Constant pressure minimization

Finally, the use of the constant pressure minimization algorithm allows searching for phenomena that can be considered as precursors of pressure-induced transitions. For example, the predicted behaviour of the anatase cell constants as a function of pressure shows that the a(P) and c(P) plots are only linear for P<4 GPa, the value that is close to both the theoretical and experimental transition pressures. At higher pressures the a constant starts to grow under compression, indicating inherent structural instability. In the case of ratile there is a different precursor effect, nami y at 11 GPa the distances between the titanium atom and the two different oxygens, axial and equatorial, become equal. Once again, the pressure corresponds closely to the phase transition point. 

The first stage of any lattice simulation is to equilibrate the structure, i.e. bring it to a state of mechanical equilibrum. The simplest procedure is to equilibrate under conditions of constant volume, i.e. with invariant cell dimensions. Extensions to the procedure were introduced by Parker (1982, 1983) who introduced the use of constant pressure minimization in the computer code METAPOCS, in which lattice energy minimization was performed with respect... 

The basis of constant pressure minimization is that both lattice vectors and coordinates are adjusted to remove forces on both the atoms and the unit cell as a whole. This is commonly performed simultaneously, using the same approach for the lattice vectors as was used in constant volume minimization for the coordinates (i.e. treating the lattice vectors as additional variables). 

We have seen that equilibrium in an isolated system (dt/= 0, dF= 0) requires that the entropy Sbe a maximum, i.e. tliat dS di )jjy = 0. Examination of the first equation above shows that this can only be true if. p. vanishes. Exactly the same conclusion applies for equilibrium under the other constraints. Thus, for constant teinperamre and pressure, minimization of the Gibbs free energy requires that dGId Qj, =. p. =... 

In both mechanical (constant S, minimize PP) and thennal (constant T, minimize G) contexts, pressure drives a system to become smaller or denser. 

Superheaters and Reheaters A superheater raises the temperature of the steam generated above the saturation level. An important function is to minimize moisture in the last stages of a turbine to avoid blade erosion. With continued increase of evaporation temperatures and pressures, however, a point is reached at which the available superheat temperature is insufficient to prevent excessive moisture from forming in the low-pressure turbine stages. This condition is resolved by removing the vapor for reheat at constant pressure in the... 

The calculation is based on the rule of thermodynamics, which states that a system will be in equilibrium when the Gibbs free energy is at a minimum. Cl The objective then is the minimization of the total free energy of the system and the calculation of equilibria at constant temperature and volume or at constant pressure. It is a complicated and lengthy calculation but, fortunately, several computer programs are now available that considerably simplify the task. PI... 

Similar schemes to the above can be used in molecular dynamics simulations in other ensembles such as those at constant temperature or constant pressure (see Frenkel and Smit, and Allen and Tildesley (Further reading)). A molecular dynamics simulation is computationally much more intensive than an energy minimization. Typically with modern computers the real time sampled in a simulation run for large cells is of the order of nanoseconds (106 time steps). Dynamical processes operating on longer time-scales will thus not be revealed. 

The papers of Wagner and Schottky contained the first statistical treatment of defect-containing crystals. The point defects were assumed to form an ideal solution in the sense that they are supposed not to interact with each other. The equilibrium number of intrinsic point defects was found by minimizing the Gibbs free energy with respect to the numbers of defects at constant pressure, temperature, and chemical composition. The equilibrium between the crystal of a binary compound and its components was recognized to be a statistical one instead of being uniquely fixed. 

Flame combustion calorimetry in oxygen is used to measure the enthalpies of combustion of gases and volatile liquids at constant pressure [54,90]. Some highly volatile liquids (e.g., n-pentane [91]) have also been successfully studied by static-bomb combustion calorimetry. In general, however, the latter technique is much more difficult to apply to these substances than flame combustion calorimetry. In bomb combustion calorimetry, the sample is burned in the liquid state and must be enclosed in a container prior to combustion. Encapsulation may be difficult, because it is necessary to minimize the amount of vaporized compound inside the container as much as possible. In addition, volatile liquids tend to burn violently under a pressure of 3.04 MPa of oxygen, which leads to incomplete combustion. These problems are avoided in flame combustion calorimetry, where the sample is carried to the combustion zone as a vapor and burned under controlled conditions at atmospheric pressure. 

The gas flow rates are adjusted to ensure a constant pressure inside the cell and to minimize degradation of the spectral resolution by the gas bubbles. Typical line-widths at half height are <4 Hz on NMR (200 MHz) and <2 Hz on (80.3 MHz). The probe fits into the room temperature shims of a wide-bore NMR magnet and the pressure and temperature range of the high pressure NMR flow are 0.1 to 20 MPa and -40 to 175 °C, respectively. 

One goal of a bulk delivery system is to provide constant pressure to a CMP polisher, so that the peristaltic metering pump in the polisher can provide constant slurry flow to the polisher platen. While many schemes have been devised to minimize the fluctuations in pressure, there are many... 

Pumps should be started slowly to prevent water hammer (a surge resulting from a sudden change in liquid velocity). Water hammer can cause cracks in the outer shell of the membrane modules as well as compaction of the membrane itself (compaction results in lower flux through the membrane at constant pressure). Also, water hammer causes the membrane modules to move in the vessel, which can cause wear to the O-rings used on standard interconnectors and lead to leaks of feed water into the permeate (see Chapter 4.3.3). An increase in pressure of no more than 10 psi per second is recommended.3 Some motors may be equipped with a "soft start" that regulates the speed with which they start up. Other considerations to minimize water hammer include ... 

Thermodynamic calculations can identify the adiabatic combustion temperature, as well as the equilibrium phases and compounds present at that temperature. The composition of the equilibrium final products is determined by minimizing the thermodynamic potential. For a system with gas and A solid number of components, at constant pressure, this may be expressed as... 

In the general iterative approach, one first determines the equilibrium state for the product composition at an initially assumed value of the temperature and pressure, and then one checks to see whether the energy equation is satisfied. Chemical equilibrium is usually described by either of two equivalent formulations— equilibrium constants or minimization of free energy. For such simple problems as determining the decomposition temperature of a monopropellant having few exhaust products or examining the variation of a specific species with temperature... 

Lower Punch Brakes. Most rotary tablet presses are equipped with lower-punch brakes that are Teflon tipped and spring loaded to apply constant pressure to the lower punches. Alternatively, some manufacturers apply pressure to a friction belt that provides resistance on the lower punches. The lower-punch brakes act as a retention system for holding the lower punches in place during press setup. More importantly, these systems help to minimize lower-punch chatter at high press speeds thus minimizing tablet weight variation. Some press manufacturers use the lower punch seals to retain the lower punches. 

At a given density, the data indicate that the solvent strength is essentially independent of temperature, consistent with the fact that the polarizability is also independent of temperature. The "thermochromic effect is minimal at constant density. However, it is significant at constant pressure in regions where density is a strong function of temperature. 

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