Table Of ContentNoname manuscript No.
(will be inserted by the editor)
Simulational study of anomalous tracer diffusion in
hydrogels
Fatemeh Tabatabaei · Olaf Lenz · Christian Holm
1
1
0
2 January27,2011
n
a Abstract In this article, we analyze different factors particle after a time t. Often, the MSD of a particle
J
that affect the diffusion behavior of small tracer par- behaves as
6
ticles (as they are used e.g. in fluorescence correlation
2 (cid:104)|∆r(t)|2(cid:105)∝tα (1)
spectroscopy(FCS))inthepolymernetworkofahydro-
] gel and perform simulations of various simplified mod- where α is the diffusion exponent. The case where the
t
f els. We observe, that under certain circumstances the MSD scales linearly with time (α=1) is called normal
o
s attraction of a tracer particle to the polymer network diffusion,andisthemostcommoncaseinphysicalpro-
t. strands might cause subdiffusive behavior on interme- cesses, for example for particles that perform a simple
a
diate time scales. In theory, this behavior could be em- random walk. A value of α(cid:54)=1 signifies anomalous dif-
m
ployed to examine the network structure and swelling fusion, with the special cases of sub-diffusion (α < 1)
-
d behavior of weakly crosslinked hydrogels with the help and superdiffusion (α > 1). During the last decades,
n of FCS. there has been considerable interest in anomalous dif-
o
fusion, and such behavior has been found in many sys-
c
[ tems. [7,8,9,10,11,12,13]
Introduction Fytas et al have employed the FCS technique to
3
study the diffusion of tracer particles in a hydrogel [5].
v
0 Fluorescence Correlation Spectroscopy (FCS) is an ex- Theyfoundevidence,thatundercertaincircumstances,
1 perimental technique that allows to study the diffusion tracer particles in a hydrogel network exhibit anoma-
5
behavioroftracerparticlesinasurroundingmediumin lous diffusion behavior.[14]
1
great detail. The goal of this work is to approach the problem
.
2
The method is based on detecting the fluctuations fromatheoreticalpointofview,toexplainhowanoma-
1
of the fluorescent light intensity in a small and fixed lous diffusion behavior can occur in such a system, and
0
1 volumeelement,usuallyformedbyalaserfocusofsub- the different processes that play a role in the diffusion.
: micron size. There is a broad range of application for From the theoretical considerations, it is shown that
v
i FCS, from studying the diffusive dynamics of simple FCS may be a valuable tool for examining the network
X
colloidsandpolymerstobiomoleculesaswellashydro- structure and swelling behavior of real hydrogels.
r gels. [1,2,3,4,5,6] Sprakel et al [15] have studied the diffusion of col-
a
Todescribethediffusionbehavioroftracerparticles, loidal particles in polymer networks via photon corre-
onetypicallyusesthemeansquaredisplacement(MSD) lation spectroscopy. They observe, that colloids that
(cid:104)|∆r(t)|2(cid:105),wherer(t)istheaveragedisplacementofthe can bind to the polymer strands of the network show
subdiffusive behavior. They explain this by the obser-
FatemehTabatabaei·OlafLenz·ChristianHolm vation, that colloids that are bound to multiple poly-
Institutfu¨rComputerphysik
mer strands behave as though they are a part of the
Universita¨tStuttgart
strand themselves, and thus show the Rouse-dynamics
Pfaffenwaldring27
D-70569Germany of a polymer segment. The authors develop a simple
E-mail:[email protected] model that explains the observations.
2
In our studies, the FCS tracer particles are smaller On the other hand, as it is well known since Flory’s
thanthecolloidalparticlesintheabovework,andthey work [18], the contour of a relaxed polymer strand it-
are assumed to be significantly smaller than a network self can be described as a random walk (RW) or a
meshcell.Furthermore,theydonotbindtothepolymer self-avoiding random walk (SAW) in three-dimensional
strands,althoughtheremightbeattractionbetweenthe space.Itfollows,thattheMSDofamonomer(cid:104)|∆r(t)|2(cid:105)
tracerandthenetworkstrands.Toourbestknowledge, toanothermonomerwithadistancecalongthecontour
therearenotheoreticalstudiesofthissituation.Inthis behaves as
work, we have focused on the behavior of tracer parti-
(cid:104)|∆r(t)|2(cid:105)∝c (3)
cles that are attracted to the polymer network and can
slide along single strands. This gives rise to a number in the case of the random walk, or as
of interesting phenomena. The following thoughts are
closely related to the biophysical problem of proteins (cid:104)|∆r(t)|2(cid:105)∝c56. (4)
sliding along DNA strands. [16,17] inthecaseoftheself-avoidingrandomwalk.Substitut-
ing equation 2 into equations 3 or 4, this results in the
MSD (cid:104)|∆r(t)|2(cid:105) of the tracer particle behaving like
Processes affecting the diffusion behavior
RW:(cid:104)|∆r(t)|2(cid:105)∝t21 (5)
Several processes that can theoretically influence the
diffusive behavior of the tracer particles in the hydro- or
gel’s polymer network can be identified (see Figure 1). SAW:(cid:104)|∆r(t)|2(cid:105)∝t53. (6)
i.e. the diffusion exponents are α= 1 (RW) and α= 3
2 5
(SAW)respectively,whichdenotesstronglysubdiffusive
behaviour.
Equations5and6areonlyvalidinthecaseofcom-
pletelyrelaxedpolymercoils.Intheotherextremecase,
when the polymer is completely stretched and rod-like,
Sliding along Hopping along Attraction to
a strand a strand kinks thespatialdistanceoftwomonomersisidenticaltothe
contourdistanceofthemonomers.Tracerparticlesslid-
ing along such a rod exhibit normal diffusive behavior
(α=1). Consequently, when stretching a polymer and
thus crossing over from relaxed coil states to stretched
rod-like polymer states, a transition from subdiffusive
Hopping between Attraction to
tonormaldiffusivebehaviormusttakeplace.Note,that
strands network nodes
inbothcasesthediffusionexponentappliestoalllength
Fig.1 Schematicsofthedifferentprocessespossiblyinvolvedin
thediffusionofasmall,non-specificallyboundtracerparticlein and time scales.
ahydrogel. When the polymer strand is neither fully stretched
nor fully relaxed, the polymer structure is becoming
more complex. Stretching the polymer chain does not
affectthepolymerstructureonalllengthscalesequally.
Sliding The structure of the stretched polymer on short length
scaleswillbeunaffectedbythestretchingandretainthe
From the point of view of this work, the most inter- random-walk or self-avoiding random walk behavior,
esting process is the process of tracer particles sliding while on longer length scales, the polymer is stretched
along a polymer strand. While tracer particles that are and becomes rod-like. A simple description of this be-
non-specificallyboundtoapolymerstrandcannoteas- haviorisgivenbyPincus’blobpicture.Inthatpicture,
ily be detached from the strand by the random kicks a polymer chain under tension can be seen as a chain
of the solvent particles, they can still perform a one- of blobs, each of which contains a distinct segment of
dimensionalrandomwalkalongthestrand.Inthatcase, thepolymer.Whilethechainofblobsbehavesrod-like,
the particles can be considered as “diffusing” along the each of the blobs itself consists of a polymer segment
contourofthepolymer.Inthisrandomwalk,themean that can still be described well by a RW or SAW. The
contour length c that the tracer has covered after the size of the blobs is determined by the stretching; the
time t behaves as larger the amount of stretching, the smaller the blobs.
(cid:104)c(cid:105)∝t12. (2) [19,20]
3
When applying this picture to the behavior of the processes play a role in the diffusion behavior of tracer
tracer particles that slide along the polymer contour, particles.
theMSDofthetracerparticlescanbeexpectedtoshow When the tracer particles are not very strongly at-
amorecomplexbehavioratdifferenttimescales,which tracted or bound to the polymer strand, they can be
would be characterized by varying diffusion exponents ripped off by thermal motion and diffuse freely in the
atdifferenttimescales.Atsmalltimescales,thetracer solventuntiltheyareadsorbedagainbythesameoran-
particles probe the random-walk-like internal structure other polymer strand. This process of escape and read-
of the Pincus blobs, and the subdiffusive behaviour de- sorptionisreferredtoashopping throughoutthiswork.
scribed above should be observable, with small expo- There are actually two hopping processes; the tracer
nents down to α= 1 (RW) and α= 3 (SAW), respec- particle can be reabsorbed by the same network strand
2 5
tively. At longer time scales, the tracer particles probe as it was ripped off, which is referred to as hopping
thestretchedstructureofthesequenceofPincusblobs, along a strand, or it is absorbed by another polymer
which leads to normal diffusive behavior and an expo- strand, which is called hopping between strands.
nent of α=1. When hopping occurs, the tracer particle diffuses
However, formally, the diffusion exponent α is only freely in the solvent. Compared to the sliding process
defined asymptotically for large timescales, therefore it depicted above, it can be expected that free diffusion
is not very useful in our case. To be able to distinguish has a significantly larger diffusion constant than the
more details of the diffusion behavior at different time sliding process and a diffusion exponent of α=1, as it
scales, we instead examine the time-dependent expo- is a normal diffusive process. Since the diffusion of the
nent α(t) hoppingparticlesismuchfasterthanoftheslidingpar-
ticles, there is the danger that it dominates the overall
dlog(cid:104)|∆r(t)|2(cid:105)
α(t)= (7) diffusive behavior and effectively hides the anomalous
dlogt
behavioroftheslidingprocess.Therelativeimportance
In the case of hydrogels, the degree of stretching ofhoppingtowardsslidingiscontrolledbythefollowing
of the polymer strands is directly influenced by the parameters:
swelling ratio of the hydrogel. While in a dry or only
– The strength of the attractive interaction governs
slightly swollen hydrogel, the strands can be expected
the hopping rate: the stronger the attraction, the
to be mostly relaxed, the strands must be stretched for
lower the probability for the tracer to escape from
strongly swollen hydrogels. This means, that the diffu-
the polymer strand, and the higher the probabil-
sion behavior of non-specifically bound tracer particles
ity to get readsorped. This parameter affects both
should exhibit the characteristics described above and
hopping processes.
should change accordingly when the gel is swollen.
– Thedensityofthepolymerstrandsdirectlycontrols
If this effect occurs in the parameter range of real
the mean free path that the tracer can cover before
hydrogelsandisnotdominatedbyotherprocessesthat
it gets absorbed by another polymer strand, i.e. it
influencethediffusionbehaviorofthetracerparticles,it
influences the effect of hopping between strands.
couldprovideanicewaytoexplorethestructureofthe
– The spatial correlations between different parts of
polymernetworkandtheswellingofhydrogelsviaFCS.
the polymer strands are unequal in the coiled and
Oneshouldnote,however,thatthismethodonlyworks
stretchedstates.Inthecoiledstate,atracerparticle
if the average length of the polymer segments (which
ismorelikelytobereadsorpedontoanotherpartof
corresponds to the average network mesh cell size) and
the same polymer strand, as in the stretched state.
theaveragetimerequiredtoslidealongthesesegments
This means that the process of hopping along a
are larger than the minimal time and length resolution
strand is suppressed by stretching the strand, while
of the FCS technique. For most real hydrogels, this is
hopping between strands should be mostly unaf-
not the case. Only in weakly crosslinked hydrogels, it
fected.
can be expected that the mesh size comes into regions
where this effect might be observable.
Sticking
Hopping It is likely that network nodes or kinks in a polymer
strand, where tracer particles can synchronously inter-
The above discussion only holds for an ideal model act with multiple polymer segments further affect the
where the tracer particles are sliding along the contour diffusive behavior. The strong attraction of the tracer
of the polymer. In reality, however, a number of other particlestothepolymerstrandattheselocationsmight
4
immobilizethetracerparticles,whichreducestheover-
Lz
all diffusion rate.[13] L
contour
Models
central periodic
image image
Starting from the assumption that tracer particles can
slidealongthepolymercontourandthereforeshowsub-
Fig. 2 Sketch of the system with a stretched periodically re-
diffusive behavior, the goal of this work was to study peatedpolymer.Thedegreeofstretchingisdefinedbythenum-
differentinfluencefactorsonthisbehaviorwiththehelp berofmonomersN andthesystemsizealongtheaxisofstretch-
of simulations, to quantify them and their relation to ingLz.
each other, and to distinguish their different origins.
To this end, we have developed a series of coarse-
To generate a polymer of seemingly infinite length
grained,simplifieddiffusionmodelsofthesystemwhere
withvariousdegreesofstretching,thepolymerwaspe-
wesuccessivelyadddetailsandallowmoreandmoreof
riodically repeated along the z-axis of the system, i.e.
the processes depicted above to happen.
the last monomer of the chain was bound to the first
All of the diffusion models used fixed polymer con-
monomer of the chain over the periodic boundary (see
formations, i.e. the polymer did not move throughout
figure 2). Note that this does not mean that any of the
the diffusion simulation. This simplifies the models to
beads was fixed in any spatial dimension throughout
a great extent, as it takes out the influencing factors
the simulations.
of the various parameters of the polymer simulation,
In this model, the degree of stretching depends on
at the cost of not describing correctly the influence of
the number of monomers N and the size of the system
the polymer dynamics on the tracer particles. We be-
along the z-axis L . Given a value of L , the system
z z
lieve that the dynamics of the polymer will not influ-
lengths L =L could be used to control the monomer
x y
ence the diffusive behavior of the tracer particles to a
density ρ of the system.
greatextent,althoughitmightshiftsomeoftheresults
Theunusualsetupoftheperiodicallyrepeatedpoly-
obtained in this work.
mer also had some effects on the Pincus blob picture
Furthermore, in the models used in this work, we
mentioned above. On the one hand, the system size L
z
did not consider the case of a polymer network with
limited the maximal size of a Pincus blob. As the vari-
connections between different polymer strands, but we
ous periodic images were always aligned chain-like, the
always used a single polymer strand of infinite length.
polymeralwaysdisplayedarod-likestructureforlength
Consequently, we did not observe the effect of network
scales larger than L . Accordingly, the time-dependent
z
nodes on the diffusion behavior. We plan to consider
diffusion exponent α(t) always approached 1 when the
both the effect of the dynamics of the polymer as well
tracer particles diffused further than this length scale.
as the effect of a polymer network with network nodes
It should be noted that this was an artifact of the sim-
in our future work.
ulation setup, which limited the maximal Pincus blob
size.Inrealsystemswithfullyrelaxedchains,itshould
in theory be possible to see the subdiffusive behavior
Polymer model
on larger length and time scales. On the other hand,
whenL waschosensmallerthantheexpectedsizeofa
All of the diffusion models used conformations of an z
completelyrelaxedpolymercoil(givenbyFlory’sargu-
coarse-grained bead-spring polymer with different de-
ment), this corresponds to an unphysical compression
grees of stretching. The conformations have been gen-
of the polymer in this direction. Therefore, such states
eratedvia astandard Langevindynamics simulationof
were avoided throughout this work.
a three-dimensional model polymer system with peri-
Two variants of the polymer model were simulated.
odic boundary conditions.
In the first variant, no excluded volume interaction be-
ThepolymerconsistedofN =200beads(or“monomers”),
tweenmonomerswasused,thereforetheconformations
each with mass m = 1, that were bound to each other
inthecoiledstatescorrespondtoarandomwalk(RW).
using a harmonic spring potential to form a chain.
The RW conformations were used only in conjunction
1
V (r)= K(r−r )2 (8) with the strongly bound tracer model described below
harmonic 2 0 to verify the theoretical considerations above. In the
The parameters were chosen to be r = 0.5 and K = second variant, a repulsive Weeks-Chandler-Anderson
0
10.0. (WCA) interaction between the monomers was used to
5
model the excluded volume interaction [21,22], which the degree of stretching (i.e. L ) and the chain length
z
is identical to the repulsive core of the Lennard-Jones N was studied. As the virtual tracer particle could not
interactioncutoffatitsminimum(equation9,ε=1.0, leave the single infinite strand, and the different peri-
σ =1.0). In this case, the conformations corresponded odicimagesofthepolymerstranddidnotinteractwith
to a self-avoiding random walk (SAW) [23]. each other, the monomer density ρ does not play a role
(cid:40) in this model.
4ε((σ)12−(σ)6),if r <(2ε)16
V (r)= r r . (9) The results were used to verify the theoretical con-
WCA 0 ,otherwise
siderationsfromabove.Therefore,themodelwassimu-
The system was simulated with help of the simula- lated both with the RW polymer conformations as well
tion package ESPResSo [24], using the Velocity Ver- as the SAW polymer conformations.
let algorithm with a time step of τ = 0.005 and a
Langevin thermostat with temperature T = 1.0 and
Freely diffusing tracer particles with nearest-neighbor
a friction parameter γ = 0.5. Various simulation runs
with up to 50×106τ in total and 500,000τ between interaction
single polymer conformations were performed, to ob-
On the next level of model refinement, the influence
tain100equilibrated,statisticallyindependentpolymer
of the hopping process onto the diffusive behavior of
conformations for each different value of the number of
the tracer particle was studied. The model was refined,
monomersN,degreeofstretching(i.e.L )andpolymer
z
andasimplemodelofasingletracerparticlewasintro-
density ρ. Note that ρ was always chosen small enough
duced. The tracer particle was represented by a bead
so that the different periodic images of the polymer
of mass m = 1 that interacted with the (fixed) poly-
strand in x or y direction did not interact with each
mer beads via a Lennard-Jones interaction (equation
other. For the given parameters, the expected end-to-
10,σ =1.0,r =2.0,εvaried).Asthepolymerbeads
end distance (and hence the value of L ) of the RW cut
z
were fixed in space, this can also be interpreted as the
polymer is about 12, while for the SAW polymer it is
diffusion of the tracer particle in a system of fixed ob-
about 20 (given the measured average bond length of
stacles.
about0.7).Thismeansthatthepolymerconformations
(cid:40)
at L = 15 (RW) resp. L = 30 (SAW) correspond 4ε((σ)12−(σ)6),if r <r
z z V (r)= r r cut. (10)
to mostly relaxed polymer chains, while polymer con- LJ 0 ,otherwise
formations at L = 100 (RW and SAW) are strongly
z On this refinement level, we still wanted to avoid
stretched.
theeffectofthetracerparticlegettingstuckatkinksof
thepolymerstrand,wherethebeadwouldfeelasignif-
Strongly bound tracer model icantly deeper potential well than close to a stretched
pieceofthepolymer.Toachievethat,wehavemodified
In the simplest diffusion model, only the process of the the interaction such, that the tracer bead only inter-
tracer particle sliding along the polymer contour was acted with the single monomer on the polymer strand
considered. To this end, the frozen polymer conforma- that was closest to the tracer particle. We call this
tionswereusedasabasisforaone-dimensionalrandom the nearest-neighbor interaction. This simple trick pre-
walk along the polymer contour, where the tracer par- vented the tracer particle from getting stuck at kinks,
ticleswerethoughttobestrictlyboundtothepolymer while it still felt the attraction of the strand.
and could not escape. The most important parameter introduced by this
To do that, a random monomer from each confor- model was the parameter ε of the interaction, which
mation was used as a first binding location of the vir- determined the strength of the attraction between the
tual,stronglyboundtracerparticle.Whenadvancinga tracer particle and the polymer and consequently the
“timestep”, either the same, the next or the previous rateofthehoppingprocess.Comparedtoasystemwith
monomer along the chain was randomly chosen as the an all-neighbor LJ interaction, the value of ε had to be
next binding location of the virtual particle, to model chosenmuchlarger,asinthismodelthetracerparticle
the diffusion of the tracer along the polymer. After t always only feels the attraction of a single monomer as
timesteps, the MSD of the binding location was mea- opposed to several monomers at once when being close
suredtodeterminethespatialdiffusionbehaviorofthe to the strand in a normal Lennard-Jones all-neighbor
virtual tracer. interaction. Another parameter that influenced the be-
The model was simulated for the various conforma- haviorofthismodelwasthemonomerdensityρ,which
tions obtained in the polymer model, and the depen- mostlycontrolledtherateofthehoppingbetweenpoly-
dence of the diffusive behavior of the virtual tracer on mer strands.
6
LangevinMDsimulations(timestepτ =0.01,tem-
perature T = 1.0, friction γ = 0.5) of the tracer were
Linear chain
performed.TheMSDandthetime-dependentexponent 10000 Stretched RW polymer (Lz=100)
Stretched SAW polymer (L=100)
z
α(t) of the tracer diffusion was measured. The model Relaxed RW polymer (Lz=15)
wasstudiedforvariousSAWpolymerconformationsat 2<r(t)> 100 Relaxed SAW polymer (Lz =30)
different degrees of stretching L and different chain
z
lengths N, as well as at different monomer densities ρ.
1
TheRWpolymerconformationswerenotconsideredin
this model.
1
It is important to note that there is no relation be-
0,8
tweenthetimescaleinthismodelandthetimescalein α(t)
0,6
the strongly bound tracer. While the time scale in the
0,4
strongly bound tracer model was a pseudo time scale
100 10000
in units of the time that the tracer particle needs to t
diffuse to the next monomer in the polymer chain, the
Fig. 3 Top: MSD of the strongly bound tracer model for both
timeunitinthismodelisdeterminedbytheLJparam- polymermodelvariants(RWandSAW)attwodifferentdegrees
eters ε and σ. ofstretching.Bottom:Time-dependentexponentofthecurvesin
thetopplot.
Freely diffusing tracer particles
SAWpolymersalikeshowvaluesofthetime-dependent
exponentα(t)atsmalltimescalesthataresignificantly
Thelastmodelconsideredinthisworkwasmostlyiden-
less than 1, and approach a value of 1 at longer time
ticaltothepreviousmodelwithnearest-neighborinter-
scales. This behavior is consistent with the picture of
action, but used a conventional all-neighbor Lennard-
the strongly bound tracer particle in a polymer that
Jonesinteraction(Equation10),sothatthetracerpar-
consists out of Pincus blobs – at short time scales, the
ticle interacted with all polymer beads within cutoff
tracerprobesthecoiledstructureofsinglePincusblobs
range. Thus, the effect of the tracer particle getting
andexhibitssubdiffusivebehavior,whileatlongertime
stuck to kinks was included in this model.
scales, the tracer particles feel the linear alignment of
The simulation parameters are similar to the pre-
multiplePincusblobs,whichresultsinnormaldiffusive
vious model. The only difference is the depth of the
behavior.
Lennard-Jones potential ε, which had to be chosen sig-
In the case of the stretched polymers (both RW
nificantly lower in this model to exhibit a comparable
and SAW), the time-dependent exponent quickly ap-
effect.
proaches a value of 1 within a few orders of magnitude
of the time scale, while in the case of the relaxed poly-
mers, it stays at values of α(t)≈0.5 (RW) or α(t)≈ 3
Results 5
(SAW)overmanyordersofmagnitude.Thisisalsowell
Strongly bound tracer model described by the Pincus blob picture, which predicts
that even while the overall structure of the polymer is
The upper subplot of figure 3 shows the MSD of the unaffected by stretching the polymer, the single blobs
tracer particles in the strongly bound tracer model, for are largest for relaxed polymer coils, while for higher
both RW and the SAW polymers as well as for two dif- degrees of stretching the blobs become smaller. Note,
ferent representative values of stretching of the chain, that even in the maximally relaxed polymer coils in
while the lower subplot shows the time-dependent ex- ourmodelthetime-dependentexponentα(t)hastoap-
ponent α(t) in the same cases. While for the relaxed proach a value of 1 for long time scales t, as the Pincus
polymer conformations Lz was close to the expected blobislimitedbythesystemsizeLz asdescribedabove.
end-to-end distance (L = 15 in the RW and L = 30
z z
in the SAW case), for the stretched conformations it
was rather large with L =100. Freely diffusing tracer particles with nearest-neighbor
z
Asexpected,theplotsshowaconstantdiffusionex- interaction
ponent α(t) = 1 for a linear chain of monomers, while
for the polymer chains it varies between 1 < α(t) < 1 In order to investigate whether it is at all possible to
2
forthepolymerintheRWcaseand 3 <α(t)<1inthe observe an effect of the polymer structure on the diffu-
5
SAW case. Interestingly, all of the curves for RW and sivebehaviorofthetracerparticlesinthetracermodel
7
est to this work, therefore none of the figures following
free diffusion figure 4 show this regime.
10000 random beads, ε=5
rraenladxoemd pboelaydms,e εr=, L10=30, ε=5 At relatively low interaction strength ε = 5, only
2<r(t)> 100 relaxed polymer, Lzz=30, ε=10 athevedryiffumsiinvoerbeeffheacvtioorfisthoebrsaernvdeodm, aonbdsttahceleMbSeaDdsploont
1 shows the same two clearly distinct regimes – a short-
timeballisticregime,andalonger-timenormaldiffusive
0.01 regime. The differences can be attributed to the beads
2
acting as simple obstacles that hinder the diffusion of
1.5
the tracer and thus reduce the diffusion constant. This
α(t) 1 also holds in the case of the relaxed polymer chain at
0.5 low interaction strength, where the MSD is virtually
0 identical to the random bead system. No effect of the
1 10 100 1000
t polymer structure can be recognized.
Fig. 4 MSD and time-dependent exponent of the freely diffus- When the interaction strength is increased, the ef-
ing tracer model with nearest-neighbor interaction in a system fect of the random beads hindering the diffusion be-
without any obstacles (“free diffusion”), in a system of random comes more pronounced, up to a level where the nor-
beadsattwodifferentinteractionstrengths,andinasystemwith
mal diffusion is so slow that between the ballistic and
arelaxedpolymerchainattwodifferentinteractionstrength.The
density of the random beads was chosen to match the polymer normal-diffusiveregimeashortintermediateregionap-
resultsasdescribedinthetext. pears,wherethetime-dependentexponentdropsbelow
1.Thisregionshouldnotbeinterpretedasarealsubdif-
fusiveregime,insteaditismoreofanartifactcausedby
with nearest-neighbor interaction, we have performed thecombinationoftwonormaldiffusionprocesseswith
a few simulations of systems that contained randomly very different length and energy scales: the diffusion
distributedbeadsasobstacles.Ifthepolymerstructure within the potential of a single bead is very fast, but
has an effect, the diffusion behavior should show dis- the tracer is strongly bound to the bead (“rattling in a
tinct differences in such a system when compared to a cage”). When the tracer particle escapes, it can diffuse
system of random beads. To be able to do such a com- fast and freely until it is caught by another attractive
parison, it is important to note that the beads in the bead(“hoppingbetweencages”).Lookingattheoverall
chainhaveasignificantoverlap,whilerandombeadsdo diffusion,thisresultsintheobservedbehaviorwiththe
not.Therefore,thevolumefractionofthesystemwhere pseudo-subdiffusive intermediate region.
thetracerparticleinteractswithanobstaclebeadissig- In the case of the relaxed polymer chain at high
nificantlylargerinthecaseoftherandombeadsthanin enough interaction strength, the effect is different. As
the case of the polymer chain at similar obstacle bead in the case of the random beads, an intermediate re-
density. To be able to see whether the actual structure gion occurs between the ballistic and normal-diffusive
of the polymer has a distinct influence on the diffusive regimes,wherethetime-dependentexponentisbelow1.
behavior, it is therefore necessary to match the volume In contrast to the random bead system, however, α(t)
affected by the obstacle beads rather than the density. is mostly constant and below 1 for several orders of
The MSD and time-dependent exponent of tracer magnitude, so that we can speak of a real subdiffusive
particlesofdifferentsystemsofrandombeadsandare- regime.
laxedpolymerchainareplottedinFigure4.Theseplots When examining the MSD plot in more detail, it
differ significantly from the corresponding plots in the canbeobservedthatinthecaseoftherelaxedpolymer
strongly bound tracer model. The most obvious differ- chain, this subdiffusive regime spans time scales of up
enceisthatatveryshorttimescalesoft<1,theMSD toatleast1000timeunitsandthecorrespondinglength
seemstoexhibitstrongsuperdiffusionwithanexponent scale of up to a few tens of bead diameters, indicating
of up to 2, that was not visible in the strongly bound that the tracer bead really slided along the polymer
tracer model. This behavior is an artifact of the tracer contour for a few tens of monomers.
MD simulation; even in the case of free diffusion, on Plotting the MSD and time-dependent exponent is
these short time scales, the tracer particles propagated very useful to look at the overall diffusion behavior. A
freelyandexhibitedballisticmotionwithadiffusionex- more detailed way to visualize what is going on in a
ponent of 2, before the Langevin thermostat destroyed systemistoplotthedisplacementprobabilitydistribu-
the velocity autocorrelation of the tracer particles and tion P(r) at different times. As we are mainly inter-
the normal diffusion sets in. This effect is of no inter- ested in the diffusion of the tracer particles along the
8
10000
t=655 t=1311 random beads Lz=100
relaxed chain, Lz=30 Lz=80
stretched chain, Lz=100 Lz=40
free diffusion (analytic) L=30
z
10-1 2<r(t)>100 Lz=15
P(z)
10-2 1,11
1
0,9
α(t)0,8
0,7
10-3 -40 -20 0 20 40 -40 -20 0 20 40 0,6
z z 1 10 t 100 1000
Fig. 5 Probability distribution of the tracer displacement in z Fig.6 MSDandtime-dependentexponentofthefreelydiffusing
direction at two different times. Other parameters: interaction model tracer with nearest-neighbor interaction for different de-
strength ε=10, density ρ=0.0084. The free diffusion case was greesofstretchingofthepolymeratconstantdensityρ=0.0084
computedanalyticallyusingEquation11. andinteractionstrengthε=10.
z-axis of the system, Figure 5 depicts the displacement the size of the intermediate regime seems to depend on
probability distribution P(z) for different systems at thedegreeofstretchingoftheattractivepolymerchain
different time scales in the intermediate regime where in the system; for mostly stretched chains, the normal-
the tracer particles exhibit subdiffusion in some of the diffusive regime is restored after a few orders of mag-
systems.Inthecaseoffreediffusion,wecaneasilycom- nitude of the time scale, while for relaxed chains, the
putethedisplacementdistributionfunctionanalytically MSD shows subdiffusive behavior for many decades. In
as in Equation 11.[25,26] When plotting the measured the most relaxed chains, the normal-diffusive regime is
displacement probability in the free diffusion case, it restored at about the length scale of L , which corre-
z
is indistinguishable from the analytically derived func- sponds to the fact that the Pincus blob size is limited
tion. by L . Therefore, this behavior is again well consistent
z
1 (cid:18)−z2(cid:19) with the Pincus blob picture.
P(z,t)= √ exp . (11)
4πDt 4Dt
In the plot, we can discern the qualitative differ-
ences between the various systems. While the free dif- ε=8
1
fusion case shows Gaussian behavior on all time scales,
in the other systems we can observe a more complex α(t)0,8
L=100 (stretched)
z
behavior.Thedistributionsintherandombeadsystem L=30 (relaxed)
z
0,6
are characterized by a sharp peak at z =0 with a very ε=9
1
small width and lower, much broader shoulders. This
behavior is consistent with a superposition of a sharp α(t)0,8
peakandabroaderGaussian,wherethepeakoriginates
0,6
ε=10
from the tracer particles being caught in the potential 1
of a random bead, while the smaller but significantly α(t)0,8
broader Gaussian comes from the free diffusion of par-
ticlesthathaveescapedthebeadpotentials.Thedistri- 0,6
10 100 1000
t
butions measured in the polymer systems look distinc-
tivelydifferentfrombothoftheseextremes.Noneofthe
curveslookslikeaclearsuperpositionoftwoGaussians, Fig. 7 Time-dependent exponent for two different degrees of
but none of them really looks like a Gaussian either. stretching (Lz =30 and Lz =100) at different values of the in-
teractionparameterεandconstantmonomerdensityρ=0.0084.
To determine the effect of the degree of stretching
of the polymer on the diffusion behavior, we have per-
formed a number of simulations for varying degrees of Further simulations have been performed for vary-
stretching at constant density ρ=0.0084 and constant ing values of the interactions strength ε (Figure 7) and
interactionstrengthε=10(Figure6).Weobserve,that the monomer density ρ (Figure 8), to be able to quan-
9
1e+06
1 ρ=0.00272 Lz=100
α(t)00,,681 ρ=0.LL0zz0==8134000 ( r(esltarxetecdh)ed) 2<r(t)>10010000 LLLzzz===843000
α(t)0,8
1
0,6
ρ=0.0148
1
α(t)0,8 0,012
0,6 ρ=0.0213 1,5
1
α(t)0,8 α(t) 1
0,5
0,6
10 100 1000
t 0
1 10 100 1000
t
Fig. 8 Time-dependent exponent for two different degrees of Fig.9 MSDandtime-dependentexponentofthefreelydiffusion
stretching (Lz = 30 and Lz = 100) at different values of the modeltracerfordifferentdegreesofstretchingofthepolymerat
monomerdensityρandconstantinteractionparameterε=10. interactionstrengthε=2anddensityρ=0.0084.
tify the influence of these parameters on the diffusion has a kink, and can easily be trapped in them for long
behavior.Fromtheplots,itcanbeconcludedthatlow- times (see Figure 10). Since the potential well caused
ering the interaction strength ε suppresses the effect of by the kink is very narrow, it affects the ballistic be-
the polymer chain, as does lowering the density ρ. As havior of the tracer particle on very short time scales,
discussed in the introduction, this can be traced to the evenbeforethevelocityautocorrelationisdestroyedby
influence of the hopping process which dominates the the Langevin thermostat.
diffusion along the polymer. Although the qualitative
effect of varying the interaction strength ε (Figure 7)
is similar to varying the density ρ (Figure 8), quantita-
tivelytheinteractionstrengthhasasignificantlyhigher
impact on the subdiffusive regime, where varying ε by
20% has a similar effect to varying ρ by factor 5. We
interpret this along the lines of what was discussed in
the introduction – while varying ρ only has an effect
onthelessimportantprocessofhoppingbetweenpoly-
merstrands,theinteractionstrengthhasaneffectonall
hoppingprocesses,bothonthemoreimportantprocess
ofhoppingalongthesamestrandaswellasonhopping
between strands.
Freely diffusing tracer particles
Figure 9 depicts the MSD and time-dependent expo-
nent of freely diffusing tracer particles at different de-
grees of stretching L at a density of ρ=0.0084. Com-
z
paring these plots to the corresponding plots in the
modelwithnearest-neighborinteractionrevealsanum-
ber of profound differences. The ballistic behavior seen Fig.10 Simulationsnapshotofthetracerparticlesgettingstuck
inthekinksofacoiledpolymer.
at short times disappears after even shorter times than
in the nearest-neighbor interaction model. We think
that the effect is caused by the introduction of kinks Curiously, the plots of the time-dependent expo-
intothemodel.Whenemployingtheusualall-neighbor nent at low density still allows to distinguish between
Lennard-Jonesinteraction,thetracerfeelsthedeeppo- the different degrees of stretching. The reason for this
tential wells along the polymer chain where the chain is, that the number of kinks is significantly larger for
10
a coiled polymer chain than for a stretched polymer teraction model, only that in this case the cages are
chain. Therefore, for very coiled polymer chains, α(t) represented by the kinks rather than by single beads.
drops much faster than for a stretched chain. This is, Unfortunately, the plots do not allow to celarly de-
however, not the effect of the tracer particles sliding cide whether subdiffusive behavior caused by the trac-
alongthepolymercontourandshallnotbefurthercon- ers sliding along the polymer contour is visible within
sidered in this work. the system. Since the densities used for producing the
The subdiffusive regime that was observed in the dataofthesefiguresareveryhigh,anexperimentalver-
nearest-neighbor model at similar densities has com- ification of these features seem to be very improbable.
pletely disappeared. From the ballistic regime at very
shorttimes,thetime-dependentexponentdirectlygoes
over into a normal diffusive regime. We believe that in
Conclusions
these systems the anomalous diffusion caused by the
tracer particles sliding along the chain is on the one
In this work, we have investigated the effect of poly-
hand completely dominated by the process of particles
mer strands (as they might be found in a hydrogel) on
being immobilized by the kinks, and on the other hand
the diffusion behavior of small tracer particles (as they
bytheprocessofhopping,andthereforeitisnotvisible
are used in FCS) that are attracted to the polymer
in these graphs.
strands. From theoretical considerations, we concluded
that when such tracer particles slide along the contour
of a polymer strand, they will exhibit subdiffusive be-
1e+06
havior due to the fractal nature of the polymer.
Lz=100 To investigate this situation, we have devised a se-
10000 Lz=80
Lz=40 ries of simplified models with increasing complexity. A
L=30
2<r(t)> 100 z stiicmlepslethnautmweerriceasltsriomnugllyatbioonunodfatomforodzeelnwcitohnftorramceartpioanrs-
1 of a polymer strand proved the soundness of the the-
oretical assumptions. In plots of the mean square dis-
0,01 placement(MSD)andthetime-dependentexponent,it
2
was observed, that when the polymer is stretched (i.e.
1,5 whenthehydrogelisswollen),thisaffectsthestructure
α(t) ofthepolymercontourandthesubdiffusivebehaviorof
1
the sliding tracer particles. This is best rationalized in
0,5 1 10t 100 1000 terms of Pincus blobs. When the polymer strands are
mostly relaxed, the blobs are relatively large, and the
Fig. 11 MSD and time-dependent exponent of the freely diffu- subdiffusive behavior of the tracer particles should be
sionmodeltracerfordifferentdegreesofstretchingofthepolymer
observable over many decades of the time scale, while
atinteractionstrengthε=2anddensityρ=0.1.
when the polymer strands are stretched, the Pincus
blobs become small and the subdiffusive behavior can
To increase the importance of tracer particles slid- only be observed on very short times.
ing along the chain over the process of hopping and In reality, tracer particles cannot be expected to
sticking, we have increased the density to significantly be so strongly bound to the polymer. Instead, thermal
higher values of ρ=0.1, which should decrease the im- kicks induced by the polymer motion most probably
portanceofhopping.TheMSDandtime-dependentex- will aid their desorption and particles can be expected
ponentareplottedinFigure11.Intheseplots,thetime- to diffuse freely between the polymer strands, which is
dependentexponentα(t)dropsbelow1inanintermedi- referred to as “hopping”. The influence of this process
ate region, which might indicate subdiffusive behavior. was investigated by the means of a simple molecular
However, the subdiffusive behavior already sets in at dynamics simulation model, where the tracer particles
veryshorttimes,wherethetracerparticleshavenotyet interacted with the frozen polymer conformations via
diffusedevenasinglebeaddiameter.Thereforeatleast a nearest-neighbor Lennard-Jones interaction. We ob-
this part of the subdiffusive region can not originate in served, that the hopping process easily dominates the
the tracer particles probing the chain contour. Instead, overalldiffusionbehavior,asithasamuchhigherdiffu-
it can be assumed that it is a result of a “hopping- sionratethantheprocessofslidingalongthepolymer.
between-cages”-process similar to the effect observed To be able to see the subdiffusion caused by the tracer
in the random bead system in the nearest-neighbor in- particlesslidingalongthepolymer,themostimportant
