Friction between Ring Polymer Brushes

4.5 MB·English

by  A. Erbas

Friction between Ring Polymer Brushes
Aykut Erbas
Northwestern University,
Department of Materials Science and Engineering
Evanston, IL 60208, USA
Jarosl(cid:32)aw Paturej
Leibniz-Institut of Polymer Research, 01069 Dresden, Germany and
Institute of Physics, University of Szczecin, Wielkopolska 15, 70451 Szczecin, Poland
(Dated: January 8, 2015)
Frictionbetweenring-polymerbrushesatmeltdensitiesslidingpasteachotherarestudiedusing
extensive course-grained molecular dynamics simulations and scaling arguments, and the results
5
are compared to the friction between linear-polymer brushes. We show that for a velocity range
1
spanning over three decades, the frictional forces measured for ring-polymer brushes are half the
0
2 correspondingfrictionincaseoflinearbrushes. Inthelinear-forceregime,theweakinter-digitation
of two ring brushes compared to linear brushes also leads to a lower number of binary collisions
n
between the monomers of opposing brushes. At high velocities, where the thickness of the inter-
a
digitation layer between two opposing brushes is on the order monomer size regardless of brush
J
topology, stretched segments of ring polymers take a double-stranded conformation. As a result,
7 monomers of the double-stranded segments collide less with the monomers of the opposing ring

brush even though a similar number of monomers occupies the inter-digitation layer for ring and
]
t linear-brush bilayers. The numerical data obtained from our simulations is consistent with the
f proposed scaling analysis. Conformation-dependent frictional reduction observed in ring brushes
o
can have important consequences in non-equilibrium bulk systems.
s
.
t
a
m INTRODUCTION taining the lubrication in tissues [11–14]. For instance,
in mammalian joints, where very low lubrication should
-
d be retained under pressures as high as 5 MPa, brush-like
If polymer chains are grafted by one of their ends to
n
structuresincombinationwiththesynovialfluidprovide
o a planar or curved surface, above a certain critical graft-
lubrication in between the articular joints [14–16]. The
c ing density ρ∗ ≈ 1/R2 [1], where R is the characteris-
[ tic equilibriumg chain s0ize, the chains0are stretched away surfaces separating the articular cartilage and the syn-

1 from the surface due to steric repulsion from surround- ovial fluid are thought to be covered by high molecular
weight molecules such as lubricin [14, 17]. In turn, these
v ing chains and form polymer brushes. A polymer brush
8 is a soft polymeric material that can deform under var- long and charged glycoproteins may function as water-
8 basedbio-lubricants. Therelationshipbetweenmorphol-
ious external forces. The external force can be due to a
4 ogy of these long and charged macromolecules on the
fluid flowing over the brush, a flow due to the relative
1
cartilagesurfacesandtheirfunctioninlubricationisstill
0 motion of a second brush, or alternatively, an external
under investigation [11, 14, 16–20]. Amphiphilic lubricin
. electrical field (if the polymers are charged). Once the
1 chains can adsorb on both hydrophilic and hydrophobic
external stimulus is completely removed, as the chains
0
surfaces[16,21],buttheirconformationsintheadsorbed
5 constituting the brush relax, the deformation of brush
state strongly depend on the type of surface. These
1 is reversed similar to the elastic deformation observed
: for solids. However, the tribological behaviour of poly- molecules can bind onto hydrophilic surfaces via their
v
chargeddomainslocatedinthecentralpartofamolecule
i meric systems more resembles that of a fluid rather than
X and form structures resembling a brush composed of lin-
a solid [2]. For instance, if two inter-digitated polymer
r brushes are slid past each other, frictional forces due to ear chains. Alternatively, they can also bind onto hy-
a
drophobic surfaces with their terminal groups to form
therelativemotionofbrushesvanishlinearlyastherela-
loop brush-like structures [16, 21, 22]. Hence, based on
tive velocity of the brushes is decreased towards zero. In
experimental observations and variations in the molec-
other words, the friction is viscous, for which no static
ular conformations in vitro, one may conclude that in
frictionoccurs,unlikethesolid-statefriction,wherefinite
reality the whole cartilage surface resembles a polymer
forces are needed to initiate motion [3].
brush composed of linear and loop-like chains. Although
The polymer brushes has attracted increasing atten-
moleculesmentionedsofararehighlycharged, andlong-
tion due to their applications in nanotechnology and
range interactions might be a dominant factor in the re-
material sciences as bio-sensors, bio-fueling, stimuli-
ducedfrictionalforces, conformationofchains–whether
responsive surfaces [4–8], or for the stabilisation of col-
they are linear or looped – can influence inter-digitation
loidal solutions [9, 10]. The most fascinating application
of molecules as well as their relaxation times, and hence,
of brush-like structures is facilitated by nature in main-
2
friction and lubrication in tissues. static interactions are known to be computationally ex-
Inter-digitation effects due to ring topology have also pensive, particularly for dense systems. Through a de-
been reported in the context of genetic material: in a tailed analysis of simulation trajectories and by employ-
computationalstudy,unconcatenatedchainsofringpoly- ingscalingarguments,wedemonstratethatthetopology
mersshowedaweaktrendtowardsinter-mixingwitheach of chain in a melt brush is an important factor in the
other compared to mixing behaviour of linear chains in reduction of frictional forces.
confined environments [23]. We should also underline In our simulations, we found that for untangled
that ring polymers in bulk and at melt densities exhibit brushes friction forces between two brushes made of lin-
different behaviour with respect to their linear counter- earchainsarealwaysroughlyafactoroftwohigherthan
parts [24–26]. The behaviour of ring polymers cannot be those for ring chains. Although this difference is small,
described by one single fractal dimension unlike linear it persists for various grafting densities and chain sizes.
chains in their melts: while size of a linear chain at melt This difference in the frictional forces of ring and linear
statescalesasR0L ∼N1/2,aringpolymerinthemeltof brushescanbeelucidatedbythesizeoftheoverlapzone,
rings is much more compact and the characteristic size whichdefinestheamountofinter-digitationbetweentwo
scales R0R ∼ N1/2 for short chains and R0R ∼ N1/3 for brushes. Itturnsoutthatthelowtendencyofringchains
sufficiently large polymerization degrees, N. In between to overlap with the opposing ring-brush leads to lower
these two limits, an intermediate regime R0R ∼ N2/5 friction forces whereas linear chains can diffuse through
arises [24, 25]. Moreover, even in the regimes where the the opposing brush more easily. Hence, brushes made of
size of ring chain is non-Gaussian, interestingly higher linearchainsexhibithigherfrictionalforces. Atveryhigh
moments of the end-to-end distance exhibit a Gaussian- velocities, on the other hand, segments of a ring brush
like behaviour [24]. take a double-stranded conformation. In turn, double-
Existing computational [27–33] and theoretical stud- strands are less efficient in momentum transfer between
ies[29,34–37]onlinear-polymerbrusheshaveshownthat the opposing brushes. The difference in friction forces
frictionalforcesactingbrushesexhibitacross-overfroma between both systems is confirmed by scaling analysis.
lineartoanon-linearregimeuponincreasingshearveloc-
The paper is organized as follows: First we briefly de-
ity. The onset of non-linear friction typically appears at
scribe the simulation methodology. Next, the results ob-
aroundshearrateswheregraftedchainsbegintostretch.
tained from non-equilibrium coarse-grained brush simu-
Ideally, one would expect a similar behaviour in the case
lationswillbediscussed. Bycarefullyanalysingourdata,
ofring-polymerbrushes. However,howthetopologyand
we will relate the difference in friction forces to the in-
peculiarscalingofringchainsalterthefrictionisanopen
trinsicpropertiesofringandlinear(grafted)chains. The
question. To the best of our knowledge there are very
scalingargumentsforbrushsystemswillalsobediscussed
few computational investigations of ring or loop poly-
toinferthedifferenceinforcesalongwiththesimulations
mer brushes out of equilibrium [38, 39]. In a previous
data. Weconcludethepaperbythesummaryofourfind-
work [39], tribolgy of loop brushes near the overlap con-
ings and future prospects.
centrations were studied, but the observed difference in
frictional forces was negligible. However, at melt den-
sities where the correlation length – distance between
two chains – is on order of monomer size, the situation
SIMULATION DETAILS
canbedifferentasthenumberofcollisionbetweenbrush
monomers can significantly increase in a denser system.
Characterizing the systematic reduction in brush fric- Simulations of polymer brush bilayers were performed
tion due to the topology of the constituting chains can using coarse-grained Kremer-Grest (KG) bead-spring
improve our understanding of how nature handles fric- model[31,40]. Eachindividualchainofapolymerbrush
tion and help the design and improvement of advanced was composed of N (or 2N) monomers (beads) con-
biomimetic lubricants. Thus, spurred by the abundance nectedbybonds. Twoadjacenteffectivechainmonomers
of brush-like structures in nature and the interesting arebondedviaa“FinitelyExtensibleNonlinearElastic”
nature of ring polymers themselves in this paper we (FENE) potential
aim to study non-equilibrium behaviour of ring-polymer
brushesatmeltdensities. Inourextensivecoarse-grained VFENE(r)=−1kr2ln(cid:34)1−(cid:18) r (cid:19)2(cid:35), (1)
MD simulations, we used neutral (uncharged) polymer 2 0 r
0
brushes. Giventhefactthatthesystemwewouldliketo
mimic is under high pressure (e.g., in joints), and ionic where bond stiffness k = 30(cid:15)/σ2, distance between the
condensation (short Debye length in physiological con- beadsr =|r|,andmaximumbondlengthr =1.5σ [40].
0
ditions) can effectively neutralize the chains, we believe Theinteractionstrength(cid:15)wasmeasuredinunitsofther-
thatthisapproximationisreasonableforthesakeofmin- mal energy k T. The non-bonded interactions between
B
imizing computational cost [28] since long-range electro- allmonomersweremodeledbythetruncatedandshifted
3
Lennard-Jones (LJ) potential f v
(cid:26) 4(cid:15)(cid:2)(σ/r)12−(σ/r)6+1/4(cid:3) r ≤r
VLJ(r)= c
0 r >r ,
c δ
(2) D δ L,R
whereσ waschosenastheunitoflengthandr iscutoff. L,R
c
Forpolymer-polymerinteractionwehaveused(cid:15)=1and
r =21/6σ. This choice of LJ potential cutoff in combi-
c
nationwithamonomerdensityofρ ≈0.6σ−3 provides v f
m
correct melt statistics [28, 29, 41]. z
g momoners
Polymerbrushescomposedofchainswitheitherlinear x
L,R
orring-liketopologywerestudied. Weperformedsimula-
tions with linear-polymer brushes composed of N = 60, FIG. 1: Scheme of brush bilayer shear simulation. The
100 monomers, and ring-polymer brushes of N = 120, force f acts in the direction opposite to velocity v.
200 monomers per grafted chain. Chains were grafted Chains at the top and bottom brush are rendered in
on a square-lattice surface with dimensions 42σ×36σ. different colors. The overlap zone (OZ) with thickness
The ring chains were grafted by one of their monomers δ is indicated by a milky-white region. The dashed
L,R
onthesurface. Noequationofmotionwassolvedforsur- circle shows the segments of chains with g monomers
L,R
facemonomers(surfacemonomerswereimmobileduring inside the penetration zone. Here N =60,
simulations). The surface and the brush monomers in- ρL =0.25σ−2 and D =50σ.
g
teract via Eq. (1), and Eq. (2). The grafting densities of
linear chains are ρL = 0.11σ−2,0.25σ−2, which respec-
g
tively corresponds to inter-anchored monomer distances numericalintegrationofequationsofmotionEq.(3). All
of 3σ,2σ at the surface. In the case of ring brushes, to simulations were performed in the N VT-ensemble, i.e.,
m
obtain an equal monomeric density, ρ ≈ 0.6 σ−3, to constant volume V, total particle number N and tem-
m m
the linear-brushes, the grafting densities were taken as peratureT. Thesystemtemperaturewassettothevalue
ρR = 0.5ρL at the same inter-plate distance D. Note T =1.68 (cid:15)/k with k =1 [41]. Periodic boundary con-
g g B B
that D and ρL,R are connected via ρ ≈ NρL,R/D. In ditionswereintroducedinthelateraldirections, i.e. inxˆ
g m g
order to construct brush bilayer systems, first two non- and yˆ whereas in the zˆ-direction fixed boundary condi-
interacting single brushes were generated as mirror im- tions were imposed. Simulations were carried out using
ages of one another. While one of the brushes is fixed molecular package LAMMPS [42]. Simulation snapshots
at z = 0, the other brush is brought into contact slowly were rendered using Visual Molecular Dynamics (VMD)
at the desired inter-plate distance z = D to obtain the [43].
samemonomerdensityforeachρL,R andN. Finally,the Thenon-equilibriumshearsimulationswereperformed
g
systems are run at v = 0 for at least 107 MD steps to as shown in Fig. 1. Both plates grafted by polymers
allow chains to relax. were moved laterally in the opposite (±xˆ-directions) at
prescribed velocities in a range of v ≈ 10−3σ/τ and
The molecular dynamics simulations were performed
v ≈1σ/τ. Theinter-planedistanceD waskeptconstant
by solving the Langevin equation of motion, which de-
during shearing. For each plate velocity, all system were
scribes the Brownian motion of a set of interacting
run for 107 MD steps until the steady state in friction
monomers, as
force was reached, i.e. error bars in time-averaged quan-
m¨r =FLJ+FFENE−Γr˙ +FR, i=1,...,N, (3) tities were independent of simulation time. Error bars
i i i i i
were calculated via block-averaging. For the purpose of
where r = [x ,y ,z ] is the position of i–th monomer. dataanalysis, additionalsimulationswererunfor106 up
i i i i
FLJ and FFENE in Eq. (3) are respectively LJ and FENE to 108 MD simulation steps depending on the velocity
i i
forcesexerted onthei–thmonomerand givenbyderiva- and in order to obtain proper plate displacements. As
tives of Eqs. (1) and (2) with respect to r . The effect of a rule each plate was displaced by at least 5 times in
i
theimplicitsolventinEq.(3)issplitintoaslowlyevolv- the corresponding ±xˆ-directions. To avoid any bias on
ing viscous force −Γr˙ and a rapidly fluctuating stochas- friction forces or vertical chain diffusion while the sys-
i
tic force FR. This random force FR is related to the fric- tem was sheared, the thermostat was applied only in the
i i
tion coefficient Γ by the fluctuation-dissipation theorem yˆ-direction [27, 41].
(cid:104)FRi(t)FRj(t(cid:48))(cid:105)=kBTΓδijδ(t−t(cid:48)). Thefrictioncoefficient In this paper, we report frictional forces that are the
used in simulations was Γ = 0.5mτ−1, where m = 1 is forces acting on the plates to keep them at a constant
the monomer mass and time was measured in units of velocities, as well as averaged brush properties that are
(cid:112)
τ = mσ2/(cid:15). The integration step was taken to be calculated from monomer trajectories by employing in-
∆τ = 0.002τ. The velocity Verlet scheme was used for house analysis programs. Unless otherwise noted, all re-
4
sults presented in this paper were averaged over time.
3
10
Linear N=60
Ring N=120
RESULTS AND DISCUSSION
σ]102 (a)
In the course of simulations, two compressed polymer ε/ 3
[
brushes were moved in the opposite (±xˆ) directions by f fR2
driving both planes grafted by chains at prescribed ve- 101 f/ L 1
locities v as shown in Fig 1. Each plane experiences a D=50σ
friction force in the opposite direction to its motion due 100-3 10-2 10-1 100
torelativemotionofbrushes. InFig.2,thefrictionforce 100
forlinearbrushesf (filledsymbols)andforringbrushes 103
L
Linear N=60
f (open symbols) are shown as a function of velocity,
R Ring N=120
v, for various brush systems with different polymeriza-
(b)
tion degrees N and chain topologies. The overall veloc- σ]102
ity dependence of friction forces is consistent with pre- ε/ 3
[
viously reported results [28–30, 44]: The frictional force f R2
increaseslinearlyuptoathresholdvelocity,whichoccurs 101 / fL 1
at different velocities for different brush bilayers. Above D=22σ f
0
a threshold velocity, a sublinear increase occurs for all 10-3 10-2 10-1 100
0
brush systems considered here. We observe this linear- 10
3
to-sublinear transition in most of our systems except the 10
Linear N=100
case with inter-plate distance D = 35σ, where only the Ring N=200
onset of the non-linearregime can be seen in Fig. 2d. As
we will discuss in more details in the following sections, σ]102 (c)
this is due to the fact that the threshold velocity sepa- ε/ 3
[
ratingthelinearandnon-linearforceregimesdependson f R2
f
segmentsizeinsidetheoverlapvolume,wheretwooppos- 101 / L 1
f
ing brushes can co-exist. Hence, as the size of the aver- D=80σ
0
age segments increases inside the overlap volume, much 10-3 10-2 10-1 100
0
slower velocities are required to observe the linear-force 10
3
10
regime. The comparison of friction forces acting on lin-
Linear N=100
ear and ring brushes, which is the main motivation of Ring N=200
this work, shows that friction forces for linear brushes (d)
are always higher than those acting on ring brushes, i.e. σ]102
ε/ 3
ftwLo>fofrRce.sTfhe/fnum≈e2riacaslcvaanlubeeosefetnheinrtahtieoinbseettwseoefnFtihg.es2e. f [ R2
The ratio hLoldsRfor a broad range of shearing velocities 101 / fL 1
f
(v ≈10−3-100σ/τ) used in simulations (see the insets of D=35σ 0
Fig. 2). 10-3 10-2 10-1 100
0
10
Average normal forces acting on the plates in the zˆ- -3 -2 -1 0
10 10 v 10 10
direction were also measured: We observed that normal [σ/τ]
pressures exhibit a weak dependence on velocity, i.e. a
three order of magnitude increase in velocity leads to a FIG. 2: Total friction forces f as a function of plate
roughly10%increaseinthenormalforcesforbothlinear velocity v for linear and ring brushes. The grafting
and ring brushes (data not shown). At high velocities, densities are a) ρL =0.25σ−2 b) ρL =0.11σ−2 c)
g g
a slight decrease in all measured pressures is observed. ρL =0.25σ−2 d) ρL =0.11σ−2 with ρR =0.5ρL for all
g g g g
Similartothefrictionalforces,thenormalpressuresmea- plots.
suredforlinearbrushesarefactoroftwohigherthanring
brushes, pL/pR ≈ 2. However, if total normal forces are
z z
rescaledbythenumberofgraftedchainsforeachsystem,
normal forces per chain are of almost equivalent in both
systems.
Inwhatfollows,wewilldiscusstheobserveddifference arately and demonstrate that the ratio f /f ≈ 2 is
L R
in the frictional forces acting on linear and ring brushes. related to the topology of chains and of their velocity-
We will consider linear and non-linear force regimes sep- dependent conformations.
5
Linear regime
In this subsection we will consider the regime where
frictional forces increase linearly with the driving veloc-
ity. If two dense polymer brushes are brought into con-
tact at the inter-plate distance D > R , as shown in
0
Fig1,whereR isequilibriumsizeofafreechaininbulk,
0
chain segments near the free ends of grafted chains can
interpenetrate through the opposing brush. Hence, the
overlap zone (OZ) where monomers of opposing brushes
can mutually interact can be defined (see a rectangular
region in Fig. 1).
From simulation trajectories, the thickness of the OZ
canbeobtainedusingthecrossproductofmonomerden-
sity profiles of top ρ (z) and bottom ρ (z) brushes.
top bott
The cross product is non-zero only if monomers from
both parts of brush coexist at the same z coordinate. FIG. 3: The equilibrium values of the width of the
The width of the OZ can be calculated from the cross- overlap zone (OZ) versus number of monomers g per
0
product-weighted averages as segment of a chain located inside the OZ for linear (full
symbols) and ring (empty symbols) brushes. Dashed
(cid:90) D/2 (cid:32)(cid:90) D/2 (cid:33)2 lines represent fitted power laws: δ ≈1.0g1/2 (for
δ(v)≡2 z2ω(z,v)dz− zω(z,v)dz , linear chains) and ≈0.625g1/2 (for r0ings). T0he inset
−D/2 −D/2 0
shows fluctuation of the end-to-end size of segments
(4)
r ≡(cid:104)r2(cid:105)1/2 inside the OZ as a function of δ . Solid line
where we defined the cross product as ω(z,v) ≡ 0 0 0
stands for fitted scaling r ∝δ .
ρ (z,v)ρ (z,v). Hence, from Eq. (4), equilibrium 0 0
top bott
value of OZ thickness reads δ(v →0)≡δ . Note that in
0
thispaperweusesubscript“zero”toindicateequilibrium
(or linear-response) values of corresponding parameters law fit is 1.6 times larger for linear brushes. The com-
(i.e. plate velocity v = 0) whereas subscripts L and R pactnessofsegmentsinringbrushesyieldstoanarrower
refer respectively to linear and ring brushes. OZ. As a remark, for very long segments in ring brushes
(g (cid:29) 1) the width of the OZ should in the asymptotic
TheequilibriumvaluesofthewidthsoftheOZforlin- R
limit converge to (cid:104)r2 (cid:105)1/2 ≈ δ ∼ g1/3. In that case:
earandringbrushesδ0L,0R areshowninFig.3asafunc- 0R 0R 0R
tion of the number of monomers g per segment of a δ /δ ∼g1/2/g1/3.
0L,0R 0L 0R 0L 0R
chaininsidetheOZforlinearandringbrushes. Asarule Sincetwoopposingbrushescanonlyinteractinsidethe
a segment is considered as being present inside the OZ if OZ,theobservedfrictionalforcesshouldberelatedtothe
z coordinate of the N-th monomer of the corresponding width of OZ. The grafted chains are not static as they
chainsatisfiesthefollowingcondition: D/2−δL,R <z < constantlydiffuseinandoutoftheoverlapvolumewhich
D/2+δL,R. As we perform our simulations nearly at isgivenbyA×δL,R whereAistheareaofagraftingplate
melt density (ρm ≈ 0.6σ−3) the random-walk statistics duetothermalfluctuations. Whentwoplatesaremoved
allows us to express square-root of the end-to-end dis- at v > 0, any chain segment that enters the OZ feels a
tance r0L,0R ≡ (cid:104)r02L,0R(cid:105)1/2 of a segment inside the OZ flow induced by the relative motion of monomers mov-
and composed of g monomers as r ∼ g1/2 . ing in the opposite direction. The force acting on each
0L,0R 0L,0R 0L,0R
The latter expression is valid for short segments. If the monomer inside the overlap volume can be most gener-
sizeofthesegmentdeterminesthewidthoftheOZ,then ally expressed via Stokes drag fm ≈ ζv, where ζ is the
the relation r ≈ δ should also hold. Indeed, monomeric friction coefficient. The total friction force
0L,0R 0L,0R
this is what we observe in Fig. 3 and its inset for both actingonρmAδ monomersinsidetheoverlapvolumecan
linearandringbrushes: r ≈δ ∼g1/2 . Inter- be expressed for both linear and ring brushes as follows
0L,0R 0L,0R 0L,0R
estingly, we found that at the same inter-plate distance
f ≈ζ vρ δ AΩ . (5)
L,R 0 m L,R L,R
Dgraftedchaininringbrushcanoccupytheoverlapvol-
ume with more monomers than a chain in linear brush In Eq. (5), we defined Ω which quantifies the number
L,R
(g > g ). This finding demonstrate that inside the of binary collisions between the monomers of two oppos-
R L
OZ segments of ring brushes are more compact as com- ing brushes. As we will see shortly, although there are
pared to segments of linear brushes. This fact is also A×δ monomers inside the overlap volume, only a frac-
supported by Fig. 3 since the pre-factor of the power- tionofthemparticipateinmomentumexchangebetween
6
opposing brushes. Hence, Ω is an important quantity
L,R
4.0
todistinguishthefrictionamongdifferentbrushsystems.
In addition, the equilibrium value of the monomeric fric- Linear N=60
3.0 Ring N=120
tion coefficient ζ is equal for monomers of both linear
0
andringbrushes. Thisisduetothefactthatonthetime D=50σ
]
scalescomparablewithtimebetweeninter-beadcollisions σ
[2.0
monomer cannot know whether it belongs to a linear or δ 1.5
ring chain. R
Based on the linear-response theory, which states that / δδL1.0 (a)
any quantity takes its equilibrium value if the pertur- 0.5
10-3 10-2 10-1 100
bation is small, we replace the quantities appearing in 1.0
Eq. (5) with their equilibrium values as v → 0, i.e., 8.0
Linear N=100
δ ≈ δ and δ ≈ δ and similarly Ω ≈ Ω and Ring N=200
L 0L R 0R L 0L
Ω ≈Ω . Thus, the frictional force ratio in the linear- 4.0
R 0R D=35σ
force regime reads ]
σ
[2.0
fL ≈ δ0LΩ0L. (6) δ 1.5
f δ Ω R
The validity of Eq. (6R) can b0Re v0eRrified from the simula- 1.0 / δδL1.0 (b)
0.5
tion data by calculating the equilibrium values of δ 10-3 10-2 10-1 100
L,R
and ΩL,R for both linear or ring brushes. From now on, 10-3 10-2 10-1 100
v
to keep the manuscript more compact we will only show [σ/τ]
data from four of our eight brush systems. All compar-
isons and discussions are valid for the systems which are
FIG. 4: The nonequilibrium thickness δ of the overlap
not shown here as well. We show our non-equilibrium
zone for linear and ring brushes plotted as a function of
simulationresultsforthethicknessofOZδ asafunc-
L,R plate velocity v. The values of δ were calculated using
tionofplatevelocityv forvariouslinearandringbrushes
Eq. (4). The insets show the ratios. The grafting
inFig.4. EachframeofFig.4comparestheOZwidthof densities are a) ρL =0.25σ−2 and b) ρL =0.11σ−2 with
g g
ring-brushes (open symbols) to those obtained for linear
ρR =0.5ρL for all plots.
brushes (filled symbols) at the same inter-plate distance g g
D. Independently of the chain topology the width of the
102
OZs exhibit a plateau for all brush systems as v → 0. (a)
The plateau values of the OZs, which are δ ≈ δ
L 0L
and δR ≈ δ0R, persist up to roughly the plate veloci- 101
ties where the linear-force regime observed in Fig. 2 also g D=50σ 0.8
ends. Indeed, as we proposed for Eq. 6, the widths of
R0.6
dOuZestδoL,tRhearfeacetquthaalttowitthheiinr ethqeuilliibnreiaurm-fovrcaelureesg.imTeh,isfoisr 100 LRiinnega Nr =N1=2600 / ggL0.4
0.2
which the frictional forces change linearly with velocity, 10-1 10-3 10-2 10-1 100
values of δ0L,0R are nothing but the size fluctuations in 102 (b)
the zˆ-direction of the chain segments inside the OZ. As
also illustrated in Fig. 1 the size of a segment with g
L,R
monomers defines the width of the OZ.
g 1 0.8
The number of monomers per segment gL,R inside the 10 D=35σ R0.6
OinZFfiogr.l5inaeasraanfudnrcitnigonbroufsthheeswplaastaenvaellyozceidtyanvduissinshgotwhne LRiinnega Nr N=2=01000 g/ gL0.4
0.2
same color code also as in Figs. 2 and 4. In the entire 0 10-3 10-2 10-1 100
10
velocity range considered here the following ratio holds 10-3 10-2 10-1 100
v
g ≈ 2g for the same value of D. At slow velocities [σ/τ]
R L
(v → 0) the ratio of monomers per segment g /g is
0L 0R
≈0.6. Theratiodropstog /g ≈0.4athighervelocities
L R FIG. 5: The number of monomers g inside the overlap
(v >10−1σ/τ).
zone per participating chain for ring and linear brush
Similar trajectory analysis was also conducted to de-
bilayersasafunctionoftheplatevelocityv. Insetsshow
termine the number of binary collisions ΩL,R inside the the ratios. The grafting densities are a) ρL =0.25σ−2
g
overlap volume: to count collisions, distances between
and b) ρL =0.11σ−2 with ρR =0.5ρL for all plots.
the monomers from opposing brushes are calculated. If g g g
7
the distance is equal or smaller than the cutoff distance 1.0
of LJ potential defined in Eq. (2), this specific pair is (a) D=50σ
countedasacollidingpair. ToobtainΩ andcompare
L,R
thenumberofcollisionsforringandlinearchains,theto-
tal number of collisions is rescaled by the corresponding
Ω0.1 3
valueofρ Aδ . InFig.6,weshowthebinary-collision
fractions mΩ L,,Ras well as their ratios in the insets. In- ΩR2 LRiinnega rN N==16200
terestingly,Lfo,Rrlinearbrushestherearemorecollisionsas /ΩL1
compared to ring brushes, i.e. ΩL > ΩR for the entire 0.0 0 10-3 10-2 10-1 100
velocity range although g ≈ 2g . An average ratio of
R L 1.0
g /g ≈ 0.5 indicate that even though a single chain in (b)
L R
ring brushes can occupy the overlap volume with more
monomers than a chain in linear brushes (g > g ) the
R L
compactness of the segments in ring brushes yield to a
Ω0.1 3
narrower OZ. This is demonstrated in Fig. 3 where the
pre-factor of the slope is bigger for linear brushes. The ΩR2 LRiinnega rN N=2=01000
compactness of segments in ring brushes also results in /ΩL1
D=35σ
a smaller number of collisions between the monomers of
0
opposing brushes. Hence, fL/fR ≈ 2. In addition, as 0.0 10-3 10-2 10-1 100
expectedvaluesofΩL andΩR areclosetotheirvaluesas 10-3 10-2v 10-1 100
[σ/τ]
v →0, which confirms that Ω ≈Ω and Ω ≈Ω .
L L0 R 0R
At the same plate separation D, the ratios of plateau
values are δ /δ ≈ 1.3 ± 0.05 from Fig. 4 and FIG. 6: The binary-collision factor Ω inside the overlap
0L 0R
Ω /Ω ≈ 1.7 ± 0.1 from Fig. 6. Plugging the later zone between two brushes as a function of the plate
0L 0R
expression into Eq. 6 gives the ratio of friction force for velocity v. See the text for details. Insets show the ratio
the linear-force regime f /f ≈ 2.3, which is consistent of linear-brush and ring-brush data. The grafting
L R
with the ratios of forces obtained in the insets of Fig. 2). densities are a) ρL =0.25σ−2 and b) ρL =0.11σ−2 with
g g
As the frictional forces increase linearly (see Fig. 2) in ρR =0.5ρL for all plots.
g g
principlenone oftermsin Eq.(5)should have anyveloc-
ity dependence. Indeed, this is what one would expect
at the linear-response level: at slow enough velocities
(<10−1σ/τ) the conformation of chain segments within ear brushes for simplicity and refer g ≡ g ≡ g and
0 0L 0R
the OZ are not affected by the velocity. As a result, the δ ≡δ ≡δ , etc. If a segment with g monomers en-
0 0L 0R 0
ratios of forces for linear and ring brushes are constant terstheOZ,theforceactingonthissegmentinthelinear
and given by Eq. (6). regime can be expressed as f ≈ζ g v. As we have dis-
ch 0 0
Our analyzes confirm that in ring-brush bilayers, the cussedintheprevioussection,belowthethresholdveloc-
amount of inter-digitation of chains is weaker compared itythechainsarestillGaussian. Hence,asexpectedfrom
tothatinlinear-brushbilayersatslowdrivingvelocities, a Gaussian chain, segment sizes in xˆ, yˆ and zˆ-directions
where the friction increases linearly with velocity. are not coupled, i.e. fluctuations in size in all direction
are independent of each other. However, as the velocity
is increased, the frictional force per segment inside the
Non-linear regime OZ reach a threshold force f∗ ≈k T/b≈ζ g v∗ at v =
ch B 0 0 I
v∗ [35]. The force on the segment given by f∗ ≈k T/b
I ch B
In this subsection we will focus on the regime where is high enough to align individual bonds in the direction
frictional forces demonstrate sublinear plate velocity v of the relative motion. Thus, vertical segment fluctua-
dependence. AscanbeseeninFig.2nonlinearregimeis tions in the zˆ-direction, which define the width of OZ,
observe for both linear and ring bilayers. The indication will be affected by the shear in the acting in xˆ-direction.
of the non-linear regime is the shrinkage of overlap vol- As a result the OZ decreases with respect to its equilib-
umewithincreasingvelocityv asshowninFig.4. Onset rium value δ < δ0 at v > vI∗ ≈ kBT/bζ0g0 ∼ 1/g0 [35].
of the non-linear regime occurs at a threshold velocity On the other hand, in a concentrated solution one may
v =v∗. Indeed,v∗ hasdifferentvaluesfordifferentchain argue that the threshold velocity occurs at slower veloc-
sizesascanbenoticedinFigs.2and4. Beforewefurther ities and is given by fc∗h ≈ ζ0gvI∗I ≈ kBT/δ0 that leads
continue our discussion on the non-linear regime, we will to vI∗I ≈ kBT/bζ0gL1+/Rν [29], where ν is the scaling ex-
briefly discuss two scaling predictions for the threshold ponent (ν = 1/2 for an ideal chain and ν = 0.588 for a
velocity, and refer them as v∗ and v∗ . swollen chain in 3D). In our simulations, hydrodynamic
I II
We will temporarily drop the indexes for ring and lin- interactions are screened and we take ν = 1/2 and ob-
8
tain v∗ ∼ 1/g3/2. The ratio of two predictions of the a) b)
II 0
threshold velocity is v∗/v∗ = g1/2. Unfortunately, in Ring brush Ring brush;
I II 0 @ Linear regime @ Non-Linear regime
our simulated systems segments sizes inside the OZ are
of the order of g ∼ 10. Thus, we cannot capture a
0L,0R
significant difference between the threshold-velocity pre-
dictions v∗ and v∗ . In principle, brush systems with
I II
largerg valuescanbedesigned. However,forlonger
0L,0R
segment sizes, e.g. g ≈ 100 chain entanglements c) d)
0L,0R Linear brush; Linear brush;
willalsocomeintoplayandintroducemorecomplexities. @ Linear regime @ Non-Linear regime
Hence, throughout this work we will leave investigation
of the threshold velocity v∗ for a separate work [45], and
refer v∗ only to distinguish between the linear (any seg-
mentinsideOZisstretchedlessthanitsequilibriumsize)
andthenon-linearforceregimes(thesegmentend-to-end
distance is more than its equilibrium size).
FIG. 7: Snapshots of the individual chains from ring (a
As discussed above, at v >v∗ the segments inside the
and b) and linear brush bilayer (c and d) simulations in
OZ are not Gaussian, and are stretched due to the rel-
the linear-force regimes (left column) and
ative motion of brushes, as illustrated by the snapshots
non-linear-force regimes (right columns).
given in Fig. 7. At the velocity range v∗ < v < v ,
max Polymerization degree of chains for linear brushes is
where the maximum velocity that we considered here
N =60 whereas for ring brushes is N =120. Monomers
v ≈ 1 σ/τ, both δ and δ decrease in a similar way
max L R of surrounding chains are rendered as points with colors
withincreasingvelocityasshowninFig.4. Thus,Eq.6is
referring to the top and bottom brushes.
stillvalidtoexplainthedifferenceinthefrictionalforces.
Ataroundv ≈1σ/τ, regardlessofthebrushtopol-
max
ogy, the width of the OZs goes to unity δL,R → σ as topreviouslydescribed”participating”chaindescription
shown Fig. 4. Hence, at v >∼ vmax, according to Eq. 6, are shown in Fig. 8 with ratios given in the insets. For
the ratio of friction forces for linear and ring brushes is slow velocities (v < 10−1σ/τ), which also corresponds
reduced to to the linear-force regime, Ψ is slightly larger than
0L
fL ≈ ΩL. (7) vΨ0>R fvo∗rtahllecdaisffeesr.enOcne tbheetwoetehnerlihnaenadr aantdhirgihngveblorucsithieess
f Ω
R R
increases and is consistent with the conditions g > g
R L
IndeedFig.6showsthattheratioofbinary-collisionfac- andδ →σ. NotethatinFig.8b, theincreaseinratio
L,R
tors for ring and linear ΩL,R is even higher at high ve- islesspronounced. Possiblereasonofthisisourcriterion
locities and reaches ΩL/ΩR ≈ 2. The limit of δL,R → σ for chains participating in the OZ which underestimates
forbothringandlinearbrushesathighvelocitiesimplies looping segments since we check only the Nth monomer
that there should be equal number of monomers inside of the corresponding chain within the OZ.
the overlap volumes ρmAδL,R. Additionally, we know Finally, at ultra-high velocities v (cid:29) 1 σ/τ, for which
gR ≈ 2gL from Fig. 5. This is only possible if stretched we cannot perform simulation since the thermostat can-
segments of ring brushes form double-stranded confor- notkeepthesystemtemperatureuniformthroughoutthe
mations inside the OZ as can be seen in the snapshot simulation box, one would expect that both δ → σ
L,R
given in Fig. 7b. In the non-linear regime a segment and Ω → 1. Since the grafted chains are completely
L,R
of a ring brush with gR monomers inside OZ is highly inclinedandgL,R →1,twoopposingbrushescanonlyin-
stretched and adopts a double-stranded conformation. teractviafewmonomersnearthefreeedgesofthegrafted
Due to double-stranding, each monomer of a segment in chains. Thisscenarioindeedyieldstoahigh-velocitylin-
a ring brush has on average three neighbours from the ear regime and f /f →1.
L R
same segment – two bonded and one nonbonded. This
leads to smaller number of collisions with the monomers
of the opposing brush. CONCLUSION
Another way of confirming that segments in the ring
brushes are double-stranded inside the OZ is to calcu- Inthisworkwedemonstratedusingscalingarguments
late the fraction of “participating” chains, Ψ . Since and MD simulations that friction forces of linear chains
L,R
linear chains has smaller number of monomers inside the are higher than those of ring brushes for a broad range
OZ(g <g )largernumberofchainsshouldoccupythe of velocities. For slow driving velocities (or shear rates),
L R
OZtokeepthemonomerdensityρ uniformthroughout segments of a linear brush can penetrate through oppos-
m
the simulation box. Calculated values of Ψ according ing brush deeper compared to segments of ring brushes.
L,R
9
1.0 was supported by the Polish Ministry of Science and
(a) Linear N=60 Higher Education (Iuventus Plus: IP2012 005072).
Ring N=120
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