Polymer networks are complex systems of molecular components. Whereas the properties of the individual components are typically well understood by most chemists, translating that chemical insight to polymer networks themselves is limited by the statistical and poorly defined nature of network structures. As a result, it is challenging, if not currently impossible, to extrapolate from the molecular behavior of components to the full range of performance and properties of the entire polymer network. Polymer networks therefore present an unrealized, important, and interdisciplinary opportu-nity to exert molecular-level, chemical control on material macroscopic properties. A barrier to sophisticated molecular approaches to polymer networks is that the techniques for characterizing the molecular structure of networks is often unfamiliar to chemists. Here, we present a critical overview of the current characterization techniques available to understand the relation between the molecular properties and the resulting performance and behavior of polymer networks, in the absence of added fillers. We highlight the methods available to characterize the chemistry and molecular-level properties of individual polymer strands and junctions, the gelation process by which strands form net-

Through synthesis, chemists can now program remarkable properties and function into individual polymer molecules, including precise conformational behavior, unusual extensional profiles, and covalent chemical responses. The level of molecular control of individual macromolecules, however, has yet to be replicated at the level of polymer networks. The challenge in translating from single macromolecular to network behavior originates from the fact that the details of molecular design have been hidden by statistical chemistry and poorly defined structures, limiting the ability to bring chemical knowledge to bear on challenges in polymer network performance and properties. Thus, there currently exists no direct and quantitative method to extrapolate from the molecular behavior of the individual components of a polymer network to the full range of physical behaviors of that network. Polymer networks therefore present an unrealized, important, and interdisciplinary opportunity for chemists of different sub-fields to exert molecular-level, chemical control of on material macroscopic properties. If polymer networks could be treated as complex chemical systems, the full power of synthetic, physical, and theoretical chemistry might be rationally directed toward overarching challenges in de novo molecular network design.

As synthetic techniques provide increasing control over a greater range of molecular structure space, an overarching challenge to realizing this vision is that of characterization. Polymer networks are not amenable to the char-acterization techniques commonly applied to small molecules, and even the language necessary to describe statistical distributions of connectivity is unfamiliar to most chemists. These barriers limit the ability of chemists to connect the creativity possible in molecular structure and function to the resulting structure and function of polymer networks. In this review, we survey techniques available for extracting molecular-level details of polymer network structure. Some of these techniques are well established over decades of use but are still rarely employed by chemists; others have been developed only recently and offer more direct and precise measures of quantities and distributions that historically have been inferred indirectly.

The focus of the review is to survey the current characterization toolkit available for extracting molecular information relevant to polymer network behavior. A comprehensive review of every instance in which a given characterization technique has been used is neither possible nor useful. Instead, we emphasize the key considerations and limits of the various techniques and provide representative examples of their use. While we cannot walk through the detailed steps required to execute each technique, we highlight key technical considerations, and we encourage the reader to seek technical details for implementation in the primary literature we cite. Our goal is that the chemist can use this review to identify techniques that are suitable to a given molecular characterization question, and in so doing address major knowledge gaps in the field. By combining contemporary characterization tools with ongoing advances in polymer chemistry, the field is now poised to update existing theories of network behavior and uncover molecular strategies to control network properties that are presently obscured.

Here, we review the methods in characterizing polymer networks. The polymer networks can be neat materials, like thermosets or elastomers, or can be gels, swollen in mobile solvent. We will refer to macromolecules that are connected to network junctions as strands. The strands can be connected as bridges be-tween different junctions, as dangling loops that connect different sites of the strand to the same junction, or as dangling ends that retain unreacted groups (Figure 1). The junctions connecting the polymer strands can be either permanent in the case of covalent bonds and trapped entanglements, or reversible in the case of supramolecular interactions, dynamic covalent associations, or transient entanglements.

We will focus our attention on amorphous networks with flexible strands (persistence length smaller than the strand length between crosslinks) and without fillers. Typically, only partial information can be obtained from individual methods; a combination of several techniques is always necessary for complete characterization of polymer networks. First, the methods available to characterize the chemistry and molecular-level properties of strands and junctions are discussed in Section 2. These methods are largely similar to traditional characterization of polymers and so we focus on only the salient techniques for comparison and connection to the analysis of crosslinked materials. We then survey analysis of the gelation process, separation of the sol and gel components, and basic molecular characterization of the sol in Section 3. We outline the main methods for characterizing the structure of the network in Section 4, particularly focusing on classical approaches for the determination of mesh size and modern methods to tease out the fraction of dangling ends and other defects. In Section 5, we thoroughly explore the main characterization techniques for describing the mechanics and dynamics of the polymer networks, with particular emphasis on the rheology and molecular spectroscopy. We highlight in Section 6 the utility of computer simulations in explaining experimental data, testing theories, and providing predictions for network structure and properties. Finally, in Section 7, the current state of polymer network characterization is summarized, best practices are discussed, and a prognosis for the future is offered.

Polymer networks can be formed from a mixture of bi-and multi-functional monomers or by cross-linking precursor chains. These precursor strands, also known as pre-polymers, are polymer chains containing reactive side chains or end groups that can form macroscopic networks under appropriate conditions, by end-linking with a multi-functional junction or by random cross-linking (i.e., vulcanization). As the basic building block, thorough physical and chemical characterization of precursor strands is vital to predictably and reproducibly form polymer networks with desirable properties. Below we discuss characterization methods applicable to the molecular and macromolecular levels. Many of these techniques are applicable before and after cross-linking; comparison of the results from these techniques before and after network formation thus provides useful information regarding the cross-links.

Nuclear magnetic resonance (NMR) spectroscopy is a highly versatile and powerful technique that has expanded into nearly all subdisciplines of chemistry and materials science. Commonly, an NMR spectrum presents peaks at different chemical shifts corresponding to nuclei of different chemical environments, allowing structural analysis and compositional characterization. More sophisticated and multidimensional NMR techniques connect NMR parameters such as chemical shift, line width, couplings, and relaxation rates to the structure and dynamics of the polymer chains at the atomic level. 1 Common one-dimensional (1D) NMR techniques (e.g., 1 H and 13 C) are often used to provide information on reaction conversion, functionalization, product purity, and monomer content. 1D NMR is particularly useful in elucidating the composition of copolymers, where recent advances of quantitative NMR measurements coupled to high-resolution mass spectrometry have enabled sequence-level characterization. 2

Infrared spectroscopy (IR) exploits the absorption of light by molecules at frequencies that match their characteristic vibrational frequencies. Specifically, Fourier-transform infrared (FTIR) and Raman are two well-established spectroscopic techniques that detect activities related to the symmetry of the molecules. A change in the dipole moment of a bond provides IR signal while a change in polarizability causes it to be Raman active. The vibrational spectra of complex molecules, such as polymers, which consist of many atoms and bonds, are correspondingly complex. However, the absorption maxima for many chemical groups have already been tabulated and enable rapid determination of the presence or absence of specific functional groups, such as carbonyls, azides, and alcohols. 3,4 Since FTIR and Raman spectroscopies are complementary to each other, these techniques collect information on the vibrational states associated with different chemical bonds. 5 Thus, IR spectroscopy to be used as a quick and low-cost (compared to NMR) method to confirm the chemical structure of a polymer by confirming the presence of the expected functional groups. 6 Common uses of FTIR and Raman spectroscopy in polymer science center around functional group determination via comparison between peaks of interest with databases. The area under a characteristic region can be translated to a molar quantity using a Beer's Law type analysis. 7 This information can then be used to determine the degree of polymerization, degree of functionalization, as well as the presence of contaminants and the morphology of polymer crystals. [8][9][10][11] In addition to the conventional bulk sampling techniques, IR has been coupled with atomic force microscopy (AFM). The combination allows high spatial resolution on a nanoscale that can hardly be achieved previously due to the optical diffraction limit on conventional IR. 12 Other molecular spectroscopies such as ultraviolet/visible absorption (UV-Vis) or fluorescence spectroscopies can also be used to identify molecular subgroups, and are particularly useful for quantifying the concentration and states of chromophores such as those necessary for opticalbased network characterization methods (Sections 4.4, 5.4.3, and 5.9).

There are a variety of experimental methods available for determining different average molecular-weights (e.g., number-average M n and weight-average M w ), molecular-weight distributions, as well as the interactions between precursor polymers and a given solvent (e.g., second virial coefficient A 2 ), which will be discussed later in this section. These methods typically utilize dilute solutions, and often require measurements at several concentrations to extrapolate to the limit of zero concentration. Different methods are applicable to different ranges of polymer molecular weights, as summarized in Table 1.

Table 1: Summary of characterization techniques used to determine different types of polymer molecular weights. [13][14][15]

Absolute Relative M n M w A 2 Range (g mol

Beyond quantifying the chemical composition (Section 2.1.1), for polymers with distinct chain-end functionalities, NMR end-group analysis provides the number-average molecular weight (M n ) of the polymer, by molar ratio of the integrated peaks from the chain-end moieties and the main-chain units. 16 However, the sensitivity of modern NMR machines is typically only sufficient to determine M n of relatively short polymers with degree of polymerization N < 100. 1 Further, other properties of the polymer are also accessible by advanced and multi-dimensional NMR techniques, such as 2D and 3D NMR. 1,17 Two-dimensional (2D) NMR techniques, which provide correlation between two parameters by plotting in the space over two frequency axes, have become increasingly popular in polymer characterization. 18,19 Throughbond correlation methods, both homonuclear (e.g., correlation spectroscopy (COSY)) and heteronuclear (e.g., heteronuclear singlequantum correlation spectroscopy (HSQC) and heteronuclear multiple-bond correlation spectroscopy (HMBC)) permit characterization of the bonded structure and are useful for polymers with complex architectures. Throughspace correlation methods (e.g., nuclear Overhauser effect spectroscopy (NOESY)) are useful in identifying stereochemistry and describing chain configurations. One advanced NMR method is particularly useful in polymer char-acterization: diffusion ordered spectroscopy (DOSY) enables the determination of size and size distribution of polymers difficult to characterize via other methods by measuring their distribution of diffusion coefficients, but only provides an estimated average molecular weight upon calibration. 19,20 NMR experiments can be performed in solution or solid-state, depending on sample solubility and information of interest, permitting complete characterization of the precursor strands prior to network formation. [21][22][23][24]

Mass spectrometry (MS) measures the mass-tocharge ratio (m/z) of a molecule by passing the sample through the following processes: ionization, separation, and detection. 25,26 Gas/liquid chromatography can be installed prior to the spectrometer to provide physical separation of molecules before their masses are measured. Owing to the discovery of soft-ionization protocols such as electrospray ionization (ESI) and matrix-assisted laser desorption/ionization (MALDI), MS has gained more popularity among the polymer chemistry community. 27 During MALDI, the polymer sample and matrix solutions are mixed and dried. The solid crystals are ionized with a laser beam, causing the polymer strands desorb into the gas phase. 28,29 The sample is then passed through a time-of-flight analyzer where ions of different m/z are dispersed while they fly through the compartment, and eventually detected at the detector. Despite its ability to produce intact molecular ions of polymers well above 10 4 g mol -1 , MALDI demonstrates best performance on polymers with low dispersity as higher-mass species could be underrepresented in the analysis. 28,29 Frequently, matrix selection plays an important role in the success of MALDI characterization; strategies on sample preparation can be found in the literature. 30 One of the most common properties obtained from a MALDI experiment is the average molecular weight and is commonly compared with results obtained from classical methods such as SEC. 31 A MALDI spectrum, however, further provides a distribution of molecular weights from which M n , M w , and Ð can be directly calculated provided that molecular ions ionize similarly across the distribution. 25 However, because higher-mass species are potentially underrepresented in the spectrum, MALDI can return biased distributions and shifted averages for polydisperse, high molar-mass samples.

In addition to providing the molecular-mass distribution, advancements such as deconvolution mass spectrometry enable finer separation. In this technique, a low molar-mass spectrometer is used to measure and deconvolute a high m/z spectrum by coupling two mass spectrometers such that analyte ions separated by m/z are then fragmented and detected again. 32 Detailed structural information can be extracted by comparing parent ions and cleaved end-groups such as shown in Figure 2 for poly(methylacrylates). 33

Osmotic pressure is a thermodynamic colligative property (that is, sensitive to the number of molecules present) that measures the free-energy difference between a polymer solution and a pure solvent. While osmometry determines the number-average molecular weight (M n ) of the material, other information like molecular distribution cannot be evaluated, which, with the advent of methods such as sizeexclusion chromatography, has resulted in a de- crease in the use of osmometry in polymer analysis. 34,35 There are two main types of osmometry techniques: membrane osmometry (MO) and vapor pressure osmometry (VPO). 36 In MO, a semi-permeable membrane separates pure solvent from a polymer solution in the same solvent. The pressure difference between the solvent and polymer solution is called the osmotic pressure Π:

where c is the concentration,

and T is the absolute temperature, permitting calculation of the polymer's numberaverage molecular weight M n and the weightaverage second virial coefficient A 2,w , a measure of the interaction between the dissolved polymers in solution. 15 The osmotic pressure at sufficiently dilute concentrations is then directly correlated to molecular weight according to the van't Hoff law:

VPO, on the other hand, is dependent on the equilibrium thermodynamics of vapor pressure, by measuring the difference in temperature (∆T ) between the two thermistor beads (which in practice is measured as a resistance difference ∆R). 35,37,38 Similar to MO, VPO directly correlates to the number-average molecular weight; unlike MO, however, VPO gives no information about absolute yield. The main advantage of VPO lies in its accuracy in determining the M n of lower molecular-weight polymers (< 10 4 g mol -1 ). 36

Viscometry is another classic technique that has been utilized in polymer science and characterization. It takes advantage of a change in the viscosity of a dilute polymer solution η in comparison to a pure solvent η s . In its simplest form, with Ubbelohde viscometers, successive measurements of the time that it takes for this dilute solution to pass between two different viscometer marks are compared to that of the pure solvent. 15,36,39 These measurements give information about the intrinsic ability of a certain polymer to increase the viscosity of a solution, which enables the calculation of its intrinsic viscosity:

where c is the polymer concentration. The Flory-Fox equation relates intrinsic viscosity to the hydrodynamic size R and viscosity-average molecular weight, M v , of the suspended polymer,

where Φ = 0.425N A = 2.5 × 10 23 mol -1 is a universal constant, in conjunction with the Mark-Houwink-Sakurada equation,

where K and a are Mark-Houwink parameters, which are tabulated for nearly all common linear polymer-solvent combinations. 15 Typically, the M v value lies between M n and M w . 36 Many other viscometric measurement geome-tries exist in addition to the Ubbelohde viscometer mentioned previously. Most common in polymer science are falling ball (or rising bubble) viscometers where the viscosity is proportional to the rate at which a ball falls (or a bubble rises) through the fluid. Rotational viscometers, which obtain viscosity from the torque needed to shear the fluid between rotating boundaries, are also common. The main advantage of viscometry is its simplicity, speed, and superior precision over a wide range of molecular weights. In order to increase the accuracy of molecular weight detection in polymer precursors, size-exclusion chromatography (below) can be combined with online viscosity detection. 40-42

Size-exclusion chromatography (SEC), including gel-permeation chromatography (GPC), is a commonly utilized characterization technique to measure polymer molecular-weight distributions (including number-average molecular weight M n , weight-averaged molecular weight M w , and dispersity Ð). 36,43 During this process, a dilute polymer solution is injected into a solvent stream and flows through a separation column packed with porous beads, resulting in an exit stream fractionated according to the pervaded volume of the material, [η]M , where [η] is the intrinsic viscosity and M is the molar mass. Detectors, most commonly differential refractometry (dRI) and ultraviolet (UV) absorption, log the separation with elution time, differentiating between different polymeric components and providing the molecular-weight distribution of the polymer when calibrated to known polymer standards such as polystyrene. One approach to overcome the limitation of calibration to a known standard is to combine SEC with viscometry (Section 2.2.4), as the absolute molecular weight of the polymer can be elucidated from its hydrodynamic size (obtained from SEC) by knowing its intrinsic viscosity, according to the Flory-Fox equation. 44 This relationship is the basis of the socalled universal SEC calibration. In a typical calibration process, the intrinsic viscos-ity is replaced by the Mark-Houwink relation-

2 . Knowing the Mark-Houwink relations for two different polymer-solvent pairs, the molecular weight of one polymer eluting at a particular volume can be calculated in terms of the other polymer, provided its elution-volume-molecular-weight relation is known. 45 The intrinsic viscosity can be measured either separately, for example, by capillary or rotary viscometry; or the SEC systems can be equipped with an online differential viscometer.

Incorporating an in-line light-scattering detector (SEC-LS) can be useful for novel systems (e.g., biopolymers and proteins) in which reference standards are unavailable and Mark-Houwink parameters are unknown. 46,47 To complement the aforementioned techniques, researchers have recently coupled SEC with Nuclear Magnetic Resonance Spectroscopy (NMR) [48][49][50] and Mass Spectrometry (MS) 51 in an effort to provide additional structural resolution and selectivity. Characterizing polymer precursors and strands prior to network formation using SEC can be extremely useful in planning the synthetic strategy of subsequent network formation 52 and accurate depiction of strand mass in subsequent gel characterization.

However, it should be noted that precise characterization of structural properties, composition, and purity of network strands, particularly of complex architectures, can be challenging and the results need to be analyzed with caution. 53 Finally, the determination of molecular weight distributions by SEC is not a suitable approach for many reversible network systems. Specifically, in supramolecular polymers, significant tailing in the distribution is normally observed due to the reversible nature of the chemical or physical bonds. 54 However, SEC also offers a significant opportunity to better understand association/dissociation kinetics in dynamic networks, by measuring the mass distribution as a function of temperature or another stimulus. [55][56][57]

In polymer solutions, light scattering occurs due to the difference in refractive indices between the polymer n and the solvent n s . There are two major modes of light scattering experiments-static (SLS) and dynamic (DLS) light scattering. The technique is dependent on the relationship between the intensity of incident light of wavelength λ that is scattered by a molecule (in this case the polymer chains) and the molecular weight and size of these molecules: 36,58 Kc R θ

) is an optical constant, dn/dc is the refractive-index increment, and R θ is the Rayleigh ratio. Structural information regarding the size and shape of the chains can be obtained by Guinier (ln R θ vs. q 2 ) or Kratky (q 2 R θ vs. q) analyses with light intensity data measured at a single (dilute) concentration as a function of angle θ (or scattering vector q = 4πn s sin (θ/2)/λ). Static light-scattering data from a series of angles and concentrations allows calculation of the weightaverage molecular weight M w , z-average radius of gyration R g , and z-average of the second virial coefficient A 2,z . A corresponding Zimm plot analysis and a schematic of the technique is shown in Figure 3. Light scattering is accurate for determination of the molecular weight in a range of ∼ 10 4 -10 6 g mol -1 . 36 In dynamic light scattering (DLS, discussed more in depth in Section 5.5.1), the change in the instantaneous scattering intensity on a time scale of the molecular motion is recorded. The scattering intensity fluctuates due to local fluctuations in concentration caused by the thermal motion (diffusion) of the polymer chains. By analyzing the time-resolved scattering intensity the diffusion coefficient D of the macromolecules can be calculated. The hydrodynamic size of the polymer can be estimated from the diffusion coefficient through the Stokes-Einstein equation:

where k B T is the thermal energy and η s is the viscosity of the solvent. Both SLS and DLS detectors can be integrated with a SEC system, completely freeing it from the necessity of calibration. 59

Thermogravimetric analysis (TGA) is a thermal analysis technique that measures mass changes of the material in response to a change in temperature. TGA is often carried out in a well-controlled atmosphere such as a vacuum or an inert gas filled environment to control the nature of the thermal decomposition by preventing oxidation. The measurements can be presented in plots that are either temperaturedependent or time-dependent where changes in mass can usually be attributed to material volatility or degradation. 60 Assessing the thermal stability of polymers is the predominant application of TGA. These data are especially valuable for polymer strands before or after formation of the network as it provides a temperature range in which the material can ultimately operate. 61 For example, a typical TGA curve is shown in Figure 4 and is used to confirm thermal stability of a solid polymer electrolyte at the operating conditions of a lithium ion battery. 62 Finally, TGA coupled to a mass spectrometer, can be useful for monitoring crosslinking that occurs by thermolysis with the loss of small molecules. 63

Differential Scanning Calorimetry (DSC) is an analytical technique that is effective at de-termining physical and chemical properties of polymers undergoing glass transitions and thermodynamic phase transformations. 14,36,64 DSC works by monitoring temperature differences between two samples, the polymer sample and a reference. The rate of change in the temperature will differ between the two samples based on compositional and physical phase changes, yielding a difference in heat flow as a function of temperature that correlates to the specific heat, C p , of the material. 65 Thus, DSC is routinely used to determine glass-transition temperatures of polymers and to evaluate melting and crystallization enthalpies. DSC has been furthermore used to study monomer 66 reactivity and precursor polymerization kinetics. 67

Force Spectroscopy Single-Molecule Force Spectroscopy (SMFS) enables quantitative investigation into the intra-molecular interactions of macromolecules, including the mechanics of proteins and other biopolymers, [68][69][70] the elastic properties and conformational changes of single polymer chains, [71][72][73] and mechanochemical structure-activity relationships. 74,75 Recently, SMFS has been used to characterize the mechanics of macromolecules from forces as low as a few pN and as high as several nN, and with a dynamic range of ≥ 1 µs. [76][77][78] Molecular properties and behaviors that have been quantified using SMFS include the force-extension relationship of a random-coiled polymer, 72,79,80 the chair-to-boat conformational transition of polysaccharides corresponding to a plateau in the force-extension curves, 71,72,81 and correlation of mechanochemical reactions with molecular structure of synthetic polymers, for which model fitting unveils reaction pathways. 75,82,83 A popular manifestation of SMFS is based on an atomic force microscope (AFM) setup (Figure 5). Polymer chains are mounted on a solid substrate, which is moved upwards until the polymer sample is brought into contact with the tip, allowing polymers to bridge the cantilever tip and the substrate through physical interactions or covalent bonding. After equilibration, the piezoelectric stage is moved downward to stretch the polymer. The deflection of the laser reflection from the cantilever is recorded with the stage translation as the force-extension relationship. Low-force regimes (usually < 200 pN) are dominated by a reduction in the chain conformational degrees of freedom, and this is often referred to as the "entropic" regime. Extensions at higher forces come with increasing contributions from so-called "enthalpic" deformations, including the extension of bond lengths and bond angles, that eventually dominate the extensional behavior. Coiled polymer chains (red) are absorbed onto a solid piezoelectric stage moving vertically. The reflected laser beam is well aligned between the cantilever tip and the center of photodiode in (a). In the force-displacement curve, the dotted line represents the process from (a) to (b) as the piezoelectric stage approaches the cantilever tip. The solid line represents the process from (b) to (e) as the piezoelectric stage retracts away from the tip. Specifically, the red plateau represents the scissions of relatively weak bonds and the releases of stored length.

While AFM-based SMFS provides high-resolution imaging at high-force regimes (10 -10 4 ) pN, optical (10 -1 -10 2 ) pN and magnetic (10 -3 -10 2 ) pN tweezers-based SMFS show advantages for low-force-range measurements. 85 Created by focusing a laser to a diffractionlimited spot, optical tweezers are ideal for the 3D manipulation of targeted molecules due to its low-noise and low-drift dumbbell geometry. 70,86,87 Manipulating magnetic particles ranging 0.5-5 µm by an external magnetic field, magnetic tweezers combine force-clamp properties with the ability to impose rotation. 88,89 A particularly useful strategy to engineer molecular responses into polymer extensional behavior is to incorporate mechanophores, chemical moieties that undergo force-coupled covalent transformations, 90,91 into the polymer strands. For example, as shown schematically in Figure 5, the scission of relatively weak bonds incorporated into the backbone occurs at low to moderate force extension (typically 400 pN 4000 nN), resulting in the release of the stored-length and enhanced singlechain toughness. SMFS enables the quantification of such mechanochemistry on a singlechain level. 92 The mechanochemical reaction rate as a function of force can be obtained from a "constant-velocity" experiment (described above and in Figure 5), in which the polymer chain is stretched at a constant velocity (usually several hundreds of nm s -1 ), but also from the "force clamp" methodology, in which the polymer is kept under a constant force. "Force clamp" experiments yield kinetic rate data from the analysis of the polymer elongation as a function of time and have been used to study polyproteins and synthetic polymers with multi-mechanophores. [93][94][95] 3 Gelation

Below the gel point, all species are soluble (in the appropriate solvent) allowing the standard dilute solution characterization methods to be utilized. Above the gel point, there is an insoluble gel fraction and a soluble sol fraction. Characterization of the sol fraction both below and above the gel point is important in under-standing the gelation process and the properties of the formed gels. The sol fraction typically consists of randomly branched polymers of various sizes and non-cross-linked precursor chains not connected to the gel, together with solvent and any other low-molecular weight reagents, catalysts, or byproducts that might be present in the reaction mixture. Of particular interest are the chemical composition of the polymer species in the sol fraction, as well as their molecular characteristics, such as molecular weight and molecular weight distribution, chain size, branching, etc. 45,96

When immersed in an excess of the appropriate solvent, the gel fraction will swell, and the sol fraction will slowly diffuse or flow out of the swollen gel into the excess solvent. A major experimental concern in any study of polymers beyond the gel point is the question of perfect separation of this sol from the gel. Soxhlet extraction, shown schematically in Figure 6, is a convenient method for complete separation of the sol and gel fractions. Solvent is boiled in the bottom flask and condenses at the top of the apparatus, dripping down on to the gel in a paper or glass thimble. The solvent carrying the sol fraction siphons back to the bottom flask, where only solvent boils. Eventually the Soxhlet extractor has all of the sol fraction in the boiling solvent and all of the gel fraction in the thimble. The swollen gel can then be dried under vacuum and weighed to determine the gel fraction, defined as the fraction of all monomers belonging to the gel. Every monomer must be either part of the sol or part of the gel, so the sum of the sol and gel fractions is unity:

The properties of the gel and sol fractions can then be characterized separately.

Owing to the very broad molar mass distributions associated with branching as the gel point is approached, one primary characterization tool is size-exclusion chromatography (Section 2.2.5). Low-angle light-scattering detectors enhance the capability to determine the molecular-weight distribution of the sol. In the sol there is a very small amount of the largest species, but they dominate the light scattering. The light-scattering intensity corresponds to the product c i M i , which corresponds to a special self-similar distribution of molecular weights of all branched or cross-linked species (Figure 7a) with fraction of N -mers P (N ) ∼ N -τ up to the characteristic degree of polymerization N * . The critical exponent τ (also called the Fisher exponent) is between 2 and 3 assuring that number-average degree of polymerization (M n ) is finite at the gel point, while the weight-average degree of polymerization (M w ) and dispersity index Ð=M w /M n diverges. The z-average degree of polymerization M z is proportional to N * and diverges faster than M w . The Fisher exponent τ = 2.5 in the mean-field theory (Flory-Stockmayer) and is close to 2.2 in 3D critical percolation. Most experimental situations are in the cross-over between mean-field and critical percolation limits. 45,97,98 A universal molar-mass distribution plot, such as Figure 7a, can then be constructed from the concentration, intrinsic viscosity, and weight-average molar mass using appropriate detectors.

The extent of reaction p can, in principle, be measured by molecular spectroscopy methods such as FTIR and NMR (Sections 2.1.1 and 2.1.2). These molecular spectroscopies further allow calculation of the distribution of reacted cross-linkers (i.e., mono-, bi-, tri-, tetrareacted) and end-group analysis. However, even with small relative error, the relative extent of reaction (i.e., distance from the gel point p c : = (p -p c )/p c ) goes to zero at the gel point and the error diverges. While this poses a problem for evaluation of the critical exponents (e.g., Fisher exponent τ ), it typically is sufficient to measure the extent of reaction and empirically find the gel point over a relatively narrow window. Figure 7b shows the determination of the gel point by plotting the measured weight-average molecular weight from SEC against the extent of reaction measured by NMR for both solutions before the gel point and the sol fraction after the gel point. The estimated window of the gel point is shaded in grey, highlighting difficulties in accurately measuring the gel point. Since the weightaverage molecular weight (M w ) diverges rapidly at the gel point, it is easy to find the approximate gel point in this fashion, although due to the large inherent error in measuring the extent of reaction, it is often more accurate to plot the size R or weight-average molecular weight as a function of N * . As such, the gel point is bet- ter determined by rheology, as will be discussed below.

Although crude, many chemists rely on the oft-used "inversion test" where a gel is identified as a sample that supports its own weight and does not flow when the vial is inverted. It is emblematic, however, of the facts that rheology (Section 5.1) is the most common manner to identify the gel point and that in many situations, most chemists care that their gel is well formed, so that they can continue to further applications and testing but are unconcerned with the exact gel point. If the reader is interested in questions such as how molecular structure influences properties in the vicinity of the gel point, then absolute characterization of the gel point or extent of network formation is likely necessary. It is also useful to know if the materials of interest are near the gel point, as very small differences in reactivity can cause small changes in the extent of reaction and thus large changes in macroscopic properties. In this situation, precise characterization of the relative extent of reaction is also necessary to appropriately compare data. In many cases, however, polymer chemists are primarily interested in whether a majority, percolating network is formed and how much material is in the network, before continuing with measurements on the isolated gel. Measurements of the gel point from oscillatory rheology (Section 5.1.4) and dynamic light scattering (Section 5.5.1) will be discussed more in depth later in this review; we will refer the interested reader to the references of this section and proceed to discuss characterization of the network structure, dynamics, and mechanical properties.

The properties of polymer networks are largely dictated by the underlying chemical and topological structure of junctions and strands. Whereas the primary chemical structure of a strand is relatively intuitive to most chemists, the statistics and topology of how strands are connected within a network is unique to polymer network chemistry relative to other synthetic targets. 99 The topological structure of networks is related to the connectivity of network sub-components, and because it is determined by synthetic methodology it presents an opportunity for chemical control and the molecular optimization of network structure for specific purposes. Characterization of topological structure (and its relationship to synthetic method), however, is in a more nascent stage than the characterization of individual strands. Polymer networks, in general, as shown schematically in Figure 8, are characterized at the molecular scale by the connectivity of components and defined by parameters such as branch functionality, percent functional group conversion, and loop identity and population. At larger length scales, polymer network topology is characterized by features such as molecular entanglements and inhomogeneities in network density (due to quenched concentration fluctuations present at gelation). Conventional scattering, microscopy, and other spectroscopic techniques are useful tools in elucidating the topology of polymer networks but lack molecular-level precision or contrast. In many cases, topological features of polymer networks (loops, dangling ends) are often teased out by fitting theoretical models to swelling experiments and other mechanical tests. However, advances such as advanced solid-state NMR and network disassembly spectroscopy are creating opportunities for probing the molecular details of the network structure directly and quantitatively. These nascent techniques can aid in the verification of existing methods and improvement of our understanding of polymer network structure and topology on their resultant properties.

The measurement of the equilibrium swelling of the polymer network in a particular sol-vent allows estimation of important network characteristics. 45 Characterization methods of network swelling can be divided into two major categories -gravimetric and size measurements. The networks need to sustain the stress induced by the swelling process without catastrophic failure. Typically, with these methods, the volume of equilibrium swelling state V eq and the volume of the dry state V dry can be obtained. Therefore, the equilibrium swelling ratio Q ≡ V eq /V dry , and the linear deformation factors λ = L eq /L dry in different macroscopic directions can be calculated (L eq and L dry are the length of equilibrium and dry state in the direction of measurement). However, estimation of molecular-level quantities, such as the average molecular-weight between cross-links M x and the number density of elastically effective chains ν eff , requires further assumptions and experimental details. The classical Flory-Rehner model 102,103 is based on the affine network elasticity model 104 with Gaussian chains and the Flory-Huggins solution theory: 105,106 ln 1 -

where vs is the molar volume of the solvent and χ is the Flory-Huggins interaction parameter between the polymer and swelling solvent. Although this model is widely used to relate Q measured experimentally to ν eff and M x , 107 many modified interaction and network elasticity models have been developed, [108][109][110][111][112][113] which can replace the original components in the Flory-Rehner model to correct for deficiencies. 114 Another challenge is that models typically assume both the elastic part of the free energy restricting swelling and the mixing part of free energy driving swelling are determined by the same elastically active parts of the network. However, in many cases there is a significant elastically ineffective part of the network (defects such as dangling ends and dangling loops) that contributes to the mixing component, but not to the elastic component of the free energy. This unknown fraction of elastically ineffective part of network that can be as large as 50% in some cases even far from gel point and makes determination of network properties from swelling much more difficult. So, instead of delving into the details of particular physical models, we herein focus on the technical details of operations for both gravimetric and size measurements; the approaches that probe the molecular details of the equilibrium swelling are discussed afterwards.

Both equilibrium swelling and swelling kinetics (if the process is slow enough) can be measured by simply weighing the gel. In a typical experiment, a network with a known mass is placed in a solvent, and then at certain time intervals, the sample is taken out and weighed. Typically, cylindrical samples with heights and diameters in the centimeter range are prepared and the swelling times range from hours to days or even weeks, 115 rendering the method convenient and low-cost, albeit slow. The gel reaches equilibrium swelling when the weight has saturated to a constant value. The method can be extended to thin gel films provided they are firmly attached to a solid, non-swellable substrate such as a glass slide. The total weight of the system (glass slide and attached gel) is usually smaller than 100 g and can be measured on standard balances with an accuracy of 0.1 mg.

Although gravimetry is arguably one of the easiest methods for studying gel swelling, several important experimental steps should be followed to ensure accurate results. Care should be taken to remove any excess solvent from the gel surface, usually by gentle patting with a paper tissue. After the solvent removal, however, the gel might start to dry immediately, potentially skewing the results. Thus, the weight of the gel should be large enough to allow the drying to introduce only a minor error in the weight measurements. For example, under ambient conditions (25 °C and 40% relative humidity) the evaporation rate of water from hydrogels can be as high as 1 mg min -1 cm -2 , 116 which could result in significant weight loss in the case of thin gel films with large surface-to-volume ratio. Another limitation of the gravimetric approach is that it only gives an average swelling for the whole sample, regardless of whether the swelling is homogeneous in all directions or not.

The physical dimensions of macroscopic, centimeter-sized gels are also easily measurable by a ruler or caliper. 117 Size measurements are especially valuable when the gel exhibits different swelling in different directions. 118,119 Size measurements, however, of gels very small in one or more dimension (e.g., thin films, microgels), require more sophisticated treatment. Direct size measurement of microgels or thin films with dimensions on the order of micrometers or larger, is usually achieved with optical microscopy. 120 One challenge with this approach, however, is the timeframe of the experiment as swelling rates increase with decreasing sample size, necessitating fast image acquisition for small samples. Optical microscopy has also been applied to measure the swelling kinetics of thin-film gels with a small number of fluorescent micrometer-sized particles trapped in them. 121 The swelling of reversible thin-film gels can also be studied by confocal microscopy 122,123 using a more complicated experimental setup. Further, network particles with dimensions smaller than micrometers are inaccessible to measurement with optical microscopy, and dynamic light scattering (DLS, Section 5.5.1) is used instead. 124 However, DLS is only suitable for the measurement of the equilibrium swelling because of limitations in time resolution and the accuracy. Nevertheless, DLS is a popular technique to study the effect of environmental factors, such as temperature and pH, on the equilibrium swelling of microgels.

Equilibrium swelling results from the balance of the elastic free energy change of the gel due to swelling of the network, ∆F el , and the free energy of the gel due to mixing between the network and solvent molecules, ∆F mix . 102 The osmotic pressure of the gel Π gel against the swelling solvent is zero at equilibrium swelling and positive for as prepared, neat, and partially swollen networks. The elastic component of the gel Π el gel is proportional to the modulus G at corresponding concentration, which can be determined by mechanical testing (Section 5.1). The mixing contribution to the osmotic pressure of the gel Π mix gel is usually considered to be equal to the osmotic pressure Π sol n of a semidilute solution of non-cross-linked chains at the same concentration; 15 an estimation of Π mix gel can be obtained by measuring Π sol n with osmometry (Section 2.2.3). Therefore, comparison of the pressure of the gel Π gel with the independently measured modulus G ∝ Π el gel and semidilute solution osmotic pressure Π sol n ≈ Π osm gel can provide an estimate of the relative contributions of tension in elastically effective strands and free energy of mixing of all chains in the swelling solvent. 15 Other than investigating the swelling behavior of gels in pure solvent, the swelling behavior of gels can be studied in media that can exert osmotic pressure on the gels as well. Experimentally, this approach is achieved by placing a network into a particle or polymer solution with known osmotic pressure and letting it swell to equilibrium against the osmotic pressure of solution. 121,125 The size of the solute particle/polymer should be much larger than the mesh size of the gel so that solutes cannot penetrate the gel, and the equilibrium swelling of the gel is suppressed in comparison to that in pure solvent. With this method, both elastic and mixing components of network free energy can be characterized. At equilibrium swelling, the sum of the mixing and elastic components of the swollen gel (Π mix gel and Π el gel respectively) should equal the osmotic pressure of the swelling media Π ext :

Furthermore, in principle, this method can be operated together with the probe penetration method-by placing networks into solutions with both dilute labeled probes, which can dif-fuse into the network, and large solutes, which are excluded by the network, the average mesh size of partially swollen gels can be measured while measuring Π ext .

The mesh size ξ mesh of polymer network is an important parameter that is related to the osmotic pressure and network swelling; molecularly, it is related to the correlation length ξ, which is defined as the average distance between the closest polymer segments of neighboring polymers. It is important to distinguish the mesh size (correlation length) from the size of the network strand between two neighboring cross-links. The former is usually smaller than the latter in a gel since polymer chains are overlapping. 15 In principle, the mesh size of a polymer network can be obtained from the modulus and the swelling of the polymer network. [126][127][128] Albeit very accessible, the analysis of these two methods is based on the choice of specific network elasticity and solution models and knowledge of the interaction parameters between polymer and solvent at the relevant gel concentration. Further, the exact mesh size of a network cannot be easily obtained without major assumption, since the mesh size of a polymer network usually has a wide distribution 123,126 and direct measurement is experimentally challenging, as explored elsewhere in this review (Sections 4.3 and 4.7). Finally, it would be useful for future studies to compare the mesh size obtained by multiple techniques, both to corroborate the experimental methodologies, but also to refine theoretical understanding of characteristic network length scales: correlation length ξ, mesh size ξ mesh , size of an elastically effective strand, distance between cross-links, etc. 129

Electron microscopy (EM) has found widespread use as a characterization tool for soft materials, where it is able to offer precise real-space identification of local structural features. In particular, transmission electron microscopy (TEM) has proven invaluable for characterizing the morphology and interfacial properties of semi-crystalline polymers, [130][131][132] polymer nanocomposites, [133][134][135][136][137][138] and block copolymers including thermoplastic elastomers (Figure 9b), but historically has been less useful for amorphous polymer networks. [139][140][141][142] Although the spatial resolution of electron microscopy techniques is theoretically smaller than length scales of structural/topological features in polymer networks, lack of electron density contrast in a homogeneous material makes it only suitable for polymer networks consisting of large mesh sizes and relatively stiff chains. 123 For example, EM techniques have imaged the supramolecular fibrillar structure of gels made from polysaccharides including gelatin, guar, and pectin (Figure 9a) to visualize structural features of ∼ 10 nm. 143,145,146 Radial integration of the Fourier transform of the obtained EM images can produce simulated scattering curves which can be compared against experimentally obtained data from small-angle X-ray or neutron scattering (Section 4.3). In the case of pectin networks (Figure 9a), the two methods were found to be broadly consistent on scales above 20 nm. 143 Another challenge is that EM images are 2D projections of a 3D sample; this issue can be partially mitigated by use of ultramicrotomy to prepare ultra-thin samples with thickness 100 nm. 147 This issue can also be addressed through the use of electron tomography, in which a series of 2D EM images are taken at different tilt angles to reconstruct the 3D sample. 148 Inhomogeneous materials can be more readily imaged by preparation techniques to enhance contrast between different regions. Staining with heavy-elements (e.g., ruthenium and osmium tetroxide) is one of the most common methods in polymers due to its transferability and ease of use (although its safety hazards must be noted). 149 Other methods, such as freeze-fracture scanning electron microscopy (FF-SEM), fracture the flash-frozen sample along weak portions of the sample (i.e., membranes, interfaces or surfaces) and have been used to image biological samples and poly-electrolyte microgels (Figure 9c). 144,150 Additionally, the absence of appreciable contrast can be useful in some cases, such as confirming the absence of precipitated iron nanoparticles in metallo-cross-linked polymer networks. 151 However, the accuracy of these methods is usually limited by fixation and staining artifacts, which prevents direct imaging of mesh size, heterogeneities, and network defects for many samples. 147 It needs to be stressed that the combination of complex samples, elaborate preparation procedures, and highly specialized techniques can lead to flaws in the final data. It is of utmost importance to distinguish genuine micrographs from artifacts. Finally, it is interesting to note the potential future of single-molecule super-resolution microscopy, which has hitherto been focused in biological research, 152 but could be realized in reversible gels and would add a significant tool to molecularly characterize polymer networks. 153

Birefringence is a property of optically anisotropic materials whose refractive index, n, depends on the polarization and direction of propagating light. 154 Birefringence of a material, ∆n, is typically measured as the difference in refractive index of the material for light polarized in two orthogonal directions. Individual polymer molecules have a polarizability anisotropy due to their three-dimensional structure, but because all of the molecules in a completely amorphous polymer material are randomly oriented, the material itself is optically isotropic. In the case of materials with some anisotropic orientation, such as in crystalline regions of a material or due to chain extension in the direction of strain, birefringence can be observed using an optical microscope containing two polarizers that are 90°to each other on either side of the sample and is closely related to the stress-optic effect to be discussed in Section 5.4. In this way, the birefringence measured through polarized optical microscopy (POM) can provide information about the strain-induced crystallization in vulcanized natural rubber, crystallinity in melt-blown or electro-spun fibers, and molecular strain and orientation during processing. [155][156][157] Further, birefringence is commonly quantified in oriented polymeric systems, especially as a signature of networks under mechanical stress and self-assembled materials such as block copolymers. 156,[158][159][160][161]

A classical technique for characterizing the structural and topological features of polymer networks is scattering, which involves exposure of a sample to incident radiation and measurement of the intensity of scattered radiation as a function of the scattering wavevector q defined as q = 4π λ sin (θ/2)

where, λ is the wavelength of incident radiation and θ is the angle between the incident radiation and the detector measuring the scattered intensity. The same basic principles apply to scattering of different forms of radiation. Sufficient scattering contrast between the background (solvent or matrix) and structures of interest is necessary to identify key features in scattering patterns. Typical forms of radiation used for studying polymers include Xrays where the scattering contrast arises from electron density differences, neutrons where the scattering contrast comes from differences in atomic nuclei, and light where the scattering contrast relies on differences in refractive index (i.e., differences in polarization of electron clouds). Small-angle scattering (SAS), including static or small-angle light scattering (SLS/SALS, 0.0003 < q < 0.002 Å -1 ) small-angle X-ray scattering (SAXS, 0.003 < q < 0.2 Å -1 ), and small-angle neutron scattering (SANS, 0.001 < q < 0.5 Å -1 ), is particularly suited to studying the structure of polymer networks on the order of 1 -1000 nm. SANS is particularly relevant to studying the most widely used types of polymer networks, due to the ability to impart contrast through the use of deuterated solvent in the case of gels or preparation with deuterated strands in the case of elastomers.

Representative SANS patterns obtained for gels typically lack peaks corresponding to strong correlations between strands or junctions due to the lack of long-range order and amorphous character of the network (Figure 10). [162][163][164][165][166][167][168][169] Although, it should be noted that scattering curves from charged gels typically do have a feature called the "polyelectrolyte peak", which corresponds to the correlation length ξ resulting from the like-charge repulsion of neighboring charged strands. [170][171][172][173][174][175][176][177][178] SANS curves are usually fit with physical models to extrapolate structural parameters, similar to other techniques discussed in this review. Broadly, the high-q region characterizes polymer chains and the low-q region highlights larger length-scale network inhomogeneities. Very-(VSANS) or ultra-small-angle neutron scattering (USANS) or SALS can be further used to extend the accessible wavevector (and associated length scales) to lower-q (longer length scales). An empirical correlation-length model describes features from both such highand low-q region:

where the power-law term describes Porod scattering from larger network inhomogeneities and the Lorentzian function describes scattering from smaller, mesh-scale structures. α and β are constants of proportionality indicative of the relative contrast and concentration, I inc is the q-invariant incoherent background, n is the Porod exponent that reflects the degree of large length scale homogeneity (i.e., disorder due to concentration fluctuations present at gelation), m is the Lorentzian exponent reflective of the polymer-solvent interactions, and ξ is the correlation length. This Lorentzian component of the empirical correlation-length model is related to the Ornstein-Zernike scattering function, which characterizes the correlation length ξ of semidilute polymer solutions, and is the same mathematical form of an exact theoretical structure factor derived from the random phase approximation (RPA) for gels in theta solvents. 179 Fitting the scattering to this scattering function, the average mesh size ξ mesh of a polymer network can be obtained (at the meanfield level). 15,180,181 Further, SAXS has been recently used to further analyze networks with is especially useful where network structural features of interest have high X-ray scattering contrast (e.g., metalcontaining junctions or counter-ions condensed on polyelectrolyte strands). The scattered intensity I(q) obtained from SAXS (but generally from any scattering technique) can be expressed as P (q) • S(q), where P (q) is the form factor de- scribing the size and shape of the network subcomponents (i.e., junctions or strands) and S(q) is the structure factor describing the spatial arrangement of the sub-components. For example, in one study on metallo-cross-linked elastomers, the size and distribution of iron crosslink clusters could be identified in this manner. 151,182 This approach has significant potential in the characterization of molecularly optimized networks of defined structure and topology.

Once formed, (i.e., after gelation or vulcanization) a polymer network is characterized by an inhomogeneous monomer density profile, which depends on temperature, solvent quality, overall polymer concentration, and externally imposed deformation fields. This bonded-in inho-mogeneity of the structure of the gel is determined by the density of cross-links in the state of preparation and in turn, dominates the structure factor. 179 Figure 10 shows SANS curves superimposed with static light scattering data at low-q as measured at different preparation conditions and after collapsing the data according to the correlation length ξ and preparation volume fraction ϕ 0 , confirming such expected behavior. 166 Small-angle scattering further provides an opportunity to connect macroscopic mechanical deformation and swelling of polymer networks to the microscopic structure. Ex situ SANS measurements on deformed (either stretching or swelling) have shown the formation of "butterfly" patterns in the 2D SANS curves, as shown in Figure 11. [183][184][185][186][187][188][189][190][191][192][193][194][195][196][197][198] This butterfly effect has been described theoretically as the existence of inhomogeneities in the equilibrium deformed state and related to the inherent anisotropy of the elastic restoring forces in the stretched network. As such, these butterfly patterns highlight the quenched disorder in the network. Further, thermal fluctuations (as observed by dynamic light scattering (DLS, Section 5.5.1 or other scattering spectroscopies) about this anisotropic equilibrium configuration are also anisotropic, but rotated 90 • . 179,197 In a related protocol, the small-angle scattering can be decomposed into contributions from thermal and quenched fluctuations, as the thermal contribution is equivalent to the scattering of a polymer solution at the same concentration and the quenched contribution results from the excess scattering of swollen gels. 184 This is analogous to the procedure discussed in Section 4.1 whereby comparing the modulus of a gel and the osmotic pressure of the non-cross-linked polymers at the same concentration, the osmotic and elastic components to the swelling could be estimated.

Recently, scattering coupled with in situ rheological measurements have become more popular to aid in the development of structure-property relationships. [200][201][202][203][204] Rheo-smallangle-neutron scattering (rheo-SANS) is most common in polymer networks due to the contrast selectivity as mentioned for static SANS, and provides a direct correlation between the structure and the macroscopic rheological response. There are a number of flow devices compatible with in situ neutron scattering measurements (flow-SANS or rheo-SANS), most notably, Couette cells, capillary flow cells, 1 -2 shear cells, and even entire oscillatory rheometers. 200 Capillary 205 and microfluidic flow devices, 206,207 for low sample volume experiments under arbitrary deformation fields, are also available. Such rheo-SANS experiments have given insights into the interplay of chain branching and alignment with the formation/breakage of reversible cross-links under different shear conditions. [200][201][202][203] Further, rheo-SAXS has shown that the glassy styrene domains (within a microphase-separated poly(styrene-b-ethyleneco-butylene-b-styrene) (SEBS) triblock thermoplastic elastomer) change shape, spacing, and ordering while undergoing uniaxial or biaxial stretching. 204

Some specific topological features of polymer networks can be quantified through network disassembly spectrometry (NDS) which precisely analyzes the proportion of vari- ous network degradation products. Initially, Johnson and coworkers were able to quantify the primary loop fraction in end-linked A 2 B 3 PEG gels through analysis of chromophore-labeled network degradation products via liquid-chromatography-massspectrometry (LC-MS). 208 A general schematic for an NDS experiment is shown in Figure 13. The concept of NDS could be extended to measure the primary loop fraction in networks with higher branch functionality (f > 3) or a dispersity in junction functionality. Further development of NDS focused on increasing ease of use and accuracy through the preparation of network precursors with isotopic or methyl group mass labels. 209,210 The quantification of secondary loops in end-linked networks, and primary loops in vulcanized networks has also been achieved with NDS. 211,212 Importantly, the direct quantification of loop defects in polymer networks through NDS has facilitated the development and validation of kinetic models to predict the topology of polymer networks as a function of their composition and synthesis conditions, as well as new theories to predict the impact of defects on polymer network elasticity. 113,[213][214][215]

Solid-state NMR is a useful spectroscopic tool to investigate the chemical structure of the gel, particularly for unique chemical environments contained in cross-links. This specificity and quantitative manner allow one to differentiate between dangling vs. elastically active crosslinks and determine the average number of repeat units between cross-links.

4.5.1 1 H, 13 C NMR Common one-dimensional (typically 1 H or 13 C, but other NMR-active nuclei are possible) NMR measurements in the solid-state can yield useful information regarding the network, beyond its chemical composition. In particular, in addition to the overall extent of reaction, the ratio of tetra-, tri-, bi-, and mono-reacted cross-link junctions, was quantified by NMR in the gelation of polyesters. 45,96 4.5.2 Multiple-Quantum NMR Multiple-quantum NMR (MQ-NMR) has become one of the versatile approaches for probing the structures and dynamics of polymer networks. 216 This technique is able to distinguish between elastically active chains and elastically inactive dangling ends due to differences in their relaxation dynamics. [216][217][218][219][220] Due to the presence of constraints (e.g., cross-links, entanglements), polymer segmental motion (at temperatures above their glass-transition temperature T g ) is anisotropic. The 1 H-1 H dipolar coupling cannot be averaged out by fast molecular motions, leading to a residual proton dipolar coupling. If the time scale of the experiment is long enough, the dipolar orientation autocorrelation function for dangling ends will eventually decay to zero, whereas that for elastically active chains will not. Therefore, the residual anisotropic 1 H-1 H dipolar coupling can be used as a probe to extract quantitative information about the constraints in polymer network. This information can further be used to determine the cross-linking density 221,222 and structural heterogeneity 218,223,224 of the polymer network. Experiments performed on too short of time scales can under-estimate the number of dangling ends, as dangling ends may exhibit slow isotropic motion, however swelling the network can increase chain mobility and lead to a more accurate measurement. Moreover, in the case of polymer networks with extremely homogeneous structures, MQ-NMR has enough resolution to distinguish higher order loop defects and their relative proportions can be distinguished through fitting procedures, as shown in Figure 14. 218 In a similar manner, MQ-NMR can also provide information about network structural heterogeneities and molecular weight between cross-links. 217,219,221,225 While more accessible and versatile than network disassembly spectroscopy for measuring the fraction of elastically effective strands, MQ-NMR is typically unable to quantify higher-order loops and more complicated defects. It would be useful to compare the results obtained from the two different techniques. Copyright © 2011 American Chemical Society.

The measurement of Darcy permeability of polymer gels in a solvent is another strategy for mesh size (characteristic pore size) characterization. 226 This can be achieved by enclosing the gel in a chamber and connect it to a tube or pipe with solvent flowing through, as shown in Figure 15. By measuring the pressure difference on both sides of the gel, the Darcy permeability can be obtained by Darcy's law, 227 which relates the low Reynold's number flow of a fluid through a porous medium, such a polymer gel:

where Q is the volumetric flow rate of fluid with a viscosity η s , induced by a difference in pressure, ∆P , over a porous medium of crosssectional area A, with thickness L, which defines a material property known as the Darcy permeability, k. 228 For flow through polymer gels, often a friction coefficient, f , is defined:

where v is fluid superficial velocity. 229 For Darcian flow, f is the ratio of fluid viscosity to medium permeability, f = η/k. The value of f can be calculated through simple permeation experiments by measuring the rate of fluid flow driven by a hydrostatic pressure through a gel of known thickness, and an average value of mesh size can be estimated. 226

Velocity 𝒗 The swelling of a gel in situ or compression of a gel under applied pressures may interfere with measurements and should be considered carefully when interpreting experimental data. However, physical confinement of gels can enable characterization of gels swollen to below their equilibrium-swollen state or under large applied pressures. [229][230][231] The picture that fluid permeates a gel by passing through microscopic pores with a width proportional to the correlation length, ξ mesh ξ, predicts f η/ξ 2 . 114 Recent SANS measurements of ξ and subsequent permeation experiments by Sakai et al. have confirmed this scaling argument for tetra-PEG hydrogels with homogeneous network structures. 229 Similar scaling arguments predict f ∼ φ 3/2 for equilibrium swollen gels in good solvents, confirming the expected concentration (volume fraction, φ) dependence of correlation length ξ ∼ φ -3/4 . 114,232 This scaling has been commonly observed in multiple techniques including SANS (Section 4.3), pulsedfield-gradient NMR (PFG-NMR, Section 5.7.2), and Darcy solvent permeability experiments using tetra-functional poly(ethylene oxide) networks cross-linked in various manners, 231,[233][234][235] although high polymer volume fraction, degree of cross-linking or gel compression can lead to deviations. 229,230 Such comparisons can be used to compare the numerical coefficients of the scaling relation and help differentiate between closely related metrics like the correlation length, hydrodynamic screening length, and mesh size.

Another relatively direct method for characterizing the mesh size and structure of the network is called probe penetration (or probe diffusion). 123,236,237 The average mesh size of a polymer network can be obtained by using tracerlabeled particles, polymers, or small molecules with different sizes as probes and measuring their permeability from dilute solution into a polymer network. The sizes of the probes should be comparable to the average mesh size of the network (typically ranges from several nanometers to tens of nanometers.) 237,238 At the equilibrium permeation, an average mesh size of the network can be obtained by plotting partition coefficients of different probes against their sizes. 123 The partition coefficient of the probe is defined as the ratio of the probe concentrations between the gel and the dilute solution outside. Note that the probes and networks used in this method should not have significant inter-molecular interactions (such as hydrogen bonding or ionic bonding). Other than the par-tition coefficient, similarly, the relative diffusivity of the probe is also frequently used for mesh size characterization, but the data analysis depends on specific models that are used. 238 Furthermore, in principle, this method can be operated together with experiments of swelling in media that exert osmotic pressure on the gel (i.e., large solutes). By placing networks into solutions with both labeled probes (dilute), which can diffuse into the network, and large solutes, which are excluded by the network, the average mesh size can be measured while measuring Π ext .

Finally, the diffusive behavior of probe particles or molecules within a polymer network can be used to characterize network structure on length scales determined by the probe size, R, and its mean-square displacement (MSD) on the time scale of the experiment. Within polymer gels, measuring the diffusion of smallmolecule or polymeric probes can provide information relating to hydrodynamic interactions and obstruction effects between probe and polymer and reduction in free volume within the gel. Probe diffusion is commonly monitored using a variety of techniques including pulsed-field-gradient NMR (PFG-NMR, Section 5.7.2), diffusion-ordered NMR (DOSY, Section 5.7.3), or forced-Rayleigh scattering (FRS, Section 5.9) as discussed elsewhere in this review. NMR-based techniques have been used to measure the diffusion of small-molecule or polymeric probes on millisecond time scales, where network heterogeneity causes the observation of multiple diffusion coefficients, typically corresponding to diffusion in high-density and low-density regions of the gel. 239,240 The length-scale of structural heterogeneities can be estimated then by identifying the experimental time scale, and corresponding probe MSD, at which homogeneous diffusion behavior is observed. Finally, particle tracking experiments of probe diffusion can be used to analyze the viscoelastic properties of a network, 241,242 which is discussed in more detail in Sections 5.3, 5.5, and 5.6.

Polymers, including gels and elastomers, are viscoelastic, exhibiting both viscous and elastic mechanical response upon deformation, which varies with the time scale and frequency of the deformation. To characterize this viscoelasticity with respect to tensile, compressive, or shear deformation, several types of experimental measurements are commonly applied, such as stress relaxation measurements, creep experiments, and oscillatory rheology.

Here, we will largely focus on linear viscoelasticity, a regime of linear response at sufficiently small values of applied strain, where the relaxation modulus is independent of strain (Figure 16). In the linear viscoelasticity regime, the elastic and viscous components of material response to deformation are independent and Boltzmann superposition states that the stress (strain) responses to successive strain (stress) stimuli are additive. All linear rheological methods (e.g., step-strain, creep, oscillatory) are equivalent and produce identical information from linear transformations. 15,36 Further, all materials have a region of linear response at sufficiently small values of applied strain; to confirm linearity one needs to check that the result is independent of the amplitude of oscillations (i.e., the measured modulus is strain-independent). Elastomers and gels have larger strain limits (typically γ 5 %) to the linear viscoelastic regime than brittle glasses or tough plastics (typically γ 1 %). Taking a step back, simple toy models of viscoelasticity can be constructed using a spring of modulus G 0 obeying Hooke's law to describe the elastic component and a dashpot of viscosity η obeying Newton's law to describe the viscous component of the deformation behavior (Figure 17). 15 The Maxwell model places the spring and dashpot in series, while the Kelvin-Voigt model places the spring and dashpot in parallel. 36 However, this only describes a single relaxation mode; complex material behavior, especially over many time scales, can

Figure 16: Schematic illustration of a strain amplitude sweep at a fixed frequency in oscillatory rheology. This experimental test is used to verify future rheological measurements are done within the linear viscoelastic regime, at strain amplitudes smaller than the strain limit.

be described by combining many such spring and dashpot elements in Maxwell ladders or Kelvin chains. It is important to emphasize that polymers are composed of many segments, contain many of such relaxation modes, and therefore display a spectrum of relaxation times τ . 15 Finally, it should be noted that the nonlinear mechanics regime is also of high interest to the study of polymer networks, particularly reversible networks, and can be probed using the same techniques discussed below, albeit at higher strains or higher strain rates γ (for reversible networks).

Tensile (or compressive) tests, in which uniaxial force is applied to elongate (or compress) mate-rials, are commonly used to determine the mechanical properties of polymers and polymeric networks by recording the stress σ as a function of imposed strain γ. Several intrinsic material properties can be obtained from the resulting stress-strain curve (Figure 18a), including: (i) the Young's modulus, E, which is the slope at small strain; 257 (ii) the tensile strength, which is the stress at failure; 258 and, (iii) the toughness, which is the area under the curve from initial state to failure. 259 Similarly, shear tests, in which force is applied to shear materials, provide the shear modulus G, 260 which is related to Young's modulus through Poisson's ratio as

where σ is the shear stress, and γ is the shear strain, and the Poisson's ratio µ is given by

where γ xx and γ yy are strains in the x and y directions. Poisson's ratios of elastomers and gels at short times (without a change in volume) are typically approximately 1/2, while Poisson's ratios for swollen gels is smaller (around 1/3) and depends on solvent quality. Two simple and common models are widely used to extract molecular details of polymer networks from the elastic behavior of unentangled gels and elastomers. One is the affine network model, so called because it relies on the primary assumption of affine deformation: specifically, that both ends (cross-link junctions) of each elastically active network strand displace identically to the macroscopic deformation of the whole network. 15 Under this assumption, molecular details such as the number density of elastically active strands 261 (ν eff ) and the number-average molecular weight of a network strand (M s ) can be related to the measured shear modulus as

where ρ is the network mass density and R is the gas constant. Since the moduli are usually obtained from tensile tests (which provide the Young's modulus), the molecular properties are obtained by first converting the Young's modulus to the shear modulus when the Poisson's ratio is known.

The other common model is the phantom network model. Unlike the affine model, which assumes the ends of each network strands are directly pinned to the elastic non-fluctuating background, the phantom network assumes the network strands are joined at fluctuating crosslinks. The phantom network model gives the shear modulus:

where f is the functionality, i.e., the number of polymer arms that are connected to a crosslinking junction. In comparison to the affine network model, the phantom network model provides a lower modulus at the same number density of elastically active strands. This difference is due to the inclusion of cross-link fluctuation, such that the distance between average positions of junctions varies by smaller factor than in affine model (i.e., changes nonaffinely). While the functionality f should be fixed and known a priori according to the chemistry of the network, polymer networks have defects including topological loops and dangling chains, which are elastically ineffective and do not contribute to the elasticity of the network.

When the fraction of these defects is low, as in networks prepared in the melt far above gel point, and the polymer strands are unentangled, the measured modulus is well predicted by the phantom model but over-predicts the modulus for other preparation conditions. 113 More recently, the real elastic network theory (RENT), was established by Olsen and Johnson to expand beyond the tree structure of the phantom network model by including the effects of loops and other topological structures on the bulk elasticity. 113 The initial RENT model treated loops of differing orders as independent without any interactions or correlations. Subsequent refinements have considered other details, such as dangling ends, pre- strain within the strands, and excluded volume interactions. 214,[262][263][264][265][266] RENT and associated models significantly improve the connection between molecular-level and bulk properties and correspondingly improve the agreement with experimental data (Figure 18b). However, more advances in characterization of loops, dangling ends, and the tree structure of polymer networks are needed to quantify these defects for regular networks and develop correlations such that RENT could be applied for an arbitrary unentangled network obtained by random cross-linking or end-linking without measurement of these defects by NDS (Section 4.4) on each sample.

Further, in order to capture more features of stress-strain curves, the curve can be fit with the phenomenological Mooney-Rivlin model, 15,267,268

(5.5) where f * is the Mooney-Rivlin function, λ is the stretch, σ true is the true stress, and σ eng = σ true /λ is the engineering stress, to obtain the Mooney-Rivlin coefficients C 1 and C 2 . For classical models (e.g., affine or phantom networks), the Mooney-Rivlin coefficients are 2C 1 = G and C 2 = 0. Many stress-strain curves, however, have C 2 > 0, and they cannot be accommodated by classical models. A value of C 2 > 0 corresponds to a stress-strain curve that exhibits strain softening behavior, which is attributed to the contributions of non-affine entanglement tubes in polymer networks. 269 In order to extract a greater level of molecular detail from stress-strain curves with non-classical behaviors (e.g., strain softening), the curves can be fit by using the Rubinstein-Panyukov relation 15,112,270

to obtain the contributions from entanglements and cross-links to the modulus, G e and G x , from which the apparent molar mass between crosslinks M x and the entanglement molar mass M e can be calculated. 271 The universal uniaxial deformation curve is shown in Figure 19, which shows the relationship between Mooney-Rivlin function f * and phantom modulus G x and entanglement modulus G e with respect to the stretch λ.

Stress relaxation experiments impose an instantaneous step-strain of magnitude γ at time t = 0 and measure the stress remaining at time t. The ratio of this stress at time t to the constant strain γ is known as the stress relaxation modulus, G(t), 15,36

Viscoelastic liquids, such as polymer melts or solutions and reversible networks, can completely relax their stress. For viscoelastic solids, such as covalent networks, however, G(t) relaxes to a finite value, the equilibrium shear modulus:

In broad terms, G eq is determined by the permanent molecular scaffolding within a network (e.g., covalent cross-links), whereas the timedependent portion of the moduli (G(0) -G eq ) is determined by the short-time-scale relaxation of network strands between cross-links, relaxation of defects (dangling strands or loops), and dynamic molecular scaffolding, such as reversible cross-linking or the slippage of physically entangled polymer strands. Thus, stress relaxation testing can be useful in investigating the molecular-level processes of relaxation under load. 275 Since the sample is kept under a constant deformation due to the imposed step strain, molecularlevel movement rather than macroscopic change dominates stress relaxation processes. Information about entanglement, chain alignment, cross-linker exchange, and, under more extreme conditions, chain scission under strain can be gleaned from stress relaxation experiments. 276,277 Additionally, stress relaxation experiments are industrially important in determining whether stress can be dissipated over accessible time scales in materials for applications that require repetitive strain or in molding and curing processes of polymers and composites. 278 This helps determine the best processing conditions and the working lifetime of a material.

Stress relaxation experiments are particularly useful in characterizing reversible networks, both ordinary reversible networks with a dissociative exchange mechanism and vitrimers, also known as covalent adaptable networks, which are a relatively new class of polymeric networks with an associative exchange mechanism of cross-linkers. 276,279 There is much discussion in the literature about the differences between these two classes of reversible networks and the impact of the cross-link exchange mechanism on the resulting mechanical properties.

Figure 20 showcases a typical stress relaxation curve of a reversible network, an ionomer with ionic clustering serving as physical transient cross-links. 280 Further, plotting observed relaxation time versus temperature can be used to calculate the activation energy of cross-link exchange in reversible networks. 281 Stress relaxation experiments are also useful in determining the topology freezing transition temperature T V in vitrimers, 279 revealing differing network structures, 282 and quantifying the effects of exchange catalyst concentration, 283 chemical structure, and (in polymeric ionic liquids) counter-ion identity.

Stress relaxation at a fixed strain and creep at a fixed stress are simple mechanical tests. Creep experiments impose a constant step-stress of magnitude σ at time t = 0 and measure the strain as a function of time t. 15 Such experiments enable the measurement of the creep compliance J(t),

which is the ratio of the time-dependent strain γ(t) to the constant stress σ. 36 Over a time longer than the relaxation time, the strain γ and creep compliance J(t) of viscoelastic materials will increase until they reach a steady state, which for solid materials becomes time-independent at an equilibrium strain γ = σJ eq = σ/G eq and for liquid materi-als becomes linearly dependent on time, ∼ t/η. J eq and G eq = 1/J eq are the equilibrium creep compliance and modulus and η is the viscosity. Upon reaching this steady state in an experiment, the stress can be removed completely (σ = 0), and the material's creep recovery can then be monitored. This part of an experiment in the linear regime gives rise to the recoverable compliance J R (t), which is the ratio of the time-dependent recovery strain γ R (t) and the stress σ. At the moment that stress is removed, γ R (t) is defined to be zero. For samples in which the steady-state creep compliance is time-dependent, only the elastic part of the compliance will contribute to the recoverable compliance, 15 Creep testing at very low stress values (within the linear viscoelastic limit) can be used to determine the zero-shear viscosity, 285 compare relative length or branching of polymer chains,, [286][287][288] and infer other important molecular information about a polymeric material, just as any other linear rheological function (G(t), etc.). Creep testing, however, is typically most relevant and easiest to utilize for measuring the long-time relaxation of viscoelastic liquids. For example, experiments under long time scale (or low frequency conditions) are important for characterizing samples with very long relaxation times, such as hydrogels with reversible protein-based cross-links. 289 These experiments can also aid in and expedite creation of rheological master curves (to be discussed below) by extending the low-frequency data from oscillatory rheology (Section 5.1.4) to include long-time data from creep. 290 In fact, creep experiments on the order of months have been used to measure rheological behavior near the gel point (where the relaxation time diverges, Section 3.2). 291,292 Both stress relaxation and creep testing can be performed using DMA, tensile testing, and rotational rheometers; specific, commercial creep-testing instruments are also available. A comparison of stress relaxation, creep, and os-cillatory rheology is shown schematically in Figure 21. Nanoindentation and AFM experiments (Section 5.1.5) to study stress relaxation and creep are also sometimes designed for particularly small samples or to study nanoscale spatial heterogeneity in material properties. 293

Dynamic mechanical analysis (DMA) and oscillatory shear rheology are two of the most common and powerful techniques for analyzing the mechanical behavior of viscoelastic materials. [294][295][296][297][298][299] DMA and oscillatory shear rheology rely on the same fundamental concepts, but there are notable differences in suitability (Figure 22a). In typical DMA measurements, a thin sample (typically with thickness (0.02 -1) mm, width (2 -5) mm, and length (5 -10) mm) is placed between two clamps, and the mechanical response of the sample under oscillatory stretching is analyzed. Therefore, this technique is suitable for solid samples, such as glassy polymers, rubbers, and stiff gels. Oscillatory shear rheology, however, accommodates both solid and liquid samples, which are typically placed either between two flat plates or between a cone and a flat plate, and the mechanical response under oscillatory shear is analyzed. Although this method can be used for samples in a glassy state, glassy samples potentially cause slipping between the sample and the plates. Thus, this method is more suitable for soft materials, such as rubbers, gels, and liquids.

During experiments, an oscillatory stress, typically tensile for DMA and shear for oscillatory shear rheology, is imposed, and the strain in the material is measured, allowing the calculation of the dynamic moduli (E * for DMA or G * for oscillatory shear rheology) at various strain amplitudes, frequencies, and temperatures. Strain-controlled instruments with applied strain and measured resulting stress are less common but also available. Cone and plate geometries in oscillatory shear rheology are most common and useful for viscoelastic samples due to the uniform shear throughout the sample. Parallel plates can also be use-ful, particularly for more elastic samples, if the sample thickness needs to be uniform or if high shear rates are needed, although a correction factor needs to be applied to convert the measured force into true rather than engineering stress. DMA has a similar problem, and samples need to be uniform and aligned correctly for accurate measurements. The respective advantages and challenges of different measuring geometries in rheometry are discussed in polymer rheology texts. 301,302 The dynamic or complex modulus G * , related to the complex viscosity η * , can be separated into storage (G ) and loss (G ) components: 15

The loss tangent is defined as

where δ is the phase shift between stress and strain responses. These storage and loss moduli characterize how much of the energy that goes into deforming the sample at a given frequency is stored as elastic energy or dissipated as heat. These moduli, as a function of frequency, are the key outputs of DMA and oscillatory shear rheology, and it is through an analysis of them that structural (e.g., molecular weight between cross-links) and dynamic (e.g., Rouse or sticker times) molecular properties can be inferred. Again, recall that the the linear viscoelastic functions (G(t), J(t), G * (ω)) are related and contain the same microstructural information; the choice of rheological method depends on convenience for the property and material of interest. Noteworthy examples of extracting molecular information in this way include the lifetime of "sticky" associations in linear ionomers from the frequency dependence 303 and the use of measured plateau modulus to characterize molecular weight between entanglements (M e ) in linear entangled polymers. 304 A common use of these techniques is a temperature sweep at constant frequency; a representative temperature-modulus curve at a single frequency is presented in Figure 22b molecular weights. The resulting curves are frequency-dependent and shift with frequency, which is the basis of time-superposition and the creation of master curves, described below. At temperatures lower than the glass transition temperature (T g ), all polymer types show similar glassy behavior: constant high modulus on the order of GPa. As the temperature increases towards T g , the moduli drop significantly as the glass transition is approached. For temperatures T > T g , the chemically cross-linked polymers (i.e., conventional covalent networks) maintain nearly constant moduli that indicate a rubbery plateau regime on the order of MPa. All non-cross-linked polymers will eventually show a decrease in modulus that is related to their so-called terminal relaxation. For linear entangled polymers, however, this terminal relaxation at high temperatures will be preceded by a rubbery plateau modulus on the order of 100 kPa due to entanglements acting as physical cross-links. Furthermore, DMA and oscillatory shear rheology can be used to characterize the relaxation behaviors of polymer materials throughout a wide range of frequencies by employing time-temperature superposition (TTS) (i.e., a master curve, Figure 22c). 15,305,306 The key concept behind TTS is that most dynamic processes have a temperature dependence; the same relaxation occurs more quickly at higher temperatures. Thus, linear rheological response data collected as frequency sweeps at a range of temperatures can be shifted to predict behavior corresponding to immeasurable frequencies. Often, the temperature dependence can be characterized as a single, Williams-Landel-Ferry (WLF) or Arrheniuslike activated process that is quantified through a so-called shift factor, a T , and a much smaller modulus scale shift factor, b T :

where ω is the angular frequency and T 0 is the reference temperature, which can be chosen to be any convenient temperature. However, TTS relies on several assumptions for validity and can fail if these assumptions are not met. The structure of the material must be homogeneous, amorphous, isotropic and not change over the temperatures used. Further, the material must be thermorheologically simple, that is, the contributing relaxation mechanisms must have the same temperature dependence. The former can be confirmed by techniques in Section 4; the latter can be detected by generating a Cole-Cole (G versus G ) or wicket plot (tan (δ) versus G ) and identifying the characteristic semi-circle curve shape. Finally, the shift factor versus temperature can be analyzed to fit an Arrhenius or WLF-like process. TTS often fails in reversible networks when the gelation process affects the phase, structure, or extent of reaction.

As shown in the master curve of Figure 22c, at high frequency, there are sharp increases in modulus due to the glass transition. A rubbery plateau regime exists at intermediate frequencies for entangled or cross-linked samples. In entangled polymers this occurs at frequencies corresponding to times intermediate to the entanglement (τ e ) and reptation times (τ rep ), providing a plateau modulus (G e ) and thus an estimate of the molecular weight between entanglements (M e ) and the chain density of effective strands. For cross-linked polymers without entanglements, the plateau region provides a plateau modulus G x and an estimate of a molecular weight between cross-links M x . However, due to the statistical nature of crosslinking, there is a large distribution of distances between cross-linking junctions and many defects such as loops and dangling ends, so this is only an estimate and an ensemble average. For cross-linked polymers with entanglements, the plateau region provides a plateau modulus that contains contributions from both entanglements and chemical cross-links. Entanglements can become trapped when occurring on a strand between a pair of cross-links, but their relative contribution to the modulus is diminished as the molecular weight between crosslinks becomes much smaller than the molecular weight between entanglements or for networks near the gel point with many dangling ends that do not trap entanglements. [307][308][309][310][311][312][313] Transient networks with physical cross-links can show a single plateau modulus that has contributions from entanglements and cross-links (as for covalent networks with trapped entanglements), or two plateaus if the reptation time is longer than the typical lifetime of the physical cross-links. At the lowest frequencies (i.e., at times longer than the longest relaxation time) is the terminal flow regime, in which both storage and loss moduli depend on frequency: G Ψ 1,0 ω 2 and G ηω. Thus, in the terminal regime, the loss modulus is proportional to the viscosity and the elastic modulus related to the zero-shear first normal-stress coefficient, Ψ 1,0 . 15,36 However, local microphase separation can alter the expected terminal frequency scaling of the moduli if the sample is in an ordered state and can provide insight into samples' mesophase structure. 244,314,315 Further, for microphase-separated polymer networks, the applied stress can affect and orient the microphase. [316][317][318][319][320] Looking specifically at the loss modulus G in the plateau region of cross-linked networks, the Curro-Pincus method can be used to understand the contributions of dangling ends, which are elastically ineffective but contribute to the viscous dissipation. The amplitude of G corresponds to the total number density of defects in the network, while the power law of G ∼ ω α where α ≈ M x /M -1 indicates the average molar mass of the defects (M ) relative to an elastically active strand (M x ). [321][322][323] Moreover, oscillatory shear rheology is particularly useful for analysis of network formation (i.e., gelation). Typical covalent networks are composed of non-reversible covalent bonds formed by chemical reactions. Reversible bonds induced by associative or "sticky" interactions, such as hydrogen bonding or ionic associations, lead to the formation of transient networks. The network formation kinetics for both networks can be studied by monitoring moduli and tan (δ) during the gelation (e.g., G , G , and tan (δ) versus time), which helps to associate the developments of the network structures with their mechanical responses. [324][325][326][327][328] In principle, these can be done as kinetic or timedependent measurements, but only the relaxation modes faster than the gelation itself can be measured and thus are unable to access the slowest relaxation modes related to large clusters near the gel point. This limits such kinetic experiments to a certain distance from the gel point. The correct manner to measure these slow modes is to quench the system (i.e., stop the cross-linking reaction) at various points and then measure the full range of relaxation times for each sample. 291,292 As such, the gel point or gelation time can be determined by oscillatory shear measurements. 294,[326][327][328] Although the criterion G = G is commonly cited, this does not rigorously correspond to the gel point. The proper identification of the gel point is when G ∼ G ∼ ω u and tan (δ) = tan (uπ/2) is frequency-independent, where u is the dynamic scaling exponent with a value of 0.5 -0.8 for unentangled gels. 291,[326][327][328][329] For entangled gels, the relaxation exponent u is lower (u ≈ 0.3), as predicted by theory and measured by experiments. [307][308][309] This criterion to determine the gelation time requires multiple measurements. Several frequency sweeps of the samples can be conducted at different curing times, and then the gel point is found when G ∼ G ∼ ω u . Alternatively, aging experiments (i.e., tan (δ) versus time t) can be conducted at several frequencies, followed by determining the gelation time as the cross point of all curves, which represent a frequencyindependent tan (δ) = tan (uπ/2).

Oscillatory shear rheology is also sensitive to and thus useful to studies of reversible gelation in associating polymers. Careful rheological studies have shown that ionomers exhibit a sol-gel transition as the average number of ionic groups (i.e., functionality for potential cross-linking) per chain is increased. Below, but close to the gel point, there is a powerlaw relaxation of the complex modulus similar to that of chemical cross-linking. 291,313 Very close to the gel point, the power-law relaxation of the complex modulus shifts from high frequency to low frequency, suggesting a transition from mean-field to critical percolation, as discussed in Section 3. However, even further above the gel point, terminal relaxation is observed in ionomers (and associating polymers or reversible networks more generally) as the dissociation of transient, physical cross-links occurs. 330 The rheological and dynamic properties of these associating polymers and reversible networks agrees well with previous theoretical predictions for gelation with effective breakup of clusters. [331][332][333] Finally, it is important to highlight that away from the gel point, these "sticky" or associative interactions form the basis of the sticky Rouse and sticky reptation models. 280,332,[334][335][336][337][338]

Nanoindentation provides a method to probe mechanical properties of materials on submicron length scales that are small enough to resolve spatial heterogeneities, providing nanoscale equivalents of bulk materials test-ing. Originally developed as an extension of depth-sensing indentation techniques that measure the hardness of materials, 339 nanoindentation has been adapted to small-scale metal, ceramic, and polymer samples. During an experiment, an indenter with a tip of known geometry is vertically pressed into a sample starting from as low of a contact force as possible (typically ∼ 5µN) and subsequently retracted. A load-displacement curve is recorded during both loading and unloading within a load range of 50-500 mN, from which the material properties of the sample can be calculated, most commonly hardness and Young's modulus. 340 Commercial nanoindentation instruments have indenters composed of diamond with tips of radius 50 -100 nm of precise geometry that move only in the z-direction. Berkovich indenters, 341 three-sided pyramid shapes with a face angle of 65.27°, are most common, but spherical, conical, and cube corner indenters are also used. 340 Determination of precise tip geometry and thus the projected contact area with the surface (which can be measured by microscopy) is an important consideration for any nanoindentation experiment. 342 In order to bring the spatial resolution of the measurements closer to truly molecular length scales, atomic force microscopes (AFMs) can be used for nanoindentation in lieu of commercial nanoindenters. The mechanical measurements by AFM typically have lower forceresolution relative to designated nanoindenters, however, due to the large compliance of the cantilever, possible tip movement in the x-y plane, and lack of well-defined geometry of conventional silicon tips. 339 Nonetheless, in addition to higher spatial resolution, AFMs provide access to lower force ranges (sub-pN), which are advantageous for the characterization of very soft polymers and gels, and it is possible to precisely characterize the real shape of the indenter-tip contact area through imaging. As a result, there is ongoing development of AFM for use in nanoindentation. 342,343 Nanoindentation characterization of polymeric materials requires the consideration of several major hurdles: (a) pile-up, viscoelasticity, and (c) adhesion. 340,344 When the inden-ter contact induces plastic deformation (as is more common in brittle networks, glasses, and semi-crystalline polymers), the sample material can either pile up or sink in at the edges of the contact area, changing the contact depth h c in relation to the total indentation depth h max , where with material pile-up, h c /h max > 1, and with material sink-in, h c /h max < 1. As such, the calculated contact area of the indenter is not accurate 340 resulting in an overestimation or underestimation of the material stiffness by up to 60%. 345 AFM or SEM can be used to observe the actual shape of the indenter contact area for fully plastic samples or more generally, numerical finite-element simulation can account for pile-up effects as well as sample adherence to the tip. 340,344,345 Spherical indenters, in particular, are occasionally used to reduce pile-up and for samples that require a small penetration depth; 346 corrections to the standard analysis procedure 341 for use with a spherical indenter are readily available. 347 Viscoelasticity further complicates nanoindentation data collection and analysis, 344,348 since the measured hardness and modulus are loading-rate dependent. 293,349 Various models have been developed to analyze the load-displacement data under both loadcontrolled and displacement-controlled conditions to account for the viscoelastic behavior. 344,350,351 For example, one recent study uses a range of loading rates to obtain the creep compliance and relaxation modulus of a sample. 352 Alternatively, oscillation of the indenter shaft can be superposed at a relatively high frequency (so-called Dynamic Indentation Testing or Continuous Stiffness Measurement), which allows the stiffness of the material to be characterized as a function of time and indentation depth. 293,353 Even with this advancement in instrumentation, obtaining reliable nanoindentation data from polymeric samples and interpreting such data remains difficult, particularly with very soft samples and samples that adhere to the indenter tip. 293,351,353 One must also be aware of the effects of sample microstructure, especially in regard to structures with dimensions on the order of the indenter tip. 293 Additionally, a related challenge is that the poroe-lastic time (characteristic relaxation time due to fluid flow through the swollen deformable medium) is reduced with the decrease in length scale from the macro-to nanoscale and can become comparable to the viscoelastic time scales of the sample for polymeric gels, contributing significantly to the observed modulus and hardness. [354][355][356] Adhesion between the sample and the tip can also complicate the analysis of nanoindentation data, but forms the basis of adhesion testing such as the Johnson-Kendall-Roberts (JKR) method, 357 which has been widely utilized for measuring the adhesive properties of an elastomer with various surfaces. It is particularly powerful for determining the modulus and adhesion energy of low-modulus solids. This method is qualitatively similar to the fracture experiments discussed below. Two surfaces (both elastomers or an elastomer with another substrate) are brought together and the force needed to pull them apart is measured. [358][359][360][361][362] Similar methods are available for characterization of the friction and lubrication between elastomeric surfaces under shear. 363

A number of molecular structural and dynamic details are correlated with the fracture energy of polymer networks. The fracture, or tearing, energy Γ is a measure of the intrinsic resistance to crack propagation in a material. This energy is equal to the strain energy release rate, G, that drives crack propagation. Despite its designation as a release "rate," G characterizes the potential energy U σ that is lost per area A, on one face of the crack:

where U w is the external work input and U el is the stored elastic energy. Griffith first proposed that crack propagation could be related to the newly created surface energy per unit area. 365 While this simple theory successfully explained the fracture of some materials, 365 it significantly underestimated the threshold for crack propagation in rubbery polymer networks. 366 Rivlin and Thomas corrected for this deficiency, proposing that the free energy per unit area for crack propagation should be the characteristic energy of tearing instead of the surface energy. 367 Evidenced by experiments, the fracture energy Γ has a critical value Γ 0 , below which the crack cannot propagate. Lake and Thomas incorporated molecular level details of polymer networks in the absence of structural heterogeneities, entanglements, or other dissipative processes. 368 The Lake-Thomas theory predicts that the critical fracture energy Γ 0 is related to the number of bridging strands per unit cross-sectional area β and the energy Ûbreak of a single bridging-strand breaking event,

where R 0 is the effective end-to-end displacement of an average elastically active network sub-chain in its undeformed state, ν eff is the number density of elastically active sub-chains, n is the number of repeat units in the average elastically active bridging strand, and U is the energy stored in each repeat unit at the time of rupture. 368 The Lake-Thomas theory has been used successfully to interpret experimental measures of Γ 0 , [369][370][371] but it is based on a set of limiting assumptions and the equations only capture a very simple ideal situation. Interest in tearresistant networks has sparked a current interest in further modifications for quantitative and predictive use in real polymer networks. For example, the energy U has historically been widely treated as the bond dissociation energy, but recent principles of covalent mechanochemistry have been applied to show that this approximation leads to a significant overestimate of U . 92 In addition, the original Lake-Thomas theory assumes that all of the energy relevant to Ûbreak is stored in the strands that bridge the propagating crack face; sub-chains that are connected to the bridging strands are not included. 368 Considering the tension is transmitted through cross-links in polymer networks, when a bridging strand is under high tension and is about to break, other sub-chains that are connected to it should also be under tension (albeit less than that of the bridging strand). The elastic energy stored in these strands is also released once the bridging strand is broken and future theories need to account for those contributions to Ûbreak .

Fracture measurements using a multitude of geometries and conditions have been developed, although rubbery polymers are usually analyzed in pre-notched thin films. 367,369,372 In the trousers test 367 (Figure 24a), the sample is cut into a trousers-like shape with cut length c larger than half the height h 0 of the undeformed state. One leg of the cut sample is clamped stationary, while the other leg is pulled away at a constant velocity v. The applied tearing force f during propagation is recorded, yielding the fracture energy: 367,372

where W (λ) is the stored elastic energy per unit volume of material when the shaded arms in Figure 24a are at stretch λ, and t 0 is the thickness of the sample in its undeformed state. The observed fracture energy Γ(v) depends on the pulling velocity v; slower extension results in lower value of Γ(v). The rate dependence can be extrapolated to give a threshold value Γ 0 -the critical fracture energy, which is the minimum energy per unit area required for crack propagation to occur, and which corresponds to the critical energy release rate G c . In the pure shear test 367 (Figure 24b), the sample is elongated in one direction (λ x ≡ λ > 1), shortened in a perpendicular direction (λ y = 1/λ < 1), and clamped in its third direction. This test requires two samples of identical dimensions with small thickness t 0 and width w 0 sufficiently (> 2×) larger than height h 0 . One specimen is un-notched and is pulled to obtain the stress-stretch curve and the stored elastic energy per unit volume W (λ) at stretch λ; the other specimen is pre-notched and is pulled to measure the critical stretch λ c at which the crack starts to propagate. The fracture energy is 367 Γ = W (λ c ) • h 0 (5.17)

where W (λ c ) is the elastic energy per volume at critical strain λ c (crack starts to propagate). The pure shear geometry is advantageous in steady-state fracture measurements for providing an energy release rate G that is independent of the cut length c when c is sufficiently (> 2×) larger than its half-height h 0 /2. Fatigue fracture measurements are also benefited from the pure shear geometry, 366,369 where a series of pre-notched specimens (i = 1, 2, 3...) are cyclically loaded to different stretch amplitudes λ i < λ c (Figure 24c). For each specimen i, the corresponding W (λ i ) and energy release rates G i = W (λ i ) • h 0 are calculated and the crack growth dc and total number of cycles n are recorded to obtain the crack growth per cycle (dc/dn) i (Figure 24d). The crack does not propagate (i.e., the crack growth rate is zero) at small G but begins to propagate above the critical fracture energy Γ 0 (Figure 24e).

The flow of polymeric or polymer-containing fluids is central to behaviors that range from industrial to biomedical. Particle image velocimetry (PIV) and particle tracking velocimetry (PTV) are optical techniques for imaging such flow fields of simple and complex fluids. [373][374][375] In these techniques, tracer particles are introduced into the sample; by following and analyzing the motion of a representative sample of particles, the flow fields can be visualized. The size, density, refractive index, and (existence or non-existence) of fluorescence of the particles must be carefully chosen to follow, but not alter, the flow and to allow sufficient imaging quality. Particles are illuminated by a thin laser sheet, and images are captured and processed.

PIV uses an Eulerian method, determining displacement and velocity by cross-correlation analysis of small subsections between each image frame. 374,376,377 PTV uses a Lagrangian method, tracking the trajectories of independent particles. 374,376,377 Regardless of the image processing method, strain and vorticity can be locally determined from the flow fields. More recently these techniques have been advanced to visualize micron-scale (µ-PIV) 376,[378][379][380] and three-dimensional (3D PIV/PTV) 381,382 flow fields.

These techniques are suitable for inhomogeneous polymer flows, 375,[383][384][385][386][387][388] and therefore useful for polymer processing applications, where polymer flows play important roles. PIV and PTV experiments have enabled investigation of the non-linear rheological behavior of entangled polymers 375,389,390 and transient networks, 391 highlighting strain localization and deformation confined to a shear zone with thickness on the order of the mesh size of the gel. For example, particle velocimetry has been used for analysis of inhomogeneous flows of associative networks, such as peptide-or proteinbased hydrogels for drug delivery: the quiescent hydrogel containing therapeutic agents is initially solid-like, flows due to shear-thinning upon injection into the body, and then solidifies again inside the target site. PIV/PTV visualized the flow profile under such a large deformation, indicating a fractured layer of the hydrogel around the interface with the shearing wall (Figure 25). 387,388 Particle velocimetry can also be used in conjunction with rheometry to verify assumptions of the deformation, such as the "no-slip" boundary condition of the complex fluid against the shearing wall. By viewing the motion of particles at or near the boundary, the existence of slip at the wall (or within a fluid layer on the order of the tracer particle's diameter) can be inferred. 301,392 Finally, information on microscopic structures and molecules inside a flow can be obtained with the aid of optical techniques (e.g., birefringence, X-ray and neutron scattering) in conjunction with PIV and PTV. 375,393,394 5.4 Mechano-Optics

The stress-optic effect, generally, is the optical activity occurring in response to stress or flow and can be measured with polarimetry to extract clues regarding the molecular origins of unique stress-strain behavior in a gel or network under strain. Specifically, polarimetry is the measurement of the polarization of transverse waves that have been reflected, slowed down (refracted), or diffracted by the material. Measurable optical activity includes birefringence, the property of having a refractive index dependent on the polarization and propagation direction of light, and dichroism, a loss of transmitted light intensity which depends on the incident polarization state. 393 A sample's optical activity can be caused by both birefringence and dichroism, as well as a variety of other phenomena. Optical techniques are nondestructive, specific, and rapid; they can be used to image the sample like in polarized optical microscopy (Section 4.2.2) or easily be combined with other characterization techniques, as covered in the following section on rheo-optics (Section 5.4.2). 158 While many systems of interest have intrinsic optical signatures, others may require clever molecular design to probe a particular length scale or process.

In polymeric gels, the gelation process is typically not due to a single, easily isolated phenomenon. In many cases, it is necessary to attempt to separate multiple effects which often take place in competition over many length scales in order to extract the physical origins of gelation, shear, or flow behavior. For example, in the case of physical gelation of gelatin, both the formation and lengthening of triple helices contribute to gelation, albeit at different time scales dependent on temperature. The contribution of each in a "fast" and "slow" gelation regime was determined by measuring the specific optical rotation during the gelation process (Figure 26). 395

Rheo-optics combines traditional rheology with optical methods to track the changes in optical properties of a material under stress, correlating rheological behavior to microstructure. These techniques are enabled by micromechanical properties with distinct optical signals resulting from material deformation. 393 Many different experimental techniques fall into the category of rheo-optics, depending on which rheological property (viscosity, modulus, characteristic relaxation time) is of interest, the kinds of optical changes which can be observed (including refraction, absorption, fluorescence, scattering, diffraction, birefringence), 158 and the relevant length scale of interest.

Due to the heterogeneous, hierarchical nature of many gels and networks, rheological data alone is often insufficient to fully understand the material. For example, rheology may show when a system reaches the gel point, but an additional optical technique is often needed to quantify the corresponding reaction conversion; or, in an associative system, rheology may demonstrate shear thinning behavior but with no clear indication of the mechanism. Combining rheological and optical techniques, often with a home-built or modified apparatus, can offer critical insight into the physical behavior of both physical and chemical gels, leading to improvements in material engineering capabilities. 393,[395][396][397][398][399] One example from a recent study on shear-thinning behavior in an associative metal-coordinated gel utilized rheo-fluorescence with a system in which un-associated bonds fluoresced, allowing for a direct measurement of physical bond dissociation behavior under nonlinear shear stress. 397

Recent advances in mechanochemistry has seen the development of myriad "mechanophores;" molecules that respond to a stress or strain stimulus through an easily decipherable chemical change. 90,400,401 Mechanochromophores have been incorporated into a variety of polymer gels, with colorimetric (changes in visible/UV light absorbance or chemiluminescence) response tracking with the breaking of bonds and the release of chain tension in response to bulk material deformation, swelling, or highintensity ultrasound. [401][402][403][404][405][406][407][408][409] Monitoring the activation of mechanophores during material deformation thus allows direct visualization of the chain tension distribution within a sample with micron-level spatial resolution. [410][411][412] One recent study examined the bond breaking patterns during swelling of PMMA using a mechanoluminescent molecule with a transient signal; unlike the signal of fluorescent mechanophores which accumulate additively over time, these short burst-like flashes allowed for visualization of singular bond breaking events. 403 Mechanochromic mechanophores are a visually pleasing method for the optical quantification and mapping of stress distribution in a material under stress, tension, or compression. As such, they are often incorporated into materials for use in rheo-optical set-ups and continue to be applied for novel uses and visualization techniques.

While stress imaging can give clues about fundamental mechanisms of stress relaxation or dissipation, it is also employed in practice to detect imminent failure by quantifying gradual bond breakage before the material macroscopically fails, which is of obvious utility in many critical load-bearing applications. On a macroscopic measurement level, stress distribution imaging may involve large, fabricated, model networks where individual "bonds" can be observed with the naked eye, 413 or a paint splatter method, where the position of paint spots or other visual markers on the surface of a material are tracked as the material deforms, imaging the strain in the material. The latter of these two methods is particularly useful in industrial coating and extrusion processes. In all of these applications, an advantage of the mechanophore approach is that it reports on the mechanical state of individual molecules within the bulk material, allowing molecular behavior (or the distribution of molecular behaviors) and its dependence on the chemical structure of the polymer network to be assessed quantitatively.

Microrheology is the study of the viscoelastic properties of small volumes of materials probed through thermal or forced excitations of colloidal particles. 414,415 The technique is divided into passive microrheology, in which the probe experiences Brownian motion caused by thermal fluctuations, typically in equilibrium; and active microrheology, in which probe motion is generated by an external force and may be extended to the non-linear viscoelastic regime. The central assumption of microrheology is that the motion of the particles is determined by the mechanical properties of the surrounding medium. Mean-square displacement data of the probe are analyzed over time to extract frequency-dependent viscosity or elastic moduli by using the generalized Stokes-Einstein relation (GSER): 416

where G(s) is the Laplace transformed relaxation modulus, ∆r 2 (s) is the Laplace transformed mean-square displacement, s is the Laplace frequency, k B T is the thermal energy, and a is the hydrodynamic radius of the spherical probe. The GSER above can be extended by making an analytic continuation (s = iω) and taking a Fourier transformation to obtain the complex modulus G * (ω): 417

where F{ ∆r 2 (τ ) } is the Fourier transformed MSD as a function of the Fourier time τ . The time-domain response of the material can also be calculated using a Laplace transformation by invoking the continuum viscoelastic identity in Laplace space s J(s) G(s) to extract the compliance J(t): 418

which is directly proportional to the meansquare displacement. The compliance J(t), as well as storage (G ) and loss (G ) moduli calculated from G * (ω), can be compared directly against macrorheology (Section 5.1.4) In some instances, the viscoelastic properties measured from microrheology agree well with those obtained from macroscopic rheological measurements, but in many situations, microrheology does not appear to fully capture the macroscopic viscoelastic properties due to lengthscale dependent dynamics in heterogeneous and non-ergodic samples. [419][420][421] Further, both the probe and soft material must meet the underlying assumptions of both components of the GSER; importantly, the probe particles should not interact with, bind to, or repel the sample, as this would affect long-time, large length-scale displacement. 422,423 One approach to mitigate artifacts and increase agreement between microrheology and macrorheology has been the introduction of two-point microrheology. This method studies the cross-correlation of two tracer particles in the same sample. Instead of measuring the MSD of individual particles, the relative displacement between two particles is measured, 424 G(s) = k B T 2πR ∆r 1 (s)∆r 2 (s) (5.21) which depends on the distance between the tracers R, assuming R a. Microrheology is often an ideal technique for studying polymeric networks due to its exceptional sensitivity and ability to characterize weak, incipient gels over a wide frequency range, track rapid kinetics of the sol to gel transition, and probe small quantities of material that would make macroscopic rheology challenging or impossible. 425,426 It has been shown that microrheology measurements on entangled and cross-linked networks can agree with macroscopic rheology, making it a valid alternative when material is limited. 420 Microrheology is particularly useful for measuring the gel point and critical exponents of physically or chemically cross-linked gels, as well as prob-Figure 27: The main underlying principle of passive microrheology is the generalized Stokes-Einstein relation, in which the Einstein component, using the fluctuation-dissipation theorem, bridges the thermal, stochastic fluctuations of a probe particle to its frequency-dependent complex mobility M Stokes (s), which the Stokes component further connects to the rheological properties of the material in which it resides. 414 ing position-dependent properties in specific regions of heterogeneous materials, such as phase separated or incipient gels. 425,[427][428][429][430][431][432] Microrheology measurements can be conducted using any experimental method which monitors fluctuations of particles as a function of time, including real-space methods, most commonly particle tracking microscopy (using high-speed optical or fluorescence microscopes) and its many variations, as well as Fourier-space methods like differential dynamic microscopy 433 or scattering techniques to be discussed below, such as dynamic light spectroscopy (DLS, Section 5.5.1), diffusing-wave spectroscopy (DWS, Section 5.5.2), and more recently X-ray photocorrelation spectroscopy (XPCS, Section 5.5.3). 414 The typical analysis pathway for passive microrheology is shown in Figure 27. While microrheology is advantageous for sensitive measurements on small volumes of valuable samples, care must be taken to ensure samples are clean and dust-free, as any additional micron-sized impurities may convolute the results.

Dynamic light scattering (DLS) excels at characterizing colloidal and nanoparticle systems; in soft matter it has been used to determine the size and diffusion coefficient of microgel particles and of interacting building blocks for supramolecular polymer networks. [434][435][436][437] DLS utilizes fluctuations in the intensity of scattered, monochromatic light to probe dynamic properties of matter at length scales ranging from 100 nm -1 mm and time scales of 10 -6 -10 s. 438 Like many of scattering spectroscopy techniques in this section, DLS measures the intensity correlation function of scattered light, normalized as:

(5.22) where I(0) and I(t) are intensities at time 0 and a delayed time t at a given scattering vector q, corresponding to the MSD ∆r 2 (t) at time t. 434 The intensity correlation function captures dynamic fluctuations experienced by the scatterer due to Brownian motion. By analyzing the time-resolved scattering intensity, the mean-square displacement of the scatterers can be calculated and related to the viscoelastic properties of the sample by the GSER (eqs. 5.18, 5.19, and 5.20). 438 DLS requires a material with strong scattering signal and analysis of DLS assumes only a single scattering event, so scatters are typically dilute in a homogeneous matrix (either polymer in solvent as in Section 2.2.6 or a microrheology experiment with tracer particles in dense networks). The scattering intensity fluctuates due to local fluctuations in concentration caused by the thermal motion (diffusion) of the polymer chains in and out of a given volume ∼ 1/q 3 . In the case of polymer networks, fluctuations may be suppressed due to entanglements or the formation of cross-links in a gel restricting motion on the size of the colloidal tracer particle (Figure 28). Particles with larger radii relative to the network mesh size undergo slower fluctuations, resulting in a longer decay time in the auto-correlation curve.

The technique has also been used to probe the sol-gel transition in a variety of systems [439][440][441][442][443][444] with a power-law behavior of the intensity correlation function

(5.23) with a single self-similar exponent 0.2 ≤ µ ≤ 0.9 at the gel point (similar to the stress relaxation exponent u observed at the gel point in macrorheology, Section 5.1.4), implying a selfsimilar or fractal structure of the clusters of the incipient gel. However, due to the nonergodic nature of solutions undergoing gelation, the time and ensemble-averaged intensity correlation functions begin to deviate at the onset of gelation as a result of significant inhomogeneities from pre-gel clusters. Because of the challenges in accurately interpreting sol-gel DLS data such as multiple scattering, detailed quantification of dynamic behavior near the gel point is typically done with a related light scattering technique, diffusing-wave spectroscopy.

Diffusing-wave spectroscopy (DWS) is a light scattering method which accounts for strong multiple scattering events. As in DLS experiments, an incoming beam travels through a sample cell, but scatters off of many particles before detection; unlike DLS, the time scale does not depend on the angle of detection but rather on the sample cell geometry and path length distribution. 445,446 The scattered light intensity is monitored over time, and the autocorrelation function is used to extract quantitative information such as the mean-square displacement of tracer particles and the complex elastic modulus of the material. DWS is a common alternative to optical tracking for microrheology measurements (Section 5.5), and is preferable in some cases due to its higher sensitivity (superb statistics from many probes and multiple scattering compared to a single probe particle) and broader range of accessible length scales (down to λ/1000). 445 While DWS is often easier to implement experimentally than DLS, the user must ensure that all light is multiply scattered, which is often done by adding probe particles. Selection of the probes is important: proper size (0.5 -2 µm), material, and concentration are necessary to minimize interactions of the particles with the system or have noticeable effects on the rheological properties being measured. 447 Static light scattering (Sections 2.2.6 and 4.3) can be used to verify that the tracer particles are the dominant source of scattering, are isotropically dispersed in the sample, and that their mean-square displacement scales inversely with their radius. DWS was originally developed as an analysis method to expand DLS into the limit of multiple scattering events. Initial studies done on colloidal glasses 446 were extended to probe viscoelastic behavior 416 and creep compliance 448 over a wide frequency range, providing characterization in good agreement with macroscopic mechanical measurements. In addition to elasticity measurements, DWS has been employed to measure the gel point in many polymeric systems; this is advantageous due the high sensitivity of the probe particles and the small sample volumes required. 449,450 DWS can also be paired with a complementary spectroscopic technique such as UV-Vis or FTIR to couple gel point and conversion measurements. 450

X-ray photon correlation spectroscopy (XPCS) utilizes a coherent synchrotron X-ray beam to measure the intensity correlation function, probing slow dynamics in condensed matter systems on 1 Å -1 µm length and 10 -6 -100 s time scales. 451 The speckle pattern formed when X-rays interact with randomly distributed scatterers (typically in soft matter systems due to added probe particles, but can also be high X-ray contrast components of the material) exhibits intensity variations which are directly related to order parameter fluctuations in real space. Analysis of the intensity correlation function yields information on nanoscale dynamics such as diffusion or relaxation processes. Advantages of the technique include no complication from multiple scattering, an opportunity to test opaque samples, and an experimental window not available with visible light or neutron scattering techniques. The sample must have adequate X-ray contrast, which in soft matter systems often requires the addition of particles. Some samples, particularly soft mat-ter or biological systems, may be badly damaged when exposed to the beam for long periods of time, limiting the longest accessible time scale.

XPCS is a relatively new technique which has been used extensively in colloids and block copolymers, while applications to polymer networks have recently emerged. [451][452][453] With the addition of tracers such as silica or gold nanoparticles, XPCS measurements have provided novel microscopic characterization of gelation and entangled dynamics. [454][455][456] The technique has also recently been recognized as a potential form of microrheology, using MSD data derived from the auto-correlation function to yield viscoelastic measurements. This is a promising avenue for future work, especially in systems which are not ideal candidates for traditional DWS or optical particle tracking methods, which preclude opaque samples. 457,458

Like its microscopic relative, nanorheology probes viscoelastic properties of a material through analyzing the motion of a nanoscale probe embedded in, or pressed onto, a material. Specifically, this technique looks at the rheological response of materials in confined geometries and involves adhesion, friction and lubri-cation between particles and surfaces. 459 Two major experimental techniques are surface force apparatus (SFA) and atomic force microscopy (AFM), which measure the normal and lateral forces between two curved surfaces (SFA) or between a tip and a flat surface (AFM) with a high force sensitivity. AFM provides improved force sensitivity while SFA measures absolute separation distance and surface deformation, which then lead to rheological properties and structure-property relationships of confined, complex soft materials. 459 The sub-nanometer length scales accessible with nanorheology (< 0.1 nm resolution with SFA) make this technique suitable for probing the mesh size, relaxation times on monomer or Kuhn length scales, and viscoelastic properties in polymer gels and networks. 460 Recent work has included improved analysis of AFM data and application to probing biomaterials; 461,462 model studies on entangled and supramolecular networks using magnetic particle nanorheology; 460,463 as well as accompanying molecular simulations probing local viscoelasticity of an entangled melt and scaling theory of storage and loss moduli as a function of length scale. 464,465 Active particle methods, (i) optical tweezer nanorheology, which employs lasers to optically trap dielectric particles and measures the meansquare displacement, as well as (ii) magnetic particle nanorheology, which employs oscillating magnetic fields to drive magnetic nanoparticle motion and measures the AC susceptibility, also enable quantification of rheological properties on the nanometer scale. 85,466 Optical tweezers can be used to exert forces in excess of 100 pN in the range of 1 -100 nm with subnanometer accuracy and sub-millisecond time resolution. However, the lasers used for optical trapping lack selectivity and exclusivity, while also resulting in local heating, which can affect the sample viscosity. Magnetic tweezers are more straightforward to implement and can impose forces in excess of 1 nN across a wider length scale but are not nearly as versatile in nanorheology experiments. 85

Emerging nuclear magnetic resonance (NMR) spectroscopies, due to the broad range of accessible time scales (10 -12 -10 3 s), are ideally suited for characterization of polymer dynamics in addition to the characterization of structure as discussed in Sections 2.1.1, 2.2.1, and 4.5.

Cycling NMR NMR relaxometry measures how an excited magnetic state returns to its equilibrium distribution and can be used to study exchange dynamics on the milliseconds to hundreds of seconds time scale. The longitudinal or spin-lattice relaxation time T 1 is related to inter-segment and intra-segment molecular motions modulating the magnetic dipole-dipole interaction of 1 H-1 H spin pairs (or other NMRactive nuclei of interest). The value of T 1 can be easily determined by fitting the signal buildup curve saturation recovery experiment with a variable relaxation delay to an exponential. The transverse or spin-spin relaxation time T 2 can be measured in an analogous manner, by quantifying the decay of the magnetization perpendicular to the magnetic field.

1 H field-cycling NMR relaxometry (FC-NMR) measures the dependence of the proton spin-lattice relaxation rate, R 1 = 1/T 1 , on the Larmor frequency, ν or ω = 2πν. This is also referred to as the nuclear magnetic relaxation dispersion (NMRD), which reflect the spectrum of reorientational and translational dynamics of polymer segments. 467,468 Time-temperature superposition (TTS, applicable for polymers at temperatures exceeding their glass transition temperature, T T g ) enables transformation of R 1 measurements to a susceptibility representation:

and the construction of master curves. Such master curves enable deconvolution of contributions from reorientational and translational motions to relaxations at different frequencies. Recently, FC-NMR has been applied in this man-ner to investigate the glassy behavior and restrictions of cross-links on the polymer dynamics of a series of industrially relevant rubbers across three decades of frequency (0.01 -100 MHz). [469][470][471] 5.7.2 Pulsed-Field-Gradient NMR Pulsed-field-gradient NMR (PFG-NMR) provides a powerful technique to probe the selfdiffusion of NMR-active nuclei on the micrometer length and millisecond time scales. PFG-NMR is based on the dependence of the Larmor frequency on the amplitude of the applied magnetic field. Magnetic-field-gradient pulses of stimulated echoes are added, which process the magnetic spins according to their Larmor frequency. If the molecules do not change their positions during the time of diffusion measurements, the signal retains its maximum intensity; otherwise, the signal loses part of its intensity and the resulting signal attenuation gives a measure of the movement of the molecules. The attenuation of the echo ψ measures the diffusion coefficient D according to 25) where γ is the gyromagnetic ratio, g is the gradient strength, δ is the gradient pulse length, ∆ is the interval between gradient pulses, t = ∆ -δ/3 is the diffusion time, and q = γgδ is analogous to a scattering vector. 472 PFG-NMR is experimentally very similar to neutron spin echo (NSE, Section 5.8) spectroscopy. Both measure dynamical information through the procession of spins in an applied magnetic field. The time and length scales of measurement and correspondingly of the information measured, however, are well-separated. NSE is capable of measuring polymer dynamics at the 1 Å -1 µm length and 1 -1000 ns time scales, while PFG-NMR excels at measuring dynamics on the 1-100 µm length and 1-100 ms time scales. The principal application of PFG-NMR to soft materials is in the measurement of self-diffusion of polymers, solvents, or ions, according to the nuclei of interest. The long-range self-diffusion of high-molecular-weight PDMS chains in a melt have been measured by PFG-NMR and the calculated monomeric friction coefficient was found to be in agreement with data from viscosity and quasielastic neutron scattering measurements. 473 Diffusion measurements of water through hydrogels or ions through solid polymer electrolytes are also common. [474][475][476][477] One interesting application for polymer networks is in measuring the diffusion of junctions or strands in reversible networks, 478 such as the motion of metal ions in metallo-cross-linked polymeric ionic liquids. 479

Closely related, diffusion-ordered NMR spectroscopy (DOSY) techniques are commonly used for facile measurements of small-molecule or polymer diffusion coefficients in solution. [480][481][482] In dilute solution, the molecular weight of polymer strands can be obtained by correlation with the measured diffusion coefficient (Section 2.2.1). [480][481][482][483] Further, as discussed previously in Section 4.7, the diffusion of molecular probes within polymer gels can provide information related to network structure on various length scales. 239,241,484 DOSY has also proven valuable for characterization of pre-gel components prior to, and during network formation, for example, in the analysis of gel point for optimization of network formation conditions. 485,486 The diffusion of the polymer strands and junctions themselves in reversible networks or the diffusion of probe particles (commonly polystyrene latex particles of different sizes) in covalent networks can be measured using this technique. The latter is closely related to nanorheology (Section 5.6), with a focus on the terminal diffusion.

Recent advances in instrumentation have enabled the simultaneous characterization of polymeric materials with NMR and rheometry (Section 5.1.4, termed rheo-NMR. Concurrent steady or dynamic rheological measurements and time-domain NMR spectroscopy can provide information about the relationship between polymer mobility, orientation or crystallization on sample deformation or mechanical properties. For example, shear induced changes in entanglement dynamics, shear history effects on crystallization, and polymer-inorganic filler interactions and their effect on crystallization have all been characterized through rheo-NMR techniques. [487][488][489] Rheo-NMR has many potential applications in the study of reversible gels to understand the coupling between macroscopic mechanical deformation and molecular cross-linking and dynamics. For example, rheo-NMR has shown that shear reduces the intermolecular linkages between κ-carrageenan, a seaweed extract that undergoes thermoreversible gelation in solution and is used as a polysaccharide thickener in the food industry. 490

Neutron spin-echo spectroscopy (NSE) is an inelastic time-of-fight scattering technique used to directly probe material dynamics at 1 Å -1 µm length and 1 -1000 ns time scales. 491 The technique measures the Fourier transform of the dynamic structure factor S(q, ω), termed the intermediate scattering function, I(q, τ ), which is a function of the momentum transfer vector, q, and Fourier time, τ . 492 The intermediate scattering function is the spatial Fourier transform of the van Hove function G(r, t), which is the real-space dynamical correlation function characterizing the spatial and temporal density distributions in the fluid. The basic principle of NSE is that if a neutron with a polarized spin rotates anticlockwise and clockwise by the same amount it should return to its original orientation provided the speed of the neutron is constant. If the velocity changes due to inelastic scattering off of a molecule, the spin will not return to the same orientation and the difference in energy will be measurable. Similar to the Hahn echo used in NMR, NSE relies on the de-phasing and re-phasing of neutron spins over time using a spin-echo-the refocusing of spin magnetization with a pulse of electromag-netic radiation. A typical experimental set up is show schematically in Figure 29. Advantages of NSE include the ability to directly measure in the time domain without the need of Fourier transforms, which avoids some of the deconvolution necessary in other scattering techniques and allows for a more direct interpretation of the data. As in SANS (Section 4.3), strategic deuterium labelling can provide information on the microscopic fluctuations of particular substructures of interest, such as the junctions in a polymer network. Disadvantages of the technique include the long counting times and large sample volumes necessary to obtain adequate signal.

As the dynamical complement to SANS (Section 4.3), NSE is ideally suited to provide detailed information on the motion of macromolecular objects on length scales several orders of magnitude smaller than those accessible with FRS (Section 5.9 below). 493 NSE studies have established microscopic evidence for entanglement tube effects and contour length fluctuations in reptation theory [494][495][496] and measured the extent of junction fluctuations in cross-linked polymer networks. [497][498][499] More recent studies have examined dynamics in novel network types, including polyelectrolyte double networks 500 and networks with slide-ring crosslinks. 501 Future studies could further delve into networks with dynamic cross-links or systems under shear, such as a recent contribution probing nanoscale dynamics in flowing liquids. 502

Forced-Rayleigh Scattering (FRS) is ideally suited for measuring slow dynamical processes such as the self-diffusion of polymer chains. Molecule displacements are quantified over micron-level length scales, enabling diffusivity measurements ranging (10 -17 -10 -5 ) cm 2 s -1 . 503 FRS is a holographic grating technique in which the user first generates a grating with two intersecting laser beams and then subsequently measures the diffracted light intensity as the gradient diffuses, as shown schematically in Figure 30. The resulting decay curve is used to determine the diffusivity of a molecule of in- terest (the tracer) through another material of arbitrary complexity (the matrix). Calculated variables such as the relaxation time constant and diffusion coefficient are extracted by fitting an exponential function to the intensity decay curve. [503][504][505][506][507][508][509][510] The matrix can be almost any material, including polymer solutions, melts, and associative or covalent networks. However, the tracer must be labelled with a photosensitive dye; tracers are usually dilute within the matrix (tracer diffusion) or selectively labeled matrix components (self-diffusion). 509,510 FRS studies on associative polymer networks have probed diffusivity behavior over four decades of mean-square-distance, resulting in the discovery of a region of anomalous super-diffusion below a specific length scale, attributable to a diffusing species in dynamic equilibrium between a fast-diffusing molecular state and a slow-diffusing associated state. 493,[509][510][511][512][513] Despite the requirement of a photochromic dye, FRS is a versatile technique uniquely capable of probing slow diffusion processes over a range of length scales in a variety of isotropic and oriented soft materials.

Dielectric spectroscopy is a sensitive technique for probing molecular relaxations and has been used extensively to develop structure-property relationships in polymers. [514][515][516] A principal advantage is the ability to measure the molecular motions over a wide temperature window and a broad dynamic range (10 -2 -10 9 Hz), covering many decades in time scales with a single bench-top technique. The technique works by measuring the electrostatic fluctuation of molecular dipoles and charges, as the complex permittivity:

as a function of angular frequency ω of the applied electric field. 514,517 With a knowledge of the molecular structure, the fluctuation of dipole moments can be related to different relaxation processes of functional groups, segments, and even entire polymer chains. This spectrum of relaxations can be further interpreted as a probe for the structure and dynamics of different components of the material. Since these molecular motions are affected by cross-linking in polymer networks, dielectric spectroscopy can be a sensitive tool to as-Figure 30: Forced Rayleigh Scattering. The writing configuration, shown on the left, uses the interference pattern of two intense, mutually coherent laser beams to generate a sinusoidal optical grating of selectively excited dye molecules. The reading configuration, on the right, monitors the diffracted intensity of the reading beam as the dye isomers diffuse. The resulting decay curves are processed to yield diffusivity measurements. sess mobility as a function of network topology and formation (Figure 31). [518][519][520][521][522] Cross-linking slows down the segmental dynamics, leading to a shift in the position of the maximum in the dielectric spectrum corresponding to segmental motion to lower frequencies. Low-frequency motion corresponding to the large-scale motion of the polymer backbone is also completely suppressed with cross-linking. As such, studies have used dielectric spectroscopy to follow the evolution of segmental dynamics during vulcanization, healing of damage, and the addition of nanoparticle fillers in natural rubbers. [523][524][525][526][527][528] Dielectric spectroscopy is also termed electrochemical impedance spectroscopy or AC impedance spectroscopy when measuring ionic or electronic conductivity. In this way, the DC conductivity (σ DC ) can be determined from the frequency-independent plateau in the spectral dependence of the real conductivity

where ε 0 is the permittivity of vacuum. 529,530 Typically, the conductivity is determined from the real component of the conductivity at the maximum in tan (δ) = ε /ε . An interesting application of impedance spectroscopy to polymer networks is in metallo-cross-linked networks formed from polymeric ionic liquids, where in conjunction with rheology (Section 5.1.4), the microscopic effects of ionic junctions and counterions mobility on the conductivity and mechanics can be unraveled. Computer simulations have proved invaluable for the study of molecular systems, with the main advantage to link macroscopic and microscopic phenomena, as well as to test predictions and assumptions of theoretical models. Molecular simulations of polymers can be crudely divided into two main methods: Molecular Dynamics (MD) and Monte Carlo (MC) simulations. In classical MD simulations, trajectories of the system are generated by numerically propagating particle coordinates in time using carefully parameterized molecular models (or "force-fields"). These trajectories can then in principle be analyzed to extract data on the various properties of the system-structural, dynamical, thermodynamical, mechanical, etc. In MC simulations, the particle coordinates are randomly transformed between microstates in a way that captures the thermodynamic ensemble, sacrificing the real time evolution in favor of efficient sampling of phase space. Both methods have many implementations and extensions that have been developed to increase computational efficiency. 535,536 Computer simulation techniques have been extensively used to study polymer solutions, melts, brushes, and networks. 536 Regardless of the method employed, the molecular model is a key ingredient in the execution and interpretation of computer simulation and is a limiting factor in the physical quantities that can be sampled. In classical simulations, all-atom (AA) simulations are the most detailed models, and as such they provide details on the atomic and molecular levels. For example, AA simulations can be used to study interactions between components, 537,538 structure, 539 dynamics and mechanical properties 540 of polymer networks. Coarse graining (CG) of the molecular model enables increased computational efficiency, which allows longer time and length scales to be explored, at the expense of limited chemical detail. Both CG and AA simulations of polymer systems that include solvent (as solutions and gels) can represent the solvent either explicitly, by incorporating solvent molecules into the system, or implicitly, by modifying the effective polymer's intra-and inter-molecular interactions. The bead-spring model by Kremer and Grest 541,542 is a powerful CG method for polymers, in which each Kuhn monomer in the chain is represented by a single bead. The beads are connected by finitely extensible springs, and inter-bead interactions are modeled with a Lennard-Jones potential that can be tuned to represent different solvent qualities at the implicit-solvent level. It can also be used for explicit-solvent simulations, which allows realistic consideration of solvent entropy and interaction with the polymer. This particular CG model has been used extensively to study polymer systems and to examine various physical theoretical predictions at and above the scale of a monomer or Kuhn length. 543 Though much more computational demanding than CG, AA models with explicit solvent can be necessary to answer questions related to molecular interactions, hydration, and structure and dynamics at the atomic or monomeric scales.

A key advantage of computer simulation techniques in the context of polymer science is their ability to conclusively determine the effect of a limited set of parameters on the physical behavior, while constraining all others. One can design specific network architecture, network strand distribution, chain stiffness and many more characteristics. With a reductionist model used in computer simulation, one can even test an "idealized" network devoid of defects to exclude their effect, 544,545 or to emulate the gelation process in order to render more realistic networks that include defects (Figure 8). 546 To create these realistic networks in silico one can follow synthetic routes similar to those employed experimentally, namely: (i) random cross-linking an equilibrated melt 547 or semidi-lute solution 548,549 of the precursor chains up to some degree of reaction, p, mimicking vulcanization, 546 (ii) end-linking of such a melt or solution by extending the MD trajectory and allowing un-reacted chain ends to bond if they are within some cutoff distance and the chain end has not yet saturated its bonding ability, 550 and (iii) emulation of free-radical polymerization of reactive monomers. 551 The model networks can be studied in the unperturbed state for structural and dynamical quantities, 550,552 or be subjected to different processes and environments such as extension and compression, shear, 553 fracture, 554 electric field, 555 or changes in solvent quality. 556 The mechanical properties of polymer networks (Section 5.1), including stress-strain curves, 274,557,558 stress relaxation, 559,560 damage evolution, 561 etc. can then be extracted from the trajectories. These quantities help evaluating the microscopic contributions to macroscopic quantities; see Figure 19 as an example of the collapse of simulation data on a universal curve for network elasticity. 112 One of the major limitations of computer simulations is that limited computational resources determine the length and time scales that can be studied. This limitation becomes more severe as the model resolution increases, i.e., from CG to AA. For networks, this problem is even worse as a chemically cross-linked polymer network is inherently a quenched system with many inhomogeneities, implying computer simulation results might differ between realizations. Therefore, one must average over several "replicas" of the system, to get meaningful reproducible results over the entire ensemble, especially if fracture is involved. 310,554 The same limitations on system size set bounds to the accessible physical quantities, as finitesize effects might become non-negligible. 562 To minimize artifacts, it is useful to simulate several system sizes, L, and extrapolate the considered property as a function of L -1 to infinite system size. It is also helpful to compare the computer simulation results with experimental measurements if available. It is common that such comparisons lead to further calibration of the molecular model, allowing for refinement of the model parameters. At the same time, for certain purposes, computer simulations can utilize non-physical mechanisms to allow sampling of longer time scales. For example, to vulcanize high-molecular-weight precursor chains to form highly entangled networks, as mentioned above, one needs to properly equilibrate melts. To avoid the long equilibration time of highly entangled melts, MC algorithms have been developed to allow large conformational changes by allowing chains to swap bonds, 547,563 or to progressively equilibrate the melt at different length scales. 564 The raw data produced by an MD run is a trajectory in the 6n-dimensional position-momentum phase-space of the system, n being the number of particles in the system. Since the scope of the applicability of computer simulation to networks is vast, it is constructive to first consider several representative examples. As mentioned earlier, the highresolution data available in computer simulations can serve as a link between theoretical models and experiments. In an illuminating example for the virtues of computer simulation, Everaers et al. have studied deformation of well-defined polymer networks with constant network-strand length; by (unrealistically) switching the ability of network strands to cross one another, i.e., making them "phantom", they were able to elucidate the effect of entanglements on various network properties including their contribution to the elastic modulus. 545 Gel swelling (Section 4.1) can also be systematically followed, for example, by varying solvent quality in an isobaric-isothermic ensemble. Designing these simulations to match changes of experimental conditions, such as changes to solvent condition (e.g., temperature, salt concentration, pH, etc.) or addition of solvent, enable a thorough exploration of the gel behavior. 556,565,566 Alternatively, gel swelling by extra solvent can be followed in the canonical ensemble by gradually changing volume. 567,568 Moreover, the ability to gauge the system at the monomer level allows for the study of the gelation process itself and comparing it to percolation and mean-field theories (Section 3). Of specific interest is the ability to directly mea-sure network strand distributions, P (M x ), 546 while in experiments it is usually only possible to determine the average M x or make predictions regarding the distribution based on the sol fraction. Furthermore, the computer simulations can be used to determine the gel point and its suppression, 569 systematically identify and quantify network "defects" (dangling ends and primary loops such as those in Figure 8), 213,570 and determine network topology including all higher-order loops (Structure 4).

More specifically, the coordinates generated by computer simulations can be analyzed for various distribution functions such as distributions of bond and chain tensions, internal distances, angular orientations, etc. The structure of networks can be analyzed in much more detail by considering the time-and ensembleaveraged distributions of networks strands' endto-end distances and orientations. Additional information is gathered through spatial or dynamical correlation functions of different network properties such as velocities, particle positions, orientations, and other degrees of freedom. All these quantities can be compared against theoretical predictions and dynamical (S(q, ω)) or intermediate structure factor (S(q, t)) data from techniques such as NSE (Section 5.8) and thereby serve as a crucial link between theory and experiments.

Perhaps the most important for networks is the stress-strain relationship, extracted by deforming the polymer network (stretching, compression, or shear) and measuring the average elastic stress. The instantaneous stress tensor is derived from the virial theorem and calculated through the sum over all beads n, 571

where m k is the mass of bead k, and v k,i , r k,i , and f k,i denote its velocity, coordinate, and force along the i direction. The stress-strain curves allow one to test elasticity theories, parsing contributions of entanglements (as mentioned above), assessing contributions of finite extensibility, and the contributions from single network strands. 274,310,540,557,572 Importantly, the shear-stress relaxation modulus 559,560 (G(t), Section 5.1.2) characterizing the linear viscoelastic response of the network can be derived from the stress auto-correlation function with the Green-Kubo formula,

where V is the volume, k B T is the thermal energy, and i, j are Cartesian coordinates with i = j. 573,574 Additional transforms of the stressrelaxation modulus to frequency (ω) space can then provide rheological information on the storage G (ω) and loss moduli G (ω), reflecting the extent of energy storage and dissipation in the network. These quantities can be compared against predictions from theoretical models for the network's dynamics as well as with rheological measurements (Section 5.1.4).

The network structure can be directly measured by examining structural distributions and spatial correlations functions. For instance, the structure of networks can be characterized directly by the size distributions of networks strands or loops. The form factor of individual chains and the structure factor of the network itself can be readily obtained and compared with scattering experiments (Section 4.3). Furthermore, the pair correlation function (also known as the radial distribution function) informs on the local spatial correlations of monomers and is defined as 575 g(r) = ρ -2 i j =i δ(r i )δ(r j -r) (6.3) where ρ is the bulk density, i, j are monomer indices, and the brackets • • • denote ensemble average. 552,575 Another important example for a correlation function accessible through computer simulations is the mean-square displacement of a particle MSD = [r(t) -r(0)] 2 , (6.4) which can, for example, inform on the anomalous diffusion of probe particles in a network (Sections 5.5, 5.6). 576,577 The probe diffusion can then shed light on the coupling between polymer relaxation modes and the particle dynamics as well as on the network structure itself, i.e., mesh-size distribution and swelling. For example, the diffusion of a probe particle moderately larger than the network mesh size is sub-diffusive at short time scales at which the coupling with polymer dynamics hinders the particle motion, whereas at longer time scales the particle behave diffusively, through a hopping mechanism. 578 The time dependence of the MSD can then be used to extract nanorheological information on the network. 464 The diffusion coefficient of probe particles can then be calculated from the MSD 15 at these long time scales. It is also worthwhile to mention the recent development of alternative methodologies to the molecular simulations discussed thus far in studying self-assembly, thermodynamic behavior, and dynamics of polymer networks. In particular, molecularly informed field theories, with their associated analytical approximations and numerical simulations, are a powerful tool in polymer physics, with applications to many categories of polymer systems, including solutions and melts of polymers of arbitrary architecture and molecular features. 579,580 Field theories, traditionally, however, have had difficulties in application to networks due to the need to explicitly enumerate all the possible structures. [581][582][583][584][585] Recently, a reformulation of the field theory in a coherent-states (CS) framework was developed to implicitly account for the full ensemble of supramolecular polymer reaction products and is expected to supplement existing methods for reversible and inhomogeneous polymer networks, 586,587 pending advancement in the numerical methods. 588 Such techniques can possibly be applied to networks with quenched disorder, 589 as well, by application of replica methods as in MD to obtain an ensemble average of the disorder. Finally, it should be noted that a plethora of mesoscopic and macroscopic methods exist to describe the continuum mechanics of gels and elastomers, albeit without the explicit connection to the molecular properties of the polymer strands and junctions; the reader is directed elsewhere in the literature. [590][591][592][593][594][595][596][597][598][599][600][601][602][603][604] 7 Perspective

The interdisciplinary nature of polymer network chemistry involves an interdisciplinary mix of characterization techniques that probe structure and dynamics at the level of individual strands, junctions, topology, and statistical mixtures. In general, each technique offers one piece of a much larger characterization puzzle, and combinations of techniques (both experimental and theoretical/computational) are often necessary in order to adequately assess the molecular structural details that are germane to a given structure-property relationship. This complexity presents a barrier to polymer network chemistry, because both the purpose and the technical details of many of the characterization techniques are unfamiliar to most chemists. In our view, this unfamiliarity obscures a tremendous opportunity. The intrinsic molecular characteristics of polymer networks can be resolved with increasing precision, and the contributions of specific molecular structure and behavior to the properties of polymer networks can be increasingly assessed. In the long run, these connections will allow chemists, through creative (macro)molecular design and synthetic control, to program properties and function into individual polymer molecules, including precise conformational behavior, unusual extensional profiles, and covalent chemical responses.

The characterization challenge presents something of a linchpin. One part of the challenge, addressed in this review, is to make the motivation for, and the limits of, available techniques increasingly accessible to the broader chemistry community. Looking ahead and beyond the scope of this review, a second set of challenges involve the further development of characterization tools and methods. Areas of particular interest to us include: (i) extensions of network disassembly methods, including those that follow the gradual evolution of disassembly with time to obtain greater insight into heterogeneities on multiple length scales; (ii) techniques for characterizing the time-dependent behavior of individual probe strands within networks, for example through approaches simi-lar to super-resolution spectroscopy; (iii) robust data collection schemes that allow primary characterization data from multiple techniques and at multiple stages of network formation (e.g., strands, junction, and ensemble network characterization) to be input directly into machine learning protocols. [605][606][607][608][609] We keenly anticipate development of advanced techniques, improvements in computational power, and informatics will enable molecular optimization of polymer networks.