Confining Potential as a Function of Polymer Stiffness andConcentration in Entangled Polymer SolutionsMasoumeh Keshavarz,*,†,‡ Hans Engelkamp,*,† Jialiang Xu,‡ Onno I. van den Boomen,‡ Jan C. Maan,†

Peter C. M. Christianen,† and Alan E. Rowan‡

†High Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules and Materials, Radboud University, Toernooiveld 7,6525 ED Nijmegen, The Netherlands‡Institute for Molecules and Materials, Department of Molecular Materials, Radboud University Nijmegen, Heyendaalseweg 135,6525 AJ Nijmegen, The Netherlands

*S Supporting Information

ABSTRACT: We directly track the tubelike motion ofindividual fluorescently labeled polymer molecules in aconcentrated solution of unlabeled polymers. We use a singlemolecule wide-field fluorescence microscopy technique that isable to determine characteristic properties of the polymerdynamics, such as the confining potential, the tube diameter,and the Rouse time. The use of synthetic polymers allows us toinvestigate the confined motion of the polymer chains not onlyas a function of polymer concentration (mesh size) but alsoversus the persistence length of the matrix polymers. Althoughthe polymers used have a persistence length much smaller than their contour length, our experimental results lead to adependence of the tube diameter on both the mesh size and the persistence length, which follows the theoretically predictedrelation for semiflexible chains.

■ INTRODUCTION

One of the key questions in polymer physics is how thecharacteristics of the individual chains in a polymeric materiallead to their collective properties. When the individual chainsbecome sufficiently long, entanglement occurs. Entanglementsgenerate topological constraints on polymer conformations anddynamics, which arise from the mutual uncrossability ofpolymers and cause a significant change in the viscoelasticbehavior of the polymer bulk. This phenomenon is generallydescribed using the tube model proposed by De Gennes1 andEdwards and Doi.2 In their model, a single chain moves in awormlike fashion within a confined tubelike pathway defined bythe transient network of entangled neighboring chains. Theentanglements restrict the accessible configuration space andimpose a tubelike domain around the test chain, therebysuppressing its transverse motion. This tube roughly follows theshape of the test chain.Despite the phenomenological character of the tube model,

its basic presumption that the polymers wriggle around in tubesin the presence of entanglements has been validated in ahandful of experiments.3−10 A tube surrounding an entangledpolymer is not permanent because it is formed by other mobilechains leading to random changes in its surface, and thereforestochastic fluctuations in the average tube diameter occur. Inorder to replace the Edwards’ static tube picture with a “soft”tube, a dynamic confining potential with a profile of theharmonic potential was introduced.11,12

The confined motion of polymers has been studied usingseveral experimental techniques from bulk to the singlemolecule level. Examples of bulk studies are rheologymeasurements,13−16 fluorescence autocorrelation spectrosco-py,17−22 neutron23−28 and light29−32 scattering where theinformation obtained is averaged over spatial distribution of themolecules, molecular heterogeneity, that is, chemical com-pounds, distribution of molecular sizes, and conformation.Therefore, it is extremely difficult to study the confiningpotential and tube width fluctuations for individual polymersusing bulk methods. In contrast, single molecule experimentsallowed to extract quantitative information on the tube widthfluctuations, as was done by imaging F-actin,12,33,34 asemiflexible biopolymer. The investigation of crowded polymerenvironments were carried on theoretically in order to furtherunderstand the confined tube-motion for semiflexible chainsystems35−39 as well as entangled rod polymer solutions.40,41

There have been a few experimental studies on the shape of theconfining potential for flexible chains such as DNA42 wheretheir results are in agreement with simulations.43−45 Until now,experimentally very little was known about the transientconfining potential through which the polymer chain moves, inparticular at the single chain level.

Received: December 16, 2016Revised: May 14, 2017Published: May 15, 2017

Article

pubs.acs.org/JPCB

© 2017 American Chemical Society 5613 DOI: 10.1021/acs.jpcb.6b12667J. Phys. Chem. B 2017, 121, 5613−5620

In this paper, we undertake an experimental study onsynthetic polymers in order to investigate the dynamics of anindividual polymer and its relation to material dynamics. Wedirectly track the motion of a labeled polymer in a matrix ofunlabeled polymers by single molecule wide-field fluorescencemicroscopy as we described previously.3 Using syntheticpolymer polyisocyanide with unique tunable properties (viz.can be very long in length in the order of micrometers and thepersistence length lp can be tuned in the range of 4−200 nm ingeneral for different derivatives (the lp of the compounds usedin this work varies between 42 and 138 nm)) allows us toinvestigate the confined motion of polymer chains versusdifferent parameters such as lp. The polymer motion can beseparated into displacement components parallel and perpen-dicular to the tube. From the probability distribution of theperpendicular displacements, we extracted the confiningpotential and the effective restoring force constant that enabledus to probe the functional dependence of the tube diameter onthe mesh size and the persistence length independently.

■ MATERIALS AND EXPERIMENT

The synthetic polymers we employed in our study arederivatives of polyisocyanopeptides shown in Figure 1a. Thepolymers used as the unlabeled matrices are tri- andtetraethylene glycol functionalized isocyano-D-alanyl-L-alanines(3,4EG-L,D-PIAA, 1), triethylene glycol functionalized isocya-no-L-alanines-D-alanyl-L-alanines (3EG-L,D,L-PIAAA, 2) andpoly(isocyano-L-alanyl-D-alanine methyl ester) (L,D-PIAA, 4).The poly(isocyano-L-alanine propanylperylene diimide) (L-PIAP, 3) was used as the labeled chain. These polymersadopt a helical conformation with four repeat units per turnstabilized by hydrogen bonds between the amide groups. Arepresentation of the schematic structure of polymer 4, forexample, is shown in Figure 1b. The polyisocyanopeptides weresynthesized following the procedure previously reported.46 Theas-prepared gelatinous solution was used as the matrix for thesingle molecule studies. The fluorescent polymer 3 wasprepared applying an established procedure47,48 and stored insolution at 4 °C. The average polydispersity index (PDI) ofpolyisocyanides is 1.6−2.49,50The polyisocyanopeptides have a high molecular weight and

therefore, unusually long chains (up to 20 μm in length) andare extremely stiff for a synthetic polymer. The persistencelengths of the studied polymers have been extracted fromatomic force microscopy (AFM) measurements51 to be lp = 42± 6 nm for 1, lp = 129 ± 6 nm for 2, and lp = 76 ± 6 nm for 4.The persistence length of the labeled polymer 3 is lp = 138 ± 6nm.3 For more information on the AFM images and the modelused for extracting the persistence length of the chains, see theSupporting Information. Although the chains are very stiff, theircontour length L is still much larger than their persistencelength L ≫ lp. Hence, our system should be considered toconsist of flexible chains supported by the fact that their end-to-end distance follows a Gaussian distribution function3 inagreement with flexible chain systems.2

In the flexible chain model, the mesh size can be calculatedusing the plateau modulus, GN

0 , obtained from a rheologymeasurement as

ξ =k TG

B

N03

(1)

where kBT is the thermal energy.1 Using the MIN method,52

the plateau modulus was extracted from the storage (G′, closedsymbols, Figure 2) and loss (G″, open symbols, Figure 2)modulus for 2 at concentrations of 5 (pink left-handedtriangles), 6.5 (blue triangles), and 8 mg mL−1 (black circles).The studied concentration range was limited to a concentrationin which we achieve entanglement as the lower limit. The upperlimit is defined when the labeled chain is observed to be static.The plateau modulus and the mesh size of matrix 1 atconcentration of 8 mg mL−1 was calculated using eq 1 and allthe mesh sizes and plateau modulus are summarized in Table 1.Moreover, the average length (Lav) of the unlabeled matriceswas estimated with AFM contour measurements53 and they arelisted in Table 1.We prepared the samples by mixing the labeled polymer

chains with the unlabeled matrix at a ratio of 1:106. The

Figure 1. Chemical structure of unlabeled (1) 3,4EG-L,D-PIAA and(2) 3EG-L,D,L-PIAAA, (3) labeled L-PIAP with perylenediimide aschromophore, and unlabeled (4) L,D-PIAA. (b) Schematic representa-tion of a polyisocyanide, for example, polymer 4 where the dotted linesrepresents hydrogen-bonds that stabilizes their secondary helicalstructure.

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samples were photoexcited with a laser (λ = 543 nm) and thefluorescence emission was collected through an objective lens(100×; NA 1.3; resolution 270 nm for emission at 580 nm) andimaged with a charge-coupled device camera. The experimentalresults were real-time movies. In fact, the motion is occurring inthree-dimension (3D) but the information we extract from themovies is in two-dimension (2D). To minimize the 2D/3Deffect, we selected the movies in which the probe polymerdisplays a visually in-plane motion (showing quasi two-dimensional dynamics) for further analysis. These movieswere analyzed using homemade software, as described in ref 3.Each raw frame was analyzed individually. The procedure isillustrated in Figure 3. Because the raw images taken via time-resolved fluorescence microscopy are noisy (Figure 3a), using amedian filter, smoothing, and Gaussian filters, their backgroundwas removed (Figure 3b). Then, a 2D Gaussian function wasused to fit the position of the fluorescent emitters in theresulting image (the fitted image in Figure 3c). With ouranalysis method, we defined the contour of the chain by spline-interpolation of the grouped fitted 2D Gaussians along thepolymer chains (black solid line in Figure 3c).3 This fittingprocedure improves our resolution down to 30−50 nm andenables us to have access to the end-to-end distance R, contourlength L (without small-scale fluctuations) and the middlepoint of the polymer chain M as a function of time (Figure 3c).For more information, about the fitting procedure and theresolution obtained, see ref 3. The resulted contour of the chainL represents a coarse-grained view of the chain where the small-scale fluctuations are omitted. When averaged over times longerthan the Rouse time, this is related to the primitive path in the

Doi−Edwards theory.2 The primitive path lies along the centerline of the tube where the chain winds around it. Thesuperposition of the primitive paths during time represents thetube like motion that can be resolved into parallel d∥ andperpendicular d⊥ displacement components. In order to extractd∥ and d⊥, we have considered 101 points along each chain outof which 50 of both ends are excluded (25 points of each side)to avoid chain free-end effects.3 The perpendicular displace-ment in time was calculated as the distance between themiddle-point of the chain in frame t2 to the nearest point of thechain in frame t1, for all pairs of frames in a movie. Thelongitudinal displacement was then calculated using the center-to-center displacement and Pythagoras’ theorem. The trajecto-ries were analyzed with a frame rate of 0.1 s in the time scalesabove the entanglement time (when the segments of the chainstart to feel the restrictions imposed by the entanglements) andfar below the disentanglement time τd (the time it takes for achain to leave its original tube completely). Figure 4a showsrepresentative snapshots of an example of a movie recordedfrom a labeled polymer chain 3 in a solution of unlabeledmatrix 2 with a concentration of 8 mgmL−1 (this movie can befound in the Supporting Information). The fluorescent polymerchain can be described as moving in an imaginary tube(indicated with lines next to the chain) formed by its unlabeledneighbors. The motion starts in the first row at t = 0. In eachrow, the motion occurs from left to right and continues to thenext row following a path indicated by the green lines in Figure4b where we superimposed the conformations of thefluorescent polymer chain contour for the indicated moviebetween t = 0 s and t = 80 s. The experimental parameters thatcan be determined from these kinds of measurements are thelength scales such as the tube diameter a and the time constantssuch as the Rouse time τR (the time it takes for a chain to movea distance in the order of its length) and disentanglement timeτd and also the number of entanglements per chain Z followingthe procedure outlined in ref 3.Our work is divided in two parts. In the first part, we study

the effect of concentration and consequently mesh size ξ on thetube diameter a using a solution of matrix 2 in tetrachloro-

Figure 2. Storage (G′, closed symbols) and loss (G″, open symbols)modules of the 3EG-L,D,L-PIAAA in tetrachloroethane (5 mgmL−1,left-handed triangles; 6.5 mgmL−1, triangles; and 8 mgmL−1, circles) asa function of angular frequency.

Table 1. Concentration, Plateau Modulus, Mesh Size, Persistence Length, Tube Diameter, Number of Entanglements PerChain, and Average Polymer Length (Lav)

a

polymer c (mg mL−1) GN0 (Pa) ξ (nm) lp (nm) a (nm) Z# Lav (μm)

3EG-L,D,L-PIAAA 5 3.0 ± 0.2 111.0 ± 0.1 129 ± 6 218 ± 25 11 ± 1 1.23EG-L,D,L-PIAAA 6.5 5.0 ± 0.3 93.8 ± 0.1 129 ± 6 173 ± 18 12 ± 1 1.23EG-L,D,L-PIAAA 8 7.3 ± 0.4 82.6 ± 0.1 129 ± 6 158 ± 30 13 ± 1 1.23,4EG-L,D-PIAA 8 10.9 ± 0.5 72.3 ± 0.2 42 ± 6 208 ± 9 8 ± 2 0.8L,D-PIAA 5 234 ± 12 26 ± 4 76 ± 6 149 ± 19 14 ± 1 0.5

aFor 3EG-L,D,L-PIAAA for different concentrations, as well as for 3,4EG-L,D-PIAA at a concentration of 8 mgmL−1, and for L,D-PIAA at aconcentration of 5 mgmL−1.

Figure 3. (a) An example of a raw image. (b) A background subtractedand smoothed fluorescence image. (c) Fitted image reconstructedfrom the fitting procedure and the chain contour (the black linerepresents the contour of the polymer neglecting its small scalefluctuations). R is the end-to-end distance and L is the coarse grainedcontour length of the chain.

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ethane at concentrations of 5, 6.5, and 8 mg mL−1 as unlabeledmatrix. In the second part, the dependence of the tube diameteron the polymer persistence length was studied applying matrix1 and 2 as the unlabeled matrices. Combining the results ofthese two independent studies and adding the extra pointextracted from the experiments on matrix 4 led to acharacteristic relation for a, lp, and ξ for flexible chains. Itshould be noted that the probe chain was always 3, which has asimilar chemical structure and properties to the matrices usedin our study, allowing genuine reptation to be observed.

■ RESULTS AND DISCUSSIONThe confining potential was extracted from the probabilitydistribution of the perpendicular component of the displace-ments (P(d⊥)) of a single chain (Figure 5, symbols). Theensemble-averaged dynamical confining potential V(d⊥) thatthe chain segments feel, was extracted using V(d⊥) = −kBT lnP(d⊥);

12 see inset of Figure 5 (symbols). Previous experimentalstudies have shown that P(d⊥) follows a Gaussian distributionfor the short-range displacements and possess an exponentialtail12 for large displacements. Therefore, a Gaussian,

π− ⊥e

wd w2

2/22 2

, was fitted to the first logarithmic decay of

P(d⊥) (black curve in Figure 5) where w is the width of theGaussian. The confining potential extracted from the Gaussianpart of P(d⊥) corresponds to the classical harmonicconstraining potential, V(d⊥) ∝ d⊥

2 (quadratic in the short-range perpendicular displacements) represented by the fittedparabola (red solid line) in the inset of Figure 5. The width ofthe fitted Gaussian (w) is the tube radius, that is, the distancebeyond which the constraining potential exceeds kBT. From the

confining potential V(d⊥), the restoring force ∝ ⊥

⊥F V d

dd ( )

d, the

restoring force constant ∝⊥

K Fd

dd

, and the tube diameter a can

be obtained. Because we have access to the full time-range of

the reptation motion of the chains from the Rouse to thedisentanglement time, one can study the time-dependence ofthe confining potential V(d⊥). Thus, we extracted the confiningpotential for several time intervals. In Figure 6b, we plot theconfining potential as a function of the perpendiculardisplacement for different time intervals of 0−0.5, 0−1, 0−1.5, 0−2.5, 0−5, 0−10, and 0−20 s for matrix 2 at theconcentration c = 6.5 mgmL−1. The confining potential wasextracted from the Gaussian part of the probability distributionof the perpendicular displacements P(d⊥) for each time interval.The fitted parabolas (solid lines) are representative examples ofthe harmonic potential, V(d⊥) ∝ d⊥

2 . The widening of theparabolas points to a weaker confinement when approachingthe tube renewal time. We think that this change in V(d⊥) ∝ d⊥

2

could be an indication for the “soft” tube, suggesting that thetube width is dynamic and changes with time. This changecontinues until the Rouse time is reached (here τR = 2.5 s) andfor larger time intervals, it seems that the concept of the tube isvanishing while approaching the disentanglement time and acompletely new tube is formed. A similar anharmonic tubesoftening for large strains on DNA has been observed byRobertson et al. using optical tweezers,42 which is in agreementwith simulations.43−45 In another study, the tube hetero-

Figure 4. (a) Snapshots of a labeled chain 3 in a solution of unlabeledmatrix 2 with a concentration of 8 mgmL−1. The overall tube wasobtained by superposition of the snapshots corresponding to the time-scales indicated. The motion starts t = 0 s and one can follow themotion from left to right in each row up to t = 80 s. The green linesare the imaginary tube defined from panel b. (b) The superimposedconformations of the fluorescent polymer chain contour for the moviebetween t = 0 and t = 80 s. The color code indicates the time.

Figure 5. (a) The probability distribution of transverse fluctuations fora polymer chain moving in matrix 2 at a concentration of 5 mg mL−1

(symbols). The solid line is a fitted Gaussian with the width w = a/2.The large range normal fluctuations deviate from the Gaussian andfollow an exponential tail. Inset: the confining potential felt by a singlechain (solid line) calculated using the transverse fluctuationsdistribution plotted as a function of the distance normal to theprimitive path (symbols, for more detail see text), (b) Confiningpotential as a function of time for matrix 2 at c = 6.5 mgmL−1

considering different time intervals (color codes). The solid lines arethe fitted parabola to the potentials: the Gaussian parts of theprobability distributions.

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geneities were explored theoretically on confined, weaklybendable rods and the shape predicted for the confiningpotential shows both the harmonic and anharmonic parts54 thatfit to our observations for polyisocyanides. In Figure 6a, we plotthe dynamical confining potential V for three concentrations ofmatrix 2 (5 (black squares), 6.5 (red circles) and 8 mg mL−1

(green triangles)). As a consequence of the higher obstacledensity present at higher concentrations, a faster decay ofcorrelations is observed. The effective restoring force constantK was extracted from the confining potential in Figure 6a andplotted as a function of concentration in Figure 6b. As shown inFigure 6c, the extracted tube diameter decreases with increasingpolymer concentration with an exponent of −0.6 as predictedby the Semenov law a ∝ c(−3/5).55 For each concentration, thesame measurement has been carried out for at least 10 samplesand all the parameters such as a, τR, and τd were extracted. Thetube diameter a presented in Figure 6c is the averaged valuewith the error bars being the standard deviation.The Rouse and disentanglement time (see Appendix) for

chains of different lengths are shown in Figure 6d,f forconcentrations of 5 (black squares in Figure 6d,f), 6.5 (redcircles in Figure 6d,f), and 8 mg mL−1 (green triangles in Figure6d,f). As predicted by the reptation theory,1,2 the Rouse time

scales with the polymer contour length as τR ∝ L2. We observeddifferent intercepts for the Rouse time versus the chain contourlength at different concentrations, pointing to an increase of τRversus concentration, which for L = 3 μm is shown in Figure 6e.The disentanglement and Rouse time follow the relation τd =3ZτR, where Z is the number of entanglements per chain.3 Zwas obtained from Figure 6f in which we plot τd as a function ofτR for three different concentrations. In Figure 6b,e, we foundan increase of K and τR with increasing concentration. τRincreases by a factor of 2.5 from 5 to 6.5 mg mL−1 and by afactor of 1.2 from 6.5 to 8 mg mL−1 (Figure 6e). This suggeststhat at higher concentrations, the Rouse and reptation timeincrease much slower. In Figure 6b, a slow increase of K is alsoobserved for higher concentrations that complements thebehavior detected for τR versus concentration. The concen-tration dependence of the Rouse time can originate from thefact that in the time regime between entanglement and Rousetime, the segments of the chain are entrapped in the tube andfeel the harmonic confining potential2 that depends on thematrix concentration as shown in Figure 6a. Moreover, weobserved that with increasing the concentration, the mesh sizedecreases while the number of entanglements per chain Zincreases, see Table 1.The functional dependence of a and ξ for matrix 2 was

studied in scenario (1) where we varied the mesh size(concentration) for the same persistence length. The resultsobtained are presented in Figure 7. A scaling law of a ∝ξ1.15±0.18 for the tube diameter and mesh size was obtained.

So far, there is no experimental data that demonstrates therelation between the tube diameter and the persistence lengthat the single chain level. Svaneborg et al. has developed amultiscale simulation method for equilibrating Kremer Grestmodel polymer melts with different stiffnesses. They haveshown that with increasing chain stiffness the entanglementtime drops rapidly.56 Using synthetic polymers in our study,rather than the biopolymers such as actin and DNA, enables usto obtain different persistence lengths for different derivativesof polyisocyanopeptides that are very similar otherwise. Theonly restriction is that we require different derivatives of thesynthetic polyisocyanopeptides forming a viscous solution inthe same solvent, tetrachloroethane, creating physical networksof the same mesh size. This turned out to be quite challenging.We found that only 1 (lp = 42 ± 6 nm) and 2 (lp = 129 ± 6

Figure 6. (a) Confining potential for a single chain in matrix 2calculated using the transverse fluctuation distributions at concen-trations of 5 (black squares), 6.5 (red circles), and 8 mg mL−1 (greentriangles) plotted as a function of the distance normal to the primitivepath. (b) The extracted effective restoring force constant K for threeconcentrations mentioned above. (c) Logarithmic plot of the tubediameter versus the concentration of the unlabeled matrix 2. The slopeof the solid line is −0.6, that is, the theoretically predicted Semenovlaw. (d) Rouse time (τR) extracted from single molecule data versuspolymer length (logarithmic scale) for concentrations of 5 (blacksquares), 6.5 (red circles), and 8 mg mL−1 (green triangles). The solidlines have a slope of 2 as predicted by reptation theory (τR ∝ L2). (e)The Rouse time for a specific length (at L = 3 μm) as a function ofconcentration. (f) Disentanglement time (τd) versus Rouse time (τR)at the concentrations of 5 (black squares), 6.5 (red circles), and 8 mgmL−1 (green triangles). The fitted lines have a slope of 3Z (Z: numberof entanglements per chain).

Figure 7. Logarithmic plot for the tube diameter versus mesh size(mesh size is calculated from the plateau modulus) where we extractan exponent of 1.15 ± 0.18.

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nm) formed viscous solutions in tetrachloroethane with similarmesh sizes among all the other derivatives of polyisocyanide.The extracted mesh sizes for polyisocyanides are much largerthan their diameter regardless of the side groups. Figure 8a

shows the confining potential for these two systems. The tubediameters for these two systems were extracted accordingly (seeTable 1). As we can see in the inset of Figure 8a, the tubediameter decreases with increasing the persistence length. Theextracted exponent from the line that connects these twopoints, is −0.2.To summarize, we have shown that the tube diameter a

depends on both the persistence length and the mesh size ofthe polymer matrix. Two power laws of a ∝ ξ1.15±0.18≈1.2 and a∝ lp

−0.2 were extracted for the tube diameter within twoindependent scenarios. Combining these two relations, resultsin a characteristic relation for a, ξ, and lp, a ∝ ξ1.2lp

−0.2 which isin agreement with the predicted relation for semiflexiblechains55,57

ξ= −a b l6/5p( 1/5)

(2)

where b is a constant (b = 0.3157 or b = 1.638). This suggeststhat eq 2 is also valid for flexible chains. In order to define theproportionality constant b for our system, we plotted the tubediameter as a function of ξ1.2lp

−0.2 for all the matrices under

study shown in Figure 8b. We added the extracted tubediameter for matrix 4 as well. As can be seen in Figure 8b, ourdata points roughly follow a linear relation that is shown with alinear fit (red solid line). The coefficient that is needed in orderto define the dependence of a on ξ and lp is determined to b =0.7 ± 0.3.

Conclusion. We have performed a single molecule study onthe dynamics of fluorescently labeled synthetic polymers in acrowded environment of unlabeled polymers. By following theshape and motion of the labeled reporter polymer in time withwide-field fluorescence microscopy, we are able to access theconfining potential of the reporter polymer as a function of theconcentration and persistence length of the surroundingpolymers. We observed that, expectedly, the disentanglement-and Rouse times increase, and the tube diameter shrinks withhigher matrix concentrations. Most interestingly, we observedthe tube diameter to also decrease with increasing persistencelength. We have extracted the functional dependence of thetube diameter on the mesh size and the persistence length atthe single chain level that nicely follows the same power law astheoretically predicted for semiflexible chains. We areconvinced that our present work will stimulate furtherexperimental research to unravel the complexity of thedynamics of entangled polymers in heterogeneous systems,which is essential for the design of novel soft matter andadaptive materials.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jpcb.6b12667.

Brief description on how the Rouse time and the tubediameter were extracted from single molecule data(PDF)Raw movie analyzed via our analysis program as anexample of the single molecule data and reptationmotion of the labeled polymer in the unlabeled matrix(AVI)Processed version of movie analyzed via our analysisprogram as an example of the single molecule data andreptation motion of the labeled polymer in the unlabeledmatrix (AVI)

■ AUTHOR INFORMATIONCorresponding Authors*E-mail: masoumeh.keshavarz@kuleuven.be. Phone: +32 16 3286 86.*E-mail: h.engelkamp@science.ru.nl. Phone: +31 24 3652440.ORCIDMasoumeh Keshavarz: 0000-0003-3685-6778Present Address(M.K.) Molecular Imaging and Photonics, Katholieke Uni-versiteit Leuven, Celestijnenlaan 200f - Box 2404, 3001 Leuven.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSFinancial support from the NanoNextNL (7A.06) (A.E.R.) andthe NWO Veni Grant (680-47-437) (J.X.) and The NationalNature Science Foundation of China (NSFC, Project No.51503143) (J.X.) are acknowledged. This work is part of the

Figure 8. Confining potential and the tube diameter versus stiffness.(a) Confining potential for a single chain in a matrix of 2 (lp = 129 ± 6nm, red circles) and 1 (lp = 42 ± 6 nm, black squares) at theconcentration of 8 mg mL−1 plotted as a function of the distancenormal to the primitive path. Inset: tube diameter versus persistencelength that shows an exponent of −0.2. (b) Tube diameter as afunction of ξ1.2lp

−0.2 for all matrices listed showing a linear relation witha (solid red line).

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research program of the “Stichting voor FundamenteelOnderzoek der Materie (FOM)”, which is financially supportedby the “Nederlandse Organisatie voor WetenschappelijkOnderzoek (NWO)”.

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