Free singular

If one assumes that tlie singular part A of the Helmholtz free energy is such a fimction... 

Beutler, T. C., Mark, A. E., van Schaik, R. C., Gerber, P. R., van Gunsteren, W. F. Avoiding singularities and numerical instabilities in free energy calculations based on molecular simulations. Chem. Phys. Letters 222 (1994) 529-539... 

Buetler T C, A E Mark, R C van Schaik, P R Gerber and W F van Gunsteren 1994. Avoiding Singularities and Numerical Instabilities in Free Energy Calculations Based on Molecular Simulations. Chemical Physics Letters 222 529-539. 

At the free edges of a laminate (sides of a laminate or holes), the interlaminar shearing stresses and/or interlaminar normal stress are very high (perhaps even singular) and would therefore cause the debonding that has been observed in such regions. 

The interlaminar shear stress, t, has a distribution through half the cross-section thickness shown as several profiles at various distances from the middle of the laminate in Figure 4-54. Stress values that have been extrapolated from the numerical data at material points are shown with dashed lines. The value of is zero at the upper surface of the laminate and at the middle surface. The maximum value for any profile always occurs at the interface between the top two layers. The largest value of occurs, of course, at the intersection of the free edge with the interface between layers and appears to be a singularity, although such a contention cannot be proved by use of a numerical technique. 

The first two terms can be evaluated right away, because the exact states tpk and are coupled by their full potentials, Vc and respectively, to unperturbed free space void states. The two other terms axe seen to become evaluable after combining them. In the difference the singular part in precisely cancels. Using G ... 

Flow of trains of surfactant-laden gas bubbles through capillaries is an important ingredient of foam transport in porous media. To understand the role of surfactants in bubble flow, we present a regular perturbation expansion in large adsorption rates within the low capillary-number, singular perturbation hydrodynamic theory of Bretherton. Upon addition of soluble surfactant to the continuous liquid phase, the pressure drop across the bubble increases with the elasticity number while the deposited thin film thickness decreases slightly with the elasticity number. Both pressure drop and thin film thickness retain their 2/3 power dependence on the capillary number found by Bretherton for surfactant-free bubbles. Comparison of the proposed theory to available and new experimental... 

When A = 0 one recovers the Lennard-Jones potential. When A = 1, the atom is annihilated smoothly and the singularity disappears progressively. The parameter a can be chosen to increase the smoothness of the free energy. A small a results in a near singularity around A = 1 while a large a results in a near singularity around A = 0. See article by Beutler et al. [55] for an algorithm to calculate an appropriate value of a. 

Simonson, T., Free energy of particle insertion. An exact analysis of the origin singularity for simple liquids, Molec. Phys. 1993, 80, 441447... 

Avoiding singularities and numerical instabilities in free energy calculations based on molecular simulations, Chem. Phys. Lett. 222 529 (1994). 

Free energy in the critical region is assumed to split into regular and singular parts (Gr and Gs respectivlely), only the latter of which obeys a scaling law,... 

The singular part of the free energy after n iterations Gsn has a value per spin which is a function of Kn and bn. Since Jo and Tc are constants the value per spin may be regarded as a function of (f )nt and (g )nB0. After n blockings the number of spins is n/(2d)n, where d is the number of dimensions. Thus... 

Reviews on water structure models include Mishima and Stanley (1998), Wallqvist and Mountain (1999), and Ludwig (2001). Mishima and Stanley (1998) concentrated their review on three relatively recent water structure hypotheses (1) the stability limit hypothesis (Speedy, 1982), (2) the singularity-free hypothesis (Sastry et al., 1996), and (3) the liquid-liquid phase transition hypothesis (Poole et al., 1992). 

Sastry, S., Debenedetti, P., Sciortino, F., and Stanley, H.E. 1996. Singularity-free interpretation of the thermodynamics of supercooled water. Phys. Rev. E53, 6144-6154. 

---