Abstract. -The flow behaviour of melts of short chains, confined in molecularly thin Couette flow geometries, is studied with molecular-dynamics simulations. The effect of wall attraction and confinement on the density and velocity profiles is analysed. In these highly inhomogeneous films, a strong correlation between the density and velocity profile is found. Sticking of the interfacial layer on the wall and slip on the wall and inside the film is manifested by changes in the velocity profile. The location of the slip is determined by the strength of the wall attraction. Even though the macroscopic phenomena of friction, lubrication and adhesion have been studied for a long time now, their molecular mechanisms have yet to be unveiled. Recent novel experimental techniques such as the surface forces apparatus together with the scanning probe microscopies are capable to give a vast amount of information on the nanometer level; these combined with computer simulations will provide further insight into the nanoscopic dynamics of friction, lubrication and adhesion. The structural and dynamical properties of ultrathin confined films between atomically flat surfaces undergoing shear are studied here with molecular-dynamics simulations. Simulations are performed on a fluid of short chains in a planar Couette flow geometry realized by confining the polymer system in the x-direction between two parallel planar f.c.c. (111) planes. Periodic boundary conditions are imposed in the other two directions. The Couette flow is introduced by moving the two walls with equal and constant velocities towards opposite directions keeping the wall-to-wall distance constant. In this way, a steady shear rate is introduced with the direction of flow parallel to the x-axis and the velocity gradient parallel to the x-axis, i.e. normal to the walls El]. The chains consist of six segments which are connected in a linear freely joined topology. Of course, the model is not expected to capture the real monomer response and segments correspond to several chemical monomers [2]. The segments of the same chain as well as segments belonging to different chains interact via a pairwise purely repulsive, shifted an

E Manias

G Hadziioannou

G Ten Brinke

I Bitsanis

CiteSeerX

EUROPHYSICS LETTERS 
Europhys. Lett., 24 (2), pp. 99-104 (1993) 
10 October 1993 
Stick and Slip Behaviour of Confined Oligomer Me 
under Shear. A Molecular-Dynamics Study. 
E. MANIAS(*), G. HADZIIOANNOU(*), I. BITSANIS(**) and G. TEN BRINKE(*) 
(*) Department of Polymer Chemistry, University of Groningen 
Nijenborgh 4, 9747 AG Groningen, The Netherlands 
(**) Department of Chemical Engineering, University of Florida 
Gainesville, F L  32611, USA 
(received 26 April 1993; accepted 6 September 1993) 
PACS. 47.55 - Nonhomogeneous flows. 
PACS. 36.20 - Macromolecules and polymer molecules. 
PACS. 68.45 - Solid-fluid interface processes. 
Abstract. - The flow behaviour of melts of short chains, confined in molecularly thin Couette flow 
geometries, is studied with molecular-dynamics simulations. The effect of wall attraction and 
confinement on the density and velocity profiles is analysed. In these highly inhomogeneous films, a 
strong correlation between the density and velocity profile is found. Sticking of the interfacial layer 
on the wall and slip on the wall and inside the film is manifested by changes in the velocity profile. The 
location of the slip is determined by the strength of the wall attraction. 
Even though the macroscopic phenomena of friction, lubrication and adhesion have been 
studied for a long time now, their molecular mechanisms have yet to be unveiled. Recent 
novel experimental techniques such as the surface forces apparatus together with the 
scanning probe microscopies are capable to give a vast amount of information on the 
nanometer level; these combined with computer simulations will provide further insight into 
the nanoscopic dynamics of friction, lubrication and adhesion. The structural and dynamical 
properties of ultrathin confined films between atomically flat surfaces undergoing shear are 
studied here with molecular-dynamics simulations. 
Simulations are performed on a fluid of short chains in a planar Couette flow geometry 
realized by confining the polymer system in the x-direction between two parallel planar f.c.c. 
(111) planes. Periodic boundary conditions are imposed in the other two directions. The 
Couette flow is introduced by moving the two walls with equal and constant velocities 
towards opposite directions keeping the wall-to-wall distance constant. In this way, a steady 
shear rate is introduced with the direction of flow parallel to the x-axis and the velocity 
gradient parallel to the x-axis, i.e. normal to the walls El]. 
The chains consist of six segments which are connected in a linear freely joined topology. 
Of course, the model is not expected to capture the real monomer response and segments 
correspond to several chemical monomers [2]. The segments of the same chain as well as 
segments belonging to different chains interact via a pairwise purely repulsive, shifted and 
EUROPHYSICS LETTERS 
truncated 12-6 Lennard-Jones (LJ) potential: 
where r is the distance between the centres of the two segments, E is the L J  energy 
parameter and o the L J  length parameter. For successive monomers of the chain a strongly 
attractive FENE (finite extensibility non elastic) spring potential is added [1-41: 
where Ro = 1.5o and k = 30&/02. This choice of parameters has been proven to prevent bond 
crossing at  the temperature used in our simulation [2]. This model has been studied 
extensively for linear chains both in the bulk [2], confined between walls [l], and under 
shear [4]. The interaction between walls and segments is modelled by a pairwise 
Lennard-Jones potential which includes the attractive tail of the potential: 
This potential is truncated at  rwc = 2.5 to reduce computational effort. In all simulations 
ow = 1 .0~ .  E,, which determines the attraction between the wall and segments, has been 
varied in order to study the influence of the strength of this interaction. 
The temperature was fmed at  kB T = 1 . 0 ~  in all simulations. The kinetic energy imparted 
from the wall to the chains is dissipated inside the film. The heat generated is removed by a 
first-order coupling of the temperature to a heat bath. This is accomplished by rescaling the 
segment velocities at  every time step [5]. For systems at  equilibrium the scaling of velocities 
is done in such a way that the centre of mass of the confined film has zero momentum. For 
systems under shear only the thermal part of the velocities parallel to the direction of the 
flow velocity are scaled. For the other two directions a first-order coupling to temperature is 
used. 
Newton's equations of motion are integrated by a variant of Verlet's algorithm with a time 
step of At = 0.00462(ma~/~)'/~. Computational needs are further reduced by the use of lists of 
nearest neighbours updated every 20 time steps [6]. The results will be presented in reduced 
TABLE I. - Systems simulated. 
Wall 
attraction 
EW(E)  
Wall-to-wall 
distance 
Mu) 
Average (a )  Wall 
segment density velocities 
v, (~/rn)~/ '  
(a) The average segment density is defined as the number of monomers per unit volume excluding the depletion zones near the 
walls. 
E. MANIAS f?t UI?.: STICK AND SLIP BEHAVIOUR OF CONFINED OLIGOMER ETC. 
distance normal to the walls 
Fig. 1. - Velocity (0) and density (---) profiles for a wall interaction (E, = 1.0~) almost identical 
imposed shear rates vw = 0.7(~/m)'/~ (a)), vw = 0.5(~/m)'/~ (b)), and different confinements a) h = 6.00 
and b) h = 4.0~. The slip between wall and system increases and the linear velocity profile in the middle 
of the film decreases with decreasing h. The imhomogeneity of the confined polymer systems is 
stronger for smaller wall-to-wall distances (h). 
units in which the energy is measured in E,  the length in o (LJ parameters) and the mass in m 
(monomer mass). The simulation parameters of the systems studied can be found in 
table I. 
Several steps and methods are employed to create initial positions and velocities. Well- 
equilibrated initial configurations of systems with linear molecules were taken from a lattice 
Monte Carlo algorithm [7]. Initial velocities from a Maxwell-Boltzmann distribution, with 
variance equal to the temperature, are subsequently randomly assigned and a constant 
temperature MD run is made. For systems under Couette flow, a non-equilibrium MD run of 
1 to 5 .  lo5 time steps is used until steady-state flow is reached. 
For relatively small wall velocities flow velocities are masked by thermal noise and 
considerably longer production runs than for the higher shear rates are required. Typically, 
for wall velocities below 0.2 (table I) runs of 8 lo6 to 1 lo6 time steps are used and for higher 
shear rates the runs are of 4 .  lo6 to 6 lo5 time steps. The dimensions of the simulation box in 
the directions parallel to the surfaces are 8.00. This size is large enough to avoid any 
interaction of a chain with its periodic image, even for chains that are close to the wall. So, for 
the simulations the volume, the temperature and the number of particles remain fixed 
(canonical ensemble, NVT MD). 
Novel experimental techniques [8,9] and computer simulations for monomeric [lo-151 as well 
as for polymeric fluids [l], indicate the existence of strong layering in the vicinity of solid 
surfaces, in the form of oscillations in density. These density variations are caused by the 
presence of a wall in combination with strongly repulsive excluded-volume interactions between 
the segments. This is similar to the layering of the pair correlation function of dense 
fluids. 
The normal layering depends on the interaction of the polymer with the wall. For a more 
attractive wall the density inhomogeneity is enhanced. The monomer density profile depends 
also on the distance between the two walls. For systems with walls relatively far apart 
(h 2 7.0~), a region with constant density is present in the middle of the film and the film 
exhibits bulklike behaviour. As soon as the walls are brought close enough together, the 
density inhomogeneity extends over the whole film, which then exhibits a strongly layered 
structure (cf. fig. 1). 
102 EUROPHYSICS LETTERS 
As theories that can predict the velocity profile for strongly inhomogeneous media are still 
developing [16-181, the first question to address concerns the shape of the velocity profile in 
confined systems under steady shear. Some characteristic examples of velocity profiles as a 
function of wall-to-wall distance are shown in fig. 1 for systems with the same shear rate and , 
wall-segment interaction. Rather than being linear throughout the slit, as predicted from 
hydrodynamics of a bulk system under steady shear, the velocity profile normal to the walls 
exhibits a strong correlation with the density profile. For a higher shear rate and a stronger 
wall-segment interaction a more detailed picture of the relation between velocity and density 
profile can be given. In all cases, the velocity profile is linear for the middle part of the 
system where the density is almost constant. Inside the first layer in contact with the wall 
the velocity profile is, for the less attractive wall, also almost linear but with a smaller slope. 
This implies that here the effective viscosity is higher than in the middle of the film. For the 
more attractive walls, however, this first layer locks on the wall (fig. 2a)) and much higher 
shear rates have to be applied to induce flow inside this interfacial area (fig. 2b)). At these 
high shear rates the low-density region between the first and second layer is characterized by 
an abrupt change in the velocity profile (interlayer slip) (fig. 3). 
In the simulations carried out for all the systems with the less attractive walls ( E ,  = 1.0~) 
some slip is present between the wall and the first layer. This slip increases with increasing shear 
rate as expected [l9,20]. The slip also increases in narrower confinements for the same 
wall-segment interactions and shear rates (fig. 1). For the weak attraction, the density profile 
does not depend markedly on shear rate, not even for the higher shear rates and the smaller 
wall-to-wall distances, in agreement with earlier simulations for monomer films [l2]. Due to 
slippage between wall and film, there is a limit on the shear rates than can be applied. 
For systems with more attractive walls ( E ~  = 2.04 much higher shear rates can be applied 
since the slip between wall and confined system is negligible even for much higher shear 
rates. In this more attractive case, for the smaller wall velocities (up to 0.9) almost the 
complete first layer sticks on the wall and moves with the imposed wall velocity and slip 
occurs between the first layer and the rest of the system (fig. 2a)). This slip is revealed by a 
very sharp curve connecting the constant velocity of the first layer with the linear velocity 
profile in the middle part of the film. The slip increasestwith shear rate. When higher wall 
distance normal to the walls 
Fig. 2. - Velocity (a) and density (---) profiles for cw = 2 . 0 ~ .  For wall velocities smaller than 0.9 the 
first layer locks on the wall and moves with the imposed v, (a)). When the shear rate is increased, a 
linear velocity profile develops inside the first layer and for even higher shear rates a slip starts to 
appear between the wall and the system (b), vw = 3.0). In all cases there is an interlayer slip between 
the adsorbed layer and the rest of the system, which increases with shear rate. 
E. MANIAS et al.: STICK A N D  SLIP BEHAVIOUR O F  CONFINED OLIGOMER ETC. 103 
distance normal to the walls distance normal to the walls 
Fig. 3. Fig. 4. 
Fig. 3. - There is a definite correlation between the density and the velocity profile for highly 
inhomogeneous confined films. The figure corresponds to h = 6.0% E ,  = 2.06 and vw = 4 . 0 ( ~ / m ) ~ / ~ .  The 
different velocity profile regions correspond to different parts of the density profile. 
Fig. 4. - For the less attractive walls ( E ,  = 1 .0~)  the density does not change much with shear rate. But 
for the most attractive walls the density profile changes systematically with shear rate. In this figure 
h = 6.0q cW = 2.08 and vw = (0.0 + 4 .0) (~ /m) ' /~ :  o 0.9, 2.0, * 3.0, + 4.0; - equilibrium. 
velocities are applied the interfacial layer is characterized by an almost linear velocity profile 
and for even higher velocities some slip appears between the wall and this first layer 
(fig. 2b)). This has been observed before for confined monomeric fluids 1151. The density 
profiles for these systems ( E ,  = 2.04 change systematically with the imposed shear rate. An 
increase in density of the first and in particular the second layer is observed, whereas the 
low-density area between these two layers decreases in density and increases in width 
(fig. 4). These density changes are due to a tendency for the chains, in contact with the wall, 
to adopt flat configurations. This is manifested by a significant increase of the fraction of 
chains with 5 and 6 contacts with the wall when shear is imposed, especially for the more 
attractive walls. This trend gets stronger with increasing shear rates and greater wall 
attractions (table 11). The adoption of flat configurations enhances the slip in the region 
between the two layers. 
For systems with very attractive walls (cw = 3.0~)  and high shear rates, as high as those 
introducing slip between wall and polymers for E ,  = 2.0$ the first layer practically locks on the 
TABLE 11. - Fractions of chains with n-contacts with the walls (normalized by the total number of 
chains douching), the walls). Quantities are in reduced units: E ~ ( E )  and v , ( ~ / m ) ~ / ~ .  
1 cont. 
2 cont. 
3 cont. 
4 cont. 
5 cont. 
6 cont. 
104 EUROPHYSICS LETTERS 
wall. The effect of increasing wall velocity is merely to introduce slip between the first layer and 
the rest of the polymer system. The behaviour of the density profile is similar to that of the 
system with E~ = 2.0~; an increase in density is observed for both the first and second layers and 
the adsorbed chains tend to adopt flat configurations on the wall for higher shear rates. ' 
In summary, non-equilibrium molecular-dynamics simulations studies of confined oligomer 
melts in a Couette flow geometry were used to investigate the response of nanoscopically 
confined short-polymer-chain films to shear. Stick conditions of a layer of the confined system on . 
the wall are observed in agreement with experiments [21]. Simultaneously, a definite correlation 
between the density profile of a highly inhomogeneous confined system and the velocity profile is 
found. This has been observed before for moderately attractive walls and monomeric systems. 
Simulation data[12,15] and theoretical analysis [l6-181 have shown that the detailed flow 
behaviour of a fluid near a solid interface is strongly affected, if not deterministically defined, by 
the equilibrium structure of the fluid in the vicinity of the wall. When more attractive walls are 
used a correlation between structural changes caused by shear and the change in flow behaviour 
of the confined system is observed. The slip, both between the wall and the fluid, and the 
interfacial layer and the middle part within the system, occurs in the low-density areas where 
different parts of the system meet (ie. wall and polymers, interfacial and middle part). 
* * *  
This research was supported by the Netherlands Foundation of Technology (SON STW) 
and the Netherlands Organization for Scientific Research (NWO). IB acknowledges support 
provided by NSF (Grant No. CTS-9015882). 
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