Accuracy of the blob model for single flexible polymers inside nanoslits that are a few monomer sizes wide

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by  Narges Nikoofard

Accuracy of the blob model for single flexible polymers inside nano-slits that are a
few monomer sizes wide
Narges Nikoofard,∗ S. Mohammad Hoseinpoor, and Mostafa Zahedifar
Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 51167-87317, Iran
(Dated: January 14, 2015)
The de Gennes’ blob model is extensively used in different problems of polymer physics. This
model is theoretically applicable when the number of monomers inside each blob is large enough.
For confined flexible polymers, this requires the confining geometry to be much larger than the
monomer size. In this manuscript, the opposite limit of polymer in nano-slits with one to several
5
monomerswidthisstudied,usingmoleculardynamicssimulations. Extensionofthepolymerinside
1
nano-slits,confinementforceontheplates,andtheeffectivespringconstantoftheconfinedpolymer
0
areinvestigated. Despiteofthetheoretical limitations oftheblob model,thesimulation resultsare
2
explained with the blob model very well. The agreement is observed for the static properties and

n thedynamicspringconstantofthepolymer. Atheoreticaldescriptionoftheconditionsunderwhich
a thedynamicspringconstantofthepolymerisindependentofthesmallnumberofmonomersinside
J blobsisgiven. Ourresultsonthelimitofapplicabilityoftheblobmodelcanbeusefulinthedesign

3 of nano-technology devices.
1

] I. INTRODUCTION confining geometry, D. Inside blobs, polymer does not
t
f feel that it is confined. The number of monomers inside
o
s Advances in nanotechnology have enabled investiga- each blob is found from the relations of the self-avoiding
1
. tions of single polymers confined in the nano-scale[1, 2]. walk (SAW) in free space, g ∼ D ν. Here, ν is the
t b
a This problem has attracted interests due to its applica- Flory exponent in 3D and b is the monomer size. In
m (cid:0) (cid:1)
tions in the design of nanotechnology devices [3–5]. It lengthscaleslargerthantheblobsize,thepolymer(with
- also helps in obtaining more knowledgeof polymers con- N realmonomers)behaveslikeaneffectivepolymerwith
d
fined in biological environments [6, 7]. Besides, it has N monomers of size D. This effective polymer has a
n g
o become possible to check accuracy of the old theories SAW in one or two dimensions for a nano-channel or a
c and improve them [8, 9]. Single polymers confined in- nano-slit, respectively. The number of monomers inside
[ side nano-spheres [10, 11], nano-channels [2, 8, 12–14] each blob, g, and total number of the blobs, N, should
g
and nano-slits (between two parallel plates) [9, 15–18]
1 be large enough, to be able to use the relations of sta-
are studied.
v tistical mechanics for SAW inside or outside the blobs,
5 The most known theory for confined polymers inside respectively [25].
5 nano-channelsornano-slitsisthedeGennes’blobmodel.
9 Forsemi-flexiblepolymers,Odijktheoryisrelevantwhen A number of studies on confined polymers are per-
2 the sizeofthe confininggeometry,D, issmallerthanthe formedinconditionsthattheabovetworequirementsare
0 not satisfied. It is shown that the blob model is applica-
persistencelengthofthepolymer,L [8,15]. Recently,it
. p ble in micro-devices, where the number of blobs is very
1 isshownthatextendeddeGennes’theoryshouldbeused
0 L2 small [26]. Also in some practical situations, the poly-
in the case of semi-flexible polymers when L <D < p
5 p w mer is confined in severe conditions that the number of
1 [19]. Here, w is the monomer width. monomersinsideeachblobisverysmall[27]. Speciallyin
v: Most of the studies on confined polymers focus on nano-devicesfor DNA barcoding, full stretching of DNA
double-stranded DNA, which is a semi-flexible polymer.
i isrequired[5]. Computersimulationsshowthattheblob
X Recently, single stranded DNA is introduced as a model
modelcanexplainthe staticsofaflexible polymerinside
r system for study on flexible polymers [20]. Also, many narrow channels [12, 14]. However, the polymer dynam-
a proteins and synthetic polymers are flexible. For flexible
icsinnano-channelsisonlydescribedwiththeblobmodel
polymers,theonlyrelevanttheoryisthedeGennes’blob
when the number of monomers inside each blob is large
model and it would be useful to better understand its
enough [13, 14].
limitations and regime of applicability. The blob model
has broad applications in different problems of polymer In this manuscript, a flexible polymer confined inside
physics, such as polyelectrolytes [21], polymers under a nano-slit is studied using molecular dynamics (MD)
tension [22], and polymers adsorbed to surfaces [23] or simulations. The extreme condition that the number of
confined near surfaces unfer external fields [24]. monomers inside each blob is small is considered. It is
Intheblobmodel,theconfinedpolymerisdividedinto shownthatourresultsfortheradiusofgyrationandcon-
smaller sections, called blobs, that have the size of the finementfreeenergyofthepolymercanbeexplainedvery
well,usingtheblobmodel. Despitepreviousstudies(Ref.
[16]and referencestherein), no correctionis made to the
slit width (distance between the two plates). Agreement
∗ [email protected] between simulation results and the blob model is ob-
2
served even for very narrow slits with 1-2 monomer(s)
width. These widths are completely below the regime of
applicability of the blob model.
Extensionof the polymer inside the nano-slithas fluc-
tuationsarounditsmeanvalue. Fromthesefluctuations,
an effective spring constant can be obtained. It is a dy-
namic variable for the polymer and is a measure of elas-
ticity of the polymer. Our results for the effective spring
constantareingoodagreementwiththeblobmodel,un-
like the caseofthe polymer inside the nano-channel[12–
14].
This discrepancy is due to the difference between the
origins of the effective spring constant of a polymer con-
fined in a nano-channel and a nano-slit. Two factors FIG. 1. (Color online) (a) Blob model for a flexible polymer
governthe fluctuations of a polymer confined in a nano- confined inside a nano-channel. The blobs of the polymer in
slit: fluctuations in the size of blobs and fluctuations in the nano-channel are arranged one after the other along the
nano-channel. (b) Top view of a polymer confined inside a
the self-avoidingwalkofthe blobs. Itisobviousthatthe
nano-slit. Theblobsof thepolymerin thenano-slit span the
latter factor is dominant. However, for a polymer con-
2Dspacebetweentheplates. Thenumberofmonomersinside
fined ina nano-channel,the blobs arearrangedone after
one blob and total number of the blobs are clear in the two
the other along the channel. So, fluctuations in the blob
cases (a) and (b).
size are the only source of fluctuations in the extension
of the polymer (see Fig. 1). This makes the internal
dynamics of the blobs important, in the dynamics of the
whole polymer. As a result, it is determining to have
the sufficient number of monomers inside each blob, for 3 D −41
a polymer confined inside a nano-channel. R|| ∼bN4 b . (1)
(cid:18) (cid:19)
Inthe nextsection, the statics anddynamics ofa flex-
Free energy of confining the polymer is k T per blob,
ible polymer confined in a nano-slit is reviewed briefly. B
F ≈k TN. Substituting g in this relation gives
In Sec III, our simulation method is described. Results conf B g
of the simulation and their compatibility with the the-
ory are discussed in Sec. IV. Finally, a summary of the 5
b 3
results is presented in the last section. F ≈k TN . (2)
conf B
D
(cid:18) (cid:19)
The force exerted on the plates is found by differenti-
ating the confinement free energy with respect to D,
II. THEORY
8
k T b 3
B
f ≈ N . (3)
Considera flexible polymerconfined betweentwo par- b D
(cid:18) (cid:19)
allel plates (Fig. 1). Confinement extends the polymer
The polymer has fluctuations around its average ex-
on the walls, to a radius of gyration larger than its free
tension. The mean square deviation from the average
radius of gyration R ∼ bNν. Here, N is the number
g extension is described by an effective spring constant
of monomers of the polymer, b is the monomer size and
ν = 3 is the Flory exponent in 3D. Sections of the chain h∆R|2|i = h R||(t)−R||av 2i ≈ kBT/keff. The simplest
5
smaller than the distance between the plates, D, do not way to calc(cid:0)ulate the spri(cid:1)ng constant, keff, is to elimi-
feel the constraints. These sections, called blobs, have a nateD betweentherelations1and2. Thisgivesthefree
statistics similarto a chaininfree space D ∼bg53, where energy as a function of extension. Differentiating the
g is the number of monomers inside each blob. polymer free energy twice with respect to its extension
andsubstituting theequilibriumextensiongivesaspring
The polymeris composedof N blobsthatcannotpen-
g constant [8]. This calculation does not give a correct re-
etrateintoeachother. So,itcanbe regardedasaneffec-
sult,becauseequation2isthefreeenergyofthepolymer
tive two-dimensional polymer with N monomers of size
g at its equilibrium extension. Indeed, the free energy of
D. We use the exponent of a self-avoiding walk in two
the polymer for any desired extension should be used in
dimensions, ν = 3, to find the extension of the polymer
4 the calculation.
parallel to the plates R|| ∼ D(Ng)43. This gives the rela- Suppose the two ends of the confined polymer are
tion between the average extension of the polymer with under tension and it is elongated to the extension R .
f
the total number of monomers and the distance between Free energy of a two- or three-dimensional polymer with
the plates, N monomers of size b which is extended to size R is
f
3
1
F ≈ k T Rf 1−ν [25]. ν is equal to 3 for 3D and
tens B bNν 5
equalto 3 for(cid:16)2D.B(cid:17)ydifferentiatingthisfreeenergytwice
4
with respect to R and substituting the equilibrium ex-
f
tension R ∼ bNν, the spring constant is obtained as
eq
k = ∂2Ftens| ≈ kBT ≈ kBT. This is in agree-
eff ∂R2f Req (bNν)2 R2eq
ment with the relation used in Ref. [28].
Here, the confined polymer is described as an effec-
tive polymer with N monomers of size D in two di-
g
mensions (ν = 3). So, the spring constant becomes
4
k ≈ kBT . Substituting for g, we get the spring
eff 3 2
D(N)4
(cid:18) g (cid:19)
constant for the confined polymer
FIG. 2. (Color online) Side view of a polymer confined in a
1 nano-slit. The surface of each plate is defined such that the
k T D 2 k T
k ≈ B N−3/2 = B . (4) closest distance of the center of a monomer to the plate is
eff b2 b R2 equal to b/2. Distance between the plates in our paper, D,
(cid:18) (cid:19) ||
is shown in the figure. An effective width for the nano-slit is
In the final relation, R is substituted from eq. 1. To also shown, to examine the effect of adding correction to the
||
derivethis relation,the confinedpolymeris simply mod- slit width.
eledasaneffectivetwo-dimensionalpolymercomposedof
blobsandtheeffectoftheexternaltensionistomovethe
in a nano-slit. A schematic of the system is shown in
confinementblobstowardalinearalignment. Thissimple
Fig. 2. The polymer is modeled as a bead-spring chain.
description is valid for small tensions with R <D(N).
f g Monomers of the polymer are connected by the FENE
The blob model is accurate when the number of potential,
monomers inside each blob, g, and total number of the
blobs, N, are considerable. This is necessary to be able 1 r 2
g U (r)=− Kr2ln 1− . (5)
to use statistical mechanics relations for a self-avoiding FENE 2 0 r
" (cid:18) 0(cid:19) #
walk inside and outside the blobs, respectively. The size
Shifted-truncatedLennard-Jonesisusedfortheexcluded
spannedbyaself-avoidingwalkerinspaceisproportional
volume interactions between the monomersand between
tothenumberofstepstothepoweroftheFloryexponent
the monomers and the plates,
in two or three dimensions, when the number of steps is
largeenough. Forg tobe largeenough,theconfiningge- 12 6
b b 1 1
ometry(nano-slitornano-channel)shouldbemuchlarger ULJ(r)=4ǫ − + , r <26b. (6)
r r 4
than one monomer size. "(cid:18) (cid:19) (cid:18) (cid:19) #
It is instructive to compare the effect of an external
ǫandbaretheenergyandlengthscalesofthesimulation.
tensiononapolymerconfinedinsideanano-channelora
K =100ǫandr =1.5bareusedfortheFENEpotential.
0
nano-slit. Prior to tension, the polymer inside the nano-
The plates are infinite and the closest distance of the
slitarrangesitsblobsintwodimensions. Theeffectofthe
center of a monomer to the plate surface is b/2. This
externaltensionistomovetheblobstowardaonedimen-
means that the spherical surface of the monomer can
sional arrangement. However, the blobs of the polymer
touch the plate, in agreement with our intuition from
inside a nano-channel have already lost two degrees of
a wall (Fig. 2). In previous studies, the center of a
freedom and lie on a one dimensional line. So, the poly-
monomers was allowed to touch the impenetrable plate
mer has to stretch its blobs to respond to the external
surface[16]. So,itwasneededtoaddthemonomersizeb
tension. One can conclude that the elasticity of a poly-
to the distance between the walls to compare the results
mer inside a nano-slit has its origin in the arrangement
with the blob model. It is important to note that this
of the blobs. This is in contrast to the elasticity of a
is not attributed to the finite size of the chains in these
polymerinside a nano-channelwhichoriginatesfromthe
simulations. Here, it will be shown that the simulation
elasticityofeachblob. Asaresult,tousetheblobmodel
results for similar chains are in agreement with the blob
for the former case, the number of blobs should be suf-
model, without any corrections in the wall distance.
ficient; but for the latter case, the number of monomers
The equations of motion are integrated using the Ve-
inside each blob is also important.
locityVerletalgorithm,withthestepsizeequalto0.01τ .
0
τ = mσ2 is the MD time scale and m is the monomer
0 ǫ
III. SIMULATION METHOD mass.qThe system is kept at the constant temperature
T =1.0 ǫ , using the Langevinthermostat with the fric-
kB
We use MD simulations to check the limits of accu- tion coefficient 1.0τ−1. The simulations are performed
0
racy of the blob model for a flexible polymer confined using ESPResSo [29].
4
1.5
26
24 1.4
22
1.3
20
18 b) 1.2
R / b||16 0.2 g (R / ||1.1
14 0.15 lo
12 z)0.1 1
(
n
10
0.05
0.9
8
0
−1 0 1
6 z
0.8
20 30 40 50 60 70 2 2.1 2.2 2.3 2.4 2.5
0.75 −0.25 log (N)
N (D / b)
FIG. 4. (Color online) log10-log10 plot of radius of gyration
FIG. 3. (Color online) Simulation results for R|| versus the of the polymer parallel to the plates R|| versus the polymer
theoretical relation N0.75D−0.25. #, 2, ×, , +, ▽, △, ⊳ length, N, for different nano-slit widths. #, 2, ×, , +,
and⊲showthesimulation resultsforD/b=1-9,in ascending ▽, △, ⊳ and ⊲ show the simulation results for D/b=1-9, in
order. Inset: Distribution of the monomers in the distance ascending order. The corresponding α values are 0.74, 0.78,
between the plates. The solid, dashed and dash-dotted lines 0.79,0.76, 0.72,0.74, 0.74,0.78and0.79. Theaverageofαis
correspondtothethreenarrowestslitsD/b=1-3,inascending 0.76.
order.
IV. SIMULATION RESULTS
A. Radius of gyration
Inoursimulations,thenano-slitwidthischangedfrom
Radiusofgyrationofthepolymerparalleltotheplates
b to 9b. Polymers with 100, 200 and 300 monomers are
versus the theoretical relation in equation 1 is shown in
investigated. Forslitswithwidth3-9b,themonomersare
Fig. 3. All data lie on a single line, in the whole range
arranged in an initial structure close to a self-avoiding
of parameters studied. This shows agreement between
walk. Then, the system is warmed up to remove any
the simulation results and the blob model. Distribution
overlap between the monomers. For slits with width b
of the monomers in the distance between the plates is
and 2b, the monomers are initially arrangedon one line.
shownintheinsetofFig. 3,forthethreenarrowestslits.
This initial configurationreaches equilibrium in a longer
Because of the soft nature of the potential, monomers
timeinterval,butitisinevitablebecauseoftheverysmall
can penetrate the walls rarely.
width of the slits.
To further investigate the correspondence of the sim-
ulation results with the theory, exponents α and β in
Allsimulationsareperformedforatotaltimeof5N2τ0. the relation R|| ∝ NαDβ are obtained from simulation.
Data points to find time averages of the static and dy- These exponents are found from the slopes of the log10-
namic properties of the polymer are collected after the log10 plots of the radius of gyration versus the polymer
time N2τ , in time intervals equalto τ . In all plots, the length and the nano-slit width, respectively. As can be
0 0
size of errorbarat each point is smaller than the symbol seen in Fig. 4, α is close to the theoretical value even
size. for very narrow slits (1-2b). α is not a decreasing or in-
creasing function of the nano-slit width. So, the small
numberofmonomersinside eachblobis notdetermining
The radius of gyration of the polymer parallel to inthis exponent. The averageofα is0.76,whichhas%1
the plates is calculated using the relation R|| = deviationfromthetheoreticalvalue. Fig. 5(a)showsthe
1 (X −X )2+(Y −Y )2. X and X are the exponentβ fordifferentlengthsofthepolymer. Itisseen
N i CM i CM i cm
x components of the positions of the ith monomer and that finite size of the chains has no effect on the simula-
p
the center of mass, respectively. Y and Y are the tion results. The results are summarized in the relation
i CM
corresponding y components. The force on each plate in R|| ∝N0.76D−0.25.
all steps of the simulation is sum of the Lennard-Jones For comparison with Ref. [16], an effective width is
forces from all monomers interacting with the plate. definedforthenano-slitbyaddingamonomersizetothe
5
1.6 300
N = 100, β = − 0.25
1.5 N = 200, β = − 0.24
250
N = 300, β = − 0.25
1.4
200
1.3
b) 150
R / ||1.2 f
og ( 1.1 100
l
1 50
0.9
(a) 0
0.8 0 5 10 15
0 0.2 0.4 0.6 0.8 1
− 2.7
log (D / b) N (D / b)
1.6
N = 100, β = − 0.33 FIG. 6. (Color online) Simulation results for f versus the
1.5 N = 200, β = − 0.34 theoretical relation ND−2.7. #, 2, ×, , +, ▽, △, ⊳ and ⊲
N = 300, β = − 0.33 show thesimulation results for D/b=1-9, in ascending order.
1.4
4.5
b) 1.3
R / ||1.2 4
g ( 3.5
o
l
1.1
3
1
0.9 og ( f ) 2.25
l
(b)
0.8 1.5
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
log [(D+δ) / b] 1
0.5
FIG.5. (Coloronline)log10-log10 plotofR|| versusthenano- 0
slit width. (a) and (b) show two different definitions for the 2 2.1 2.2 2.3 2.4 2.5
nano-slitwidth(seeFig. 2). Ascanbeseen,omittingδresults log ( N )
in an excellent improvement in thescaling exponent.
FIG.7. (Coloronline)log10-log10plotoff versusthepolymer
nano-slit width (see Fig. 2) in Fig. 5(b). The exponent length, N, for different nano-slit widths. #, 2, ×, , +,
β deviates largely from the theoretical value, with this ▽, △, ⊳ and ⊲ show the simulation results for D/b=1-9, in
change in the definition of the wall surface. ascendingorder. Thecorrespondingγvaluesare1,0.99,0.99,
0.98, 0.99, 0.96, 0.98,0.95 and0.99. Theaverageofγ is0.99.
B. The confinement force theory and simulation.
The exponents γ and ω in the relation f ∝NγDω are
It is not easy to measure the free energy of confining obtainedfromsimulation. AccordingtoFig. 7,deviation
the polymer in the nano-slit. Instead, the force that the in γ from the theoretical value (=1) is less than %1.
confined polymer exerts on the plates is studied in the In Fig. 8(a), the value of ω is found from the log -
10
simulation. The results are compared with equation 3 log plot of force versus the nano-slit width. Deviation
10
from the theory. The simulation results for the force are of the exponent from the theoretical value (-2.7) is less
plottedversusthe theoreticalrelationinFig. 6. Alinear than %8. The simulation results are summarized in f ∝
behavior is observed, which shows agreement between N0.99D−2.9. Again, to check our definition for the nano-
6
2.5 x 10−3
N = 100, ω = − 2.9 3.5
N = 200, ω = − 2.9
N = 300, ω = − 2.9 3
2
2.5
1.5
og ( f ) keff 2
l 1 1.5
1
0.5
0.5
(a)
0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00 0.1 0.2 0.3 0.4 0.5 0.6
log(D / b) N − 1.5 (D / b)− 0.5

2.5
N = 100, ω = − 3.5 FIG. 9. (Color online) Simulation results for the effective
N = 200, ω = − 3.6 springconstantversusthetheoreticalrelation N−1.5D0.5. #,
N = 300, ω = − 3.6 2, ×, , +, ▽, △, ⊳ and ⊲ show the simulation results for
2
D/b=1-9, in ascending order.
1.5
g ( f ) verTshues stihmeutlhaetoiornetriceasullrteslafotriotnhe(eeqffueacttiiovnes4p)rainrge cdoenpsictatendt
o in Fig. 9. The linear curve confirms the excellent agree-
l 1
ment between theory and simulation. This agreement is
observed even when the nano-slit width is very small.
The exponents η and ζ in the relation k ∝ NηDζ
0.5 eff
are found from the simulation results. Fig. 10(a) shows
the log -log plot of k versus the polymer length.
10 10 eff
(b) The average slope of the plots is -1.3. This exponent
0 has the largest deviation, %13, among the exponents in-
0.5 0.6 0.7 0.8 0.9 1
log [(D+δ) / b] vestigated in this manuscript. The exponent is not a
monotonic function of the nano-slit width and it cannot
be deduced that an improvedagreement between theory
FIG. 8. (Color online) log10-log10 plot of f versus the nano- and simulation would be observed with increasing the
slit width. (a) and (b) show two different definitions for the
nano-slit width.
nano-slitwidth(seeFig. 2). Ascanbeseen,omittingδresults
Dependence of the effective spring constanton the slit
in an excellent improvement in thescaling exponent.
width is investigated in Fig. 10(b). The average of the
exponentis0.48. Althoughdeviationoftheaveragefrom
the theory is around %4, the exponents themselves are
slit width, the exponent ω is found from the log -log
10 10
not very close to it. Overall, a good agreement between
plot of force versus D+δ (Fig. 8(b)). Deviation of the
theory andsimulationis observedand the simulationre-
exponent becomes more than %30.
sults are summarized in the relation k ∝N−1.3D0.48.
eff
To check the effect ofcorrelationsin the calculationof
h∆R2i, a different averaging scheme was also examined.
C. The effective spring constant ||
Data points with a time interval of 100τ were collected
0
fromthewholedata. Startingfromdifferentinitialtimes,
Fluctuations in the radius of gyration h∆R2i can be
|| 100 different series of data points can be defined. The
usedto findaneffective springconstantfor the polymer,
valuesofthespringconstantobtainedfromeachofthese
keff ≈ h∆kBRT2i. Thiseffectivespringconstantisameasure seriesareveryclosetotheonesshownonFigs. 10(a)and
||
of the dynamics of the confined polymer. For example, 10(b).
the relaxation time of the confined polymer is found by Furtherinvestigationisneededtoexplainthelargerde-
dividing the friction coefficient from the solvent to this viations of the dynamic exponents from theory, relative
effective spring constant [15]. to the static ones. One can attribute these deviations to
7
−0.2 V. SUMMARY AND DISCUSSION
(a)
−0.4 In this manuscript, we checked the accuracy of the de
Gennes’ blob model for a flexible polymer inside a nano-
−0.6 slit, using MD simulation. Theoretically,the blob model
is accurate for wide slits and long polymers. However,
)eff−0.8 wNeaneox-asmlitisnewditthhewmidotdhesl ienquthale tliom1it-9oftvimereysnaarmroownosmlitesr.
k
g ( −1 size were investigated. Simulation results both for the
o
l static properties and the dynamic spring constant of the
polymer were in excellent agreement with the theory.
−1.2
Our results showed that the blob model can describe
theeffectivespringconstantoftheconfinedpolymereven
−1.4 for very narrow slits. This is in contrast to a recent
study on single flexible polymers inside nano-channels.
−1.6 Indeed, the blob model is only applicable for the spring
2 2.1 2.2 2.3 2.4 2.5
constant of the polymer in nano-channels with at least
log ( N )
10monomerswidth. Here,itwasexplainedthatthisdis-
crepancyisaresultofthedifferentoriginsofthepolymer
0.2
N = 100, ζ = 0.40 (b) dynamics in the two cases. The internal dynamics of a
blob is determining in the overalldynamics of the chain,
0 N = 200, ζ = 0.46
N = 300, ζ = 0.58 for a polymer confined in a nano-channel. But, for a
−0.2 polymer confined in a nano-slit, the dynamics is domi-
nated by the two-dimensional arrangement of the blobs
−0.4
between the plates.
)eff−0.6 weIrne ouuserds.imIutlwataiosnosb,speorvlyemdetrhsatwitthhe1e0ff0e-c3t00ofmtohneofimneitres
k
g ( −0.8 size of the chains is not considerable. Our results on
lo the accuracy of the blob model evenfor narrowslits and
−1 shortchainscanbeveryusefulinthestudiesofpolymers
in confinement. Polymer confinement occurs in nano-
−1.2
technologydevices,suchasnano-poresequencing[3,30],
DNA barcoding [5] and polymer separation devices [4].
−1.4
The main result of this manuscript was that the blob
−1.6 modelissometimesaccuratebeyonditstheoreticallimits,
0 0.2 0.4 0.6 0.8 1
in practical situations. Considering that the blob model
log (D / b)
isusedinmanyproblemsofpolymerphysics[25],thisre-
sult is applicable in different circumstances. A polymer
FIG.10. (Coloronline)(a)log10-log10 plotofkeff versusthe adsorbed to a surface [23] or compressed on a surface
polymer length, N, for different nanoslit width. #, 2, ×, ,
by an external field [24] is confined to a region near the
+, ▽, △, ⊳ and ⊲ show the simulation results for D/b=1-9,
surface. For these problems with 2D confinement of the
inascendingorder. Thecorrespondingη valuesare-1.6,-1.6,
polymer, the above result can be used directly. How-
-1.7, -1.4, -1.1, -1.5, -1.3, -1.3, -1.6. The average η is -1.3.
ever, for a polymer under external tension more caution
(b)log10-log10 plot of keff versustheslit width, for different
polymer lengths. The average of ζ is 0.48. isneeded,sincetheblobsarearrangedin1Dandthe dy-
namicsmaybemoresensitivetothenumberofmonomers
inside one blob.
the finite length of the chains or insufficient simulation
times for equilibration of the dynamic quantities. How- ACKNOWLEDGMENTS
ever, larger fluctuations may be an intrinsic property of
the dynamic quantities and persist even in simulations The authors would like to thank Dr. Hossein Fazli for
with longer chains and longer run times. useful discussions.
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