Melts of single-chain nanoparticles: A neutron scattering investigation

The impact of purely intramolecular cross-linking on the properties of a polymer melt is studied by neutron diffraction and quasielastic incoherent and coherent neutron scattering on a system composed exclusively of single-chain nanoparticles. As a reference, a parallel study is presented on the melt of the linear precursor chains ’ counterpart. Associated with structural heterogeneities provoked by the internal compartmentalization due to cross-links, a dramatic slowing down of the relaxation of density fluctuations is observed at intermediate length scales.


I. INTRODUCTION
Single-chain nanoparticles (SCNPs) are nano-objects obtained by intramolecular cross-linking of individual macromolecular chains. Due to their ultrasmall size, softness, and internal compartmentalization, they are believed to be a fundamental ingredient in the field of nanotechnology. 1 Concerning practical applications of SCNPs, several proof-of-concept experiments have confirmed their potential use in catalysis, biosensing, bioimaging, protein mimicry, drug delivery, and all-polymer nanocomposites, among other applications. For instance, metalloenzyme mimetic SCNPs showed enhanced catalytic activity 2 and selectivity 3 toward a variety of chemical compounds, as well as in some cases catalyst recyclability. 4 Moreover, SCNPs can be used as nanoreactors for the synthesis of other nanomaterials (e.g., gold nanoparticles 5 and quantum dots 6 ) and even synthetic macromolecules (e.g., watersoluble polymers 7 and intrinsically conducting polymers 8 ) or as templates for the synthesis of fluorescent carbon nanodots 9 as well as reversible nanocontainers for CO 2 storage. 10 Several SCNPs are able to sense metal ions 11 and even proteins. 12 A variety of SCNP systems have been designed and constructed for bioimaging (e.g., magnetic resonance imaging, 13 single photon emission computerized tomography, 14 and fluorescence imaging 15 ) and drug delivery applications (e.g., controlled delivery of chiral amino acid derivatives, 16 peptides, 17 vitamins, 18 and drugs 19 ). Also, SCNPs have been employed for the development of all-polymer nanocomposites. 1,20 In addition to such applied interest, SCNPs can be considered model macromolecules to address basic questions on the effect of chain topology on the material properties. For instance, a system composed exclusively of SCNPs realizes the situation of a purely intramolecularly cross-linked polymer bulk. Thus, investigations on a melt of SCNPs can help address one fundamental question: What is the impact of the chain topology on the properties of a polymeric material and, in particular, how does it behave if cross-links are of purely intramolecular character? This basic question also arises motivated by previous results obtained on a nanocomposite consisting of SCNPs embedded in a matrix of flexible linear chains. [21][22][23] There, a clear impact of the presence of SCNPs was found on the dynamics of the matrix at different levels (glasstransition, Rouse dynamics, and, particularly striking, on the entanglements).
In this work, we present a neutron scattering investigation on melts of SCNPs. Neutron scattering provides spatial resolution through the scattering vector (Q) dependence of the measured magnitudes. Energy-resolved experiments have been applied to both protonated and perdeuterated systems to follow self-atomic (hydrogen) and collective motions, respectively. Complementary neutron diffraction has revealed the structural features. A bulk sample composed of the long linear counterpart chains ( precursor macromolecules without cross-links) has also been investigated as a reference. Our study demonstrates that the structural and dynamical properties at local length scales including the intermolecular distances (in particular, the structural relaxation) are hardly sensitive to chain topology. Its impact, as revealed by this microscopic technique, consists of a slowing down of collective dynamics occurring in the region of the so-called intermediate length scales (ILSs). The ILS region corresponds to length scales larger than the typical intermolecular distances but not yet in the hydrodynamic regime. This is an almost unexplored region in glass-forming systems, and here we apply a recently developed ansatz 24,25 to describe it in the linear precursor melt. In the proposed scenario, a crossover from diffusivelike collective dynamics at intermolecular length scales toward the viscoelastic coupling of stress and density fluctuations at mesoscales is invoked. The behavior of the melt of SCNPs indicates a strong shift of such a crossover to larger length scales. The slower collective dynamics appearing at ILS in the SCNPs-apparently dominated by the slow diffusive component in this framework-is associated with emerging structural heterogeneities with a characteristic length of about 1 nm. We discuss these findings also in the light of recently reported results on macroscopic investigations by rheological and dielectric spectroscopy techniques of these systems. 26

A. Samples
Protonated and deuterated SCNPs (hNP and dNP) based on tetrahydrofuran (THF) were obtained from protonated or completely deuterated copolymer precursors (hPrec and dPrec) via intramolecular azide photodecomposition at high dilution conditions. Details about the chemical reactions involved and characterization of hNP and dNP can be found in Refs. 26 and 27. The chemical formula of the deuterated precursor (dPrec) is shown in Fig. 1. The THF-fraction x was 0.81 for hPrec and 0.82 for dPrec. Molecular weights M w were 33.2 kg/mol and 36.8 kg/mol with polydispersities M w =M n of 1.55 and 1.23 for hPrec and dPrec, respectively. The glass-transition temperatures T g , 202 K (hPrec) and 199 K (dPrec), increased by 1 K in the corresponding SCNPs' bulk.

B. Neutron scattering experiments
The QENS experiments on the protonated samples were carried out by combining the time-of-flight FOCUS instrument 28 31 In fully deuterated samples, the scattered neutron intensity is dominated by the coherent contribution, where all atomic pair correlations are approximately equally weighted. 32 Three temperatures well above T g were investigated for both kinds of samples. Accessing low Q-values, the structural properties at large length scales were explored on the deuterated systems by small angle neutron scattering (SANS) in the D11 instrument at the ILL. 33 The integrated intensity of FOCUS spectra complemented this information for high Q-values in the neighborhood of the amorphous halo. Figure 2 shows QENS results after Fourier transformation into the time domain and deconvolution from instrumental resolution. The panels on the left correspond to the hPrec sample and on the right to the hNP sample. FOCUS (circles) and SPHERES (diamonds) data are combined.

III. QENS RESULTS: H-SELF-MOTIONS FROM PROTONATED SAMPLES
Nowadays, it is well established 34 that in glass-forming polymers above T g , the results on protonated samples-corresponding to the intermediate incoherent scattering function of the hydrogens-can be well described above % 2 ps by means of a Kohlrausch-Williams-Watts (KWW) or stretched exponential with a stretching exponent β close to 0.5. To check that the melts investigated follow this "standard" behavior, we imposed the constant value of 0.5 for the β-exponent in the fits of the results shown in Fig. 2. As shown, this description works rather well for both sets of data, at least in the accessed dynamic window covered by the experiments. The average characteristic time for H-self-motions hτi ¼ τ H s Γ(1=β)=β, represented in Fig. 3, shows a strong dispersion in Q, being well approximated by a power law hτi / Q À2=β (i.e., /Q À4 ) in the Q-range Q & 1 A À1 . This relation between dispersion and stretching is indicative of Gaussian diffusive behavior with an associated sublinear increase of the mean squared atomic displacement. 35 Deviations from this asymptotic law at higher Q-values can usually well be accounted for by assuming a distribution of local jumps as the underlying microscopic mechanism for such an anomalous diffusion, which can be resolved by the QENS experiments when exploring small enough length scales. 36,37 In the framework of the anomalous jump diffusion (AJD) model, based on these ideas, the characteristic time for the self-correlation function can be expressed as Here, ' o is the preferred jump length. The resulting values of the parameters involved in the description by this model are provided in Table I. Note that, in this case, hydrogen dynamics is followed by the experiments and consequently, the obtained parameters correspond to these particular nuclei.

IV. COLLECTIVE FEATURES FROM DEUTERATED SAMPLES
The neutron results on the deuterated samples reflect collective features. Figure 4 shows the differential cross section measured on the samples-revealing static properties, in particular, the structure factor S(Q)-and in Fig. 5, we can see representative NSE curves representing the normalized dynamic structure factor S(Q, t)=S(Q). We first consider the results corresponding to the linear precursor chains. As can be seen in Fig. 4, the structure factor presents a well defined amorphous halo centered at around In main-chain polymers, this peak can usually be attributed to correlations between atoms belonging to nearest neighbor chains, 38 separated by an average interchain distance of about d % 2π=Q max . Here, d % 4:5 A . To describe the low-Q region of S(Q), a constant function has been considered, as expected for a homogenous system at large length scales with respect to the intermolecular distances. There, long-range density fluctuations are reflected by the structure factor. A low-Q contribution that can be accounted for by a law /Q Àx , with x ¼ 3:5, has also been considered. This additional scattering is usually found in fully deuterated polymer melts and could be attributed to the presence of microbubbles. As shown in the inset of the figure, the power law persists down to the lowest-Q values accessed in the SANS experiment.
For the intermediate temperature investigated, Fig. 6 shows the NSE results on the normalized S(Q, t) at different Q-values. Let us remind that, for Q % Q max , these data directly reveal the temporal evolution of interchain correlations. In the terminology of the Mode Coupling Theory (MCT), 39,40 these measurements would constitute the direct microscopic observation of the structural relaxation in the system. As shown in Fig. 6, the curves can be well described by KWW functions The values obtained for the β parameter are displayed in Fig. 7 as a function of the scattering vector. They show a kind of modulation with the structure factor around an average value of 0.47 for 320 K.
With increasing temperature, we observe a tendency of this value to increase, reflecting a less stretched feature of the dynamic structure factor. To determine the characteristic times of the NSE curves, we have assumed a Q-independent β-value equal to the average obtained for each temperature, namely, 0.  Approaches merely based on diffusive mechanisms predict for τ c a continuous slowing down with decreasing scattering vector in the low-Q region well below the structure factor peak. Therefore, they obviously fail in reproducing the experimentally observed acceleration of the collective dynamics at intermediate length scales. Around a given Q c -value located in the ILS region, the collective response of glass-forming systems crosses over from being dominated by diffusion to a region where it is dominated by the viscoelastic coupling of stress and density fluctuations. Only   beyond a certain length scale-somehow related to Q c -the system would appear as viscoelastically homogeneous. This crossover is captured by a model proposed by Novikov et al. 24 that was later reformulated for the case of glass-forming polymers or systems showing stretched relaxation behavior in general. 25 Such an extended approach was applied 25,42,43 to the case of polyisobutylene (PIB) using the NSE data previously reported in Ref. 44. The application of this model to the dPrec results is explained in detail in the Appendix. It provides the description of the collective times shown as dashed-dotted lines in Fig. 8. For the precursor melt, Q c % 0:6 A À1 .
We now move to the bulk made of SCNPs. Regarding the local structure, Fig. 4 shows that the amorphous halo of the SCNPs' bulk is exactly in the same position as that of the precursor chains. The average intermolecular distances and short-range order are thus hardly disturbed by internal cross-links. The collective motions involved in the structural relaxation as revealed by NSE in the neighborhood of Q max also take place in a very similar way in the two melts, as can be seen in Figs. 5(b) and 5(d). On the contrary, the decay of the S(Q, t) at ILS is dramatically affected by the presence of intramolecular cross-links. These features are directly evidenced in Fig. 5. While close to Q max , the NSE results on the SCNPs can be very well described by KWW functions with the same β-values as for the melt of precursor chains-and actually with very similar characteristic times-at lower Q-values (below % 0:6 A À1 ) the decay becomes more stretched and its characteristic time is longer. As shown in Fig. 5 and imposing the β-values describing the precursors' data at higher Qs. The average characteristic times obtained in this way are represented in Fig. 8 with filled symbols. For Q & 0:6 A À1 -approaching the ILS-they tend to continuously increase toward larger length scales, reminding a diffusion dominatedlike dynamics in a wide Q-range. In fact, they can be approximately accounted for by the purely diffusive component deduced for dPrec from the previous analysis. In the above commented framework for collective dynamics at ILS, for the melt of SCNPs, the crossover to a nondiffusive characteristic time at Q c would be strongly shifted to low Q-values (even below those accessed by the IN11c window), implying that the SCNPs bulk would behave viscoelastically homogeneous only at extremely large length scales. Is there any structural evidence for the existence of heterogeneities responsible for such a dynamical behavior at ILS? SANS experiments provide the answer to this question. As can be seen in Fig. 4, in the Q-region around 0:1 A À1 , SANS results on the melt of SCNPs show indeed a pronounced increase of the scattered intensity with respect to that in the precursor. Such an excess can be well accounted for by an Ornstein-Zernike expression, A value of about 9 Å for the characteristic length ξ is obtained. To fully describe the SANS results in the whole range investigated, this function is added to a low-Q contribution. The latter follows a similar power law as that found for the melt of precursor chains and might, therefore, be attributed to the same origin-the presence of microbubbles in the samples.

V. DISCUSSION
The dynamics of glass-forming polymers above T g is very complex and involves different processes and mechanisms for relaxation. The relevance of each of these contributions depends on the length scale of observation. At relatively local length scales of the order of intermolecular distances (equivalently, Q-values in the range 1À2 A À1 ), the main dynamical process is the α-relaxation related to the vitrification phenomenon. Let us first focus on the results obtained in such a Q-range. As aforementioned, collective motions reveal nearly identical features for the melt of SCNPs and the melt of linear precursor chains at this level. Regarding the selfmotions, the Fourier transformed QENS results on the hNP sample are also very similar to those on hPrec, as can be seen from Fig. 2.
After evaluation in terms of the AJD model, we can see in Fig. 3 and Table II that the values of the characteristic times and AJD parameters used to fit them are indistinguishable within the uncertainties. This implies that the atomic motions in the α-relaxation regime can be qualified as corresponding to an anomalous sublinear diffusion with underlying discrete primary steps and are very much the same for both melts. The differences in the behavior of the two systems start when smaller Q-values (equivalently, larger length scales) are explored. They are clearly patent in the coherent structure factor. The deviations are set in at Q % 0:6 A À1 . This is the range where also the crossover from α-relaxation to the so-called Rouse dynamics is expected to occur in glass-forming polymers. At large length scales, connectivity plays a crucial role in the dynamics of interconnected objects like macromolecules. The universal predictions of the "bead-spring"-like model proposed by Rouse 45 are expected to describe the dynamics for a melt of linear chains at sufficiently large length scales, as long as entanglement effects are absent (or for times shorter than the entanglement time, for entangled chains). The identification and characterization of the crossover between α-relaxation and Rouse dynamics are nontrivial problems that were subject of studies in the past. 34,46 To check the validity and explore the crossover regime at a microscopic level, two kinds of correlation functions can be used. One of them is the incoherent scattering function corresponding to self-motions of atoms within the segment. This function has been addressed in this work by QENS on hydrogenated polymers. In principle, within the uncertainties, the results on both samples at low-Q values are compatible with the Rouse predictions 45 for incoherent scattering of a KWW functional form with β ¼ 0:5 and characteristic time following a Q À4 -dependence. At the same time, as shown in Fig. 3, the results can also be reasonably described by the extrapolation of the above introduced AJD model toward the lowest Q-values reached. This indicates that, as in other polymers, 22 the crossover between both dynamical regimes takes place in a very subtle way and it is not easy to clearly define their limits. A detailed scrutiny of this crossover is beyond the scope of this work, where we want to focus on the impact of the cross-links on the observed dynamical features. Conversely, due to the strong Q-dependence of the incoherent scattering in the low Q-regime and the limited energy resolution of the spectrometers, the accuracy in the determination of the characteristic times prevents a precise quantification of their differences. From a direct comparison of the FOCUS results at low Q-values, a small difference could be envisaged between the two melts (see Fig. 9). The spectra corresponding to the SCNPs are slightly narrower, pointing to a subtly slower dynamics at large length scales. The other correlation function for which the Rouse model formulation is provided is the single chain dynamic structure factor. This is available by NSE on labeled samples where hydrogenated and deuterated chains are mixed and relates to pairs of atoms intramolecularly connected. Studies on such melts of polymers have demonstrated the validity of the Rouse model in its range of applicability. 34 This is, however, severely limited for long chains at long times. Long linear chains can easily interpenetrate. At long times, the topological constraints emerging from interpenetration together with chain uncrossability-the entanglements-lead to deviations from standard Rouse-like dynamics that can be accounted for invoking the tube concept and related reptation mechanism proposed by deGennes and Edwards. 34 The single-chain dynamic structure factor of the melts here investigated will be the subject of future work, where we will address the impact of intramolecular cross-linking on the entanglement formation with a direct microscopic insight. Here, we comment that in a recent work 26 including a rheological study of the melts here presented, an almost complete disappearance of the rubbery plateau has been found for the melt of SCNPs with respect to the precursor system. Such a striking effect suggests a strong suppression of the entanglements and was attributed to the internal multiloop topology of the SCNPs. [47][48][49] It is well known that, in ring polymers, the entanglement formation is strongly hindered with respect to their linear counterparts. [50][51][52] In the present paper, the NSE experiments were carried out on fully deuterated samples. They thus address the collective features involving all atomic pair correlations, independently of belonging to the same or different chains. There is no theoretical framework that expresses the accessed observable-the dynamic structure factor-in terms of the Rouse model or similar approaches considering chain dynamics as key ingredients. We have thus applied a recently proposed ansatz to describe the reference results on the dPrec melt sample. It is worth noting that such an approach had only been considered until now for the case of PIB and seems to work reasonably well also for PTHF from the present study. In this framework, the NSE results on the SCNPs point to a strong shift of the crossover to viscoelastically homogeneous behavior toward much larger length scales than in the linear precursors. In fact, our SANS experiments show that internal crosslinking induces an excess of density fluctuations at nanometric length scales with respect to the melt of linear topology. The presence of cross-links also manifests in a more complex relaxation of the dynamic structure factor at such length scales. Its functional form becomes markedly more stretched than in the melt of linear chains and the characteristic times increase dramatically with decreasing Q-value-at least in the window covered by the NSE experiments. These effects could be attributed to the emergence of new constraints in the SCNPs melt related to the internal domain topology and leading to certain correlations to persist over longer times. Additional mechanisms for relaxation would thus need to appear in the internally cross-linked melt in order to reach the complete decay of the dynamic structure factor. Close to cross-links and, in particular, within small domains, chain conformation and packing would differ from the average in the bulk material. Neighboring segments chemically connected by the internal crosslinks induced during SCNPs synthesis could feel an extra friction to relax, provoking the retardation of the decay of the density fluctuations at the corresponding length scales. The topological complexity induced by intramolecular bonding would be expected to require the occurrence of a hierarchy of relaxation mechanisms to reach the full relaxation of the SCNPs in the melt. In fact, the dielectric spectroscopy investigation reported in the above referred work 26 points to the appearance of an additional dynamic contribution to dipolar relaxation slower than the α-process that would be related to the relaxation mechanism involving the internal domains in the SCNPs.
Finally, we compare our results on the melt of SCNPs with those previously obtained on nanocomposites consisting of mixtures of poly(methyl methacrylate) (PMMA)-based SCNPs and linear chains of polyethylene oxide (PEO). [21][22][23] In those works, the mutual influence of both components on the dynamics was addressed by neutron scattering on labeled samples. Due to limitations of the experimental window, the most thorough studies were carried out on the PEO component. For this component, a shift in the glass-transition temperature revealing slower segmental dynamics as well as a slowdown of the Rouse modes was observed upon mixing with the more rigid SCNPs. These effects were also found in blends of the linear counterparts and can in principle be attributed to the large dynamic asymmetry displayed by the two components of the mixture; therefore, they would not be expected in our SCNPs melt, where all the structural units are chemically identical. We find, nevertheless, hints for slowdown and stretching of the dynamical response at large length scales that are not related to enhanced intermolecular friction due to a neighboring more rigid component. On the other hand, the main effect observed when the topology of the PMMA-like component in the nanocomposite changes from a linear chain to a SCNP is a striking tube dilation for PEO chain dynamics. As shown by the rheological measurements on the SCNPs melt above commented, 26 an effective disentanglement is also observed upon internal cross-linking of the PTHF chains. Thus, the creation of internal loops in the macromolecules seems to induce a release of the topological constraints for chain dynamics in both kinds of systems, despite their very different intrinsic properties.

VI. CONCLUSIONS
SCNPs are fascinating nano-objects not only for their potential use in a large variety of fields in the nanotechnology arena, but also as model systems to investigate fundamental problems of polymer dynamics. In particular, here we have addressed the question of how intramolecular cross-linking affects the atomic motions and the collective features at length scales of the order of the intermolecular distances and larger, exploring the regime of the so-called intermediate length scales region. With this aim, we have performed a parallel investigation on the melt composed by linear chains as a reference. For this reference sample, we have found "standard" self-motions describable by the anomalous jump diffusion model in the whole Q-range investigated, pointing thus to a very smooth crossover from α-relaxation toward Rouse-like dynamics. We have also shown that the recently proposed ansatz to describe the collective dynamics at ILS for glass-forming polymers 25 based on that developed by Novikov et al. 24 works rather well for the linear precursor melt. We note that this kind of investigation is very scarce in the literature and, therefore, this work has also contributed in this direction. The impact of intramolecular cross-linking is hardly resolvable at length scales of the order of the intermolecular average distances as microscopically explored by both the dynamic structure factor and the incoherent scattering function of hydrogens addressed by our quasielastic neutron scattering experiments. The effects set in at larger length scales, toward the intermediate length scales regime, where also the crossover from segmental to Rouse dynamics is expected. A subtle slowdown of the hydrogen self-motions can be inferred from the low-Q results on the hNP-samples. The main effect is observed on the coherent scattering: structural heterogeneities are revealed by SANS, and the decay of the dynamic structure factor exhibits marked extra-stretching as well as much longer characteristic times than the precursor melt. The emergence of a hierarchy of complex additional mechanisms for the relaxation of internal loops in the SCNPs would be at the origin of these observations for collective dynamics. Future experiments on the single chain dynamic structure factor (isotopically labeled macromolecules in a melt of SCNPs) may elucidate the impact of intramolecular cross-links on chain dynamics at a molecular level for these intriguing systems.

APPENDIX: MODEL TO DESCRIBE THE COLLECTIVE RELAXATION AT MESOSCALES IN GLASS-FORMING SYSTEMS-APPLICATION TO THE LINEAR PRECURSOR MELT
The window spanned by NSE in our experiments covers the neighborhood of the structure factor peak and the so-called intermediate length scales regime. A model has been recently proposed to describe the collective dynamics of glass-forming polymers there and successfully applied to the case of existing data on polyisobutylene (PIB). 25,42,43 This model consists of an interpolation formula that embeds the mesoscopic (nondiffusive) and the high-Q (diffusive) limits of the collective times in an analytical expression as proposed by Novikov et al., 24 1 The nondiffusive (Q-independent) time τ Q!0 c should reflect the viscoelastic coupling of stress and density fluctuations on scales long enough compared to atomic dimensions, but not yet in the hydrodynamic limit. 24 Its contribution to the total collective time is affected by a Gaussian cutoff factor e ÀQ 2 ξ 2 c to ensure that it is present only on length scales beyond a characteristic length ξ c . This characteristic length is expected to be ξ c 2π=Q max . On the other hand, the collective diffusive time τ D c (Q) can be obtained in the spirit of a Sköld 53 -like renormalization, The approximation for the diffusive contribution was generalized to the case of the observed anomalous diffusion. In particular, to account for the full phenomenology observed for glass-forming polymers, the incoherent scattering function was assumed to be that corresponding to the AJD model above introduced, i.e., Eq. (1) together with the expression for the characteristic time [Eq. (2)]. Then, the collective diffusive counterpart of τ s (Q) can be expressed as The model for collective times of the precursor was applied using the information on self-atomic motions obtained by QENS on the protonated sample. Figures 10 and 11 show the experimental times obtained using the same Q-independent β-parameter to fit both correlation functions. It was chosen as the one better describing on average the collective results. The solid lines in Fig. 10 are fits of the hydrogen self-correlation times to Eq. (2). The resulting parameters are given in Table III. We note that the characteristic time τ s (Q) involved in the calculation of τ D c corresponds to the self-part of the dynamic structure factor, in which all nuclei, not only the hydrogens, are involved. Therefore, it might differ from the characteristic time experimentally determined from QENS measurements on protonated samples. In fact, in the case of PIB, where molecular dynamics simulations were available, a proportionality factor was found to be about 2 between τ s (Q) calculated averaging overall the nuclei and τ H s (Q). Therefore, we have assumed that τ D c (Q) can be approximated by Eq. (A3) with the same ' o as that determined for τ H s (Q) and τ s,o ¼ Aτ H s,o (for the collective results at 280 K, the information extracted from the QENS measurements on the protonated sample at 285 K was used). Then, the only free parameters involved in the application of the model [combining Eqs. (A1) and (A3)] were A, hτ Q!0 c i, and ξ c . The resulting fitting curves are shown by the dashed-dotted lines in Fig. 11, and the values of the parameters are listed in Table III. For illustrating the model, in Fig. 12, the two contributions to the total model curve are represented: that from the mesoscopic time (indicated by the arrow) affected by the cutoff factor, and the collective (anomalous) diffusive characteristic time. The implicitly used selfcorrelation time is also depicted.
FIG. 11. Collective characteristic times measured on dPrec (filled symbols) and characteristic times for H-self motions determined from the QENS experiments on hPrec (empty symbols) for the three temperatures investigated: the red diamonds correspond to 360 K, the green squares correspond to 320 K, and the blue circles correspond to 280 K (dPrec) and 285 K (hPrec). Solid lines are fits of Eq. (2) to the incoherent times and dashed-dotted lines of the model for collective times.