Application of time constants

The errors in the present stochastic path formalism reflect short time information rather than long time information. Short time data are easier to extract from atomically detailed simulations. We set the second moment of the errors in the trajectory - [Pg.274]

From Fig. 2.35 it may be seen that for the Maxwell model, the strain at any time, t, after the application of a constant stress, Cg, is given by... 

Figure 5.2. NEMCA and its origin on Pt/YSZ catalyst electrodes. Transient effect of the application of a constant current (a, b) or constant potential UWR (c) on (a) the rate, r, of C2H4 oxidation on Pt/YSZ (also showing the corresponding UWR transient)3 (b) the 02 TPD spectrum on Pt/YSZ4,7 after current (1=15 pA) application for various times t. (c) the cyclic voltammogram of Pt/YSZ4,7 after holding the potential at UWR = 0.8 V for various times t.

Integrated Mnvino Average Model tIMAl. Proper application of time series analysis requires that the variance of the series be constant and that there be no major trend. Any segment of the time series should be very much like any other segment. If this is not the case then the inferences will depend... 

Complex models are often slow in execution owing to the large number of equations involved and the large range of time constants. Under these circumstances it is often useful to approximate the transient behaviour of the full model by a simpler model representation which is faster to compute. Such simplifications are commonly achieved by a combination of first-order lags and time delays and are often represented in Laplace transform form, especially when the sub-model is to be used as part of a control engineering application. 

In the controlled (constant) potential method the procedure starts and continues to work with the limiting current iu but as the ion concentration and hence its i, decreases exponentially with time, the course of the electrolysis slows down quickly and its completion lags behind therefore, one often prefers the application of a constant current. Suppose that we want to oxidize Fe(II) we consider Fig. 3.78 and apply across a Pt electrode (WE) and an auxiliary electrode (AE) an anodic current, -1, of nearly the half-wave current this means that the anodic potential (vs. an RE) starts at nearly the half-wave potential, Ei, of Fe(II) - Fe(III) (= 0.770 V), but increases with time, while the anodic wave height diminishes linearly and halfway to completion the electrolysis falls below - / after that moment the potential will suddenly increase until it attains the decomposition potential (nearly 2.4 V) of H20 -> 02. The way to prevent this from happening is to add previously a small amount of a so-called redox buffer, i.e., a reversible oxidant such as Ce(IV) with a standard... 

Figure 21. Change of potential of aluminum in 2 M NaCl solution upon application of two constant-current pulses (10 mA/cm2) with shortening of the time interval between them.

The presence of the electron acceptor site adjacent to the donor site creates an electronic perturbation. Application of time dependent perturbation theory to the system in Figure 1 gives a general result for the transition rate between the states D,A and D+,A. The rate constant is the product of three terms 1) 27rv2/fi where V is the electronic resonance energy arising from the perturbation. 2) The vibrational overlap term. 3) The density of states in the product vibrational energy manifold. 

Now a shear rate represents the rate of change of strain so for the application of a constant shear rate to the sample the strain is the product of the rate and the time ... 

In general it will be necessary to measure via impedance measurements using a four electrode cell. A schematic diagram of the cell which would be used for such measurements is shown in Fig. 10.15. The expected behaviour will be as described in Eqn (10.3) except that Warburg impedances can arise from either or both phases. An example of an impedance spectrum of the H2O/PVC interface is shown in Fig. 10.16. The application of a constant overpotential will, in general, lead to a slowly decaying current with time due to the concentration changes which occur in both phases, so that steady state current potential measurements will be of limited use. 

Time characterizing the response of a viscoelastic liquid or solid to the instantaneous application of a constant strain. 

Change in strain with time after the instantaneous application of a constant stress. or... 

Since all electrophoretic mobility values are proportional to the reciprocal viscosity of the buffer, as derived in Chapter 1, the experimental mobility values n must be normalized to the same buffer viscosity to eliminate all other influences on the experimental data besides the association equilibrium. Some commercial capillary zone electrophoresis (CZE) instruments allow the application of a constant pressure to the capillary. With such an instrument the viscosity of the buffer can be determined by injecting a neutral marker into the buffer and then calculating the viscosity from the time that the marker needs to travel through the capillary at a set pressure. During this experiment the high voltage is switched off. 

This expression, known as Sand s equation, gives the variation of the interfacial concentration of M"+ with time after application of a constant current density. But one seeks also to know the time variation of the potential difference across the interface at which the electronation reaction M"+ + ne — M is occurring. To obtain this information, one recalls that the charge-transfer reaction across the interface is assumed in the present treatment to be virtually in equilibrium and therefore the Nenist equation (7.177) can be used to relate the potential difference to the concentration at the interface. That is, by substituting (7.181) in (7.177),... 

In electrochemical techniques based on current excitation, a current signal is impressed on a quiescent electrochemical cell, and a response signal, usually the potential of the working electrode, is measured as a function of time. The most commonly used excitation signal is the application of a constant-current-step function to the cell as shown in Figure 4.1. 

However, it is not ideal to use cyclic or linear-sweep voltammetry as a method for analytical purposes. A more suitable method is chronoamper-ometry, which in fact is the application of a constant potential located in the limiting-current plateau and measurement of the limiting-current as a function of time. With this method, it is possible to measure continuously, and the required equipment setup becomes much more simplified. 

An example of the concentration profiles of the oxidized species O, calculated for different times and corresponding to the application of a constant potential under linear diffusion conditions, is shown in Fig. 1.20. The electrode reaction at the interface leads to the depletion of species O at the solution region adjacent to the electrode surface. As the time increases, the layer in the solution affected by the diffusive mass transport becomes thicker, which indicates that linear diffusion is unable to restore the initial situation (for a more detailed discussion on concentration profiles and their relation with the current, see Sects. 2.2.1 and 2.2.2). 

Fig. 2.3 Experimental current-time curve (a) and logarithmic curves (b) for the application of a constant potential to a graphite disc electrode of radius 0.5 mm (planar electrode) for the reduction of Fe(CN)g. ...

Fig. 2.15 (Solid line) Current-time curves for the application of a constant potential to a spherical electrode calculated from Eq. (2.142). D0 = Dr = 10-5 cm2 s 1, co = cr = 1 mM, rs = 0.001 cm, (E — E ) = -0.2 V, 7=298 K. (Dashed line) Current-time curves for the application of a constant potential to a planar electrode of the same area as the spherical one calculated from Eq. (2.28). (Dotted line) Steady-state limiting current for a spherical electrode calculated from Eq. (2.148). The inner figure corresponds to the plot of the current of the spherical electrode versus j ft...

Fig. 6.18 Dimensionless current-time (a) and charge-time (b) curves corresponding to the application of a constant potential Ei — /ic° = —0.2 V to an electro-active monolayer calculated from Eqs. (6.116) and (6.115) assuming a Butler-Volmer kinetics with a = 0.5. The values of (k°-r) are 0.05 (black), 0.1 (red), 0.25 (green), 0.5 (blue),...

Creep measurements involve the application of a constant stress (usually a shearing stress) to the sample and the measurement of the resulting sample deformation as a function of time. Figure 9.6 shows a typical creep and recovery curve. In stress-relaxation measurements, the sample is subjected to an instantaneous predetermined deformation and the decay of the stress within the sample as the structural segments flow into more relaxed positions is measured as a function of time. 

Crazing Creep Fine cracks on surface of a material. Compressive deformation occurring over time in both cured and uncured polyurethane, resulting from the application of a constant load or stress. 

Fig. 5.7. Variation of potential with time (chronopotentiogram) in a system controlled by diffusion and with application of a constant current, x is the transition time. Er/4 is the potential when t = t/4 (Section 10.5).

Boundary value — A boundary value is the value of a parameter in a differential equation at a particular location and/or time. In electrochemistry a boundary value could refer to a concentration or concentration gradient at x = 0 and/or x = oo or to the concentration or to the time derivative of the concentration at l = oo (for example, the steady-state boundary condition requires that (dc/dt)t=oo = 0). Some examples (dc/ dx)x=o = 0 for any species that is not consumed or produced at the electrode surface (dc/dx)x=o = -fx=0/D where fx=o is the flux of the species, perhaps defined by application of a constant current (-> von Neumann boundary condition) and D is its diffusion coefficient cx=o is defined by the electrode potential (-> Dirichlet boundary condition) cx=oo, the concentration at x = oo (commonly referred to as the bulk concentration) is a constant. 

The elastic properties discussed so far relate to stresses applied at relatively low rates. When forces are applied at rapid rates, then dynamic moduli are obtained. The energy relationships and the orders of magnitude of the data are much different [570]. Because of the experimental difficulties, only little work at rapid rates has been carried out with cotton fiber compared to that done with testing at low rates of application of stress. In contrast, cotton also responds to zero rate of loading, i.e., the application of a constant stress. Under this condition the fiber exhibits creep that is measured by determining fiber elongation at various intervals of time after the load has been applied. Creep is time-dependent and may be reversible upon removal of the load. However, even a low load applied to a fiber for a long period of time will cause the fiber to break. 

Figure 28. Chronopotentiometry potential-time relationship after application of a constant current. Ej/4 is the potential acquired by the electrode at time r/4.

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