Noninvasive Experimental Evidence of the Linear ... - ACS Publications

Letter pubs.acs.org/JPCL

Noninvasive Experimental Evidence of the Linear Pore Size Dependence of Water Diffusion in Nanoconfinement Houria Chemmi,† Dominique Petit,*,† Pierre Levitz,‡ Renaud Denoyel,§ Anne Galarneau,∥ and Jean-Pierre Korb† †

Physique de la Matière Condensée, Ecole Polytechnique-CNRS, Palaiseau 91128, France Physicochimie des Electrolytes et Nanosystèmes Interfaciaux, CNRS-UMR 8234, Université Pierre et Marie Curie, 4 place Jussieu, 72522 Paris Cedex 5, France § MADIREL, Aix-Marseille Université, CNRS-UMR 7246, Centre de St Jérôme, 13397 Marseille Cedex 20, France ∥ Institut Charles Gerhardt Montpellier, UMR 5253 CNRS-UM-ENSCM, ENSCM, 8 rue de l’Ecole Normale, 34296, Montpellier Cedex 05, France ‡

ABSTRACT: We show that nuclear magnetic relaxation experiments at variable magnetic fields (NMRD) provide noninvasive means for probing the spatial dependence of liquid diffusion close to solid interfaces. These experiments performed on samples of cylindrical and spherical nanopore geometries demonstrate that the average diffusion coefficient parallel to the interface is proportional to the pore radii in different dynamics regimes. A master curve method allows extraction of gradients of diffusion coefficients in proximity of the pore surfaces, indicative of the efficiency of coupling between liquid layers. Due to their selectivity in frequency, NMRD experiments are able to differentiate the different water dynamical events induced by heterogeneous surfaces or composed dynamical processes. This analysis relevant in physical and biological confinements highlights the interplay between the molecular and continuous description of fluid dynamics near interfaces.

H

Here, we propose an original method based on the nuclear magnetic relaxation dispersion (NMRD) technique for probing the spatial dependence of the water diffusion and the efficiency of the coupling between fluid layers specifically at or close to pore surfaces. Basically, the NMRD profile measures the longitudinal spin relaxation rate R1 as a function of magnetic field strength or Larmor frequency.30 It gives a direct probe of the liquid dynamics because it scans a large range of applied magnetic fields and yields unique information about to which a liquid is dynamically correlated with the specific relaxation process at the pore surface.31,32 Contrary to pulsed field gradient (PFG) and magnetic resonance imaging (MRI) techniques that mix the fluid diffusion contribution from molecular to hundreds of micrometers or more due to gradient intensity limitation, the NMRD technique uses a shorter accessible time scale and focuses on shorter length scales well adapted for exploring the fluid dynamics within nanopores. To illustrate this analysis, we have synthesized well-defined, calibrated porous silica materials like MCM-4133,34 and SEOS35 in the nanoscale range. From these syntheses, we obtained two MCM-41 samples of 3.3 and 11.8 nm diameters with cylindrical pores and two SEOS samples of 10 and 14 nm cavity diameters with spherical pores.34−37 Moreover, we have

ow does water diffusion occur close to a solid interface? Answering this question is vital in life sciences,1,2 physical chemistry in porous media,3−5 and civil engineering.6,7 Several examples of hydric transport exist in multiscale systems such as catalysis,8,9 cement-based materials,10,11 and living tissues.12−14 However, in such multiscale contexts, other questions remain on how to relate the macroscopic and microscopic fluid properties15 and how to probe experimentally the characteristic distances of interest. For water at the nanoscale range, there is always interplay between the efficiency of diffusive transport and the impact of confinement.16−18 However, the description of water dynamics by an isotropic or anisotropic and spatially dependent diffusion is questionable, as recently shown by numerical simulations.3,19−21 Several experimental attempts have been proposed to address the question. Optical techniques probing the rotational water diffusion at the interface,22−24 X-rays and neutron scatterings characterizing the local water diffusion,25,26 and infrared with other related techniques are also useful.27 However, most of these techniques are invasive, especially for biological media, and are limited for exploring dynamics deep inside of materials. An experimental technique for characterizing in situ and noninvasively such anisotropy and spatial dependence is thus absolutely necessary. Nuclear magnetic resonance (NMR) is well-known as a noninvasive technique widely used for probing molecular dynamics in multiscale domains,28,29 but the influence of anisotropy and spatial dependence on relaxation observables is yet to be explored. © XXXX American Chemical Society

Received: December 7, 2015 Accepted: January 11, 2016

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DOI: 10.1021/acs.jpclett.5b02718 J. Phys. Chem. Lett. 2016, 7, 393−398

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The Journal of Physical Chemistry Letters carefully controlled the relative humidity (RH) in our NMRD experiments using saturated water solutions with different salts.37 The hydrated MCM-41 and SEOS samples were obtained after equilibrium conditions at 25 °C with RH = 90 and 97%, respectively, for ensuring water saturation in pores. In all of our experiments, the relaxation is monoexponential, leading to a single R1. On the basis of an original proton NMRD data analysis considering an anisotropic diffusion induced by the interface, we found a linear relation between pore radii and radially averaged diffusion coefficient fluids parallel to interfaces in either cylindrical or spherical pore geometries and independently of the material. Not only the confinement does limit the diffusion of the fluid, but the average parallel diffusion coefficient is proportional to the pore size. This astonishing linearity between pore size and diffusion will be explained below. The proton NMRD profiles at 25 °C of water-saturated MCM-41 samples with large and small cylindrical pores of 11.8 and 3.3 nm diameter, labeled M2 and M1, respectively, have been compared (Figure 1). The two NMRD raw data sets were

Figure 2. 1H NMRD profiles of SEOS-saturated samples with 10 (circle) and 14 nm (triangle) cavity diameters (main figure). In the inset, the master curve obtained by scaling both the relaxation rate and frequency by the factors f R1 = 14/10 and f freq = 1/f R1, respectively, is superposed on the SEOS with a 10 nm cavity diameter.

the ratio of pore radii for renormalizing the whole NMRD profiles. Moreover, this unique ratio of radii is a strong indication of a single molecular surface dynamics process, as pointed out below. Therefore, in SEOS samples also, the frequency renormalization confirms that the surface dynamics of water is directly controlled by the pore size. As the same results on rescaling frequencies (insets of Figures 1 and 2) have been obtained for cylindrical and spherical pores, we can affirm that the water surface dynamics in a pore is only dependent on the pore size independently of the pore geometry. A first step of analysis, based on the basic principles of NMR relaxation, allows progress in the understanding of the origin of these frequency renormalizations. When considering translational diffusion of a fluid over an average distance dc between two relaxation sinks bound to the surface and defined by the physical chemistry of the material, one can introduce a correlation time τc for which the molecular diffusion coefficient D verifies dc 2 ∝ Dτ c . Then, the corresponding characteristic frequency is ωc = 1/τc ∝ D for a constant dc. The NMRD results revealed that the rescaling frequency factor is proportional to the pore size leading to ωc ∝ Rp, where Rp is the pore radius. Therefore, the conservation of the physical chemistry of the studied system (i.e., conservation of dc) and the relaxation mechanism based on the translational diffusion lead directly to a linear dependence between the diffusion coefficient and the pore radius, D = νc Rp, where νc scales as a gradient of the diffusion coefficient. This linear dependence between diffusion and pore size for cylindrical or spherical geometry is consistent with previous chromatographic results39 for similar nanoporous materials. As this first step of analysis does not take into account the spatial averaging induced by the NMRD measurement, the analysis has to be completed by two successive steps, the anisotropy and the spatial averaging of the diffusion. The fluid confinement implies an intrinsic anisotropy in the water diffusion close to the pore surface. This effect requires indeed an axial diffusion tensor with two different diffusion coefficients D⊥ and D∥ in the perpendicular and parallel directions of the pore surface, respectively. Because the diffusivity D⊥ homogenizes the relaxation rate R1 to a single experimentally observed value, it homogenizes any other observables like D∥ to a spatial averaged value ⟨D∥⟩. This shows that the characteristic time for radial homogenization is shorter than the time scale limit explored by the NMRD

Figure 1. 1H NMRD profiles of MCM-41-saturated samples with 3.3 (diamond) and 11.8 nm (square) pore diameters (main figure). In the inset, the master curve obtained by scaling both the relaxation rate and frequency by the factors f R1 = 1.6 and f freq = 3.3/11.8, respectively, is superposed on the MCM-41 with a 3.3 nm pore diameter.

renormalized in a single master profile (inset of Figure 1), independently of any model of relaxation, first by rescaling the frequency axis with the f freq = 3.3/11.8 corresponding exactly to the ratio of the two pore radii and second by multiplying the relaxation rate values R1,M2 by a scaling relaxometry efficiency factor f R1 = 1.6 due to fast exchange.38 As explained below, the presence of two kinds of molecular surface dynamics does not lead to a ratio of radii for f R1 when applying the fast exchange modeling. The frequency scaling factor, which is the key experimental result, proves unambiguously that the water dynamics is directly controlled by the pore size. The proton NMRD profiles at 25 °C of water-saturated SEOS samples featuring spherical pores of 10 and 14 nm cavity diameters, labeled S1 and S2, respectively, are shown in Figure 2. The two NMRD raw data sets were renormalized in a single master profile (inset of Figure 2), independently of any model of relaxation, by multiplying the relaxation rate values R1,S2 by a scaling relaxometry efficiency factor f R1 = R1,S2/R1,S1 = 14/10, corresponding to the ratio of the cavity sizes, and by multiplying the associated frequency dependencies by the frequency scaling factor f freq = 1/f R1. For spherical pores, it is worthwhile that the two scaling factors are dependent only on 394

DOI: 10.1021/acs.jpclett.5b02718 J. Phys. Chem. Lett. 2016, 7, 393−398

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The Journal of Physical Chemistry Letters

series is obtained by summing the weighted inverse of the diffusion in each phase. In this fast exchange situation, the relaxation efficiency becomes a weighted sum of the individual relaxation rates,43 leading to a monoexponential behavior. These serial diffusive states are typical of an intermittent dynamics between interfaces. The weighted sum is valid, providing a sufficiently long molecular residence time in each phase for which physical observables such as the diffusion coefficient and relaxation times can be defined. Moreover, the molecular residence time should be short enough to induce an average of these physical observables. In our experiments, the relaxation being monoexponential, this proves the validity of the fast exchange regime for a biphasic case, leading to the average sum of the inverse of diffusivity

technique. As a consequence, this technique does not give any information on the value of D⊥ and gives only an average value ⟨D∥⟩. Now, the question is how to take into account this spatial averaging. For a homogeneous fluid with an average direction of flux, the modeling of the diffusive transport should be equivalent when considering a whole set of layers in series perpendicular to the average flux or an ensemble of layers parallel to the average direction of flux. An elementary calculation shows that the description in series involves an average of ⟨1/D∥⟩, while the description in parallel involves an average of ⟨D∥⟩.40,41 In the following, we describe separately and formally the monophasic and biphasic situations ending in each case to a linear pore size dependence of the average parallel diffusivity. Then, these two approaches are applied successfully to the real porous systems MCM41 and SEOS. For a monophasic situation, the parallel description respects the local symmetry and becomes the relevant approach for calculating the average diffusion. In the presence of a coupling between the different parallel layers, the coefficient of diffusion close to the pore surface DI changes progressively to the one D0 of the free liquid due to the influence of the solid−liquid interface. As a consequence, a radial dependence D(r) should be considered explicitly. As the NMRD experiments show, this radial dependence should lead to the experimental observation ⟨D∥⟩ ∝ Rp after spatial integration over the whole pore. The following a priori exponential radial dependence, D∥(a) = D0 − δD·exp(−a/lc), where δD = (D0 − DI) is a diffusion barrier, fulfills this requirement. Here, lc is a characteristic length, and a is the distance from the pore surface. This exponential form represents a first attempt expressing that the spatial variation of D∥(a) is proportional to the deviation from equilibrium value D0. It is similar to the radial behavior obtained when considering the release of steric hindrance on molecular reorientations in nanopores.42 Integrating over a uniform distribution of liquid molecules in the pore and developing in series on Rp/lc, we find ⟨D ⟩ ≈ DI +

δD ⎛ 1 Rp ⎞ R p ⎜1 − ⎟ nlc ⎝ n + 1 lc ⎠

1 1 εS ⎛ 1 1 ⎞ = + ⎜ − ⎟ ⟨D ⟩ Db V ⎝ Ds Db ⎠

where S and V are the surface area and the volume of the pore, respectively. Ds and Db correspond to D∥ in the surface and center pore phase, respectively, and ε is the thickness of the surface liquid phase, which is on the order of a few water molecules’ sizes.37,44,45 For Ds much smaller than Db, eq 3 simplifies to ⟨D ⟩ ≈

δD R p ≡ vcR p nlc

Ds R p ≡ vcR p mε

(4)

In eq 4, the gradient of the diffusion coefficient becomes νc = Ds/(mε), where m = 2 and 3 for cylindrical and spherical pores, respectively. A confined fluid can be in both monophasic and biphasic situations inside of pores because the interface can present heterogeneities inducing several coupling properties at the solid/fluid surfaces. For instance, this is the case for water in the presence of hydrophilic and hydrophobic heterogeneities. For these systems, the average ⟨D⟩ involves both the sum of ⟨D∥⟩ for coupled and ⟨1/D∥⟩ for uncoupled layers. Contrary to the standard PFG techniques that probe an average mixing micro-, meso-, and macrodiffusivity, the NMRD technique can separate the mesoscopic heterogeneities due to its frequency selectivity. In the following, we will prove the validity of this approach on a quantitative understanding of our NMRD experiments of our two different classes of samples. For the MCM41 materials, the quantitative analysis provides evidence of a mix of monophasic and biphasic situations. For the SEOS materials, the quantitative analysis allows one to discriminate several steps of the composed dynamical process. Figure 3 shows another single master curve that takes into account all of the 1H NMRD data of the MCM-41 samples. This master curve realized by the dual renormalization described above has been built in order to correspond to the saturated MCM-41 with a 3.3 nm pore diameter. Not only does this master curve drastically increase the number of data points, but the perfect continuity of the experimental points of these samples to a single master curve between 10 kHz and 10 MHz proves that the physical chemistry is conserved for both samples. This allows extension of the frequency range in a domain to 2.8 and 10 kHz for the sample with a 3.3 nm pore diameter. Thus, this induces better precision for extracting the dynamical parameters from our modeling. It is well-known that the proton longitudinal relaxation rate 1/T1 is a linear combination of spectral densities J(ω) at both Larmor and twice Larmor frequencies.46 One observes in Figure 3 two

(1)

where n = 3 and 4 for a cylindrical and spherical pore, respectively. Providing that DI ≪ (δDRp/nlc) and Rp < (n + 1) lc, eq 1 simplifies to the following linear pore size dependence of the average parallel diffusivity: ⟨D ⟩ ≈

(3)

(2)

Here, the already introduced gradient of the diffusion coefficient, νc = (δD/nlc), is now directly related to the liquid and interface properties through the diffusion barrier δD and the characteristic length lc. This length comes from a coupling between each layer parallel to the interface. This modeling that requires lc > Rp/(n + 1) implies a length scale of lc larger than the molecular distances. For a biphasic situation, there are two different cases when considering the efficiency of the exchange between the interface and center pore phase. For a very slow exchange, the average diffusion involves a parallel transport in the two phases where the overall diffusion is the weighted sum of diffusion in each phase. In this slow exchange situation, the NMR relaxations evolve separately in each phase, leading to a biexponential behavior. For a fast exchange, the diffusive transport involves serial states between the two phases. Here, this transport in 395

DOI: 10.1021/acs.jpclett.5b02718 J. Phys. Chem. Lett. 2016, 7, 393−398

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The Journal of Physical Chemistry Letters

between these two latter situations shows that the monophasic model is still relevant to describe the dynamics of the fast motion. Therefore, the monophasic model explains the proportionality between D and Rp. From these experimental values, an upper limit for lc can be computed using the expression νc ≈ (δD/3lc) at the pore interface (eq 2). The values of ⟨D∥⟩ show that DI ≪ D0 and νc ≈ (D0/3lc), where D0 = 2.3 × 10−9 m2/s is the free water diffusion, yielding lc = 7 nm. As the coupling is not sufficiently efficient, the value of lc is greater than the pore radius; however, it is strong enough to induce a radial diffusion dependence. The same approach for the slow motion characterized by τm2 = 5.4 μs leads to an extremely slow diffusion ⟨D2∥⟩ ≈ 1.0 × 10−14 m2/s with a diffusive meaning only valid between 2.8 and 10 kHz and a very weak νc = 6.1 μm/s leading to lc = 0.12 mm. Obviously, these meaningless weak and large values of νc and lc, respectively, show that only the biphasic fast exchange can be used to take into account the slow motion necessary for explaining the proportionality between D and Rp. The evaluation of νc is thus another key result of this study allowing probing of the efficiency of coupling between the surface and bulk fluids in this heterogeneous system. It is precisely the analysis of the master curve of Figure 3 that proves the heterogeneity of the interface. The frequency dependence allows separation of the slow and fast contributions coming from this heterogeneity. Hence, NMRD experiments are able to take into account the average diffusion coefficient involving both the sum of D∥ for coupled and 1/D∥ for uncoupled layers. The value lc = 7 nm verifies the condition lc > Rp/(n + 1) required for justifying eq 1, showing that the water proton dynamics is described beyond the molecular scale. In Figure 4, we build another master curve with all of the 1H NMRD data of the SEOS samples. The continuous line in

Figure 3. Master curve (diamond) builds with all of the MCM-41 1H NMRD profiles in order to correspond to the data of the saturated MCM-41 with a 3.3 nm pore diameter. Short and long dashed curves present the asymptotic behaviors, discussed in the text, at low and high frequency range, respectively. The continuous line shows the best fit of the whole master curve, as described in the text.

frequency regimes below and above 100 kHz represented by dashed lines and induced by slow and fast motions characterized by short and long translational correlation times τm1 and τm2 on the pore surface, respectively. These two contributions are analyzed using the spectral density, J(ω) = τm ln[1 + 1/(ωτm)2], where the best fit leads to τm1 = 0.32 ns and τm2 = 5.4 μs. This form of J(ω) comes from the Fourier transform of a pairwise autocorrelation function G(τ) = (τm/ |τ|)[1 − exp(−|τ|/τm)]37,43 describing the persistence of the dipolar correlation during molecular surface diffusion. Among the different possibilities to find the long time behavior 1/|τ| for G(τ),1,43,47 only the translational diffusion in a layered fluid in the spherical or cylindrical domain1 fulfills the diffusive transport in MCM-41. For the fast motion where ωτm1 < 1 in the whole frequency range studied, the asymptotic form J(ω) = −2τm1 ln(ωτm1), typical of surface diffusion, describes precisely the master curve. For the slow motion, the whole expression of J(ω) is needed, and the two asymptotic regimes, above and below ωτm2 = 1, are present in the frequency range below 100 kHz. Below 10 kHz, the Larmor period is longer than τm2; therefore, the surface diffusive regime is wellestablished and follows the previewed logarithmic asymptotic behavior. Now, near 100 kHz, the Larmor period is about two times τm2, and the molecular dynamics reduces to a simple hopping regime, where the spectral density J(ω) behaves proportional to 1/(ω2τm2), similarly with a Lorentzian when ωτm > 1. The continuous line in Figure 3, which represents the sum of these two dynamics contributions, fits the overall frequency dependencies of 1/T1(ω). The value of τm1 = 0.32 ns obtained for the fast motions leads to ⟨D∥,1⟩ = 1.8 × 10−10 m2/s with the hopping distance dm = 0.472 nm, which is typical of a silanol distance on silica surfaces.48 From the value of ⟨D∥,1⟩, one obtains an estimation of the diffusion gradients νc = 0.11 m/s at the pore interface. This diffusion gradient ν c characterizes indeed the efficiency of the coupling between layers in proximity of the pore surface. This coupling is strong enough to induce a radial dependence of D∥. However, one notes that such efficiency for confined water is smaller than the value, 1.33 m/s, necessary for reaching free water diffusion on a radius range size. The reduction of an order of magnitude

Figure 4. Master curve (diamond) builds with all the SEOS 1H NMRD profiles in order to correspond to the data of the saturated SEOS with a 10 nm pore diameter. I (intermittent) and D2D (surface diffusion) dashed curves present the asymptotic behaviors, discussed in the text, at low and high frequency ranges, respectively. The continuous line shows the best fit of the whole master curve, as described in the text.

Figure 4 is the best fit obtained through the frequency dependence of R1 ∝ R1,2D + R1,I, where R1,2D has logarithmic behavior above 3 MHz similar to the one used above for MCM41 and R1,I is an intermittent process induced by loops in water and adsorption steps at the interface below 3 MHz.49 This latter process has been useful to explain NMRD profiles obtained when exploring water dynamics in plaster settings.50 Figure 4 396

DOI: 10.1021/acs.jpclett.5b02718 J. Phys. Chem. Lett. 2016, 7, 393−398

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discrimination of several steps of a complex dynamics process composed by both loops in water and surface diffusion during adsorption events. On the basis of our experimental and theoretical results, we believe that the proposed noninvasive method allows exploration of the interplay between molecular and a continuous description of fluid dynamics in physical and biological confinements.

shows the efficiency of the NMRD technique to discriminate among the several steps of a complex dynamics process composed of both loops in water and surface diffusion during adsorption events. At low frequency, the experiment captures the complete statistics of this composed process as described by the intermittent theory.50 At high frequency, the relaxation is driven only by the surface dynamics during the adsorption events. At high frequency above 3 MHz, we can use the logarithmic spectral density J(ω) = τm ln[1 + 1/(ωτm)2] and obtain the surface translational correlation time τm = 77 ps and a jump size of 0.3 nm, leading to a diffusion coefficient of D2D = 2.9 × 10−11 m2/s and a estimation of diffusion gradient νc = 5.7 mm/s that is very small compared to the one of free water. This value of νc implies a very weak coupling, which justifies a biphasic modeling. At low frequency below 3 MHz, the characteristic frequency of the intermittent modeling is ωA = δ2/(2DIτ2A),50 where δ is the size of the smallest loop, DI is the coefficient of diffusion close to the pore surface, and τA is the average adsorption time. The best fit obtained with δ = 0.3 nm and the spectral density given in ref 50 lead to ωA = 3 × 105 rad/s, τA = 7.6 ns, and a diffusion coefficient DI similar to the one of free water. This low-frequency regime involves all of the adsorption and loop steps included in the intermittent dynamical processes. It becomes difficult to define an average ⟨D∥⟩ because the size and time scales characterizing these loops involve power-law probability density functions. However, Figure 4 shows that the master curve approach stays valid for this intermittent dynamics, which implies that ωA is linearly proportional to the pore radius Rp. Starting from DI ∝ Rp, τA must be proportional to 1/Rp for homogeneity reason, ensuring the linear proportionality between ωA and Rp. The ratio τA/τm, giving the average number of jumps on the pore surface during adsorption, generalizes the dynamical surface affinity previously introduced.51 As this ratio is a physical chemistry parameter characterizing the surface, it is independent of the pore size. Therefore, τA has the same pore size dependence as τm, which follows effectively a 1/Rp dependence, as already seen above. As a consequence, the master curve approach is sufficiently robust to take into account the intermittent dynamics process. In conclusion, we have proposed an original experimental method based on NMR at variable magnetic fields experiments (NMRD) and a theoretical analysis of the data that allows probing of the spatial dependence of the diffusion coefficient of liquids specifically in proximity of the pore surfaces. One of the key results found from these experiments is the linear relationship between average parallel diffusion coefficients and pore radii. Another result is the robustness of the frequency scaling of the master curve approach able to take into account the complexity of the water dynamics at the pore surface for samples of different geometries. This approach has proven useful for evaluating the efficiency of the coupling between liquid layers within nanopores by extracting gradients of diffusion coefficients. The application of this method to water confined in synthesized calibrated nanopores has been successful to deal with several dynamical processes on pore surfaces for different materials. This shows the ability of the proposed method to discriminate between the influence of the geometrical confinement on intrapore dynamics and the chemistry of the interface induced by different synthesis of the materials. For instance, the frequency selectivity of NMRD profiles has been able to separate the different couplings coming from the spatial heterogeneities on the pore surfaces. This frequency selectivity of NMRD has also allowed

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank E. Bloch and V. Hornebecq for SEOS synthesis, C. Tourné-Péteilh and J.-M. Devoisselle for useful discussions, and the ATILH association for funding support.

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REFERENCES

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DOI: 10.1021/acs.jpclett.5b02718 J. Phys. Chem. Lett. 2016, 7, 393−398

Letter

The Journal of Physical Chemistry Letters

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DOI: 10.1021/acs.jpclett.5b02718 J. Phys. Chem. Lett. 2016, 7, 393−398