Numerical integration calculation of collision integrals for exponential attractive potential of gase

Mason, E. A.

Munn, R. J.

Smith, F. J.

English

NASA Technical Reports Server

UNIVERSITY OF MARYLAND 
INSTITUTE FOR MOLECULAR PHYSICS 
L 
m----- -  - - 
R .  J. M u n n ,  E. A. Mason, and Francis J. Smith 
GPO PRICE s 
CFSTI PRICE(S1 s 
IHP -NASA -44 
November 30, 1964 
https://ntrs.nasa.gov/search.jsp?R=19660024125 2020-03-16T20:22:09+00:00Z
COLLISION INTEGRALS FOR THE EXPONENTIAL ATTRACTIVE POTENTIAL* 
R .  J. Munn, t E .  A .  Mason, and Franc is  J. Smith z 
I n s t i t u t e  f o r  Molecular Physics, Univers i ty  o f  Maryland 
College Park, Maryland 
ABSTRACT 
C o l l i s i o n  i n t e g r a l s  f o r  the exponent ia l  a t t r a c t i v e  p o t e n t i a l  a r e  
ca l cu la t ed  by accu ra t e  numerical  i n t eg ra t ion .  There a re  some 
unexpected d i f f e rences  from what had been expected by analogy wi th  
inve r se  power a t t r a c t i v e  po ten t i a l s .  Some impl ica t ions  of t hese  
r e s u l t s  f o r  t h e  c a l c u l a t i o n  of high-temperature gas t r a n s p o r t  
p r o p e r t i e s  a r e  discussed.  
I -. 
I :  
~ 
* Supported i n  p a r t  by t h e  National Aeronautics and Space Adminis t ra t ion  
(Grant NsG-5-59) . 
t Harkness Fellow of the Commonwealth Fund, 1963-1964; on leave  from the 
Department of Physical  Chemistry, The Univers i ty ,  B r i s t o l ,  England. 
1 On leave  from the Department of Applied Mathematics, Queen's 
Univers i ty ,  B e l f a s t ,  Northern I r e l and .  
I. INTRODUCTION 
Simple a t t r a c t i v e  o r  r epu l s ive  p o t e n t i a l s  a r e  use fu l  f o r  
c a l c u l a t i o n s  of t he  t r a n s p o r t  p rope r t i e s  of gases ,  e s p e c i a l l y  a t  high 
S ing le  term inve r se  power laws have been a f a v o r i t e  r ep resen ta t ion  
of such p o t e n t i a l s ,  p a r t l y  because the eva lua t ion  of t h e i r  t r a n s p o r t  
c o l l i s i o n  i n t e g r a l s  r e q u i r e s  only a s i n g l e  se t  of quadra tures .  The 
numerical  r e s u l t s  a v a i l a b l e  f o r  inverse  power a t t r a c t i v e  and r epu l s ive  
p o t e n t i a l s  have been summarized and extended by Kihara, Taylor ,  and 
H i r ~ c h f e l d e r . ~  However, it o f t en  happens t h a t  a s i n g l e  term exponent ia l  
g ives  a b e t t e r  r ep resen ta t ion  of the i n t e r a c t i o n  than  does an  inve r se  
power, bu t  the eva lua t ion  of the c o l l i s i o n  i n t e g r a l s  f o r  an  exponent ia l  
p o t e n t i a l  r equ i r e s  a l a r g e  amount of numerical  quadrature .  The 
c o l l i s i o n  i n t e g r a l s  f o r  the repuls ive  exponent ia l  have been accu ra t e ly  
c a l c u l a t e d  by Monchick, bu t  the only r e s u l t s  a v a i l a b l e  f o r  the 4 
a t t r a c t i v e  exponent ia l  a r e  an estimate by Brokaw’ bqsed on a s c a l i n g  
analogy with a t t r a c t i v e  inverse  powers. 
The purpose of t h i s  paper is t o  r e p o r t  the c a l c u l a t i o n  of the 
c o l l i s i o n  i n t e g r a l s  f o r  the exponent ia l  a t t r a c t i v e  p o t e n t i a l  by 
accu ra t e  numerical i n t e g r a t i o n .  Such a c a l c u l a t i o n  i s  now comparatively 
easy  because of the ex is tence  of an e f f i c i e n t  computer program f o r  the 
automatic  c a l c u l a t i o n  of c o l l i s i o n  i n t e g r a l s .  6 
2 
Comparison of the r e s u l t s  wi th  those f o r  inverse  power a t t r a c t i v e  p o t e n t i a l s  
and w i t h  Brokawls estimates shows that the c o l l i s i o n  i n t e g r a l s  f o r  
v i s c o s i t y  c o r r e l a t e  well (i -e  .,Brokaw's e s t ima tes  are reasonably accu ra t e )  
but t h a t  the  c o i i i s i o n  i n t e g r a i s  f o r  d i f f u s i o n  do not  c o r r e i a t e  weli and 
i n  f a c t  sksr~ a rather d i f f e r e n t  temperature dependence. Some impl ica t ions  
of these r e s u l t s  f o r  the c a l c u l a t i o n  of high-temperature gas t r a n s p o r t  
p r o p e r t i e s  a r e  discussed.  
11. NUMERICAL CALCULATIONS 
It is  first necessary t o  decide on a condi t ion  a t  the o r i g i n  f o r  
an a t t r a c t i v e  p o t e n t i a l ,  inasmuch a s  d i f f e r e n t  assumptions can l e a d  t o  
q u i t e  d i f f e r e n t  c o l l i s i o n  i n t e g r a l s  .3 We chose t h e  r epu l s ive  core 
condi t ion ,  i n  which t h e  p o t e n t i a l  has a small r epu l s ive  core  a t  the 
o r i g i n  t o  d e f l e c t  outwards any t r a j e c t o r i e s  of r e l a t i v e  motion tha t  reach  
the o r i g i n ,  s i n c e  th i s  i s  the most r e a l i s t i c  model i n  the  problems of 
phys ica l  i n t e r e s t .  I n  t h e  l i m i t  of a core  of i n f i n i t e s i m a l  s i z e ,  the 
exac t  na tu re  of the r epu l s ion  is expected t o  be unimportant and can be 
chosen f o r  mathematical convenience. The usual  choice of a rigid-sphere 
core  was imprac t ica l  because of the d i s c o n t i n u i t y  involved; a s h o r t -  
ranged exponent ia l  was t h e r e f o r e  used, and the c o l l i s i o n  i n t e g r a l s  f o r  the 
a t t r a c t i v e  exponent ia l  were ca l cu la t ed  a s  the low-temperature l i m i t s  of 
the Morse p o t e n t i a l ,  
where E i s  the depth of the po ten t i a l  well, re i ts  p o s i t i o n , d t h e  
3 
. 
-- 
s e p a r a t i o n  f o r  whichy(U) = 0, and C i s  a parameter which g ives  a 
measure of the width of t h e  well ,  r e l a t e d  t o  r a n d u b b y  the r e l a t i o n  
8 
C [ ( re /u)  - 11 = I n  2. If  w e  write the a t t r a c t i v e  exponent ia l  i n  the 
form 
y ( r )  = - yo exp (-r/p) , 
t hen  yG = 2~ exp (Cre/r) and p=C/C. 
I 
The conventional def i n i t i o n 7  of the reduced c o l l i s i o n  i n t e g r a l s  
de,"'* t o  be equal  t o  u n i t y  f o r  r i g i d  spheres  of diameter 6, and t o  
* 
be func t ions  of t he  reduced temperature T = kT/E, a r e  inconvenient 
f o r  exponent ia l  p o t e n t i a l s ,  4,5 and we have adopted the d e f i n i t i o n s  5 
a = ln(yo/kT) = C - In (T*/4). 
The procedure was t o  p ick  a value of C and then  e v a l u a t e  a (49s)* by the 
previous ly  descr ibed computer program 6 a t  success ive ly  lower values  of 
T* u n t i l  t h e  r e s u l t s  appeared t o  be independent of the r epu l s ive  core.  A s  
a check t h e  process  was repeated wi th  a new value of C.  
a l l  these r e s u l t s  t o  a p l o t  of J C a y s )  vs. a ,  we f i n d  a fami ly  of curves,  
one f o r  each value of C,  which approach a common l i m i t i n g  curve 
c h a r a c t e r i s t i c  of the a t t r a c t i v e  p a r t  of the p o t e n t i a l  a lone ,  as  shown i n  
F ig .  1 f o r  J (',')with C = 10, and C = 20. 
On convert ing 
The unce r t a in ty  i n  the numerical i n t e rg ra tdons  i s  estimated t o  
be no worse than  1 o r  2 p a r t s  in  1000. Values of J ('") and J were 
read  o f f  la rge-sca le  graphs a t  round values  o f a ,  and a r e  given i n  
Table I. This process d i d  no t  apprec iab ly  inc rease  the numerical  
4 
unce r t a in ty .  The c o l l i s i o n  i n t e g r a l  r a t i o s  A*, B*, and C*, which occur  i n  
gas  mixture  formula^,^ a r e  a l s o  given i n  Table I. The r a t i o  A* was 
c a l c u l a t e d  d i r e c t l y  from the  d e f i n i t i o n ,  but B* and C* were ca l cu la t ed  
from the i d e n t i t i e s  
The r a t i o  E+ which occurs i n  some of the h igher  approximations,  was a l s o  
c a l c u l a t e d  from an  i d e n t i t y ,  
The d e r i v a t i v e s ,  which cont r ibu te  less than  one percent  t o  t h e  r e s u l t s ,  were 
eva lua ted  by numerical d i f f e r e n t i a t i o n .  Because of t h e  smallness  of the 
d e r i v a t i v e s ,  the r a t i o s  B*, C*, and E* can be given t o  one more s i g n i f i c a n t  
f i g u r e  than  t h e  o the r  q u a n t i t i e s  i n  Table I. 
111. DISCUSSION 
A remarkable r e s u l t  not iced i n  the  present  c a l c u l a t i o n s  is t h a t  
T* must be very low indeed before t h e  e f f e c t  of t h e  r epu l s ive  core has 
disappeared,  of t h e  order  of 10 o r  I n  view of t h i s  r e s u l t ,  we -2 
have not  extended the t abu la t ions  t o  low values  o f a ,  which would correspond 
t o  l a r g e r  values  of T*. For such cases , the  f u l l  r e s u l t s  f o r  the Morse 
p o t e n t i a l 6  should probably be  used. 
The r e s u l t s  f o r  t he  repuls ive  exponent ia l  and f o r  Brokawls 
e s t ima tes  of t he  present  values  of J (’”) and J (”‘) a r e  shown i n  F ig .  1 
f o r  comparison wi th  t h e  present  c a l c u l a t i o n s .  A l l  three r e s u l t s  a r e  i n  
reasonable  accord f o r  J (2J2), but f o r  J (’”) Brokawfs e s t ima tes  a r e  q u i t e  
f a r  o f f  and the  r epu l s ive  exponent ia l  values  a r e  reasonably c lose .  This  
i s  s u r p r i s i n g ,  and i n  f a c t  i s  j u s t  the  reverse  of what w e  had expected. 
We conjec ture  t h a t  t h i s  i s  due t o  the d i f f e r e n t  condi t ions  a t  the 
p o t e n t i a l  o r i g i n  f o r  an  exponent ia l  p o t e n t i a l  and f o r  the inverse  power 
p o t e n t i a l s  upon which Brokaw based his  estimates. The exponent ia l  Is 
f i n i t e  a t  t k e  s r i g i n ,  vhereas  t h e  inverse  pewer is s i n g u l a r .  It i s  
a l r e a d y  known from the r e s u l t s  f o r  inverse  power p o t e n t i a l s  t h a t  the 
A?= 1 i n t e g r a l s  a r e  s e n s i t i v e  t o  the condi t ions  a t  the  o r i g i n  , but t h a t  
the e =  2 i n t e g r a l s  a r e  independent of them. 3 
I n  Brokawfs scheme, the a t t r a c t i v e  exponent ia l  i s  matched i n  
va lue  and s lope  by an inverse  power p o t e n t i a l ,  a t  a s p e c i f i e d  mean value 
of s e p a r a t i o n  d i s t ance  which va r i e s  wi th  temperature .  Since t h i s  scheme 
turned  out t o  be poor f o r  J (”’), it is  worthwhile t o  cons ider  o the r  
p o s s i b i l i t i e s  f o r  c o r r e l a t i n g  the c o l l i s i o n  i n t e g r a l s  f o r  exponent ia l  and 
inve r se  power p o t e n t i a l s  i n t o  a s i n g l e  family group. I n  t h i s  connection 
the c o r r e l a t i o n  scheme used by Baroody f o r  r epu l s ive  p o t e n t i a l s  i s  of 
i n t e r e s t .  Baroody analyzed the s c a t t e r i n g  i n  terms of two energy- 
dependent parameters,  the  d i s t ance  of c l o s e s t  approach i n  a head-on 
c o l l i s i o n ,  r c ,  and the s lope  of the p o t e n t i a l  a t  t h i s  d i s t ance ,  
7 =  - ( d l n y / d l n r )  . These choices obviously a r e  inappropr i a t e  f o r  a n  
a t t r a c t i v e  p o t e n t i a l ,  which has rc = 0, but  the obvious analogue i s  t h e  
c r i t i c a l  d i s t ance  of c l o s e s t  approach f o r  o r b i t i n g ,  ro. 
po in t  f o r  a n  a t t r a c t i v e  p o t e n t i a l .  Since o r b i t i n g  tends t o  produce 
i s o t r o p i c  s c a t t e r i n g ,  which is  a l so  c h a r a c t e r i s t i c  of r i g i d  spheres ,  i t  
might be expected t h a t  nr: would be a good measure of the c ross  s e c t i o n s .  
8 
P C  
This is a unique 
6 I 
1 
Such an  estimate f o r  the d i f f u s i o n  c ross  s e c t i o n  of Ions i n  n e u t r a l  gases  
has been used by Dqlgarno, McDowell, and Williams,' who c a l l  i t  a "capture"  
c ros s  s e c t i o n .  
This  c o r r e l a t i o n  scheme can be c a r r i e d  through without  d i f f i c u l t y ,  but  the 
r e s u l t s  are very  s i m i l a r  t o  Brokawfs. 
The s lope  parameter would then  be Y o  = ( d  l n y / d  I n  P ) ~  . 
0 
I n  p a r t i c u l a r ,  the p red ic t ed  J 
13 not  " u a d ,  bLit the gredicted W )  is t e e  h igh  and has the  wrnng 
temperature  dependence. 
We conclude t h a t  c o l l i s i o n  i n t e g r a l s  f o r  a t t r a c t i v e  exponent ia l  
p o t e n t i a l s  a r e  no t  n e c e s s a r i l y  s i m i l a r  t o  c o l l i s i o n  i n t e g r a l s  f o r  a t t r a c t i v e  
inve r se  power p o t e n t i a l s ,  a l though there does seem t o  be a c lose  s i m i l a r i t y  
f o r  r e p u l s i v e  p o t e n t i a l s ,  a s  shown by Brokaw. 
I n  view of the foregoing r e s u l t s ,  i t  is  worthwhile t o  i n v e s t i g a t e  
the e r r o r s  t o  be encountered i n  ca l cu la t ions  of high-temperature gas  t r a n s p o r t  
p r o p e r t i e s  when the " t a i l s "  of p o t e n t i a l  energy curves are represented  by 
a t t r a c t i v e  exponent ia l s .  A simple procedure is t o  assume some p o t e n t i a l  
model t o  be t h e  exact  p o t e n t i a l ,  curve- f i t  t h e  " t a i l "  of the  model curve by 
an exponent ia l ,  and see how much e r r o r  i n  the c o l l i s i o n  i n t e g r a l s  r e s u l t s .  
We have s e l e c t e d  the Morse p o t e n t i a l  t o  r ep resen t  a n  exac t  in te ra tomic  
po ten t i a l . 1 °  For C = 20, w e  f ind g r a p h i c a l l y  that the Morse p o t e n t i a l  can 
be well  represented  f o r y ( r )  = 0 . 5 ~ :  t o  0.01~ by the express ion  
cp( r )  = -(1.7-,1014~)exp (-r/0.033 ) .  ( 7 )  
The q u a n t i t i e s  which d i r e c t l y  determine the t r a n s p o r t  c o e f f i c i e n t s  a r e  the 
products  CT fi 
values  of ddJs)*  with  ( 0 . 0 3 3 ~  ) J 
a comparison i s  shown i n  Table I1 f o r  s e v e r a l  va lues  of ~ T / E .  
= a  p 2J("s)o We should t h e r e f o r e  compare the Morse 
Such 
2 (-&SI* 
"), where a = I n  (1 .7x lOl4~ /kT) .  
All the 
8 
. 
TABLE I 
C o l l i s i o n  i n t e g r a l s  f o r  the exponent ia l  a t t r a c t i v e  p o t e n t i a l  
ct 
18 
20 
22 
24 
26 
30 
32 
34 
36 
40 
42 
44 
46 
48 
50 
52 
54 
56 
28 
38 
J1,O 
0 -903 
o .910 
o .916 
0.922 
0.98 
0 -933 
0 -937 
0.941 
0.944 
0 -947 
0 -950 
0 0953 
0 -955 
0 -957 
0 -959 
o .961 
0.9625 
0.964 
0 .  9665 
0 -9655 
p,2) 
1.006 
1.005 
1.003 
1.003 
1.003 
1.004 
1.004 
1.003 
1.002 
1.002 
1.002 
1.002 
1.002 
1.002 
1.002 
1.002 
1.002 
1.001 
1.001 
1.001 
A* 
1.114 
1 .lo4 
1.089 
1 -082 
I .096 
1.076 
1.071 
1.066 
1.061 
1.058 
1 -055 
1.052 
1.049 
1 -047 
1 .Ob5 
1 .Ob3 
1 .Ob1 
1.039 
1 *037 
1.036 
B* 
1 .om 
1.0669 
1.0611 
I ,0565 
1.0522 
1.0483 
1 .Ok?O 
1 .Oh21 
1.0396 
1.0374 
1.0355 
1.0338 
1.0320 
1.0306 
1.0292 
1.0281 
1.0269 
1.0260 
1.0247 
1.0237 
C* 
0.9613 
0.9636 
0.9686 
0 -9711 
0 -9734 
0 9754 
0 *9771 
0.9786 
0 9799 
0.9810 
0.9820 
0.9838 
0.9852 
0.9858 
0.9864 
0.9869 
0 -9875 
0.9880 
0.9829 
0.9845 
E* 
0 -9723 
0.9751 
0 -9774 
0 -9793 
0.9809 
0.9821 
0 -9833 
o .9853 
0.9861 
o .9868 
0 -9875 
0.9881 
o .9886 
0.9891 
o .9896 
o .9908 
0.9844 
0.9900 
0.9904 
0 3911 
9 
Exact Present Brokaw E 
. 
Exact Present  Brokaw 
- -  , -  
0.01 1.5739 1.443 1.556 
0.04 1.4109 1.336 1.443 
0.10 1.3117 1.267 1.372 
0.40 1.1800 1.165 1.266 
Comparison of c o l l i s i o n  i n t e g r a l s  c a l c u l a t e d  e x a c t l y  f o r  the Morse p o t e n t i a l  
with C = 20, and f o r  approximations i n  which the p o t e n t i a l  " t a i l "  i s  f i t t e d  
by an exponent ia l .  
1.6986 1.524 1.569 
1.5246 1.413 1.457 
1.4171 1.342 1.385 
1.2692 1.238 1.279 
c. 
t 
10 
FIGURE CAPTION 
.- 
Fig. i. C o l l i s i o n  i n t e g r a i s  f o r  exponent ia l  
p o t e n t i a l s .  S o l i d  curves a r e  f o r  a n  
a t t r a c t i v e  p o t e n t i a l ,  and dashed curves 
f o r  a repuls ive  p o t e n t i a l .  The branches 
marked C = 10  and C = 20 show how the 
c o l l i s i o n  i n t e g r a l s  f o r  Morse p o t e n t i a l s  
approach those f o r  the a t t r a c t i v e  
exponent ia l  a t  low temperatures .  Brokaw 1s 
curves are estimates based on a n  analogy 
with inve r se  power p o t e n t i a l s .  
c 
c 
8 .  
1.2 
1 . 1  
I .o 
09 
.8 
R 1 I I I 
Brokaw v 
i 
I 
1. attractive 
c = 2 0  
I 
/ 
I 
I /repulsive 
I 
I 
I 
I 
I 
0 20 40 60 
CY 
4 
11 
REFERENCES 
. 
c 
c 
1 
c 
2 
3 
0 
9 
10 
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V .  T T. Vanderslice, s. !.leissI!l2r,, E. A .  Y2scn, aad I?. 2. Fallen, %ys. 
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L. Monchick, phss. Flu ids  2, 695 (1959). 
R .  S.  Brokaw, Phys. F lu ids  944 (1961). 
F.  J .  Smith and R .  J. Munn, J. Chem. Phys. 41, 0000 (1964). 
J. 0. Hirschfe lder ,  C .  F. Cur t i s s ,  and R .  B. Bi rd ,  Molecular Theory 
of Gases and Liquids  (John Wiley and Sons, Inc. ,  New York, 1954). 
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---
E. M. Baroody, Phys. F lu ids  1 4 1182 (1961); Phys. Rev. -’ 124 745 (1961); 
Phys. F lu ids  2, 925 (1962). 
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This  procedure was suggested t o  us by D r .  W .  L .  Bade. 


Collision Integrals for the Exponential Attractive Potential

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