results are apphed to A l - based binary alloys { A lC u , A l N i ) ^ C u - based binary alloys { C i i A l ) and N i - based binary ( N i A l ) and compared with the exper imental data.
I INTROD UCTIO N
It is known th a t if the free energy of a system is known, we can find the thermo dynamic properties So it is vory important to dpterrnine the free energy V' although it is
At first, wo shall restrict ourselves to the simplest case of pairwise interactions
To calculatp Ự’ of biliaiv A B alloys we shall use the quasi-chemical approximation for
the multi - conipoiu'ut systeiiis [1, 2 3] We denoto th r potential of interaction of an
atom of ( oinponeiit A with th at of componoiit B ^ a n d the numbers of atoms in these
G = Y ^ N A f i A A + N a b M a b - k B T l n W { { N A ] { N a b } ) , (1)
th aiui B - t h C'ornpononts:
, f - ụ -A A - ụ -B B
where nj is the (first) coordination number; HAS = tí\ụ>AB\ is the chemical potential of the pure A B subsystem, i.e., of the imagine one - component system in which all atoms interact with each other by the potential ipAB and k'B is the Boltzman’s constant.
41
Trang 242 Vu Van Hung, H o a n g Van Tick, N g u y e n Van Quang
(3)
(4)
In the consideml case where the concentration of on<‘ component is small (for
of the pairs and we have
Substituting (5) into (1) aiui taking into account (4) we get
G - G,\ + N b { 2 h a b - fiAA) - k s T l n—^7 \jịỊ \ ’ ( 6 )
wheiP G a is Gibbs free energy of metal A In this paper, using the obtained in [4] results for the Gibbs free eiieig.v GA of niPtal A and (6) WP investigate the thermodynamic properties
of b i n a r y A ~ B a l lo y s with face -centered cubic structures The analytic expressions for the thennociyuainic quantities as the thermal expansion coefficient Q, the specific heats
Cv and Cp, Pte are obtaiupcl The obtained results are compared with the experimental
data
II T H E EXPRESSION OF T H E THER M O D Y N A M IC QU ANTITIES
FOR THE BINARY A - D ALLOYS WITH F c c STRUCTURES.
At first wr can find tlif> Gibbs flPO riif'igy of alloys ill the approximation form analogous to (6)
N„y-is the entropy of mixing
If only take into’ account the interaction of particles being on first coordination sphere, we can find
where A \ ' is the volume change on substituting a particle 5 , Ui is the num ber of particles
Trang 3atom B with atoms A boiiig on two first and second coordination spheres, i.e., of an atom
of th(' iniagiiu'd one - coinpoiiont system in which all atoms interact with each other bv the
tli(' tirst coordination sphero in the imagined system in which all atoms interact with each
with otht’i oiH*s by tli(‘ Ị)otíMitial ự>AB
Siiỉ>stitutiiifj (9) into (7) we have found the Helmholtz free energy V’ of binary A B
allovb (in the case of the pressure p — 0)
I n v e s t i g a t i o n o f T h e r m o d y n a m i c P r o p e r t i e s o f B i n a r y A - B Alloys by., 43
0.4.4 = 3 { ^ + ỡ [ , T + / n - ( l - e - 2 " ) ] } ,
Ĩ
r*fìí-ì»» f i <1 I Z 1 1 - * 4-K «
-rA.Ao i i i T t ' i a c t i o n poteMitial energy betwoen zero-th and i - th particles of A
i i H ' i a l , i ' Ị ị .\ has H form aiialoí2,ous to ( 1 1 ) but p a r a n i e t o r k in this case has the form
( 12 )
111 t l i i ’ a Ị ) ] ) i o x i n i H t ( ' form
(13)
]>aiauK'tf‘i /.■ aiiil //{) c\rc (‘fpial to
ớ'V4/A(ri)
th(' ùist c o o i d i i i a t i o n s p l i e r o ( i n t h e cas(' o f f.c.c l a t t i c e Uk e q u a l t o Ỉ^I = l , ỉ/2 = a / 2 ) and
If tho displaceniont of the particlf' from equilibrium position of perfect metal A is
A B subsystem i.e., of the imagined one - component system in which all atoms with the
Trang 444 Vu Van Hung, H oang Van Tick, N guy en Van Q uang
corresponding nearest neighbors distances at tem p eratu re T are equal to
(15)
By the moment method, the displacement, of the particle of the metal A is considered
and is equal to [4
VỐ =
2 7 0 2
2 q 2 3f)3 ^ 4 0
«1 = 1 +
Ả-4
.TCth.T
0-2 +
Ẳ-6 « 3 + k.^
■0-4
(16)
ư
0.3 =
- T + T
i -x c i n x - t — X c m X -f- - X cTii X,
— + ^^.Tcth.T + — x cWi X + — r cth T + ^.T^cth X ,
: ^ c t h ^ r + ^ T ^ c t h ^ T + ^ T ^ c t h ' ‘ T + i , T ^ c t h ^ x ,
x t n x H -X
1 V - ( d ^ ^ A , A o \ , J d ^ i p A A o \
the aid of (13) and (17) the expression of the displacement of the particle yi (or ya) has
a form analogous to (16), but the parameter k in this case has the form (12) or (14) and
I B A = Ạ l A + 1 b )
l A B = 1 A +
d u L
V d v ĩ n d u l
d ^ < ^ A A { r \ ) ^
/ e g j
We notice th a t the nearest neighbor distance a of the binary A B alloy is approxi mately equal to the distances ữA.QAB Oĩ a s A- Besides, from (10) we see th a t the Helmholtz
Trang 5free ('Ii( i f>,v i,' is H f iuu tioii of t h e Ii(>an'st n o i g h h o i ( li s ta nc p a. T h u s , p x p an c l i n g t hi s f u n c t i o n
on tlic ncaiPst iK'iglihoi distance a in spcond order approximation, wo find the follov/ing ('XjJK'ssious
I n v e s t i g a t i o n o f T h e r m o d y n a m i c P r o p e r t i e s o f B i n a r y A - B A l l o y s by 45
da'^ J t x ( a - cia Ý,
= i Ị' b a {<’ b a ) + 2 d'h'>HA
V' a b ( o ) = ^ ' a b { « a b ) + ^ “ ‘ ^ a b Ý
-Fioiii the (lefiuitioii of the isothennal bulk modulo B'r with B y = y o { ^ ) r j , , the K'siilts (10) (19) and miniiiiiziiig Ụ' : ( |^ ) y y, ^ = 0 we can find the pquilibrium distance
( Ì —
u.siuii, thv theniiodyiianiic relations and the expression of the Helmholtz free energy
oxpaiision coi'fficii'ut o the sp('cific heats (TV and Cj, of binary A B alloy Where, the
isoĩhrnnal conipiossiiíiliĩy has tho fonn
(2 1)
hon
* l /í _ * i
\ I = rrrrí
: i N
)/■ ^>“ 1 = í r
Fruin lli(' (lofinitioii o f tho t hornial (‘x p a i i s i o n cơ(*ífic-ient, it is e a s y t o d e r i v e t h e
t’t'lluwiiiii f u i n m l a
wIk'K' a ' is the linear thcinml expansion coefficient of the metal A [4];
ri BA
and
\
d'^tBA da.dtì d'^ỶAB
Trang 64G Vĩi Van Hung, Ho ang Vail Tich, N g u y e n Van Quang
A p p l y i n g th(' Gil)l)s - l l t ' l i n h o l t z i ( ‘l a t i o n a i u l Tisiii^ (10) \V(‘ Hiui i!io ( ' x p r r s s i ) n f o r
ĩ h o (MU'r^y o f biliarv A Ỉ Ỉ a l l o y a n d s o t h e s p oc iH c h e a t at c o n s t a n f v ol unu' r v lia,s t \\0 f o r m
— 1 ~ C i i i Ỉ - f /M ) "t” ( ' ỉ ỉ -f ( 2 5 ) ill w l i i c h r'^.' is t l i r l ieai at c o n s i a i i t voliiiiH’ o f iiH'tal A [4j A c c o i d i i i ^ ‘ 0 tlì(' al)o\'(' o h t a i i H ' d i r s i i l t s in 0](li*r t o f ind ' Oi \ \ v Iimst US(' íli(' ('xpi'i'ssioiis )ĩ t h ( '
p a i a i i H ' t e r s Ả- d ( ‘fiu('(l l>y ( 1 2 ) ( 11) a i u l ( 1 8 ) c o i K ' s p o i u l i i i ^ t o t h(' tV(’(' ('iK'i'^y o r
t ' \}ỉ T h ( ' s p o c i í i c h e a t at c o n s t a n t p i ( ‘s s i n0 c.'f> a n d t h ( ‘ a í l i a h a ĩ i c c o i n p i o s s i h i l i t y ; a i ( ‘
n i i i u ' d f r o m f h o k n o w n t l i o n i i o d v i i a i u i c I f ' l a t i o n s
9 T \ ' a ' C r
c , - f ' r -f
\ 7
( 2 6 )
A t l a s t t h ( ‘ i s o l l u n n i a l a n d a d i a b a t i c l ) ul k i nochi li D ị a n d o f l ) i u a i \ ‘ A B a l l o \ '
ar( ' ('(Ịual t o
I I I N U M E R I C A L R E S U L T S F O R A l C u A l X i C u A l A N D X i A l A L L O Y S
T h ( ' i n t i ' i a c t i o n p o t ( ‘n i i a l l)(‘t \ v e e n t w o a t o m s o f a iiK'tal is list'd ill t h e f o i u i
of t l i c n — fif oiii' [7j
I)
f { r ) =
Ì Ì Ỉ
IỈ / /■() Vrn
\vh<*r(' D /■() a i ( ‘ f ioiii tli(' ( ' x p o r i n u M i t a l ( l a t a a n d li i ii a r o ( l o t on ii ii K ’s hv t l i i ‘ ('iu])irii al w a y (ill T ii\)\v 1) [7
{ A l C i i A I N i ) , C i i - l)as('(l h i n a n ’ a l l o y s ( C / / 4 / ) a n d N i - hasf'cl h i n a i v a l l o y s ( N i A l ) w i t h
f c c s t n i c t u n ' {ii\ = 12) U s i i i ^ ( 2 8 ) , Tal)l(' 1 a n d ( Ỉ I ) ( 12) ( 1 4 ) ( 1 7 a n d 18) \V(‘ o b t a i n
t h (' v a l u o s o f ] >a ia ui (' t( ns A T l i r n ' f o r ( ' , fVoiii I'i'snlts iU)(l ( 11) ( 1 5 ) ( IG) ( 20 ) ( 2 7 )
\ vr o h t a i i i í h ( ' \ a l u ( ' s o f till' cuii iỊ íi os si hi l it x' \ /- liiK^ai' t li oi ii ia l ('xj)asicjn c o ( ‘fiic-i('nt a aiiil
c o n s t a n t - p K ' s s u n * spfH’itic lu'iit C/> a t p i r s s u i o p = 0 T h o K ' s u l t s f or A l C i i A Ỉ N Ỉ c t i AỈ
a n d A ' / 4 / a l l o y s a i r sunii iiari z(Hl in T a b l i ' s 4.
In t h ( ' c a s o o f A i C n a n d N i p u r o m e t a l s {tlì(' c o i i c o i i t i a t i o i i o f a t o i i i s B : c Ịi —
())-í l u ' oỉ)taiiKHÌ K ' s u l f s vvi'll coincidi* w i t l i tli(' (' xpiniiiUMital ( l a t a (lal >l (' s 2 1).
F o r Al C' f i , A ! ! \ / ( ’i i Al a n d N i A Ỉ a l l o \ ' s ị C ỉ i < 0 1) t h e c a l c u l a t i H l l i ' s u l i s for (\ a n d
C' r a l s o c o i u c i d i ’ \V('11 w i t h t h í ' (\\p<niiiKMital ( l a t a (Tal)l<‘s 3 4).
Table 1: E x p erim en tal v a lu e s of p a ra m e te rs D, To [7]
Trang 7lììv es ti g a ti o n o f Therrtiodynamic P r o p e r t i e s o f B i n a r y A - B A lloys by , 47
A1 Cp (Cal/niol.K) 2.99 Õ.07 5.69 5.99 6.18 6.34 6.65
Cu Cp (Cai/mol.K) 3.80 5.37 Õ.78 5.97 6.08 6.17 6.32 6.47 6.63
Ni Cp (Cal/mol.K) 3.40 5.20 5.68 5.89 6.01 6.10 6.24 6.37 6.62
Table 2: The specific h e a t at c o n s ta n t p re ssu re Cp of m etal
Table 3: The specific h e a t a t co n stan t p re ssu re Cp of alloys a t T em p era tu re 400‘^K
Allovs
Table 4: T herm al expansion coefficient a of alloys
Trang 8In conclusion, it should be noted th at the moment m ethod really to iiiv^sngate tho therniodynainic prop(*rtios of binary allovs with face - rentered cubic structin Theso results are light still for other cubic ones However, we must notice th at thoi) paraiiietei are (leterinined by other formulae
In the following paper we shall use the results of this paper for the invetgation of
th e therniodynaiiiic propeities of alloys witli other cubic* structure
R EFER EN C ES
[1] J.L Hill Statistical Mechanics, Me Gray - Hill Book Company, NewYork 7oronto/Lon
1956: Izd in Lit., Moskva, 1960
3 V.B Magalinsky Izv vuzov, Ftzika, No- 9(1997)17.
4 Nguven Tang and Vu Van Hung Phys Stat Sol (b), 149(1988) 511.131(1990)
165
5] Mark Mostoller and all Phys Rev B, Vo 19, No8(1979) 3938.
Press, New York/London, 1961
7] Zhenshu, G J Daries Phys S t a t S o l (a) V A78, No2(1983) 596.
9] Metal Hand Book EdifAon 1948 (American Society fore metals) 811 ~ 143.
T A P CHÍ KHO A HOC OHQGHN, K H T N t x v n°2 - 1999
N G I I I Ẻ K C Ứ Ư C Á C T Í N I 1 C H A T N I 1 I Ệ T Đ Ộ N G C Ủ A I Ĩ Ợ P K I M e M
THAY T H E A- B BANG PH Ư Ơ N G PH Á P MÔMEN
Khoíĩ Vật Ịý - Đại học Sư phạm - DH QG Hà Nội
Bằng phương pháp mòmen, các đại lượng nhiệt động của hợp kim tha>thế AB có cấu trúc lập phư ang tâm diện đ ã được nghiên cứu Biểu thức giải tích của cá'đại lương nhiệt động như hệ số nén đ ằn g nhiệt XTi hệ số dản nờ nhiệt a , nhiệt d u n g iêng đằng tích C y , của hợp kim thay thế AB đ ã thu nhận được Các két quả lý thuyt được áp dụng cho các hợp kim đôi AI (AlCu, AlNi) hợp kim đòi gốc Cu (Cu Al) v à ^ (Ni Al)
và so sánh với số liệu thực nghiệm