Polymeric materials exhibit a rich hierarchy of dynamics from fast sub-molecular motions to collective segmental relaxations and slow chain diffusion. Such dynamical hierarchy dictates that the performance of polymeric materials is tightly linked to fast molecular dynamics, necessitating a thorough understanding of the dynamic behavior of polymers on the nanoscale. Recent advances in the synthesis of polymer composites with nanoscale fillers further amplify the need to probe polymer dynamics over spatial and temporal nanoscales to achieve reliable engineering of materials with well-defined properties. This tutorial focuses on the efficacy of neutron spectroscopy techniques, combined with judicious hydrogen/deuterium labeling, in selectively probing local and collective dynamics that underlie macroscopic properties in polymeric materials with varying degrees of complexity.

## I. INTRODUCTION

Advances in polymer engineering over the past few decades have resulted in a widespread use of simple and sophisticated polymeric materials in various applications. In particular, polymer–nanoparticle composites have taken center stage as candidates for multifunctional materials, combining the light weight of polymers with the exquisite functionality of the embedded nanoparticles.^{1–5} These composite materials have changed the landscape of technological and practical applications^{6–8} but still suffer from design-shortcomings specifically related to the dynamic interactions between polymers and nanoparticles.^{9} Besides the obvious effects on the dispersion and rheological behavior of polymeric materials,^{2,10} polymer dynamics strongly influence polymer aging and long-term material performance.^{11–14} In fact, one can easily see that the success or failure of polymer nanocomposites most likely originates from molecular level interactions that consequently influence higher length and time scale dynamics and eventually determine functional material properties. However, in order to make the leap from simple nano-filled polymers to nano-enhanced materials, and ultimately nano-engineered composites with precise patterns and functionality, it is imperative to understand the hierarchy of structure and dynamics that emerge in various classes of polymer nanocomposites.^{15} Most often, a full characterization of complex polymeric systems involves determining the assembly, function, and cooperation of various material components, structurally and temporally.^{16–18} While structural properties in such materials are (in general) readily accessible, the dynamics are experimentally more challenging to discern due to the wide range of length and time scales over which they manifest. This tutorial focuses on current and future applications of neutron spectroscopy methods in selectively probing polymer dynamics in complex polymeric systems on sub-molecular and molecular scales. The selectivity in measuring specific dynamic modes is enabled by the inherent characteristics of neutron scattering techniques and their ability to provide spatial and temporal resolution at relevant scales.^{19,20} Implementing judicious isotope labeling further allows selective studies of molecular subgroups and specific components within polymeric materials of interest.^{21}

## II. NEUTRON SCATTERING: BASICS AND PRINCIPLES

### A. Neutron interactions with matter

A unique property of neutrons is that they have no charge and can thus penetrate matter far deeper than charged particles, allowing for bulk measurements of reasonably thick samples. Given their chargeless state, neutrons interact with the nuclei of atoms rather than with the electronic cloud,^{22} which has two important consequences that are discussed below.

First, this implies that neutrons scatter from atomic nuclei with a diameter on the order of 10^{−5} Å and atomic spacing of a few Å's. To put this in context, we compare these distances to the wave properties of neutrons. Based on wave–particle duality, neutrons can be thought of as waves with a de Broglie wavelength $\lambda =h/mnv$, where *h* is the Planck constant, $mn$ is the mass of the neutron, and *v* is the neutron speed. For slow neutrons (energy ≈ few meV),^{23} typically used in scattering experiments, $\lambda $ is on the order of a few Å’s. In this regime, the scattering of neutrons from atomic nuclei is akin to the scattering of low-energy incident waves from point-like scatterers. The scattering potential, in this case, is adequately given by the Fermi pseudopotential $V(r)=2\pi \u210f2bmn\delta (r)$, where $\delta (r)$ is the Dirac delta function at the atomic positions and *b* is the neutron scattering length, which denotes the strength of the neutron–nucleus interaction and is related to the scattering cross section, $\sigma $, as $\sigma =4\pi b2$. The Fourier transform of $V(r)$ yields a constant scattering potential, $V=2\pi \u210f2b/mn$, in a reciprocal space (commonly used in discussing wave scattering). To describe scattering from molecules or atomic ensembles rather than individual atomic nuclei, a coarse-grained version of the potential should be used, such that $V\xaf=2\pi \u210f2\rho /mn$, where $\rho $ is the neutron scattering length density (SLD) defined as $\rho =1Vmolecule\u2211ibini$, and the sum is carried out over all atoms with a given occurrence $ni$. For example, knowing the neutron scattering length of hydrogen and oxygen nuclei, respectively, given by $bH=\u22123.74fm$ and $bO=5.81fm$, one can easily calculate the SLD of water, $\rho H2O=2bH+bOVH2O=2(\u22123.74fm)+5.81fm30\xc53=\u22120.56\xd710\u22126\xc5\u22122.$ Fortunately, there is no need to perform these calculations for every type of molecule in an experiment. User websites and interfaces, such as the neutron activation and scattering calculator developed by the NIST Center for Neutron Research, have the required tools for SLD calculations of various materials, mixtures, and solutions.^{24}

### B. Isotope sensitivity

The second important consequence of the interaction of neutrons with atomic nuclei, as opposed to the interaction of x-rays with the electronic cloud, is that the neutron scattering cross section has a non-monotonic dependence on the atomic number of elements within a scattering specimen.^{25} This implies that the presence of heavier elements, such as silicon (Si), does not necessarily dominate the scattering signal when lighter elements such as hydrogen are present. This also implies that nuclear isotopes can have significantly different neutron scattering cross sections^{26} [Fig. 1(a)]. A classic example is that of hydrogen (H) and deuterium (D), which exhibit the largest difference in neutron scattering length for common isotopes ($bH=\u22123.74fm$ vs $bD=3.99fm$). In fact, the isotope sensitivity of neutrons is one of the most attractive and unique features of neutron scattering in soft materials,^{27–29} in general, and in polymeric materials,^{21,30} in particular. The rich abundance of H in polymeric materials (and solvents) and the relative ease of deuterium labeling enable selective structural and spectroscopic measurements of various moieties within simple and complex polymeric materials, as highlighted in multiple examples throughout this tutorial.

### C. More on the neutron scattering length density

Neutron SLD can be thought of as the analog of the optical index of refraction. Since optical phenomena are easier to visualize, it is informative to explain the significance of the SLD in the language of optics. For instance, light traversing the boundary between two isotropic materials refracts according to Snell's law: $n1sin\u2061\theta 1=n2sin\u2061\theta 2$. If the two materials have identical indexes of refraction, $n1=n2$ {e.g., Pyrex rod in vegetable oil [Fig. 1(b)]}, then the incident light traverses the boundary without getting refracted (or scattered), rendering the immersed part of the rod invisible to the human eye. The same scenario applies to neutron scattering from materials with identical SLDs. For example, silica nanoparticles ($\rho SiO2=3.55\xd710\u22126\xc5\u22122$) are almost identical in SLD to an H_{2}O:D_{2}O mixture with a mixing ratio of 41:59, which results in a null neutron scattering signal from such a dispersion. Notice that the mixture SLD can be calculated as $\rho mixture=x\rho H2O+(1\u2212x)\rho D2O=3.54\xd710\u22126\xc5\u22122$, where *x* is the volume fraction of H_{2}O and $\rho D2O=6.39\xd710\u22126\xc5\u22122$. On the other hand, a dispersion of the same silica nanoparticles in D_{2}O yields a very strong neutron scattering signal due to the significant SLD difference, $\Delta \rho $, between silica and D_{2}O. This type of contrast amplification or contrast matching scheme is unique to neutron scattering and is often implemented in experiments to highlight the components (or molecular groups) of interest or to make other components invisible without largely changing the physiochemistry of the sample.^{31}

### D. Elastic vs inelastic scattering

So far, the discussion has been focused on the strength of neutron–nuclei interactions. In this section, we will consider the consequence of scattering events on the properties of the neutron. Based on scattering theory, the scattering of a neutron (or particle) by an object can alter its momentum and energy. Drawing from analogy with classical collision theory, scattering events are classified into two types: elastic and inelastic.^{32}

Under this classification, *elastic scattering* refers to the scenario where there is no change in the neutron energy ($E=hc/\lambda $) but a possible change in its wavevector, $k\u2192$. Notably, the wavevector $k\u2192$ indicates the direction of propagation of the neutron, and its magnitude ($k=2\pi /\lambda $) determines the neutron energy. The conservation of energy in elastic scattering implies that the scattered neutron maintains the same incident wavelength, $\lambda $. This necessitates that any change in $k\u2192$ is limited to a change in the wavevector direction not its magnitude. In this case, the scattering intensity is sufficiently expressed in terms of the momentum transfer $Q\u2192=k\u2192\u2212k\u2192\u2032$, where $k\u2192$ and $k\u2192\u2032$ are wavevectors of the incident and scattered neutrons, respectively. The magnitude of $Q\u2192$ is expressed in terms of the angular deflection of the neutron, also referred to as the scattering angle $\theta $, such that $Q=2ksin(\theta /2)=4\pi sin(\theta /2)/\lambda $. Notice that *Q* has dimensions of inverse length; hence, *Q*-space is commonly referred to as reciprocal space. Broadly speaking, *Q* is related to real-space structures of size, *d*, as $Q\u22482\pi /d$.

In contrast, when energy is exchanged between the incident neutron and the sample, it causes a change in both the direction and magnitude of the wavevector of the scattered neutron, referred to as *inelastic scattering*. This type of scattering is harnessed in neutron spectroscopy techniques, which are designed to detect small differences in the neutron energy as a result of scattering from a sample. Assuming conservation of total energy during scattering events, i.e., the energy gained by the neutron is lost by the sample, inelastic scattering measurements can, thus, provide valuable information about the dynamics of the scattering entities within the sample.

### E. Coherent vs incoherent scattering

Another important aspect of neutron scattering is that it has two components: a coherent and an incoherent component^{23,33} [see Fig. 1(a)]. The first results from the interaction of the neutron wave with *all* nuclei in the sample such that the scattered waves from different nuclei have well-defined relative phases. In other words, coherent scattering depends on the relative distances of the nuclei or atomic ensembles. When combined with energy considerations, this implies that *elastic coherent* scattering contains information about the equilibrium structure of the sample and is, thus, adequately used in structural characterization, whereas *inelastic coherent* scattering encodes information about relative motions of atoms or atomic ensembles and can be effectively utilized in probing collective dynamics. In incoherent scattering, on the other hand, the neutron wave interacts independently with nuclei, resulting in random relative phases of the scattered waves. Incoherent scattering may result from the interaction of a neutron wave with the same nucleus at different positions or times, providing information about diffusive atomic motions, as explained later. Notably, hydrogen (H)—commonly present in non-deuterated polymers—has a very large incoherent neutron scattering component [Fig. 1(a)]. In structural measurements, this often results in a rather undesirable incoherent background signal that obscures weaker coherent scattering signals. However, the large incoherent scattering of H is an advantage in studies of self-dynamics of H-rich moieties in polymeric systems (see Sec. III). It is important to point out that when coherent scattering is required, the isotopic substitution of H by D can be implemented to reduce the incoherent contribution.

## III. QUASI-ELASTIC NEUTRON SCATTERING (QENS)

The master expression used to describe neutron scattering is the double differential scattering cross section $\u22022\sigma /\u2202\Omega \u2202E$, which resembles the number of neutrons scattered into a solid angle between $\Omega $ and $\Omega +\u2202\Omega $ with an average energy change $\u2202E=\u210f\omega $.^{34} As discussed above, scattering events are classified as elastic when there is no change in the neutron energy ($\u210f\omega =0$) or when the change in energy is smaller than the resolution of the used scattering technique. On the other hand, inelastic scattering events signify a change in the neutron energy ($\u210f\omega \u22600$) due to an exchange of energy between the neutron and the sample. To differentiate between scattering events that involve an excitation in the sample and those that do not, the latter are usually referred to as *quasi-elastic* scattering events.^{35} This type of scattering usually manifests in broadening around elastic lines or in a change in dynamic relaxations, as will be discussed in more detail throughout this tutorial.

Taking into account the coherent (coh) and incoherent (inc) contributions to the scattering, the double differential scattering cross section can be written as^{35}

where the sums run over all nuclei of types $\alpha $ and $\beta $. $S\alpha \beta coh(Q\u2192,E)$ and $S\alpha inc(Q\u2192,E)$ are the coherent and incoherent scattering functions, respectively. As inferred from Eq. (1), the coherent contribution involves scattering from pairs of nuclei (or atoms), and the incoherent contribution involves scattering from the same nucleus. The temporal Fourier transforms of these functions are the intermediate scattering functions, $S\alpha \beta coh(Q\u2192,t)$ and $S\alpha inc(Q\u2192,t)$. Further Fourier transformation to real space yields the van Hove correlation and self-correlation functions,^{36} $G\alpha \beta (r\u2192,t)$ and $G\alpha self(r\u2192,t)$, respectively, given by

In the expressions above, $r\u2192i\alpha (t)$ is the position vector of the *i*th atom of type $\alpha $ at time *t*. The sum is carried out over all available atoms of type $\alpha $, where $N=\u2211\alpha \u2061N\alpha $. A closer look at these functions indicates that $G\alpha \beta (r\u2192,t)d3r\u2192$ represents the probability of finding a particle of type $\alpha $ at position $r\u2192$ and time *t*, given that a particle of type $\beta $ is at the origin at time$t=0$. In the static case, $G\alpha \beta (r\u2192,t=0)=\delta \alpha \beta (r\u2192)+g\alpha \beta (r\u2192)$, where $g\alpha \beta (r\u2192)$ is the static pair-distribution function, which provides structural correlations of atomic distributions within the sample. The time-dependence of $G\alpha \beta $ provides further information about the relative motions or dynamics of correlated moieties. Conversely, $G\alpha self(r\u2192,t)$ indicates correlations in the positions of a single particle at different times and thus describes diffusive dynamics.

Going back to Eq. (1), one can see that the coherent scattering function carries information about correlated structures or motions, whereas incoherent scattering is related to single particle dynamics or self-motions. The weights of the two contributions to the total scattering cross section are determined by the coherent and incoherent scattering lengths of the present isotopes. This implies that, in protiated polymer samples, the scattering signal is dominated by the incoherent scattering from H atoms (see Fig. 1), which is ideal for studies of self-dynamics. However, partial or full replacement of H by D can have compelling uses in other studies of polymer dynamics.^{37–39} For instance, partial D-labeling is an effective approach to obscure incoherent scattering from D-labeled moieties, allowing selective studies of self-motions in the unlabeled moieties, whose incoherent scattering still dominate the signal in the high-*Q* range. Partial deuteration can also be used to amplify the SLD between different scattering entities, enabling studies of coherent correlated dynamics within the sample. The choice of the deuteration scheme in a given experiment is dictated by the polymer dynamics of interest and the used neutron spectroscopy technique.

In the rest of this tutorial, multiple examples will be provided on partial and full D-labeling in polymeric systems to access selective polymer dynamics on molecular and sub-molecular levels. Access to such dynamics is enabled by the inherent capability of quasi-elastic neutron spectroscopy to detect energy changes that are commensurate with corresponding dynamic processes such as side-group rotations, segmental relaxations, molecular diffusion, etc. Examples of these dynamic modes are highlighted in Fig. 2 along with the QENS techniques that can access the corresponding length and time scales. Here, we will focus on two types of QENS methods in the studies of polymer dynamics, namely, neutron backscattering spectroscopy (BSS) and neutron spin-echo (NSE) spectroscopy.

### A. Neutron backscattering spectroscopy (BSS)

Neutron backscattering spectroscopy was proposed by Maier-Leibnitz^{40} in 1966 based on the concept of backscattering from monochromator and analyzer crystals to achieve a high energy resolution (≈ few *μ*eV), which enables measurements of fast atomic or molecular motions.^{41} The geometry of backscattering spectrometers (Fig. 3) is such that the scattering of the incident neutron beam from the sample is intercepted by a panel of analyzers [e.g., Si (111) crystals] and is backscattered into a detector array. The value of the momentum transfer, *Q*, is calculated from the known values of the scattering angles covered by the analyzer panels and the detectors. BS spectrometers generally cover a limited *Q*-range, $0.2\xc5\u22121\u2272Q\u22722\xc5\u22121$, and a limited dynamic range with Fourier times $0.3ns\u2272t\u22723ns$. However, more extended dynamic ranges are accessible on time-of-flight backscattering spectrometers, with typical energy resolutions on the order of 10–100 *μ*eV (and equivalent Fourier times of ≈5–100 ps), with the exception of the BASIS spectrometer at the Spallation Neutron Source (SNS), which can reach an energy resolution of ≈3 *μ*eV.^{42} Faster ps time scales (≈0.5–30 ps) can be probed by time-of-flight chopper spectrometers^{43} (TOF, not discussed in this tutorial), whereas slower time scales (up to ≈few 100 ns) can be accessed with neutron spin-echo (NSE) spectroscopy^{44} (discussed in Sec. III B). BSS thus uniquely bridges the gap between the dynamic ranges accessible by TOF chopper spectrometers and by NSE. For studies of dynamic phenomena that span a wide dynamic range, combining results from multiple spectrometers with complementary, and preferably overlapping, dynamic ranges is usually required. However, studies as such should be pursued with caution as it is important to take into account instrumental resolution effects, which are inherently convoluted with the scattering function. This means that direct comparison of the scattering intensity, $I(Q,\omega )\u221d\u22022\sigma /\u2202\Omega \u2202E$, obtained from experiments on the same sample at different spectrometers or with different energy resolutions might not be adequate. To mitigate this complication, two different approaches can be considered. In the time domain, resolution effects can be eliminated by normalizing the Fourier transformed scattering spectrum at a given *Q* by the Fourier transform of the resolution function at the corresponding *Q*, such that

In the frequency domain, the measured scattering intensity is analyzed as a convolution of a model scattering function and the resolution function; i.e., $S(Q,\omega )\u2297R(Q,\omega )$. Both approaches are commonly utilized in the analysis of neutron backscattering data on polymeric materials.

In practice, neutron backscattering spectrometers can be operated in two different modes: elastic and quasi-elastic. When operated in an elastic mode, BS spectrometers measure the elastic intensity from the sample, such that changes in the elastic intensity signify emergent dynamics within the sample. For polymers, such changes are usually probed as a function of external factors, e.g., temperature, pressure, etc., allowing for studies of dynamic transitions in response to sample environment. In the case of non-deuterated polymers, the primary contribution to the scattering signal is the incoherent scattering from the H-rich moieties. Thus, a common and effective application of BS elastic scans (or fixed-window scans) entails studies of polymer dynamics through probing the self-motions of H-atoms within polymeric materials. Typically, the elastic incoherent scattering intensity, $Ielinc$, is obtained by summing the scattered intensity over all detectors, i.e., over all accessible momentum transfers *Q*. Figure 4(a) shows an example of the temperature dependence of the elastic incoherent scattering intensity from poly(methyl methacrylate) (PMMA) for temperatures between 50 K and 460 K at a heating rate of 1 K/min. The intensity is normalized by $Ielinc$ at 50 K, at which most of the hydrogen dynamics are expected to be “frozen.” The figure clearly shows that as the temperature increases, the elastic intensity decreases. Keeping in mind that elastic scattering is associated with immobile atoms (or atoms moving over time scales slower than the instrumental resolution), the observed loss in intensity thus indicates increased polymer mobility as the sample is heated, which is expected. The extent of the mobility of the probed H-motions can be measured by investigating the *Q-*dependence of $Ielinc$ at each measured temperature. Using the Debye–Waller approximation, one can calculate the amplitude, $\u27e8u2\u27e9$, of the mean square displacements (MSDs) as

This equation dictates that the fits should intercept the y axis at the origin. However, it is not uncommon to observe vertical shifts in the intensity patterns^{45–47} due to possible coherent and multiple scattering effects,^{48} in which case a temperature-dependent *y*-intercept should be added to the right-hand term of Eq. (5). An example of such shifts is illustrated in Fig. 4(b), which shows *Q*-dependent elastic incoherent scattering measurements on PMMA. The obtained MSD patterns are shown in Fig. 4(c) over the measured temperature range. Although based on a rather crude approximation, MSD measurements can be extremely useful in identifying the temperatures at which specific dynamical processes activate.^{47} This is obtained by investigating the onset of thermally induced dynamics from the thermal dependence of the MSD. This type of analysis is based on the harmonic model which assumes that the internal energy of the system is purely vibrational in the low-*T* regime, where $\u27e8u2\u27e9=3kBT/\kappa $, with $\kappa $ being the effective material stiffness. Obviously, applying this model outside the low-*T* regime is purely heuristic. However, it is fair to assume that thermally induced motions introduce additional degrees of freedom, which result in a change of the coefficient in the expression of $\u27e8u2\u27e9$ vs *T* but maintain a linear *T*-dependence of the MSDs. This approach offers a qualitative framework for identifying the onset of different polymer dynamic modes, as shown for PMMA in Fig. 4(c). Particularly important is the highest transition associated with segmental relaxations which denotes the glass transition temperature, $Tg$. In this example, *T _{g}* ≈ 380 K, which is in excellent agreement with literature values.

^{49}

On the other hand, when operated in a quasi-elastic mode, backscattering spectrometers allow measurements of the energy spread over each of the available detectors, i.e., at well-defined accessible *Q*-values. The energy sensitivity is achieved by different means depending on the neutron source. At reactor sources, the energy of the incident beam is set by a monochromator, and the energy variation is achieved either by varying the temperature of the monochromator^{50} or by Doppler-driving the monochromator.^{51} For backscattering spectrometers at spallation sources, the incident energy is determined from the time-of-flight between the source and the detector.^{42} Depending on the configuration of the instrument and the wavelength distribution, energy transfers of several 100 *μ*eV are achievable. In all cases, the dynamics of the sample are obtained from the energy spread in intensity as a function of accessible *Q*-values. The broadening of the energy spectra, relative to the instrument resolution, is associated with relaxations of polymer dynamics within the accessed energy window, such that faster dynamics cause stronger broadening. This is clearly illustrated in Fig. 5 which shows increased broadening of the quasi-elastic spectra on PMMA (at $Q\u22481\xc5\u22121$) with increasing sample temperature.

The fits to the data in Fig. 5 are obtained using a convolution of the instrumental resolution function with a model function $S(Q,\omega )$, such that the Fourier transform of $S(Q,\omega )$ is given by Kohlrausch–Williams–Watts (KWW) function^{52–53} in the time domain,

In Eq. (6), *DWF* is the Debye–Waller factor, $\tau R$ is the temperature-dependent *Q*-dependent relaxation time of the measured dynamics, and $\beta $ is the stretching exponent. The average relaxation time is then calculated using the time-temperature superposition principle as $\tau av=(1/\beta )\Gamma (1/\beta )\tau R$. Stretched exponential functions of the KWW form are commonly used in describing polymer dynamics. The stretching exponent *β* $(0<\beta \u22641)$ describes the level of dynamic heterogeneity in the sample. In other words, *β* describes the distribution of decay modes of individually relaxing species,^{54} such that *β*-values close to zero indicate lack of dynamics within the measured dynamic range and *β*-values approaching unity correspond to a single decay mode. The stronger deviation from unity of *β*-values indicates increased dynamical heterogeneities in the sample. In polymer melts, $\beta \u22480.5$ is commonly observed.^{55}

### B. Neutron spin-echo (NSE) spectroscopy

Neutron spin-echo spectroscopy was first proposed by Mezei^{56} in 1972 and later developed by Hayter and Penfold and Pynn^{57,58} The technique is based on the spin-manipulation of a polarized neutron beam using spin-flippers and magnetic precession coils, as depicted in Fig. 6. A full understanding of NSE spectroscopy requires a quantum mechanics treatment of the neutron wavefunction throughout the different components of the spectrometer. Instead, this tutorial will present a simplified semi-classical explanation for the mode of operation of NSE spectrometers and the measured signals.

To explain the design of NSE spectrometers, we describe the magnetic-field effect on the neutron spin in each magnetic component of the NSE instrument. Neutrons exiting the π/2-flipper at the beginning of the flight path experience a magnetic field as they enter the upstream coil (to the left of the sample position), resulting in a precession of the neutron spin $s\u2192$ about the magnetic field $B\u2192$. The spin precession is described by the Bloch equation,^{59}

where $\omega L$ is the Larmor frequency given by $\omega L=\gamma B$ and $\gamma $ is the neutron gyromagnetic ratio. The precession angle is then calculated as

This expression indicates that the precession angle depends on the neutron speed, *v*, and, consequently, on the duration of time that the neutron takes to traverse the magnetic field region of length *l*. In other words, the amount of precession of the neutron spin depends on the energy of the neutron. A similar treatment is carried out in the downstream coil (to the right of the sample position) designed to have the same length, *l*, and the same magnetic field strength, *B*.

The net neutron polarization is obtained from the total precession angle (or phase) acquired throughout the magnetic field regions such that

where $\phi in$ and $\phi out$ are the precession angles acquired in the upstream and downstream coils, respectively. The negative sign in the expression of $\phi tot$ is due to the π-flip incurred before the downstream precession coil, which reverses the direction of precession.

In the absence of a sample or in the event of elastic scattering from the sample, the neutron does not experience a change in energy, i.e., $v1=v2$. In this case, the total acquired phase is zero and the net polarization is $P=1$, which implies full retrieval of the initial neutron spin state. This is referred to as “spin echo.”

However, if the neutron quasi-elastically scatters from the sample, the energy exchange with the sample alters the speed of the scattered neutron (downstream part of the instrument) such that the change in the neutron energy is $\Delta E=\u210f\omega =12mnv22\u221212mnv12\u2245mnv\Delta v$. Substituting into Eq. (9) results in a modified expression of the total phase, such that

where *t* is the Fourier time defined as $t=\gamma Bl\u210f/mnv3=0.186Bl\lambda 3$. Note that the Fourier time is solely dependent on the strength of the magnetic field, coil geometry, and neutron wavelength. In experiments, the Fourier time is varied by varying the strength of the magnetic field (by changing the current in the electromagnetic coils) or by considering different neutron wavelengths (inherently accessible on spallation sources). For example, for a neutron of wavelength $\lambda =8\xc5$ and a magnetic field setup with $B=0.4T$ and $l=1.2m$, the Fourier time is calculated to be $t\u224850ns$ and the corresponding energy change is $\Delta E\u22480.01\mu eV$. This provides a rough estimate of the time scales and energy changes associated with sample dynamics that are accessible in NSE experiments.

According to Eq. (9), the change in the neutron speed due to energy exchange with the sample should result in a loss of polarization. Indeed, NSE signals are generally represented in terms of polarization (or intensity) loss, which contains information about dynamic relaxations within the sample, as discussed later. In NSE measurements, the polarization is measured in terms of the intermediate scattering function $S(Q,t)$ obtained from the echo scans at each detector pixel (see Fig. 7). The measured polarization of the scattered beam is given by $P\u221dS(Q,t)/S(Q,0)\u221d2A/(Nup\u2212Ndown)$, where $Nup$ and $Ndown$ are the count rates in the non-spin-flip and spin-flip conditions measured with the π/2-flippers off and the π-flipper off and on, respectively. Notice that the physical information about the sample is encoded in the amplitude, *A*, which is obtained from fitting the echo signal, i.e., the number of neutron counts, *N*, to a damped cosine function of the form

In Eq. (11), $\zeta $ is the period of the cosine function, $\sigma $ is the width of the Gaussian envelope determined by the characteristics of the incoming beam (mean wavelength and wavelength spread), $\varphi c$ is the phase current, and $\varphi 0$ is the echo point which depends on the field path experienced by the neutron [see Fig. 7(b)].

For proper data normalization, two additional measurements should be performed: resolution (*R*) measurement on an elastic scattering reference (e.g., carbon) and background (BKG) measurement performed on an empty sample cell or a solvent-containing cell depending on the conditions of the actual sample. In both cases, fits of the echo signals are performed using the same procedure following Eq. (11). Data correction for resolution and background effects is then performed according to the following equation:

where $T\xaf$ is the transmission of the sample relative to background and $Vf$ is the volume fraction (mostly applicable in the case of solutions). Finally, the detector pixels are grouped into a limited number of *Q* arcs to obtain the *Q*-dependence of the normalized intermediate scattering function [see Fig. 7(a)]. Larger variations in *Q* required for measurement of *Q*-dependent dynamics are obtained by considering different neutron wavelengths and/or by moving the downstream arm of the NSE spectrometer to access different scattering angles, $\theta $.

It is important to point out that another basic characteristic of NSE spectroscopy that differentiates it from other QENS techniques is that, unlike standard QENS methods which additively account for coherent and incoherent contributions to the scattered intensity [see Eq. (1)], NSE is predominantly sensitive to coherent scattering. The NSE normalized intermediate scattering function is given by

where $Icoh(Q,t)=\u2211\alpha ,\beta \u27e8b\alpha b\beta \u27e9S\alpha \beta coh(Q,t)$ and $Iinc(Q,t)=\u2211\alpha \u2061\u27e8(\Delta b\alpha )2\u27e9S\alpha inc(Q,t)$. This equation clearly shows that the incoherent scattering (mainly contributed by H-atoms) is suppressed by a factor of 1/3 in the NSE signal. NSE is thus more adequately suited to measure coherent scattering associated with *collective dynamics*. The suppression of the incoherent signal also favors the use of deuterated samples for enhanced signals in NSE measurements.

An example of NSE data is shown in Fig. 8 for fully deuterated PMMA (dPMMA, M_{w} = 25 kg/mol) collected at the main diffraction peak, i.e., at $Q=0.9\xc5\u22121$, which is a characteristic of the overall periodicity set by the inter-chain distance. At this *Q*-value, NSE selectively probes segmental relaxations or chain–chain relaxations.^{47} However, measurements at a higher *Q* can access subgroup dynamics and correlated mainchain-subgroup dynamics as demonstrated in the earlier work by Genix *et al*.^{60} This emphasizes the importance of structural characterization of the sample prior to NSE measurements due to the spatiotemporal nature of polymer dynamics. In either case, the normalized intermediate scattering function can be fitted to a KWW stretched exponential function [Eq. (6)].

Figure 8 shows NSE data collected on dPMMA at temperatures above the glass transition temperature, $Tg$, along with the optimal stretched-exponential fits. It is important to realize that in the KWW model, $\tau R$ and *DWF* are temperature dependent, whereas the stretching exponent *β* is, in principle, independent of temperature. In the course of fitting, the value of *β* should be extracted from the high-*T* dataset that shows the most pronounced decay and used as an initial value for the fits of the lower-*T* data sets. An iterative approach should be adopted until the *β*-parameters from all *T*-sets on a given sample converge to within 15%. This step is necessary to ensure reliable extraction of $\tau R$ and a proper description of the relaxation dynamics. The *β*-value in Fig. 4 was found to be 0.71 ± 0.13, and the resultant temperature-dependence of $\tau R$ was found to exhibit an Arrhenius temperature dependence such that $\tau R=\tau 0exp[EA/kBT]$.^{47} This data treatment offers an estimate of the activation energy of the observed dynamics; in this case $EA\u224838.5kBT$. Other forms of temperature dependence have been observed for segmental relaxations ($\alpha $-process) in polymeric systems, mostly following the Vogel–Fulcher–Tammann equation.^{61}

## IV. POLYMER DYNAMICS MEASURED BY NEUTRON SPECTROSCOPY

The main objective of this tutorial is to show the unique capabilities of neutrons as a spatiotemporal probe that can access molecular motions and provide selective information about self- and collective dynamics in polymeric systems. This is illustrated through select examples in which QENS techniques have been utilized to extract physical and dynamical properties of polymeric materials with various degrees of complexity. The examples highlighted here focus on important dynamical phenomena in polymeric materials that are particularly suited for QENS studies and, by no means, constitute a comprehensive review of the literature on this topic.

### A. Polymer melts

#### 1. Polymer confinement

A common approach for studying confined polymer dynamics is the use of templates with nanoscale pores or cavities;^{62–65} e.g., porous silicon or anodized alumina. In such templates, the surface-normal orientation of the pores can be utilized to selectively probe molecular motions along the pore and radial directions by choosing the appropriate direction of the momentum transfer, $Q\u2192$. Kusmin *et al*.^{66} used this approach in NSE experiments to study the effect of confinement on poly(ethylene oxide) (PEO) chains in porous silicon with a mean pore diameter of 13 nm. In their study, they used a mixture of deuterated and protiated chains such that the average SLD of the mixture matched that of the template. This choice of SLD serves to minimize elastic off-specular scattering from the structural periodicity of the template and further amplifies single chain coherent scattering due to the large SLD difference between the deuterated and protiated chains. The measured spectra were analyzed in terms of a two-state model in which free chains in the center of the pore diffuse within an infinite cylinder and exhibit bulklike internal dynamics, whereas chains adsorbed to the pore walls have immobile center-of-mass and exhibit much slower internal dynamics. For short chains (M_{w} = 3 kg/mol), the radius of the infinite cylinder was found to be 1.4 nm and the thickness of the adsorbed layer approximately equal to the Flory radius. For longer chains (M_{w} = 10 kg/mol), the diffusion rate along the pore was found to be smaller than that in the radial direction, in agreement with the later work by Tung *el al*. using elastic recoil detection.^{67} For both molecular weights, however, the internal dynamics of the free chains were not affected by confinement.

In another NSE experiment, Martín *et al.* studied entangled PEO chains subjected to a stringent confinement set by the cylindrical nanopores of anodized alumina templates.^{64} The experiment was performed with short and long polymer chains, with chain dimensions either much larger or smaller than the lateral pore size. Figure 9 shows the NSE data collected on both samples. In the Rouse regime, i.e., at intermediate time scales, the data show a slowdown in relaxations compared to the bulk behavior. This effect was explained by pinning events at the pore walls. However, at long Fourier times, the relaxations tend to plateau at values determined by the spatial extent of the molecular motions. Further analysis of the data showed that the weakly confined short polymer exhibited bulklike entanglement density, whereas the strongly confined long chains displayed tube dilation. These measurements presented unequivocal microscopic evidence for the dilution of the total entanglement density under strong confinement, a phenomenon that had been previously hypothesized solely based on macroscopic observations.

#### 2. Chain architecture

Among the earlier investigations of the effect of chain architecture, studies of poly(*n*-alkyl methacrylates) (PnMAs) received significant attention due to the intriguing behavior of PnMAs characterized by the possible existence of two different glass-transitions associated with main chain dynamics and alkyl nanodomains.^{37,61,68,69} Selective deuterium labeling and spatiotemporal resolution enabled detailed measurements of the rich dynamics associated with the various subgroups within the polymer chain.^{37,61} Guided by neutron diffraction measurements to identify the length scales at which different dynamic modes are manifested, QENS and NSE measurements yielded collective relaxations of the main chain [collected at the *Q*-value of the main diffraction peak, see Fig. 8(a)] as well as dynamics within the alkyl nanodomains (collected at higher-Q values of the secondary peak).^{70} As expected, main chain relaxations showed typical stretched-exponential decays characterized by a stretched exponential $\beta \u22480.5$ and temperature-dependent structural relaxation times which scale with viscosity. On the other hand, self-motions of H-atoms in the alkyl side-groups (collected on samples with deuterated main chain and protiated side group) showed anomalous structural relaxations especially in high-order members of the polymer family.^{71} The importance of such measurements is that they bridge the gap between local dynamics dictated by the mobility of the side groups with the rheological properties set by the chain architecture.

Another notable study on the effect of chain architecture on polymer dynamics was performed by Zamponi *et al.* on branched polymer melts.^{72} Using NSE, combined with selective D-labeling schemes, they studied three-arm polyethylene stars with different symmetries: symmetric stars with long well-entangled arms and asymmetric stars where one of the arms was shortened to a single entanglement length. In either case, the segments nearest to the branch point were left protiated, whereas the rest of the star was deuterated, allowing for molecular measurements of the branch-point. A linear “two-arm” chain was studied as a reference (Fig. 10). The driving hypothesis was that the introduction of a branch point would restrict the reptation of a star polymer until the arms have fully relaxed. Indeed, the measured dynamic structure factor showed a clear plateau at longer Fourier times, indicative of spatial confinement. As discussed earlier, the plateau values are directly related to the confinement distances. In this study, the measured plateau values indicated that the symmetric star is confined to a space of ≈ 42 Å, smaller than the tube diameter (47 Å). In comparison, the asymmetric star (with one short arm) displayed the same topological confinement effects of the branch-point within the accessible time range of NSE. However, macroscopic rheological measurements showed that the drag of the short arm was stronger than expected and only at longer times does the asymmetric star display the flow behavior of linear chains.

In a more recent study, Middleton *el al.* performed QENS measurements on a series of linear polyethylenes (PEs) with precisely spaced functional groups to understand the effect of the polymer morphology on fast dynamics.^{73} The measurements revealed two dynamic processes with different composition and length-scale (i.e., *Q*) dependence. Faster dynamics were attributed to bond rotations and hydrogen vibrations, whereas slower dynamics were identified as structural relaxations of the polymer backbone. Both processes were modeled as stretched exponential functions of the KWW form. QENS measurements showed that increasing the acid content or shortening the PE spacers between side groups resulted in a significant slowdown of the structural relaxations along with an increase in the dynamic heterogeneity manifested in deviations in the stretching exponent (from ≈0.7 to ≈0.4). By further examining specific hydrogen atom positions along the backbone, the study showed that structural heterogeneity imposed by associating acid groups correlates with a mobility gradient along the polymer backbone. The results were compared with MD simulations that validated the observed molecular motions and relaxations, setting the stage for studies on more complex ion-containing polymers.

### B. Polymer nanocomposites

#### 1. Interfacial dynamics

A classic example of neutron spectroscopy studies on interfacial polymer dynamics is reported in the study of Kropka *et al.* on polymer-C60 composites.^{74} Using elastic backscattering scans, they showed that the dispersion of 1 wt. % C60 nanoparticles in PMMA and polystyrene (PS) resulted in noticeable shifts in the total elastic incoherent scattering intensity relative to the pure polymer, indicating a decrease in H-mobility and a concomitant reduction in the hydrogen mean-square displacements [see Fig. 11(a)]. The reduction in H-mobility was attributed to a slower interfacial polymer region around the nanoparticles, which exhibits suppressed dynamics on the nanosecond time scale. Analysis of the intermediate scattering function on PMMA and PMMA–C60 composite provided an estimate of the fraction of the interfacial immobilized polymer chain segments of ∼ 2.5% [Fig. 11(b)]. The results enabled nanoscale-level explanation of the shifts in glass transition temperature independently measured by calorimetry.

Similar experiments were performed by Akcora *et al*. on PMMA-grafted silica nanocomposites with different dispersion states.^{75} Using isotopic labeling, they prepared miscible and immiscible composites with protiated matrix and deuterated grafts, and vice versa. Elastic scattering scans in backscattering measurements on these samples allowed selective investigations of the hydrogen mobility (or MSD) within the matrix and grafted chains, respectively, in both dispersion states. The similarity in the obtained matrix MSD in dispersed and aggregated composites indicated that the macroscopic phase separation does not influence the local matrix chain dynamics. However, small differences in the brush MSD suggested that the segmental mobility of grafted chains is somewhat affected by the dispersion state and decreases slightly with particle aggregation. Combined with rheological measurements, they concluded that mechanical reinforcement in composites is primarily dictated by the formation of a particle network that presents an easy pathway for the propagation of stress, as opposed to chain bridging between the glassy polymer layers at the particle–polymer interfaces, which could also result in an enhanced stress response of the polymer nanocomposite.

The effect of interfacial effects and particle percolation was also studied by Ashkar *et al*.^{47} on PMMA composites with carbon nanotube (CNT) networks characterized by strong interfacial interaction between PMMA and CNTs. Elastic incoherent scattering measurement and NSE spectroscopy experiments were performed on PMMA composites with different CNT loading to study polymer self-motions and collective dynamics at various degrees of CNT percolation. An upward shift in the elastic incoherent intensity of the composites, relative to data in Fig. 4(a), indicated that the inclusion of nanotubes results in a fraction of immobilized interfacial polymer segments that appear static within the 2 ns resolution of the spectrometer. MSD measurements showed that all composites experienced decreased mobility relative to neat PMMA. However, the extent of mobility suppression was found to be independent of CNT loading (1 vol. % up to 15 vol. %), indicating that the percolation of the slow interfacial polymer regions around the CNTs kinetically restricts the mobility of the fast non-interfacial segments. Complementary NSE measurements showed that the segmental relaxations in the composites are an order of magnitude slower than in neat PMMA. These findings suggest that particle percolation could alter the mechanical properties in nanocomposites in ways that transcend stress propagation along the particle network where concomitant reduction in polymer mobility and relaxations could significantly impact macroscopic material properties.

Other examples of the use of neutron spectroscopy in the study of interfacial dynamics include the work by Roh *et al.*^{76} on poly(butadiene) (PBD) nanocomposites. Backscattering QENS spectra were collected on neat and composite PBD with high volume fractions of carbon-black prepared by extracting unattached or free chains. Comparably, the QENS spectra on the composite showed significant narrowing relative to the neat polymer indicating that the binding of PBD chains to the carbon black particles results in a slowdown of the segmental motions and an increase in the dynamical heterogeneity (as inferred from the extracted values of the stretching exponent of the KWW fits).

Another recent study by Senses *et al*.^{77} used neutron spectroscopy to investigate the dynamics of the interphase region in nanocomposites with dynamically asymmetric interphases formed by silica nanoparticles with adsorbed PMMA, a high-*T _{g}* polymer, dispersed in PEO, a low-

*T*polymer. Utilizing selective isotope labeling of the chains, both PEO and PMMA were contrast-matched to silica to render the particles invisible to neutrons. Backscattering and NSE spectroscopy measurements were performed to study the role of an interfacial polymer on the segmental and collective dynamics of the matrix chains over time scales of sub-ns to 100 ns. Due to the weak enthalpic interaction of PEO chains with adsorbed PMMA, PEO-SiO

_{g}_{2}/PMMA composites were thought of as akin to nanocomposites with neutral interfaces. In this case, the NSE results show that the Rouse relaxation remains unchanged and are identical to PEO homopolymer. This observation is contrary to PEO-SiO

_{2}composites, characterized by strong interfacial interactions between PEO and bare silica, which show much slower relaxation rates (see Fig. 12). The measurements also provided evidence of tube dilation in nanocomposites with weakly interfacial interactions. Complementary backscattering measurements revealed that the glassy and rubbery states of the bound polymer, at different temperatures, underpin the collective relaxations and the mechanical properties. As a whole, the results suggest that the presence of well-dispersed weakly attractive particles is likely to be the cause of macroscopic stiffening in composites.

#### 2. Processing effects

Neutron spectroscopy techniques have also been effectively utilized in understanding the impact of nanocomposite processing on polymer dynamics and resultant rheological properties. In a recent study, Senses *et al*.^{78} used a combination of neutron spectroscopy and x-ray photon correlation spectroscopy (XPCS) to investigate chain dynamics and nanoparticle motion in attractive polymer nanocomposites subjected to large deformations. PEO-silica composites were measured before and after shear. QENS measurements revealed substantial reduction in polymer mobility in the presence of the silica nanoparticles. Additional measurements after shear revealed further reduction in the polymer mobility due to increased pinning of interfacial polymer segments at the surface of the attractive silica particles (see Fig. 13). Supporting NSE experiments showed slowdown of the Rouse rates at high particle loadings, i.e., in the strongly confined state, but no noticeable change at low particle loading, i.e., lightly confined system. These shear-induced changes were used to explain the slow nanoparticle motion measured by XPCS. This study provided an elegant approach to understand the effects of shear on segmental dynamics, setting the stage for new possibilities for designing mechanically responsive nanocomposites.

### C. Polymer solutions

#### 1. Polymer–particle solutions

Concentrated and semi-dilute polymer–particle solutions are of great interest in a variety of industrial and pharmaceutical applications. They also serve as a model system for studies of complex flow and transport processes in biological environments and porous media. Depending on the relative size of the particles and the polymer chains, these solutions can have various properties. Predicting and controlling these properties require hierarchical characterization of the structure and dynamics exhibited over relevant length and time scales. Neutron spectroscopy, combined with judicious contrast matching schemes, can be of significant impact in such investigations. This is clearly illustrated in a recent study^{79} which utilized neutron contrast variation to selectively probe polymer dynamics in semidilute solutions of high-M_{w} PS and SiO_{2} nanoparticles in 2-butanone, obtained by matching the solvent SLD to that of SiO_{2} [Fig. 14(a)]. Taking advantage of the inherent differences in x-ray scattering cross sections from the same materials (see Fig. 1), complementary XPCS measurements were synergistically used to investigate the dynamics of the nanoparticles in solution. These complementary measurements provide a clear picture of the effect of long-range interparticle interactions on the coupling between particle and polymer dynamics. In all cases, the polymer dynamics follow the Zimm model [Fig. 14(b)]. When the particles are comparably sized to the radius of gyration of the polymer, the dynamic coupling was found to result in subdiffusive particle dynamics. However, the coupling is more complex over interparticle distances, manifested by particle dynamics not fully described by the relaxation of the surrounding polymer chains.

#### 2. Brush dynamics

For polymer grafted nanoparticles, the structure and dynamics of the polymer brush can have significant impact on their dispersion state in solution or in polymer matrices for nanocomposite applications. Due to the highly curved nature of nanoparticles and the geometric configuration of grafted brushes, the behavior of polymer grafts on nanoparticles is more complex than their analogs on planar substrates. To understand this complex behavior, Wei *et al*.^{80} synthesized PMA-grafted nanoparticles with selectively deuterated blocks for the concentrated polymer brush (CPB) region and the semi-dilute polymer brush (SDPB) region. Using NSE, they were able to selectively study segmental dynamics within the CPB and SDPB regions in solutions of 1,1,2,2-tetrachloroethane prepared with the two types of PMA-grafted nanoparticles. Their NSE measurements showed that, in both regions, the segmental relaxations exhibit the standard stretched exponential behavior of the KWW form. However, the relaxation in the CPB region was found to be significantly slower than the SDPB region due to confinement effects in the CPB region (see Fig. 15). Interestingly, the dynamics in the SDPB region were found to follow Zimm-like behavior at all length scales within the measurement window.

Brush dynamics were also measured using NSE by Poling-Skutvik *et al*.^{81} Their measurements were performed on high molecular weight PS-grafted silica nanoparticles dispersed in semidilute solutions of linear polymers. The structural characterization of the samples showed that the linear free chains do not penetrate the grafted corona but rather serve to increase the osmotic pressure of the solution, which eventually results in the collapse of the grafted polymer and subsequent aggregation at high matrix concentrations. Their NSE measurements showed that segmental relaxations follow the Zimm model on short time scales and are controlled by the solvent viscosity. At longer time scales, the measurements showed signs of confinement effects by neighboring grafted chains, preventing full relaxation over the accessible time scale. Furthermore, the addition of free linear chains to the solution was found to have no effect on the Zimm relaxations but increased the confinement effects on the grafted chains.

## V. SUMMARY AND OUTLOOK

This tutorial highlights the applications of neutron spectroscopy methods, particularly QENS and NSE, in studying selective dynamics in simple and complex polymeric systems at length and time scales, which are relevant to dynamic modes that dictate important material properties. Multiple examples were used to illustrate the possibilities of neutron spectroscopy studies on polymeric systems in melt, solution, and composite forms. These studies are enabled by the inherent properties of neutrons to show “where atoms are and what atoms do”—as stated in the 1994 Nobel Prize announcement in recognition of the pioneering work of Shull and Brockhouse for using neutrons to study the structure and excitations in crystals. Since then significant strides have been made in expanding the application of neutron scattering to soft materials, which seldom exhibit the regular structural patterns of solids and crystals. Developments in neutron instrumentation have led to significant advances in polymer science, from early studies^{82} of Rouse^{83} and Zimm^{84} dynamics in simple polymers to elaborate investigations of detailed structures and dynamics in a plethora of neat and composite polymeric systems.^{18,30} Looking into the future, neutron scattering and spectroscopy techniques will continue to be uniquely positioned to uncover detailed structural and dynamical properties in various forms of polymeric materials. The need for selective spatiotemporal characterization, enabled by neutron spectroscopy and isotope substitution, will continue to grow with the increasingly advanced designs of multi-component polymeric materials for various technological applications. Fortunately, concomitant advances in materials simulations and their synergetic integration with neutron scattering^{85} are setting the stage for more sophisticated experiments that will further push the boundaries of neutron scattering applications in advanced material characterization.

## ACKNOWLEDGMENTS

The author is grateful to her Ph.D. advisor, Professor Roger Pynn, for introducing her to the fascinating field of neutron scattering, its fundamentals, and its many applications. She is also thankful to Professor Ramanan Krishnamoorti and Dr. Paul Butler for many stimulating discussions on polymeric materials and the exciting opportunities that neutrons offer in their nanoscale characterization. The author thanks Dr. Saptarshi Chakraborty for assistance with some figures. She also acknowledges vibrant and enlightening discussions at the Macromolecular Innovation Institute and the Center for Soft Matter and Biological Physics at Virginia Tech. This work was supported by faculty startup funds from Virginia Tech.

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