Solution NMR methods for structural and thermodynamic investigation of nanoparticle adsorption equilibria

Characterization of dynamic processes occurring at the nanoparticle (NP) surface is crucial for developing new and more efficient NP catalysts and materials. Thus, a vast amount of research has been dedicated to developing techniques to characterize sorption equilibria. Over recent years, solution NMR spectroscopy has emerged as a preferred tool for investigating ligand–NP interactions. Indeed, due to its ability to probe exchange dynamics over a wide range of timescales with atomic resolution, solution NMR can provide structural, kinetic, and thermodynamic information on sorption equilibria involving multiple adsorbed species and intermediate states. In this contribution, we review solution NMR methods for characterizing ligand–NP interactions, and provide examples of practical applications using these methods as standalone techniques. In addition, we illustrate how the integrated analysis of several NMR datasets was employed to elucidate the role played by support–substrate interactions in mediating the phenol hydrogenation reaction catalyzed by ceria-supported Pd nanoparticles.


Introduction
Nanoparticle (NP) catalysts have found many applications in numerous elds such as energy, 1,2 petrochemical, 3,4 and medicine, 5,6 due to their high surface-area-to-volume ratio and sizedependent properties that make them highly efficient and tunable catalysts. Since sorption equilibria play an essential role in determining the efficiency and selectivity of NP catalysis, many analytical techniques have been utilized to characterize ligand-NP surface interactions, including ultraviolet-visible (UV/Vis) spectroscopy, 7-10 uorescence, 11,12 vibrational spectroscopy, [11][12][13] and isothermal titration calorimetry (ITC). 14 However, the outcomes of these analytical methods are generally dependent on the specic system under investigation and experimental conditions. In recent years, solution NMR spectroscopy has emerged as a preferred tool to investigate the interaction between the substrate and the NP surface. Indeed, due to its ability to probe dynamic processes occurring over a wide range of timescales (ps to hours), NMR spectroscopy is applicable to a broad range of NP-ligand pairs. In addition, the ability of NMR to provide data with atomic resolution allows a comprehensive characterization of the NP-substrate interaction, returning structural, kinetic, and thermodynamic information on the sorption equilibrium.
This contribution gives a brief overview of non-NMR techniques to investigate small molecule-NP interactions and the types of information these approaches can provide. In addition, we review solution NMR methods commonly employed in the characterization of small molecule-NP interactions, and we provide examples of their applications to various ligand-NP systems. Finally, we discuss how the combined analysis of multiple solution NMR methods was used to characterize the contribution of substrate-support interactions toward the hydrogenation of phenolic compounds catalyzed by ceria-supported palladium (Pd/ CeO 2 ) NPs.

Langmuir's theory
Experimental data on ligand-NP interactions are commonly analyzed within the framework of Langmuir's theory. [15][16][17] The Langmuir model relies on four primary assumptions: (i) the adsorbent surface is homogeneous, (ii) each binding site can only hold one adsorbent molecule (i.e. monolayer coverage), (iii) adsorbed molecules do not interact with one another, and (iv) the adsorption process is reversible. Although modied models that address the limitations introduced by such assumptions were proposed, [18][19][20][21] the original Langmuir isotherm model is, so far, the most common approach for modelling sorption equilibria.
The interaction between a ligand (L) and a vacant binding site on the surface (S) to form an adsorbed species (LS) can be represented in the form of a chemical equation: the Langmuir model, the concentration of binding sites is generally unknown and has to be part of the equation. The fraction of surface binding sites occupied by ligands is dened as the surface coverage (q): where g LS is the grams of adsorbed ligand per gram of adsorbent and g max is the maximum adsorption capacity (in grams) that corresponds to the monolayer formation in the specic ligand-NP system under investigation. Therefore, the fraction of vacant sites will be expressed as 1 À q.
In the original gas-phase Langmuir model, rates of adsorption (r ads ) and desorption (r des ) are derived by means of the ligand pressure (p) and the surface coverage q (r ads ¼ k ads p(1 À q) and r des ¼ k des q, respectively). However, when applied to the liquid phase systems, the ligand amount is quantied in terms of molar concentration. Taking this and the fact that at equilibrium r ads ¼ r des into account, the Langmuir isotherm model is generally represented in one of the two forms shown below: 17 where C L and q LS are the concentrations of desorbed and adsorbed ligand at equilibrium, respectively, q max is the maximum adsorption capacity (in mol L À1 ), and K is the Langmuir equilibrium constant that corresponds to the inverse of the equilibrium concentration of free ligand when half of the available adsorption sites are occupied (Fig. 1). Therefore, like any equilibrium constant of an association process, the higher the value of K, the stronger the ligand-NP interaction.
In an ideal adsorption study, q LS and C L are measured at increasing ligand concentrations, and the experimental isotherm resulting from plotting q LS versus C L is modelled using eqn (3b) to obtain q max and K (Fig. 1). Although powerful in providing information on the number of adsorption sites (q max ) and strength of the NP-ligand interaction (K), the latter approach suffers from several experimental limitations. Indeed, accurate modelling of q max and K can only be obtained if a range of ligand concentrations can be spanned so that q LS is measured at C L values that are much smaller and much larger than K À1 . In the case where the range C L ( K À1 cannot be sampled (i.e. the analytical technique used to measure q LS and/or C L is not sensitive enough), the experimental isotherm will grow too steeply nearby C L $ K À1 to obtain an accurate estimate for the Langmuir constant (red and orange curves in Fig. 1). On the other hand, if conditions at which C L [ K À1 cannot be analyzed (i.e. the ligand and/or NP aggregate at a high concentration of small molecule), the experimental isotherm never reaches saturation (i.e. q LS $ q max ) and q max cannot be accurately determined by modelling the experimental data (light blue and dark blue curves in Fig. 1). If the size and morphology of the investigated NP are known, the latter limitation can be circumvented by estimating q max as the highest theoretical ligand density in a monolayer on the NP surface. Oentimes, the value of K obtained by modelling adsorption data with eqn (3a) and (3b) is treated as a conventional equilibrium constant and used for the estimation of the free energy of adsorption (DG ads ) by means of the following relationship: where T is the temperature and R is the universal gas constant. In such cases, care must be taken to the fact that the Langmuir equilibrium constant obtained by tting experimental adsorption data is usually not dimensionless. 22 A recent publication reviews the topic in further details and summarizes mathematical approaches to transform the Langmuir constant into a dimensionless standard equilibrium constant compatible with eqn (4). 23 In addition to providing estimates for thermodynamic parameters, the Langmuir model can be expanded for the investigation of adsorption kinetics. Generally, adsorption of ligands to NPs is described using either a pseudo-rst-order (eqn (5)) 24 or a pseudo-second-order (eqn (6)) 25 kinetic model: In these models, A is an empirical constant that depends on the actual values of k ads and k des , k obs is the observed rate constant of the corresponding pseudo-rst or pseudo-second kinetic process, and q t is the adsorbed amount of the ligand at each time point measurement. In many cases, experimental limitations require the introduction of a time delay factor (t 0 ) that compensates for the time gap between the addition of ligands to the system and the collection of the data. As a result, the pseudo-rst-order kinetic model is more oen represented by the following equation: 26,27 q(t À t 0 ) ¼ A(1 À e Àk obs (tÀt 0 ) ) Depending on the analytical method employed to investigate the adsorption kinetics, eqn (5)- (7) are oen modied to accommodate substitution of q with a directly measured observable. 27 For processes that follow pseudo-rst-order kinetics, an accurate determination of k ads and k des can be obtained if k obs is determined at several concentrations of ligands (L ¼ C L + q LS ). Since the rate constants are connected by the linear function k obs ¼ k ads C L + k des , the slope of the experimental k obs over C L plot provides the value of k ads while the intercept with the y-axis yields k des . Knowledge of the adsorption and desorption rate constants can be used to calculate the Langmuir equilibrium constant by means of the following equation: 23 Of note, the latter approach to the determination of K does not require estimation of q max and is, therefore, the preferred method to obtain the Langmuir equilibrium constant for weak ligand-NP interactions. 26 However, it should be noted that eqn (8) only applies to sorption equilibria based on elementary reactions. In the case of more complex equilibria involving one or more intermediate states, the ratio of the rate constants in eqn (8) would not provide an accurate estimate of K. 23 Overview of analytical methods for characterizing sorption equilibria Spectroscopic analytical techniques such as UV/Vis, IR, Raman, and uorescence are the most developed methods for studying the binding of small molecules to NP surfaces, and a large number of studies are available in the literature that use these methods to obtain a qualitative description of binding as well as the necessary parameters for quantifying adsorption via the Langmuir theory. In addition, a variety of other analytical methods (i.e. X-ray, 28,29 microscopy, 30,31 SS-NMR, 32-36 etc.) have been sparsely applied to specic NP-ligand systems, but they have not found a general application for the characterization of sorption dynamics.

Ultraviolet-visible (UV/Vis) spectroscopy
UV/Vis is a sensitive and cost-effective spectroscopic method for the detection and characterization of diverse analytes, ranging from small molecules 7-10 to large biopolymers. 11,12 The UV/Vis adsorption wavelength of an analyte is highly sensitive to the immediate environment, and thus, different UV/Vis absorption maxima are expected for an adsorbed ligand and for the same molecule in the bulk solution. Due to this potential ability to differentiate among adsorbed and desorbed species, UV/Vis spectroscopy has been oen used as a standalone technique for detecting ligand-NP adducts and, in some instances, to obtain structural and thermodynamic information on the interaction. However, the low resolution of UV/Vis spectra exposes a major drawback of this technique that oen requires separation of the desorbed and adsorbed species via precipitation or ltration procedures for quantitative analysis. This sample preparation step hampers the characterization of weak ligand-NP interactions by UV/Vis spectroscopy.
Systems that involve the adsorption of proteins onto NPs are generally excessively complex for conventional UV/Vis and are normally characterized by Surface Plasmon Resonance (SPR). 12 Formation of the NP-ligand conjugate results in a shied and broadened adsorption spectrum. Since these changes are related to the concentration of bound species, SPR could yield quantitative binding parameters. Although this method is applicable to nearly any type of ligand, it is restricted to metal NPs capable of producing SPR spectra.
Quantitative analysis of binding. The most common approach to quantitative analysis of small molecule-NP systems by UV/Vis involves exposing a standard ligand solution of known concentration (L) to a known amount of NPs and monitoring the changes in the concentration of free (C L ) and surface adsorbed (q LS ) ligand over time. Although C L and q LS could be in principle determined directly by observing the UV/ Vis signals of the free and bound small molecule, respectively, the intrinsic low-resolution of UV/Vis spectroscopy hampers a direct quantication of the two species from a single spectrum. Therefore, C L is oen determined from UV/Vis spectra measured on samples in which the adsorbed species is ltered or precipitated out of solution. The q LS value can be derived indirectly by subtracting C L from L, and the obtained data can be t using the Langmuir adsorption model (eqn (5)-(8)) to yield the kinetics and thermodynamics of sorption. Employment of UV/Vis spectroscopy in this manner was reported for a few protein-NP and small molecule-NP systems, where the equilibrium association constant 37-41 or a range of kinetic and thermodynamic parameters 9,10,40-47 were determined.
In applications examining adsorption on noble metal NPs (generally gold), UV/Vis spectroscopy can be conveniently complemented by SPR. 48 In this hybrid method, C L is obtained by UV/Vis as described above, while q LS is measured directly by observing the changes in the SPR spectrum of the NP caused by ligand binding.
The recent interest in DNA-based nanotechnologies ignited further development in the UV/Vis-based methods for quantifying NP adsorption. In particular, a novel quantitative colorimetric approach was introduced to assess the binding strength of DNA bases and nucleosides to silver 39 and gold 38 NPs. The approach relies on following colorimetric changes induced by aggregation of NPs upon ligand binding. As it was shown that the NP aggregation rate is directly correlated to the concentration of adsorbed ligand, C L and q LS can be obtained by the colorimetric assay.
Qualitative analysis of binding. Interpretation of UV/Vis spectral changes can also provide valuable non-quantitative information about ligand binding. Indeed, a change in peak position in the spectrum of a ligand upon the addition of NP can indicate the occurrence of ligand-NP interaction. In addition, analysis of these spectral changes in comparison to known references could return structural insight into the mode of interaction. For example, de Haan et al. found that the changes in the UV/Vis spectrum of alizarin in the presence of ZnO NPs are consistent with the absorbance spectrum observed for deprotonated alizarin, suggesting that alizarin binds to ZnO NPs in its deprotonated form. 8 Fluorescence Fluorescence spectroscopy and its variations, such as Fluorescence Correlation Spectroscopy (FCS), are convenient techniques for investigating adsorption processes due to their high intrinsic sensitivity and their ability to provide input experimental data for the Langmuir isotherm model. The uorescence of ligands is highly sensitive to quenching by the formation of NP-ligand conjugates. Therefore, analysis of the changes in uorescence spectra (i.e. intensity, broadening, a shi in emission maxima) provides qualitative, and in some cases quantitative, insight into sorption processes.
Although uorescence was used to study the adsorption of proteins and small uorescent molecules (dyes) onto NPs, 11,12 the need for the ligand or the NP to be either naturally uorescent or to carry a covalently attached uorescent label has limited its broader applicability to the study of sorption equilibria. In addition, if NPs are not removed from the solution (by ltration or centrifugation) before the uorescence measurement is taken, they have the potential to induce inner lter and light scattering effects that complicate the analysis of the data. 49,50 Quantitative analysis of binding. Three main uorescencebased approaches to derive quantitative information for NPligand systems were described for inherently uorescent small molecules and proteins. The Fluorescence Quenching Titration (FQT) method exploits the ability of NPs to quench the uoresce of bound ligands. Therefore, the solution of a uorescent ligand is titrated with NP to achieve complete quenching of the signal. Plotting the normalized uorescence intensity over concentration of NP yields a nonlinear decay that is modelled to provide the estimated binding constant. 51 This methodology found wide application in examining both protein-NP 12 and small molecule-NP 52 interactions. In addition to estimation of binding constants, the uorescence quenching method could serve as a tool for examining the accessible surface area. For example, the Molecular Probe Adsorption (MPA) method demonstrated the convenience of uorescence spectroscopy for estimating the available adsorption surface in NPs containing a corona phase. 53 The Fluorescence Correlation Spectroscopy (FCS) method provides several advantages over FQT. FCS relies on measuring variations of uorescence produced by species that diffuse through a part of a sample that is actively monitored by a laser beam. This technique is highly sensitive and can detect exceptionally low concentrations of uorophores, as low as nM. In addition, it does not suffer from complications that arise when measurements are performed with NPs present in the solution. Since the uorescence of the free ligands can be measured independently from the molecules bound to NPs, the primary outcome of FCS is C L , while q LS is derived mathematically based on the measured concentration of the desorbed ligand. A wellillustrated example that studied the adsorption of uorescent compounds rhodamine 6G and calcein on colloidal silica and alumina NPs showed the applicability of FCS to small molecule-NP systems. 54 Finally, if the ligand and the NP of interest are not intrinsically uorescent, a competition assay can be applied to characterize NP adsorption. This method is based on exposing the NP loaded with a surface-adsorbed uorescent molecule to a solution of the non-uorescent ligand. The release of the uorescent probe from the NP surface is then monitored to determine the concentration of the non-uorescent ligand that is adsorbed on the NP surface. The approach was demonstrated using uorescently labeled oligonucleotides bound to gold NPs through an alkanethiol linker and mercaptoethanol as a displacing agent. 55 Qualitative analysis of binding. Qualitative analysis of uorescence spectra was utilized to examine surface characteristics of ligand-NP systems such as charge density, 56 functional group density, dielectric constants, 57 and interfacial pH. 58 Despite the very high sensitivity of uorescence-based techniques and their ability to provide in situ measurements allowing for characterization of dynamic adsorption processes, the inherent challenges (among them the requirement for the uorescently active ligands and a complicated experimental setup) dramatically limited the extent of studies of small molecules-NP binding thus far.

Vibrational spectroscopy
Interpretation of changes in the vibrational spectra of molecules upon adsorption on a surface was shown to be a highly versatile and convenient approach to yield experimental data necessary for qualitative and quantitative analysis of surface sorption. The two primary vibrational spectroscopy techniques are Fourier-Transform Infrared (FTIR) and Raman spectroscopies. The relative simplicity of sample preparation and data collection resulted in an extensive utilization of these methods for investigating sorption equilibria.

Fourier-transform infrared (FTIR) spectroscopy
The use of FTIR to characterize ligand-NP interactions relies on the ability of IR spectroscopy to detect the establishment of new functionalities in the sample (i.e. a ligand-NP bond) or the spectral perturbation of an IR-active functional group within the ligand or NP upon binding. [11][12][13] Although the high complexity of FTIR spectra has limited their application to the characterization of small molecule-NP interactions, 59-66 the more recent introduction of Attenuated Total Reectance Fourier Transform Infrared (ATR-FTIR) 13,67-75 and Surface-Enhanced Infrared Absorption (SEIRA) 13,76-79 has expanded the use of IR-based methods to ligand-NP systems of higher complexity.
Quantitative analysis of binding. Qualitative assessment of capping layers and functionalization at the NP surface is the primary outcome of IR-based techniques. Nonetheless, a carefully designed experiment that includes the necessary standard calibrations could provide a quantitative description of surface coverage as well as the thermodynamic and kinetic parameters of sorption. 67 In the most common quantitative application, the time-dependent intensity change in the ATR-FTIR peaks corresponding to the adsorbed ligand is used to asses timedependent changes in concentrations of the bound species and obtain the adsorption and desorption rate constants. This approach works best for slow adsorption processes that reach equilibrium at a slow enough rate to allow for data collection at multiple time points (typically minutes to hours), and was successfully applied to investigate the binding of small molecules on TiO 2 , 80-87 iron (oxyhydr)oxides, 88 hematite, 89,90 Pt/ Al 2 O 3 , 91 and CeO 2 NPs. 92 In some of these studies, the adsorption and kinetic data were used to derive the Langmuir equilibrium constant using eqn (8). [83][84][85]92 Qualitative analysis of binding. Two-dimensional infrared spectroscopy (2D-IR) 93,94 provides a high-resolution picture of the IR-active functionality in the investigated sample and has been utilized in several qualitative investigations of surface adsorption. For example, 2D-IR spectroscopy allowed the characterization of the structure and mobility of the capping layer comprised of amide and carboxylate-containing ligands on gold NPs. 95 In another application, Bian et al. demonstrated the ability of multiple-mode multi-dimensional IR to obtain a complete picture of the conformations sampled by 4-mercaptophenol bound to gold NPs. 96 2D-IR was also employed to study the conformations sampled by the tripeptide glutathione (used a capping layer) on the surface of silver NPs 97,98 and their dependency on the NP size. 97 In another example, 2D-IR was used to detect the Pt-H bond formed upon H 2 activation on Pt 99 and the aggregation of a complex organometallic dye on the surface of TiO 2 NPs. 100

Raman spectroscopy
One of the main advantages of Raman over IR spectroscopy for the analysis of sorption equilibria is its ability to perform measurements in aqueous environments since water does not interfere with Raman spectra. However, its lower sensitivity has hampered a broader application of the technique. The more recently introduced Surface-Enhanced Raman Scattering (SERS) 101-103 overcomes the low sensitivity limitation of conventional Raman spectroscopy and, under certain circumstances, could even provide single-molecule resolution. [104][105][106][107][108] The number of reported studies that employ SERS to examine surface processes has been rising in the last few years, and we expect the utilization of this technique to study NP adsorption will be much broader in the near future.
Quantitative analysis of binding. There is no standard methodology for the utilization of Raman-based techniques to quantify sorption processes on NPs. However, several recent illustrations include carefully designed experimental protocols capable of providing quantitative data for the chemisorption of small molecules. Similar to the ATR-FTIR technique mentioned above, change in intensity for well-observed SERS bands of ligands interacting with NP can be plotted as a function of time and t to a Langmuir adsorption model to result k obs for the interaction, as was illustrated by adsorption of 4-aminothiophenol and 2-thio-5-nitrobenzoic acid on gold NPs. 27 One of the most comprehensive examples of SERS application to the ligand-NP system was illustrated by a very detailed investigation of the inuence of surface curvature of nanotextured gold NPs on the binding of thiolated ligands. 109 In this study, a wide range of structural, kinetic, and thermodynamic data were obtained using plasmon-enhanced Raman scattering. A robust methodology for determining ligand packing density and concentrations of bound and free substrates by SERS relies on utilizing isotopically labeled versions of the ligands as internal standards. Under this protocol, SERS can deliver the equilibrium constant by tting the experimental data with the Langmuir adsorption model. 110,111 However, the requirement for the labeled substrates has signicantly limited the broader application of this method. A newly introduced approach called "Hot Spot"-Normalized Surface-Enhanced Raman Scattering (HSNSERS) not only eliminates the need for additional internal standards by utilizing surface-enhanced elastic scattering for calibration purposes but also allows for real-time observation of ligand exchange process in situ. 112 Application of the HSNSERS technique was initially demonstrated by examining the interaction between several chloroanilines and gold NPs. 112 A few additional studies expanded the utilization of this methodology. 113,114 Qualitative analysis of binding. SERS has found widespread application in qualitative analysis of ligand-NP interactions, providing important information on the binding mode of ligands on the NP surface. 45,46,48,111, However, this technique nearly always has to be accompanied by other supporting methods, either theoretical (i.e. DFT, surface selection rules) [127][128][129][130][131] or experimental (i.e. XPS, FTIR, UV/Vis), 118,127 to aid in the interpretation of the results.

Isothermal titration calorimetry (ITC)
Isothermal Titration Calorimetry (ITC) is a versatile technique for examining thermodynamic and, in some cases, kinetic parameters on molecular interactions. [138][139][140] The ability of ITC to directly measure DH of binding makes it stand out from the rest of the available methods. ITC data collection is performed by measuring the heat adsorbed or released when small aliquots of ligand solution are injected into a sample containing the NP receptor, and binding takes place. Fitting the titration curve (heat over ligand/receptor molar ratio) delivers a complete set of thermodynamic (DH, DS, and DG) and stoichiometric parameters of binding.
Quantitative analysis of binding. ITC has been extensively utilized to study biomacromolecular interactions 141 and, more recently, extended to the examination of other types of ligand-receptor systems. In particular, ITC was used to quantify the affinity of ligands towards the NP surface and guide the rational design of NP receptors with regulated binding strength towards specic targets. 142,143 In the drug-delivery eld, ITC derived data are used to quantify the loading of active pharmaceutical ingredients on NP delivery vehicles. 144 In addition, ITC was reportedly used to characterize the thermodynamics of a few other ligand-NP systems. 52,145,146 Qualitative analysis of binding. Although quantitative analysis of binding is seen as the main strength of the ITC technique, a quick qualitative assessment of relative affinity trends could be performed on a series of compounds without the need for theoretical modelling of the experimental data. For example, using ITC data recorded for several DNA bases and their peptide analogs in the presence of gold NPs, Gourishankar et al. determined relative binding affinities of the ligands toward gold NPs. 147 Despite the ability of ITC to yield near-complete thermodynamic characterization of binding, its broader application to the small molecule-NP system has been hampered due to experimental challenges and complexities involved in data processing. Among these, relatively long experimental times, interference from the aggregation of the investigated substrates and NPs, signicant complications in assessing very weak and very strong bindings (K < 10 3 M À1 and K > 10 8 M À1 , respectively), and the requirement to account for every contribution to heat in the studied system are generally considered the principal challenges of the ITC technique.
Solution NMR methods for characterizing sorption equilibria NMR signals originate from low-energy transitions that relax slowly compared to other spectroscopic methods (ms-s). Although this property dooms NMR to the rank of a poorlysensitive technique, it also provides NMR with several advantages over other analytical methods. Indeed, the low relaxation rate of its signals makes NMR a high-resolution method capable of returning different spectroscopic peaks for the different atoms composing the analyte under investigation. In addition, the fact that NMR excited states are long-lived allows extensive spectroscopic manipulations of the NMR signals that resulted in the creation of an unmatched portfolio of NMR experiments able to characterize dynamic processes occurring on timescales ranging from picoseconds to hours. This ability to probe with atomic-resolution dynamic interactions occurring on a large range of timescales makes NMR an ideal method for studying sorption equilibria that involve multiple adsorbed and intermediate states.
In this section, we review solution NMR methods for the investigation of surface adsorption. Before diving into specic NMR experiments, we will introduce the concept of chemical exchange as applied to solution NMR. Due to their large molecular size, NPs exhibit a large transverse relaxation rate (R 2 ). Because of the large R 2 , NMR signals of NPs are broadened out beyond the detection level of conventional solution NMR spectroscopy (discussed in detail below). For this reason, the study of surface adsorption by direct investigation of NPs themselves using solution NMR techniques is oen impossible, and solid-state NMR techniques are more oen applied to this end. 32,148,149 However, in the presence of a chemical exchange equilibrium between a free NMR-visible state and a NP-bound NMR-invisible state, several solution NMR experiments (reviewed below) can be used to imprint information on the structure and dynamics of NP surface-bound species into the spectrum of the free ligand. Therefore, the concept of chemical exchange is crucial for solution NMR studies of surface adsorption.

Chemical exchange in NMR
A process that involves a nucleus exchanging between different chemical environments is called a chemical exchange. In the context of this review, the binding of a small molecule to a NP surface is an example of chemical exchange. The chemical exchange induces changes in the NMR spectrum, and the appearance of the spectrum depends on the timescale in which the exchange occurs. [150][151][152][153] There are three main timescales associated with NMR when describing chemical exchange: slow, intermediate, and fast. Each exchange regime can be dened in terms of a chemical shi difference (Du), relaxation rates of the exchanging species, and exchange rate constant (k ex ). In this review, we will use a two-state exchange model to illustrate how exchange affects the NMR spectrum and will refer to this model to simulate data (Fig. 2).
Consider a two-state model where a molecule is in an exchange between state A and state B: The two states are assumed to have different chemical shis (i.e., Du s 0). k 1 and k À1 are the forward and reverse rst-order rate constants for the equilibrium (corresponding to k ads and k des in eqn (1), respectively), and k ex ¼ k 1 + k À1 . [150][151][152][153] For simplicity, we will rst consider a two-state model where the two exchanging species have similar longitudinal (R 1 ) and transverse (R 2 ) relaxation rates (R A 1 $ R B 1 and R A 2 $ R B 2 , respectively) such as in equilibria describing a conformational change. It is important to note that this is not the case for the small molecule-NP system, which will be discussed later. Fig. 2a illustrates the effect of the exchange on the NMR spectrum of a molecule containing a single NMR-active nucleus. In the fast exchange regime (Du ( k ex ), a single peak at the population-averaged chemical shi is observed. The observed relaxation rates will also be population-weighted: where R obs 1 and R obs 2 are the observed longitudinal and transverse relaxation rates, respectively (note that one single peak is detected in the fast exchange regime), and p A and p B are the fractional populations of states A and B, respectively. On the contrary, in the slow exchange regime (Du [ k ex ), two peaks are observed, one at the chemical shi of state A and the other at the chemical shi of state B, each relaxing at its own relaxation rates. When the conditions for fast or slow exchange do not apply (i.e. Du $ k ex ), the system is considered to be in the intermediate exchange regime, and the R obs 2 is enhanced by the exchange contribution, R ex : where R obs,A 1 and R obs,A 2 are the R obs 1 and R obs 2 , respectively, for the NMR peak of species A in the slow-intermediate exchange regime.
The appearance of the NMR spectrum is chiey dependent on R obs 2 , as the linewidth (peak width at half height measured in Hz) of each NMR signal (Dn 1/2 ) is given by: Therefore, the increase in R obs 2 caused by R ex results in broadening of the detected signal and consequential reduction in intensity (note that the integral of the NMR signal is not affected by the exchange and, therefore, a larger linewidth is associated with a lower peak height). As shown in Fig. 2a, starting from the slow exchange regime and increasing k ex , the two peaks will gradually broaden as a result of the exchange. When Du $ k ex the peaks will coalesce into one single NMR signal that is highly broadened by the chemical exchange. Further increase in k ex will shi the system to the fast exchange regime with a consequent increase in resolution and peak intensity.
We will now consider a two-state model where a molecule is in equilibrium between the free and NP-bound states. Since the two states under chemical exchange differ considerably in molecular size, the dependency of the NMR relaxation rates to simulate a sorption equilibrium in which a small molecule adsorbs on a small NP. In the left panel, the populations of desorbed (state A) and adsorbed (state B) species are considered equal (p A ¼ p B ¼ 0.5). In the right panel, the equilibrium is skewed toward state A (p A ¼ 0.99 and p B ¼ 0.01). In (c), R A 2 ¼ 10 s À1 and R B 2 ¼ 1000 s À1 to simulate a sorption equilibrium in which a small molecule adsorbs on a large NP. In the left panel, the populations of desorbed (state A) and adsorbed (state B) species are considered equal (p A ¼ p B ¼ 0.5). In the right panel, the equilibrium is skewed toward state A (p A ¼ 0.99 and p B ¼ 0.01). In all simulations, the chemical shift difference between states A and B (Du) is 600 rad s À1 . Note that the larger is R B 2 and the more apparent is the effect of the small molecule-NP interaction on the NMR spectra of the ligand. Therefore, the larger is the NP size and the higher is the sensitivity of solution NMR in detecting and characterizing sorption. Nanoscale Advances from the molecular rotational tumbling needs to be taken into account when discussing the effect of the exchange on the measured NMR spectra. As shown in Fig. 3, the R 1 reaches a maximum when the rate of rotational tumbling (1/s C , where s C is the rotational correlation time) is on the same order as the Larmor frequency of the observed nucleus. Although this condition depends on the nucleus under investigation (i.e. 1 H and 13 C have different Larmor frequencies) and on the static eld of the spectrometer, the fact that R 1 is small for very small and very large molecules makes R A 1 $ R B 1 in the majority of the small molecule-NP systems (Fig. 3). On the other hand, the R 2 always increases by increasing s C (Fig. 3), and R A 2 ( R B 2 for small molecule-NP sorption equilibria. The large R B 2 oen results in line broadening beyond detection level and, as a result, the NPbound state becomes invisible by NMR (see Fig. 2b and c). However, despite its invisibility, the surface-bound state can affect the relaxation rate and chemical shi of the NMR-visible free molecule via chemical exchange (Fig. 2b and c), and a number of NMR experiments were developed to use the perturbation of the NMR signals of the NMR-visible state to obtain information on the NP-bound species. Fig. 2b and c show how the chemical exchange between two states with a large difference in R 2 affects the NMR spectra at different timescales. In the fast exchange regime, a single peak is observed with population-weighted chemical shi and relaxation rates (eqn (10) and (11)). However, since the transverse relaxation data measured in the presence of NP are oen analyzed in comparison to the R 2 of the free state, the following equation is more commonly used: where the lifetime line broadening (R llb ) describes the increase in the observed transverse relaxation of state A caused by the exchange with a state with a higher R 2 . In the intermediate exchange regime, eqn (12) and (13) still apply for R 1 . On the other hand, the R 2 of the NMR-visible peak is given by: It is important to note that while R llb is caused by the exchange between states with a large difference in R 2 , R ex is due to the exchange between states with different chemical shis. Therefore, R llb occurs even for exchange processes with Du ¼ 0.
In the slow exchange regime, when the transverse relaxation rate of the NMR-invisible state is faster than dissociation from the bound to the free state (i.e. R B 2 > k À1 ), the observed R 2 is increased by the apparent rate of the forward reaction (i.e. This is because any binding event results in an irreversible loss of magnetization as the signal is completely relaxed before the small molecule returns to the NMR-visible free state. When this limiting condition does not apply, R llb < k 1 (see below).
In summary, the theoretical considerations discussed in this section indicate that although NP-bound species are not observed directly by solution NMR, the contrast between the fast R 2 of the bound state and the slow R 2 of the free state can be utilized to imprint information about the NP-bound state onto the spectrum of the NMR-visible free state. Given that the R 2 of the visible state is affected differently depending on the timescale of the free-bound equilibrium, the types of solution NMR methods one can use to characterize NP adsorption are also dependent on the timescale in which the exchange occurs. In the following sections, we discuss solution NMR methods for characterizing small molecule-NP interactions at different exchange regimes.

Preparation of stable NMR samples containing nanoparticles
Solution NMR methods to investigate sorption equilibria rely on the direct observation of the ligand-NP adduct (in the case the investigated complex is small enough to be directly detected in a NMR experiment) or on the indirect observation of the NMRinvisible ligand-NP interaction on the spectra of the NMRvisible free ligand (see above). In either case, to obtain accurate and reproducible data, it is important that the ligand and the NP remain homogeneously suspended within the sample throughout the experiment. For soluble NPs, no special procedure is required to prepare NMR samples since the ligand-tosurface ratio will stay consistent throughout the sample and will not change over time. On the contrary, insoluble NPs quickly sediment in solution, changing the concentration of ligand-NP adduct within the NMR active volume over time. Therefore, establishing a generally applicable sample preparation method that results in homogeneously dispersed stable suspensions of insoluble NPs was a crucial step toward increasing the applicability of solution NMR methods to the characterization of sorption equilibria. The method consists in preparing the NMR samples in a gel matrix that prevents NP Fig. 3 Simulation of 13 C relaxation rates as a function of correlation time at 800 MHz. Simulations were performed for a C-H spin system of fixed bond length (1.09Å) using eqn (51) and (52). The longitudinal relaxation rate (R 1 ) increases as s C increases, reaches a maximum when the rate of rotational tumbling matches that of Larmor frequency (i.e. when s C ¼ 1/n Larmor ), and decreases as the tumbling rate becomes slower than Larmor frequency (blue curve). The transverse relaxation rate (R 2 ) increases as s C increases (red curve). Transparent gray boxes highlight the ranges of s C expected for small molecules (left) or proteins and NPs (right). sedimentation. 155,156 Such gel matrix should (i) undergo rapid gelation (which allows stabilization of a homogeneous suspension of NPs), (ii) not interfere with the ligand-NP interaction of interest, and (iii) have a minimal ngerprint in the NMR spectrum used for characterizing the ligand-NP system. A library of small molecule gelators that satisfy these requirements was recently developed to stabilize NP suspensions in a number of NMR-friendly aqueous 155 and organic solvents. 156 When discussing sample preparation protocols, it is also important to mention that indirect detection of the NMRinvisible ligand-NP adduct on the spectra of the NMR-visible free ligand is usually performed on samples in which the sorption equilibrium is highly skewed toward the free state (p A $ 90%). Indeed, the large R 2 of the NP-bound state results in line broadening of the NMR signals of the free state even when p B # 1% (note that the exact detection limit depends on R B 2 , with higher R B 2 's resulting in higher sensitivity), and a p B > 10% oen results in a complete loss of signal (Fig. 2b and c).

Analysis of chemical shi changes (CSCs)
The NMR chemical shi is very sensitive to the local chemical environment of the nucleus. Therefore, chemical shis are very sensitive reporters of intermolecular interactions. In a conventional CSC experiment, the NMR-visible interaction partner (i.e. the ligand) is kept at a constant concentration, while the NMRinvisible component of the interaction (i.e. the NP) is titrated into the solution, and the chemical shi change is monitored as a function of the titrant concentration. Fig. 4 illustrates how the NMR spectrum of a small molecule ligand changes upon the titration experiment within the fast, intermediate, and slow exchange regimes. To construct the spectra in Fig. 4, a simple two-state binding model was assumed. Analysis of this gure reveals that no chemical shi change is detected for processes occurring on the slow exchange regime since separate peaks are observed for the free and bound states with changes in peak intensities (discussed below) as the NP is titrated into the ligand (Fig. 4e). At the slow-intermediate exchange regime, a low population of the bound state (#20%) results in a small CSC, while a high population of the bound state ($50%) causes extreme line broadening due to R ex . These effects make monitoring the changes in chemical shi challenging for systems with binding kinetics in the slow-intermediate regime (Fig. 4bd). Analysis of chemical shi change works best for weak binding interactions that occur on a fast timescale. In the latter case, the observed chemical shi (d obs ) is a weighted average of the free and bound states chemical shis (d A and d B , respectively): The CSC is measured in reference to the chemical shi of the free state, as described by the following equation: Given that p A + p B ¼ 1, a rearrangement of eqn (19) gives: Although the ability of the CSC analysis to return the concentration of bound ligand makes these experiments a potential input for the Langmuir isotherm modelling (e.g., p B $ q in eqn (3a)), CSC experiments are associated with several practical problems that hamper their widespread use in quantitative studies of sorption. Indeed, as p B approaches 1, the NMR signal may become invisible due to the large R 2 of the bound state. Moreover, since most NPs are sparingly soluble, obtaining near saturation of binding (required for accurate modelling of the Langmuir isothermsee above) can be challenging, especially for the weak ligand-NP interactions that can be analyzed quantitatively by CSC. As an alternative, one can titrate the ligand into a xed concentration of NP. In this case, the population of the bound state will decrease as the ligand is titrated in, eventually reaching close to 0 for concentrations of ligand much higher than K. However, it should be noted that the latter approach still suffers from the low sensitivity at low ligand concentrations (when most of the ligand is in the NMRinvisible bound state) and cannot be applied to ligands that oligomerize or precipitate at a concentration higher than K.
Despite all the challenges of CSC, analysis of the NP-induced changes in chemical shis can still be used to characterize the functionalities involved in the interaction and the binding mode. For example, Calzolai et al. determined the location of the interaction site between the ubiquitin and gold NP by analyzing CSCs. 157 They observed that the chemical shis of only specic residues of ubiquitin changed upon the addition of gold NPs, revealing that the interaction is specic. Similarly, the binding sites for C 60 fullerene in proteins were identied using CSC analysis. 158,159 Also, by monitoring the interaction between L-lysine and water-soluble ruthenium NPs by CSC analysis, Martinez-Prieto et al. revealed that the binding orientation of Llysine on the NP surface is relevant to the deuteration reaction catalyzed by the NP. 160 In some cases, the study of surface adsorption on a metal surface using CSC can be hindered by the extreme broadening and peak shi caused by the Knight-shi effect. 161,162 One way to overcome this disadvantage is to have a spacer atom between the observed nucleus of the adsorbed species and the metal surface. For example, using oxygen as the spacer atom, Tedsree et al. investigated the chemisorption of formic acid on various metal colloid catalysts and found that the 13 C chemical shi values of the adsorbates were directly correlated to the chemisorption strength. 163

Peak intensity
The NMR signal intensity (intended as the integral of the NMR peak) is related to the number of nuclei resonating at a given chemical shi. Therefore, the intensity of an NMR signal can be used to determine the concentration of a species in the analyzed sample. If a ligand undergoes chemical exchange between the free and NP-bound states on the slow timescale, separate peaks are observed for the adsorbed and desorbed species with intensities that are proportional to the populations of each state ( Fig. 2 and 4). Due to its large R 2 , the signal of the bound state is line broadened and oen invisible in the NMR spectrum. However, the signal for the desorbed ligand is NMR-visible and with an intensity that is proportional to its population in solution. Therefore, in the slow exchange limit, if a sample of a ligand is titrated with known amounts of NP, the decrease in signal intensity can be used to experimentally quantify C L and q LS (Fig. 4) and model the interaction with the Langmuir theory (eqn (3a)). For example, Wang et al. determined the binding capacity of various proteins to gold NPs by monitoring the decrease in the intensity of the free protein signal at different NP concentrations. 164 In another example, Wang et al. showed that by using a 1D half-lter experiment, one could investigate the competitive binding of multiple proteins, each labeled with different isotopes, to gold NPs. 165 Xu et al. used a similar approach to investigate the competitive adsorption of two different proteins, GB3 and ubiquitin, and two GB3 variants differing by only one residue to gold NPs. 166 In this study, the authors developed an external referencing system using 15 N labeled tryptophan to facilitate the accurate conversion of signal intensity into the concentration of free ligand.
When using the change in signal intensity of the desorbed state in quantitative analysis of sorption equilibria, it is advisable to measure signal intensities by peak integration algorithms rather than by peak height. Indeed, when a ligand is in equilibrium between free and NP-bound state in the slow exchange regime (Du [ k ex ), if R B 2 > k À1 , the R 2 of the desorbed state is increased by k 1 (eqn (17)) with a consequential reduction of peak height due to line broadening. In addition, in the more common case of intermediate-slow exchange between desorbed and adsorbed species (Du > k ex ), additional line broadening is caused by the exchange contribution to R 2 . As R ex is maximum when the exchanging species are equally populated (i.e. p B $ 0.5) and is minimum at p B $ 0 or 1, the presence of an intermediate-slow exchange results in additional modulations of peak height that are not linearly dependent on the concentration of desorbed and adsorbed species (Fig. 4).

Diffusion ordered spectroscopy (DOSY)
DOSY is a useful NMR technique to measure the diffusion coefficients (D) of molecules. 167 The most basic DOSY pulse sequence is shown in Fig. 5. 168 A 90 pulse is applied to excite the nuclear spins, followed by the rst gradient, which labels the spins with a phase change that depends on their positions in the NMR tube. A 180 pulse is then applied to invert the sign of the phase change, and a second gradient is applied to refocus the signal. When a sample molecule diffuses during the delay period (shown as D in Fig. 5), its NMR signals will not be completely refocused by the second gradient, resulting in reduced NMR signal intensity. Therefore, free small molecules that diffuse fast in the sample will experience greater signal attenuation than the NP-bound molecules that diffuse slowly due to the large size of the NP.
The intensity change from DOSY experiments can be quantitatively described by the Stejskal-Tanner equation: 169 where I is the observed intensity, I 0 is the initial intensity, D is the diffusion coefficient, g is the gyromagnetic ratio, G is the gradient strength, d is the gradient duration, and D is the delay period. To obtain the value of D, a series of measurements are taken by varying either the gradient strength, the gradient duration, or the delay period and tting the intensity decay using eqn (21). 2D DOSY spectra are generated by plotting chemical shis on one axis and diffusion constants on the other. 170 Each line in the NMR spectrum will give one or more lines in the diffusion dimension at the values corresponding to the diffusion coefficient of the associated species. Therefore, DOSY makes a powerful tool for analyzing mixtures containing more than one ligand-NP species. Moreover, for a spherical molecule, the diffusion coefficient can be used to determine the molecular size using the Stokes-Einstein equation: where k B is the Boltzmann constant, T is the temperature, h is the viscosity of the solution, and r H is the hydrodynamic radius of a molecule. Therefore, an estimate of the NP size can be obtained from the value of D obtained for the bound species. One very important aspect to notice is that the exchange regimes of a DOSY experiment are dened by the diffusion time (D). Considering a simple two-state exchange model where a molecule is in an exchange between states A and B (eqn (9)), the fast exchange regime in the diffusion timescale is dened when D [ 1/k ex (i.e. several exchange events between states A and B occur during D), and the slow exchange regime in the diffusion timescale is dened when D ( 1/k ex (i.e. very few exchange events between states A and B occur during D). 171-173 Therefore, by varying D, one can observe the system in either fast or slow exchange in the diffusion timescale. If the exchange is fast in the diffusion timescale, a single population weighted diffusion coefficient will be observed: where D obs is the observed diffusion coefficient, and p n and D n are the fractional population and diffusion coefficient of state n, respectively. If the exchange is slow in the diffusion timescale, two different diffusion coefficients will be observed, resulting in a bi-exponential decay of signal intensity as a function of the varying parameter in the DOSY experiment (G, d, or D in eqn (21)see above). Given that D denes the exchange regime in diffusion experiments, DOSY can be applied to exchange systems occurring on a wide range of timescales. However, if D ( 1/k ex and Du [ k ex (i.e. the system is in slow exchange in both the diffusion and chemical shi timescales), no change in D will be observed by tting the intensity of the NMR-visible signal of the free ligand with eqn (21). This limitation, together with the fact that D cannot be larger than 1/R 2 (to avoid excessive intensity loss during the experiment), hampers the application of DOSY to ligand-NP systems for which R 2 [ k ex .
DOSY experiments have been used to characterize several ligand-NP systems. For example, Hens et al. investigated trioctylphosphine oxide (TOPO) capped colloidal InP nanocrystals using solution NMR. 174 From the diffusion coefficient obtained from DOSY measurements and comparing it to the nanocrystal diameter, the authors determined that the broad resonance superimposed on the free TOPO signal was coming from TOPO molecules adsorbed at colloidal nanocrystals. Similarly, Moreels et al. used DOSY measurements to identify the ligands that are tightly bound to the surface of the PbSe nanocrystals. 175 In another example, using DOSY, Hassinen et al. determined that octylamines are dynamic ligands for CdSe quantum dots and that free and bound octylamines exhibit fast exchange. 176

Saturation transfer methods
When a small molecule is in an exchange between the free and NP-bound states, the large molecular size of the NP increases the R 2 of the bound small molecule. The increase in R 2 is oen large enough to cause broadening of the NMR signal beyond detection level. Saturation transfer experiments achieve indirect detection of the bound state on the spectra of the NMR-visible desorbed state by selective saturation of the NP or the bound state itself. The saturation is then transferred to the visible state via chemical exchange. This section will discuss the two main saturation transfer experiments applied to ligand-NP systems: saturation transfer difference (STD) and dark-state saturation transfer (DEST).
Saturation transfer difference (STD) NMR. The STD NMR method has been widely used to study ligand-protein interactions. [177][178][179][180] The standard pulse sequence for 1D 1 H STD NMR is shown in Fig. 6a. It involves a train of Gaussian-shaped pulses to achieve the saturation of selected resonances before applying a hard 90 reading pulse. The STD NMR experiment requires taking the difference between an on-resonance spectrum obtained by selectively saturating the 1 H signals of the NP and a reference off-resonance spectrum in which the saturation eld is moved to a region that does not contain peaks of the NP or the ligand. In the on-resonance experiment, the saturation applied to selective NP resonances is quickly transferred to the other 1 H spin on the NP as well as to protons of any ligand bound to the NP surface via spin diffusion (which is highly efficient among 1 H spins in high molecular weight systems such as proteins and NPs). Therefore, when bound ligands dissociate into the solution, they cause a decrease in the signal intensity of the free state due to the partial transfer of saturation from the NP-bound state. As a result, when the on-resonance spectrum is subtracted from the off-resonance reference, only signals from ligands that bind to the NP will persist (Fig. 6b).  168 The black narrow and wide rectangular-shaped pulses represent 90 and 180 pulses, respectively. The white rectangles represent gradient pulses. D is the delay period (i.e. time of diffusion), d is the gradient duration, and G 1 is the gradient strength.
Quantitative interpretation of the STD experiment can be obtained by analyzing the magnitude of the STD effect measured as: where I 0 and I sat are the signal intensities in the off-and onresonance spectrum, respectively. The STD effect is then normalized by multiplying it by the ligand excess, yielding the STD amplication factor. 178

STD amplification factor
where [L] and [R] are the concentrations of ligand and receptor, respectively. A titration curve of the STD amplication factor as a function of the ligand concentration can be t using a binding isotherm to determine the equilibrium constant for the binding process. 178 STD NMR protocols and theoretical modeling have been extensively developed and tested for screening projects aimed at identifying ligands for protein drug targets. 177,[181][182][183] More recently, several STD NMR applications to the investigation of NP adsorption have appeared in the literature. For example, STD NMR has been used to screen libraries of small molecules against NP targets, 184,185 to determine the affinities of NP-ligand adducts, 184,186 and to investigate the forces driving small molecule adsorption. 187 It is important to note that when using the STD NMR for NPs, the concentration of the receptor (noted as [R] in eqn (25)) should be the total concentration of binding sites. As discussed above, when introducing the Langmuir model, this quantity is not easily measurable for NP receptors and introduces ambiguities in the interpretation of the experimental data. To overcome this issue, Zhang et al. used a constant proportional to the surface area of the NP in place of the concentration of the binding sites to calculate the relative STD amplication factor. 184 STD NMR can also give structural insight into the adsorption process. For example, by analyzing the pattern of STD amplication factors measured for the 1 H nuclei of rhodamine B, it was observed that the small molecule binds to polystyrene NPs via its benzoic acid ring. 186 Despite the successful applications described above, a number of factors limit the widespread use of STD NMR in the characterization of sorption equilibria. Indeed, while STD NMR is effective for k À1 > R 1 (note that a dissociation rate slower than the longitudinal relaxation would cause the ligand to relax back to equilibrium before returning into the NMR-visible free state), a very large k À1 may result in a population of bound state too low to be detected by the experiment. 179 Another limitation of the STD NMR experiment is that the NP must be protonated in order to permit efficient saturation transfer via 1 H spin diffusion. Such condition hampers the applicability of STD NMR to the analysis of adsorption on metal or metal oxide NPs.
Dark-state exchange saturation transfer (DEST). The DEST experiment was originally developed to obtain thermodynamic, kinetic, and structural information on the interaction between small, NMR-visible proteins and large aggregates. 154,[188][189][190] Later, the application of DEST was expanded to the investigation of ligand-NP interactions. 191,192 The DEST experiment takes advantage of the fact that the NMR resonances of the NP-bound state can be saturated selectively by applying a weak radiofrequency (RF) eld far off-resonance from the resonances of the free state. Indeed, although broadened out beyond detection level, the bound state resonances are still present within the NMR spectrum. Given their large linewidth, the bound state peaks can be selectively saturated by an RF eld positioned at a large offset compared to the resonances of the NMR-visible state. The saturation is then transferred from the bound state to the free state via chemical exchange and observed as a reduction in the signal intensity of free species (Fig. 7). In a typical setup, the DEST experiment is repeated at several RF eld offsets to obtain a DEST prole reporting the signal intensity of the NMR-visible state as a function of the offset. The presence of an NMR-invisible bound state in chemical exchange with the NMR-visible free state will be detected as a broadening of the DEST prole compared to the one measured in the absence of NP (Fig. 7). The width of the DEST prole depends on the R 2 of the bound state and the strength of the applied RF eld, as shown in Fig. 8. The higher the R 2 of the bound state and the stronger the RF eld, the broader the DEST prole will be.
The DEST prole can be modelled using the Bloch-McConnell equation to obtain populations of free and bound states Fig. 6 (a) Standard pulse sequence for 1D STD NMR. The saturation is achieved by applying a train of n Gaussian-shaped pulses. After the saturation period, a 90 excitation pulse (represented by the rectangular-shaped pulse) is applied. d 1 identifies the recycling delay. (b) Schematic representation of the STD NMR protocol. Selective saturation is applied to a receptor proton signal. The saturation is transferred to the entire receptor and to any NMR-invisible attached ligand via spin diffusion. When the ligand dissociates from the receptor, it carries the saturation in the NMR-visible state, reducing the intensity of the observed NMR signal. Therefore, when the saturated spectrum is subtracted from a reference spectrum (acquired in the absence of saturation), only residual signals originating from small molecules that bind to the receptor are visible. and exchange rates. 189 For a two-site exchange system, the signal intensities for different RF offsets can be simulated by solving the following differential equation: where M is the matrix representing the magnetization of each state in chemical exchange, E is the matrix representing the equilibrium magnetization, R is the matrix representing the relaxation rates of each state, K is the matrix representing the rate constants for the exchange kinetics, U is the matrix representing the saturation offset, and S is the matrix representing the saturation eld strength. 193 The matrices for each term are shown below: Fig. 7 (a) In the DEST experiment, the NMR-invisible surface-bound species is selectively saturated by an off-resonance RF field. The saturation is then transferred to the visible free species via chemical exchange, resulting in reduced signal intensity. (b) Simulated NMR signals for a free small molecule with R 2 ¼ 10 s À1 (black line) and for a NP-bound small molecule with R 2 ¼ 100 (blue line) or 1000 s À1 (red line). The broad resonance of the NP-bound state can be selectively saturated by an off-resonance RF. (c) A DEST profile is constructed by plotting the intensity of the NMR-visible peak as a function of the RF offset. The black line is the DEST profile simulated for a small molecule with R 2 ¼ 10 s À1 in the absence of exchange. The blue and red lines are the DEST profiles simulated in the presence of exchange with an NPbound state with R 2 ¼ 100 and 1000 s À1 , respectively. In all simulations, u rf ¼ 150 Hz, p A ¼ 0.95, and k ex ¼ 600 s À1 . Note that, in the absence of exchange, a sharp DEST profile is obtained since saturation of the visible signal is achieved at small RF offsets only. In the presence of exchange with a species with large R 2 , a broader DEST profile is obtained since saturation of the NP-bound state at large RF offsets is transferred to the NMR-visible state by chemical exchange. Fig. 8 Simulated DEST profiles for a two-site exchange equilibrium between the NMR-visible (p A ¼ 0.95 and R A 2 ¼ 10 s À1 ) and NMRinvisible (p B ¼ 0.05) states over a range of k ex values (0 to 10 000 s À1 , top to bottom). R B 2 was set to 100 (blue line) or 1000 s À1 (red line) to simulate adsorption to a small or large NP, respectively. The DEST curve simulated at k ex ¼ 0 s À1 (corresponding to the data obtained in the absence of NP) is colored black. The saturation field strengths are 150 and 500 Hz for the left and right panels, respectively. Note that the DEST experiment is most efficient in detecting ligand-NP adduct with large R B 2 that exchange slowly with the desorbed state (k ex ( R B 2 À R A 2 ). In the fast exchange limit, the shape of the DEST profile is no longer dependent from k ex .
S ¼ where E is unity, M n k represents the magnetization on the k-axis of state n, U n represents the difference between the resonance frequency of state n and the frequency of the applied saturation eld, and u k is the strength of the saturation eld applied to the k-axis.
When modelling a DEST prole, R A 1 , R A 2 , u x , and u y are usually provided as input parameters and not optimized (note that R A 1 and R A 2 can be measured experimentally on samples of the ligand prepared without adding the NP). Therefore, data modelling returns R B 1 , R B 2 , U A , U B , k 1 , and k À1 . The thermodynamics of the equilibrium can be calculated from the modelled parameters using the equations: The DEST experiment is an excellent method for detecting and characterizing large systems that are NMR-invisible due to their fast relaxation rates. Since the DEST experiment uses a relatively high saturation eld strength (larger than 100 Hz) to maximize the detection of the NMR-invisible state (Fig. 8), the difference in chemical shi between the states in chemical exchange is usually not resolved 154 (note that this is the main difference between DEST and CEST, 194 another saturation transfer NMR experiment that uses low saturation eld strengths to resolve chemical shi differences between chemical states with similar R 2 's). Therefore, when modelling DEST proles, U A and U B are oen considered to be equal in order to reduce the number of variable parameters. Also, it is important to highlight that if the R 2 of the bound state is smaller than the applied saturation eld, p B and R B 2 cannot be determined independently with the DEST experiment alone, and additional experimental data are needed to decorrelate the two parameters. 195 Alternatively, the two parameters can be decorrelated by calculating R B 2 based on the rotational correlation time of the NP (s NP ) that can be estimated by using the average size of the NP and the Stoke-Einstein equation.
DEST can characterize the exchange between a small molecule and a large receptor occurring on the slow timescale (ms-s) at the condition that k ex is faster than R 1 (which implies that the exchange from free to bound and back to the free state has to occur before the longitudinal magnetization is lost via R 1 relaxation). For completeness, it should be mentioned that DEST data can also be acquired for systems exchanging on a fast timescale (k ex > 1000 s À1 ). 154 However, as in a fast exchanging system, one single peak is observed with a population-averaged R obs 2 (eqn (11)), the R 2 of the bound state must be small to avoid the complete loss of the NMR signal.
Compared to STD NMR, the main limitation of DEST is the need to use heteronuclei for the saturation step, which limits its sensitivity and increases the experimental acquisition time. Indeed, when using 1 H DEST, the fast 1 H spin diffusion can cause unwanted intra and intermolecular magnetization transfers that complicate modelling of the experimental data. 189,192 To avoid this issue, several 2D and 1D proton detected pulse sequences were developed to perform 13 C or 15 N DEST experiments on proteins and small molecules. 190,192,196 The pulse sequence used for 1D 13 C DEST is shown in Fig. 9. Although less sensitive than STD, the DEST experiment holds the advantage of not requiring a transfer of saturation from the NP to the ligand via spin diffusion, which allows investigation of non-protonated receptors such as metal and metal oxides NPs.
In recent years, several applications of DEST to the investigation of ligand adsorption on NPs were reported. Ceccon et al. determined the kinetics and thermodynamics of adsorption of the huntingtin peptide over titanium(IV) oxide (TiO 2 ) NPs by simultaneous analysis of 15 N DEST and DR 2 (difference in R 2 of the free state in the absence and the presence of NPsee below) data. 197 Similarly, using 15 N DEST, DR 2 , and exchange-induced chemical shi, Hassan et al. characterized the dynamics and exchange kinetics of the interaction between ubiquitin and mercaptosuccinic acid-capped CdTe quantum dots. 198 The DEST experiment has also been used to investigate the structures of the surface-bound species. For example, we used 13 C DEST in combination with other NMR relaxation methods to characterize not only the kinetics and thermodynamics of binding but also the structure of phenol (PhOH) adsorbed on the surface of ceria (CeO 2 ) and Pd on ceria (Pd/CeO 2 ) NPs. 192 With the combination of 1 H DEST and molecular dynamics (MD) simulations, Xue et al. determined the adsorption mode of different amino acids on the surface of TiO 2 NPs. 199

NMR relaxation
NMR relaxation is one of the most prevalent methods to study macromolecular dynamics due to its extreme sensitivity to molecular motions. [200][201][202][203][204][205] Since the overall tumbling and internal motions of a molecule are oen signicantly perturbed by the surface interaction (e.g., decrease in molecular mobility of the ligand upon binding to NP surface), NMR relaxation makes an ideal method for detecting and characterizing adsorption processes in solution. This section will discuss how relaxation-based NMR methods have been applied to small molecule-NP systems. We will focus our attention on the R 1 , R 2 , and relaxation dispersion experiments.
Transverse relaxation (R 2 ). As discussed above, the transverse relaxation rate of a ligand is enhanced by chemical exchange with an NMR-invisible NP-associated state. Since the increase in R 2 of the free ligand reects the properties of the bound state (e.g., population, binding kinetics, and internal exibility), the measured transverse relaxation can be used to characterize the ligand-NP interaction. In particular, the relevant observable is the increase in R 2 upon addition of NP (DR 2 ) measured as: where R NP 2 and R noNP 2 are the R 2 values measured for the free ligand NMR signals in the presence and in the absence of the Fig. 9 (a) Pulse sequences used to measure proton-detected 1D 13 C DEST, CPMG, and R 1 experiments. Each experiment is measured by inserting the blocks shown in (b)-(d) into the dashed box shown in the double refocused-INEPT based sequence. Full details on the pulse sequences are available in Egner et al. 192 In the DEST experiment, CW represents the continuous wave used for saturation. During the saturation period, a train of 180 pulse (represented by the thicker rectangular-shaped pulse) is applied in the 1 H channel to remove relaxation artifacts coming from the 1 H-13 C dipolar interaction. In the CPMG experiment, s CP is a variable parameter that determines the CPMG field (n CPMG ¼ 1/4s CP Hz). The CW applied in the CPMG experiment is for 1 H decoupling. The phase cycling employed is: f 1 ¼ (x, Àx); f 2 ¼ 2(x), 2(Àx); f 5 ¼ 8(x), 8(Àx); f 6 ¼ 4(y), 4(Ày); f 8 ¼ (y, Ày); f 10 ¼ 16(x), 16(Àx); f rec ¼ 2(x), 4(y), 2(x), 2(y), 4(x), 2(y). The duration and strength of the gradients are as follows: G 1 , 1 ms, 30 G cm À1 ; G 2 , 0.05 ms, 35 G cm À1 ; G 3 , 0.1 ms, 40 G cm À1 .

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Nanoscale Advances NP, respectively. In the fast exchange regime, DR 2 ¼ R llb (eqn (15)) and can be linked to the fractional populations of desorbed and adsorbed species and to their transverse relaxation rates using the formula: Therefore, R B 2 and the concentrations of free and bound ligand (C L and q LS in eqn (3)) can be obtained by modelling DR 2 data with eqn (36). As discussed above, in the slow exchange (17)). However, if the transverse relaxation rate of the NP-bound state is not sufficiently fast to destroy all magnetization (i.e. if R B 2 < k À1 ), then the enhancement to the transverse relaxation rate will be less than k 1 . In the case p A [ p B (which is very common in solution NMR investigation of sorptionsee above), it can be shown that: 206 which is valid for both the fast and slow timescale regimes. However, it is important to note that in the most common scenario of the intermediate exchange regime, the experimental DR 2 is enhanced by R ex (eqn (12) and (13) 191 Using the distribution of 15 N DR 2 along the amino acid sequence, they showed that ubiquitin undergoes fast rigid-body rotation about an internal axis while bound to the surface of the NP. In addition, using paramagnetically labeled NPs, they used DR 2 to measure the dipolar interaction between the protein and the paramagnetic center, which allowed the determination of the ubiquitin residues in direct contact with the NP surface. Xie and Brüschweiler used 13 C DR 2 to investigate how ligand modications affect binding to silica NPs. 207 In particular, they used the fact that in the fast exchange regime p B $ DR 2 /R B 2 (eqn (36)) to obtain the free energy of adsorption for a variety of ligand-NP interactions. We have shown that, if predicted binding modes are available for a ligand-NP system, a standalone interpretation of DR 2 can be used to obtain a semi-quantitative analysis of sorption based on the use of scaling factors that depend on the amount of small molecule in each binding mode. 208 Using this approach, we determined that functionalization of PhOH with an oxygen at position 2 or 4 greatly enhances adsorption on the Pd or CeO 2 component of Pd/CeO 2 nanorods, respectively.
Carr-Purcell-Meiboom-Gill (CPMG) relaxation dispersion (RD). RD relies on measuring the transverse relaxation at increasing repetition rate of 180 refocusing pulses to progressively suppress the R ex contribution to the R 2 of a system in chemical exchange on the intermediate timescale (k ex $ Du s 0). 154 In CPMG RD, the CPMG eld is introduced as a series of refocusing 180 pulses spaced by a constant delay (2s CP ) (Fig. 9). As the CPMG eld is equal to 1/4s CP , an RD prole can be obtained experimentally by plotting the R 2 measured at different s CP values as a function of the CPMG eld. Fig. 10a shows an ideal RD prole simulated for the NMR-visible signal of a small molecule in an exchange between the desorbed and NPadsorbed state on the intermediate timescale. The R 2 of such signal will be enhanced compared to the one measured in the absence of NP by both R ex and R llb : Progressive suppression of R ex by increasing the CPMG eld introduces a curvature in the RD prole that plateaus when R ex $ 0 s À1 (note that only in this condition the experimental DR 2 ¼ R llb ). Fig. 10b shows the RD proles simulated for two-state sorption equilibria occurring on different timescales. If the exchange is fast or slow on the chemical shi timescale (k ex [ Du or k ex ( Du, respectively), R ex $ 0 s À1 and R obs 2 ¼ R A 2 + R llb for any CPMG eld. However, within these limits, the shape of the RD prole is very sensitive to the timescale of the equilibrium, and modelling of experimental RD datasets can provide information on the kinetics and thermodynamics of sorption. The R 2 at any specic value of s CP can be calculated using the formula: where M(0) and M(t 2 ) are the transverse magnetization calculated for a duration of the CPMG block of 0 and t 2 s, respectively. M(0) and M(t 2 ) are the rst elements of vectors M(0) and M(t 2 ), respectively: where M A and M B represent the transverse magnetizations of states A and B, respectively, n is the number of CPMG cycles used (see Fig. 9), A* the complex conjugate of A, and A is given by: R is given by: with Modelling an RD prole returns R B 2 , Du, k 1 , and k À1 (note that R A 2 can be measured experimentally on samples of the ligand prepared without adding the NP). p A and p B can be obtained from the modelled parameters using eqn (33) and (34).
Although CPMG RD is commonly applied to characterize chemical exchange with k ex values between 200 and 2000 s À1 , the method performs best for processes occurring on the intermediate to slow timescale. 154 Indeed, in the intermediate to fast regime, Du and p B cannot be accurately modelled by a single CPMG data set, and it is common to acquire RD data at multiple static elds and external temperatures to improve the accuracy of modelling. 154,209 Also, when possible, it is preferable to measure RD proles on 1 H spins. On conventional NMR spectrometers, the proton channel can sustain higher CPMG elds compared to channels dedicated to heteronuclei, which allows more efficient suppression of R ex for intermediate to fast exchange processes. Also, given the high gyromagnetic ratio, 1 H spins tend to have larger Du, which increases the ability of the RD method to detect and characterize exchange processes with small p B and large k ex . A pulse sequence for acquisition of 1D 1 H CPMG experiments on small molecule-NP systems is shown in Fig. 11.
Although CPMG RD has been one of the most popular methods to characterize protein conformational equilibria, [210][211][212][213][214][215][216] its standalone application to the characterization of ligand-NP systems has been quite limited. Recently, CPMG RD has been combined with DEST, DR 2 , and R 1 data to investigate sorption equilibria involving both protein and small molecule ligands. 155,192,217 Longitudinal relaxation (R 1 ). Compared to R 2 , the longitudinal relaxation is less sensitive to chemical exchange. Therefore, changes in R 1 upon adsorption are rarely analyzed. Nonetheless, as an estimate for R B 1 is obtained from the analysis of DEST (see above), measuring the R 1 of the NMR-visible ligand peaks in the absence (R noNP 1 ¼ R A 1 ) and in the presence (R NP 1 ) of NPs can provide additional experimental restraints for modelling sorption equilibria. R 1 can be obtained from inversion recovery experiments. 218,219 A pulse sequence for the 1D proton-detected 13 C R 1 experiment applied to small molecule-NP systems is shown in Fig. 9. In the presence of exchange between the free and NP-   11 PROJECT pulse sequence used for 1 H CPMG spectra of small molecules. 229 The narrow and wide rectangular-shaped pulses represent 90 and 180 pulses, respectively. The 90 pulse sandwiched between the refocusing CPMG pulses is used to suppress modulation from homonuclear scalar coupling. s CP is a variable parameter that determines the CPMG field (n CPMG ¼ 1/4s CP Hz). The phase cycling employed is: f 1 ¼ 2(x), 2(y), 2(Àx), 2(Ày); f 2 ¼ y, Ày, y, Ày, x, Àx, x, Àx; f rec ¼ 2(x), 2(Àx), 2(y), 2(Ày).
bound state, the experimental R NP 1 of the ligand can be modelled according to the following equation: where M(0) and M(t 2 ) are the detected magnetizations when the relaxation delay (2nD in Fig. 9d) is set to 0 and t 2 s, respectively. M(0) and M(t 2 ) are the rst elements of vectors M(0) and M(t 2 ), respectively: Here, M A and M B represent the initial magnetizations of states A and B, respectively, A* represents the complex conjugate of A, and A is given by: where R is given by the summation: Modelling the ligand-NP adduct. In the sections above, we have discussed how modelling DEST, RD, R 1 , and/or R 2 data measured for the NMR-visible signals of a free ligand in the presence of chemical exchange with a NP-bound state returns the thermodynamic (p A and p B ) and kinetic (k 1 and k À1 ) parameters of sorption. In addition, such analysis returns information on the structure and dynamics of the surface-bound species encoded in the modelled R B 1 and R B 2 values. Indeed, when the exchange contribution is completely suppressed, NMR relaxation is induced by the modulation of magnetic interactions caused by molecular motions occurring on the ps-ns timescale. In the case of spin-half nuclei in a diamagnetic system, the relevant magnetic interactions are dipolar couplings and chemical shi anisotropy (CSA).
For an isolated 1 H-13 C (or 1 H-15 N) bond vector, the 13 C (or 15 N) R 1 and R 2 are linked to the molecular dynamics using the model-free method developed by Lipari and Szabo: 220,221 Here, m 0 is the magnetic permeability, ħ is the reduced Planck's constant, g n is the gyromagnetic ratio of the n nucleus, r is the internuclear bond distance, and Ds is the chemical shi anisotropy. J(u) represents the spectral density function which describes how much local eld is oscillating at the given frequency: where S 2 is a generalized order parameter representing the spatial restriction of the internal motion, s LNP is the correlation time of the ligand in a rigid interaction with the NP, and s e is the effective correlation time of the C-H (or N-H) bond vector given by: 191 where s NP and s loc are the correlation time of the NP and of the local motion, respectively. In the model-free approach, if the motion of the C-H bond vector is restricted (i.e. the C-H bond is rigidly attached to the NP surface), S 2 $ 1 and the overall tumbling of the NP and binding kinetics are the only contribution to relaxation (i.e., s LNP is the only correlation time modulating the magnetic interactionseqn (58)). On the other hand, if the C-H bond vector is highly exible, S 2 $ 0 and relaxation is determined by both s LNP and s loc . Modelling 13 C and 15 N relaxation rates using the model-free approach was used to investigate the molecular dynamics of PhOH bound to CeO 2 and Pd/CeO 2 NPs 192,208 and of the protein ubiquitin adsorbed on liposome NPs. 191 However, it is important to notice that eqn (54)-(60) only apply to the spin-half heteronuclei covalently attached to single hydrogen (i.e. protein backbone amides, CH groups in aromatic rings, etc.), and more complex models are required for more complex systems.

Combining solution NMR experiments
Several solution NMR experiments have been described above that investigate chemical exchange processes occurring on different timescales. In particular, chemical shi analysis is very powerful for characterizing fast chemical exchange, CPMG RD is particularly suited for intermediate to slow exchange, while saturation transfer methods (STD and DEST) and the analysis of peak intensities work best for processes in the slow exchange regime. In addition to these methods, analysis of DOSY, DR 1 and DR 2 experiments can report on the chemical exchange between desorbed and NP-adsorbed states on a broad range of timescales (Fig. 12). Although using these methods as standalone techniques can provide important insight into sorption equilibria, the utility of combining two or more NMR experiments sensitive to different timescales for investigating surface adsorption was demonstrated in several applications. Indeed, adsorption on a heterogeneous surface is oen a complex process involving multiple intermediate states and adsorbed species in chemical exchange (among themselves and with the desorbed state) on different timescales, and the combined analysis of different NMR observables can help to deconvolute the contribution of each elementary step to surface adsorption. For example, Libich et al. investigated the interaction of the polypeptide amyloid b (Ab40) with the molecular chaperone GroEL by using a combination of 15 N DR 2 , 15 N DEST, 15 N CPMG RD, and exchange-induced chemical shi change analysis. 217 They showed that since these methods depend differently on the kinetics and thermodynamics of adsorption, their combined analysis increased the stability of data modelling and the precision of the tted parameters. In addition, by combining 15 N DR 2 , 13 C and 15 N DEST, and 1 H, 13 C, and 15 N CPMG RD, Libich et al. investigated the chaperonin activity of GroEL, demonstrating that the molecular machinery accelerates the interconversion between the native and partially folded states of a triple mutant of the Fyn SH3 domain. 222 Another series of studies explored the combination of 1 H R 1 and R 2 for investigating sorption in porous materials. 223 It was found that the 1 H R 2 /R 1 ratio of ligands in saturated porous media provides information on the residence time of the NMR-visible small molecule inside the pore and on the energy of adsorption. 224 In the following section, we highlight how the combination of 13 C R 1 , 13 C R 2 , 13 C DEST, and 1 H and 13 C CPMG RD was used to investigate small molecule-NP sorption with atomic resolution. In particular, using the interaction between PhOH and Pd/ CeO 2 nanocubes as the model system, we will illustrate the combined use of multiple NMR methods to obtain kinetic, thermodynamic, and structural information on adsorption/ desorption equilibria involving multiple elementary steps occurring on multiple timescales.
Application to the PhOH-Pd/CeO 2 system Pd/CeO 2 NPs catalyze the hydrogenation of PhOH under mild reaction conditions (1 bar H 2 at room temperature) more efficiently than other Pd-based catalysts. 225 However, it is unclear whether the outstanding performance of Pd/CeO 2 is due to a higher dispersion of the Pd catalyst, better adsorption of PhOH on the catalyst surface, benecial electronic effects of ceria on the supported metal, or a direct contribution of ceria in catalysis via H 2 spillover. To help address this question, in the last few years, we conducted a series of studies aimed at characterizing the adsorption of PhOH on Pd/CeO 2 NPs by solution NMR. 155,192,208,226 Initially, we focused on studying the binding of PhOH onto bare CeO 2 NPs of cubic morphology using 1 H DEST and 1 H CPMG RD. 155 We noticed that the data could not be modelled using a two-site exchange mechanism and that expansion of the Bloch-McConnell matrices to account for the existence of two adsorbed species in chemical exchange with the desorbed state was required in order to fully reproduce the 1 H DEST prole. However, due to the high sensitivity of 1 H DEST to spin diffusion that overemphasizes contributions from surface-bound species and complications in modelling 1 H relaxation, an accurate determination of the kinetic, thermodynamic, and structural parameters of sorption was not obtained in this initial study.
In a follow-up study, we combined 1 H and 13 C CPMG RD, 13 C DEST, and 13 C R 1 experiments to investigate the adsorption of PhOH on CeO 2 and Pd/CeO 2 NPs of cubic morphology. 192 Consistent with our preliminary investigation using 1 H DEST, modelling of the data required a two-step adsorption process in which the desorbed species is in equilibrium with tightly adsorbed PhOH via a weakly associated intermediate state (Fig. 13). It is important to note that simultaneous analysis of all datasets was key to the determination of the two-step binding mechanism. Indeed, a one-step binding model was sufficient to describe the 1 H CPMG RD, 13 C DEST, and 13 C R 1 (but not the 13 C CPMG RD) measured for PhOH-CeO 2 , or the 1 H and 13 C CPMG RD and 13 C R 1 (but not the 13 C DEST) measured for PhOH-Pd/

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Nanoscale Advances CeO 2 (Fig. 13a). Removing spin-diffusion artifacts from the DEST experiment allowed for accurate modelling of the kinetics and thermodynamics of adsorption as well as for determination of the 13 C R 1 and R 2 values for PhOH in the NP-associated states (Fig. 13b). Next, we used the modelled R 1 and R 2 to calculate the order parameter for each C-H bond in the weakly and tightly associated state by eqn (54)-(60) (Fig. 13c). Analysis of the S 2 values indicated that irrespective of the NP, all C-H groups are highly mobile in the weakly associated state (S 2 < 0.01), which suggested that this intermediate species consists of PhOH hydrogen-bonded to the CeO 2 component of the NP (Fig. 13c,  le). On the other hand, we noticed that the S 2 values obtained for PhOH tightly associated with CeO 2 are position-dependent, with very low S 2 values (<0.01) for the ortho and meta positions, and S 2 $ 1 for the para position. This S 2 trend (which is linked to the high 13 C R 2 measured for the para carbon - Fig. 13a) revealed that while the ortho and meta C-H groups reorient fast relative to the external magnetic eld, the para C-H bond reorients with the same correlation time as the NP (see eqn (58)). These considerations brought us to conclude that PhOH associates tightly to CeO 2 by inserting the O atom into Ovacancies present on the NP surface. Indeed, the latter interaction mode keeps the para C-H group rigid with respect to the NP surface while allowing fast reorientation of the meta and ortho C-H bonds (Fig. 13c, middle). Finally, the modelling of the relaxation rates returned an S 2 $ 1 for all C-H bonds of PhOH tightly associated with Pd/CeO 2 . Such observation indicated that the molecule is rigidly attached to the NP and conrmed Fig. 13 Combined analysis of 13 C R 1 , 13 C R 2 , 13 C DEST, and 1 H and 13 C CPMG RD experiments on the PhOH-Pd/CeO 2 system. (a) 13 C DEST (saturation field strength ¼ 1 kHz) and 1 H and 13 C CPMG RD profiles of PhOH in the presence of 1 wt% CeO 2 (left) and 1 wt% Pd/CeO 2 (right) NPs. Data were collected at 800 MHz. Experimental data for the para, meta, and ortho C-H bonds are represented in blue, green, and red circles, respectively. The dotted and solid lines represent the best fit to a two-site and three-site exchange model, respectively. (b) Modelled kinetic and thermodynamic parameters for PhOH adsorption on CeO 2 (left) and Pd/CeO 2 (right) NPs. PhOH is a blue oblate, CeO 2 is an orange square, and Pd sites are gray spheres. (c) Structural modelling of PhOH adsorbed on the CeO 2 and Pd/CeO 2 surface. The order parameter (S 2 ) obtained for each C-H group is shown. Adapted from Egner et al. 192 a "at" binding of PhOH to Pd hypothesized by ab initio calculations (Fig. 13c, right). 227 In the same work, we have tested the proposed binding models by reacquiring the NMR data in the presence of 20 mM inorganic phosphate, which binds strongly to CeO 2 but does not interact with Pd. 228 We found that the addition of phosphate reduced the population of PhOH weakly bound to CeO 2 and Pd/CeO 2 , and completely abolished the interaction of PhOH with O-vacancies on CeO 2 . However, the population and exchange kinetics of PhOH tightly bound to Pd/CeO 2 were unaffected by the presence of phosphate, con-rming that the tight adsorption of PhOH on Pd/CeO 2 is mediated by contacts between PhOH and the Pd component of the NP. Finally, in order to address the possible direct involvement of CeO 2 in catalytic conversion of PhOH, we compared the kinetics of substrate hydrogenation measured in the presence and in the absence of 20 mM phosphate with the adsorption data obtained by solution NMR. The rationale for this comparison was that reaction rates are proportional to the concentration of active substrate-catalyst complex in solution. Therefore, we expected the reaction rate to be reduced in the presence of phosphate if catalysis is mediated by interactions between PhOH and CeO 2 (Fig. 14). The results of our experiment conrmed this scenario and revealed an active role for the support in catalytic hydrogenation of PhOH by Pd/CeO 2 .

Conclusions
As applications of NP catalysts continue to expand, understanding sorption equilibria has become of utmost importance since the efficiency and selectivity of NP catalysis are highly dependent on the interactions that substrates, intermediates, and products establish with the NP surface. Over recent years, solution NMR spectroscopy is emerging as a powerful method for detecting and characterizing small molecule-NP interactions due to its exceptional sensitivity to molecular motions and capability of observing dynamic processes over a wide range of timescales. Through this comprehensive review of various solution NMR techniques for characterizing small molecule-NP interactions, we illustrated procedures for stable NMR sample preparation of insoluble NPs, introduced theoretical and practical aspects of the relevant NMR experiments, discussed how the combined analysis of multiple NMR methods can provide mechanistic insight into sorption equilibria involving multiple elementary steps, and reviewed a number of practical applications. We believe solution NMR spectroscopy has a bright future ahead in the investigation of ligand-NP interactions, and we expect it to become a crucial investigative tool for the development of new generations of heterogeneous catalysts.

Conflicts of interest
There are no conicts to declare. Fig. 14 Graphic representation of how the direct participation of CeO 2 in catalysis was revealed by the addition of inorganic phosphate (Pi). PhOH is a blue oblate, the product of the reaction (cyclohexanone or cyclohexanol) is a green circle, the CeO 2 surface is shown as an orange rectangle, the Pd site is a gray half sphere, and Pi is a red circle. The NMR data revealed that while the addition of 20 mM Pi reduced the population of weakly bound PhOH-CeO 2 intermediate by $50%, it did not affect the population and exchange kinetics of PhOH tightly bound to Pd. Since the rate of PhOH hydrogenation catalyzed by Pd/ CeO 2 is also reduced by $50% by the addition of 20 mM Pi, the results reported in Egner et al. 192 suggest that the weak PhOH-CeO 2 interaction is catalytically relevant.