Unravelling mass transport in hierarchically porous catalysts

Bio-derived platform chemicals and fuels are important for the development of sustainable manufacturing. However, their e ﬃ cient production from biomass necessitates new catalysts and processes optimised for the selective transformation of large molecules. Mesoporous and hierarchically porous functional materials are promising catalyst candidates for biomass valorisation, but quantitative relationships between pore dimensions/connectivity, mass transport, and corresponding catalytic performance are poorly de ﬁ ned. A family of hierarchical macroporous – mesoporous SBA-15 sulfonic acids were prepared with tunable macropore diameters for carboxylic acid esteri ﬁ cation. Turnover frequencies for long-chain (palmitic and erucic) acids were proportional to macropore diameter ( # 370 nm), whereas propanoic acid esteri ﬁ cation was independent of macropore size. Pulsed ﬁ eld gradient NMR di ﬀ usion experiments reveal that larger macropores enhance esteri ﬁ cation of bulky carboxylic acids by conferring superior pore interconnectivity and associated mass transport.


Introduction
Porous solids nd widespread practical application in catalysis, sorption, and separation science wherein pore dimensions and network connectivity control the attendant internal surface area and accessibility. 1,2Micro-and mesopores (0.5-50 nm diameter) confer high areas that can maximise the density of surface chemical functions such as catalytically active sites, while macropores (>50 nm) enhance the rate and extent of uid permeation through a pore network.Hierarchical porous solids, typically comprising interpenetrating bimodal mesoporemicropore or macropore-mesopore networks, extend control over molecular transport, adsorption, and reaction. 3For example, the introduction of mesoporosity into zeolites improves their lifetime, 4,5 and activity and selectivity for (bio) fuels 6,7 or ne chemicals 8 production, while macropores promote esterication 9 and selective alcohol oxidation 10 over sulfonic acid and Pd mesoporous SBA-15 silicas respectively, CO 2 hydrogenation over Cu alumina, 11 and transesterication over Mg-Al hydrotalcites. 12This is conceptually akin to fractal networks in Nature, 13 wherein the transport of matter is optimised by rapid convective ow through large channels, that in turn supply a network of smaller channels through which diffusion is constrained.Although chemical transport through hierarchical pore networks is extensively modelled, natureinspired design principles 14 have yet to be implemented to synthesise materials with architectures 'tailored for purpose'.
7][18][19] In practice, such enhanced accessibility is also facilitated by truncation of mesopore channels in ordered hierarchical porous solids (commonly prepared by dual-templating routes) compared with their mesoporous counterparts; 20,21 shorter mesopores may mitigate any attendant diffusional resistance.The interdependence of pore hierarchy and individual pore dimensions in experimental studies hampers validation of transport models and necessitates direct measurement of diffusion and reaction in systematically related families of hierarchical porous solids.
Routes to interpenetrating bimodal macro-mesopore networks using microemulsion colloidal polystyrene microspheres or co-surfactant templating routes are particularly attractive for liquid phase heterogeneous catalysis using bulky substrates.3][24][25] However, in the context of catalysis, systematic studies of porosity are largely conned to mesoporous silicas, wherein e.g.pore-expansion increased the turnover frequency (TOF) of sulfonic acid functionalised SBA-15 and KIT-6 for carboxylic acid esterication, 26 or hierarchically porous silicas in which only the mesopore diameter was varied for Fischer-Tropsch synthesis. 27In all cases, improved mass transport is inferred from superior catalytic performance.Nevertheless, very few experimental studies have correlated molecular diffusion through mesopores and corresponding reactivity, 28,29 and to our knowledge diffusion has never been experimentally measured for an ordered hierarchical porous catalyst.][32] Here we assess the impact of macropore size on molecular diffusion and carboxylic acid esterication for propylsulfonic acid functionalised, hierarchical macroporous-mesoporous SBA-15 (PrSO 3 H/MM-SBA-15) catalysts.PFG NMR is used to probe diffusive restriction and pore connectivity within these catalyst frameworks.Larger macropores reduce pore network tortuosity and increase diffusion through mesopore channels, accounting for the >4-fold increase in TOF observed for palmitic and erucic acid esterication with methanol.

Hierarchically porous SBA-15
Hierarchically porous silicas were prepared by a modied true liquid crystal templating technique incorporating a range of polystyrene nanospheres as macropore-directing hard templates. 34Pluronic P123 (2 g, Aldrich average M n $ 5800) was sonicated with HCl-acidied deionised water (2 g, pH 2) at 40 C to form a homogeneous gel.Tetramethoxysilane (4.08 cm 3 , Acros 99%) was added to the gel and rapidly stirred for 5 min (800 rpm) until a homogeneous liquid formed.Immediately following this phase change, polystyrene nanospheres of a particular size (6 g, independent of bead diameter, as a ne powder) were added with agitation (100 rpm) for 1 min.The resulting viscous, homogeneous mixtures were heated in vacuo (100 mbar) at 40 C for 2 h to remove reactively-formed methanol.The solids thus obtained were exposed to air at room temperature for 24 h to complete condensation of the silicate precursor, and then calcined at 550 C for 6 h (ramp rate 5 C min À1 ) to remove organic templates.

Acid functionalisation
The preceding hierarchically porous silicas were functionalised with propylsulfonic acid by an alkoxide graing method.In each case, silica (1 g) was added to a 100 mL round-bottomed ask with anhydrous toluene (30 mL) under N 2 at 90 C with stirring (700 rpm).A (3-mercaptopropyl)trimethoxysilane thiol precursor (1 mL) was subsequently added and the reaction mixture stirred for 24 h.The resulting solid was ltered, washed with ethanol, and dried at 80 C overnight.Surface thiols were then oxidised to sulfonic acids (1 g solid in 20 mL of 30 vol% H 2 O 2 ) with stirring (400 rpm) overnight.Propylsulfonic acid functionalised materials were recovered, washed with deionised water and ethanol, and dried overnight at 80 C.

Material characterisation
Powder XRD was performed on a Bruker D8 ADVANCE tted with a Cu K a X-ray source with a Ni lter and LynxEYE 192channel high speed strip detector.Wide angle scans were performed between 10-80 with a step size of 0.02 and dwell time of 0.6 s, and low angle scans between 0.5-8 with a step size of 0.01 and a dwell time of 1 s.Crystalline phases were determined by Rietveld renement using Profex v3.11.1, and crystallite sizes determined using the Scherrer equation.N 2 adsorption isotherms were measured using a Quantachrome 4000 surface area analyser.Surface areas and pore size distributions were determined through application of the BET and BJH equations respectively to the recorded isotherms.XPS analysis was performed using a Kratos Axis HSi spectrometer equipped with a monochromated Al K a (1486.8eV) X-ray source and charge neutraliser, operating <10 À9 Torr.Survey spectra were recorded with a pass energy of 160 eV, and high-resolution spectra with a pass energy of 40 eV.Spectra were Shirley background-subtracted, calibrated to adventitious carbon (284.8 eV), and tted using Casa v2.3.15.S 2p peaks were tted using a Gaussian-Lorentzian (70 : 30) lineshape and a doublet separation of 1.16 eV.Sulfur loadings were determined by CHNS analysis using a Thermo Flash 2000 organic elemental analyser calibrated against a sulphanilamide standard.SEM was performed using a JEOL 7800F Prime FEG SEM tted with a BSE and SE detector.Samples were affixed to carbon tape, and images recorded at a probe current of 3 mA and an accelerating voltage between 1-15 kV.STEM images were recorded using an aberration-corrected JEOL 2100F STEM with an accelerating voltage of 200 kV, a HAADF detector and Gatan bright eld camera.Electron micrographs were processed with ImageJ 1.46r, and particle sizes determined from samples of >200 particles.

Catalytic testing
Esterication reactions were performed in a 50 mL roundbottomed ask at 90 C with stirring (700 rpm).25 mg of catalyst was tested with 5 mmol of propanoic acid, palmitic acid, or erucic acid, and 12.5 mL methanol with 0.1 mmol dihexylether as an internal standard.0.25 mL aliquots were periodically removed, diluted with methanol, and analysed using a Varian 450 GC with 8400 autosampler tted with a ZB-50 column.

PFG NMR
Pulsed-eld gradient (PFG) NMR diffusion measurements were performed to assess the diffusive behaviour of liquids in the unrestricted bulk and within PrSO 3 H/MM-SBA-15 catalysts.Samples for NMR analysis were prepared according to the following procedure: PrSO 3 H/MM-SBA-15 exhibiting 150 nm and 430 nm macropores were pressed into 1 cm diameter pellets.The pellets were broken apart into approximately 4 mm Â 4 mm Â 2 mm pieces and dried at 105 C for 12 h.The catalysts were soaked in excess cyclohexane (Sigma Aldrich, $99.5%) or dodecane (Fischer Scientic, $99%) for at least 48 h under ambient conditions, with each liquid absorbed through capillary imbibition.Prior to NMR analysis the saturated materials were removed from the liquids and rolled over presoaked lter paper; this process removed any excess liquid on the outer surface of the pellets without drawing liquid from the pores.The saturated materials were then transferred to sealed 5 mm NMR tubes to a sample depth of $10 mm. 1 H PFG NMR experiments were performed using a Bruker DMX 300 NMR spectrometer equipped with a 7.1 T magnet (300.13MHz for 1 H) and a Bruker Diff30 diffusion probe capable of producing pulsed magnetic eld gradients up to 11.6 T m À1 .Samples were placed at the centre of the radiofrequency coil and le for at least 15 minutes to thermally equilibrate.Diffusion of unrestricted liquids was analysed using the pulsed gradient stimulated echo (PGSTE) sequence (Fig. 1a). 35To minimise the effects of background magnetic eld gradients (so-called internal gradients 36 ) the diffusion of liquids conned to the porous catalysts was analysed using the alternating pulsed gradient stimulated echo (APGSTE) pulse sequence (Fig. 1b). 37In each case, measurements were performed by holding the gradient pulse duration d constant and linearly varying the magnetic eld gradient strength g.Trapezoidal gradient pulses were used to ensure reproducible pulse shapes across all values of g.The recycle time was 2T 1 (as measured using the inversion recovery method) and 2 dummy scans were used to equilibrate the sample magnetisation before the acquisition of each data point.All measurements were performed at 20 AE 0.1 C and under ambient pressure.A summary of the typical PFG NMR acquisition parameters used is reported in Table 1.

Results and discussion
Synthesis of macroporous SBA-15 Macropore size control was achieved through the synthesis of monodispersed polystyrene nanospheres of tunable diameter (200 to 580 nm) as sacricial hard templates and their subsequent compaction into a colloidal crystal assembly.Ordered, hexagonal close-packed mesoporous SBA-15 silica networks were subsequently formed throughout the crystal assembly by a true liquid crystal templating method. 34Calcination resulted in hierarchically macroporous-mesoporous silicas (MM-SBA-15) with uniform macropore diameters of 150 to 430 nm  (proportional to template size) shown in Fig. 2; macropore shrinkage relative to the polystyrene template was consistent with previous reports for hierarchically porous alumina. 39N 2 porosimetry of the as-prepared porous frameworks revealed a common type-IV isotherm with H1 hysteresis indicative of mesoporous SBA-15, with common BJH mesopore diameters of 4 nm (Fig. S2 †).Low angle XRD conrmed the d 10 reection characteristic of p6mm SBA-15 was present for all materials (Fig. S3 †).
The preceding silica frameworks were functionalised with a common, and uniformly distributed, propylsulfonic acid (PrSO 3 H) loading ($1 wt% S, Table S1 †) to yield a family of hierarchically porous solid acid catalysts (PrSO 3 H/MM-SBA-15).S 2p XP spectra (Fig. S4 †) conrmed complete oxidation of the thiol precursor, and N 2 porosimetry conrmed retention of the parent mesopore networks following PrSO 3 H functionalisation (Fig. S5 †).The density of sulfonic acid groups was approximately independent of macropore diameter (0.69 nm À2 AE 0.05), and in a regime where the strongest acidity obtains due to lateral interactions between neighbouring sulfonic acid head groups. 40HAADF-STEM showed that mesopores were highly ordered in all cases, with a common mesopore diameter and macropore/mesopore interface (Fig. S6 †).

Catalytic enhancement through macropore tuning
The impact of macropore diameter on the activity of PrSO 3 H/ MM-SBA-15 catalysts was studied for free fatty acid (FFA) esterication with methanol as a function of chain length.In the case of propanoic acid, turnover frequencies (TOFs) per sulfonic acid site were independent of macropore size (and similar to that reported for mesoporous PrSO 3 H/SBA-15), 26 whereas those for bulky palmitic (C 16:0 ) and erucic (C 22:1 ) acids both signicantly increased with macropore size (Fig. 3).For palmitic acid, per site activities increased four-fold as the macropores expanded from 150 to 430 nm, while negligible activity was observed for the unsaturated, bent erucic acid for 150 nm macropores.Although erucic acid can readily access 150 nm macropores, only 2-8% of all active sites reside in the macropores (Table S1 †) and hence negligible catalysis occurs within them.The question arises as to why increasing the macropore size promotes erucic acid esterication.A plausible explanation is that the increased pore connectivity for larger macropores (observed by PFG NMR below) is at least partially associated with break-up of the mesopore channel network, 20,21 which in turn increases the number of mesopore openings and opportunities for the carboxylate group to react over active sites at mesopore entrances.This is akin to the mechanism invoked for the oxidation of n-alkanes (up to C 12 ) over molecular sieves, in which an end-on approach into small pore entrances favours terminal methyl attack. 41he maximum TOFs for propanoic and palmitic acids are in accordance with those reported for PrSO 3 H/SBA-15 and PrSO 3 H/ MM-SBA-15 with 340 nm macropores and a similar acid loading. 21,26The inuence of macropore diameter on esterication activity is solely attributable to internal pore architecture, since particle size distributions across the hierarchical catalysts were essentially constant (modal values $12 mm, Fig. S7 †).This sensitivity of TOF to macropore and molecular size suggests that in-pore diffusion may be rate-limiting for bulky reactants; indeed the linear relationship between palmitic acid esterication and macropore diameter (Fig. S8 †) is reminiscent of that for anticipated for Knudsen diffusion through porous media.Direct evidence of in-pore transport limitations for palmitic acid esterication was obtained from mechanical grinding of the 210 nm macropore catalyst.This reduced the mean particle size (and implicitly diffusion pathlength) approximately 3-fold without changing the porosity (Fig. S9 and Table S2 †), resulting in a corresponding two-fold increase in TOF from 12 to 22 h À1 .Apparent activation energies for palmitic acid esterication over the 210 and 430 nm macropore hierarchical catalysts also indicate a switchover from diffusion limited to reaction-rate limited esterication with increasing macropore size, rising from 34 AE 3 kJ mol À1 (210 nm) to 45 AE 3 kJ mol À1 (430 nm).The latter barrier is in fair agreement with the homogeneous acid catalysed value of 56 kJ mol À1 , 42 suggesting that esterication over the 430 nm macropore catalyst is largely free from mass transport limitations.

PFG NMR diffusion studies
Insight into the preceding dependence of TOF on catalyst structure and carboxylic acid size was subsequently derived by investigating the changes in pore connectivity with macropore diameter.PFG NMR experiments are uniquely suited to such studies, enabling the direct and non-invasive evaluation of molecular self-diffusion coefficients within liquid-saturated catalyst structures. 43For long observation times D, the effective diffusion coefficient of the restricted liquid D eff is reduced from that of the unrestricted liquid D 0 (in the bulk media) by the tortuosity of the porous network s.This parameter denes the overall interconnectivity of accessible pore structures and may be estimated from: Here D eff (D / N) is the effective self-diffusion coefficient of the restricted liquid observed in the long-time diffusion limit, such that D eff is invariant to further increases in D. Such measurements probe average molecular displacements (or root mean square displacements, RSMD ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ), which are signicantly greater than the modal pore sizes present.5][46][47][48] For liquids exhibiting non-negligible interactions with the porous matrix, s will vary with molecular structure and functionality, giving an effective tortuosity factor that may no longer represent the pore connectivity. 28,46,48n the present work, the diffusivities of cyclohexane and dodecane were explored for the preceding 150 nm and 430 nm macroporous sulfonic acid silicas.The signicant difference in molecular size between these probe molecules is an approximate analogue for the structural differences between the shortand long-chain carboxylic acids in Fig. 3.The use of alkanes was necessitated by the relatively high melting points of erucic and palmitic acids (33.8 C and 62.9 C respectively) which prohibited self-diffusion measurements using our diffusion probe.Solubilising the acids in methanol is problematic due to: (i) competitive in-pore diffusion between methanol and the acid (diffusion of 2-component systems depends on their molar fraction); (ii) strong hydrogen bonding networks throughout the imbibed methanol/acid mixture and with the pore surfaces, which prevent determination of the true tortuosity of the porous catalyst (only possible when probe molecules do not chemically interact with pore surfaces or each other); and (iii) catalytic reaction resulting in changing molecule populations which is not congruent to obtaining useful diffusion information. 46Ideal probe molecules should therefore be chemically inert, free of polar functional groups, exhibit sufficiently high diffusivities that the long-time diffusion limit can be accessed (wherein probe molecules can access multiple pores during a sensible observation time of $100 ms), and express long relaxation times; criteria which favour relatively 'small' probe molecules, with dodecane the largest such probe typically investigated.
Self-diffusion coefficients D 0 and D eff (D / N) are readily obtained through appropriate PFG NMR diffusion experiments. 31Log-attenuation plots for unrestricted liquids were obtained using the PGSTE pulse sequence (Fig. 4); D 0 was determined by tting the acquired data points to the Stejskal-Tanner equation: 35

SðgÞ
Here S 0 is the NMR signal in the absence of any applied eld gradient and S(g) is the acquired signal in the presence of magnetic eld gradients of magnitude g.The b-factor for the PGSTE pulse sequence is: where g is the gyromagnetic ratio of 1 H and d is the effective gradient pulse length in Fig. 1.D 0 values for unrestricted liquids were obtained by non-linear least squares tting of the PGSTE data according to eqn ( 2) and (3), and were 1.291 AE 0.002 Â 10 À9 m 2 s À1 and 7.76 AE 0.01 Â 10 À10 m 2 s À1 for cyclohexane and dodecane respectively, in excellent agreement with literature values. 49og-attenuation data for the self-diffusion of cyclohexane and dodecane when restricted within PrSO 3 H/MM-SBA-15 catalysts possessing 150 nm and 430 nm macropores were subsequently obtained (Fig. 5); these materials were selected as limiting small and large pore solid acid catalysts.The APGSTE data points were acquired using four different observation times ranging from 25-200 ms.It is apparent that these signal attenuations do not follow a simple exponential decay of the form described in eqn (2).Indeed, the attenuation data exhibit signicant curvature on a log-scale, which is qualitatively independent of the observation time over the range of D values explored.Extracting physically meaningful diffusion coefficients from these data requires careful consideration of the origin of this complex signal.Curved log-attenuation data are typically attributed to multicomponent exponential behaviour of the form SðgÞ=S 0 ¼ X i p i expfÀbD i g.Here, the diffusion coefficients D i describe the mobility of different molecular ensembles i with effective populations p i .Such an approach is generally valid if the mean lifetime of molecules within each population exceeds the observation time over which diffusion is measured (the so-called slow exchange condition), 31 and if appropriate molecular environments exhibiting different length scales and connectivity exist throughout the sample to generate multiple diffusion coefficients.Normal (or Fickian) self-diffusion must be assumed within each environment, such that a Gaussian distribution of molecular displacements evolves over time. 43,50Typical examples include the long-and shortrange behaviour of gases and volatile liquids imbibed in microporous crystalline solids, wherein different diffusion coefficients may be assigned to inter-and intracrystalline diffusion. 51Differentiation between these transport phenomena primarily depends on the RMSD of the diffusing probe molecules, and hence the observation times and experimental temperatures employed.Relevant studies in which multicomponent exponential data ttings of this form have been applied include investigations of diffusion within hierarchical meso/ microporous zeolites [52][53][54][55] and aggregates of mesoporous SBA-15. 56Galarneau et al. examined n-hexane diffusion in zeolite FAU-Y and mesoporous Al-MCM-41, 57 demonstrating that selfdiffusion within a mechanical mixture of these materials could be described as a simple superposition of the diffusivities within the individual materials, whereas the self-diffusion coefficient from a hierarchical combination of these materials lay in an intermediate regime between that characteristic of mesopores or micropores.Furthermore, Adem et al. investigated the self-diffusion of n-hexane during the pseudomorphic synthesis of MCM-41 58 ) to explore time-dependent pore structures.Multicomponent exponential ttings distinguished between long-range interparticle displacement and intraparticle displacement within the resulting pore structures, revealing a notable dependence of the intraparticle effective self-diffusion coefficient on both pore size and connectivity.
A biexponential tting of the form S(g)/S 0 ¼ p fast exp{ÀbD fast } + p slow exp{ÀbD slow } provided good agreement with our acquired attenuation data (Fig. S10 †), where D fast describes the initial, steep signal decay at low b, while D slow describes the limiting gradient of the shallow decay observed at high b.Recalling the APGSTE sequence for the observation of restricted selfdiffusion, the relevant b-factor is: where s e is the spin echo time dened in Fig. 1.For cyclohexanesaturated materials we obtain D fast z 2.6 Â 10 À10 m 2 s À1 and D fast z 4.0 Â 10 À10 m 2 s À1 for catalysts exhibiting 150 nm and 430 nm macropores respectively (Table S3 †), with associated maximum RMSDs (D ¼ 200 ms) of 10 and 13 mm probed by our cyclohexane diffusion experiments respectively.Corresponding values for dodecane are D fast z 1.4 Â 10 À10 m 2 s À1 and D fast z 2.2 Â 10 À10 m 2 s À1 for the two catalysts (Table S3 †), with maximum RMSDs (D ¼ 200 ms) of 7 and 9 mm probed respectively.These RMSD are adequate for probing intraparticle diffusion in the present materials as our PFG NMR measurements will be dominated by diffusion through the largest catalyst particles (Fig. S7 †); the signicant reduction in D fast (relative to D 0 ) further suggests that any spin population outside of the catalyst particles is small.The corresponding signal intensity p fast comprises over 95% of the acquired NMR signal in each case, such that D fast may be assigned to bulk-pore diffusion throughout the material.It is noteworthy that these RMSD values signicantly exceed the macropore domains present in either material, suggesting that D fast characterises diffusion occurring through both the macropores and mesopores.This postulate is supported by chemical cascade reactions using the same macroporous-mesoporous SBA-15 framework, 34 wherein a high degree of connectivity between mesopores and macropores was evidenced.The slowly decaying signal attenuation at high b in Fig. 5 is more difficult to assign.Biexponential tting yields D slow $ 10 À12 m 2 s À1 for both the cyclohexane-and dodecane-saturated catalysts.Considering the above discussion of RMSDs and the assignment of D fast to diffusion through the entire pore network, it is rather surprising to identify a small spin population exhibiting such a signicantly different diffusion coef-cient.Such an observation might imply the existence of congurational diffusion, where liquid mobility is dominated by interactions with the pore walls within one of the domains. 43owever, in the present materials mesopore diameters are too large for congurational diffusion to occur, and no micropore population was identied by N 2 porosimetry t-plot analysis (data not shown), suggesting this slow component cannot be to congurational diffusion through structural micropores.Previous studies of self-diffusion within an adsorbed surface layer reported similar PFG NMR attenuation curves to the present data. 59,60However, covalent modication of the surface with alkyl groups removed this component from the attenuation data. 60The existence of a slowly diffusing surface component can therefore be discounted in this work due to presence of surface propyl sulfonic acid groups.Furthermore, the aforementioned studies required thousands of repeat scans to observe the slowly diffusing surface layer, 59,60 and hence such a population is unlikely to be detected by the limited repeat scans in this work.It is pertinent then to consider the isotropy of the pore structures under study.2][63] For a powderaverage of saturated anisotropic pores, the PFG signal is expected to attenuate according to: 63 where D par and D perp are self-diffusion coefficients describing the rates of displacement parallel and perpendicular to the direction of the mesoporous channels, respectively, and erf(/) is the error function.Recalling that the hierarchical structure of the materials explored here comprises both isotropic macropores and anisotropic mesopores, we suggest an appropriate attenuation equation will be of the form Similar expressions have been used to quantify self-diffusion within SBA-15 aggregates 56 and aluminosilicate nanotubes. 64,65ere, D iso is an isotropic diffusion coefficient similar to D fast described above; we propose that D iso represents diffusion throughout the entire catalyst, and is dominated by the isotropic macropores.The signal fractions p iso and p aniso describe the magnitude of attenuated signals due to the isotropic and anisotropic components of eqn ( 6) respectively, and in the following tting processes have been set such that p iso ¼ 1 À p aniso .We propose that p aniso represents a small spin population which remains within mesoporous channels for the entire observation time D, and which attenuates according to eqn (5), while p iso represents the remainder of the spin ensemble.
It is appropriate here to highlight the recent development by Splith et al. 66 of a complex model to address diffusive exchange between isotropic and anisotropic environments.The authors considered the diffusion and exchange of water within and between crystals of the anisotropic microporous metal-organicframework Al fumarate.The resulting model improved tting of PFG echo attenuation data relative to that achieved using eqn (5) but required that population of the isotropic environment was negligible.Such an approximation is appropriate for systems containing saturated anisotropic materials with only vapour phase within the intercrystalline voids, but is clearly not valid for our hierarchical catalysts which comprise fullysaturated isotropic and anisotropic pores.A different approach was therefore adopted in which APGSTE data for liquid-saturated PrSO 3 H/MM-SBA-15 pellets were t to eqn (4) and ( 6) using a non-linear least squares tting.The resulting two-component t is in good agreement with the acquired data; representative curves are shown in Fig. 5, where the isotropic and anisotropic contributions of eqn (6) are deconvoluted for clarity.This deconvolution visualises the slowly decaying signal at high b resulting from a small spin population exhibiting anisotropic diffusion.A small decrease in the signal obtained from this population (p aniso ) is evident with increasing D. There are two possible explanations for this decrease: (i) nuclear spin T 1 relaxation during the longitudinal storage period T ¼ D À 2s e , facilitating additional signal attenuation with increasing D; (ii) diffusive exchange, reducing the number of spins experiencing purely anisotropic self-diffusion during D. Since longitudinal relaxation time constants T 1 (Table S4 †) were at least an order of magnitude greater than the maximum D value employed, molecular exchange appears the dominant mechanism underpinning the decrease in p aniso .A decrease in p aniso with increasing macropore size is also apparent, as would be expected from the concomitant increase in p iso .The inherent differences in p aniso for the two probe liquids may be rationalised by considering the enhanced T 1 relaxation rates (1/T 1 ) exhibited by the dodecane-versus cyclohexane saturated catalysts (Table S4 †).
Unfortunately, the ts in Fig. 5 are of insufficient quality to robustly estimate D perp ; we attribute this to dominance of the acquired data by isotropic signal attenuation at small b, and hence focus discussion on D iso and D par .These diffusion coef-cients (Table S5 †) are shown as a function of observation time in Fig. 6, and are of equivalent orders of magnitude ($10 À10 m 2 s À1 ) in contrast to the diffusion coefficients assigned through the preceding biexponential tting (Table S3 †).Self-diffusion coefficients obtained from the isotropic component of eqn ( 6) appear in Fig. 6a.For the dodecane-saturated catalysts, and cyclohexane-saturated analogue with 150 nm macropores, a small decrease in D iso occurs with increasing D over the range 25-100 ms.Since the observed D iso values are all signicantly reduced from D 0 , we interpret these diffusivities as indicative of an intermediate regime between unrestricted and restricted diffusion. 31Molecules therefore experience signicant interactions with the pore walls but do not displace through a large enough sample of the pore space to be considered within the long-time diffusion limit.Above D ¼ 100 ms D iso is invariant, with dodecane presenting asymptotic D iso (D / N) y D iso (D ¼ 100 ms) values of $1.37 Â 10 À10 m 2 s À1 and $2.27 Â 10 À10 m 2 s À1 within the 150 nm and 430 nm macropore catalysts respectively, and cyclohexane presenting an asymptotic value of $2.86 Â 10 À10 m 2 s À1 within the 150 nm macropore catalyst.These diffusivities may therefore be interpreted as a probe of pore connectivity.It is interesting to note that we do not observe the same trend when considering cyclohexane-saturated PrSO 3 H/MM-SBA-15 with 430 nm macropores, which exhibits by far the largest D iso of the alkane/silica systems investigated (measurements were conducted multiple times to conrm this anomaly).In this latter case, a slight but steady increase in D iso with increasing D was observed suggesting that D iso observed at small D is already indicative of the long-time diffusion limit.The small increase in D iso with increasing D suggests an increased propensity for the imbibed cyclohexane to diffuse out of the particles and into the intercrystalline voids during large observation times.For the purposes of assigning a long-time diffusion limit value to this system we take a D iso (D / N) y D iso (D ¼ 25 ms) value of $4.35 Â 10 À10 m 2 s À1 .
Values of the anisotropic diffusion coefficient D par are presented in Fig. 6b.Despite greater uncertainty in these D par values, tting clearly reveals that D par > D iso , suggesting diffusion is faster through the mesopore network than the overall hierarchical pore structure.This nding is in agreement with Adem et al., who reported a distinct decrease in the effective self-diffusivity of n-hexane imbibed within MCM-41 upon the addition of large, secondary mesopores embedded in the structure which create 'stagnation zones'. 58More detailed insight is obtained from the ratios D 0 /D iso and D 0 /D par (Table S6 †), which we interpret as providing direct insight into pore connectivity within the catalyst structures studied.In particular, D 0 /D iso reects the pore connectivity of the overall catalyst structure, while D 0 /D par reects that of the anisotropic mesopores.These ratios are plotted as a function of observation time in Fig. 7.The ratio D 0 /D iso is shown in Fig. 7a and b for each catalyst/alkane combination, and is invariant at D $ 100 ms, consistent with the preceding analysis of their effective self-diffusion coefficients.These values are therefore considered indicative of the long-time diffusion limit, and hence a probe of pore connectivity.The anomalous behaviour of cyclohexane within PrSO 3 H/MM-SBA-15 with 430 nm macropores is no longer apparent when interpreted as a D 0 /D iso ratio and is essentially invariant to increasing observation time (Fig. 7a).Values of this ratio for the four alkane/catalyst systems are summarised in Table 2 and reveal two notable features.First, D 0 /D iso is greater for the catalyst with 150 nm macropores than that with 430 nm macropores, irrespective of the probe molecule considered, i.e. overall pore network tortuosity decreases with increasing macropore size.Second, D 0 /D iso values for cyclohexane and dodecane are signicantly different for the same macropore size; dodecane values are approximately 20% larger for both macropore sizes.A consequence of the latter is that these values represent an effective tortuosity which is inuenced by molecular size and/shape.However, since the change in D 0 /D iso with macropore size is qualitatively independent of the probe liquid, we can remain condent that an increase in macropore size is indeed associated with a decrease in overall tortuosity.S5 †).Nevertheless, there is clearly substantial overlap between the four datasets,  particularly clear at short observation times, where the RMSD is signicantly less than the modal catalyst particle size (Table S5 †).This overlap strongly suggests that the interconnectivity (and pore diameter and curvature) of the mesopores is similar in both catalysts, as expected given that the only difference in their syntheses was the size of sacricial polymeric bead used to form the macropores, and further evidences that the tortuosity of our PrSO 3 H/MM-SBA-15 catalysts is solely controlled by macropore size.General agreement (within experimental error) between cyclohexane and dodecane D 0 /D par values (Fig. 7b and  c) indicates that diffusive displacement within the mesopores of both catalysts is largely independent of molecular size/shape.

Structure-reactivity correlations
We rst note that macropores were varied independently of the size or tortuosity of mesopores (Fig. S5, S6 † and 7c), enabling the inuence of macropores on carboxylic acid esterication by the hierarchical catalysts to be rationalised.For propanoic acid, esterication was independent of macropore size, as may be anticipated from its small kinetic diameter ($0.48 nm) and hence minimal diffusive restriction by the pore network.A recent PFG NMR study on organic acid diffusion through mesoporous SBA-15 and KIT-6 silicas 47 reported that the effective tortuosity increased with acid chain length, and that pore walls enhanced diffusion possibly by disrupting the dynamic hydrogen bonding network present throughout the liquid mixture (as has previously observed for polyols and polyol/ methanol mixtures [41][42][43][44] ).It follows that propanoic acid selfdiffusion may be dominated by hydrogen-bonding dynamics, rather than by pore network tortuosity, consistent with the present ndings that propanoic acid TOF and D 0 /D iso (equated here to macropore size) are uncorrelated.Conversely, the preceding SBA-15 and KIT-6 PFG-NMR study showed that longchain acid diffusion was dominated by (meso)pore tortuosity and molecular size. 45We might therefore expect the TOFs for palmitic and erucic acids to follow the tortuosity of our hierarchical PrSO 3 H/MM-SBA-15 catalysts, as indeed is observed; long-chain acids TOFs are inversely correlated with D 0 /D iso .
PFG NMR analysis of the diffusion of solvated acids was not possible due to competing catalytic esterication.It is therefore insightful to consider the difference in effective tortuosity experienced by cyclohexane and dodecane within the small and large macropore catalysts (Table 2).Dodecane exhibits greater D 0 /D iso values than the smaller cyclohexane probe molecule, irrespective of macropore size.It follows that a secondary, sizedependent transport resistance must be present, which further reduces the mobility of large molecules in addition to restrictions imposed by the inherent pore network tortuosity.Since dodecane lacks any polar functionality, this secondary transport resistance cannot be attributed to e.g.enhanced interactions with surface sulfonic acid groups or hydroxyls, but rather reects topological constraints of the pore system.This nding agrees with pore-scale diffusion simulations, 67,68 which reveal that the effective tortuosity of mesoporous SBA-15 and KIT-6 silicas depends on the size of the probe molecule.Note that this contrasts with simulations 67,68 and PFG-NMR studies of amorphous mesoporous silicas, for which effective tortuosity is independent of molecular size. 46Relative D 0 /D iso values for cyclohexane versus dodecane through PrSO 3 H/MM-SBA-15 also depend on macropore size, being 23% greater for 150 nm macropores and 15% greater for 430 nm macropores (Table 2), i.e. the secondary transport resistance identied above decreases with increasing macropore size.This observation manifests our chemical intuition that larger macropores do indeed enhance mass transport of long-chain acids through hierarchically porous catalysts.In contrast D 0 /D par , which reects diffusion through only the mesopores, is approximately independent of probe molecule.This suggests that the size-dependent transport resistance seen for D 0 /D iso arises from: (i) topological effects at the macropore/mesopore interface, i.e. bulkier molecules (dodecane, palmitic or erucic acid) struggle to enter mesopores, but once correctly oriented can diffuse relatively freely along mesopore channels (recall that D par > D iso ); and/or (ii) stagnation zones created by small macropores that are embedded in the silica framework and hinder transport through the mesopore network.Constricted pore openings are reported for mesoporous MCM-41 and SBA-15. 67,69A detailed analysis of secondary transport resistances within these hierarchically porous catalysts is the subject of future studies.

Conclusions
The synthesis and characterisation of hierarchical macroporous-mesoporous SBA-15 sulfonic acid catalysts exhibiting tunable macropore sizes is reported, facilitating new insight into transport phenomena through hierarchically porous materials.Catalytic performance of the resulting PrSO 3 H/MM-SBA-15 family towards the liquid phase esterication of short and long-chain carboxylic acids with methanol revealed that turnover frequencies for propanoic acid were structure invariant, whereas those for palmitic and erucic acids exhibit striking increases with macropore size.Complementary PFG NMR diffusion measurements enable the non-invasive assessment of pore connectivity and diffusive restrictions for liquids imbibed within these catalysts.Effective self-diffusion of cyclohexane and dodecane within PrSO 3 H/MM-SBA-15 catalysts containing 150 nm and 430 nm macropores reveals complex signal attenuation data, attributed to a superposition of isotropic and anisotropic contributions.Diffusivities extracted from these measurements provide an estimate of the structural tortuosity through comparison with bulk selfdiffusion coefficients.Tortuosity of the overall pore network decreases with increasing macropore size, indicating that larger macropores improve pore connectivity throughout the hierarchical framework.Larger macropores enhance longchain carboxylic acid esterication by improving pore structure connectivity and hence active site accessibility.Propanoic acid esterication is indifferent to the tortuosity of our hierarchically porous catalysts, consistent with the diffusive behaviour of short-chain carboxylic acids in SBA-15, 47 and presumably reecting its small kinetic diameter.Molecular size is an important factor in determining the extent of diffusive restriction imposed by the hierarchical pore networks, and increased for larger probe molecules resulting in larger effective tortuosities for dodecane than cyclohexane.This additional restriction was more pronounced for smaller macropores exacerbating the poor catalytic turnover for palmitic and erucic acids within small (150 nm) macropore PrSO 3 H/MM-SBA-15 catalysts.In summary, PFG-NMR analysis shows that diffusion limitations are not related to a single textural parameter, but rather depend on a convolution of molecular size, pore connectivity, and pore diameter.Pore size and connectivity are themselves not mutually exclusive, since we have shown that altering the macropore diameter also change the effective tortuosity throughout hierarchical pore networks (a measure of connectivity).Trends in catalytic esterication are explicable by considering the enhanced mass transport attainable through tailoring macropore size (and attendant pore connectivity) to the reactant size.This work highlights the signicant capability of PFG NMR experiments for elucidating pore structure and connectivity within complex porous solids, and more specically as a powerful tool to guide the design of hierarchically porous catalysts for maximal activity.

Fig. 1
Fig. 1 NMR diffusion pulse sequence diagrams for (a) PGSTE and (b) APGSTE experiments.Radiofrequency (RF axis) pulses are indicated by vertical bars; thin and thick bars represent 90 and 180 RF pulses respectively.Gradient pulse timings (g axis) are specified according to the notation of Tanner. 38Trapezoidal gradient pulses of incremental magnitude g are shown with effective pulse durations d and d/2 for the PGSTE and APGSTE sequences, respectively.Homospoil gradients are also shown and are applied during the longitudinal storage period T to remove any residual transverse magnetisation.The observation time is D.

Fig. 5
Fig. 5 APGSTE data for PrSO 3 H/MM-SBA-15 catalysts: PFG NMR signal attenuation curves for (a and b) cyclohexane saturated and (c and d) dodecane-saturated PrSO 3 H/MM-SBA-15 catalysts with respective 150 nm and 430 nm macropores.Two-component fits to the acquired data points utilised eqn (6); isotropic and anisotropic fitted components are shown in red and blue respectively.
Fig. 7b also shows D 0 /D par as a function of observation time.The uncertainty in these values is signicant, and stems from errors in assigning D par during tting (Table

Fig. 6
Fig. 6 (a) D iso and (b) D par values obtained by fitting APGSTE data to eqn (6).Macropore diameters are indicated in parentheses.Errors represent AE1 standard deviation of three measurements of three different samples and are too small to be observed in (a).

Fig. 7
Fig. 7 (a and b) Effective tortuosity parameters as a function of observation time, and (c) comparative summary for PrSO 3 H/MM-SBA-15 with different macropores and probe molecules (D ¼ 100 ms for D 0 /D iso values and D ¼ 25 ms for D 0 /D par values).

Table 1
Summary of typical PFG NMR acquisition parameters