SUPPLEMENTARY MATERIAL
Solution of the complete Curtiss-Bird model for polymeric liquids subjected to
simple shear flow
Pavlos S. Stephanou,* and Martin Kröger
Department of Materials, Polymer Physics, ETH Zurich, CH-8093 Zurich, Switzerland
*Author to whom correspondence should be addressed; electronic mail:
[email protected]
A. Real basis of the spherical harmonics
We define the real basis of the spherical harmonics (RSH) ylm based on the spherical
harmonics Ylm as
 1 *
 2 Ylm  Ylm  if m>0

ylm  Yl 0
if m=0 .
 1

Ylm*  Ylm  if m<0
 i 2
Note that here the spherical harmonics are defined without the
Ylm  u   Alm Pl m  cos   eim , Alm 
(A1)
 1
m
factor:
2l  1  l  m !
m
where Pl  x  are the associated
4  l  m !
Legendre polynomials and the asterisk denotes a conjugation (Ref. 1, p. 788). These
definitions are identical with the following ones:
1
yl m  2 Al m Pl cos  m  
m
yl 0  Yl 0
.
yl , m  2 Al , m Pl
m
(A2)
sin  m  
The RSH can be shown to obey the following orthogonality relations:
 duy
y   l   m .
(A3)
lm 
As it will be made evident in Appendix B we need to consider the rotational operator
defined as
L  u

.
u
(A4)
For the action of the rotational operator on the RSH we obtain the following:
Lz ylm    1 myl ,  m
m
L y ylm 
Lx ylm

,


for
m

1



 alm yl , m1  alm yl , m1 
1 
 alm yl ,m1  alm yl ,m1 
2
 1

2
m
(A5a)
whereas for m  1
1
L y yl , 1  u    al1 yl , 2 ;
2
1 
L y yl 0 
al 0 yl1 ;
2
1
L y yl1  u   al1 yl 2  2al1 yl 0 ;
2


where alm

l

Lx yl , 1  u   
1
2

2al1 yl 0  al1 yl 2

1 
al 0 yl , 1
2
1
Lx yl1  u    al1 yl , 2
2
Lx yl 0 
,
(A5b)
m  l  m  1 (also note that al , m  alm ). The Laplacian is given as

L  L ylm  L2 ylm  l  l  1 ylm .
(A6)
Finally, the following forth-rank matrix needs to be defined for Section B.
Blm   duylm  u  uz u y  Ly y  u    u y2  Lz y  u   .
2
(A7)
B. Methodology to obtain the RSH expansion of the single-link distribution function
We consider the steady-state limit of Eq. (1) which we rewrite as:
 2 f  , u 
 

3
(B1)
  N 2

f  , u     Γf  , u   
κ : uu  0 ,
2

u u
u
4
1
f  , u   f  , u  
, Γ   δ  uu  κ  u with κ  κ , normalization
4
1   
where
1
 ds  duf  , u   0
and vanishing boundary conditions at chain ends for f  , u  . We
0
next consider the following expansion in spherical harmonics with an upper cut-off for the
spherical harmonics index l equal to L:
L

l
 
f  , u       alm
cos      blm  sin      ylm  u  .
(B2)
l  0 m  l   0
where  are yet unknown. By first considering normalization and the boundary conditions
Eq. (B2) simplifies to:
L
l

1   
 
f  , u  
b00 sin       blm
sin  ylm  u  .
(B3)

2   0
l 1 m  l   0
even
Our next step is to use expansion (B3) in eq. (B1) to obtain:

1
2
L



b  sin   1     



00
0
even
l


2

   b  sin    1    
l 1 m  l  0
2
lm


Γ
u 

 3
  N 2l  l  1  ylm  u     Γylm  u    
κ : uu  0

u
 4
We then multiply with sin    ynp  u and integrate over u and  to get
1

2 
L
l

1

b  sin    1       duy



0
even
 
2
00

0
1
y   duynp
np 00
.
(B4)


  Γy00   
u


2

blm   sin    sin    1        N 2l  l  1   duynp  u  ylm  u  , (B5)





l 1 m  l   0
0
  duynp  u 

 3
  Γylm  u    
 sin    duynp  u κ : uu  0
u
 4 0
1
or by using
3

  Γylm  u      duynp  u L   u  κ  uylm  u 
u 
,
  duylm  u L   u  κ  uynp  u    WiBnplm
  duynp  u 
(B6)
then,


1  ,  even
2
 b00 
1      n0 p 0  WiBnp 00 
2
L
l
1
3 
2


  1
 1         N 2 n  n  1  bnp
 Wi   blm  Bnplm 
1   1  Wi  duynp u xu y  0
2 



2
2 l 1 ml
4
.
(B7)
Note that already in the last equation in Eq. (B6) and in Eq. (B7) we have specified our
treatment to the case of shear flow. Now by taking cases separately we find that
b00   0,  and bnp    0,  even,n, p . For the rest,

L
l
  b   A
l 1 m  l
lm
nplm
, for  odd
 Cnp
Anplm  1         N 2 n  n  1   nl mp  1   nl mp WiBnplm ,


2
Cnp 
6

Wi
(B8)
1
 n 2 p 2
15
where  nl is Kronecker's delta (if n=l  nl  1 , otherwise  nl  0 ). By solving this we

actually find that for odd l then blm  0 , and by making a choice for the upper cut-off of
the spherical harmonics, L we may obtain the expansion, with regards to the dimensionless
shear rate Wi, up to Wi L / 2 . Without further ado we give here the final expression up to
fourth order in Wi obtained using Mathematica:
4
1
4
f  , u  
4Wi 3
4Wi 4



1 5
 

 
3
5
g1   y2,2  4Wi 2
y20  y22   g3   
y40 
y44  
1  4Wi
 g 2    
15
15 
21

 7

7 3
 
  2
1 7
 
5
5 3
35
g 5    y4  4 
g 6   y4 2  5 g 7   
y6 2 
y6 6  
  g 4    y2  2 
15  49
539
286
21
 11 26
 
  2 3
2
30 3
15
5
g8   y20 
g9   y22 
g10   y42 
g11   y40 
g12   y44

15  3773
147
539
1618617
231231
 g13  
 1 175

1 15  1
35

y60  y64   g14   
y62  5
y66 

121 91  7
286 

 11 234
 21 15
 
6 105
75
 g15   
y80 
y84  21
y88 
  O Wi 5 

13 187
2431  
 143 17
,(B8)
where


6 sin  
6 sin  
6 sin  
; g 2    
;
g



3 
2
K2
K2
K2 K4
 1 
 1 
 1 

g1    
odd
odd
odd

6 12 K 2  23K 4  sin  
6  2 K 2  K 4  sin  
;
g





5
3
K2 K4
K 22 K 42
 1 
 1 

g 4    
odd
odd

6 10 K 2  11K 4  sin  
6 sin  
;
g





7
2 2
K2 K4
 1 
 1  K 2 K 4 K 6

g 6    
odd
odd
6  50 K  187 K 2 K 4  253K  sin  
K 24 K 42
 1 

g 8    
2
2
2
4
odd
2
2

6  70 K 2  71K 2 K 4  69 K 4  sin  
6  4 K 2  3K 4  sin  
g 9    
;
g





10
4 2
K2 K4
K 22 K 43
 1 
 1 

odd
odd
2
6  48020 K K 4  17550 K K 6  18447 K 2 K 4 K 6  72358 K 4 K 6  sin  
g11    
K 23 K 43 K 6
 1 

2
2
2
2
odd
5
2
2
2
6 113876 K 2 K 4  307138K 2 K 6  190619 K 2 K 4 K 6  72358K 4 K 6  sin  
g12    
;
K 23 K 43 K 6
 1 

odd
6  49 K 2 K 4  50 K 2 K 6  55K 4 K 6  sin  
;
K 22 K 42 K 62
 1 

g13    
odd

6  3K 2 K 4  2 K 2 K 6  K 4 K 6  sin  
6 sin  
;
g





15
2 2 2
K2 K4 K6
 1 
 1  K 2 K 4 K 6 K 8

g14    
odd
odd
, (B9)
where we have used the shorthand notation K j  1        N 2 j  j  1 .
2
It would be instructive to check whether we obtain eq. (19.5-17) in Ref. 4 when    0 ; up
to third order we have the simplifications g1    34 E2   , g2    g3    161 E4   ,
and
7
1
1
g4     967 E6   , g5     160
E6   , g6     160
E6   , g7     480
E6  
where E2n   are the even Euler polynomials (see Ref. 2 p. 2024, and Ref. 3 p., 805)
defined via,

E2 n     1 4  2n  !
n
 1
odd
sin  
 
2 n 1
.
(B10)
.
(B11)
The first three are expressed as [Ref. 3 p., 809]
E2     1   
E4      2 3   4
E6      3  5 3  3 5   6 
With this in mind we obtain for    0
6
1 

Wi 2 
f  , u  
 1    y2,2  u  
  2 3   4  

1  3Wi
4 
15
4 15
 3
 Wi 3 
1 5
5

y

y

y

y
3  5 3  3 5   6   ,


20
22
40
44  
7 3
21  120 15
 7
(B12)
10

3
15 3
5 7
35
y4  2 
y62  5
y6 6 
 y 2  2  5 y4  4 
7
77
11 26
286
 7

which is the same as (19.5-17) in Ref. 4 (the third order term given in Eq. (7.18) in Ref. 2)
in the case of shear flow.
C. Results of the BD simulations
In this section we provide the results of the BD simulations for the material functions
without making them dimensionless with the zero-shear-rate values. This is done for the
following two reasons: a) the fidelity of our BD algorithm is thus verified, and b) the
capability of the theoretical results, Eq. (6) in the main text, to capture both the zero-shearrate values at Wi=0 and the downturn at larger Wi is also verified. This comparison is
provided in Figs. S1-S4.
7
FIG. S1: Predictions for the shear viscosity (a), first (b) and second viscometrc function
(c) when   0 and N  0.5 .
FIG. S2: Predictions for the shear viscosity; parameter values same as in Fig. 1.
8
FIG. S3: Predictions for the first viscometric function; parameter values same as in Fig. 2.
FIG. S4: Predictions for the second viscometric function; parameter values same as in
Fig. 3.
9
D. Thermodynamic admissibility of the complete CB model
Here we provide the GENERIC framework5-7 for the complete CB model. It
actually stands as an extension of the GENERIC framework for the simplified CB model
(with    0 ) provided by Öttinger [Ref. 7, Section 4.3.2]. Concerning the total energy E,
total entropy S, and the Poisson matrix L they need not be modified and we immediately
adopt expressions (4.141), (4.142), and (4.145 to 6), respectively. Adopting the
T
factorization for the friction matrix proposed by Edwards,8 M f  CM  DM  CM , we here
consider
0
0 0

0 
0
CM  
 0   v  

0 0
0

0
0


0  T 0
, CM 
0
0


0
 1 

0
0
 
 v
0
 
0
0
0

0 ,
0

 1
(D1)
where XT is the transpose of X. Here v is the velocity gradient tensor. This particular
T
form for C M , and thus CM , is the one dictated by the degeneracy requirement
CMT   E  x   0 . Next we make the choice
0
0

 0 2  I CB 
DM  
0
0

0
A



0 A 
,
0 0

0 M f ,44 
0
0
(D2)
which leads to the following form for the friction matrix (note that we use Einstein’s
implicit summation convention for repeated Greek indices)
10
0
0
0
0



   A 
 0  2   I CB      I CB   
.
Mf 
1
 0     I CB  
   I CB       v A 
2


0


A

v
A
M
 
  
f ,44


(D3)
Here
1
I CB 
3
nk BT 2 N    1    d  du f  u,  , t  uuuu ,
2
0
(D4)
M f ,44  , u,  , u  

, (D5)


 

2 




1


f

,
u


N

f

,
u





u

u










nkB  N 


u
u 
1
A    1  N 2 1    f  u,  , t  u   u u 

,
u
(D6)
Expression (D4) is simply eq. (4.150) of Ref. 7, (D5) is a special case of eq. (8.169) of Ref.
7 when
  
s
  2 s  1    
2
and
x
 s u s u
x
  2 s    N 2 I , whereas
expression (D6) is proposed here for the first time and is needed to obtain the extra terms
in the stress tensor missing from (4.151) of Ref. 7. Using the above the evolution equation
for the single-link distribution function and the stress tensor expression reads
f

1   2
 N 2  
 v f    κ   I  uu   uf  
f


f,
t
u
  2
 u u
(D7)
and
1
1

τ  nk BTN 1      d  d 2uf  u, , t  I  3uu    N 2   1    d  d 2uf  u, , t  I  3uu 
,
0
0

1

3 κ :   1    d  du f  u, , t  uuuu 
0

(D8)
11
which is the same as Eqs. (19.3-26) of Bird et al. (for a strictly monodisperse system) and
eq. (19.4-18), respectively4. Finally, requiring that DM is positive semi-definite (a property
inherited to Mf as well) requires that the two averages above have to be non-negative or
that 0     1 and   0 .
References
1. G. B. Arfken, and H. J. Weber, Mathematical Methods for Physicists, 6th Ed.,
Elsevier Academic Press, (2005)
2. C. F. Curtiss and R. B. Bird, J. Chem. Phys., 74, 2016 (1981).
3. M. Abramowitz, and I. A. Stegan, Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables; 9th ed. (Dover Publications, 1965).
4. R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids; Vol.
2, Kinetic Theory, 2nd ed. (John Wiley & Sons, New York, 1987).
5. M. Grmela, and H. C. Öttinger, Phys. Rev. E, 56, 6620 (1997)
6. H. C. Öttinger, and M. Grmela, Phys. Rev. E, 56, 6633 (1997)
7. H. C. Öttinger, Beyond Equilibrium Thermodynamics (Wiley-Interscience, 2005).
8. B. J. Edwards, J. Non-Equilib. Thermodyn., 23, 301 (1998).
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