solutions we have placed the Debye. Bueche equations :+~) on a microscopic basi~; by showing thai d~ey ca,~. be derived as a mean-field approx{ma.tio~ to ...
Pfgvsica tt0A (i975) 63-'75 {) Nor.~h.4]oh2++,d P~eD¢L,¢D+(r) ~ 1 - (Air) + ( B , ' ? ) ,
z(r) ~ A . i F ,
as
r - > ~c.
(2.10)
where A and B are constants to be determined, The ffictio~l coefficient j.d can be expressed directly in terms oi the constant A, We ~had show that the tOt,:t] force exerted on the fluid is given by ~
= j" F(r) dr = -- 4X'qoA Vo,
Sincefa is defined by ~ j~ = 6~'~oa,
(2.1~)
= --/~vo we find (2. t2)
where a = ~A can be interpreted as an effective radius, F o r a hard sphere a would coincide with the sphere radius.
66
B.U. FELDERHOF In order to derive (2.11) we consider the stress tensor
Pt,
o ::~, ~]o ( V V),, -
(2.13)
.... - Then (2.3) can be written where (V V),. , R and r < R are 4 ( 0 = 1 -- (A/r) + (B/tf . 3 ),,
Z(r) = A / r e ,
R < r <
.....
~=Orq:) ' ~4>
) d r
0
, . , f ~,p , + _9. o"< #.... )" + 6(8g>') ~ .~_ ~I. 8~b,' 0
+ U(r) [(r 84,' 4- ~: a~) 2 + 2 (a45>'1} ,.2 d,']. /
(14.6)
The lhst t ~ o lines sho~ that W is stationary when Ldp = 0 ~or v .riations 3 4 for which the b o u n d a r y terms vanish. The expression in the thh ";~ line is positive defiuite. Hence the stationary point is actually a minimum. This statement can be traced back to the m o l e e,r~ ...... l principle o f minimum energy, ,'issipation valid for the general tbrm o f [4~give:: it, (4,1),
FRICTION
...... t>Cq, _ ~f ~ \~~ ,:::'~:C> ¢'1I: *~" I N
COEFFICIENT
SOLUTION
71
Suppose one considers a trial function &,(r) = ~/~,+ &15 (r) v;hich is regular at the origin and has asymptotic behaviour ?~4r) "~ I -
A,/r,
as
r-,
.,~ ' 7 (.,. ,)
c,~,
x~'ith a tri:;:l value A~. If & is the desired exact .-.o~utlon o, L= 0 with asymptotic behaviour (2. I0). then the first t,,vo lines in (4.6) vanish a~ld one has t¥[4) ] ~ W[4h]. This can be expressed in two ways. using either (4,2} or (4.3). The second gives the variational r~rinciple a
t
~::,:~a~
-k 7
f
q:h
(L~tl dr.
(4.8)
o
Usin~ the asymptotic behaviour (4.7) one ca:a integrate the second term in ( 4. z ~) by paris a;~d cast the integral in a s o m e w h a t nicter form~ Hence from (4.2) one obtah~s the variational principle j.
a
~
~
t" [r-.o~ " " tr' 2
+ 4#.,~ -~ + U t r ) ( r
2 q'h" ''~'
,,I . . . . -1. 6~?:. 2 ,~J -r2 4o ~,,',o~d.;,
dr
(4.9)
0
,-,,r-, }
J Le !atter form is the most convenient one. Once one has found the best trial fur~ction one ca~ substitute back into ( 2 . t 8 ) a n d c o m p a r e the values R>r the friction coefficient found from (2.j2) and (2.18).
5, Perturbation theory It is straightforward to do ordinary p e t t m b a t i o n theory on the pair of differential equations (2.8a, b) with the strength of the poter~tial U(r) as expansion parameter. We shall carry this scheme ordy to first order. The unperturbed flow gives as the zeroth-order solution 4)o(r) = 1 and ;%0) = 0. Hence the first-order perturbation ,.b~(r) satisfies r"~'~,.~'', + 4rcy~ - r 2 U (r) + raZ"~. = O.
(5.~)
F r o m (2.19) we find as the first-order result f~r Z(r) £
z!(r) = r -2 j ' r ' 2 U ( / ) dr:.
(5.2)
o
Provided tile potential U(r) falls off more rapidly than 1/r 3 the asymptotic behaviour is given by no
zl(r) = r -'2 .[ r ' 2 U ( r ') dr' + ~: (1/r2), o
as
r - ~ rF.;.
(5.3)
72
K I L FEILDERHOF
C o m p a r i n g with (2. t0) this is seen to imply A~ ~:~,,;~7/4r.~h.:0,and hence one obtains the f'ree~dr.aini~~g limit result f~ ~. n,'L Subs,ituting (5,2) in (5.1) one car, sol~-e h:)r ~]q(r), using the b o t m d a r y conditions .at zero and iafinity. T h u s ol:m firtd~ ~,~(r) = -.([~r) f r ~te ,r ~ dr' + (1/3r a) : 0
:!~
O
¢5.4}
¢
The firstoorder correction. Io ,:he free-draining limit can be found by a.,~.i,}~e ~hi~ function in (Z 18). The result can be cast in lhe form
.,~
" + j,;
,~,2.---:£,:{,~./~,~.~..f j" ~.f... r......t'(*)?0"). ,:it,:.,~-..
~t'~.:~)
which is equivalent Io an approxima~e expre_,, h:m derived ~}3,' ~,Jr~.~;~.,t, > rk:fe r.-(r>) is the le:sser (larr, er) o f r and ~'. P'o~ ~h~ ,, ..... - V - disiribt~{on 5.6 )
,he integral in {5.5) can be evaluated explicitly. 1,,~ ~erm>.~of K.irkwood':, din-~en;~.~ono less parameter X .......................~.,..!, = r,~X~ ;~.2(0.1 , 6'=a;2 ~jo (.r°/
,,
one rinds f = . 2 [1
~ ~" + ((X2}]
(gaussian).
5.8)
This shoald be c o m p a r e d with the res'At obtained ir~ {he Kir'kwood-..R{~¢eman {{}eorv 7 ; ~,G; ]
The corresponding resul~ ff~r the sphere from eq. (3~ 1! } b; .f~l = n~ [1 -,- 23/23}.J(" + C(X2')]
(sphere).
(5. ~0)
in principle the above perturbation theory can be carried systematically to any tesired order. ~n practice one is particularly interested in tb~ effects for strong 1.ote~fial, and in this case a different type of perturbation e:' i?ansion is required. In setting up such a perturbation procedure we are guided b'~, the results obtained for the unitbrm sphere.
FlaACI:ION COEFFiCiENT OF POLYMER:S IN SOLL.FF|ON
73
We R~fow the sxme strategy as i:: t!~e cas~: .~>: r~:>~a~i[~,,:~a~ i:Yicti,m':} a~:d do ordinary ix~:fturbafio~ theory' h~ ~he regi Rr:, v,her~: {:~, po:c.ntial is 'weak and a differem e×pans;[om e u d i n e d hero,a, in :}m regio~:~0 < ;~ < .R~ v.&e~e the potemial is sirong. The .appro×imate solutions a~e fitted by condm~i{y at R~> a M s u b s e q u e n t l y the best value o f R:, is determined by applying the varia:io~:a: pr:4c~ple ( 4 9 ) . Finaliy the f-ricdon c o e N d e m f ~ is; f i ~ g d eitJ?er f:'om (4.9) or f r o m (2. i8L 7he trial fur:c:irma+ expansion p a r a m e t e r e arid now write d"m pair o f different{al e q u a u { m s (ZSa°b.) in the t b r m r~-(ik + 4r~&- -
r ~ b%Sc + r~£,.~' = O,
tt
r'Zc + 2rz~- - 2 Z c -
)
~:r:"U'~s
9~
Zc
:
~"Z¢', ~ +
g2Xc. 2 + " " ,
,~e find agai~ ~uccessr~,e equatiom, £o~- the di:f]%ren{: orders
'
~o
,
order Zc. o .... 0 a~(1 . . o. + . . 4r~Oc, o - ,.e Uq!Jc, ~> = r"@c,
O.
{5
20) rl
One caa st~bstitute J~e ~oi~Jlioi~ in ~I~e cqtmticmf, fbr (%!k< ~ ~Z~, ~) a~@ ~oive ~hc~e~ taking a c c o u n t o f the b o t m d a w conditio~lso S u b s e q u e m t y o~e ¢a~ co.~tim~e sys!ema|icaHy to h~gher order. Aga.ir~ we :.dmfl be satisfied wi~i~ ~he }o~ves>order result. The solution o f (5,20) can be writter~ C~4k< o (r) whec~ 4c~ o(~') is normaUzed to c~,c,c,(0) = ~. Eq. (5.20) c a n n o t be solved expliciUy for arbitrary potenliat a e d one is obliged to i n t r o d u c e furflmr approximations. T h e eqe: i.ion is identical, however, with (I.3.4), for which we have developed a W N B ~ {ution ie article I. We refier to I for the dem~ ' " s o f ~his solutior~.
KR H ' 7 I K ) N C O E F b : (~:!:!! "./3 ='.d. P
