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Frictional properties of dilute polymer solutions
Pfgvsica tt0A (i975) 63-'75 {) Nor.~h.4]oh2++,d P~eD¢L,¢D+
F R I C T I O N A L P R O P E R T I E S OF D I L U T E P O L Y M E R S O L U T I O N S IlL TRANSLATIONAL-FR|CTION COEFFICIENT [L LL F E L L ) E R H O F +
D~v~,rtmee+e of Pia~~tc~. Ouee,+ a:lusy C:?dtes~-e+l+.)+i~:e~'~it.v+g" Lom&~e, #,.+:n~d~:+;~, E] 4N .....ET~gta;~d
~n the framework of our ge{ae++a~izatio~ of the I)eb?e--Bueche vheo.,-y for the frictiomd rJ.~op<;o ties o f dilute polymer solulioas ~se smdv {t~e m m ~ a t i o n a b f + i c t i o n coeflicicm o!~ polymers ,.vith spherically symmetric segment distribution. A varkt++ona~ principle o f minimum energy dissipation i~ formu~;atedv&ich i:+s~.+,i~ahge~:c>rnumericai work.
I. Introduction
in the first article +} ([) of this series on the f~ic~ic;nal properties of dilute polymer 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 the equations describing the hydrodynamic in~eractioris between polymer segments. Thus ,ae have created an attractive alternative ~o the co~wentionally accepted Kirkwood+ Riseman (KR) theory'+). We have exter+ded the analysis of Debye and Bueche io deaf with more general po[y:ner models that,, the unif'k~rm sphere, in 1 we have presented the detailed theory of" the rotational friction coefficientf.. In the second article% (tl) we have outlined a simplified version of o~,r theory involving an angular average (PAA) , ~",he hydrodynamic interaction+ similar to an approxi+ minion made in the KR theory, in ti~is article we return to the exact mean-fieid theory and study it:~ consequences %r the +rans}atio~at-friction coeiScient j~. The present article can be read independently o f ! and tI+ We start directly t?om the macroscopic equations; for a discussion of the microscopic basis we rei~er to 1. In section 2 we apply ~he Debye+ Bueche equations to the flow situation of translationa{ drag on a spherically s~mmetric polymer and derive two general expressions for the friction coefticientf+. In section 3 we give the explicit solution for the uniform sphere and a spherical-shell segmem distribution. In section 4 we study # Presel'~t address: Instimt f~.i+'Theoretische Pi~ysik, F
63
64
B.U. FELDERHOF
the energy dissipation and derive a variational principle which will be useful for numerical work. In section 5 we develop perturbation methods which are applicable to general segment distributions.
2. Translational friction of a spherically symmetric polymer We consider a polymer in the usual picture o f a large number o f segments connected by bonds in a flexible structure that can adopt many conformations. The polymer is immersed in a fluid which is assumed to satisfy the linear NavierStokes equations for steady, incompressible flow. In article I of this series it was shown that in a mean-field approximation the basic equations of the theory describing the hydrodynamic imeractiont; between polymer segments are v ev -
(r) ( V -
u) -
W
= 0,
v.
V = 0,
(2.
where *jo is the solven! viscosity, V(r) is the average fluid flow velocity, g is the friction coefficient of a segment, ~)(~,) is the segment density distribution, u(r) gives the rigid-body motion o / t h e polymer, and ?(r) is the average pressure. The averages arc over the statistical distribution that characterizes the corfformations of the polymer segments. "['he second term in (2.1) accounts for the friction between polymer and fluid. The average force density exerted on the fluid is given by F(r) = - @ (r) [ V(r) -- u(r)].
(2.2)
Hencefbrth we shalt assume the segment distribution Off) to be spherically symmetric. It is convenient to choose a coordinate frame in which the polymer is at rest with the centre of mass at the origin, in which case u(r) = 0. Introducing the inverse shielding lengff~ z by ~a(r) = ~t? (r)/'t~o, one then has the basic equatioas V 2V - :?(r) V -
~ * VP = 0,
V. V = 0,
(2.3)
which must be solved with tile condition that the solution tends to the u ~ e r t u r b e d flow pattern at large distances from the polymer. The unperturbed flow velocity ~%(r) and pressure po(r) in the absence of the polymer satisfy the equations ~}o V%o - Vpo = 0,
V. % = 0.
(2.4)
/n the case of translational drag one considers Vo(r) = Vo constant, and po(e) = P o constant. We choose Vo to be in the z direction so that in spheri ::al coordinates
(r, 0, ,p) vo = voe~ = vo (cos 0er - sin 0co)
(2.5)
FRICTION COEFFICIENT OF POLYMERS IN SOLUTION
65
with unit vectors e=, e, and e0. In order to solve (2,3) we look for a velocity field V(r) with the same angular dependence. Thus we put V(r) = vo {q~(r) cos 0G - [qS(r) + ½,'&' (r)] sin 0e0} = ~b(r) vo -
(2.6)
- ~ r - ~ 6 ' (r) r × (r x ,,o)
and seek an equation for q)(r). In (2.6) a prime indicates differentiation with respect to r and in the second term we have used the condition o f incompressibility V . V = 0. CIearly the unperturbed flow vo is o f this form with qS(r) = t, The angular dependence o f the pressure must be given by the scalar ( r , vo) and we put (2.7)
P(r) = Po .... qovoZ (r) cos 0 = Po -- ~jo2 (r) r , vo,/r,
~t then follows by substitution s) that eqs. (2.2) are satlsfled provided 4~(r)and y(r) satisfy the set o f coupled equations re0" + 4r(// -- r-'U(r)4) + * Z = 0,
(2.8a)
r2z '' + 2rz' - 27. - r 2U' (r){/) = 0,
(2 8b'~
where U(r)
=
~2(r.),
(2.9)
which for brevity we shall cal~ the potential. The differential equations (2 8a, b) must be solved with the b o u n d a r y conditions that 4~(r) and ;((r) are regular at the origin and that (~(r) tends to unity and z(r) to zero at infinity, At large distances the potential U(O tends to zero rapidly and (2.8a, b) becomes a pair of free-field equations. Fitting to the unperturbed flow we can therefore write f'or the asymptotic "t:ehaviour 4>(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),. , <,# ~=- (c," ~.,/c;x#) + (r.~ . V~ i ~c,.~,).
V-,~ = - g ( r ) ,
(2.14).
V , V = 0.
Integrating the first equation over a volmne O bounded by a surface ~" and applying Gauss2~ theorem one finds -f~.
n d S = .[ g ( r ) &'.
2-"
(2.15)
~2
This shows the equality o1" two alternative expressions for the force exer{ed on the fluid in ,t2. The totai fbrce is found by extending the integral on the right over all space, .~- = j' F0") dr = - g J'e(r) V ( r ) dr.
( Z 16)
Substituting ~2.6) one obtains
.,*~
=
-v
o"4,=," J'"
[(hit) + !W'~b' 0 9 t ~j(r)r ~ dr.
(2.17)
0
From Ll~e left-hand side o1' (2.15) and the asymptotic behaviom" (2.10) one verifies eq. (2. t i ). Thus we have two expressions fo~" the translational friction coefficient, ri::, (2.12),,,1~ - 4~k>d = brWoa and L~ = 4_,~ - " f [4~(
(2.18)
These are to be compa, red with the corresponding expressions (11.4,2) /ound in the PAA theory4), To conc!ude this section we note that eq. (ZSb) can be solved explicidy lbr z(r), The homogeneous equation has solutions r and t/r e. With the bom-~dary conditions dmt Z(r) must be regular at the origin and tend to zero at infinity the solution o f (2,8b) is therefore given by ct>
~.
z(r) = - ~ xr .t L (r .} q~(r') dr' r 1
'
f~
¢
.__
. - [r'-~U ' (r") 4 ( / ) dr'. -~r d l
-3
(2.19)
Hence once the velocity function ~/)(r) is known one can c a h u i a t e lhe pressure using (2.7).
FR1CTION
COEFFICIENT
OF
POLYMERS
IN SOLUTION
67
3, Explicit solutions for uniform sphere and spherical shell F o r a few simple potentials U(r) the differential equations (2.8a, b) ca~ be solved explicitly. We consider first the uniform sphere for which U(r) = [ ~ i
= (~'°/~'°) " f oc°nstant R r < r < :x,.l°r
0~r
Hence eq, (2.8b) becomes h o m o g e n e o u s in the two regioT~s ap, d one has the simple solutions
z(O
=
[
A/r:,
for
R < r < ,~,
k D~',
:for
0 ~ r < R.
(3.2)
In the exterior region r > R the asymptol.ic solution (2.[0} is exact, so that 4@) = 1 -. (A/r) + (B/r~).
r > R.
(3.3)
In the interior region r < R the differential equation (Z8a) becomes r2~" + 4r~])' - ~ r : ~
+ .Dr 2 = O,
r < R.
(3.4}
The solution which is regular at the origin is given by qb(r) :: C ( c o s h xr \,
. . r2
.
sinh ~r') .
. y ,~
/
1)
+ --.-, ~2
r <
1~.
(.3,5)
We note that the velocity and pressure corresponding to the terms multiplied b} D satisfy Darcy's law :79 V = - V P . The coefficients A, B, C and D must be found by fitting the solutions ai r = R with the aid of b o u n d a r y conditiot~s. The boundary conditions for the original problem are that all c o m p o n e n t s of V and the n o r m a I - n o r m a t and normal-tangential c o m p o n e n t s of ~!,e stress tensor c; are continuous at the boundary surface. The continuity of V ~mpties from (2.6) that 409 and 4,'(r) must be con-. tinuous at r = R. The derivative; of the velocity are given by s) VV = vo [qb' cos 0e:er - ½ (r4;' + 3~5') sin Oe~eo + ½q}' sin 0e0e, - ½~' cos 0 !'.e0e~ + e~e:, )].
(3.6)
Hence the stress tensor is = qovo [2q~' cos OerG - ½- (rq;' + 2<]/) sin 0 (Ge0 + e0e,.)
- 4.' c,~:., 3 (ee,e,j + e,-~G)] -- (p~) - ~jovoz cos 0) I.
(3.7)
68
B, U. FELDERHOF
The continuity o f o,, at r = R requires that z ( r ) be c o n t i n u o u s . Finally, the continuity o f a,0 requires that 4/'(r) be c o n t i n u o u s at r = R. I m p o s i n g the f o u r b o u n d a r y conditions and solving for the coefficients one finds A = ~-R
1 +
Go(o') (o31 2'
B == R 2 (A + 2R),
C = 2 A R / [ ( r z (cosh (r) Go(c0],
D =:
A/R 3,
(3,8)
w M r e a :-- x R is D e b y e ' s shielding ratio and Go(o) = I - tanh or/or.
(3.9)
];'.'lonl (2, 12) we therefore find i~or the friction coefficic ~t _f~ = 6::qo R
(sphere~.
(3,10)
This resul~ is idenlical with that o b t a i n e d by D e b y e and BuecheZ). In fl~e high-densily limit ~ - ~ o~, one obtains for the effective radius a = R , whicl~ leads t:o the hard-sphere resultf~ = 6=~h~R. F o r low density z ~ 0 one has 2 ~ 8 4 ~:,~ = R [-,:;(r" - -c.~f + e~J(o'~')].
;.* -, 0.
(3. i 1 )
The first term gives the free-draining limit for fl, e ffictic.n c o e N c i e n t . f~ ::: 6r:.'qoa and with the n u m b e r o f segments ~, = 4~Ra 0 one finds ,/£~ = n g
(free draining),
With
(3.12)
a restfl~ a!so obtain~ed from (2.18) by using f o r t h the u n p e r t u r b e d value ~b(r) == 1. W h e n !.he segments are distributed in a :;phericat shell o f radius R one has U(/') "-= z2(r) =
/.t~)(r
-- R),
(3.13)
where ¢~ = ,~r:/~to aim T :is the u n i f o r m surface density, r. = n / 4 = R e, The solutions o f (2.Sa, b) in ihe regions r > R and r < R are 4 ( 0 = 1 -- (A/r) + (B/tf . 3 ),,
Z(r) = A / r e ,
R < r < <:.,
(3.14) ,
.- T d D r ' ,
z(r) =: D r ,
0 ~ r < R.
The conditions at r = R are that V m u s t be c o n t i n u o u s , and that .he c o m p o n e n t s o f e¢ involvil~g a normal direction must j u m p by ,~~,(I,' +
)
.-- (r,.,(R-) = '?o# V~ (R),
(3.15)
FRICTION COEFFICIENT OF POLYMERS IN SOLUTION
69
where ex runs through (r, 0, q:). Hence one finds the j u m p condi{ions for ¢(r) ana ;Z(r) ¢ ' ( R + ) = ¢ ' ( R - ),
4,(R + ) = 4)(R - ),
R [C(R+) z(R+)
- 6"(R-)]
- z(R-)
t~R
(3.~6)
= t*~. ( R ) .
The cc,efficients A . . . . A == R
= # [2¢ (R) + R<,5' (R)],
D are easily found to be ,
B = !~R2A,
C :--. 1 -- ~ ( A / R ) ,
D = 0.
(3.17)
Note that the pressure perturbation inside the shell wmisbes. In the limit/.t .-, ,z, one finds again the hard-sphere result a = ~A = R, while the free drahfing limit (!~ .-~ 13) is A ~ ~ R z, which imptiesj~ = n~. In the general case the m, nstationalfriction coefficient of the shel! can be expressed as J~l .... 6r:rioR
!,R _3 + # R
= 6=~7oR , °'2 ~ + o-a
(shell},
i3.18)
where ,J'- = 3}~R = 3n~/4=~/oR° as in the case of the uniform sphere. For a discussio~ of these results and a comparisor~ with the results of PAA theory we re[i_'r to article II of this series4).
4. Energy dissipation and variational principle In order to handle more realistic segment density distributions, e.g., a gaussian, it is necessary to look t'oi" perturbation methods allowing one to find approximate solutions o f the differential eq .....tions (2.8a, b) and a corresponding expression for the friction coefficient fd. As in the case of rotational drag a study of the energy dissipation leads to the formulation of a variational principle. It has been shown in I that the total energy dissipation is given by W = '70 j" [} (VV).~ + x 2 V2I dr,
(4.1)
where the integral is over all space, F o r the case o f translational friction one can subs:itute (2.6) and (3.6). Performing the angular in|egrations one finds ~3
w=
~ o ~ , , ; .f [r~¢ "~ + 4r#4," + m4, '~ I)
+ U(r) (,"-6" T' 4 r ~ b ' + 64-')1 r= dr.
(4.2)
70
B. U, F E [ , D E R H O F
Integral on b / p a r t s !cads to the {brm ~t/'r =
~",a ~t<>t' O"-
( ---- t'4~fl (1~
"k r a ¢~. q5 . - - .4¢3r/)4) . " +. 4,'2rb~/>' + 2r"q5-
~':
(4.3)
_I
o
where L is a linear operator defined by =r
(/~
+ ~r'9
+ 8re4~ ' - . 8 r % '
U' (r'*~,b' + 2r'b/J).- U (r~
.-
4r:~,b').
+
(4.4)
Ii is easily seen that by eliminating the function z(r) from (he pair oF differential equations (2.8a,b) one arrives m the equation Lq5 =- O, Hence the integral in (4,3} vanishes. The boundary terms in (4.3t can be calculated from the asymptotic behaviour (2.10) and one finds (4.5)
W = 4=qovg A = ./-5v;,
as was to be expected. As in the case of rotationa! l:Yiction one finds t " a t L¢ = 0 is also the guler equation for the functional W as given in (4.2) upo~ variatioe of ~/),. so that ~he energy dissipation is stationary !'or the acluaI flow, °io second order it~ variation,~ 4>-+ 4 + 84 one has
/ ~: ~=,/oro ., 2 ~ [ - 2 , "-~ (r2% " + 4,'(F' + 4 ¢ '
-
~ ' ' Ur"q>
.....
~=Orq:) ' ~4>
<5
'
~ < h " -~' 9 ' ) ~4 ~" b " ][o ~ + tr-< ~.r+
-r
2 1 "2 "
.-'~i
(&fl) (L(/>) 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<:m r :> 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:i
.............................
'
::
o,-.
.,
3t'
,, dr
arid {}~e corresponding approxhr, a{e ......a, ...
r' '~'U ' ''~ d r"
~S l 1 t
.:~,~.
X
2&'} -':::r :: ,,A: - i r':U,~r'):b"
........
The coefficients ,f, arid B: v,i~! be ibu::d by appiying :he bo:u:d:.::'y co::dhio::s at: We ~;ou: from the exp~ici: soiut:o:; for d:e urdL?:m sphere ob:ained iu see:ion 3 that lhe so:~tio~! it1 the h-::erio: :reg og i:::.c~-:aracte:ri;~ed h? :wo coefiiciems d a:,d O. (:'.orresponding a p p r o x i m a : e sotutkmv :-::nay {~.e fi:yur:d b) ?<:tim: <.%~t>,~ u,b,,:n~dor: scbeme,:¢ v.hich a:.,toma:ica::y generate ~v,o :::::::::::::::::::::::: .,:>i~..,:it:,,:~.,.,,o~-dif~c-re:~: :ype, wl.dch '~i!:ibe denob~:d by {#.h,...£< ~ :,:.r:.d~(~..'.:,. ZpL %':'.first r:~btai.r~:M:: e:'xpa::sion Ibr :he p:Hr (qt,v, Z:~.~. To {i:i.s¢:~d :~::::n:rod~.:c-e a fo:-:::M exp::ns{cm }:?~.::'umevev:. which fim4:y is set equa: ::: ¢mi:y. and w:::c the: p::ir o[ di~:%.rc-:::i;~::e.q::,a:i
{5.13)
su[::,s~i:u~e
@£:,
ti::{, solu{io~:~,
:::',:
&,
°,~:......
-
~
.c,s ~'i~'
"
,
(< I4" Zv
=
Zg:.. o +
~:Zoo :
, :"Zo.
a +
" ' "
and s:.tve successiveiy fi)r (dh)./ ,7~m9 by equating coeffidents ~}i~ ,.~;,,. o~'~:-~.:::o zero. To lowest order one finds - U?&:. ,~ + ZD. o =
0, { 5 . : 5)
r~x'r~, e + ~.rT~o, o
..... 2ZD, e = 0
74
B~U~ FEH:]ERfiOF
< <-:or
•
,
J .
,,
where D, is an arbitrary consta[H, m~d ~-hete :!br Z,>. e~ we have ~ak.e~~ ~'.e r e g ~ a r solution. T o first order ~_ne:~ ~ has r°q}m o + 4rq~m <. . . .
,
~z< ~
(,}.
'.-
( 5,. I';% "
9
'
r2
r,~
Substituting f r o m (5. f6) one can ea.~ity solve d-@i pair o f eqm~iom> v, ith accorma+ 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'<&. : O.
S u b s t h m i n g the ~;eries expansk>~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<,,Q% M f£RS f:N ;S soL{.~........ 1 ION:
@, ~": C j : 5 ~ - o {r}. . . -+ . . . D~.4:-;~ .~, t~'},,.
,:~, :,::
t)~"/,,,.~,...
{. ......
oi' R o i:~ opl.imiz¢~l b y u s i ~ s ~l~e %;:~Natio.~m,t p.','ir~cip~c_
A ckt~|ed~men~s
Rcfetence~
3} A G K irkv,:ood a~@ J. Ri~:n;~J.m 3 ~:h<:n:~, Phi.~ 16 (} 94;I.;;, % 5
p. 2~]0 J. G. K kk.~ood. ,L Po!}'m¢~ S c i t 2 (~)34~ i.
%:, ....
2',..: I }
F R I C T I O N A L P R O P E R T I E S OF D I L U T E P O L Y M E R S O L U T I O N S IlL TRANSLATIONAL-FR|CTION COEFFICIENT [L LL F E L L ) E R H O F +
D~v~,rtmee+e of Pia~~tc~. Ouee,+ a:lusy C:?dtes~-e+l+.)+i~:e~'~it.v+g" Lom&~e, #,.+:n~d~:+;~, E] 4N .....ET~gta;~d
~n the framework of our ge{ae++a~izatio~ of the I)eb?e--Bueche vheo.,-y for the frictiomd rJ.~op<;o ties o f dilute polymer solulioas ~se smdv {t~e m m ~ a t i o n a b f + i c t i o n coeflicicm o!~ polymers ,.vith spherically symmetric segment distribution. A varkt++ona~ principle o f minimum energy dissipation i~ formu~;atedv&ich i:+s~.+,i~ahge~:c>rnumericai work.
I. Introduction
in the first article +} ([) of this series on the f~ic~ic;nal properties of dilute polymer 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 the equations describing the hydrodynamic in~eractioris between polymer segments. Thus ,ae have created an attractive alternative ~o the co~wentionally accepted Kirkwood+ Riseman (KR) theory'+). We have exter+ded the analysis of Debye and Bueche io deaf with more general po[y:ner models that,, the unif'k~rm sphere, in 1 we have presented the detailed theory of" the rotational friction coefficientf.. In the second article% (tl) we have outlined a simplified version of o~,r theory involving an angular average (PAA) , ~",he hydrodynamic interaction+ similar to an approxi+ minion made in the KR theory, in ti~is article we return to the exact mean-fieid theory and study it:~ consequences %r the +rans}atio~at-friction coeiScient j~. The present article can be read independently o f ! and tI+ We start directly t?om the macroscopic equations; for a discussion of the microscopic basis we rei~er to 1. In section 2 we apply ~he Debye+ Bueche equations to the flow situation of translationa{ drag on a spherically s~mmetric polymer and derive two general expressions for the friction coefticientf+. In section 3 we give the explicit solution for the uniform sphere and a spherical-shell segmem distribution. In section 4 we study # Presel'~t address: Instimt f~.i+'Theoretische Pi~ysik, F
63
64
B.U. FELDERHOF
the energy dissipation and derive a variational principle which will be useful for numerical work. In section 5 we develop perturbation methods which are applicable to general segment distributions.
2. Translational friction of a spherically symmetric polymer We consider a polymer in the usual picture o f a large number o f segments connected by bonds in a flexible structure that can adopt many conformations. The polymer is immersed in a fluid which is assumed to satisfy the linear NavierStokes equations for steady, incompressible flow. In article I of this series it was shown that in a mean-field approximation the basic equations of the theory describing the hydrodynamic imeractiont; between polymer segments are v ev -
(r) ( V -
u) -
W
= 0,
v.
V = 0,
(2.
where *jo is the solven! viscosity, V(r) is the average fluid flow velocity, g is the friction coefficient of a segment, ~)(~,) is the segment density distribution, u(r) gives the rigid-body motion o / t h e polymer, and ?(r) is the average pressure. The averages arc over the statistical distribution that characterizes the corfformations of the polymer segments. "['he second term in (2.1) accounts for the friction between polymer and fluid. The average force density exerted on the fluid is given by F(r) = - @ (r) [ V(r) -- u(r)].
(2.2)
Hencefbrth we shalt assume the segment distribution Off) to be spherically symmetric. It is convenient to choose a coordinate frame in which the polymer is at rest with the centre of mass at the origin, in which case u(r) = 0. Introducing the inverse shielding lengff~ z by ~a(r) = ~t? (r)/'t~o, one then has the basic equatioas V 2V - :?(r) V -
~ * VP = 0,
V. V = 0,
(2.3)
which must be solved with tile condition that the solution tends to the u ~ e r t u r b e d flow pattern at large distances from the polymer. The unperturbed flow velocity ~%(r) and pressure po(r) in the absence of the polymer satisfy the equations ~}o V%o - Vpo = 0,
V. % = 0.
(2.4)
/n the case of translational drag one considers Vo(r) = Vo constant, and po(e) = P o constant. We choose Vo to be in the z direction so that in spheri ::al coordinates
(r, 0, ,p) vo = voe~ = vo (cos 0er - sin 0co)
(2.5)
FRICTION COEFFICIENT OF POLYMERS IN SOLUTION
65
with unit vectors e=, e, and e0. In order to solve (2,3) we look for a velocity field V(r) with the same angular dependence. Thus we put V(r) = vo {q~(r) cos 0G - [qS(r) + ½,'&' (r)] sin 0e0} = ~b(r) vo -
(2.6)
- ~ r - ~ 6 ' (r) r × (r x ,,o)
and seek an equation for q)(r). In (2.6) a prime indicates differentiation with respect to r and in the second term we have used the condition o f incompressibility V . V = 0. CIearly the unperturbed flow vo is o f this form with qS(r) = t, The angular dependence o f the pressure must be given by the scalar ( r , vo) and we put (2.7)
P(r) = Po .... qovoZ (r) cos 0 = Po -- ~jo2 (r) r , vo,/r,
~t then follows by substitution s) that eqs. (2.2) are satlsfled provided 4~(r)and y(r) satisfy the set o f coupled equations re0" + 4r(// -- r-'U(r)4) + * Z = 0,
(2.8a)
r2z '' + 2rz' - 27. - r 2U' (r){/) = 0,
(2 8b'~
where U(r)
=
~2(r.),
(2.9)
which for brevity we shall cal~ the potential. The differential equations (2 8a, b) must be solved with the b o u n d a r y conditions that 4~(r) and ;((r) are regular at the origin and that (~(r) tends to unity and z(r) to zero at infinity, At large distances the potential U(O tends to zero rapidly and (2.8a, b) becomes a pair of free-field equations. Fitting to the unperturbed flow we can therefore write f'or the asymptotic "t:ehaviour 4>(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),. , <,# ~=- (c," ~.,/c;x#) + (r.~ . V~ i ~c,.~,).
V-,~ = - g ( r ) ,
(2.14).
V , V = 0.
Integrating the first equation over a volmne O bounded by a surface ~" and applying Gauss2~ theorem one finds -f~.
n d S = .[ g ( r ) &'.
2-"
(2.15)
~2
This shows the equality o1" two alternative expressions for the force exer{ed on the fluid in ,t2. The totai fbrce is found by extending the integral on the right over all space, .~- = j' F0") dr = - g J'e(r) V ( r ) dr.
( Z 16)
Substituting ~2.6) one obtains
.,*~
=
-v
o"4,=," J'"
[(hit) + !W'~b' 0 9 t ~j(r)r ~ dr.
(2.17)
0
From Ll~e left-hand side o1' (2.15) and the asymptotic behaviom" (2.10) one verifies eq. (2. t i ). Thus we have two expressions fo~" the translational friction coefficient, ri::, (2.12),,,1~ - 4~k>d = brWoa and L~ = 4_,~ - " f [4~(
(2.18)
These are to be compa, red with the corresponding expressions (11.4,2) /ound in the PAA theory4), To conc!ude this section we note that eq. (ZSb) can be solved explicidy lbr z(r), The homogeneous equation has solutions r and t/r e. With the bom-~dary conditions dmt Z(r) must be regular at the origin and tend to zero at infinity the solution o f (2,8b) is therefore given by ct>
~.
z(r) = - ~ xr .t L (r .} q~(r') dr' r 1
'
f~
¢
.__
. - [r'-~U ' (r") 4 ( / ) dr'. -~r d l
-3
(2.19)
Hence once the velocity function ~/)(r) is known one can c a h u i a t e lhe pressure using (2.7).
FR1CTION
COEFFICIENT
OF
POLYMERS
IN SOLUTION
67
3, Explicit solutions for uniform sphere and spherical shell F o r a few simple potentials U(r) the differential equations (2.8a, b) ca~ be solved explicitly. We consider first the uniform sphere for which U(r) = [ ~ i
= (~'°/~'°) " f oc°nstant R r < r < :x,.l°r
0~r
Hence eq, (2.8b) becomes h o m o g e n e o u s in the two regioT~s ap, d one has the simple solutions
z(O
=
[
A/r:,
for
R < r < ,~,
k D~',
:for
0 ~ r < R.
(3.2)
In the exterior region r > R the asymptol.ic solution (2.[0} is exact, so that 4@) = 1 -. (A/r) + (B/r~).
r > R.
(3.3)
In the interior region r < R the differential equation (Z8a) becomes r2~" + 4r~])' - ~ r : ~
+ .Dr 2 = O,
r < R.
(3.4}
The solution which is regular at the origin is given by qb(r) :: C ( c o s h xr \,
. . r2
.
sinh ~r') .
. y ,~
/
1)
+ --.-, ~2
r <
1~.
(.3,5)
We note that the velocity and pressure corresponding to the terms multiplied b} D satisfy Darcy's law :79 V = - V P . The coefficients A, B, C and D must be found by fitting the solutions ai r = R with the aid of b o u n d a r y conditiot~s. The boundary conditions for the original problem are that all c o m p o n e n t s of V and the n o r m a I - n o r m a t and normal-tangential c o m p o n e n t s of ~!,e stress tensor c; are continuous at the boundary surface. The continuity of V ~mpties from (2.6) that 409 and 4,'(r) must be con-. tinuous at r = R. The derivative; of the velocity are given by s) VV = vo [qb' cos 0e:er - ½ (r4;' + 3~5') sin Oe~eo + ½q}' sin 0e0e, - ½~' cos 0 !'.e0e~ + e~e:, )].
(3.6)
Hence the stress tensor is = qovo [2q~' cos OerG - ½- (rq;' + 2<]/) sin 0 (Ge0 + e0e,.)
- 4.' c,~:., 3 (ee,e,j + e,-~G)] -- (p~) - ~jovoz cos 0) I.
(3.7)
68
B, U. FELDERHOF
The continuity o f o,, at r = R requires that z ( r ) be c o n t i n u o u s . Finally, the continuity o f a,0 requires that 4/'(r) be c o n t i n u o u s at r = R. I m p o s i n g the f o u r b o u n d a r y conditions and solving for the coefficients one finds A = ~-R
1 +
Go(o') (o31 2'
B == R 2 (A + 2R),
C = 2 A R / [ ( r z (cosh (r) Go(c0],
D =:
A/R 3,
(3,8)
w M r e a :-- x R is D e b y e ' s shielding ratio and Go(o) = I - tanh or/or.
(3.9)
];'.'lonl (2, 12) we therefore find i~or the friction coefficic ~t _f~ = 6::qo R
(sphere~.
(3,10)
This resul~ is idenlical with that o b t a i n e d by D e b y e and BuecheZ). In fl~e high-densily limit ~ - ~ o~, one obtains for the effective radius a = R , whicl~ leads t:o the hard-sphere resultf~ = 6=~h~R. F o r low density z ~ 0 one has 2 ~ 8 4 ~:,~ = R [-,:;(r" - -c.~f + e~J(o'~')].
;.* -, 0.
(3. i 1 )
The first term gives the free-draining limit for fl, e ffictic.n c o e N c i e n t . f~ ::: 6r:.'qoa and with the n u m b e r o f segments ~, = 4~Ra 0 one finds ,/£~ = n g
(free draining),
With
(3.12)
a restfl~ a!so obtain~ed from (2.18) by using f o r t h the u n p e r t u r b e d value ~b(r) == 1. W h e n !.he segments are distributed in a :;phericat shell o f radius R one has U(/') "-= z2(r) =
/.t~)(r
-- R),
(3.13)
where ¢~ = ,~r:/~to aim T :is the u n i f o r m surface density, r. = n / 4 = R e, The solutions o f (2.Sa, b) in ihe regions r > R and r < R are 4 ( 0 = 1 -- (A/r) + (B/tf . 3 ),,
Z(r) = A / r e ,
R < r < <:.,
(3.14) ,
.- T d D r ' ,
z(r) =: D r ,
0 ~ r < R.
The conditions at r = R are that V m u s t be c o n t i n u o u s , and that .he c o m p o n e n t s o f e¢ involvil~g a normal direction must j u m p by ,~~,(I,' +
)
.-- (r,.,(R-) = '?o# V~ (R),
(3.15)
FRICTION COEFFICIENT OF POLYMERS IN SOLUTION
69
where ex runs through (r, 0, q:). Hence one finds the j u m p condi{ions for ¢(r) ana ;Z(r) ¢ ' ( R + ) = ¢ ' ( R - ),
4,(R + ) = 4)(R - ),
R [C(R+) z(R+)
- 6"(R-)]
- z(R-)
t~R
(3.~6)
= t*~. ( R ) .
The cc,efficients A . . . . A == R
= # [2¢ (R) + R<,5' (R)],
D are easily found to be ,
B = !~R2A,
C :--. 1 -- ~ ( A / R ) ,
D = 0.
(3.17)
Note that the pressure perturbation inside the shell wmisbes. In the limit/.t .-, ,z, one finds again the hard-sphere result a = ~A = R, while the free drahfing limit (!~ .-~ 13) is A ~ ~ R z, which imptiesj~ = n~. In the general case the m, nstationalfriction coefficient of the shel! can be expressed as J~l .... 6r:rioR
!,R _3 + # R
= 6=~7oR , °'2 ~ + o-a
(shell},
i3.18)
where ,J'- = 3}~R = 3n~/4=~/oR° as in the case of the uniform sphere. For a discussio~ of these results and a comparisor~ with the results of PAA theory we re[i_'r to article II of this series4).
4. Energy dissipation and variational principle In order to handle more realistic segment density distributions, e.g., a gaussian, it is necessary to look t'oi" perturbation methods allowing one to find approximate solutions o f the differential eq .....tions (2.8a, b) and a corresponding expression for the friction coefficient fd. As in the case of rotational drag a study of the energy dissipation leads to the formulation of a variational principle. It has been shown in I that the total energy dissipation is given by W = '70 j" [} (VV).~ + x 2 V2I dr,
(4.1)
where the integral is over all space, F o r the case o f translational friction one can subs:itute (2.6) and (3.6). Performing the angular in|egrations one finds ~3
w=
~ o ~ , , ; .f [r~¢ "~ + 4r#4," + m4, '~ I)
+ U(r) (,"-6" T' 4 r ~ b ' + 64-')1 r= dr.
(4.2)
70
B. U, F E [ , D E R H O F
Integral on b / p a r t s !cads to the {brm ~t/'r =
~",a ~t<>t' O"-
( ---- t'4~fl (1~
"k r a ¢~. q5 . - - .4¢3r/)4) . " +. 4,'2rb~/>' + 2r"q5-
~':
(4.3)
_I
o
where L is a linear operator defined by =r
(/~
+ ~r'9
+ 8re4~ ' - . 8 r % '
U' (r'*~,b' + 2r'b/J).- U (r~
.-
4r:~,b').
+
(4.4)
Ii is easily seen that by eliminating the function z(r) from (he pair oF differential equations (2.8a,b) one arrives m the equation Lq5 =- O, Hence the integral in (4,3} vanishes. The boundary terms in (4.3t can be calculated from the asymptotic behaviour (2.10) and one finds (4.5)
W = 4=qovg A = ./-5v;,
as was to be expected. As in the case of rotationa! l:Yiction one finds t " a t L¢ = 0 is also the guler equation for the functional W as given in (4.2) upo~ variatioe of ~/),. so that ~he energy dissipation is stationary !'or the acluaI flow, °io second order it~ variation,~ 4>-+ 4 + 84 one has
/ ~: ~=,/oro ., 2 ~ [ - 2 , "-~ (r2% " + 4,'(F' + 4 ¢ '
-
~ ' ' Ur"q>
.....
~=Orq:) ' ~4>
<5
'
~ < h " -~' 9 ' ) ~4 ~" b " ][o ~ + tr-< ~.r+
-r
2 1 "2 "
.-'~i
(&fl) (L(/>) 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<:m r :> 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:i
.............................
'
::
o,-.
.,
3t'
,, dr
arid {}~e corresponding approxhr, a{e ......a, ...
r' '~'U ' ''~ d r"
~S l 1 t
.:~,~.
X
2&'} -':::r :: ,,A: - i r':U,~r'):b"
........
The coefficients ,f, arid B: v,i~! be ibu::d by appiying :he bo:u:d:.::'y co::dhio::s at: We ~;ou: from the exp~ici: soiut:o:; for d:e urdL?:m sphere ob:ained iu see:ion 3 that lhe so:~tio~! it1 the h-::erio: :reg og i:::.c~-:aracte:ri;~ed h? :wo coefiiciems d a:,d O. (:'.orresponding a p p r o x i m a : e sotutkmv :-::nay {~.e fi:yur:d b) ?<:tim: <.%~t>,~ u,b,,:n~dor: scbeme,:¢ v.hich a:.,toma:ica::y generate ~v,o :::::::::::::::::::::::: .,:>i~..,:it:,,:~.,.,,o~-dif~c-re:~: :ype, wl.dch '~i!:ibe denob~:d by {#.h,...£< ~ :,:.r:.d~(~..'.:,. ZpL %':'.first r:~btai.r~:M:: e:'xpa::sion Ibr :he p:Hr (qt,v, Z:~.~. To {i:i.s¢:~d :~::::n:rod~.:c-e a fo:-:::M exp::ns{cm }:?~.::'umevev:. which fim4:y is set equa: ::: ¢mi:y. and w:::c the: p::ir o[ di~:%.rc-:::i;~::e.q::,a:i
{5.13)
su[::,s~i:u~e
@£:,
ti::{, solu{io~:~,
:::',:
&,
°,~:......
-
~
.c,s ~'i~'
"
,
(< I4" Zv
=
Zg:.. o +
~:Zoo :
, :"Zo.
a +
" ' "
and s:.tve successiveiy fi)r (dh)./ ,7~m9 by equating coeffidents ~}i~ ,.~
0, { 5 . : 5)
r~x'r~, e + ~.rT~o, o
..... 2ZD, e = 0
74
B~U~ FEH:]ERfiOF
< <-:or
•
,
J .
,,
where D, is an arbitrary consta[H, m~d ~-hete :!br Z,>. e~ we have ~ak.e~~ ~'.e r e g ~ a r solution. T o first order ~_ne:~ ~ has r°q}m o + 4rq~m <. . . .
,
~z< ~
(,}.
'.-
( 5,. I';% "
9
'
r2
r,~
Substituting f r o m (5. f6) one can ea.~ity solve d-@i pair o f eqm~iom> v, ith acco
r ~ b%Sc + r~£,.~' = O,
tt
r'Zc + 2rz~- - 2 Z c -
)
~:r:"U'<&. : O.
S u b s t h m i n g the ~;eries expansk>~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<,,Q% M f£RS f:N ;S soL{.~........ 1 ION:
@, ~": C j : 5 ~ - o {r}. . . -+ . . . D~.4:-;~ .~, t~'},,.
,:~, :,::
t)~"/,,,.~,...
{. ......
oi' R o i:~ opl.imiz¢~l b y u s i ~ s ~l~e %;:~Natio.~m,t p.','ir~cip~c_
A ckt~|ed~men~s
Rcfetence~
3} A G K irkv,:ood a~@ J. Ri~:n;~J.m 3 ~:h<:n:~, Phi.~ 16 (} 94;I.;;, % 5
p. 2~]0 J. G. K kk.~ood. ,L Po!}'m¢~ S c i t 2 (~)34~ i.
%:, ....
2',..: I }