Direct Observation of Two Distinct Diffusive Modes for Polymer Rings

Article Cite This: Macromolecules XXXX, XXX, XXX-XXX

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Direct Observation of Two Distinct Diffusive Modes for Polymer Rings in Linear Polymer Matrices by Pulsed Field Gradient (PFG) NMR Margarita Kruteva,* Jürgen Allgaier, and Dieter Richter Jülich Centre for Neutron Science (JCNS-1) and Institute for Complex Systems (ICS-1), Forschungszentrum Jülich GmbH, 52425 Jülich, Germany ABSTRACT: Using high resolution PFG NMR spectroscopy, we have studied the diffusion of well-characterized polymer rings in linear host matrices at various observation times Δ, varying both ring and host molecular weights. For the first time, to our knowledge, for higher Mw rings in entangled melts it was possible to directly distinguish two different diffusive modes: (i) fast diffusion that scales inversely with the host chain length and (ii) much slower diffusion depending much more strongly on the host molecular weight. Furthermore, we studied the diffusion of the linear chains in the host melts. The diffusion data were analyzed in terms of existing theories and compared to simulations. The two-mode structure directly verifies the hypothesis of qualitatively different mechanisms for ring diffusion in linear melts. The fast mode quantitatively agrees with the assertion of a special diffusion channel for once threaded rings, while the suggested diffusion of unthreaded rings was not found. The slow mode scales more weakly with the host chain length than predicted by the constraint release (CR) mechanism. However, considering an interdependence of constraints (Macromolecules 1986, 19, 105), the slow mode was quantitatively related to tube renewal processes. Using this concept also the molecular weight dependence of the matrix diffusion is described naturally. In contrast to the explicit observation of fast and slow diffusive modes simulations reveal broadly distributed heterogeneities leading to a prevailing CR mechanism only. The strong size dependent ring diffusion in an entangled matrix remains unintelligible. Finally, even though we have distinguished two well-defined significantly different diffusive modes with characteristic times in the millisecond range that would be expected to interchange, a detailed analysis in terms of a two-state diffusion model allowing for state changes reveals that within the experimental sensitivity no such exchanges take place.

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INTRODUCTION Ring polymers are characterized by a unique topologyamong all polymer systems, rings stand outthey do not have ends.1 For polymers, this attribute is particularly important since for all other topologies, relaxation processes like reptation or branch retraction take place via the chain ends. Also nature exploits the particularities of ring topology, for instance in packing chromatin rings in nucleosomes, providing easy access to genetic informationfor rings, interpenetration is entropically costly.2 Recently as a consequence of large available computer power and progress in synthesis, after a long intermission, increased research activities both by simulation3,4 and experiment1 were reported. One intriguing scientific challenge is posed by blends of ring and linear polymers. While in such blends the dynamics of the linear chains is hardly affected, minute concentrations of linear chains have a large effect on ring rheology.5 Dilute rings in linear hosts were proposed as probes to study the entanglement dynamics of the linear host.6 MD simulation revealed that in linear matrices ring diffusion is strongly affected.7−9 The so far most instructive experimental results were reported by the Kramers group10 using forward nuclear recoil spectroscopy on blends of dilute PS rings in linear matrices. Both were varied over a large range of molecular weights. The authors observed a strong slowing down of ring diffusion with increasing host molecular weight as well as for increasing ring size in identical matrices. The observed diffusion data were interpreted © XXXX American Chemical Society

as a result of three basic diffusion mechanisms: (i) nonthreaded rings diffuse by reptation in the confining tubes provided by the host; (ii) among the rings threaded by host molecules, the once threaded species are thought to move “like a smoke ring” following the contour of the long linear chain; (iii) finally multiple threaded rings are supposed to diffuse by constraint release enacted by the linear host. It turned out that all three processes were needed to reasonably well describe the average diffusion data. Cosgrove et al.11 studied equimolar low molecular weight PDMS ring/linear blends and found that threading occurred beyond a molecular weight of Mw = 2500 Da corresponding to 33 monomers. Von Meerwall investigated short alkane ring−linear blends by pulsed field gradient (PFG) NMR.12 The molecular weight dependence was found to be overwhelmed by free volume effects; cyclic alkanes were observed to diffuse more slowly than the linear counterparts, and two distinct diffusion coefficients were revealed in blends with cycloalkanes; the diffusion coefficients were consistent with the Rouse expectation. Thereby, the NMR signal from the blends was always in the fast exchange limit. PFG NMR studies on low molecular weight ring/linear blends of poly(ethylene oxide) (PEO)13 were evaluated in terms of simple mixing rules− Received: August 27, 2017 Revised: November 7, 2017

A

DOI: 10.1021/acs.macromol.7b01850 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

a two-state diffusion model for the self-correlation function is displayed; (iii) then materials and samples are introduced. (iv) The main part of the paper presents the experimental results and their evaluation; (v) in the Discussion, we compare different existing theories with simulation results; (vi) finally, the paper closes with a Conclusion.

deviations were attributed to threaded ring fractions. At a molecular weight of 1500 g/mol the authors derived that 50% of the rings (volume fraction 25%) would be threaded. The resulting diffusion coefficient of the ring is about 2.5 times below that of the linear melt. Also single-molecule studies both on bulk polymer melts14 as well in DNA solutions15 were reported. Recently, by small angle neutron scattering the conformation of dilute rings in linear blends were studied and compared with the corresponding conformations in the pure ring melt.6,16,17 While compared to the Gaussian prediction in a 2 kg/mol (R2) ring melt, the ring conformation is significantly compressed, in a blend with 2 kg/mol (L2) linear chains the ring swells close to the Gaussian expectation. Similarly, for R20 dilute rings in a linear L20 host the ring exhibits a Gaussian conformation with an Rg much larger than that in the ring melt (Rmelt g = 3.05 nm; =3.65 nm). Rblend g Lately the number of simulations on ring/linear blends employing either Monte Carlo (bond fluctuation model), coarse grained MD or atomistic MD techniques took off. Here we mention only some exemplary results. Atomistic simulations on blends of 20 kg/mol (R20) PEO rings in linear chains with molecular weights of 2, 10, and 20 kg/mol (L2, L10, and L20) were presented by the Mavrantzas group.9 They found some swelling of the R20 ring in the L2 linear blend, while in L10 and L20 linear hosts the rings exhibit the same conformation as in the ring melt. In all cases the matrix remained unaffected. The center of mass diffusion constant for the R20 ring in the L2 linear matrix agrees perfectly with experiment (Dsim= 5.0 × 10−12 m2/s; Dexp= 4.9 × 10−12 m2/s). In the longer matrices, the relaxation times slow down, an effect that was connected to threading. Finally, a topological analysis revealed: (i) all rings are threaded by at least by one molecule, (ii) average threading numbers are 27 in L2, 20 in L10, and 16 in L20 blend; (iii) in a L20 blend, 80% of the linear chains participate in threading events; (iv) no ring is threaded by less than six chains. Furthermore, the survival times of threading was analyzed on a symmetric R5/L5 blend at a linear concentration of 75%.18 The simulations yielded survival times up two orders longer than the Rouse time of the ring. Thereby, two-thirds of the ring-linear entanglements live longer than the average ring-relaxation time. Compared to that, ring−ring entanglements live much shorter. Heavily threaded rings show strongly reduced dynamics, and the system becomes strongly heterogeneous. Shanbhag and co-workers performed Monte Carlo simulations on ring/linear blends applying the bond-fluctuation model.8,19 In agreement with the MD simulations they also found that the linear chains in the blend are hardly affected, while the ring diffusion in a symmetric 300 beads blend changes from the pure ring melt to the diluted ring/linear blend by a factor of about 20. In asymmetric blends with molecular weights of the host smaller than that of the ring, they reported an unusual maximum of the ring diffusivity at intermediate host size, which is explained by the increased number of threadings with decreasing Mw of the matrix.20 This effect is stronger than the increase of the matrix mobility. Finally, the Kremer group performed coarse-grained MD simulations on ring/linear blends7 for the symmetric case. Compared to pure ring melts their simulations revealed a decrease of the ring tracer diffusion by 2 orders of magnitude (400 beads; corresponding to 14.3 entanglements); at the same time, the viscosity increased by similar factors. The paper is arranged as follows: (i) first we present predictions from scaling analysis that we quantify and a hypothesis that was based on MC simulations. (ii) Thereafter,

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THEORETICAL BASIS The nomenclature that will be used for this work is defined in the Nomenclature section. We start presenting the main outcome of the scaling theory for ring diffusion in a linear matrix21 and quantify the predictions by properly inserting prefactors following Graessley.22 The dynamics scales by the elemental kT diffusivity D0 = ξ , where ξ is the monomeric friction. For PEO at the measuring temperature of 413 K D0 = 2.26 × 10−9 m2/s.23 According to Graessley, in the reptation limit, the diffusion of a probe chain with n monomers in in long chain p-matrix follows n 4 D* = D0 e2 (1) 15 n On the other hand, a long n-chain in a shorter p-chain matrix diffuses by the constraint release (CR) mechanism n3 * = 4 αD0 e DCR 15 np3

(2)

with α close to 1. For the further discussion, we need a few definitions: tube size and primitive step d = l 2 pe1/2 with l the monomer length; the curve linear diffusion along the Gaussian tube follows Rouse D behavior with a diffusion coefficient Dc = p0 ; the contour length

of the tube is Lp(P) ≡ Lp = Pd. Including the proper prefactors eq 1 the longest relaxation time of an entangled linear melt, the reptation time, amounts to τrep(p) ≅

Lp 2 2Dc

.

2 3

τrep(p) =

5l p 8D0pe

(3)

and finally, neglecting corrections like CR and contour length fluctuations (CLF) for diffusion by reptation we have Drep(p) =

R2(p) 6τrep

(4)

Constraint release occurs via tube renewal: The relaxation time of an effective constraints is given by the reptation time of the confining chains: τrep(p) ≡ τ′c allowing for a local tube change by a step size d. The confining tube of the n-chain is thought to move as a Rouse chain. Thus, the time for tube renewal corresponds to the Rouse time of this tube τ′tube(N) ≅ N2τ′c ; Klein considered interdependent constraints and concluded a reduction of τc′ by a factor of p1/2 resulting in τc = τrep(p)/p1/2. Then the tube renewal time becomes τtube(N )R ≅ N 2τc =

N 2τrep(p) p1/2

=

5n2l 2p5/2 8ne 3D0

(5)

Finally, diffusion by tube renewal is derived to DR (N ) = B

R2(n)p1/2 6N 2τrep(p)

=

ne2 2np1/2

Drep(p) (6) DOI: 10.1021/acs.macromol.7b01850 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules where R2(n) = nl2 is the size of the confined probe chain; for a 1 Gaussian ringR2(n) = 2 nl 2 . Equation 6 connects the ring diffusion by CR or tube renewal with the reptation diffusion of the host. This has the advantage that we do not need to relate to theoretical expressions for reptation diffusion with all the necessary corrections not discussed here, but employing the measured host p-diffusion includes all effects of CR and CLF and explicit predictions are possible. Further mechanisms for ring diffusion: (i) If a ring is not threaded then it may move by reptation in the tube provided by the host. The probability for nonthreading is given by PR(N) = const × e−γN (γ = 0.6−0.7); however, this includes all ring conformations obeying these conditions; for flat rings, able to reptate in the tube, γ changes to β ≅ 1.1, and the ring reptation diffusion would then be DR =

4n 4 D0 2e 15 n

a PFG NMR experiment, the diffusion coefficient D is usually determined from the dependence of spin echo decay ⎛ ⎛ δ ⎞⎞ A(q2)/A(0) = exp⎜ −q2D⎜Δ − ⎟⎟ ⎝ ⎝ 3 ⎠⎠

on the strength of the field gradient pulses g through the momentum transfer q = γHδg where γH is a proton gyromagnetic ratio, δ is a gradient pulse length. A(q2) is the echo amplitude, and A(0) contains information about T1 and T2 relaxation processes. Δ is the observation time. If different diffusive states are possible and coexisting, the selfcorrelation function has to be formulated accordingly. For fast exchange the situation is simple and PFG NMR measures the average diffusion. However, if the exchange is slow, then the different states, their occupation and the exchange between them have to be explicitly considered. Following the general concept of Singwi and Sjölander25 for a self-correlation function originating from statistically independent states including exchanges between them, Kärger derived the appropriate self-correlation function to describe diffusion related spin echo decay.26,27 It presents a generalized description of the echo decay in a system containing two subsystems with different diffusion coefficients allowing for state changes between them. In the intermediate and slow exchange range, the diffusion spin echo decay, in addition to the diffusion constants characterizing the different states, contains information on the population of these states and the exchange rate between them. According to Kärger, the spin echo decay then is given by

(7)

Note that this type of diffusion does not depend on the matrix size as long as the matrix is strongly entangled. (ii) Kramer suggested also a special case for once threaded rings, where the threaded ring diffuses by Rouse type motion following the contour of the threading chain. For this, the diffusion coefficient follows DR1 ≅

n 8 D0 e 15 np

(8)

Recently Shanbhag and co-workers studied the self-diffusion in symmetric ring/linear blends by Monte Carlo simulations.8,19 The results were rationalized in a model somewhat different to Klein’s approach. According to these simulations the entanglements (threadings) on a ring are

N (ϕL) P

=

ϕL ϕ

where ϕ is the total

concentration and ϕL the concentration of host chains. The length of the primitive path becomes

LN (ϕL) LP

= ϕL /ϕ. The

primitive path of the rings rearranges as it relaxes by CR. For the ring diffusion DR(ϕL) Shanbhag hypothesizes 1 1 1 = + DR (ϕL) DR (0) DCR (ϕL)

⎛ ⎛ δ ⎞⎞ A(q2)/A(0) = p′f exp⎜ −q2D′f ⎜Δ − ⎟⎟ ⎝ ⎝ 3 ⎠⎠ ⎛ ⎛ δ ⎞⎞ + ps′ exp⎜ −q2Ds′⎜Δ − ⎟⎟ ⎝ ⎝ 3 ⎠⎠

(12)

D′f , s = C1 ± C2

(13)

C1 =

(9)

where DCR(ϕL) is the Rouse diffusivity of the tube relaxing by CR and DR(0) is ring diffusion constant in the ring melt. The ring diffusion by constraint release was treated as the Rouse diffusivity of the confining tube; the local hopping time governing CR was set to linear polymer relaxation time τc. The effective drag on a ring in the blend is dominated by entanglements (threadings) τc = Rp2/Dp(ϕL). Inserting this in eq 9 leads to N (ϕL) 1 1 = + DR (ϕL) DR (0) DL(ϕL)

(11)

⎛ ⎞ 1⎜ 1 ⎛⎜ 1 1 ⎞⎟⎟ + + + D D s f ⎜ τf ⎟⎠⎟⎠ 2 ⎜⎝ q2 ⎝ τs

(14)

⎛ ⎞2 1 ⎜ 1 ⎛⎜ 1 1 ⎞⎟⎟ 1 4 C2 = Ds − Df + 2 ⎜ − ⎟⎟ + 4 2 ⎜⎝ τ f ⎠⎠ q τf τs q ⎝ τs

(15)

1 (p Ds + pf Df − Ds′); ps′ = 1 − p′f Ds′ − D′f s

(16)

p′f =

2

This complex decay of the echo amplitude with q depends on the populations (ps, pf), the mean life times (τs, τf), and the diffusion coefficients (Ds, Df) of the slow and fast ring diffusion states, respectively. eqs(12-16 may be rewritten in terms of the τf thermal occupation of the fast diffusive state thf = τ + τ and τf

(10)

This is an approach different from Klein: There it was assumed that rings of different threading states behave differently and add up with their respective probabilities. Here CR and diffusive processes in the corresponding ring melt happen in series. This approach has later been extended to asymmetric ring/linear blends24 with the result of a strong reduction of DR with increasing length of the host p. Like incoherent neutron scattering, proton PFG NMR measures directly the self-correlation function of the protonated species, collective phenomena do not contribute to the signal. In

f

s

mean lifetime of the fast componentas the independent parameters.

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EXPERIMENTAL SECTION

Methods. The PFG NMR measurements were performed using a magnetic resonance analyzer Bruker Minispec (mq20) that operates at a 1 H frequency of 20 MHz and equipped with a permanent magnet. Furthermore, highly viscous samples with slower diffusion coefficients were studied using a Bruker Avance 600 MHz equipped with Diff30 1H C


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Macromolecules

All data follow a single exponential decay and were fitted with eq 11 where Δ was 15 ms for 2K rings (Figure 1a) and went from 15 to 300 ms for 5K rings, revealing the diffusion coefficient of the rings in the different matrices. All data are in agreement with the observation of a single average diffusion coefficient. The results including the pure ring melt data30 are presented in Table 2. We note that the diffusion spin echo decay for the 2K rings is not dependent on the observation time Δ. Larger rings display spin echo decays that depend on the observation time Δ indicating dynamic heterogeneities in an entangled matrix (Mmatrix > 10K). Table 2 presents the diffusion coefficients for the 5K rings obtained at maximal observation times only. We also measured the diffusion coefficients of the linear chains in the investigated matrix melts with molecular weights between 2 kg/ mol and 80 kg/mol (L2 to L80) (1 ≤ P ≤ 40) . For P = 1 the diffusion result is very close to the Rouse expectation (5.4 × 10−11 m2/s compared to 5.56 × 10−11 m2/s). Between L5 and L80 we find an effective power law for DL = 1.4 × 10−6/p2.4 somewhat steeper than the average power law behavior for different polymers of DL ∝ p−2.28 (Figure 2).31 Interestingly the R2 pure ring melt displays a diffusion coefficient that amounts to about double of that in the L2 matrix; thus the ring in its own melt diffuses faster than what the Rouse model predicts.16 In the L2 linear host, we find DR = 5.6 × 10−11 m2/s, very close to the expected Rouse value of 5.35 × 10−11 m2/ s. Thus, in a host of not-entangled linear polymer the ring diffusion follows the Rouse prediction. For longer entangled hosts the R2 diffusion is remarkably constant. Between host molecular weights of 10 kg/mol to 80 kg/mol (5 to 40 entanglements) DR does not change but is constantly a factor of 4 below the Rouse value. Compared to the R2 ring, R5 behaves differently. While for R2 the ring melt diffusion was nearly double than that in the L2 host, R5 in L2 moves fairly similar (2.3 × 10−11 m2/s) in the blend compared to 2.4 × 10−11 m2/s in a pure ring environment. For higher Mw hosts, we varied the observation time Δ and found that though all spectra appeared to be single exponential, with increasing Δ the slope of the exponential decreased. For R5 in L20 the corresponding diffusion coefficient saturated at Δ = 100 ms; for R5 in L80 the decrease persisted until 600 ms (see inset in Figure 4). In Table 2, we present the saturated or lowest diffusion coefficients for both long matrices. With the exception of the R5/ L80 blend the R5 diffusion largely follows the diffusion behavior of the host (Figure 2). The diffusion coefficient in the R5/L80 blend is only slightly slower than that in the R5/L20 blend. The bulk of the observations were performed with the Bruker Avance 600 MHz PFG NMR instrument. With this instrument, measuring at long observation times Δ and at larger q2 values the slowly diffusing components became visible and a line shape analysis of the NMR spectra could be performed. Figure 3 directly shows the coexistence of well separated fast and slow diffusion modes. This implies that during time Δ at most a very limited number of exchanges between the different diffusion modes can happenotherwise the fast exchange limit would apply, where the NMR signal would be characterized by a single exponential decay showing the average diffusion coefficient. The data were analyzed in terms of Kärger’s two state diffusion model, eqs 12−16, allowing for two diffusive modes and exchanges between them (see above). The data for different observation times Δ were jointly fitted; parameters were the two diffusion coefficients Df and Ds, the lifetime of the fast process τf and the thermal occupation thf of this process. Using different starting parameters, the fit was tested for stability. The results are

probe head. The attenuation of the echo signal from a pulse sequence containing a magnetic field gradient pulse is used to measure the translational diffusion of the molecules (protons) in the sample at the time scales from ten to thousand milliseconds. During this time the protons are able to overcome the distances of order of hundreds of nanometers. Diffusion spin echo decays were measured using a standard stimulated echo pulsed-field-gradient (STE) sequence28 at the temperature 413 K. Observation times Δ were varied from 15 to 600 ms on Minispec and from 30 to 1000 ms on Bruker Avance 600. The gradient pulse length δ was 5 ms for the Minispec instrument and 3 or 5 ms at Bruker Avance 600 system. The relaxation times T1 and T2 have typical values of 1 and 0.5 s for all sample types allowing diffusion measurements at the time scale of order of 1 s. Synthesis and Characterization. Cyclic hydrogenous PEOs were obtained from linear precursors under conditions of high dilution.29 The cyclization raw products were fractionated using chloroform/heptane as solvent/nonsolvent pair, in order to remove higher molecular weight condensation byproducts. Residual linear precursor was removed by first oxidizing its alcoholic chain ends to carboxylic groups, followed by fixation on a basic ion-exchange resin. The details are given in reference.29 This purification boosted the purity of the cyclic product to >99% for R10 and >99.5% for the others as was obtained from nuclear magnetic resonance (1H NMR) for the hydrogenous material. The characterization yielded Mn = 20100 g/mol (R20), Mn = 10900 g/mol (R10), and Mn = 5280 g/mol (R5) and 1860 g/mol (R2) using 1H NMR.6,16,17 The deuterated linear chains were synthesized by anionic polymerization. For the deuterated linear polymers, molecular weights corresponding to hydrogenous polymers are given: Mn = 1790 g/mol (L2), Mn = 5010 g/mol (L5), Mn = 8900 g/mol (L10), Mn = 21300 g/ mol (L20),Mn = 39400 (L40), and Mn = 82500 g/mol (L80) were obtained by size exclusion chromatography (SEC) with PEO calibration. The polydispersity indices Mw/Mn were 1.01 for the hydrogenous PEO and 1.03 for the deuterated PEO, respectively. Blends of hydrogenous ring and deuterated linear polymers (h/d) were prepared in solution and freeze-dried from benzene. The volume fraction of the hydrogenous polymers in h/d blends was 10% in all but one experiment. All materials are listed in Table 1a; the investigated blends in Table 1b.

Table 1. Overview of the materials used and the R/L blends indicating the ring fraction in volume (a) molecular weights of the rings and linear chains always in h-equivalents 2K [g/mol] rings (R) linear (L)

5K [g/mol]

10K [g/mol]

20K [g/mol]

40K [g/mol]

1860 5280 10900 20100 − 1790 5010 8900 21300 39400 (b) percent composition of the investigated blends

80K [g/mol] − 82500

R/L

L2

L5

L10

L20

L40

L80

R2 R5 R20

20 10 10

− 10 −

10 10 −

10 10 10

− − 10

10 10 10

The materials are coded as Rxx or Lxx where R or L denotes the topology either ring or linear and xx the nominal molecular weight in kilodaltons. The ring concentrations are always below the overlap concentration of the rings. For instance, for the largest ring (20K) the overlap concentration is 16 vol %. The samples were infiltrated in 5 mm NMR tubes during one or 2 days depending on the viscosity and sealed under high vacuum conditions to avoid the contact of the polymer with air.

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RESULTS Figure 1 displays Minispec data on the hydrogenated R2 and R5 rings in different deuterated hosts such that the experiment observes the motion of the rings. D


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Macromolecules

Figure 1. Diffusion spin−echo decays obtained for the rings 2 kg/mol R2 (a) and 5 kg/mol R5 (b) in different linear matrices (see legend). The observation time in each experiment was set to Δ = 15 ms. The diffusion coefficients are presented in the Table 2.

τrep ∼ p3.5

displayed in Figure 2 and Table 3. We emphasize the overall good quality of the data description, indicating that the picture of two different modes of diffusion is well supported by the data. The data were fitted applying the Levenberg−Marquardt algorithm. The quoted statistical errors were calculated in evaluating the covariant matrix. The two-state diffusion description surprisingly does not indicate any exchange between the two diffusive modes: τf ≫ 1 s. The parameter thf presents the thermal occupation of the fast mode that varied between thf = 0.1 for R20/L20 and thf = 0.33 for R20/L40; R20/L80 with thf = 0.28 is somewhere in between. Thus, the R20 ring largely is diffusing in the slow mode. For R5/ L20, the situation is the opposite: basically all rings diffuse in the fast mode, but the slow mode is clearly visible on the diffusion spin echo decay (Figure 4). The insert presents the fit results for the diffusion coefficients from the Minispec data for R5/L20 and R5/L80 as a function of observation time Δ. For the L20 host a clear saturation of the diffusion coefficient with Δis visible, while for the L80 host the apparent D seems to go on dropping. Considering the very different ranges in q2 that are covered by Minispec and the 600 MHz instrument: qmax2 = 9.5 × 1012 m2/s and qmax2 = 5.2 × 1014 m2/s, respectively, both instruments are sensitive to quite different regimes of the fast process. The plotted lines in the Figure 4 represent fit results, where we have fixed Df to the Minispec result. The comparison with the data shows, that apparently, a distribution of faster processes persists. Comparing the values of the diffusion constant in the 2 diffusive modes, we note that they differ by 1 to 2 orders of magnitude with an increasing difference toward higher host Mw. Figure 2 displays the results as a function of matrix length. Clearly the fast mode shows a much weaker host length dependence than the slow mode.

as expected for entangled linear melts in the range (2.5 pe ≤ p ≤ 40 pe). Only the L2 melt is off as Mw = 2 kg/mol corresponds to the entanglement molecular weight. The characteristic times τR for both the R5 as well as the R20 ring follow weaker power laws, while the τR2 is constant for larger p. For R20, we find τR20 ∼ p2.2; for R5, τR5 ∼ p1.7 holds. As for the diffusion coefficient for the R5 ring in various matrices, we observe a saturation of τR5 for the longest host L80. For the discussion of the results on the ring/linear blends, we start with the observations on R2 and R5 before we shall occupy us with the larger R20. Finally, we briefly discuss the diffusion results on the linear host. R2 Ring. An earlier NSE investigation,16 found that compared to the R2/L2 ring blend the pure ring melt displays a surprisingly fast center of mass diffusion. These results agree qualitatively with atomistic MD simulations.32 The fast diffusion turned out to be an explicit violation of the Rouse model. The smaller ring size was argued to “obviously generate at least partly a shielding of the monomers from the heat bath resulting in a smaller friction per monomer compared to the linear chain and the Rouse prediction”, respectively. At the same time, the ring/linear mixture, where, as found by SANS experiments, the ring size increases, displays close to standard Rouse diffusion as expected for a nonentangled melt: Calculated Rouse diffusion wouldbe 5.56 × 10−11 m2/s very close to the value 5.6 × 10−11 m2/s observed. For longer entangled hosts we found the ring diffusion to be remarkably constant: between host molecular weights of 10 kg/ mol to 80 kg/mol (5 to 40 entanglements) DR does not change but is constantly a factor of 4 below the Rouse value. We conclude the following. 1. Apparently, there are no long-living threading events, since they would impose a dependence on the size of the host. E.g., for once threaded rings, eq 8 would predict D ∼ 1/p. Another explanation would be a collapse of the ring. However, SANS data from a blend R2/L20 showed a perfectly Gaussian conformation of the ring.33 Finally, we may rationalize this finding by the observation, that the Kuhn length of PEO (8.3 Å) and the ring radius of gyration (11 Å)16 are of similar order preventing threading. 2. The ring is not confined by entanglements and performs Rouse type diffusion.

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DISCUSSION Aside of the diffusion results as a further means of discussion we also present the characteristic times (Figure 5) for the slow mode (so far it could be separated) τrep / R =

RL / R 2 6DL / R

(18)

(17)

Figure 5 and Table 4 very nicely show the consistency of all results; the characteristic times for the relaxation of the linear melts over a large range follows a power law E


Article a In addition, the diffusion coefficients for the pure linear melts (L bulk) are shown. bDiffusion coefficient measured at maximal possible observation times (Δ = 300 and 600 ms for 20K and 80K matrix respectively). c600 MHz Instrument. dExtrapolated value. eTaken from ref 16.

L80

− − − − (5.20 ± 0.07) × 10−14 c

(1.45 ± 0.05) × 10−11 (3.2 ± 0.3) × 10−13 b − − 1 × 10−14d

L40 L20

1.44 ± 0.05× 10−11 (6.7 ± 0.1) × 10−13 b − − (1.75 ± 0.08) × 10−13 1.37 ± 0.05× 10−11 (2.5 ± 0.01) × 10−12 − − (1.7 ± 0.1) × 10−12

L10 L5

− (7.5 ± 0.1) × 10−12 − − (8.7 ± 0.5) × 10−12

L2

(5.6 ± 0.08) × 10−11 (2.3 ± 0.1) × 10−11 − (4.9 ± 0.01) × 10−12 (5.4 ± 1.5) × 10−11 e

R bulk

(10.0 ± 0.1) × 10−11 (2.4 ± 0.01) × 10−11 (1.06 ± 0.17) × 10−11 (2.6 ± 0.01) × 10−12 − R2 R5 R10 R20 L bulk

Table 2. Diffusion Coefficients [m2/s] Obtained from the Slope of the Spin Echo Decays Measured by Minispec for Different Rings in Various Linear Matrices (See Figure 1)a

Macromolecules

Figure 2. Diffusion coefficients of the rings as a function of molecular weight of the matrix. Where not explicitly shown, the error bars for the diffusion coefficient are smaller than the size of the symbols: dashed red line, matrix independent diffusion of R2; full black line, power law behavior of linear chain diffusion p−2.4; solid blue line, slow R20 diffusion (guide to eye); dashed black line, prediction of eq 7 for the reptation diffusion of a flat R20 ring in the tube provided by the matrix; dashed-dot blue line, prediction of eq 8 for the diffusion of a once threaded R20 ring; dashed blue line, prediction of eq 2 for diffusion by constraint release of a R20 ring.

3. However, the effective friction is a factor of 4 higher than the Rouse expectation. This observation reminds on the diffusion of POSS particles of similar size in long chain PEO-melts.34 There an increase of the effective friction by a factor of 4 was found. Lungova et al. concluded that the microrheological viscosity on a scale close to the tube size is not only determined by the Rouse friction but includes further resistance most likely relating to the topological constraints imposed by the other chains. However, for POSS the increase in friction depends on the Mw of the entangled host. R5 Ring. The R5 diffusion came out to be equal or faster than the diffusion of the linear host. The characteristic times exhibit a power law τR ∼ p1.7 in between the predictions for once threaded rings and CR. For the longest host chain (L80) the power law levels off, which may be related to the time dependent effective diffusion coefficient not saturated yet (Figure 4, insert). Most of the data were obtained by Minispec, showing always a single exponential decay. However, the apparent average diffusion coefficient decreases, if the observation time is prolonged (Figure 4 insert). This indicates the presence of a slower diffusive mode that reduces the average diffusion with observation time Δ. The R5/L20 sample was also studied with the 600 MHz instrument (see fit results in brackets in Table 3). Figure 4 shows that these data display a two-mode structure with the slow mode only weakly occupied. Inspecting the quality of the fit with the twostate model, we realize that other than for the R20 ring (Figure 3) the description of the initial fast mode is not satisfactory, while the slow mode is well described. The significant difference between the Minispec result that covers only a small q2Δ range F


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Figure 3. Diffusion spin echo decays for R20 in L40 (a) and R20 in L80 (b) measured at different observation times (see legend). The lines present fits with the Kärger’s model eqs 12−16.

Table 3. Results of Fitting with Two-State Kärger’s Modela R20/L2 2

Df [m /s] Ds [m2/s] thf τf [s]

R20/L20 −12

(4.9 ± 0.01) × 10 − − −

R20/L40 −13

(3.2 ± 0.03) × 10 (1.33 ± 0.01) × 10−14 0.100 ± 0.004 ≫1

R20/L80 −13

(1.26 ± 0.10) × 10 (2.54 ± 0.13) × 10−15 0.33 ± 0.03 ≫1

R5/L20 −14

(5.05 ± 0.14) × 10 (6.86 ± 0.12) × 10−16 0.28 ± 0.03 ≫1

−13

(6.7 ± 0.1) × 10 [(4.1 ± 0.1) × 10−13] (7.4 ± 0.01) × 10−14 [(5.6 ± 0.1) × 10−14] 0.96 ± 0.001 (0.975 ± 0.001) ≫1

Fast and slow diffusion coefficients Df and Ds, the thermal occupation of the fast diffusing state thf and the lifetime of the fast process τf for the R20 ring in various matrices. Values in bracketssee Discussion.

a

and the outcome from the 600 MHz instrument (6.7 × 10−13 m2/ s vs 4.0 × 10−13 m2/s) also originates from the apparent distribution of fast processes evident in Figure 4. For the L20 host we may now compare the experimental results with the predictions for the various modes of diffusion: (i) ring reptation in the tube of the host eq 7 DR = 7.6 × 10−12 m2/s; (ii) once threaded rings eq 8 DR = 9.42 × 10−13 m2/s; (iii) diffusion by constraint release eq 2 DR = 4.16 × 10−15 m2/s; (iv) constraint release using the measured linear diffusion coefficients eq 6 Dtube,R = 6.83 × 10−14 m2/s; (v) and finally the Shanbhag mechanism eq 10 DR = 1.94 × 10−14 m2/s. We note that the experimental result for slow diffusion of DR = 7.4 × 10−14 m2/s is close to identical to the prediction of eq 6 suggesting that the slow mode is driven by CR; eq 2 comes up with a much too slow diffusion indicating that corrections to reptation that are inherent in eq 6 are highly important. Finally, the Shanbhag mechanism is also too slow. The experimental result for the fast mode 6.7 × 10−13 m2/s is not too far away from the prediction for once treaded rings (DR = 9.42 × 10−13 m2/s). However, the obvious distribution of diffusion events evident in Figure 4 suggests a superposition of a number of processes that may relate to important corrections necessary for the relatively small ring. We may also compare with Kramer’s results on PS. For that purpose, we need to rescale the data by the ratio of the very PEO different entanglement molecular weights (MPS = e = 18K; Me 2K); thus, the 5K PEO ring corresponds to a 45K PS ring; Kramer’s data are compatible with an even weaker power law with an exponent of about 1.5. As an explanation, the PS paper

hypothesized about contributions from ring reptation eq 7) and once threaded rings eq 8. R20 Ring. Other than for the two smaller rings and thanks to the higher resolution measurements relating to significantly larger values for q2 and longer observation times Δ for R20, we explicitly could identify different fast and slow diffusion modes (Figures 2 and 3). In view of very recent simulationscoarse grained, Monte Carlo and atomisticthe finding of two different well-defined and well separated diffusion modes is rather unexpected. By topological analysis on atomistic ring/linear PEO blend simulations the Mavrantzas group reported a broad distribution of threading events.18,35 For the case of R10 rings in a L20 linear matrix, they find at least six threadings per ring, which would inhibit the observed fast process. Similarly Shanbhag’s bond fluctuation MC results indicate multiple threading and constraint release motion only.36 In a R5/L5 blend, Mavrantzas’ threading lifetime analysis reveals very longlived threadings that last for up to two orders longer than the Rouse time of the ring. For our system, we found τRouse (R5) = R 1.78 × 10−8 s; τR(L5) = 4.3 × 10−7 s a factor of about 24 slower than the Rouse time. We note that τR(L5) reflects the average slowdown of motion including all different states of threading. The simulations found important heterogeneities with no particular evidence for a fraction of once treaded rings with distinctly different diffusion properties. Similarly, in the various simulations, Shanbhag’s group did not find explicitly separated diffusive modes but at high linear concentrations concluded completely CR related ring motion. G


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Macromolecules

Figure 4. Diffusion spin echo decay for 5K rings in the L20 matrix at Δ = 200 and Δ = 400 ms. The lines represent fitting with the two-state Kärger’s model eqs 12−16, where the fast component is fixed to the average diffusion coefficient 6.7 × 10−13 m2 s−1 measured by Minispec. The insert presents the dependence of the apparent average diffusion coefficient on the observation time Δ measured at Minispec (black symbols, R5/L20; red symbols, R5/L80).

Figure 5. Characteristic times of R2, R5, and R20 rings in blends as a function of molecular weight of the linear matrix. As a reference the characteristic times of the linear matrices are presented (black circles). The solid lines represent fits with a power law. The dashed line is an extension of the p−1.7 power law to the longer matrices (>20 000 g/mol). Note, the last point for R5/L80 (in brackets) relates to the unsaturated value (Figure 4, inset).

In contrast the evaluation of earlier experimental results in particular by Kramer’s group anticipated different well-defined diffusive modes but could never explicitly separate them in an experiment. We proceed in quantitatively comparing our results with Klein’s scaling analysis. In Figure 2, we display the predictions for (i) reptation of a nonthreaded ring eq 7, (ii) for conventional constraint release (CR) eq 2, and (iii) for the motion of once threaded rings eq 8. The dashed black line in Figure 2 presents the expectation for diffusion of unthreaded rings. Obviously, this process is not observed. Even though entropically this process is unlikely, we note that the most detailed earlier investigations by Kramer’s group on PS ring/linear blends assumed this mechanism, in order to rationalize their results on the observed average diffusion behavior. The dashed-dot line displays the prediction for ring motion of once threaded rings eq 8: Astonishingly this prediction nearly quantitatively agrees with our results for the fast diffusive process. Not only the magnitudes are very close but also the dependence on the matrix length “p” is well compatible with the predicted p−1 power law. We note, (i) this agreement might be coincidence (ii) nevertheless this process was inferred by Kramer, in order to properly describe the average diffusion data− here we seem to observe it directly. The thermal occupation of the fast mode is between 10% and 33%. The PS systems in terms of the entanglement numbers P are well comparable with the PEO-blends. For the interpretation of the diffusion results Kramer needs up to 2/3 of once threaded rings. Figure 6 compares the experimental data on the slow diffusive mode with various theoretical predictions. The full blue line shows the prediction of eq 2 for diffusion originating from CR. The experimental results are more than 1 order of magnitude above this prediction. Furthermore, also the dependence on the

host length “p” appears to be weaker. The dashed blue line displays the CR-prediction including Klein’s p1/2 correction (eq 6; middle part): The correction goes in the right direction, the weaker slope in p is now very similar to the experiment, while the absolute values are about a factor of 2 too high. Since we have measured independently also the host center of mass diffusion, we may follow eq 6 (right part) and express the slow diffusion mode in terms of the measured reptation diffusion of the host Drep(p). As mentioned in the theory section the measured host diffusion coefficients contain all corrections to reptation such as CLF or CR that need to be considered for a proper theoretical description of the host motion. In our theoretical section these corrections were neglected so far but are included in the experimental results for Drep (p). Application of eq 6 in Figure 6 leads to the crosses and the black solid line in near quantitative agreement with experimental results. Table 5 lists the resulting Dtube,R for 20K rings. Finally, based on simulation results, very recently the Shanbhag group came up with a different hypothesis about the diffusive motion of rings in a linear polymer melt (see also theory section). eq 10 states a simple relationship between the ring diffusivity in the corresponding ring melt and the chain diffusion in the host. Figure 6 includes the prediction of eq 10 as red squares and a red line. Both the predictions of eq 6 and 10 are relatively close to the experimental results. However, the host chain length dependence of the latter is stronger and therefore appears to be less compatible with the experimental data. The Klein approach essentially predicts Dtube,R ∼ 1/p5/2 while the H


Article

Macromolecules Table 4. Characteristic Times for Linear Chains (a) and Rings (b) (a) linear chains R2(p) [m2]

chain

−17

1.53 × 10 3.83 × 10−17 7.76 × 10−17 1.53 × 10−16 3.07 × 10−16 6.14 × 10−16

L2 L5 L10 L20 L40 L80

τrep(p) [s]

Drep(p) [m2/s] 5.9 × 10−11 8.7 × 10−12 1.7 × 10−12 1.75 × 10−13 5.2 × 10−14 1 × 10−14 b) rings in linear matrices

6.8 × 10−8 7.3 × 10−7 7.7 × 10−6 1.5 × 10−4 9.8 × 10−4 0.01

τR [s] ring

R2(n) [m2]

L2

L5

L10

L20

L40

L80

R2 R5 R20

7.65 × 10−18 1.91 × 10−17 7.65 × 10−17

2.3 × 10−8 8.2 × 10−8 2.6 × 10−6

− 4.3 × 10−7 −

9.3 × 10−8 1.3 × 10−6 −

8.8 × 10−8 4.8 × 10−6 9.6 × 10−4

− − 5.0 × 10−3

8.8 × 10−8 1.0 × 10−5 0.02

Table 5. Diffusion Coefficients for 20K Rings in Different Hostsa host 20K 40K 80K

Ds [m2/s] −14

1.33 × 10 2.54 × 10−15 6.86 × 10−16

Drep [m2/s] −13

1.75 × 10 5.1 × 10−14 1 × 10−14

Dtube,20K [m2/s] −14

1.79 × 10 3.84 × 10−15 5.21 × 10−16

DR(0.9) [m2/s] 1.93 × 10−14 2.85 × 10−15 2.78 × 10−16

a

Experimental values for the slow ring diffusion Ds; the reptation diffusion of the linear host Drep; the predicted Dtube,20K for the 20K ring (eq 6 and the prediction of Shanbhag DR at ϕL = 0.9 (eq 10).

Figure 7. Diffusion coefficients of the rings vs molecular weight in 2K (L2) matrix and 20K (L20) matrix. The lines are power laws n−1.0 (red line) and n−3.0 (blue line).

Figure 6. Diffusion coefficients of 20K rings (slow component, see Table 3) vs molecular weight of linear matrix (blue squares). The solid blue line is the diffusion originating from CR eq 2). Triangles/dashed line indicate corrected CR prediction (eq 6. Crosses/solid black line show the prediction in terms of the Klein correction involving the reptation diffusion of the host Drep(p) eq 6; right part). Red squares/ solid red line represent Shanbhag’s hypothesis (eq 10. Cartoon in the inset represents CR mechanism for the threaded rings.

predictions, eqs2 and 6. Our result qualitatively agrees with the earlier results on PS ring/linear blends (Dring(n) ∼ n−3.2) for large matrices. As we are aware of, so far simulation have not yet studied this phenomenon. The two-state model also informs about the rate of state changes between the fast and slow diffusive modes: All data fits consistently resulted in very large life times (τf ≫1 s) for the fast state very much beyond what could be detected in the PFG NMR data. This result surprises and is difficult to rationalize. E.g. the characteristic times for the R20/L40 blend (Table 3) are τrep(40K) = 9.8 × 10−4 s and τR = 5 × 10−3 s. One would think that after the time τR has elapsed, a ring has completely freed itself from its original constraints. We wonder: why for the much longer observation times compared to these intrinsic times is

Shanbhag conjecture suggests DR(ϕ)L ∼ 1/p3. The data support the first approach better. Now we turn to the dependence of the characteristic times on the ring size: Figure 7 displays the ring diffusion coefficients as a function of ring size for different host lengths. For the L2 host, DR(n) ∼ 1/n, as expected for Rouse diffusion of the ring in the short matrix, where only the number of friction exerting beads is counting. However, for the rings in L20, the n-dependence is much stronger DR(n) ∼ n−3 in contradiction to the scaling I


Article

Macromolecules ⎛ pe2 ⎞ ⎟ = Drep(1 + δ) Deff = Drep + DCR = Drep⎜⎜1 + 3/2 ⎟ p ⎠ ⎝

there not the possibility of a change between the two diffusion modes, or why is the experiment not in the fast exchange regime? In order to better understand the data, we tested them against shorter life times τf. Figure 8 displays such a test for the R20/L40

Table 6 presents the values for Drep, δ and Deff. Figure 9 compares Drep, Deff, and DL with each other. While pure reptation leads to a slope of −2 instead of the experimental

Figure 8. Diffusion spin echo decay for the R20/L40 blend measured at Δ fixed to 700 and 1000 ms. The lifetime of the fast component was fixed to τf = 1 s (dashed lines) and τf = 0.1 s (solid lines). In the inset the magnification of the small q-range is shown.

Figure 9. Diffusion coefficient of linear matrix Dexp, the reptation prediction Drep eq 1, and the effective diffusion Deff eq 19. Black line corresponds to the slope p−2.4, and the red line is a fit to Deff resulting in a power law p−2.3.

blend, where for fixed τf = 1 s and τf = 0.1 s a best fit of the data was undertaken. The comparison of the fitting results with fixed lifetimes with the data in the observation window 700 ms ≤ Δ ≤ 1000 ms clearly demonstrates a significantly stronger dispersion in Δ for the predicted PFG NMR spectra than observed experimentally. We need to conclude that the data do not give a hint for diffusion mode exchanges within the accessed time range. These results pose questions that need to be answered in the future. Finally, in order to investigate the consistency of the Klein correction (p1/2) also for the diffusion of the linear host, we interpret the matrix diffusion data in assuming two different diffusion channels, one by reptation (Drep; eq 1) and the other one by CR, including the Klein correction DCR =

(19)

value of −2.4 and underestimates the diffusion constants for the shorter chains, the reptation diffusion augmented by diffusion through CR including Klein’s correction nearly quantitatively agrees with the experimental results. Furthermore, the slope of −2.3 for Deff is very close to the experimental molecular weight dependence. Considering CR as a second diffusion channel, we can describe the experimental data rather well. Thus, using the same approach for the description of the ring and the host diffusion respectively, we are able to describe both in a consistent way. We conclude that the simple heuristic consideration of an interdependence of constraints leads to a nearly quantitative prediction for both the slow mode of the ring motion as well as for the diffusion within the linear matrix. Compared to the heavy calculations needed, in order to properly calculate the effect of contour length fluctuations and constraint release,37 this is a remarkable result.

pe 3 4 D . Then 0 15 p7/2

the effective diffusion coefficient for the combined mechanisms would be

Table 6. Chain Length in Units of Monomers p, Correction Factor (1 + δ), Reptation Diffusion Drep, and Combined CR and Reptation Diffusion Deff Compared to the Experimental Results for the Matrix Diffusion (DL) from Table 2a

a

Mw

5K

10 K

20 K

40 K

80 K

p Drep 1+δ Deff Dexp

114 2.11 × 10−12 2.66 5.61 × 10−12 8.7 × 10−12

227 5.32 × 10−13 1.59 8.46 × 10−13 1.7 × 10−12

455 1.33 × 10−13 1.21 1.61 × 10−13 1.75 × 10−13

909 3.31 × 10−14 1.07 3.54 × 10−14 5.2 × 10−14

1818 8.28 × 10−15 1.03 8.53 × 10−15 1.0 × 10−14

The units for the diffusion coefficients are m2/s. J


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ORCID

CONCLUSIONS Using high resolution PFG NMR spectroscopy, we have studied the diffusion of polymer rings in linear host matrices at various observation times Δ for different ring and host molecular weights. In order to focus on the ring motion, the linear hosts were deuterated. The data for the larger rings in entangled hosts explicitly display two different diffusive modes: (i) fast diffusion that depends inversely on the host chain length and (ii) much slower diffusion depending much more strongly on the host molecular weight. In addition, we studied the diffusion of the linear chains in the host melts. For the two lower molecular weight rings the behavior is different: The diffusion coefficient of the R2 ring for entangled hosts does not depend on the host molecular weight. The R5 ring displays an intermediate behavior with a weaker dependence on the host Mw. A high resolution analysis of R5 in L20 showed that also this smaller ring displays fast and slow modes, with the slow mode very close to the prediction of tube renewal. First of all, our explicit observation of two well-separated diffusion modes contradicts simulation results that all find a broad distribution of heterogeneities compatible with a CR picture with no further evidence of a fast mode. In contradiction to these results the observed two-mode structure directly verifies the hypothesis of qualitatively different possibilities for the rings to diffuse. Quantifying as much as possible the scaling predictions, we find that the fast mode quantitatively agrees with the assertion of a special mode of diffusion for once threaded rings, while the suggestion of unthreaded rings undergoing reptation in the tubes provided by the host could not be supported. For the slow mode, a number of different predictions are presented in the literature. A detailed comparison with the data showed that diffusion by a conventional CR process would be too slow. On the other hand, following Klein’s suggestion of a mutual dependency of constraints together with employing the measured diffusivities of the host for calculating the diffusion coefficient for the slow mode, we arrive at a nearly quantitative agreement between prediction and results. The achieved quality of data description may be taken as a direct confirmation of the proposed interdependence of constraints. We note that with this approach also the Mw-dependence of the matrix diffusion may be described naturally. Thus, the simple Klein suggestion of interdependent constraints leads to a consistent picture for both ring probe diffusion in a linear matrix and for the linear matrix itself. Using the semiempirical theory of Shanbhag that was based on MC simulations, leads to a slow mode that, though also close to the experimental results, indicates a stronger dependence on the host chain length as observed. As found earlier by the Kramer’s group the strong size dependent ring diffusion in an entangled matrix remains unintelligible. Here new simulations might deliver a clue. Finally, experimentally we have distinguished two well-defined significantly different diffusive modes with characteristic times in the millisecond range or below that would be expected to interchange. However, a detailed analysis in terms of a twostate diffusion model allowing for state changes clearly shows that within the experimental sensitivity of several seconds no such exchanges take place. Again, a theoretical understanding is still missing.

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Margarita Kruteva: 0000-0002-7686-0934 Jürgen Allgaier: 0000-0002-9276-597X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank S. Gooßen for the synthesis of the ring polymers,6,30 C. Hövelmann for the sample preparation, and M. Monkenbusch for the help with a fitting program.

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NOMENCLATURE number of monomers in the ring entanglement length (in monomers) for the corresponding linear chain N = n/ne entanglement number for the ring Ln length of the primitive path for the ring p number of monomers per linear chain pe ≡ ne entanglement length (in monomers) of the linear chain P entanglement number for the linear chain Lp length of the primitive path for the linear chain n ne

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REFERENCES

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*(M.K.) E-mail: [email protected]. K


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Direct Observation of Two Distinct Diffusive Modes for Polymer Rings

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