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Mesoporous CuO nanostructures for low-temperature CO oxidation

SOURAV GHOSH1,2, SUKANYA KUNDU1,3and MILAN KANTI NASKAR1,3,*

1CSIR - Central Glass and Ceramic Research Institute, Kolkata 700032, India

2Technical Research Centre, S.N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700106, India

3Academy of Scientific and Innovative Research (AcSIR), Ghaziabad 201002, India

*Author for correspondence (milan@cgcri.res.in)

MS received 1 December 2020; accepted 14 February 2021

Abstract. Preparation of CuO nanostructure was reported by oxalic acid-assisted wet-chemical method in aqueous medium. As-prepared sample was identified as C2CuO4nH2O, which was further transformed into CuO after heat treatment. To enhance the textural property, CTAB was employed as soft-templating agent. DTA–TGA characterization was performed to investigate the thermal stability of as-prepared C2CuO4nH2O sample, whereas Raman and XPS measurements confirmed the presence of CuO. FESEM and TEM studies revealed porous architecture with shelled interior for CuO sample. The formation of porous network could be demonstrated by oxidative decomposition of C2CuO4nH2O due to high temperature calcination. The BET surface area and pore volume were found to be 51 m2g-1 and 0.4492 cc g-1, respectively. The catalytic activity of sample was investigated for CO oxidation and achivedT50and T100at 133 and 175°C, respectively, which were further compared with commercialized CuO sample and previously reported data.

Keywords. CuO; mesoporosity; ligand-assisted synthesis; nanostructure; CO oxidation.

1. Introduction

Toxicity of carbon monoxide (CO) is well known since it is often able to eliminate oxygen from blood and form com- plex with haemoglobin [1]. The industrial demand for low- temperature catalytic oxidation of CO has raised steadily in context of fuel production, gas sensors, automobile indus- try, air quality measurement and gas mask [2]. CO oxida- tion is also important for proton exchange membrane fuel cells (PEMFCs) where trace level CO (below 10 ppm) within H2could act as a poison for platinum (Pt) electrode [3]. Precious metal-based catalysts particularly palladium (Pd), platinum (Pt) and gold (Au) have enormously used for the same purpose. Excellent performance of precious metal catalyst could be explained on the basis of their efficiency for low-temperature molecular oxygen dissociation, strong affinity towards atomic oxygen and CO, and uniform active site distribution [4]. Nevertheless, low availability and high price limit their bulk scale application in commercial mar- ket. Among non-precious transition metal oxides, copper oxide (CuO)-based composite network has been widely exploited for low-temperature CO oxidation [5]. In this regard, CuO–CeO2composite was broadly employed due to

their synergetic redox dynamics between Cu?2/Cu?1 and Ce?4/Ce?3[6]. Incorporation of porous supporting materi- als along with CuO–CeO2 catalyst increased active site dispersion throughout the interfacial framework, which further resulted significant improvement in catalytic per- formance [7]. Luo et al[8] found that higher surface area and pore diameter of silica as catalytic support played the decisive role to enhance the catalytic performance of CuO–

CeO2/SiO2catalyst for CO oxidation. Lunin and co-workers [9] described the importance of wood sawdust-assisted bio- templating route to design biomorphic catalyst and showed promising catalytic activity for CuO–Ce0.8Zr0.2O2 via enhancement in oxygen mobility and CO adsorption over the catalytic surface. Recently, CuO/TiO2 composite has achieved propitious efficiency as a splendid catalyst for the same issue [10]. Despite the numerous studies on CuO- based composite network, application of unsupported mesoporous CuO nanostructures, particularly for low-tem- perature CO oxidation is rarely reported in literature [11,12].

In this report, we have illustrated ligand-assisted soft- templating method for the preparation of mesoporous CuO nanostructures using cetyltrimethyl ammonium bromide

Supplementary Information: The online version contains supplementary material available at https://doi.org/10.1007/s12034-021- 02475-6.

https://doi.org/10.1007/s12034-021-02475-6

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(CTAB) as soft-templating agent. Briefly, CuO nanostruc- ture was prepared at 80°C for 2 h in aqueous medium fol- lowed by annealing at 400°C with 1°C min-1heating rate.

Further, catalytic oxidation of CO was investigated by mesoporous CuO nanostructures and compared with com- mercial CuO as reference. Herein, the role of CTAB on the textural property (such as surface area, pore volume and pore diameter) of CuO has been discussed and directly correlated with catalytic performance for CO oxidation. The synthetic approach is significant in terms of shorter reaction time, aqueous-based method and controllable mesoporous architecture. Again, mesoporous CuO displayed 50 (T50) and 100% (T100) CO oxidation at 133 and 175°C, respec- tively. This study primarily unlocks the strategy to use unsupported mesoporous CuO nanostructures as a potential catalyst, which is reusable for low-temperature CO oxidation.

2. Materials and methods

2.1 Materials preparation

All reagents in this report were used without any additional purification.

Synthesis of CuO nanostructure was performed by one- pot wet-chemical method. Twenty millimoles of CTAB and 4 mmol of copper nitrate (Cu(NO3)23H2O) were dissolved in 140 ml of deionized water (DI) and stirred at 80°C for 2 h. Sixty millilitres of aqueous solution containing 12 mmol oxalic acid was added dropwise and stirring was continued for 30 min. Then, subsequent precipitate was centrifuged, followed by washing with DI and dried at 60°C for 12 h. The as-prepared sample was calcined at 400°C with 1°C min-1heating rate and 2 h soaking period. The calcined sample is indexed as CMCO-4. To investigate the role of CTAB, the same experiment was repeated without CTAB and assigned as MCO-4.

2.2 Characterization

The structural investigation was examined by powder X-ray diffraction method (PXRD) with Philips X’Pert Pro PW 3050/60 (CuKa, nickel filter setup, 30 mA and 40 kV), X-ray photoelectron spectroscopy (XPS, Thermo Fisher Scientific), Fourier transform infrared spectroscopy (FTIR, PerkinElmer Spectrum Two instrument), differential ther- mal analysis with thermogravimetry (DTA–TG, Netzsch STA 449C, Germany), Raman spectroscopy (RENISHAW equipment, Argon 514 nm laser) and H2-TPR measurement (Micromeritics Chemisorb 2720 instrument). Textural properties (surface area and porosity) were studied by multi-point BET measurement (ASIQ MP, Quantachrome).

Morphological study was carried out by field emission scanning electron microscope (FESEM, SUPRA 55-VP,

10 kV) and transmission electron microscope (TEM, JEM 2100F field emission, 200 kV).

2.3 Catalytic experiment

A fixed-bed glass based mechanical reactor (i.d. = 4 mm) was constructed with continuous flow facility. The pre- treatment of catalyst was performed with a gas mixture containing 80 vol% He and 20 vol% O2 for 1 h. Next, a standard gas composition (1 vol% CO, 20 vol% O2 and balance with N2) was passed through the sample and con- version was checked at different temperatures using on-line GC (gas chromatography, Varian CP3800) equipped with thermal conductivity detector (TCD). To attain the steady state, catalyst bed was stabilized at each temperature for 30 min. For this experiment, catalyst loading, flow rate and weigh hourly space velocity (WHSV) were maintained as 50 mg, 40 ml min-1and 48,000 ml g-1h-1, respectively, throughout the process. The conversion efficiency was calculated from the area under the curve [13]:

%Conversion ¼ CO2 peak area=

CO2 peak area þ CO peak area

ð Þ:

The conversion rate from CO to CO2 was calculated using the following equation:

Conversion rate mol s1m2

¼CCOðXCO=100Þ=ð0:05SAÞ;

where SA refers surface area of the catalyst. Furthermore, the parameter (0.059SA) signifies specific surface area for 50 mg catalyst loading. CCOand XCOattribute to moles of CO passed per second through catalyst bed and per- centage of CO conversion, respectively. TheEa(energy of activation) value was determined from the slope of ln (conversion rate) versus 1/T plot.

3. Results and discussion

3.1 Structural studies

As shown in figure 1a, PXRD spectrum with 2h= 22.9°, 36.2°, 38.8°, 42.4°, 46.8° and 52.0° can be ascribed as C2CuO4nH2O (JCPDS 21-0297) for as-prepared sample.

After the calcination at 400°C, diffraction peaks around 2h= 32.5°, 35.4°, 35.55°, 38.7°, 48.7°, 53.4°, 58.3°, 61.5°, 66.2° and 68.1°clearly represents the appearance of mon- oclinic CuO (JCPDS 05-0661). No other phase, such as metallic Cu or cuprous oxide (Cu2O) could be detected for both as-prepared and calcined samples. Thermal stability of as-prepared C2CuO4nH2O sample was investigated by DTA–TG measurement (figure 1b). In DTA–TG analysis, the conspicuous exothermic peak around 310°C manifests decomposition of oxalate and organic residue with distinct

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residual mass loss up to 46%. No mass loss was further conceived within 300–500°C indicating complete removal of H2O and CO2from C2CuO4nH2O. On the basis of DTA–

TG study, annealing temperature was selected at 400°C for the complete transformation of CuO from C2CuO4nH2O precursor. Figure1c displays Raman peaks around 274, 319 and 610 cm-1 indicating existence of phase-pure CuO for calcined samples. The peak position at 274 cm-1 corre- sponded the Ag mode, while 319 and 610 cm-1 appeared for Bg mode of phonon vibration [14]. It is worthy to be stated that only three Ag?2Bg phonon modes are active among twelve 4Au?5Bu?Ag?2Bg modes for CuO (space group: C62h). Furthermore, shifting the Raman peaks to lower wavenumber indicates decrease in particle size for as-prepared CMCO-4 compared to MCO-4. Elemental composition and phase purity of calcined sample we further characterized by XPS study. For Cu 2p spectrum, peak position allocates as 932.3 and 952.3 eV conveying Cu 2p3/2and Cu 2p1/2transitions, respectively (figure1d). The difference between Cu 2p3/2and Cu 2p1/2is 20.0 eV, which further supports the presence of phase-pure CuO according to the previous study [15]. In addition, appearance of satellite signals at 940.3 and 943.3 eV for Cu 2p3/2 and 961.8 eV for Cu 2p1/2, confirmed the existence of Cu(II) state in calcined sample. As-displayed in O 1s spectrum, broad peak around 529.18 and 530.6 eV can be recognized as lattice and adsorbed oxygen, respectively, for CuO sample (figure 1e). However, it is already reported that the coordination environment of surface adsorbed oxygen spe- cies (Ob) was chemically different from structural oxygen (Oa) of lattice site [16,17]. The oxygen vacancy was further

calculated from (Ob/Oa) ratio and found to be *0.74, indicating the enhanced contribution of lattice oxygen compared to adsorbed oxygen species for catalytic network [18].

For calcined sample, surface area and porosity mea- surement were carried out by nitrogen adsorption–desorp- tion experiment. The isotherm is type IV in nature with H-3 hysteresis loop stating mesoporosity and slit-like pore geometry for CuO (figure2a) [19,20]. The external surface area and pore volume were obtained to be 51 m2g-1and 0.4492 cc g-1, respectively. The broader pore-size distri- bution renders interparticle porosity with asymmetric size and shape (figure 2b). For the controlled experiment pur- pose, textural property was also measured for as-prepared CuO sample without using CTAB (MCO-4) and compared with commercialize CuO (reference) sample. The surface area value was found to be 33 and 0.65 m2g-1for MCO-4 and commercialized CuO sample, respectively. The enhancement in surface area for CMCO-4 could be attrib- uted to the templating behaviour of CTAB. This synthetic strategy is advantageous in terms of surface area and porosity compared to commercialized CuO sample.

icrostructural analysis was performed by FESEM and TEM for both C2CuO4nH2O and CuO. For as-prepared C2CuO4nH2O sample, FESEM image shows mostly micro- spheroid like architecture with non-uniform size and shape (figure3a). The structural assembly was 0.5–1.5lm in size and surface smoothness was clearly visible throughout the sample. TEM study further supports the same and no such distinct porosity was observed within as-prepared C2CuO4 nH2O (figure3b and c). Furthermore, higher magnification

20 30 40 50 60 70

CuO

C2CuO4-nH2O JCPDS File: 21-0297

(121,130)

(220)

(111)(011,101)(120,210)

(110) (020) (202) (-113) (-311,310) (220,113)

(-202)

(111,200)

(002,-111)(110)

JCPDS File: 05-0661

In te nsity (a.u.)

2 Theta (degree)

0 100 200 300 400 500

50 60 70 80 90 100

Temperature (°C)

R e sidual m ass (%)

Ex o

200 400 600 800 1000

Bg Bg

Ag

Raman shift (cm

-1

)

In te nsity (a .u.)

(a) (b) (c)

925 930 935 940 945 950 955 960 965 (20.0 eV)

(Sattelite) (Sattelite)

Cu 2P1/2

Cu 2P3/2

(952.3 eV) (932.3 eV)

In te nsity (a .u.)

Binding Energy (eV)

524 526 528 530 532 534 536 530.6

Spectrum Fit Peak 1 Fit Peak 2 Cumulative Fit Peak 529.18

In te nsity (a .u.)

Binding Energy (eV)

(d) (e)

(i) (ii)

Figure 1. (a) XRD patterns of as-prepared (i) C2CuO4.nH2O and (ii) CuO. (b) DTA–TG and (c) Raman plot of CuO sample. XPS plot of (d) Cu 2p and (e) O 1s pattern for as-prepared CuO sample.

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image discloses that nanoparticles were self-aggregated together to form micro-spheroid nanostructures along with the appearance of numerous stacking faults at the edge of architecture (figure 3d). HRTEM image visibly depicts 0.247 nm interplanner spacing for (120) plane of C2CuO4nH2O (figure3e). After calcination, mostly porous micro-spheroid like architecture was obtained for CuO sample (figure 4a). TEM image renders porous nanostruc- tures with shell-like interior and a significant contrast between solid and hollow counterpart was encountered (figure4b and c). Close inspection manifests stone-wall like assembly with particle size of about 25–40 nm (figure4d). The inter-layer spacing was accounted as 0.25 nm representing (002,-111) planes of CuO sample (figure4e). The formation of micro-spheroid like nanostructure with shell-like interior can be demonstrated by oxidative decomposition of C2CuO4nH2O due to high temperature heat treatment [21,22].

3.2 Catalytic studies

The experimental results for catalytic oxidation of CO to CO2is manifested in figure5a. For the CMCO-4, 25% (T25) and 50% (T50) conversions reached at 94 and 133°C, respectively, whereas 175°C was required for 100% (T100) conversion. For comparison, catalytic activity was also measured for MCO-4 and commercialized CuO (reference) sample. The performance sequence was obtained as: com- mercialized CuO (reference)\MCO-4\CMCO-4. For the MCO-4 and commercialized CuO sample, 100% conversion (T100) occurred at around 224–363°C, respectively. The bar chart for T25, T50 andT100 for CO conversion is also pre- sented in figure 5b. Performance of catalyst is typically dependent onEafor that catalytic reaction. Usually, lower Ea conveys the higher potential catalytic performance for catalyst. To measureEa, Arrhenius diagram is plotted for all 0.0 0.2 0.4 0.6 0.8 1.0

0 50 100 150 200 250

300 CMCO-4

MCO-4 Reference CuO

Volumeadsorbed(cm3g-1STP)

Relative pressure (p/po) 0 20 40 60 80 100 120 140 160 180 200 0.0000

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007

CMCO-4 MCO-4 Reference CuO

dV/dD(cm3g-1Å-1)

Pore diameter (Å)

(a)

(i)

(b)

(ii) (iii)

(i) (ii) (iii)

Figure 2. (a) Nitrogen adsorption–desorption isotherms and (b) pore-size distributions of the as-synthesized sample as (i) CMCO-4, (ii) MCO-4 and (iii) commercialized CuO (reference).

Figure 3. (a) FESEM and (b–e) TEM images of as-prepared C2CuO4nH2O sample.

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Figure 4. (a) FESEM and (b–e) TEM images of as-prepared CuO sample.

0 50 100 150 200 250 300 350 400 450 0

20 40 60 80 100

CMCO-4 MCO-4 Reference CuO

Temperature (°C)

%COConversion

CMCO-4 MCO-4 Reference CuO 0

50 100 150 200 250 300 350

400 T25

T50 T100

Temperature(°C)

10-3* 1/T (K) ln(conversionrate)[mol/m2/sec]

1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 -19

-18 -17 -16 -15 -14 -13 -12

-11 CMCO-4 (21.90 KJ/mol)

MCO-4 (25.88 KJ/mol) Reference CuO (34.52 KJ/mol) (ii)

(i) (iii)

(ii) (i) (iii)

(a) (b)

(c)

Figure 5. (a) Percentage CO conversion into CO2, (b) bar chart corresponding to 25% (T25), 50% (T50) and 100%

(T100) CO conversion into CO2, and (c) Arrhenius plot for different catalyst samples as (i) CMCO-4, (ii) MCO-4 and (iii) commercialized CuO (reference).

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the catalysts (figure 5c). The catalytic conversion rate sig- nificantly increased with increase in temperature, which could be accounted for the generation of subsurface oxygen species at high temperature in O2 contact [23]. The Ea values were calculated as about 21.90, 25.88 and 34.52 kJ mol-1for CMCO-4, MCO-4 and commercialized CuO (reference) samples, respectively, evidencing previous catalytic trend for all the samples. The required temperature for CO oxidation is correlated with surface area, porosity and Ea of catalysts in tabular format (table 1). Tang et al [24] proposed that with increase in temperature, the density of Cu?1 species is rapidly increased via the reduction in Cu?2. In addition, high surface area and pore volume of catalysts enable better dispersion of surface sites with rapid mobility of oxygen species. CO, as ap-acidic ligand pref- erentially adsorbs on Cu?1to form Cu?1(CO)2complex via r-donation and p-back bonding interaction [25,26]. It is worth noting that with increase in the surface area of cat- alyst, surface availability of Cu?1as an active site increases that further reduces the Eavalue via favouring the desorp- tion of CO to CO2, which is rate-determining step (RDS) of catalytic reaction [27].

To garner further structural inside, post-catalytic sample was subjected for XRD and FTIR measurements. From as- depicted XRD spectrum, no significant change in elemental phase was encountered thereby indicating stability of CuO phase after catalysis (supplementary figure S1a). FTIR spectrum also shows signature peaks around*452,*520 and *592 cm-1, manifesting stretching of Cu–O for as- prepared CuO sample before catalysis (figure S1b). Inter- estingly, peaks are slightly shifted to *476, *537 and

*604 cm-1for post-catalytic sample indicating improved crystallinity during high-temperature catalytic processes.

However, appearance of conspicuous peak at*2130 cm-1 signifies the presence of Cu?1(CO)2on the surface of CuO sample [24]. The presence of reduced copper species was further authenticated by XPS analysis of post-catalytic sample. For Cu 2p spectrum, binding energy value around 933.1 and 953.2 eV refers to the existence of Cu(II) state, whereas the value between 931.1 and 951.3 eV corresponds to reduced copper species from post-catalytic network (supplementary figure S1c) [28]. The change in binding energy for Cu 2p further supports the elemental in-situ transformation from Cu(II) to Cu(I) during catalytic con- version of CO for CuO sample. Surprisingly, O 1s spectrum arrives at 528.3 and 531.6 eV along with distinct binding

energy value at 530.08 eV indicating significant improve- ment in chemisorbed oxygen species for post-catalytic sample (supplementary figure S1d). The oxygen vacancy (Ob/Oa) was significantly increased from 0.74 to 3.88 dur- ing the pre-treatment of catalyst. The significant enhance- ment in chemisorbed oxygen species renders defect-rich crystal lattice site that further assistsin-situtransformation from Cu(II) to Cu(I) which leads to the formation of Cu?1(CO)2on the surface of CuO sample. Lunin and co- workers [9] precisely demonstrated the role of optimized textural property and oxygen mobility for catalytic oxida- tion of CO. For CuO sample, facile electron exchange between Cu(II) and Cu(I) leads to improved surface oxygen mobility that have a pivotal role for chemisorption of oxygen species for catalytic network [29]. Furthermore, formation of defective lattice (oxygen vacancy) during pre- treatment improves bulk oxygen mobility which assists rapid redox reactions on the catalytic surface [30,31]. The surface redox capability can be further clarified by charac- teristic reducibility of catalysts by TPR (temperature pro- grammed reduction) measurement (supplementary figure S2) [32]. Interestingly, reducible nature is more facile and prominent at low temperature for CMCO-4 compared to MCO-4 and reference CuO. The increase in reducibility of catalyst indeed lowers down theEafor oxygen migration that could lead to improved catalytic performance for CO oxidation at low temperature [7].

The reusability of best catalyst was checked for four cycles (figure 6), and performance for complete CO con- version (T100) of catalyst declined from 175 to 218°C. To further elucidate the recyclable performance, surface area

Table 1. Textural properties of CuO samples applying as a catalyst for CO oxidation.

S.

no. Sample name

Surface area (m2g-1)

Pore volume (cc g-1)

Pore diameter (nm)

T25

(°C) T50

(°C)

T100

(°C)

Energy of activation (kJ mol-1)

1 CMCO-4 51 0.4492 35.61 94 133 175 21.90

2 MCO-4 33 0.3637 44.35 119 158 224 25.88

3 Reference CuO 0.65 0.006 40.22 227 302 363 34.52

100 200 40

80

100 200 40

80

100 200 40

80

100 200 40

80

T100= 200°C

T100= 178°C

4th 2nd 3rd

1st

T100= 218°C

T100= 175°C

Temperature (°C)

% CO Conversion

Figure 6. Percentage CO conversion into CO2over CMCO-4 for consecutive cycles.

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and porosity measurement were performed by N2 adsorp- tion–desorption method for the post-catalytic sample (supplementary figure S3). The surface area and pore vol- ume were found to be 43 m2g-1and 0.33 cc g-1, respec- tively. The significant loss of surface area and pore volume could be the reason behind decay in catalytic performance over a period of time. The catalytic efficiency of the syn- thesized sample (CMCO-4) was also compared with those reported in previous literatures and displayed comparable efficiency for CuO-supported catalytic samples (table 2) [7,9–11,33]. This synthetic approach could be relevant for the fabrication of other porous transition metal oxide for different applications in catalysis and clean energy context.

4. Conclusions

CuO nanostructure was prepared by ligand-assisted method using CTAB as soft-template in aqueous solution. As-pre- pared sample showed C2CuO4nH2O phase, whereas CuO was formed after calcination at 400°C. The use of CTAB has significant influence to enhance the surface area and pore volume of sample. The surface area and pore volume of catalyst were found to be 51 m2g-1and 0.006 cc g-1, respectively, which were much higher than commercialized CuO sample. For the calcined sample, porous nanostructure with shell-like interior was encountered due to oxidative decomposition of as-prepared C2CuO4nH2O. The sample with highest surface area and pore volume rendered excel- lent efficiency towards catalytic conversion of CO to CO2. CTAB-assisted CuO nanostructure achieved 100% CO conversion at 175°C, whereas commercialized CuO required 363°C for the same. Improved catalytic activity for CTAB-assisted CuO nanostructure is attributed to high surface area and pore volume, as well as low Ea (21.90 kJ mol-1) for CO to CO2conversion. The variation in catalytic activity could be ascribed to the availability of active sites and mobility of oxygenated species over the mesoporous framework of catalyst.

Acknowledgements

This research work was financially supported by DST- SERB sponsored project, GAP 0616 (grant no. SR/S3/ME/

0035/2012), Government of India. Dr Sourav Ghosh acknowledges Technical Research Centre Project (AI/1/

64/SNB/2014) of S N Bose National Centre for Basic Sciences, Kolkata and Council of Scientific and Industrial Research (CSIR) (grant no. 31/015(0104)/2012-EMR-1) for financial support. He also thanks Dr Subhra Jana, Dr Venkataramanan Mahalingam, Dr Mohua Chakraborty and Ruma Ghosh for their helpful suggestions. SK is thankful to Academy of Scientific and Innovative Research (AcSIR).

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The specific surface area, average pore size and measured pore volume of VC@C were 135 · 46 m 2 /g, 4 · 443 nm and 0·180 cm 3 /g, respectively indicating that the as-obtained

Effect of CuO addition on structure and electrical properties of low temperature sintered quaternary piezoelectric ceramics..

BET specific surface area and pore volume are observed to increase with increase of the calcination temperature up to 700 ◦ C followed by a further decrease with

Infrared spectra, epoxy number, pore volume and surface area of these copolymers are compared with copoly- mers of similar composition synthesized by traditional

The pore structure characteristics evaluated are: porosity, pore specific surface area, mean pore diameter, pore size distribution, mean distribution pore radius, pore shape

The effective hydrodynamic radius of the porous medium ‘r eff ’ has been derived and correlated both to the geometry of the porous medium, viz., minimum pore size and maximum pore