Molecular Origin and Dynamic Behavior of Slip in Sheared Polymer Films
Nikolai V. Priezjev and Sandra M. Troian
Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544, USA
(Received 21 April 2003; published 7 January 2004)
The behavior of the slip length in thin polymer films subject to planar shear is investigated using
molecular dynamics simulations. At low shear rates, the slip length extracted from the velocity profiles
correlates well with that computed from a Green-Kubo analysis. Beyond chain lengths of about N 10,
the molecular weight dependence of the slip length is dominated strongly by the bulk viscosity. The
dynamical response of the slip length with increasing shear rate is well captured by a power law up to a
critical value where the momentum transfer between wall and fluid reaches its maximum.
DOI: 10.1103/PhysRevLett.92.018302 PACS numbers: 83.50.Lh, 83.10.Rs, 83.50.Rp, 83.80.Sg
Slippage at liquid-solid, polymer-solid, and polymer-
polymer interfaces can strongly influence hydrodynamic
behavior in microscale and nanoscale flows. Contributing
factors include poor interfacial wettability or weak mo-
lecular attraction between phases [1,2], surface roughness
[3], high shear rates [4], a reduction in polymer interfa-
cial viscosity [5], and nucleation of nanobubbles at hydro-
phobic surfaces [6]. Despite the enormous interest in
slip behavior, there is yet no consensus on which parame-
ters control the degree of slip in simple fluids and poly-
meric systems nor how the slip length depends on the
local shear rate.
Navier [7] first proposed that the slip velocity V
s
at a
wall-fluid interface varies linearly with the shear rate
_
according to V
s
L
o
s
_
. He assumed a constant (shear-
independent) slip length L
o
s
, defined as the extrapolated
distance into the solid phase where the tangential velocity
vanishes. More than a century later, Tolstoi [1] used
Frenkel’s [8] molecular kinetic theory to link L
o
s
to the
liquid-solid equilibrium contact angle in vacuo through
the activation energy for molecular displacements. This
derivation determined that L
o
s
a=
wall
1, where a
is the center-center distance between adjacent molecules,
is the bulk fluid viscosity, and
wall
is the reduced
viscosity of the first fluid layer near the wall. Since
then, a number of molecular dynamics (MD) studies
have focused on the functional dependence of the slip
length at low shear rates on molecular parameters affect-
ing momentum transfer at a wall-fluid interface [9–16].
Stratification of the fluid layers near the wall plays an
especially significant role [10] in determining the degree
of slip. Recent equilibrium studies of simple fluids [17,18]
have elucidated the dependence of the slip length on the
fluid-wall contact density, the interaction energy, the in-
plane diffusion coefficient, and the structure factor of the
first fluid layer.
Even less is known about the dynamic behavior of
the slip length with increasing shear rate. MD simulations
[4] of simple liquids in Couette flow have shown that the
slip length for Newtonian fluids interacting through
a Lennard-Jones (LJ) potential depends nonlinearly on
the shear rate according to L
s
L
s
1
_
=
_
c
1=2
. Here
_
c
represents the maximum shear rate the fluid can sus-
tain beyond which there is no additional momentum
transfer between the wall and fluid molecules.
How generic this behavior is and whether there exists a
comparable scaling for polymeric fluids remain open
questions.
In this Letter we use MD simulations to examine the
dependence of the slip length on molecular parameters
and shear rate in thin polymer films subject to planar
shear. The emphasis on polymeric fluids is timely since
most experiments devoted to slip phenomena rely on
polymers in order to diminish evaporative effects and to
enhance the degree of slip for measurement purposes
[19–22]. Results of our simulations support the view
that the low [17] and high shear [4] behavior of the slip
length reported earlier for simple fluid systems is more
generally applicable to polymeric systems. It is also
shown that beyond chain lengths of about N 10,the
net molecular weight dependence of the slip length at low
shear rates is dominated by the bulk fluid viscosity.
The simulation cell consisted of a simple or polymeric
fluid (3456 monomers) subject to planar shear in the
^
xx
direction. The fluid monomers interacted through the LJ
potential V
LJ
r4
r
12
r
6
with a cutoff distance
r
c
2:5, where and represent the energy and length
scales of the fluid phase. The wall-fluid parameters were
chosen to be
wf
0:6,
wf
0:75,andr
c
2:5;
the fluid phase density and temperature T were held
fixed at 0:81
3
and 1:1=k
B
, respectively. The polymer
fluid was modeled as a collection of bead-spring units
(N 2–16 beads) interacting through a finitely extensi-
ble nonlinear elastic (FENE) potential [23] V
FENE
r
k
2
r
2
0
ln1 r
2
=r
2
0
with k 30
2
and r
0
1:5 [24].
This fluid represents a semidilute polymer melt far below
the entanglement regime.
The upper and lower walls of the shear cell each con-
sisted of 1152 atoms distributed between two (111) planes
of an fcc lattice of density
w
4. The direction of
shear was oriented along 11
22. The fluid was confined
to a gap width h 24:57 and the entire cell (xyz)
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measured 25:03 7:22 h. Periodic boundary con-
ditions were enforced in the
^
xx and
^
yy directions. Constant
temperature of the fluid phase was maintained by a
Langevin thermostat with friction coefficient
1
at-
tached to the
^
yy degree of freedom for each monomer
[10,25]. The equations of motion were integrated using a
fifth-order Gear predictor-corrector scheme [26] with a
time step t 0:002, where
m
2
=
p
.
The fluid was sheared by translating the upper wall at a
constant speed U while the lower wall remained sta-
tionary. After an equilibration period exceeding 10
5
,
the velocity profile within the fluid, V
x
z==U,was
obtained by averaging the instantaneous monomer speeds
in bin widths of 0:5 (^zz direction) for a period of about
4 10
5
. The sample velocity profiles shown in Fig. 1
correspond to the maximum wall velocity imposed for
various chain lengths N. The velocity fields for N>1
remained linear at all shear rates examined. A slight
curvature in the velocity profile near the walls was noted
for simple fluids (N 1) at wall speeds exceeding U
4:5= (or equivalently for shear rates exceeding
_
0:1
1
). This curvature introduced a maximum error of
0:5 in estimating the slip length. An example of the
linear velocity profile for simple fluids (N 1) is also
shown in Fig. 1 for U 1:0=. In this study, the maxi-
mum slip velocity at the interface was about 2=.The
Reynolds number, based on the maximum fluid velocity
and gap height, was of order 10 indicating laminar flow
conditions even at the highest shear rates.
Figure 2 shows the viscous response of sheared films
with increasing shear rate and chain length (inset). The
normalized shear viscosity, =
3
, was computed
from the Kirkwood relation [23]. The maximum shear
rate the fluid can sustain before completely slipping de-
creases with increasing N since longer chain lengths
undergo more slippage. Chain lengths below N 6 be-
have as Newtonian liquids throughout; chain lengths
N 8 undergo shear thinning. The inset in Fig. 2
shows a monotonic increase in the zero shear viscosity,
_
! 0, with increasing chain length [23].
The dynamic response of the slip length with increas-
ing shear rate for N-mers ranging from N 1–16 is
shown in Fig. 3. The slip length, which is nearly constant
at low shear rates, exhibits a strong, nonlinear increase
with
_
for both Newtonian and shear thinning fluids.
Beyond a critical shear rate the wall can no longer impart
additional momentum to the fluid layer. The inset dem-
onstrates that the asymptotic value of the slip length, L
o
s
(obtained by extrapolating the leftmost three points to
zero shear rate), increases monotonically with increasing
chain length.
-10
-5
0
5
10
z
/
σ
0
0.2
0.4
0.6
0.8
1
V
x
/ U
N=1 U=6.5 σ/τ
N=4 U=5.0 σ/τ
N=12 U=4.5 σ/τ
N=1 U=1.0 σ/τ
FIG. 1. Sample velocity profiles, V
x
=U, within the shear
cell. Slip lengths were estimated from the relation L
s
U=
_
h=2 where the shear rate
_
dV
x
=dz.
0
0.05
0.1
0.15
γ
.
τ
1
2
3
4
5
6
7
8
9
η / ετσ
−3
04812
16
N
2
4
6
8
10
η(γ
.
=0) / ετσ
−3
N=1
2
3
4
6
12
16
8
FIG. 2. Behavior of the shear viscosity, =
3
, as a func-
tion of increasing shear rate for chain lengths ranging from
N 1–16. Inset: dependence of the bulk viscosity in the limit
of low shear rates as a function of N.
0.001 0.01 0.1
γ
.
τ
10
20
30
40
50
60
70
80
90
L
s
/ σ
0481216
N
10
20
30
40
L
s
o
/ σ
N=1
2
3
4
6
12
16
8
FIG. 3. Behavior of the slip length with increasing shear rate
for chain lengths ranging from N 1–16. As shown for
N 6, the errors bars are larger at the lower shear rates
because of thermal fluctuations. Inset: variation of the slip
length, L
o
s
, as a function of chain length N in the limit of
low shear rates.
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The data in Fig. 3 can be made to collapse onto a
common curve L
s
=L
s
1
_
=
_
c
1=2
, as shown in
Fig. 4. The values of L
s
N and
_
c
N, which are tabulated
in the inset legend, were obtained by fitting a straight line
to the data L
2
s
versus
_
. The vertical and horizontal
intercepts of the straight line identify the values L
s
and
_
c
[27]. The auxiliary slip lengths, L
s
, were slightly
smaller than the extrapolated values L
o
s
in Fig. 3, defined
by the low shear regime. This discrepancy indicates that
the dynamic collapse holds at the shear rates exceeding
_
* 5 10
3
1
. All cases examined demonstrated a
sharp increase in the slip length as
_
=
_
c
! 1.Thiscriti-
cal behavior suggests that the slip condition at the
wall-fluid interface can affect flow at large distances
from the wall.
In addition to the dynamic response, it is interesting to
probe exactly what molecular parameters control the
degree of slip as
_
! 0. Barrat and Bocquet investigated
this issue [17] for simple fluids (N 1) by appealing to
the fluctuation-dissipation theorem. They developed a
Green-Kubo analysis for computing the friction coeffi-
cient of the first fluid layer near the wall and showed that
the slip length at low shear rates, , scales according to
D
q
jj
=Sq
jj
c
for constant temperature and attrac-
tive wall-fluid interactions. The wave number, q
jj
, repre-
sents the first reciprocal lattice vector of the wall in the
direction of shear. D
q
jj
, the collective diffusion coeffi-
cient, Sq
jj
, the structure factor, and,
c
, the contact
density, are all quantities computed within the first fluid
layer.
We computed by MD simulations the analogous pa-
rameters for polymeric fluids. The location of the first
fluid layer was estimated from the first minimum in the
monomer density profile in the direction perpendicular to
the wall. The value of the first peak in the profile
defined the contact density,
c
. The results show that
c
decreases with increasing N by about 10%. In our
simulations, the reciprocal lattice vector was chosen to
be q
jj
6:024
1
, which probes distances of about a
molecular size. The in-plane structure factor, Sq
jj
j
P
N
‘
1
e
iq
jj
x
j
2
=N
‘
, where N
‘
is the total number of mono-
mers in the first fluid layer, decreases by about 30% from
N 1 to 16. This behavior is expected since polymer
chains cannot pack as densely or as orderly as simple
fluids near a wall. Computation of the collective diffusion
coefficient, D
q
jj
, which for polymers reflects internal re-
arrangements of the chains (and not the N-mer center of
mass diffusion coefficient), requires an estimate of the
decay rate of the density-density autocorrelation func-
tion. The data were fit by a Kohlrausch-Williams-Watts
stretched exponential according to h
q
jj
t
q
jj
0i
expAq
2
k
t
&
h
q
k
0
q
k
0i, where A and & are fitting
parameters and
q
jj
P
N
‘
1
e
iq
jj
x
. The parameter &N
varied from 0:89 0:03 for N 1 to 0:81 0:03 for
N 16. The slip length, , as computed from the ex-
pression for the friction coefficient (see Ref. [17]), is
proportional to
R
1
0
dth
q
jj
t
q
jj
0i, from which the col-
lective diffusion coefficient is calculated to be D
q
jj
A&=&
1
, where is the gamma function.
Each of the separate variables D
q
jj
, 1=Sq
jj
,and1=
c
exhibit a similar dependence on N characterized by a
sharp increase at small N that asymptotes to a constant
value at large N (not shown). Figure 5 shows the behavior
of the combined ratio, D
q
jj
=Sq
jj
c
, normalized by its
value for N 1. This result indicates that the slip length
N is dominated by the bulk viscosity above chain
lengths of about N 10. Tolstoi’s model for simple fluids
(N 1) also predicts that the slip length at low shear
rates is linearly proportional to the bulk viscosity. The
0.01 0.1 1
γ
.
/γ
.
c
1
2
3
4
5
L
s
/ L
s
*
γ
.
c
(1) = 0.145 τ
−1
L
s
*
= 3.9 σ
γ
.
c
(2) = 0.091 τ
−1
L
s
*
= 6.5 σ
γ
.
c
(3) = 0.072 τ
−1
L
s
*
= 8.2 σ
γ
.
c
(4) = 0.061 τ
−1
L
s
*
= 10.1 σ
γ
.
c
(6) = 0.048 τ
−1
L
s
*
= 13.2 σ
γ
.
c
(12) = 0.035 τ
−1
L
s
*
= 24.5 σ
γ
.
c
(16) = 0.032 τ
−1
L
s
*
= 31.2 σ
γ
.
c
(8) = 0.041 τ
−1
L
s
*
= 16.7 σ
FIG. 4. Master curve describing the dynamic behavior of the
slip length for fluid chain lengths ranging from N 1–16.The
dashed line represents the function L
s
=L
s
1
_
=
_
c
1=2
.
The legend lists the fitted values of L
s
N and
_
c
N used to
normalize the data.
FIG. 5. The dependence of the ratio, D
q
jj
=Sq
jj
c
(normal-
ized by its value for N 1), on chain length N. Inset: The
correlation between the slip lengths obtained from the shear
flow profiles, L
o
s
N, versus the equilibrium measurements of
N. The solid line y 223:5x 0:24 is plotted for reference.
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inset in Fig. 5 shows a strong correlation between L
o
s
(inset of Fig. 3) and predictions of the slip length eval-
uated from D
q
jj
=Sq
jj
c
. This indicates that the
Barrat-Bocquet analysis accurately predicts the depen-
dence of the slip length on the parameters described even
for polymeric chains that behave as shear thinning fluids.
In summary, we have conducted an MD study of the
slip response of N-mer polymer chains (1 N 16
subject to planar shear. The first part of the study estab-
lishes that the dynamic behavior for the slip length versus
shear rate is well described by the function L
s
=L
s
1
_
=
_
c
1=2
. The dynamic collapse holds at shear rates
exceeding
_
* 5 10
3
1
. The second part of the
study demonstrates that there is a strong correlation be-
tween the slip length at low shear rates, obtained from the
actual velocity profiles and that predicted from a Green-
Kubo analysis of the friction coefficient between the wall
and the first fluid layer. The simulations also indicate that
beyond chain lengths of about N 10, the molecular
weight dependence of the slip length is mostly dominated
by the bulk fluid viscosity. This linear proportionality
lends support to Tolstoi’s early slip model based on
Eyring dynamics.
Further studies will determine how general are these
findings. In particular, how sensitive is the slip exponent,
1=2, to the chain length and MD parameters used in this
study and how robust are these results to higher molecular
weight films subject to shear? The fact that the dynamic
behavior of the slip length is captured by a simple func-
tional form for both simple and polymeric fluids suggests
the possibility of universal scaling for the slip velocity
V
s
L
s
_
_
.
The authors thank Dr. P. A. Thompson of PeerMedia,
Inc. for kindly sharing his source code and for helpful
discussions. The authors also thank Professor Roberto Car
for computational support and useful conversations.
Financial support by the National Science Foundation
(CTS and DMR divisions), by Unilever Research U.S.,
by the U.S. Army (TACOM, ARDEC), and NASA is
gratefully acknowledged.
[1] D. M. Tolstoi, Dokl. Akad. Nauk SSSR 85, 1089 (1952).
[2] T. D. Blake, Colloids Surf. 47, 135 (1990).
[3] S. Richardson, J. Fluid Mech. 59, 707 (1973).
[4] P. A. Thompson and S. M. Troian, Nature (London) 389,
360 (1997).
[5] F. Brochard, P. G. DeGennes, and S. M. Troian, C. R.
Acad. Sci., Ser. II 310, 1169 (1990).
[6] O. I. Vinogradova, N. F. Bunkin, N.V. Churaev, O. A.
Kiseleva, A.V. Lobeyev, and B.W. Ninham, J. Colloid
Interface Sci. 173, 443 (1995).
[7] C. L. M. H. Navier, Mem. Acad. Sci. Inst. Fr. 6, 839
(1827).
[8] J. I. Frenkel, Theory of Liquids (Oxford University Press,
Oxford, 1946).
[9] I. Bitsanis, J. J. Magda, M. Tirrell, and H. T. Davis,
J. Chem. Phys. 87, 1733 (1987).
[10] P. A. Thompson and M. O. Robbins, Phys. Rev. A 41, 6830
(1990).
[11] P. A. Thompson, G. S. Grest, and M. O. Robbins, Phys.
Rev. Lett. 68, 3448 (1992).
[12] R. Khare, J. J. de Pablo and A. Yethiraj, Macromolecules
29, 7910 (1996).
[13] P. A. Thompson, M. O. Robbins, and G. S. Grest, Isr.
J. Chem. 35, 93 (1995).
[14] M. J. Stevens, M. Mondello, G. Crest, S. T. Cui, H. D.
Cochran, and P. T. Cummings, J. Chem. Phys. 106,
7303 (1997).
[15] E. Manias, G. Hadziioannou, I. Bitsanis, and
G. ten Brinke, Europhys. Lett. 24, 99 (1993).
[16] A. Jabbarzadeh, J. D. Atkinson, and R. I. Tanner, J. Chem.
Phys. 110, 2612 (1999).
[17] J.-L. Barrat and L. Bocquet, Phys. Rev. Lett. 82,4671
(1999); Faraday Discuss. 112, 119 (1999).
[18] L. Bocquet and J.-L. Barrat, Phys. Rev. E 49,3079
(1994).
[19] R. G. Larsen, The Structure and Rheology of Complex
Fluids (Oxford University Press, New York, 1999),
Chap. 1 and references therein.
[20] Y. Zhu and S. Granick, Phys. Rev. Lett. 87, 096105 (2001).
[21] K. B. Migler, H. Hervet, and L. Leger, Phys. Rev. Lett. 70,
287 (1993).
[22] R. G. Horn, O. I. Vinogradova, M. E. Mackay, and
N. Phan-Thien, J. Chem. Phys. 112, 6424 (2000).
[23] R. B. Bird, C. F. Curtiss, R. C. Armstrong, and
O. Hassager Dynamics of Polymeric Liquids (Wiley,
New York, 1987), 2nd ed.
[24] K. Kremer and G. S. Grest, J. Chem. Phys. 92, 5057
(1990).
[25] G. S. Grest and K. Kremer, Phys. Rev. A 33, 3628 (1986).
[26] M. P. Allen and D. J. Tildesley, Computer Simulation of
Liquids (Clarendon, Oxford, 1987).
[27] Fluids consisting of longer chains showed a slight down-
ward curvature at the low shear rates when plotted as
L
2
s
versus
_
. The deviation from linearity may be due to
finite size effects since fluids consisting of longer chains
produced slip lengths exceeding the cell gap width h.
Adjustment of the inverse exponent of L
s
(the ordinate
axis) to produce the best line fit yielded a slip exponent
variation of 0:5 0:05.
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