Nanoscale Study of Polymer Dynamics - ACS Nano (ACS Publications)

Nanoscale Study of Polymer Dynamics Masoumeh Keshavarz,†,‡ Hans Engelkamp,*,†,§ Jialiang Xu,‡ Els Braeken,§ Matthijs B. J. Otten,‡ Hiroshi Uji-i,§ Erik Schwartz,‡ Matthieu Koepf,‡ Anja Vananroye,§ Jan Vermant,∥,⊥ Roeland J. M. Nolte,‡ Frans De Schryver,§ Jan C. Maan,† Johan Hofkens,*,§,# Peter C. M. Christianen,† and Alan E. Rowan*,‡ †

High Field Magnet Laboratory (HFML - EMFL), Radboud University, Toernooiveld 7, NL-6525 ED Nijmegen, The Netherlands Institute for Molecules and Materials, Department of Molecular Materials, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands § Division of Molecular Imaging and Photonics, Department of Chemistry, Katholieke Universiteit Leuven, Celestijnenlaan 200 F, B-3001 Heverlee, Belgium ∥ Department of Chemical Engineering, Katholieke Universiteit Leuven, de Croylaan 46, B-3001 Heverlee, Belgium ⊥ Department of Materials - Hönggerberg, ETH Zürich, Wolfgang-Pauli-Strasse 10, CH-8093 Zürich, Switzerland # Nano-Science Center/Department of Chemistry, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark ‡

S Supporting Information *

ABSTRACT: The thermal motion of polymer chains in a crowded environment is anisotropic and highly confined. Whereas theoretical and experimental progress has been made, typically only indirect evidence of polymer dynamics is obtained either from scattering or mechanical response. Toward a complete understanding of the complicated polymer dynamics in crowded media such as biological cells, it is of great importance to unravel the role of heterogeneity and molecular individualism. In the present work, we investigate the dynamics of synthetic polymers and the tube-like motion of individual chains using timeresolved fluorescence microscopy. A single fluorescently labeled polymer molecule is observed in a sea of unlabeled polymers, giving access to not only the dynamics of the probe chain itself but also to that of the surrounding network. We demonstrate that it is possible to extract the characteristic time constants and length scales in one experiment, providing a detailed understanding of polymer dynamics at the single chain level. The quantitative agreement with bulk rheology measurements is promising for using local probes to study heterogeneity in complex, crowded systems. KEYWORDS: reptation, single molecule studies, time-resolved fluorescence microscopy

T

entanglements. Furthermore, the concentration of the polymer solution should be larger than the critical concentration at which entanglement starts to play a role. The fundamental work on reptation theory has stimulated numerous theoretical and experimental studies to investigate the tube-like motion in entangled media. Several experimental techniques such as nuclear magnetic resonance,3−5 neutron6−9 and light10−13 scattering as well as fluorescence autocorrelation spectroscopy14−18 have been used to probe the dynamics of entangled polymers in bulk. However, with the use of scattering techniques, it is extremely difficult, if not impossible, to obtain information on the level of individual polymer chains, regarding their temporal and spatial dynamics and the role of heterogeneity and molecular individualism. To this end, several

he rheology of polymers in melts and concentrated solutions that arises from chain entanglement effects has been described by the reptation model, proposed by de Gennes1 and extended by Edwards and Doi.2 In this model, a polymer chain is confined by the surrounding matrix and moves inside an imaginary tube defined by the transient network of entangled neighboring chains. The polymer molecule cannot make transverse movements and may relax only along the tube in a snake-like fashion (reptation). A detailed observation of polymer motion in an entangled environment including the small scale fluctuations (that is, in the order of a few milliseconds and a length scale of several monomer units) is practically almost intractable. Instead of looking at the local dynamics of each monomer, the reptation model focuses on the dynamics at a higher level, of the primitive chain. Reptation occurs if several criteria are fulfilled, one being that the polymer chain should exceed a critical length, which depends on the molecular weight between © XXXX American Chemical Society

Received: November 3, 2015 Accepted: December 20, 2015

A

DOI: 10.1021/acsnano.5b06931 ACS Nano XXXX, XXX, XXX−XXX

Article

www.acsnano.org

Article

ACS Nano groups have developed optical imaging techniques that are capable of monitoring single polymer dynamics. Perkins and co-workers19 were the first to directly visualize reptation using standard fluorescence microscopy. They followed the relaxation of a labeled individual DNA chain after stretching it with optical tweezers. It was shown that the only way the polymer chain could relax was by following its own path without making transversal movements. Texeira et al.20 studied the dynamics of individual entangled DNA chains under the effect of a bulk flow deformation. For different chains in the same flow experiment, very different behavior, which is evidence for molecular individualism, was found. Käs and coworkers21 followed the motion of labeled actin filaments and imaged the confining tube in which the polymer chain could move by superimposing a large number of images and plotting the contours of the observed polymer chains. The restricted motion could be seen as undulations of the actin filament in a tube formed by its neighbors. The tube diameter was determined and shown to decrease with increasing concentration. Further studies on actin filaments and DNA chains,22−27 and more recently the synthetic single-walled carbon nanotubes in a gel, have been examined in order understand the effect of flexibility on the diffusion.28 However, in all these studies, the parameters which can be extracted are obscured by the physical properties of the materials studied and the analytical approach. In this work, we present a comprehensive study on the dynamics of individual polymer chains by means of fluorescence microscopy. We make use of a unique, fluorescently labeled, synthetic polymer that is sufficiently long (μm) to be followed during its motion through concentrated solutions of nearly identical, but unlabeled polymers. We adopt concepts routinely used in single molecule tracking experiments29,30 and localization-based superresolution microscopy31−33 (fitting emitters with a 2D Gaussian function) as well as concepts used in polymer modeling (approximating the chain as a string of beads34) to extract nanometer-scale information from diffraction limited images (for details see Methods and Supporting Information). We show that in a single experiment all relevant length scales and, for the first time, time constants can be accessed and that it is possible to probe the properties of both the labeled chain and the surrounding matrix. By measuring the single chain dynamics in their longitudinal relaxing modes, we are able to demonstrate the validity of the scaling laws in the reptation theory for flexible chains.

Figure 1. Schematic representation of the single polymer reptation experiment. (a), Chemical structure of unlabeled L,D-PIAA (2) and labeled L-PIAP with perylenediimide as chromophore (1) and cartoon of the helical structure of a polyisocyanopeptide (backbone in blue, side groups in red). (b), 3D presentation of the reptation. A fluorescently labeled polymer, 1 (red line), moves within an imaginary tube (wireframe purple tube with diameter a) formed by the entangled neighboring unlabeled polymer chains, 2, (the gray lines). (c), Schematic representation of the experiment. The fluorescent polymer (1, red lines) in a transparent matrix of unlabeled polymers (2, gray lines) was excited with a laser (green light). The fluorescence emission (red light) was collected by an objective lens and imaged on a CCD camera.

ization of the polymers can be found in the Methods. Briefly, polyisocyanopeptide has a high molecular weight (leading to unusually long chains, up to 20 μm in length) and is extremely stiff for a synthetic polymer. The persistence length lp of the matrix polymer L,D-PIAA amounts to 76 ± 6 nm,39 whereas the lp of the probe chain L-PIAP was determined to be 120 ± 30 nm in solution or 138 ± 10 nm on mica (see Supporting Information). These unique properties permit us to investigate the different aspects of polymer chain dynamics in an in-depth single molecule optical study. Nevertheless, the contour length of the chains is still (much) more than ten times the persistence length, and therefore, the polymers should be treated as being flexible, in contrast to, for instance, actin and tubuline.21 Due to the extreme length and relative stiffness of these polymers, entanglement is reached even at low concentrations in terms of weight per volume of solvent. The molecular weight (Mw) of the unlabeled polymers can be varied using different ratios of nickel(II) initiator to monomer. In our study, we used ratios of [initiator]/[monomer] = 1/10 000 and 1/2000. The studied tube motion is illustrated in Figure 1b, where a fluorescent polymer (1, red line) moves within an imaginary tube (the purple wireframe) characterized by its tube diameter (a) defined by the surrounding polymer chains (2, gray lines). The diameter of the tube, a, corresponds to the average distance between the nearest entanglement points.1 The tube motion of the fluorescent polymer, 1, was directly visualized using a wide-field fluorescence microscopy setup (Figure 1c). The sample containing mainly unlabeled polymers and only a few labeled ones (concentration ratio 106:1) was photoexcited with a laser (λ = 543 nm). The focal plane was placed at the middle of the 100 μm thick sample cell to avoid any possible side wall effects. The fluorescence emission was collected through an objective lens (100×; NA 1.3; resolution 270 nm for emission at 580 nm) and imaged with a CCD camera. Movies with visually predominantly in-plane motion, showing quasi two-dimensional dynamics, (more than 20 movies out of 100) were selected for further analysis. These movies were analyzed using homemade software, as exemplified in Figure 2. Each raw frame (Figure 2a) was analyzed

RESULTS AND DISCUSSION The synthetic polymer that we used in our study is a polyisocyanopeptide that can be synthesized by a nickel(II) catalyzed polymerization of isocyanopeptides.35 This type of polymer has been characterized extensively.36−38 The polymer has a rigid helical backbone with approximately four monomer units per helical turn. In our approach, we directly visualized reptation by tracking the motion of a poly(isocyano-L-alanine propanylperylene diimide) (L-PIAP, 1, fluorescent polymer), in a matrix of unlabeled poly(isocyano-L-alanyl-D-alanine methyl ester) (L,D-PIAA) molecules,36 using single molecule wide-field microscopy. The basic molecular structures of the unlabeled and labeled polyisocyanopeptide (backbone in blue, side groups in red) are shown in Figure 1a. The polymer helical conformation is stabilized by hydrogen bonds between the amide groups of the side chains.36 Details of the characterB


Article

ACS Nano

then calculated using the center-to-center displacement. Both d⊥ and d∥ were calculated using only monomers close to the middle of chains to avoid chain free-end effects. Figure 3a shows representative snapshots of a movie recorded from a labeled L-PIAP polymer chain (1) in a

Figure 2. Data analysis. (a and b) An example of a raw, and a background subtracted, smoothed fluorescence image. (c) Reconstructed image from the fitting procedure and the chain contour (without its small-scale fluctuations, the black line in the image). (d) An example of a simulated chain (blue line with small-scale fluctuations), and the contour (black, smooth curve) that has been recovered with the analysis program. (e) The parameters extracted from the fitting procedure, such as end-to-end distance (R), center of mass (C.O.M.), middle point of the contour (M), the contour length (L) and the tube diameter (a). The points along the polymer chain indicate the fitted Gaussians. (f) d⊥ between the chain at t1 and t2 is calculated as the smallest distance between the curve at t1 and M at t2; d∥ is calculated using the center-to-center displacement and d⊥ (figure not to scale).

individually. First, a background, which was calculated using a 51 × 51-kernel median filter, was subtracted and the image was smoothed using a 5 × 5-kernel Gaussian filter. Using a 5 × 5kernel turned out to yield the best consistent fit. The resulting picture (Figure 2b) was then fitted with a string of 2D Gaussians along the contour of the polymer (Figure 2c, for more details on image fitting procedure see Supporting Information). The coordinates of the Gaussians belonging to a single polymer chain were grouped together and splineinterpolated, leading to the contour of the chain (black solid line in Figure 2c). To estimate the resolution of this procedure, we simulated many (>50) polymer chains with lp = 120 nm, convoluted them with the 3D point spread function of our setup, and included realistic noise. An example is shown in Figure 2d. Subjecting those synthetic images to our analysis procedure allowed us to extract the chain contours to a subwavelength precision of about 30−50 nm (black line in Figure 2d; for more details, see Supporting Information). The contours obtained from the movies represent the conformations of the chains with the small-scale fluctuations omitted. When averaged over times larger than the Rouse time, this corresponds to the primitive path in the Doi−Edwards theory. From the fitting procedure, we obtained, as a function of time, the end-to-end distance, contour length L (without small-scale fluctuations), center of mass position and motion, and the middle point of the polymer chain (Figure 2d). This method permits us to study the polymer motion as a function of the properties of the reptating polymer and the characteristics of the local surrounding matrix, which is only possible in a single molecule experiment. To this end, we calculated the perpendicular (d⊥) and longitudinal displacements (d∥) from the movies, as shown in Figure 2e. The perpendicular displacement in time (perpendicular to the initial contour and related to the tube diameter, see below) was calculated as the distance between the middle-point of the chain in frame t2 to the nearest point of the chain in frame t1, for all pairs of frames in a movie. The longitudinal displacement (parallel to the initial chain contour and related to forward motion) was

Figure 3. Movie snapshots showing the reptation motion (Movie 1, Supporting Information). (a) Snapshots of a labeled L-PIAP (1) chain in a concentrated solution of unlabeled L,D-PIAA (2). The overall tube was obtained by superposition of the snapshots corresponding to the time scales indicated. In the second column (10−15 s), the polymer molecule started to escape the yellow tube (0−9 s) represented by the dashed lines in the first column since the new tube (green solid line in the second column) did not follow the same path of the original tube. This can be seen in all the following images. (b) The superimposed conformations of the fluorescent polymer chain contour for the entire movie from t = 0 to t = 84 s. The color code indicates the time. The tube diameter was extracted to be a = 160 ± 10 nm and the disentanglement time (the time where all overlap with the previous primitive path has vanished) of the first tube motion (the U shape in the right part of the figure) was estimated to be τd = 50 ± 5 s.

concentrated solution of unlabeled L,D-PIAA (2) molecules. It is clearly visible that the fluorescent polymer chain can be described as moving in an imaginary tube (indicated with lines next to the chain) formed by its unlabeled neighbors (see also Movie 1 in the Supporting Information). The overall tube in which the single polymer molecule is moving for short time periods is shown in the lower part of Figure 3a. It is obtained by superimposing the snapshots corresponding to the time scales indicated. For example, in the second column (10−15 s), it can be seen that the polymer has started to escape the yellow tube (0−9 s) represented by the dashed lines in the first column since the new tube (green lines in the second column) does not follow completely the same path of the original tube. This tube escape can be followed in the consecutive images. Looking at the snapshots it is evident that after about 50 s the polymer chain has left its original tube completely. C


Article

ACS Nano

limiting the lateral motion with respect to the tube. The tube diameter (a = 0.16 ± 0.03 μm) can be estimated from this image (arrows in the inset of Figure 4b), and also by plotting a histogram of the perpendicular displacements (d⊥, solid symbols, Figure 4b) which can fit to either a Gaussian possessing an exponential tail40 or two Gaussians (solid lines in Figure 4b). The latter data revealed a very narrow Gaussian distribution, the width of which corresponds to a tube diameter of a = 0.15 ± 0.03 μm. The d⊥ histogram of the dilute matrix showed a very broad peak (symbols in Figure 4a) with a width of a = 1.00 ± 0.03 μm, 1 order of magnitude larger than the tube diameter in the concentrated matrix (see Movies 2 and 3 in Supporting Information). The persistence length, lp, and contour length, L, of the fluorescent polymer without its smallscale fluctuations can also be estimated from our fluorescence data (see Supporting Information). The experimentally determined contour length can be taken, in a first approximation, to be related to the primitive chain length of the Doi−Edwards model. Our real time data (Movies) unambiguously demonstrate that the polymer molecules display constrained motion. Moreover, from the data we can obtain detailed and quantitative information about the space- and time-dependence of the polymer dynamics. This information becomes apparent when the mean value of perpendicular displacement, ⟨d⊥⟩, and the mean square value of longitudinal displacement, ⟨d2∥⟩, is calculated as a function of the lag time, for a single labeled chain in a concentrated matrix. d⊥ is the perpendicular displacement distance (d⊥ > 0), which was calculated and grouped for each pair of frames (for all time intervals) and then averaged for each time interval (lag time t). We show a situation in which the polymer chain changes its original tube several times within 80 s of its motion (Figure 5a). To calculate the tube diameter of this chain within the different time periods and spatial regions, we divided the movie (Movie 4, see Supporting Information) into four parts, such that each part contains at least one tube. The perpendicular displacement, d⊥, as a function of lag time is plotted in Figure 5b (1−4), along with ⟨d⊥⟩ (t) for the entire movie (Figure 5b (5)). We observed two regimes where ⟨d⊥⟩ (t) follows a power law.41,42 The slope of the fits was observed to change from 1/4 to 1/2 for each plot. This gradient change occurs when the polymer segments face tube constraints. The average perpendicular displacement in the crossing point corresponds to the tube diameter, which is extracted for all consecutive time periods (1 to 4), as well as for the entire movie (5) (Figure 5c). The average value is the same, a = 0.08 ± 0.03 μm, for all parts of the movie, highlighting the remarkable homogeneity of the matrix over the studied area. We emphasize that with this method we can probe directly the local homogeneity or the heterogeneity of the matrix by tracking the tube diameter of the labeled single polymer over time. Alternatively, the same information can be extracted from histograms of the observed perpendicular displacements extracted from the entire movie. In the inset of Figure 5d, the histogram of the transverse fluctuations of time traces of chain contours is shown. The distribution of the transverse fluctuations (solid line, Figure 5d) allows the determination of the tube diameter.40 The tube diameter extracted from this method (a = 0.102 ± 0.030 μm) is in good agreement with the tube diameter extracted from Figure 5b. To obtain information about the dynamics of the reptation motion of individual chains, we plotted the mean square value of longitudinal displacement, ⟨d2∥⟩, versus lag time (Figure 5e)

In Figure 3b we have plotted the superimposed chain contours to monitor the traversed path of the fluorescent polymer for the entire movie from t = 0 to t = 84 s, leading to an estimated tube diameter of a = 160 ± 10 nm (indicated by the arrows). The imaginary tubes in Figure 3a are clearly visible in Figure 3b. It can be seen that at long time scales the most left blue lines (Figure 3b) define a completely different tube. The polymer chain has left the original tube (the purple U shaped tube in the right part of Figure 3a). This gives an approximation of the disentanglement time for the first tube, which is estimated to be τd = 50 ± 5 s. Additional movies are given in the Supporting Information (Movies 3, 5), including the full analysis. The polymer dynamics is found to be strongly dependent on the concentration of the surrounding matrix, 2 (Figure 4). The

Figure 4. Polymer motion in (a) dilute and (b) concentrated solutions. Insets (both have the same scale) show (a) the polymer motion in a dilute matrix (concentration = 3 mg/mL) during a time of t = 80 s. The polymer chain explores a large area representing Brownian motion. (b) The polymer motion in a concentrated matrix (concentration = 5 mg/mL), which by contrast represents a confined motion referred to as reptation with a tube diameter of a = 0.16 ± 0.03 μm (indicated by arrows in the figure). The occurrence of perpendicular displacements, d⊥ (solid symbols in (a) and (b)) is indicated. A narrow Gaussian distribution (the green and red solid lines are the individual Gaussian fits and the blue line is the sum of the green and red fits, in (b)) is observed for the concentrated 2 matrix, the width of which corresponds to the tube diameter (a = 0.15 ± 0.03 μm), while the dilute matrix shows a broad peak (symbols in (a)) with a width of a = 1.00 ± 0.03 μm.

inset of Figure 4a displays the polymer 1 motion within a dilute polymer 2 matrix, (concentration = 3 mg/mL), with [initiator]/[monomer] = 1/10 000. In a period of 80 s, the polymer chain explores a rather wide area (4 × 4 μm2) in a diffusive manner typical for Brownian motion; in contrast, a 1 chain in a more concentrated 2 matrix (concentration = 5 mg/ mL, Figure 4b inset), with the same initiator to monomer ratio as the dilute matrix, is restricted by the neighboring chains D


Article

ACS Nano

time required for one chain to move out of its original tube. (iv) For times longer than τd, normal diffusive behavior takes place, and ⟨d2∥⟩ ∝ t. In Figure 5e, we observe two regimes (the solid lines separated with a dashed line in Figure 5e) where ⟨d2∥⟩ relates to a power law to the lag time. The first regime, with an exponent of 0.28 ± 0.05, refers to local reptation, and the second regime, with an exponent of 0.50 ± 0.05, corresponds to the reptation regime. The crossing point of these two regimes is the Rouse time, which equals to τR = 0.4 ± 0.1 s for the polymer in Figure 5e. To the best of our knowledge, our result is the first experimental observation of the specific time dynamics of polymers within the Rouse (ii) and reptation (iii) regimes, and is in full agreement with simulations.41,42 Due to photobleaching of the fluorescent polymer, it was difficult to obtain enough data points to determine the disentanglement time, τd (start of regime (iv)) from this plot. Instead, as described above, we derived the disentanglement time from the movie snap shots, see Figure 3b. Furthermore, since the time resolution of our experimental setup is set to 0.1 s (in order to capture images with the desired signal-to-noise ratio), we do not have access to region (i). This implies that we cannot determine the entanglement time and the corresponding tube diameter from the parallel displacement (⟨d2∥⟩) data ((a/2)2 = ⟨d2∥⟩(t = τe)41,42). The entanglement time, however, was calculated using the Rouse and disentanglement times, and the tube diameter was estimated from the perpendicular displacement data (Figure 5b and Supporting Information). So far we have presented the space and time dependence of one single polymer. To link our single molecule data to bulk polymer properties, we have analyzed more than 20 chains of different lengths and used the parameters (τd, τR and a) to explore the analytic expressions for time- and space-dependent behavior of entangled polymers at the single chain level. The Rouse time, τR, disentanglement time, τd, and entanglement time, τe, are related by τR = Z2τe and τd = 3Z3τe.44 As a consequence, τd = 3ZτR, where Z is defined as the number of entanglements per chain. The relationship between disentanglement time and Rouse time is illustrated in Figure 6a where the slope of the linear fit (solid line) corresponds to 3Z, thus, Z = 14.0 ± 0.1. Knowing Z enables us to calculate the entanglement time for the labeled chain (using the relationship τd = 3ZτR), which in our system is in the order of 10−3 s. To substantiate our approach, we performed rheology measurements on the L,D-PIAA polymer solutions (see Methods) (inset of Figure 6a). The plateau modulus, G0N, was extracted for the polymer matrix using MIN method45 to yield G0N = 300 Pa. Using the relationship G0N = (4/5)(ρφRT/Me) for a polymer solution with ρ = 1600 kg/m3 and φ = 0.003 (φ is the volume fraction of the polymer in solution and ρ the density of the polymer solution), gives Me = 31 kg/mol, where Me is the average molecular weight between topological constraints within a tube.46 The polymer average molecular weight (Mw) was estimated with AFM contour measurements47 and amounted to 380 kg/mol. The ratio between the average polymer weight and the entanglement molecular weight, Z = (5Mw/4Me)44 was calculated to be Z = 15, which is very close to the value Z = 14 found for the single molecule data. This correlation shows that our single molecule derived data agree very well with the bulk rheology experiments.

Figure 5. Space- and time-dependence of the polymer motion. (a) The motion of a fluorescent polymer chain, 1, in the concentrated 2 matrix where the polymer chain changes its path several times within 80 s. The color bar shows the time. This movie is divided into four parts (solid lines below the figure). (b) ⟨d⊥⟩ versus lag time for all parts of the movie, 1 to 4, and for the entire movie, 5. The average value of perpendicular displacement at the crossing point (which we define as the point where the slope of the fitted lines changes from 1/4 to 1/2) gives an estimate of tube radius, a/ 2. (c) The tube diameter for movie parts 1 to 4 and the entire movie 5 versus plot numbers (the dashed lines is the tube diameter of the entire movie) indicating the homogeneity of the matrix over this period (d), histogram of the transverse fluctuations of time traces of chains (the inset) and its probability density fitted with a Gaussian function possessing an exponential tail (solid line) from which a tube diameter of a = 0.102 ± 0.030 μm is extracted. (e) ⟨d2∥⟩ as a function of lag time, indicating the regimes of local reptation (solid line with a slope of 0.28 ± 0.05) and reptation (solid line with a slope of 0.55 ± 0.05). The time point between the two regimes is the Rouse time τR = 0.4 ± 0.1 s.

(⟨d2∥⟩ was calculated using a similar procedure as for ⟨d⊥⟩ described above). We expected from the reptation model to see four dynamic regimes:43 (i) t < τe: In this regime, the chain does not feel the constraints imposed by topological constrictions. The time scale on which a defect diffuses along the chain is referred to as entanglement time τe. In this regime, the mean square longitudinal displacement scales with time as ⟨d2∥⟩ ∝ t1/2. Though this time scale is very short, we have access to it indirectly within our experimental approach (see below). (ii) τe < t < τR: This regime is referred to as local reptation, and ⟨d2∥⟩ ∝ t1/4. The polymer diffuses over a distance of the order of its size during a characteristic time called the Rouse time τR. (iii) τR < t < τd: This is the regime of reptation where the longitudinal creep through the tube is possible, and ⟨d2∥⟩ ∝ t1/2. The parameter τd, the disentanglement time, is the E


Article

ACS Nano

reptating chain nor the molecular weight of the surrounding matrix showed any correlation with the tube diameter. It was observed that the scaling laws as derived from our single molecule experiments are in excellent agreement with the De Gennes−Doi−Edwards reptation model. In our work, we demonstrate that all characteristic time constants and length scales of polymer dynamics can be obtained within one experiment. We are able to observe the characteristic dynamics of polymers in a gel in the Rouse, reptation and diffusive regimes. Our analytical approach permits us to determine the scaling laws of polymer motion with respect to contour length, even in an extremely polydisperse polymer matrix. Somewhat surprisingly, the results, obtained at the nanoscale, are in complete agreement with bulk rheology measurements suggesting a homogeneous material. In contrast to biopolymers, which usually have properties that cannot be easily changed, synthetic polymers have the advantage that the parameters of the reptating polymer chains such as the persistence length and the surrounding matrix (material, density, orientation, morphology and chirality) can be independently varied. The effect of changing these parameters is currently under study. The approach outlined in this contribution paves the way for a better understanding of the role of single chain dynamics in relatively poorly understood phenomena such as gelation or polymer crystallization. This local probe technique will allow extraction of the same parameters in highly heterogeneous samples that are inaccessible for bulk techniques, especially crowded cytoskeletons in the cell, as a function of the stage of the cell cycle and response to stress.

Figure 6. Scaling laws. (a) Disentanglement time (tube renewal time) as a function of the Rouse time. The slope of the linear fit is 3Z, in agreement with Z extracted from the rheology measurements (see text). Inset: Storage (G′) and loss (G″) modulus of the 2 matrix L,D-PIAA in tetrachloroethane (5 mg/mL). (b) Tube diameter, a, versus contour length, L, of the labeled chains (symbols) for polymers of different molecular weight, showing no correlation confirming the fact that the tube diameter depends on the surrounding chains. No relationship between the tube diameter and molecular weight is observed (see text). (c) τR and τe as a function of contour length L confirming the predictions that τR, τe ∝ L2. (d) Relationship between disentanglement time and the contour or primitive path length. The slope of the curve is in agreement with reptation theory, which predicts τd ∝ L3.

METHODS Synthesis of Polyisocyanides. Polyisocyano-L-alanine-D-alanine methyl ester (L,D-PIAA, 2) was synthesized following the procedure as previously reported,36 using 1,1,2,2-tetrachloroethane (TCE) as a solvent and nickel(ll) perchlorate as the catalyst. The catalyst/ monomer molar ratio was 1:10 000 or 1:2000 depending on the experimental requirement and the as-prepared gelatinous solution was used as the matrix for the reptation measurements. The fluorescent polymer (L-PIAP, 1) was prepared according to an established procedure48 with a monomer/catalyst molar ratio of 10 000:1 and stored in solution at 4 °C. The polymer was diluted with L,D-PIAA as indicated for the fluorescence measurements. The average polydispersity index (PDI) of polyisocyanides is 1.6−2.35,49 Rheology Measurements. The unlabeled polymer matrix L,DPIAA was physically characterized as published before.36,50 In addition, rheological measurements were performed using a stress-controlled rheometer (Discovery HR-1, TA Instruments) with aluminum parallel plate geometry (40 mm diameter) and a gap of 200 μm. The storage and loss modulus of the material in the linear regime were obtained by applying an oscillatory strain of 2% with a frequency range of 0.01− 100 rad/s and measuring the sinusoidal stress response at T = 20 °C. Time-Resolved Fluorescence Microscopy Experiments. The motion of the fluorescent L-PIAP chain was visualized with a wide field microscopy setup. The fluorescent polymer chains were excited at a wavelength of 543 nm focused in the back focal plane of a 100× oil objective lens (N.A. = 1.3, ZEISS). The emission signal was collected with the same objective, separated from the excitation light by a dichroic mirror (DCLP 560) and filtered (LP-565). The movies were recorded with a Princeton Instruments ProEM 512B charge coupled device (ProEM 512B CCD, model: 7555-0005) camera using an integration time of 100 ms. The temporal and spatial resolutions of the experimental setup were 0.1 s and 270 nm, respectively. Fitting multiple Gaussian point spread functions using the Matlab program

In Figure 6b we have plotted the tube diameter a of the polymer chain versus the contour length L of the reptating polymer for a matrix consisting of polymers (2) with different molecular weights (Mw) ([Initiator]/[monomer] = 1/10 000 (red round symbols) and 1/2000 (black square symbols)). As can be seen, no correlation is observed since the tube is defined by the density of neighboring obstacles, which, as long as the matrix polymers are long enough, solely depends on the polymer concentration. In Figure 6c both τR (left axis, square symbols) and τe (right axis, round symbols) are plotted as a function of the contour length L. We observed the power laws τR and τe ∝ L2.07±0.05, which are in good agreement with the theory of reptation (the relationship should scale with L2). Additionally, we plotted τd as a function of L (Figure 6d) revealing a power law τd ∝ L2.9±0.01, again in full agreement with reptation theory, which predicts τd ∝ L3.1

CONCLUSIONS In the above studies, we have shown that direct visualization of the tube dynamics of a synthetic organic polymer using wide field microscopy is possible. With the use of different analytical methods, the tube diameter was extracted, and by carefully analyzing the space- and time-dependence of the dynamics of the polymer chains, we have also probed the local homogeneity/heterogeneity of the polymer matrix. The local reptation and reptation regimes have been observed and the Rouse time extracted. Neither the contour length of the F


Article

ACS Nano allowed us to extract the polymer contour to a resolution of ∼30−50 nm (see Supporting Information). Sample Preparation. The samples were prepared by putting a drop of the polymer solution in tetrachloroethane in between two coverslips, using a perforated Teflon sheet with a thickness of 100 μm as a spacer. This closed sample configuration enabled us to limit the evaporation of tetrachloroethane during the measurement time.

(7) Richter, D.; Butera, R.; Fetters, L. J.; Huang, J. S.; Farago, B.; Ewen, B. Entanglement Constraints in Polymer Melts. A Neutron Spin Echo Study. Macromolecules 1992, 25, 6156−6164. (8) Schleger, P.; Farago, B.; Lartigue, C.; Kollmar, A.; Richter, K. Clear Evidence of Reptation in Polyethylene from Neutron Spin-Echo Spectroscopy. Phys. Rev. Lett. 1998, 81, 124−127. (9) Wischnewski, A.; Richter, D.; Monkenbusch, M.; Willner, L.; Farago, B.; Ehlers, G.; Schleger, P. Reptation in Polyethylene-Melts with Different Molecular Weights. Phys. B 2000, 276, 337−338. (10) Kanematsu, T.; Sato, T.; Imai, Y.; Ute, K.; Kitayama, T. Mutualand Self-Diffusion Coefficients of a Semiflexible Polymer in Solution. Polym. J. 2005, 37, 65−73. (11) Pajevic, S.; Bansil, R.; Konak, C. Dynamic Light Scattering Study of Linear Polyelectrolyte Diffusion in Gels. Macromolecules 1995, 28, 7536−7542. (12) Goetter, R.; Kroy, K.; Frey, E.; Baermann, M.; Sackmann, E. Dynamic Light Scattering from Semidilute Actin Solutions: A Study of Hydrodynamic Screening, Filament Bending Tiffness, and the Effect of Tropomyosin/Troponin-Binding. Macromolecules 1996, 29, 30−36. (13) Tsunashima, Y.; Nemoto, N.; Kurata, M. Dynamic Light Scattering Studies of Polymer Solutions. 2. Translational Diffusion and Intramolecular Motions of Polystyrene in Dilute Solutions at the Theta Temperature. Macromolecules 1983, 16, 1184−1188. (14) Zettl, H.; Häfner, W.; Böker, A.; Schmalz, H.; Lanzendörfer, M.; Müller, A. H. E.; Krausch, G. Fluorescence Correlation Spectroscopy of Single Dye-Labeled Polymers in Organic Solvents. Macromolecules 2004, 37, 1917−1920. (15) Zettl, H.; Zettl, U.; Krausch, G.; Enderlein, J.; Ballauff, M. Direct Observation of Single Molecule Mobility in Semidilute Polymer Solutions. Phys. Rev. E 2007, 75, 061804. (16) Zettl, U.; Hoffmann, S. T.; Koberling, F.; Krausch, G.; Enderlein, J.; Harnau, L.; Ballauff, M. Self-Diffusion and Cooperative Diffusion in Semidilute Polymer Solutions as Measured by Fluorescence Correlation Spectroscopy. Macromolecules 2009, 42, 9537−9547. (17) Liu, R.; Gao, X.; Adams, J.; Oppermann, W. A Fluorescence Correlation Spectroscopy Study on the Self-Diffusion of Polystyrene Chains in Dilute and Semidilute Solution. Macromolecules 2005, 38, 8845−8849. (18) Wiśniewska, A.; Sozański, K.; Kalwarczyk, T.; Kędra-Królik, K.; Pieper, C.; Wieczorek, S. A.; Jakieła, S.; Enderlein, J.; Hołyst, R. Scaling of Activation Energy for Macroscopic Flow in Poly(Ethylene Glycol) Solutions: Entangled − Non-Entangled Crossover. Polymer 2014, 55, 4651−4657. (19) Perkins, T. T.; Quake, S. R.; Smith, D. E.; Chu, S. Relaxation of a Single DNA Molecule Observed by Optical Microscopy. Science 1994, 264, 822−826. (20) Texeira, R. E.; Dambal, A. K.; Richter, D. H.; Shaqfeh, E. S. G.; Chu, S. The Individualistic Dynamics of Entangled DNA in Solution. Macromolecules 2007, 40, 2461−2476. (21) Käs, J.; Strey, H.; Sackmann, E. Direct Imaging of Reptation for Semiflexible Actin Filaments. Nature 1994, 368, 226−229. (22) Le Goff, L.; Hallatschek, O.; Frey, E.; Amblard, F. Tracer Studies on F-Actin Fluctuations. Phys. Rev. Lett. 2002, 89, 258101. (23) Kumaki, J.; Kawauchi, T.; Yashima, E.; Yashima. “Reptational” Movements of Single Synthetic Polymer Chains on Substrate Observed by in Situ Atomic Force Microscopy. Macromolecules 2006, 39, 1209−1215. (24) Glaser, J.; Chakraborty, D.; Kroy, K.; Lauter, I.; Degawa, M.; Kirchgeßner, N.; Hoffmann, B.; Merkel, R.; Giesen, M. Tube Width Fluctuations in F-Actin Solutions. Phys. Rev. Lett. 2010, 105, 037801. (25) Romanowska, M.; Hinsch, H.; Kirchgeßner, N.; Giesen, M.; Degawa, M.; Hoffmann, B.; Frey, E.; Merkel, R. Direct Observation of the Tube Model in F-Actin Solutions: Tube Dimensions and Curvatures. Europhys. Lett. 2009, 86, 26003. (26) Nishinaka, T.; Doi, Y.; Hashimoto, M.; Hara, R.; Shibata, T.; Harada, Y.; Kinosita, K., Jr.; Noji, H.; Yashima, E. Visualization of Reca Filaments and DNA by Fluorescence Microscopy. J. Biochem. 2007, 141, 147−156.

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.5b06931. Description of additional results, details of the analysis algorithms and simulation details (PDF) Example movies (ZIP)

AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS Financial support from the Ministry of Education, Culture and Science (Gravity program 024.001.035) (R. Nolte, A. Rowan), the European Research Council (ERC Advanced Grant, ALPROS-Grant Agreement No. 290886, R. Nolte., ERC Advanced grant FLUOROCODE-Grant Agreement No. 291593, J. Hofkens, ERC Starting grant PLASMHACAT Grant Agreement No. 280064, H. Uji-i), the NanoNextNL (7A.06) (A. Rowan), the NWO Veni Grant (680-47-437) (J. Xu) and the Flemish government in the form of long-term structural funding “Methusalem” grant METH/08/04 CASAS (J. Hofkens) are acknowledged for this work. We further acknowledge the support from HFML-RU/FOM, member of the European Magnetic Field Laboratory (EMFL). Part of this work has been supported by EuroMagNET II under the EU contract number 228043, the Stichting Fundamenteel Onderzoek der Materie (FOM) with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). REFERENCES (1) de Gennes, P. G. Reptation of a Polymer Chain in the Presence of Fixed Obstacles. J. Chem. Phys. 1971, 55, 572−579. (2) Doi, M.; Edwards, S. F. Dynamics of Concentrated Polymer Systems. Part 1.Brownian Motion in the Equilibrium State. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1789−1801. (3) Herrmann, A.; Kresse, B.; Wohlfahrt, M.; Bauer, I.; Privalov, A. F.; Kruk, D.; Fatkullin, N.; Fujara, F.; Rössler, E. A. Mean Square Displacement and Reorientational Correlation Function in Entangled Polymer Melts Revealed by Field Cycling 1h and 2h NMR Relaxometry. Macromolecules 2012, 45, 6516−6526. (4) Brown, W.; Zhou, P. Solution Properties of Polyisobutylene Investigated by Using Dynamic and Static Light-Scattering and Pulsed Field Gradient NMR. Macromolecules 1991, 24, 5151−5157. (5) Cosgrove, T.; Sutherland, J. M. Self and Mutual Diffusion Measurements in Dilute and Semi-Dilute Polystyrene Solutions. Polymer 1983, 24, 534−536. (6) Monkenbusch, M.; Richter, D. High Resolution Neutron Spectroscopya Tool for the Investigation of Dynamics of Polymers and Soft Matter. C. R. Phys. 2007, 8, 845−864. G


Article

ACS Nano (27) Smith, D.; Perkins, T.; Chu, S. Self-Diffusion of an Entangled DNA Molecule by Reptation. Phys. Rev. Lett. 1995, 75, 4146−4149. (28) Fakhri, N.; MacKintosh, F. C.; Lounis, B.; Cognet, L.; Pasquali, M. Brownian Motion of Stiff Filaments in a Crowded Environment. Science 2010, 330, 1804−1807. (29) Saxton, M. J.; Jacobson, K. Single-Particle Tracking: Applications to Membrane Dynamics. Annu. Rev. Biophys. Biomol. Struct. 1997, 26, 373−399. (30) Rocha, S.; Hutchison, J. A.; Peneva, K.; Herrmann, A.; Müllen, K.; Skjøt, M.; Jørgensen, C. I.; Svendsen, A.; De Schryver, F. C.; Hofkens, J.; et al. Linking Phospholipase Mobility to Activity by Single-Molecule Wide-Field Microscopy. ChemPhysChem 2009, 10, 151−161. (31) Betzig, E.; Patterson, G. H.; Sougrat, R.; Lindwasser, O. W.; Olenych, S.; Bonifacino, J. S.; Davidson, M. W.; Lippincott-Schwartz, J.; Hess, H. F. Imaging Intracellular Fluorescent Proteins at Nanometer Resolution. Science 2006, 313, 1642−1645. (32) Flors, C.; Hotta, J.-i.; Uji-i, H.; Dedecker, P.; Ando, R.; Mizuno, H.; Miyawaki, A.; Hofkens, J. A Stroboscopic Approach for Fast Photoactivation−Localization Microscopy with Dronpa Mutants. J. Am. Chem. Soc. 2007, 129, 13970−13977. (33) Vandenberg, W.; Leutenegger, M.; Lasser, T.; Hofkens, J.; Dedecker, P. Diffraction-Unlimited Imaging: From Pretty Pictures to Hard Numbers. Cell Tissue Res. 2015, 360, 151−178. (34) Li, Y.; Abberton, B.; Kröger, M.; Liu, W. Challenges in Multiscale Modeling of Polymer Dynamics. Polymers 2013, 5, 751− 832. (35) Cornelissen, J. J.; Donners, J. J.; de Gelder, R.; Graswinckel, W. S.; Metselaar, G. A.; Rowan, A. E.; Sommerdijk, N. A.; Nolte, R. J. Beta -Helical Polymers from Isocyanopeptides. Science 2001, 293, 676−680. (36) Cornelissen, J. J. L. M.; Graswinckel, W. S.; Adams, P. J. H. M.; Nachtegaal, G. H.; Kentgens, A. P. M.; Sommerdijk, N. A. J. M.; Nolte, R. J. M. Synthesis and Characterization of Polyisocyanides Derived from Alanine and Glycine Dipeptides. J. Polym. Sci., Part A: Polym. Chem. 2001, 39, 4255−4264. (37) Schwartz, E.; Palermo, V.; Finlayson, C. E.; Huang, Y. S.; Otten, M. B.; Liscio, A.; Trapani, S.; Gonzalez-Valls, I.; Brocorens, P.; Cornelissen, J. J.; et al. ″Helter-Skelter-Like″ Perylene Polyisocyanopeptides. Chem. - Eur. J. 2009, 15, 2536−2547. (38) Kouwer, P. H.; Koepf, M.; Le Sage, V. A.; Jaspers, M.; van Buul, A. M.; Eksteen-Akeroyd, Z. H.; Woltinge, T.; Schwartz, E.; Kitto, H. J.; Hoogenboom, R.; et al. Responsive Biomimetic Networks from Polyisocyanopeptide Hydrogels. Nature 2013, 493, 651−655. (39) Samorı, P.; Ecker, C.; Gossl, I.; de Witte, P. A. J.; Cornelissen, J. L. M.; Metselaar, G. A.; Otten, M. B.; Rowan, A. E.; Nolte, R. J. M.; Rabe, J. P. High Shape Persistence in Single Polymer Chains Rigidified with Lateral Hydrogen Bonded Networks. Macromolecules 2002, 35, 5290−5294. (40) Wang, B.; Guan, J.; Anthony, S. M.; Bae, S. C.; Schweizer, K. S.; Granick, S. Confining Potential When a Biopolymer Filament Reptates. Phys. Rev. Lett. 2010, 104, 118301. (41) Stephanou, P. S.; Baig, C.; Tsolou, G.; Mavrantzas, V. G.; Kroger, M. Quantifying Chain Reptation in Entangled Polymer Melts: Topological and Dynamical Mapping of Atomistic Simulation Results onto the Tube Model. J. Chem. Phys. 2010, 132, 124904. (42) Kremer, K.; Grest, G. S. Dynamics of Entangled Linear Polymer Melts - a Molecular-Dynamics Simulation. J. Chem. Phys. 1990, 92, 5057−5086. (43) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986. (44) Larson, R. G.; Sridhar, T.; Leal, L. G.; McKinley, G. H.; Likhtman, A. E.; McLeish, T. C. B. Definitions of Entanglement Spacing and Time Constants in the Tube Model. J. Rheol. 2003, 47, 809−818. (45) Liu, C.; He, J.; Ruymbeke, E. v.; Keunings, R.; Bailly, C. Evaluation of Different Methods for the Determination of the Plateau Modulus and the Entanglement Molecular Weight. Polymer 2006, 47, 4461−4479.

(46) Huang, Q.; Mednova, O.; Rasmussen, H. K.; Alvarez, N. J.; Skov, A. L.; Almdal, K.; Hassager, O. Concentrated Polymer Solutions Are Different from Melts: Role of Entanglement Molecular Weight. Macromolecules 2013, 46, 5026−5035. (47) Sheiko, S. S.; da Silva, M.; Shirvaniants, D.; LaRue, I.; Prokhorova, S.; Moeller, M.; Beers, K.; Matyjaszewski, K. Measuring Molecular Weight by Atomic Force Microscopy. J. Am. Chem. Soc. 2003, 125, 6725−6728. (48) Schwartz, E.; Kitto, H. J.; de Gelder, R.; Nolte, R. J. M.; Rowan, A. E.; Cornelissen, J. J. L. M. Synthesis, Characterisation and Chiroptical Properties of ’Click’able Polyisocyanopeptides. J. Mater. Chem. 2007, 17, 1876−1884. (49) Schwartz, E.; Le Gac, S.; Cornelissen, J. J.; Nolte, R. J.; Rowan, A. E. Macromolecular Multi-Chromophoric Scaffolding. Chem. Soc. Rev. 2010, 39, 1576−1599. (50) van Buul, A. M.; Schwartz, E.; Brocorens, P.; Koepf, M.; Beljonne, D.; Maan, J. C.; Christianen, P. C. M.; Kouwer, P. H. J.; Nolte, R. J. M.; Engelkamp, H.; et al. Stiffness Versus Architecture of Single Helical Polyisocyanopeptides. Chem. Sci. 2013, 4, 2357−2363.

H


Nanoscale Study of Polymer Dynamics - ACS Nano (ACS Publications)

Recommend Documents