Size-effects

Irrespective of all uncertainties it seems clear that a particle size reduction down to some tens of a nanometer leads to a quick lithium infiltration provided the lithium (Li and e ] has been transported rapidly to the particle surface (and provided the transfer reaction is quick enough]. Hence it is the network addressing these small particles and for high currents the Li transfer resistance which become decisive for the performance of the electrode. The second half of the contribution is devoted to exactly this theme. 

There are two important consequences of reducing the particle size in our context (i] faster charging and discharging and 

Measurements of electrode performance are usually carried out in the galvanostatic mode, therefore, the charging time also depends on the magnitude of the current. If the current value is relatively high, then the particles will not be fully charged at all (within a given potential window]. 

Schematic presentation of two extreme types of eiectro-chemicai contacting. Left curve point-iike eiectrochemi-cai contacting corresponding to Eq. 8.10. Right curve ideai eiectrochemicai contacting corresponding to Eq. 8.9. 

we explain theoretically the origin of the square law observed experimentally in Fig. 8.14b. We show that such a law naturally occurs when solving the equations for transport of mass and charge inside a particle. The detailed analysis depends on the boundary conditions, that is, on the way and geometry the particles are contacted. For the present purposes we will only consider two limiting cases ideal contact (particles are uniformly coated by an ionic and electronic conductor] and point-like electrochemical contacting (Fig. 8.15, on left]. 

10 Electron density contours in sodium chloride. Numbers indicaie the electron density (electrons A = 10 electrons pm ) along each comour line. The boundary of each ion is defined as the minimum in electron density between the ions. The intemudear distance is 281 pm ( Z8l A). [Modified from Schoknecht, C. Z. Naturforsch. 1957, I2A, 983. Reproduced with permission.] 

Although not many simple ionic compounds have been studied with the requisite aixutai to provide data on ionic radii, there are enough to provide a basis for a complete set oT ionic ladiL Such a set has been provided in the crystal tadU of Shaimon and Prewitt. Values of these radii are given in Table 4.4. 

Factors Affecting A comparison of the values given in TaUe 4.4 allows one to make some conclusions 

For transition metals the multiplicity of the spin state affects the way in which the ara ons can approach the cation this alters the effective radius. Although this is an important factor in determining cationic radii, it is beyond the scope of the present chapter and will be deferred to Chapter 11. 

For both cations and anions the crystal radius increases with the increase in cO ordination number. As the coordination number increases, the repulsions among the coordinating counterions become greater and cause them to back ofT a bit. Alternatively, one can view a lower coordination number as aDowing the counter-ions to compress the centra] ion and reduce its crystal radius. 

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As on previous occasions, the reader is reminded that no very extensive coverage of the literature is possible in a textbook such as this one and that the emphasis is primarily on principles and their illustration. Several monographs are available for more detailed information (see General References). Useful reviews are on future directions and anunonia synthesis [2], surface analysis [3], surface mechanisms [4], dynamics of surface reactions [5], single-crystal versus actual catalysts [6], oscillatory kinetics [7], fractals [8], surface electrochemistry [9], particle size effects [10], and supported metals [11, 12]. 

Near critical points, special care must be taken, because the inequality L will almost certainly not be satisfied also, cridcal slowing down will be observed. In these circumstances a quantitative investigation of finite size effects and correlation times, with some consideration of the appropriate scaling laws, must be undertaken. Examples of this will be seen later one of the most encouraging developments of recent years has been the establishment of reliable and systematic methods of studying critical phenomena by simulation. 

Challa MSS, Landau D P and Binder K 1986 Finite-size effects at temperature-driven Ist-order transitions Phys. Rev. B 34 1841 -52... 

Kitakami O et al 997 Size effect on the crystal phase of cobalt fine particles Phys. Rev. B 56 13 849 Cullity B D 1978 Elements of X-ray Diffraction (Reading, MA Addison-Wesley)... 

Rossetti R, Nakahara S and Brus L E 1983 Quantum size effects In the redox potentials, resonance Raman spectra and electronic spectra of CdS crystallites In aqueous solution J. Chem. Phys. 79 1086... 

Rossetti R ef al 1984 Size effects In the excited electronic states of small colloidal CdS crystallites J. Chem. Phys. 80 4464... 

Buffat P and Borel J P 1976 Size effect on the melting temperature of gold particles Phys. Rev. A 13 2287... 

Valov P M and Leiman V I 1997 Size effects in the melting and crystallization temperatures of copper chloride nanocrystals in glass JETP Lett. 66 510... 

Borel J P 1981 Thermodynamical size effect and the structure of small clusters Surf. Sc/. 106 1... 

Skripov V P, Koverda V P and Skokov V N 1981 Size effect on melting of small particles Rhys. Status Solidi A 66 109... 

Quantum efficiencies Quantum efficiency Quantum electronics Quantum fluids Quantum mechanics Quantum size effect Quantumwell... 

Our communication describes grain size effect in XRF of powder and powder slurry-like substances in terms of the generalized model ... 

Specimen size effects have been attributed to rising R-curve behavior for most graphites studied. 

G. R. Romanoski and T. D. Burchell, Specimen Size Effect an Fracture Toughness of Nuclear Graphites, Extended Abstracts and Program - 20 Biennial Conference on Carbon, Pub. Electrochemical Society, 1991 pp 584 585... 

Band gap engineetring confined hetetrostruciutres. When the thickness of a crystalline film is comparable with the de Broglie wavelength, the conduction and valence bands will break into subbands and as the thickness increases, the Fermi energy of the electrons oscillates. This leads to the so-called quantum size effects, which had been precociously predicted in Russia by Lifshitz and Kosevich (1953). A piece of semiconductor which is very small in one, two or three dimensions - a confined structure - is called a quantum well, quantum wire or quantum dot, respectively, and much fundamental physics research has been devoted to these in the last two decades. However, the world of MSE only became involved when several quantum wells were combined into what is now termed a heterostructure. 

An intriguing recent review of size effects in materials due to microstructural and dimensional constraints with a focus on mechanical properties, including those of multilayers, is by Arzt (1998). 

Large deformation contacts and finite size effects... 

The finite size effects in the contact between a spherical lens of polyurethane and a soft flat sheet of crosslinked polyfdimethyl siloxane) (PDMS) has been addressed by Falsafi et al. [37]. They showed that for deformations corresponding to contact diameters larger than the sheet thickness, the compliance of the system was affected by the glass substrate supporting the soft sheet. In order to minimize the finite size effects in the adhesion measurement of small elastomeric lenses, Falsafi et al. [38] and Deruelle et al. [39] used relatively thick elastic sheets to support their samples. 

Figure 30. Particle size effects on minimum fluidization and bubbling velocities.

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