Surface theoretical considerations

Opeka, M. M., Talmy, I. G., Zaykoski, J. A. (2004). Oxidation-based materials selection for 2000 °C -I- hypersonic aero-surfaces theoretical considerations and historical experience. Journal of Materials Science, 39, 5887-5904. doi 10.1023/ B JMSC.0000041686.21788.77. 

It is now well established that in lithium batteries (including lithium-ion batteries) containing either liquid or polymer electrolytes, the anode is always covered by a passivating layer called the SEI. However, the chemical and electrochemical formation reactions and properties of this layer are as yet not well understood. In this section we discuss the electrode surface and SEI characterizations, film formation reactions (chemical and electrochemical), and other phenomena taking place at the lithium or lithium-alloy anode, and at the Li. C6 anode/electrolyte interface in both liquid and polymer-electrolyte batteries. We focus on the lithium anode but the theoretical considerations are common to all alkali-metal anodes. We address also the initial electrochemical formation steps of the SEI, the role of the solvated-electron rate constant in the selection of SEI-building materials (precursors), and the correlation between SEI properties and battery quality and performance. 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

In order to attempt a more quantitative description one may start from the early theoretical considerations of Boudart9 who was first to tackle the problem of predicting the change in heats of adsorption with changing work function O. According to his early semiempirical electrostatic model when the work function of a surface changes by AO then the heat of adsorption, -AHad, of covalently bonded adsorbed species should change by ... 

Two approaches were taken to provide Insight to the uniformity of these samples 1) perform extensive visual Inspection of the materials to ensure that no large palladium crystallites were present, which was the case for both samples, and 2) perform surface area measurements using hydrogen TPD to establish the relationship between the observed data by STEM and the estimated surface area from theoretical considerations. 

Vermaak JS, Mays CW, Kuhhnann-Wilsdorf D. 1968. On surface stress and surface tension I. Theoretical considerations. Surf Sci 12 128-133. 

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

AW Hixson, JH Crowell. Dependence of reaction velocity upon surface and agitation. I. Theoretical consideration. Ind Eng Chem 23 923-931, 1931. 

Hartmann T, Ober D (2000) Biosynthesis and Metabolism of Pyrrolizidine Alkaloids in Plants and Specialized Insect Herbivores. 209 207-243 Haseley SR, Kamerling JP, Vliegenthart JFG (2002) Unravelling Carbohydrate Interactions with Biosensors Using Surface Plasmon Resonance (SPR) Detection. 218 93-114 Hassner A, see Namboothiri INN (2001) 216 1-49 Hauser A (2004) Ligand Field Theoretical Considerations. 233 49-58 Hauser A (2004) Light-Induced Spin Crossover and the High-Spin Low-Spin Relaxation. 234 155-198... 

Volume approximation (when the surface contribution to the free energy of a globule is neglected) works the better the farther the system is from the point of the coil-to-globule transition. In the framework of this approximation, it coincides with the -point, whereas under the theoretical consideration where the surface layer is taken into account, a gap appears separating these two points. The less is the length of polymer chain l, the more pronounced is this gap. Hence, the condition, imposed on the thermodynamic and stoichiometric parameters of the system by the equation of the -point,... 

Burke, M. Lindow, S. E. Surface properties and size of the ice nucleation of active bacteria Theoretical considerations. Cryobiology 27, p. 80-84, Copyright 1990 by Academic Press Inc. 1990... 

The greater the latter, the greater also is the decrease in surface tension per degree, and the ratio temperature coefficient/coefficient of expansion is approximately the same—between 2 and 3—for a very large number of liquids. Some explanation of this fact, as well as many other connections between surface tension and various physical constants will be suggested by theoretical considerations, to which we now proceed. 

The relations between the intrinsic pressure and other physical constants developed in the foregoing paragraphs have been found from theoretical considerations based on Laplace s theory, that is, on the assumption of cohesive forces acting over very small distances. They are of interest to us inasmuch as there is a necessary connection between intrinsic pressure and surface tension. While no numerical expression has so far been found for this, it is obvious that high intrinsic pressures must be accompanied by high surface tensions, since the surface tension is a manifestation of the same cohesive force as causes intrinsic pressure. (See, however, equation 3, p. 27, for an empirical relation between the two.)... 

An important qualitative conclusion, which agrees with experience, can immediately be drawn from the theoretical considerations we have developed. A small quantity of dissolved substance may reduce the surface tension very considerably, but can only increase it slightly. Thus, sodium chloride increases the surface tension of water to a small extent the concentration in the surface layer is accordingly smaller than in the bulk and the effect of the solute is thus counteracted. On the other hand, many organic salts, e.g., the oleates, reduce the surface tension and therefore accumulate in the surface layer, so that, in extreme cases, the whole of the solute may be collected there and produce a considerable effect, although the absolute quantity may be exceedingly slight. 

After several cycles of the compression and expansion, the dynamic jc-A curve becomes a single closed loop, somewhat distorted from a genuine ellipsoid. In order to analyze the forms of the hysteresis loop under stationary conditions, we have measured the time trace of the dynamic surface pressure after five cycles of the compression and expansion, and then Fourier-transformed it to the frequency domain. The Fourier-transformation was adapted to evaluate the nonlinear viscoelasticity in a quantitative manner. The detailed theoretical consideration for the use of the Fourier transformation to evaluate the nonlinearity, are contained in the published articles [8,43]. 

Theoretical considerations show that bacterial adhesion to solid, liquid, or sorbed substrates is a powerful mechanism to improve substrate mass transfer. According to Fick s law, the diffusive mass flux of a substrate towards the cell surface J is strongly affected by the space coordinate in direction of the transport ... 

Siloxane bonds are opened readily with strong bases. Nucleophilic attack at the silicon atoms is involved in the reaction mechanism. Attempts at complete hydrolysis of the siloxane bonds on the surface of Aerosil by Boehm and Schneider 267) met with limited success. Partial hydrolysis was achieved with cold water over long periods, with boiling water in limited time, and by the action of ammoniacal pyrocatechol solutions. However, the number of silanol groups never exceeded 3.3OH/100A. Ashasbeenshownonpage228, avalueof ca. 5OH/100A would be expected from theoretical considerations. 

Considerable literature developed around the ability of numerical indices derived from graph theoretical considerations to correlate with S AR data. This was a source of mystery to me for some time. A colleague, loan Motoc, from Romania, with experience in this arena and a very strong intellect, helped me understand the ability of various indices to be useful parameters in QSAR equations [19-21]. loan correlated various indices with more physically relevant (at least to me) variables such as surface area and molecular volume. Since computational time was at a premium during the early days of QSAR and such indices could be calculated with minimal computations, they played a useful role and continue to be used. As a chemist, however, I am much more comfortable with parameters such as surface area or volume. 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

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