Electrospun Carbon Fibers: Promising Electrode Material for Abiotic

Article pubs.acs.org/JPCC

Electrospun Carbon Fibers: Promising Electrode Material for Abiotic and Enzymatic Catalysis Adriana Both Engel,†,§ Yaovi Holade,‡,§ Sophie Tingry,*,† Aziz Cherifi,† David Cornu,† Karine Servat,*,‡ Teko W. Napporn,‡ and Kouakou B. Kokoh‡ †

Institut Européen des Membranes, UMR 5635, Place Eugène Bataillon CC 047, 34095 Montpellier, Cedex 5, France Université de Poitiers, IC2MP, CNRS UMR 7285, Équipe SAMCat, 4 rue Michel Brunet-B27, TSA 51106, 86073 Poitiers, Cedex 9, France

‡

S Supporting Information *

ABSTRACT: Carbon nanofibers (CNFs) are a promising material as conducting support of catalysts in (bio)electrochemical applications. Self-standing CNFs result from the carbonization at 1200 °C of electrospun polymer fibers with mean fiber diameter of 330 ± 50 nm. Such felts present interesting properties like fibrous and porous morphology to relieve the mass-transfer limitation of substrates and to provide high loadings of catalysts to enhance the electrochemical performances of the resulting electrodes. We show the beneficial feature of the CNFs as support compared with carbon dense structure for efficient immobilization of either abiotic catalysts based on metal nanoparticles or enzyme as biological catalyst. More specifically, palladium or platinum modified gold nanocatalysts remarkably boost the glucose electro-oxidation when deposited onto CNFs. Similarly, the immobilization of the bilirubin oxidase enzyme on the porous CNFs induces significant improvement of the mediated oxygen electroenzymatic reduction. The advances presented in this work show the high performance of the electrospun carbon fiber electrodes as promising materials for abiotic and enzymatic catalysis for the development of hybrid biofuel cells. industrialized.14 Starting from polymeric fibers obtained directly by electrospinning, a convenient heat treatment has the ability to transform the polymeric chains in carbon chains,15 thus enriching the fiber mat with several properties like electrical conductivity and chemical stability. Apart from electrospinning, other techniques can be employed for achieving carbon fibers, such as chemical vapor deposition (CVD),16 extrusion, and other types of spinning techniques (wet, dry, melt, or gel spinning for instance).17 Except for the CVD technique, all of the others are based on the synthesis of polymeric fibers, followed by transformation to carbon fibers via a carbonization process. The greatest advantage of electrospinning technique in preference to all of the other techniques presented is the possibility of obtaining fibers with very low mean diameter (in the range between some nanometers to some micrometers18) and long enough to form a free-standing paper or mat of fibers. Extrusion and spinning, for example, may create long fibers, but they are way thicker than electrospun ones, in the range of micrometers.19 By CVD, on the contrary, it is possible to obtain fibers with diameters in the range of tens of nanometers but with fiber lengths of only some millimeters.20 Electrospun

1. INTRODUCTION Different methods have been proposed to synthesize carbonbased materials for energy conversion and storage systems.1 With these synthetic routes, various types of carbon nanocomposites were successfully tailored, such as carbon nanotubes (CNTs), buckypaper, and carbon nanofibers (CNFs).2−4 The exceptional electrical, physical, and thermal properties of these advanced carbon-based materials make them a preferential choice in electronics, bionanotechnology, energy conversion, and storage.2,3,5 In catalysis, it was reported that the direct immobilization of metal nanoparticles onto CNFs induced an improvement on the catalytic ability of the NPs attributed to the strong interaction between the nanoparticles and the support.4,6 Besides, the 3D structure of these materials was found to be a promising choice of electrode material for enzymes immobilization in biofuel cells (BFCs).5,7 In this context, electrospinning can be considered as the technique of choice for the synthesis of such CNFs suitable as catalytic electrode materials. Electrospinning is a well-known and developed technique that appeared on the 1930s, and ever since it has found its way on countless applications. Examples go from filtration,8 biosensors,9 wound healing,10 composites reinforcement,11 drug delivery,12 energy generation devices,13 among others. So important is the relevance of the technique that it has surpassed laboratory prototypes and started being © 2015 American Chemical Society

Received: May 6, 2015 Revised: June 8, 2015 Published: June 11, 2015 16724

DOI: 10.1021/acs.jpcc.5b04352 J. Phys. Chem. C 2015, 119, 16724−16733

Article

The Journal of Physical Chemistry C carbon fibers possess interesting properties for applications as electrode materials because the small mean fiber diameter leads to high specific surface, which is crucial for the immobilization and the accessibility of high amount of species on its surface. Moreover, the long fibers obtained allow the creation of a freestanding electrode, with no need of supporting material. We present the structural and electrochemical characterizations of electrospun CNFs. The aim is to show the advantage of the electrospun CNFs as suitable 3D conducting support for catalysts immobilization, compared with a conventional carbon dense electrode composed of glassy carbon (GC). The electrochemical performances of the two different supports are compared for both abiotic and biological catalysts. All of the tests are performed in a phosphate-buffered solution at pH 7.4, which is a physiological condition suitable to future perspectives in implanted devices. First, the supports are tested regarding the immobilization and the catalytic activity of palladium or platinum modified gold nanocatalysts (AuxPty and AuxPdy), deposited on Vulcan carbon, toward the glucose oxidation reaction. Prior to the electrochemical tests, different physicochemical characterizations of these nanocatalysts are performed. The efficiency of the CNF support toward GC is pointed out with Au/C and Pd/C nanoparticles. Second, bilirubin oxidase enzyme (BOD) with the mediator ABTS is entrapped in Nafion matrix deposited on the supports and the as-prepared biocathodes are applied to the bioelectrocatalytic reduction of O2. The optimization of both abiotic anode and biocathode shows that electrodes based on electrospun carbon fibers will be excellent candidates for developing hybrid BFCs.

2000 rpm. PAN solution was pumped with a feed rate of 2.4 mL h−1 through a needle with inner diameter of 800 μm. The tip-to-collector distance was established as 14 cm, and the process was carried out for 3 h under high voltage of 20 kV. By the end of the process, a felt of PAN nanofibers was removed from the apparatus and stabilized under an air environment at 250 °C for 2 h with a heating rate of 2 °C min−1 in an ashes furnace from Vecstar. The final step for the achievement of CNFs was the carbonization at 1200 °C (1 h dwell) in a highpurity nitrogen atmosphere, with a heating rate of 2 °C min−1 in a tubular furnace (Vecstar VTF-4, England). The obtained CNFs mats were freestanding and easy to handle and were cut into strips to serve as electrodes for further tests toward glucose oxidation and O2 electroreduction. 2.2. CNF Morphological Characterizations. The morphology of the CNFs synthesized via electrospinning was analyzed by X-ray diffraction (XRD) with a PANalytical XpertPRO diffractometer equipped with an X’celerator detector using Ni-filtered Cu Kα-radiation at wavelength of 1.5406 Å. The specific surface of the fibers was measured by BET method with a Micrometrics ASAP 2010 instrument.21 Finally, fibers structure was observed by scanning electron microscope (SEM) on a Hitachi S4800 microscope after metallization by gold sputtering. Mean fiber diameter was measured with ImageJ free software after the analysis of at least 50 fibers on several SEM images. 2.3. Abiotic Nanocatalysts Synthesis from Bromide Anion Exchange Method. The synthesis method used herein was the so-called bromide anion exchange (BAE) method.22,23 As previously shown, this synthetic route gives well-dispersed nanoparticles with high electrochemical active surface area and good catalytic properties.22,24 The preparation of the different kinds of metal nanoparticles deposited onto Vulcan XC 72R was carried out in a thermostated glass reactor (500 mL) with magnetic stirring. To prepare the supported nanomaterials, we dissolved a suitable amount of metal(s) salt(s) in 100 mL of ultrapure water (Milli-Q Millipore, 18.2 MΩ cm at 293 K), initially thermostated at 25 °C. Under vigorous stirring, an amount of potassium bromide (KBr, ≥ 99%) purchased from Sigma-Aldrich was added with a molar ratio φ = n(KBr)/ n(metal(s)) fixed to 1.5. This ratio was chosen from different studies by varying φ from 0 to 20.4. After a few minutes of homogenization, a suitable amount of Vulcan XC 72R carbon black (thermal-annealed at 400 °C under a nitrogen atmosphere for 4 h) was added under constant ultrasonic homogenization to obtain a metal loading of 20 wt %. Two reduction agents were used to synthesize the nanomaterials: sodium borohydride (NaBH4, 99%) and L-ascorbic acid (AA, ≥ 99%) provided by Sigma-Aldrich. For Pt/C, Au/C, and AuPt/C bimetallics, 15 mL of NaBH4 (0.1 mol L−1) was used. Besides, for Au/C, Pd/C, and AuPd/C, 13 mL of AA (0.1 mol L−1) has been employed. This latter hydrogen-less reducing agent (AA) allows avoiding the hydrogen insertion in the palladium crystallographic network. After this reduction step, the reactor temperature was elevated to 40 °C for 2 h. Finally, the supported materials were filtered, rinsed several times with ultrapure water, and dried in an oven for 12 h. Herein, various atomic compositions of bimetallics Au 100−x Pt x /C and Au100−xPdx/C have been synthesized and tested toward electrochemical reactions. 2.4. Physicochemical Characterizations of the Abiotic Nanocatalysts. Thermogravimetric analysis (TGA) was used to determine the metal loading. The thermodynamic reaction

2. EXPERIMENTAL SECTION 2.1. Electrospinning Setup for Carbon Nanofibers Preparation. An electrospinning solution was prepared by mixing a necessary amount of polyacrylonitrile (PAN, purchased from Goodfellow, reference AN316010/3, 50 μm copolymer 99.5% AN/0.5% MA) in previously heated DMF (N,N-dimethylformamide from Sigma-Aldrich) at 70 °C to obtain a 10 wt % solution. The mixture was kept under constant stirring at 70 °C for 6 h until a translucent yellowish solution was obtained. After the cooling of the solution to room temperature, electrospinning was executed on an IME Technology apparatus equipped with a rotating drum collector, as illustrated in Figure 1. The drum (diameter of 10 cm) was covered with aluminum foil, and its rotating speed was fixed to

Figure 1. Electrospinning apparatus employed for electrospun fibers fabrication. Photographs of electrospinning apparatus courtesy of Adriana Both Engel. Copyright 2015. 16725

DOI: 10.1021/acs.jpcc.5b04352 J. Phys. Chem. C 2015, 119, 16724−16733

Article

The Journal of Physical Chemistry C

purchased from Sigma-Aldrich) was added thereafter in a concentration of 5 vol % and rapidly mixed together, thus finalizing the enzyme solution preparation. 40 μL of the final solution was dropped on the electrode (geometric surface of 0.7 × 0.7 cm2 each face, that is, 0.98 cm2) and left to dry overnight at 5 °C. Nafion acts as an insoluble polymer electrolyte to hold the species on the electrode surface. The enzyme loading was varied on the electrode: 2.8, 5.6, 8.4, 11.2, 16.8, and 22.4 nmol cm−2 were deposited relative to the electrode geometric surface. 2.6. Electrochemical Measurements. Support Characterization. Tests were realized on CNFs or GC electrodes for comparison of the effect of porous and dense structure on the kinetics of Fe(CN)63−/Fe(CN)64− redox reaction. Five mM of K4Fe(CN)6·3H2O and 5 mM of K3Fe(CN)6 (potassium hexacyanoferrate(II) trihydrate and potassium ferricyanide (III), purchased from Sigma-Aldrich) were added to PBS 0.1 M at pH 7.4, with Ag/AgCl/Clsat− as reference electrode, stainless steel as counter electrode, and either the synthesized CNFs or GC as working electrode support. Blanks were realized with only PBS solution, and the measurements were performed with a potentiostat (Ametek, Versa Stat 3). Anode Characterization. An analogical potentiostat EG&G PARC model 362 (Princeton Applied Research) was used to characterize the anode catalysts in a conventional threeelectrode cell. Typically, a homemade reversible hydrogen electrode (RHE) was used as reference electrode. To avoid any pollution that could affect the potential control, it was separated from the solution by a Luggin capillary tip. GC with 6.48 cm2 geometric surface area served as counter electrode, while the CNF mat (1 cm × 1 cm) was used as conducting support to fabricate the working electrode. The catalytic ink was prepared similarly to our previous paper,24,25 by mixing water, Nafion suspension, and a catalytic powder. For GC electrode, previously well-polished with alumina powder, 3 μLink cm−2 was deposited for electrochemical tests. For CNFs, different volumes of the catalytic ink were tested: 20, 25, 30, 40, and 50 μLink cm−2 (immobilization on each side), and only the optimized ink volume was used for the following studies. The cyclic voltammetry experiments were fulfilled in 0.2 mol L−1 PBS at pH 7.4 and 37 °C. Before performing any electrochemical experiment, the working solution was deoxygenated by bubbling nitrogen for 30 min. In a typical CO stripping experiment, CO (Air Liquide, ultrapure) was adsorbed at 0.10 V versus RHE (Pt/C) or 0.3 V versus RHE (Pd/C) for 5 min, followed by the solution degassing for at least 30 min before CV experiments. Then, 10 mmol L−1 of glucose (D-(+)-glucose, Sigma-Aldrich, 99.5%) was used as substrate to investigate the catalytic activity of the as-prepared anode materials. Biocathode Characterization. O2 reduction tests were carried out on a Teflon open cell under magnetic stirring on a heating plate. The temperature of the cell containing a preheated O2-saturated PBS solution was kept constant at ca. 35 °C. The reference electrode was Ag/AgCl/Clsat−, and the results were later referred to the RHE scale. The counter electrode was a stainless-steel slab, and the working electrode was composed of the freestanding CNFs mat (ca. 1 cm2 of dipped surface) with BOD-ABTS couple immobilized, as previously mentioned.

was also studied by coupling differential thermal analysis (DTA) to TGA. Both analyses were conducted on TA Instruments SDT Q600 apparatus equipped with TA Universal Analysis software. In standard test, ca. 7 mg of the sample was put in an alumina crucible and thermally heated under air flow of 100 mL min−1 from 25 to 900 °C following a heating rate of 5 °C min−1. A second empty crucible was used as reference. To determine the overall atomic composition of metals in the bimetallic catalysts, we used the inductively coupled plasma optical emission spectrometry (ICP-OES) technique. For the analysis, ca. 10 mg of the catalyst powder was introduced into a reactor, followed by the addition of aqua regia. The mixture (acid + catalyst powder) was then heated in the microwave for complete dissolution/mineralization of the metal. Chemical analyses were finally performed by using a spectrometer Optima 2000 DV (PerkinElmer). XRD was used to determine the crystallographic structure and the crystallite size of the as-synthesized nanostructures. The raw XRD pattern data were recorded using ENPYREAN (PANalytical) diffractometer in Bragg−Brentano (θ−θ) configuration with a copper tube powered at 45 kV and 40 mA (Cukα1 = 1.54060 Å and Cukα2 = 1.54443 Å). The kβ component interference was avoided by installing a nickel filter in a secondary optic. The diffractogram acquisition was realized from 15 to 140° in diffraction angle (in 2θ) every 0.05° with respect to a time of 2 min/step. Finally, information on interest was extracted from the patterns by using the common pseudoVoigt function on HighScorePlus software; however, XRD patterns shown herein are not fitted. Furthermore, particles dispersion on Vulcan carbon and the mean particles size assessment were investigated by transmission electron microscopy (TEM). A TEM/STEM JEOL 2100 UHR (200 kV) equipped with a LaB6 filament was used for the micrographs observation. The ImageJ free software was used to get the size distribution and mean particle size by measuring the diameter of at least 600 isolated particles. Then, this mean particles size was compared with those determined by considering the statistic methods. We also investigated the homogeneity of the catalysts by performing the energydispersive X-ray (EDX) analysis. Indeed, during the highresolution TEM (HRTEM) analysis it is possible to quantify the elementary particle composition. The JED Series AnalysisProgram provided by JEOL Company was used during this EDX analysis. 2.5. Biocathode Preparation. CNF-supported biocathodes to be employed for O2 enzymatic electroreduction were prepared via the drop-casting technique, resulting in an electrode with enzyme, mediator, and carbon particles immobilized on the surface of CNFs entrapped in a Nafion matrix. The preparation protocol consisted of, first, ultrasonicating Super P carbon particles (0.04 μm grain size, specific surface area 62 m2 g−1 purchased from Timcal) with an optimized concentration of 7.5 mg mL−1 in phosphate-buffered solution (PBS) for 30 min in a sonication bath. PBS used in this work was prepared from sodium dihydrogen phosphate monohydrate (NaH2PO4·2H2O, Acros Organics; ≥ 99%) and disodium hydrogen phosphate (Na2HPO4, Fluka Biochemika; ≥ 99.5%) salts using ultrapure water. Then, BOD enzyme (33.6 mg mL−1) (bilirubin oxidase, activity 3.04 U mg−1 purchased from Amano) and ABTS (3.4 mg mL−1) (2,2′-azinobis(3ethylbenzothiazoline-6-sulfonate) diammonium salt, from Sigma-Aldrich) were added to this solution and mixed in a vortex for 20 min. Nafion (Nafion 117 suspension 5 wt %,

3. RESULTS AND DISCUSSION 3.1. Carbon Nanofibers: Morphological and Electrochemical Characterizations. In this work, electrospinning 16726

DOI: 10.1021/acs.jpcc.5b04352 J. Phys. Chem. C 2015, 119, 16724−16733

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The Journal of Physical Chemistry C

Figure 2. (a) Image of a strip of the resulting free-standing electrospun CNFs mat, measuring ∼2.5 × 1 cm. (b) SEM image of the carbon fibers with mean fiber diameter of 330 ± 50 nm. Magnification factor of 10 000; sample was metallized with gold sputtering prior to observation. (c) XRD pattern of the CNFs (carbonization temperature = 1200 °C). (d) Cyclic voltammograms of 5 mM K3Fe(CN)6 + 5 mM K4Fe(CN)6 at a CNFs electrode in comparison with a glassy carbon electrode, in PBS 0.1 M, pH 7.4, scan rate 10 mV s−1.

Table 1. Physicochemical Data of the As-Prepared Nanocatalysts from DTA-TGA, ICP-OES, XRD, TEM, and EDX Analyses technique/parameter DTA-TGA XRD

TEM

ICP-OES

EDX

synthesis yield: ϖ (%) a

Tpeak (°C) wt % LV (nm) a(111) (Å) aBragg (Å) Dm (nm) Ds (nm) Dv (nm) Dm,p (nm) Pt (at %) Pd (at %) Au(at %) Pt (at %) Pd (at %) Au (at %)

Pt

Aua

Pd

Au80Pt20

Au60Pt40

Au90Pd10

428 18 3.9 3.916 3.907 3.1 3.5 3.7 3.1

610 21 7.0 4.074 4.064 5.8 7.4 8.1 5.7

538 19 12.0 3.889 3.880 9.0 11.6 12.7 8.5

627 20 5.2 4.060 4.059 3.6 4.7 5.2 3.4 26

584 17 4.3 4.056 4.050 3.7 4.6 5.0 4.7 42

622 19 9.4 4.053 4.057 7.7 10.2 11.1 7.6

74 22

58 43

78 94

57 95

91

94

97

90 10 6 94

14 86 98

Au: it is Au/C-NaBH4, using NaBH4 as reducing agent.

As shown in Figure 2d, both electrodes exhibit well-defined redox peaks. The value of the peak separation (difference between anodic peak potential and cathodic peak potential, ΔEp) is 0.13 V for CNFs electrode and 0.14 V for GC electrode; however, a current amplification by a factor of 6 of both anodic and cathodic peaks is observed when CNF electrode is employed instead of GC. This behavior is ascribed to the porous nature of the CNFs mat offering a higher accessible electroactive area. 3.2. Glucose Oxidation on Abiotic Catalysts. 3.2.1. Structural and Physicochemical Characterization of the Metallic Nanoparticles. Because a chemical method was used to synthesize the nanomaterials, the chemical yield (ϖ) of the synthesis was first checked. It is defined as the ratio (or percent) between the experimental mass of the catalyst powder and the theoretical expecting weight according to the initial composition on the reactor. The obtained values of ϖ are reported in Table 1. Furthermore, the reproducibility of the synthesis is well-guaranteed in terms of chemical yield and electrochemical results herein and in our previous reports.22 As indicated in Table 1, ϖ goes from 91 (Pt/C) to 98% (Au90Pd10/C). In the entire synthesis, the chemical yield was always superior to 90%, providing evidence of a reliable synthetic route.28 Because the value of ϖ gives only qualitative information about the metal content, TGA analysis was

technique was employed for the synthesis of PAN nanofibers that, after convenient thermal treatment under a nitrogen atmosphere at 1200 °C, yielded carbon fibers with suitable properties for the proposed application. A photograph of the free-standing CNFs mat obtained is presented in Figure 2a. Figure 2b shows a SEM image of the CNFs, characterized by a smooth surface and a mean fiber diameter of 330 ± 50 nm. The specific surface of the carbon fibers was measured by the BET method (fulfilled on Micrometrics ASAP 2010 instruments), and a value of 14 m2 g−1 was found, in good agreement with literature values for electrospun carbon fibers.3,26 Figure 2c presents the XRD pattern of the synthesized electrospun CNFs. The broad peak in the region of 2θ near 25° corresponds to the (002) planes, and the smaller peak near 44° corresponds to the (100) planes. This result is in good agreement with the literature and indicates the presence of graphitic domains, even though the peak being broad indicates that fibers are more turbostratic than crystalline.27 Full widths at half-maximum (fwhm, β) and the scattering angles of (002) planes for the CNFs are of β = 0.0977 rd (5.6°) and 2θ = 24.62°. The crystallite size parameter Lc of 1.45 nm was calculated with the aid of the Scherrer equation (Lc = 0.9λ/(β cos θ)). Cyclic voltammetry technique using Fe+III/Fe+II redox probe was used to assess the electron-transfer properties on the surface of the CNFs electrode compared with a GC electrode. 16727

DOI: 10.1021/acs.jpcc.5b04352 J. Phys. Chem. C 2015, 119, 16724−16733

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The Journal of Physical Chemistry C

induces a high catalytic activity.32 For Au80Pt20/C, a small shift was also observed in the XRD patterns. Quantitative basic data were extracted from the patterns, like crystallite size (LV) and lattice parameter by considering XRD fundamental laws.33 Because the three kinds of metals (Au, Pt, Pd) have cubic symmetry, the determination of LV was realized by applying the Debye−Scherrer equation.33 The lattice parameter was determined by using the interplanar spacing of crystallographic plane (hkl) method and Bragg’s law.21 The different parameters are summarized in Table 1. The lattice parameter for Pd was ca. 3.89 Å for all of the (hkl) planes. This indicates clearly that there is no Pd crystalline structure modification by hydrogen insertion. Indeed, the hydrogen insertion in Pd structure inducing the lattice parameter dilation is a well-documented science,34−36 and it can happen during the nanoparticles synthesis when using a hydrogen-source agent like NaBH4.22,31 Both Au/C-AA and Au/C-NaBH4 have the same lattice parameter, which is close to 4.07 Å, in agreement with the reported bulk and nanomaterial value.23 The value of the lattice parameter of the Pt nanoparticles (aBragg = 3.907 Å and a(hkl) = 3.916 Å for (111) plane) is significantly different and especially smaller than the bulk (3.924 Å).37 This contraction phenomenon is not surprising because it has been observed in the literature for Pt nanoparticles and has been attributed to quantum size effect or surface tensions.37,38 According to Vogel,38 this effect of lattice contraction can be explained by the decrease in the number of nearest neighbors of the surface atoms, which will further affect the valence electrons distribution in a smaller number of metallic bonds. Figure S1a−f in the Supporting Information (SI) shows a TEM micrographs of the obtained structures. Particle agglomerations were noticed when using AA as reducing agent. As aforementioned, the mean particle size (Dm,p, by assuming a spherical particle) was evaluated by considering the most representative number (>600 isolated particles) and fitting the histograms by LogNormal function. The particle size distribution for the different materials is represented by histograms in Figures S1g−l in the SI. The values of Dm,p were compared with the statistical values,21 and the resulted values are summarized in Table 1. For all particles, Dm,p ≈ Dm. This means that the particles are “quasi-spherical”. For palladium nanoparticles, we could expect a slight difference because the Br− ion used herein as “surfactant” induces preferential orientation toward (100) crystallographic plane in free-support environment. The particles with (100) crystallographic planes in fcc symmetry are cubic, not spherical. On the contrary, the presence of the support (actually, Vulcan XC 72R) will strongly affect the shape of the obtained nanoparticles. Nevertheless, many Pd cubic nanoparticles were observed during TEM micrographs acquisition of Pd/C materials. (See the inset in Figure S1c in the SI.) The presence of low coordination indices (100) or (111) has been found to boost the catalytic activity of palladium.21 In general, LV must be substantially higher than Dm,p, and the gap between the two parameters is progressively more pronounced as the particle size decreases (