Crystallization behavior of a novel poly(aryl ether ketone): PEDEKmK

Isothermal melt and cold crystallization kinetics of PEDEKmK linked by meta-phenyl and biphenyl were investigated by differential scanning calorimetry in two temperature regions. Avrami analysis is used to describe the primary stages of the melt and cold crystallization, with exponent n = 2 and n = 4, respectively. The activation energies are − 118 kJ/mol and 510 kJ/mol for crystallization from the melt and the glassy states, respectively. The equilibrium melting point T⁰_m is estimated to be 309°C by using the Hoffman-Weeks approach, which compares favorably with determination from the Thomson-Gibbs method. The lateral and end surface free energies derived from the Lauritzen-Hoffman spherulitic growth rate equation are σ = 8.45 erg/cm² and σ_e = 45.17 erg/cm², respectively. The work of chain folding q is determined as 3.06 kcal/mol. These observed crystallization characteristics of PEDEKmK are compared with those of the other members of poly(aryl ether ketone) family. © 1997 John Wiley & Sons, Inc. J Appl Polym Sci 64: 1451–1461, 1997     

Morphology and spherulitic growth kinetics in poly(ethylene oxide)/poly(n‐butyl methacrylate) blends

Results of a study concerning the morphology and the spherulite growth rates of poly(ethylene oxide) (PEO) in binary blends with poly(n‐butyl methacrylate) (PnBMA) are reported. Microscopic observations show that blending causes the spherulite structure to become coarser and less birefringent and confirms that the spherulitic growth rates of PEO were reduced by the addition of PnBMA. X‐ray diffraction studies show no change in the unit cell dimensions and a decrease in the degree of crystallinity upon blending. Analysis of the spherulite radial growth rate data by using Lauritzen–Hoffman theory indicates that crystallization in the range of 300 to 330 K occurs solely within regime III. The calculated surface free energy of folding, σ_e, for pure PEO is 57 erg cm^?2 and decreases with increasing the content of PnBMA in the blend. Copyright © 2004 Society of Chemical Industry     

The morphology of poly(aryl-ether-ether-ketone)

The morphology and related properties are described for the aromatic thermoplastic poly(aryl-ether-ether-ketone) (PEEK) [C₆H₄OC₆H₄OC₆H₄CO]_n. Topics covered include crystallinity, crystallization and melting behaviour, Iamellar thickness and spherulitic structure. The data are used to derive the following material parameters $\text{T1}{}_{\mathrm{<mtext>m</mtext>}}\text{= 395°}\text{C}$, σ_e = 49 erg cm^?2, σ_s = 38 erg cm^? and ΔH_F = 130 kJ kg^?1. PEEK is closely analogous to poly(ethylene terephthalate) in its crystallization behaviour except that the main transitions occur about 75°C higher.     

Isothermal and nonisothermal crystallization kinetics of nylon-11

Analysis of the isothermal, and nonisothermal crystallization kinetics of Nylon-11 is carried out using differential scanning calorimetry. The Avrami equation and that modified by Jeziorny can describe the primary stage of isothermal and nonisothermal crystallization of Nylon-11. In the isothermal crystallization process, the mechanism of spherulitic nucleation and growth are discussed; the lateral and folding surface free energies determined from the Lauritzen–Hoffman equation are ς = 10.68 erg/cm² and ς_e = 110.62 erg/cm²; and the work of chain folding q = 7.61 Kcal/mol. In the nonisothermal crystallization process, Ozawa analysis failed to describe the crystallization behavior of Nylon-11. Combining the Avrami and Ozawa equations, we obtain a new and convenient method to analyze the nonisothermal crystallization kinetics of Nylon-11; in the meantime, the activation energies are determined to be −394.56 and 328.37 KJ/mol in isothermal and nonisothermal crystallization process from the Arrhonius form and the Kissinger method. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 70: 2371–2380, 1998     

Isothermal and nonisothermal melt‐crystallization kinetics of syndiotactic polystyrene

Analyses of the isothermal and nonisothermal melt kinetics for syndiotactic polystyrene have been performed with differential scanning calorimetry, and several kinetic analyses have been used to describe the crystallization process. The regime II→III transition, at a crystallization temperature of 239°, is found. The values of the nucleation parameter K_g for regimes II and III are estimated. The lateral‐surface free energy, σ = 3.24 erg cm^?2, the fold‐surface free energy, σ_e = 52.3 ± 4.2 erg cm^?2, and the average work of chain folding, q = 4.49 ± 0.38 kcal/mol, are determined with the (040) plane assumed to be the growth plane. The observed crystallization characteristics of syndiotactic polystyrene are compared with those of isotactic polystyrene. The activation energies of isothermal and nonisothermal melt crystallization are determined to be ΔE = ?830.7 kJ/mol and ΔE = ?315.9 kJ/mol, respectively. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 83: 2528–2538, 2002     

Polypentadecalactone prepared by lipase catalysis: crystallization kinetics and morphology

BACKGROUND: This work addresses the need to better understand the crystallization kinetics and morphology of poly (ω‐pentadecalactone) (PPDL). This polyester has promising mechanical properties and a unique structure that resembles that of polyethylene. PPDL is a member of the poly(ω‐hydroxy fatty acid) family, which can be derived from biobased feedstocks. RESULTS: PPDL (M_n = 34 000 g mol^?1 and dispersity D = M_w/M_n = 2.7) was synthesized using enzyme catalysis. Equilibrium melting enthalpy and equilibrium melting point were determined using extrapolation techniques, being 227 J g^?1 and 101 °C, respectively. In addition, the equilibrium melting point ( ) was found to be 109.3 °C by the nonlinear Hoffman‐Weeks plot. For , the lateral surface free energy (σ), fold surface free energy (σ_e) and fold work (q) are 10.4 erg cm^?2, 47.5 erg cm^?2 and 2.6 kcal mol^?1, respectively; while for , they are 25.1 erg cm^?2, 46.6 erg cm^?2 and 2.6 kcal mol^?1, respectively. The results indicated the existence of a regime I to regime II transition during crystallization at about 80 °C. Polarized optical microscopy and AFM provided further evidence for the regime I–II transition. In regime I, coarse spherulites were formed through splaying out and occasional branching of lamellae, as well as stacking of lamellae through screw dislocation. In contrast, in regime II, banded spherulites were formed through crystal twisting. CONCLUSION: Morphological changes in PPDL at spherulitic and lamellar levels in regimes I and II were confirmed by differential scanning calorimetry, POM and AFM. Copyright © 2009 Society of Chemical Industry     

Miscible blends of syndiotactic polystyrene and atactic polystyrene. Part 2. Depolarized light scattering studies and crystal growth rates

Crystallization kinetics and morphology in miscible blends of syndiotactic polystyrene (sPS) and atactic postyrene (aPS) have been investigated by means of time-resolved depolarized light scattering (DPLS), polarized optical microscopy (POM) and scanning electron microscopy (SEM). Two different weight-average molecular weight of aPS, i.e. M_w=100k and 4.3k, were used to prepare the blends and denoted sPS/aPS(H) and sPS/aPS(M), respectively. Owing to a dilution effect, addition of aPS reduces the crystal growth rate and the overall crystallization rate of sPS; the reduction is more significant in sPS/aPS(M) of which a depression of equilibrium melting temperature is found due to the enhanced mixing entropy. Linear crystal growth is always observed in sPS/aPS(H) at the temperatures studied (240-269 °C) and results in an interfibrillar segregation morphology revealed by SEM, whereas sPS/aPS(M) with high aPS content exhibits non-linear growth behavior at low supercooling and gives an interspherulitic segregation morphology. Based on the Lauritzen-Hoffman theory, the fold surface free energies (σ_e) of sPS lamellae derived from DPLS and POM are in fair agreement, being 15.1 erg/cm² from the former and 12.6 erg/cm² from the latter. The peculiarly low values of σ_e and the derived work of chain folding are discussed briefly. On addition of aPS, the lateral surface free energy of lamellae remains intact (9.9 erg/cm²) regardless of aPS molecular weight used, which is ascribed to the absence of specific interaction between sPS and aPS components. Moreover, it seems that the activation energy for sPS chains to diffuse from the miscible melt to the crystal growth front is slightly increased in sPS/aPS(M), plausibly attributable to the extra energy required for the demixing process.     

Crystallization and microstructure of poly(butylene oxalate)

Crystal structure and morphology development of poly(butylene oxalate) (PBOX) during isothermal crystallization were studied with X-ray diffraction, time-resolved simultaneous small-angle X-ray scattering, differential scanning calorimetry, and optical microscopy. Results indicate that the decrease in the long period at low crystallization temperature indicated the occurrence of secondary crystallization in the interlamellar space. Meanwhile, the lamellar thickness slightly increased with crystallization temperatures due to the formation of thicker crystalline layers at high temperatures. The crystal growth rate of PBOX was analyzed by optical microscopy. Using values of the equilibrium melting temperature of 117.4 °C and the fold surface free energy of 32.37 erg/cm² obtained by the Gibbs–Thomson theory, the nucleation parameter, K_g, of 97264 K² and the lateral surface free energy of 17.68 erg/cm² were determined from the Lauritzen and Hoffman equation. These values are comparable to various semicrystalline polymers previous reported and are not available up to now for PBOX in the literature.     

Isothermal crystallization of iPP in model glass-fiber composites

Isothermal crystallization of iPP in model glass-fiber composites is studied by DSC, and the basic energetic parameters of crystallization are determined. Unsized untreated and thermally treated glass fibers are used in model composites to determine the role of the surface on nucleation and crystallization processes. Thermally treated glass fibers are found to exhibit a predominant nucleating effect as compared to unsized untreated ones, and the crystallization proceeds faster, resulting in lower values for the half-time of crystallization (10–120 s). The energy of formation of a nuclei of critical dimensions at a given T_c is also lower, and it decreases as the content of the fibers in the composite increases. The surface free energy of folding, σ_e = 140 × 10⁻³ J/m², was determined for iPP in the composite containing 50% glass fibers, while for pure iPP, σ_e = 170 × 10⁻³J/m² was found. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 69: 381–389, 1998     

Activity of substrates in the catalyzed nucleation of poly(ethylene terephthalate) melts

The activity, Φ of AgBr, AgI, PbF₂, Ag₂S, LiF, and CaF₂ in the catalyzed nucleation of poly(ethylene terephthalate) (PET) melts was determined using a nonisothermal differential scanning calorimetry (DSC) technique. A comparison with existing experimental data was made. It is established that the higher the melting temperature of the substrate the lower its activity as a crystallization core in the heterogeneous nucleation of PET. The lateral surface energy, σ, the end surface energy, σ_e, the adhesion energy, β, and the difference between the surface energies at the substrate/melt, σ_sf, substrate/deposit, σ^*, and the total energy of misfit dislocations, E_d [i.e., σ_sf - (σ^* - E_d)] were calculated. © 1997 John Wiley & Sons, Inc. J Appl Polym Sci 63: 349–353, 1997     

Effect of lignin particles as a nucleating agent on crystallization of poly(3‐hydroxybutyrate)

The effect of lignin fine powder, as a new kind of nucleating agent, on the crystallization process of poly(3‐hydroxybutyrate) (PHB) was studied. The kinetics of both isothermal and nonisothermal crystallization processes from the melt for both pure PHB and PHB/lignin blend was studied by means of differential scanning calorimetry. Lignin shortened the crystallization half‐time t_1/2 for isothermal crystallization. The activation energy ΔE for PHB/lignin and pure PHB in the isothermal crystallization process was ?237.40 and ?131.22 kJ/mol, respectively, clearly indicating that the crystallization of the PHB/lignin blend was more favorable than that of pure PHB from a thermodynamic perspective. At the same time, according to polarized optical microscopy, the rate of spherulitic growth from the melt increased with the addition of lignin, which is ascribed to the reduction of surface fold energy σ_e, that is, σ_e is 59.2 × 10^?3 and 41.6 × 10^?3 J m^?2 for pure PHB and PHB/lignin, respectively. Polarized optical microscopy also showed that the spherulites found in PHB with lignin were smaller in size and greater in number than those found in pure PHB. The wide‐angle X‐ray diffraction indicated that an addition of lignin caused no change in the crystal structure and degree of crystallinity. These results indicated that lignin is a good nucleating agent for the crystallization of PHB. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 94: 2466–2474, 2004     

Quiescent crystallization kinetics and morphology of isotactic polypropylene resins for injection molding. I. Isothermal crystallization

The quiescent isothermal crystallization kinetics of polypropylene was studied as a function of molecular weight (M_w), amount of ethene, and amount of maleic anhydride and acrylic acid grafting. Differential scanning calorimetry and polarized light optical microscopy were used to follow this kinetics. It was observed that the linear growth rate, G, decreased with the increase of M_w, but increased with the amount of ethene. In the grafted polymers, as the amount of grafting increased, G decreased. The fold surface free energy, σ_e, was found to increase with the increase in M_w. The heterophasic and grafted polymers had σ_e values higher than the homopolymers. All samples showed spherulitic morphology, except the acrylic acid-grafted polypropylene that showed axialitic morphology. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 68: 1159–1176, 1998     

The effect of end groups of PEG on the crystallization behaviors of binary crystalline polymer blends PEG/PLLA

The effect of end groups (2OH, 1OH, 1CH₃ and 2CH₃) of poly(ethylene glycol) (PEG) on the miscibility and crystallization behaviors of binary crystalline blends of PEG/poly(l-lactic acid) (PLLA) were investigated by differential scanning calorimetry (DSC) and polarizing optical microscopy (POM). A single glass-transition temperature was observed in the DSC scanning trace of the blend with a weight ratio of 10/90. Besides, the equilibrium melting point of PLLA decreased with the increasing PEG. A negative Flory interaction parameter, χ₁₂, indicated that the PEG/PLLA blends were thermodynamically miscible. The spherulitic growth rate and isothermal crystallization rate of PEG or PLLA were influenced when the other component was added. This could cause by the change of glass transition temperature, T_g and equilibrium melting point, T⁰_m. The end groups of PEG influenced the miscibility and crystallization behaviors of PEG/PLLA blends. PLLA blended with PEG whose two end groups were CH₃ exhibited the greatest melting point depression, the most negative Flory interaction parameter, the least fold surface free energy, the lowest isothermal crystallization rate and spherulitic growth rate, which meant better miscibility. On the other hand, PLLA blended with PEG whose two end groups were OH exhibited the least melting point depression, the least negative Flory interaction parameter, the greatest fold surface free energy, the greatest isothermal crystallization rate and spherulitic growth rate.     

Well‐functioned photovoltaics based on nanofibers composed of PBDT‐TIPS‐DTNT‐DT and graphenic precursors thermally modified by polythiophene,polyaniline and polypyrrole

Reduced graphene oxide nanosheets modified by conductive polymers including polythiophene (GPTh), polyaniline (GPANI) and polypyrrole (GPPy) were prepared using the graphene oxide as both substrate and chemical oxidant. UV–visible and Raman analyses confirmed that the graphene oxide simultaneously produced the reduced graphene oxide and polymerized the conjugated polymers. The prepared nanostructures were subsequently electrospun in mixing with poly(3‐hexylthiophene) (P3HT)/phenyl‐C71‐butyric acid methyl ester (PC₇₁BM) and poly[bis(triisopropylsilylethynyl)benzodithiophene‐bis(decyltetradecylthien)naphthobisthiadiazole] (PBDT‐TIPS‐DTNT‐DT)/PC₇₁BM components and embedded in the active layers of photovoltaic devices to improve the charge mobility and efficiency. The GPTh/PBDT‐TIPS‐DTNT‐DT/PC₇₁BM devices demonstrated better photovoltaic features (J_sc = 11.72 mA cm^?2, FF = 61%, V_oc = 0.68 V, PCE = 4.86%, μ_h = 8.7 × 10^?3 cm² V^–1 s^?1 and μ_e = 1.3 × 10^?2 cm² V^–1 s^?1) than the GPPy/PBDT‐TIPS‐DTNT‐DT/PC₇₁BM (J_sc = 10.30 mA cm^?2, FF = 60%, V_oc = 0.66 V, PCE = 4.08%, μ_h = 1.4 × 10^?3 cm² V^–1 s^?1 and μ_e = 8.9 × 10^?3 cm² V^–1 s^?1) and GPANI/PBDT‐TIPS‐DTNT‐DT/PC₇₁BM (J_sc = 10.48 mA cm^?2, FF = 59%, V_oc = 0.65 V, PCE = 4.02%, μ_h = 8.6 × 10^?4 cm² V^–1 s^?1 and μ_e = 7.8 × 10^?3 cm² V^–1 s^?1) systems, assigned to the greater compatibility of PTh in the nano‐hybrids and the thiophenic conjugated polymers in the bulk of the nanofibers and active thin films. Furthermore, the PBDT‐TIPS‐DTNT‐DT polymer chains (3.35%–5.04%) acted better than the P3HT chains (2.01%–3.76%) because of more complicated conductive structures. © 2019 Society of Chemical Industry     

Effect of molecular weight on spherulitic growth rate of poly(ϵ‐caprolactone) and poly(ϵ‐caprolactone)‐b‐poly(ethylene glycol)

The spherulitic growth rates of a series poly (?‐caprolactone) homopolymers and poly(?‐caprolactone)‐b‐ poly(ethylene glycol) (PCL‐b‐PEG) block copolymers with different molecular weights but narrow polydispersity were studied. The results show that for both PCL homopolymers and PCL‐b‐PEG block copolymers, the spherulitic growth rate first increases with molecular weight and reaches a maximum, then decreases as molecular weight increases. Crystallization temperature has greater influence on the spherulitic growth rate of polymers with higher molecular weight. Hoffman–Lauritzen theory was used to analyze spherulitic growth kinetics and the free energy of the folding surface (σ_e) was derived. It is found that the values of σ_e decrease with molecular weight at low molecular weight level and become constant for high molecular weight polymers. The chemically linked PEG block does not change the values of σ_e significantly. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci 2007     

Struktur und elektrische Leitfähigkeit von polymeren organischen Halbleitern

In this review article we want to give information about low molecular and polymer organic semiconductors, which were recently synthesized in our institute. Specific electric conductivities up to σ_298°K = 9.0 · 10^?5Ω^?1 · cm^?1 and thermic activation energies of E = 0.30 eV of polyenearylenes, respectively -heteroarylenes were measured. Polyazomethines have a maximum σ_298°K = 3.3 · 10^?9Ω^?1 · cm^?1 and E = 0.35 eV. Polymers with indophenine units have conductivities up to σ_298°K = 1.1 · 10^?4Ω^?1 · cm^?1 and E = 0.39 eV. A maximum of σ_298°K = 5.0 · 10^?2Ω^?1 · cm^?1 and E = 0.05 eV was found for bis-(1.2-dicyanoethylenedithiolo)-metal salts. Polymers with a phthalocyanine- or hemiporphyrazine-like structure achieve a conductivity of σ_298°K = 2.3 · 10^?2Ω^?1 · cm^?1 and E = 0.15eV. Coordination polymers of dimercaptomaleic acid, respectively their monoamide show a maximum of σ_298°K = 3.2 · l0^?lΩ^?l · cm^?1 and E = 0.20 ev. Polymers with σ_298°K ≤1.5 · 10^?5 Ω^?l · cm^?l and E ≥ 0.5 eV were obtained by the polymerization of succinonitrile. All the investigated substances show an electronic conductivity. The existence of an ionic conductivity could, in all cases, be excluded by using direct current measurements over a long period of time.     

Crystallization kinetics of a PEEK/LCP blend

The crystallization kinetics of a polyetheretherketone (PEEK)/liquid crystalline polymer (LCP) blend was studied by using differential scanning calorimetry. Nonisothermal runnings were performed on heating and on cooling at different rates. Isothermal crystallization experiments at 315, 312, 310, and 307°C, from the melt state (380°C) were performed in order to calculate the Avrami parameters n and k and the fold surface free energy, σ_e. Polarized light optical micrographs were also obtained to confirm the Avrami predictions. It was observed that the LCP retarded the PEEK crystallization process and that the PEEK melting temperature decreased with the amount of LCP, but the LCP melting temperature increased with the amount of PEEK. Probably the PEEK improves the perfection of the LCP crystalline domains. A spherulitic morphology in pure PEEK and its blends was predicted by the Avrami analysis; however this morphology was only observed for pure PEEK and for the 80/20 composition. The other compositions presented a droplet and fibrillar-like morphology. The overall crystallization rate was observed to decrease with the crystallization temperature for all compositions. Finally, σ_e was found to decrease with the increase of LCP in the blends, having unrealistic negative values. Thus, calculations were made assuming σ_e constant at all compositions. It was observed that δ, the interfacial lateral free energy, decreased but still remained positive. It was concluded that in these blends neither σ_e nor σ could be considered constant. © 1995 John Wiley & Sons, Inc.     

Mechanical properties and superstructure of poly(ethylene terephthalate) fibers zone-drawn and zone-annealed by CO2 laser heating

A laser-heating zone-drawing and zone-annealing method using a continuous-wave carbon dioxide laser was applied to poly(ethylene terephthalate) (PET) fiber to improve its mechanical properties. The as-spun fiber was zone-drawn under a applied tension (σ_a) of 4.44 MPa at a laser power density (PD) of 6.08 W cm⁻², and then the laser-heated zone-drawn fiber was zone-annealed. The laser-heating zone-annealing was carried out in three steps: the first annealing was carried out under σ_a = 139 MPa at 4.83 W cm⁻²; the second annealing was carried out under σ_a = 283 MPa at 4.83 W cm⁻², and the third annealing was carried out under σ_a = 432 MPa at 3.45 W cm⁻². The surface temperature distribution of the fiber irradiated with the CO₂ laser was measured by using an infrared thermographic camera equipped with a magnifying lens. The relation between the laser power and the surface temperature of the fiber became clear in the laser-heating zone-drawing and the laser-heating zone-annealing. The fiber obtained finally had a birefringence of 0.239, a degree of crystallinity of 55%, a tensile modulus of 19.8 GPa, and a storage modulus of 25.7 GPa at 25°C. In FTIR measurements, a trans conformation increased with the processing, but a gauche one decreased. The laser-heating zone-drawing and zone-annealing method was found to be effective in producing the PET fiber with high modulus and high strength. © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 82: 2775–2783, 2001     

Effect of poly(l‐lactide)/poly(d‐lactide) block length ratio on crystallization behavior of star‐shaped asymmetric poly(l(d)‐lactide) stereoblock copolymers

Effect of Poly(l ‐lactide)/Poly(d ‐lactide) (PLLA/PDLA) block length ratio on the crystallization behavior of star‐shaped poly(propylene oxide) block poly(d ‐lactide) block poly (l ‐lactide) (PPO–PDLA–PLLA) stereoblock copolymers with molecular weights (M_n) ranging from 6.2 × 10⁴ to 1.4 × 10⁵ g mol^?1 was investigated. Crystallization behaviors were studied utilizing differential scanning calorimetry (DSC), polarized optical microscopy (POM), and wide‐angle X‐ray diffraction (WAXD). Only stereocomplex crystallites formed in isothermal crystallization at 140 to 156°C for all samples. On one hand, the overall crystallization rate decreased as PLLA/PDLA block length ratio increased. As PLLA/PDLA block length ratio increased from 7:7 to 28:7, the value of half time of crystallization (t_1/2) delayed form 2.85 to 5.31 min at 140°C. On the other hand, according to the Lauritzen–Hoffman theory, the fold‐surface energy (σ_e) was calculated. σ_e decreased from 77.7 to 73.3 erg/cm² with an increase in PLLA/PDLA block length ratio. Correspondingly increase in nucleation density was observed by the polarized optical microscope. Results indicated that the PLLA/PDLA block length ratio had a significant impact on the crystallization behavior of PPO–PDLA–PLLA copolymers. POLYM. ENG. SCI., 55:2534–2541, 2015. © 2015 Society of Plastics Engineers     

Electrodeposition from a fluidized bed electrolyte. II. Effects of bed porosity and particle size

Mass transfer from a fluidized bed electrolyte containing inert particles has been found to depend on bed porosity and particle size. The optimum porosity was found to vary from 0.52 – 0.57 with decreasing particle size but mass transport increased with particle size.A mass transfer entry length effect was observed on the cylindrical cathode but its position within the bulk of the bed was found not to be critical, thus indicating that the hydrodynamic entry length was small. The limiting current density was found to vary as (d _e/L _e)^0.15 whered _e is the annular equivalent diameter andL _e the electrode length.List of symbols Re_I modified Reynolds No. =U _o d _p /v(1–) - Re_II particle Reynolds No. =U _o d _p /v - Re_O sedimentation Reynolds No. =U _i d _p v (constant value) - Re_t terminal particle Reynolds No. =U _t d _p /v - Sc Schmidt No. =v/D - St_I modified Stanton No. =k _L /U _o - C _b bulk concentration, M cm^–3 - D diffusion coefficient, cm² s^–1 - d _t tube diameter, mm - d _e electrode equivalent diameter, mm - d _p particle diameter, mm - bed porosity - zF Faradaic equivalence - cd current density - i _L limiting current density, mA cm^–2 - i _LO limiting current density in the absence of particles - k _L mass transfer coefficient, cm s_–1 - L _e electrode length, mm - m, n constants or indices - v kinematic viscosity, cm² s^–1 - U _o superficial velocity, cm s^–1 - U _i sedimentation velocity, cm s^–1