Confining Potential as a Function of Polymer Stiffness and

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Confining Potential as a Function of Polymer Stiffness and Concentration in Entangled Polymer Solutions Masoumeh Keshavarz, Hans Engelkamp, Jialiang Xu, Onno I. Van Den Boomen, Jan C. Maan, Peter C. M. Christianen, and Alan E. Rowan J. Phys. Chem. B, Just Accepted Manuscript • Publication Date (Web): 15 May 2017 Downloaded from http://pubs.acs.org on May 20, 2017

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The Journal of Physical Chemistry

Conning Potential as a Function of Polymer Stiness and Concentration in Entangled Polymer Solutions Masoumeh Keshavarz,

Boomen,

†High

‡

∗, †, ‡

Jan C. Maan,

Hans Engelkamp,

†

∗,†

Jialiang Xu,

Peter C. M. Christianen,

†

‡

Onno I. van den

and Alan E. Rowan

‡

Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules and Materials, Radboud University,Toernooiveld 7, 6525 ED Nijmegen, The Netherlands

‡Radboud

University Nijmegen, Institute for Molecules and Materials, Department of

Molecular materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands.

E-mail: [email protected]; [email protected]

1

ACS Paragon Plus Environment


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Abstract We directly track the tube-like motion of individual uorescently labeled polymer molecules in a concentrated solution of unlabeled polymers. We use a single molecule wide-eld uorescence microscopy technique that is able to determine characteristic properties of the polymer dynamics, such as the conning potential, the tube diameter and the Rouse time. The use of synthetic polymers allows us to investigate the conned motion of the polymer chains not only as a function of polymer concentration (mesh size) but also versus the persistence length of the matrix polymers. Although the polymers used have a persistence length much smaller than their contour length, our experimental results lead to a dependence of the tube diameter on both the mesh size and the persistence length, which follows the theoretically predicted relation for semiexible chains.

Introduction One of the key questions in polymer physics is how the characteristics of the individual chains in a polymeric material lead to their collective properties. When the individual chains become suciently long, entanglement occurs. Entanglements generate topological constraints on polymer conformations and dynamics, which arise from the mutual uncrossability of polymers and cause a signicant change in the viscoelastic behavior of the polymer bulk. This phenomenon is generally described using the tube model proposed by De Gennes Edwards and Doi.

2

1

and

In their model, a single chain moves in a worm-like fashion within a con-

ned tube-like pathway dened by the transient network of entangled neighboring chains. The entanglements restrict the accessible conguration space and impose a tube-like domain around the test chain, thereby suppressing its transverse motion. This tube roughly follows the shape of the test chain. Despite the phenomenological character of the tube model, its basic presumption that the polymers wriggle around in tubes in the presence of entanglements has been validated in a

2


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handful of experiments.

310

A tube surrounding an entangled polymer is not permanent since

it is formed by other mobile chains leading to random changes in its surface, and therefore stochastic uctuations in the average tube diameter occur. In order to replace the Edwards' static tube picture with a soft tube, a dynamic conning potential with a prole of the harmonic potential was introduced.

11,12

The conned motion of polymers has been studied using several experimental techniques from bulk to the single molecule level. ments,

1316

Examples of bulk studies are rheology measure-

uorescence autocorrelation spectroscopy,

1722

neutron

2328

and light

2932

scat-

tering where the information obtained is averaged over spatial distribution of the molecules, molecular heterogeneity,

i.e. chemical compounds, distribution of molecular sizes and confor-

mation. Therefore, it is extremely dicult to study the conning potential and tube width uctuations for individual polymers using bulk methods. In contrast, single molecule experiments allowed to extract quantitative information on the tube width uctuations, as was done by imaging F-actin,

12,33,34

a semiexible biopolymer. The investigation of crowded polymer

environments were carried on theoretically, in order to further understand the conned tubemotion for semiexible chain systems

3539

as well as entangled rod polymer solutions.

40,41

There have been a few experimental studies on the shape of the conning potential for exible chains such as DNA

42

where their results are in agreement with simulations.

4345

Until

now, experimentally very little is known about the transient conning potential through which the polymer chain moves, in particular at the single chain level. In this paper we undertake an experimental study on synthetic polymers, in order to investigate the dynamics of an individual polymer and its relation to material dynamics. We directly track the motion of a labeled polymer in a matrix of unlabeled polymers by single molecule wide-eld uorescence microscopy as we described previously.

3

Using synthetic

polymer polyisocyanide with unique tunable properties ( viz. can be very long in length in the order of

µm

and the persistence length

general for dierent derivatives (the

lP

lp

can be tuned in the range of 4-200 nm in

of the compounds used in this work varies between

3



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42 nm and 138 nm)) allows us to investigate the conned motion of polymer chains versus dierent parameters such as

lp .

The polymer motion can be separated into displacement

components parallel and perpendicular to the tube. From the probability distribution of the perpendicular displacements, we extracted the conning potential and the eective restoring force constant which enabled us to probe the functional dependence of the tube diameter on the mesh size and the persistence length independently.

Materials and experiment The synthetic polymers we employed in our study are derivatives of polyisocyanopeptides shown in Fig. 1a. The polymers used as the unlabeled matrices are tri- and tetraethylene glycol functionalized isocyano-( d)-alanyl-(l)-alanines (3,4EG- l,d-PIAA,

1), triethylene gly-

col functionalized isocyano-( l)-alanines-(d)-alanyl-(l)-alanines (3EG- l,d,l-PIAAA, poly (isocyano- l-alanyl-d-alanine methyl ester) ( l,d-PIAA, propanylperylene diimide) ( l-PIAP,

4).

2) and

The poly(isocyano- l-alanine

3) was used as the labeled chain.

These polymers adopt

a helical conformation with four repeat units per turn stabilized by hydrogen bonds between the amide groups. A representation of the schematic structure of polymer

4 e.g.

is shown in

Fig. 1b. The polyisocyanopeptides were synthesized following the procedure previously reported.

46

The as-prepared gelatinous solution was used as the matrix for the single molecule

studies. The uorescent polymer stored in solution at 4 1.6-2.

°C.

3 was prepared applying an established procedure 47,48 and

The average polydispersity index (PDI) of polyisocyanides is

49,50

The polyisocyanopeptides have a high molecular weight and therefore, unusually long chains (up to 20

µm

in length) and are extremely sti for a synthetic polymer. The per-

sistence lengths of the studied polymers have been extracted from atomic force microscopy (AFM) measurements nm for

4.

51

to be lp

= 42 ± 6

nm for

1, lp = 129 ± 6 nm for 2 and lp = 76 ± 6

The persistence length of the labeled polymer

3 is lp

4


= 138 ± 6

nm.

3

For more

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(a) 1

2

R2

3

N 0.5

N n

O N n

NH

N n

R1

R1

O

N

O

O

N

O

O

R2

HN

O

O

HN

HN

O

NH

O

0.5

O O

O O

O

O C 6H 13

3

3

O

O 4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


C 6H 13

(b) 4

O N H

n

N

O O

O

O NH O

O

N H

O

N H

O

O

O

1

O

2

Figure 1: Chemical structure of unlabeled ( ) 3,4EG-l,d-PIAA and ( ) 3EG-l,d,l-PIAAA,

3

4) l,d-PIAA. (b) 4 where the dotted lines represents

( ) labeled l-PIAP with perylenediimide as chromophore and unlabeled ( Schematic representation of a polyisocyanide

e.g. polymer

hydrogen-bonds that stabilizes their secondary helical structure.

information on the AFM images and the model used for extracting the persistence length of the chains, see the supporting information. Although the chains are very sti, their contour length

L

is still much larger than their persistence length

L lp .

Hence, our system

should be considered to consist of exible chains supported by the fact that their end-to-end distance follows a Gaussian distribution function

3

in agreement with exible chain systems.

5


2


In the exible chain model, the mesh size can be calculated using the plateau modulus,

G0N ,

obtained from a rheology measurement as

s ξ=

where

kB T

is the thermal energy.

1

3

kB T , G0N

(1)

Using the MIN method,

52

the plateau modulus was

0 00 extracted from the storage ( G , closed symbols, Fig. 2) and loss ( G , open symbols, Fig. 2) modulus for 8 mg ml

-1

2 at concentrations of 5 (pink left-handed triangles),

6.5 (blue triangles) and

(black circles). The studied concentration range was limited to a concentration

10 G’, G” (Pa)

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G’_8 (mg/ml) G”_8 (mg/ml) G’_6.5 (mg/ml) G”_6.5 (mg/ml) G’_5 (mg/ml) G”_5 (mg/ml)

1

1 10 angular frequency (rad/s) 0 00 Figure 2: Storage ( G , closed symbols) and loss ( G , open symbols) modules of the 3EGl,d,l-PIAAA in tetrachloroethane (5mg/ml left handed triangles, 6.5 mg/ml triangles and

8 mg/ml circles) as a function of angular frequency.

in which we achieve entanglement as the lower limit. The upper limit is dened when the labeled chain is observed to be static. The plateau modulus and the mesh size of matrix at concentration of 8 mg ml

-1

1

was calculated using Eq.1 and all the mesh sizes and plateau

modulus are summarized in Table 1. Moreover, The average length ( Lav ) of the unlabeled matrices was estimated with AFM contour measurements

53

and they are listed in Table 1.

We prepared the samples by mixing the labeled polymer chains with the unlabeled matrix at a ratio of

1 : 106 .

The samples were photo-excited with a laser ( λ = 543 nm) and the

uorescence emission was collected through an objective lens (100 ×; NA 1.3; resolution 270

6


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Table 1: Concentration, plateau modulus, mesh size, persistence length, tube diameter, number of entanglements per chain and average polymer length ( Lav ) for 3EG-l,d,l-PIAAA for dierent concentrations, as well as for 3,4EG- l,dPIAA at a concentration of 8 mg/ml, and for l,d-PIAA at a concentration of 5 mg/ml.

3EG-l,d,l-PIAAA 3EG-l,d,l-PIAAA 3EG-l,d,l-PIAAA 3,4EG-l,d-PIAA l,d-PIAA

G0N

ξ

lp

a

Z

Lav

(mg ml )

(Pa)

(nm)

(nm)

(nm)

#

( µm)

5 6.5 8 8 5

3.0 ± 0.2 5.0 ± 0.3 7.3 ± 0.4 10.9 ± 0.5 234 ± 12

111.0 ± 0.1 93.8 ± 0.1 82.6 ± 0.1 72.3 ± 0.2 26 ± 4

c

Polymer

-1

129 ± 6 218 ± 25 11 ± 1 00 173 ± 18 12 ± 1 00 158 ± 30 13 ± 1 42 ± 6 208 ± 9 8 ± 2 76 ± 6 149 ± 19 14 ± 1

1.2 00 00 0.8 0.5

nm for emission at 580 nm) and imaged with a CCD camera. The experimental results were real-time movies.

In fact, the motion is occurring in 3D but the information we extract

from the movies is in 2D. To minimize the 2D/3D eect, we selected the movies in which the probe polymer displays a visually in-plane motion (showing quasi two-dimensional dynamics) for further analysis. These movies were analyzed using home-made software, as described in Ref.

3

Each raw frame was analyzed individually. The procedure is illustrated in Fig. 3. Since

the raw images taken via time-resolved uorescence microscopy are noisy (Fig. 3a), using a median lter, smoothing and Gaussian lters, their background was removed (Fig. 3b). Then, a 2D Gaussian function was used to t the position of the uorescent emitters in the resulting image (the tted image in Fig. 3c).

With our analysis method, we dened the

contour of the chain by spline-interpolation of the grouped tted 2D Gaussians along the polymer chains (black solid line in Fig. 3c).

3

This tting procedure improves our resolution

down to 30-50 nm and enables us to have access to the end-to-end distance length

R, contour

L (without small-scale uctuations) and the middle point of the polymer chain M

as a function of time (Fig. 3c). For more information about the tting procedure and the resolution obtained see Ref.

3

The resulted contour of the chain

L represents a coarse-grained

view of the chain where the small-scale uctuations are omitted. When averaged over times longer than the Rouse time, this is related to the primitive path in the Doi-Edwards theory.

7


2


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Page 8 of 25

The primitive path lies along the center line of the tube where the chain winds around it. The superposition of the primitive paths during time represents the tube like motion that can be resolved into parallel extract

dk

and

d⊥ ,

dk

and perpendicular

d⊥

displacement components. In order to

we have considered 101 points along each chain out of which 50 of both

ends are excluded (25 points of each side) to avoid chain free-end eects.

3

The perpendicular

displacement in time 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 was then calculated using the center-to-center displacement and Pythagoras' theorem. The trajectories were analyzed with a frame rate of 0.1 s in the time scales above the entanglement time (when the segments of the chain start to feel the restrictions imposed by the entanglements) and far below the disentanglement time

τd

(the

time it takes for a chain to leave its original tube completely). Fig. 4a shows representative

M

(b) 1 µm

(a) Figure 3:

R

L (c)

(a) An example of a raw image, (b) A background subtracted and smoothed

uorescence image. (c) Fitted image reconstructed from the tting procedure and the chain contour (the black line represents the contour of the polymer neglecting its small scale uctuations).

R is the end-to-end distance and L is the coarse grained contour length of the

chain.

snapshots of an example of a movie recorded from a labeled polymer chain of unlabeled matrix

2

3 in a solution

with a concentration of 8 mg/ml (this movie can be found in the

Supplemental Material). The uorescent polymer chain can be described as moving in an imaginary tube (indicated with lines next to the chain) formed by its unlabeled neighbors The motion starts in the rst row at

t = 0.

In each row, the motion occurs from left to

right and continues to the next row following a path indicated by the green lines in Fig. 4b where we superimposed the conformations of the uorescent polymer chain contour for the indicated movie between

t=0

s and

t = 80

s. The experimental parameters that can be

8


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determined from these kind of measurements are the length scales such as the tube diameter

a

and the time constants such as the Rouse time

τR

(the time it takes for a chain to move

a distance in the order of its length) and disentanglement time entanglements per chain

Z

τd

following the procedure outlined in Ref.

and also the number of

3

(a) 2 µm

(b)

t=0 s

1 µm

t=0 s

t=80 s

t=80 s Figure 4: (a) Snapshots of a labeled chain

3

in a solution of unlabeled matrix

2

with a

concentration of 8 mg/ml. The overall tube was obtained by superposition of the snapshots corresponding to the time-scales indicated. The motion starts motion from left to right in each row up to

t = 80

t = 0 s and one can follow the

s. the green lines are the imaginary tube

dened from Fig. 4b. (b) The superimposed conformations of the uorescent polymer chain contour for the movie between

t=0

and

t = 80

s. The color code indicates the time.

Our work is divided in two parts: In the rst part we study the eect of concentration and consequently mesh size

ξ

on the tube diameter

a

chloroethane at concentrations of 5, 6.5 and 8 mg ml

using a solution of matrix

-1

2 in tetra-

as unlabeled matrix. In the second

part the dependence of the tube diameter on the polymer persistence length was studied applying matrix

1

and

2

as the unlabeled matrices.

Combining the results of these two

independent studies and adding the extra point extracted from the experiments on matrix

4 led to a characteristic relation for a, lp , and ξ for exible chains. 9


It should be noted that


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

the probe chain was always

Page 10 of 25

3 which has a similar chemical structure and properties to the

matrices used in our study, allowing genuine reptation to be observed.

Results and discussion The conning potential was extracted from the probability distribution of the perpendicular component of the displacements ( P (d⊥ )) of a single chain (Fig. 5, symbols). The ensembleaveraged dynamical conning potential using

V (d⊥ )

that the chain segments feel, was extracted

V (d⊥ ) = −kB T ln P (d⊥ ); 12 see inset of Fig. 5 (symbols).

have shown that

P (d⊥ )

Previous experimental studies

follows a Gaussian distribution for the short range displacements

and possess an exponential tail

12

d2

for large displacements. Therefore, a Gaussian,

was tted to the rst logarithmic decay of

P (d⊥ )

⊥ 2 √ e− 2w2 , w 2π

(black curve in Fig. 5) where

w

width of the Gaussian. The conning potential extracted from the Gaussian part of

V (d⊥ ) ∝ d2⊥

corresponds to the classical harmonic constraining potential,

is the

P (d⊥ )

(quadratic in the

short range perpendicular displacements) represented by the tted parabola (red solid line) in the inset of Fig. 5.

The width of the tted Gaussian ( w ) is the tube radius,

distance beyond which the constraining potential exceeds

V (d⊥ ),

the restoring force

diameter

a

F ∝

can be obtained.

kB T .

i.e. the

From the conning potential

dV (d⊥ ) , the restoring force constant dd⊥

K ∝

dF and the tube dd⊥

Since we have access to the full time-range of the reptation

motion of the chains, from the Rouse to the disentanglement time, one can study the timedependence of the conning potential for several time intervals.

V (d⊥ ).

Thus, we extracted the conning potential

In Figure 6b, we plot the conning potential as a function of

the perpendicular displacement for dierent time intervals of

0 − 2.5

s,

0−5

s,

0 − 10

s and

0 − 20

s for matrix

0 − 0.5

s,

0−1

s,

0 − 1.5

2 at the concentration c=6.5 mg/ml.

s,

The

conning potential was extracted from the Gaussian part of the probability distribution of the perpendicular displacements

P (d⊥ )

for each time interval. The tted parabolas (solid

lines) are representative examples of the harmonic potential,

10


V (d⊥ ) ∝ d2⊥ .

The widening of

Page 11 of 25

(a) V(d ) (*kBT)

0.10

2

T

P(d )

0.15 T

1

w

0.05

0 0.0

0.1

0.2 0.3 d (µm) T

0.00 0.0

0.2

0.4 0.6 d (µm)

0.8

1.0

T

V(d ) (*kBT)

3 2

(b)

0-0.5 s 0-1 s 0-1.5 s 0-2.5 s 0-5 s 0-10 s 0-20 s

T

1 0 0.0

0.1 0.2 d (µm)

0.3

T

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 5:

(a) The probability distribution of transverse uctuations for a polymer chain

moving in matrix

2

at a concentration of 5 mg ml

Gaussian with the width

w = a/2.

-1

(symbols).

The solid line is a tted

The large range normal uctuations deviate from the

Gaussian and follow an exponential tail. Inset: the conning potential felt by a single chain (solid line) calculated using the transverse uctuations distribution plotted as a function of the distance normal to the primitive path (symbols, for more detail see text), (b) Conning potential as a function of time for matrix

2

at

c=6.5

mg/ml considering dierent time

intervals (color codes). The solid lines are the tted parabola to the potentials: the Gaussian parts of the probability distributions.

the parabolas points to a weaker connement when approaching the tube renewal time. We think that this change in

V (d⊥ ) ∝ d2⊥

could be an indication for the `soft' tube, suggesting

that the tube width is dynamic and changes with time. This change continues until the Rouse time is reached (here

τR =2.5

s) and for larger time intervals, it seems that the concept of

the tube is vanishing while approaching the disentanglement time and a completely new tube is formed.

A similar anharmonic tube softening for large strains on DNA has been

11



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observed by Robertson

Page 12 of 25

et al using optical tweezers, 42 in agreement with simulations. 4345

In another study, the tube heterogeneities were explored theoretically on conned, weakly bendable rods and the shape predicted for the conning potential shows both the harmonic and anharmonic parts

54

that ts to our observations for polyisocyanides. In Fig. 6a, we plot

the dynamical conning potential 6.5 (red circles) and 8 mg ml

-1

V

for three concentrations of matrix

2 (5 (black squares),

(green triangles)). As a consequence of the higher obstacle

density present at higher concentrations, a faster decay of correlations is observed. eective restoring force constant

K

The

was extracted from the conning potential in Fig. 6a

and plotted as a function of concentration in Fig. 6b. As shown in Fig. 6c, the extracted tube diameter decreases with increasing polymer concentration with an exponent of predicted by the Semenov law

a ∝ c(−3/5) . 55

For each concentration, the same measurement

a, τ R

has been carried out for at least ten samples and all the parameters such as were extracted.

The tube diameter

−0.6 as

and

τd

a presented in Fig. 6c is the averaged value with the

error bars being the standard deviation. The Rouse and disentanglement time (see Appendix) for chains of dierent lengths are shown in Fig. 6d,f for concentrations of 5 (black squares in Fig. 6d,f ), 6.5 (red circles in Fig. 6d,f ) and 8 mg ml

-1

(green triangles in Fig. 6d,f ). As predicted by the reptation theory,

the Rouse time scales with the polymer contour length as

τR ∝ L2 .

1,2

We observed dierent

intercepts for the Rouse time versus the chain contour length at dierent concentrations, pointing to an increase of

τR

versus concentration, which for

The disentanglement and Rouse time follow the relation of entanglements per chain. of

τR

Z

τd = 3ZτR

is shown in Fig. 6e.

, where

Z

was obtained from Fig. 6f in which we plot

τd

for three dierent concentrations. In Fig. 6b,e, we found an increase of

increasing concentration. of

3

L = 3 µm

τR

1.2 from 6.5 to 8 mg ml-1

increases by a factor of

2.5 from 5 to 6.5 mg ml-1

is the number as a function

K

and

τR

with

and by a factor

(Fig. 6e). This suggests that at higher concentrations, the Rouse

and reptation time increase much slower. In Fig. 6b a slow increase of higher concentrations that complements the behavior detected for

12


τR

K

is also observed for

versus concentration.

Page 13 of 25

5 6.5 8

4

(a)

(b)

200 K (nN/m)

V(d ) (*10-9) J

5

T

150

3 2

100

1 0 0.0

50 0.1 0.2 d (µm)

5

0.3

6 7 c (mg/ml)

250

(c)

200 τR (s)

tube diameter (nm)

T

1

5 6.5 8

8

(d)

slope= 2

150 slope = -0.6 0.2 5 1.5

6 7 c (mg/ml)

8

3 L (µm) 120 100 80 60 40 20 0

(e)

1.0

4 (f )

5 Z=11 ± 1 6.5 Z=12 ± 1 8 Z=13 ± 1

τd (s)

τR (s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.5 5

6 7 c (mg/ml)

8

0.5

Figure 6: (a) Conning potential for a single chain in matrix

1.0

1.5 τR (s)

2.0

2.5

2 calculated using the transverse

uctuation distributions at concentrations of 5 (black squares), 6.5 (red circles) and 8 mg ml

-1

(green triangles) plotted as a function of the distance normal to the primitive path. (b)

The extracted eective restoring force constant

K

for three concentrations mentioned above.

(c) Logarithmic plot of the tube diameter versus the concentration of the unlabeled matrix

2.

The slope of the solid line is

−0.6 i.e. the theoretically predicted Semenov law.

(d) Rouse

time (τR ) extracted from single molecule data versus polymer length (logarithmic scale) for concentrations of 5 (black squares), 6.5 (red circles) and 8 mg ml solid lines have a slope of 2 as predicted by reptation theory ( τR

-1

(green triangles). The

∝ L2 ).

(e) The Rouse time

L = 3 µm) as a function of concentration. (f ) Disentanglement time (τd ) versus Rouse time ( τR ) at the concentrations of 5 (black squares), 6.5 (red circles) and 8 -1 mg ml (green triangles). The tted lines have a slope of 3Z (Z : number of entanglements

for a specic length (at

per chain).

13



The concentration dependence of the Rouse time can originate from the fact that in the time regime between entanglement and Rouse time, the segments of the chain are entrapped in the tube and feel the harmonic conning potential

2

which depends on the matrix concentration

as shown in Fig. 6a. Moreover, we observed that with increasing the concentration, the mesh size decreases while the number of entanglements per chain The functional dependence of

a

and

ξ

for matrix

Z

increases, see Table 1.

2 was studied in scenario (1) where we

varied the mesh size (concentration) for the same persistence length. The results obtained are presented in Fig. 7. A scaling law of

a ∝ ξ 1.15±0.18

for the tube diameter and mesh size

was obtained.

tube diameter (nm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 25

250 200 150 slope=1.15 ± 0.18 100 90 mesh size (nm)

110

Figure 7: Logarithmic plot for the tube diameter versus mesh size (mesh size is calculated from the plateau modulus) where we extract an exponent of

1.15 ± 0.18.

So far, there is no experimental data that demonstrates the relation between the tube diameter and the persistence length at the single chain level. Svaneborg

et al has developed

a multiscale simulation method for equilibrating Kremer Grest model polymer melts with dierent stinesses. They have shown that with increasing chain stiness the entanglement time drops rapidly.

56

Using synthetic polymers in our study, rather than the biopolymers

such as actin and DNA, enables us to obtain dierent persistence lengths for dierent derivatives of polyisocyanopeptides which are very similar otherwise. The only restriction is that we require dierent derivatives of the synthetic polyisocyanopeptides forming a viscous so-

14


Page 15 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


lution in the same solvent, tetrachloroethane, creating physical networks of the same mesh size. This turned out to be quite challenging. We found that only

2 (lp

= 129 ± 6

1 (lp = 42 ± 6 nm) and

nm) formed viscous solutions in tetrachloroethane with similar mesh sizes

among all the other derivatives of polyisocyanide.

The extracted mesh sizes for polyiso-

cyanides are much larger than their diameter regardless of the side groups. Fig. 8a shows the conning potential for these two systems. The tube diameters for these two systems were extracted accordingly (see Table 1). As we can see in the inset of Fig. 8a, the tube diameter decreases with increasing the persistence length. The extracted exponent from the line that connects these two points, is

−0.2.

To summarize, we have shown that the tube diameter

a depends on both the persistence

length and the mesh size of the polymer matrix. Two power laws of

a ∝ lp−0.2

a ∝ ξ 1.15±0.18≈1.2

were extracted for the tube diameter within two independent scenarios. Combining

these two relations, results in a characteristic relation for

a, ξ

and lp ,

in agreement with the predicted relation for semiexible chains:

a ∝ ξ 1.2 lp−0.2

b

is a constant ( b

= 0.31 57

or

b = 1.6 38 ).

(2)

This suggests that Eq.2 is also valid for

exible chains. In order to dene the proportionality constant the tube diameter as a function of

ξ 1.2 lp−0.2

which is

55,57

a = bξ 6/5 lp(−1/5)

where

and

b

for our system, we plotted

for all the matrices under study shown in Fig. 8b.

We added the extracted tube diameter for matrix

4 as well.

As can be seen in Fig. 8b, our

data points roughly follow a linear relation that is shown with a linear t (red solid line). The coecient that is needed in order to dene the dependence of to

b = 0.7 ± 0.3.

15


a on ξ

and lp is determined


5

lp=129 ± 6 nm lp= 42 ± 6 nm

slope= -0.2

a (nm)

200

4 V(d ) (*10-9) J

150

3

50

100 lp (nm)

2

T

1

(a)

0 0.0

0.1

0.2 0.3 d (µm)

0.4

0.5

T

240 220 200 a (nm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 25

slope=0.7 ± 0.3

180 160 140 120

(b) 20

80 60 40 -0.2 1.2 lp ×ξ (nm)

100

Figure 8: Conning potential and the tube diameter versus stiness. (a) Conning potential for a single chain in a matrix of

2 (lp = 129 ± 6 nm, red circles) and 1 (lp = 42 ± 6 nm, black

squares) at the concentration of 8 mg ml

-1

plotted as a function of the distance normal to

the primitive path. Inset: tube diameter versus persistence length that shows an exponent 1.2 −0.2 for all matrices listed showing a linear of −0.2. (b) Tube diameter as a function of ξ lp relation with

a

(solid red line).

Conclusion We have performed a single molecule study on the dynamics of uorescently labeled synthetic polymers in a crowded environment of unlabeled polymers. By following the shape and motion of the labeled reporter polymer in time with wide-eld uorescence microscopy, we are able to access the conning potential of the reporter polymer as a function of the concentration and persistence length of the surrounding polymers. We observed that, expectedly, the disentanglement- and Rouse times increase, and the tube diameter shrinks with

16


Page 17 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


higher matrix concentrations. Most interestingly, we observed the tube diameter to also decrease with increasing persistence length. We have extracted the functional dependence of the tube diameter on the mesh size and the persistence length at the single chain level which nicely follows the same power law as theoretically predicted for semiexible chains.

We

are convinced that our present work will stimulate further experimental research to unravel the complexity of the dynamics of entangled polymers in heterogeneous systems, which is essential for the design of novel soft matter and adaptive materials.

AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected]. Phone: +32 16 32 86 86. *E-mail: [email protected]. Phone: +31 24 3652440

Present Address M.K.: Molecular Imaging and Photonics, Katholieke Universiteit Leuven, Celestijnenlaan 200f - box 2404, 3001 Leuven.

Acknowledgement Financial support from the NanoNextNL (7A.06) (A.E.R.) and the NWO Veni Grant (68047-437) (J.X.) and The National Nature Science Foundation of China (NSFC, Project no. 51503143) (J.X.) are acknowledged. This work is part of the research program of the 'Stichting voor Fundamenteel Onderzoek der Materie (FOM)', which is nancially supported by the 'Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)'.

17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 25

Supporting Information Available The following les are available free of charge.

Supporting Information: brief description on how the Rouse time and the tube diameter were extracted from single molecule data.

raw Movie and Movie_processed: A raw movie that has been analyzed via our analysis program (whose snapshots are shown in Fig. 4) together with its processed version are presented as an example of the single molecule data and reptation motion of the labeled polymer in the unlabeled matrix.

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24


Page 25 of 25

Graphical TOC Entry V(d ) (*10-8) J

0.10

T

w

w

0.5

0 0.0

0.05

t=200 s

t=0 s

1

0.1

0.2 0.3 d (µm) T

0.00 0.0

0.2

0.4 0.6 d (µm)

0.8

1.0

T

P(d )

0.15 T

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


25


Confining Potential as a Function of Polymer Stiffness and

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