Gradient, time dependent

As is evident from the fomi of the square gradient temi in the free energy fiinctional, equation (A3.3.52). k is like the square of the effective range of interaction. Thus, the dimensionless crossover time depends only weakly on the range of interaction as In (k). For polymer chains of length A, k A. Thus for practical purposes, the dimensionless crossover time is not very different for polymeric systems as compared to the small molecule case. On the other hand, the scaling of to is tln-ough a characteristic time which itself increases linearly with k, and one has... 

Analytic teclmiques often use a time-dependent generalization of Landau-Ginzburg ffee-energy fiinctionals. The different universal dynamic behaviours have been classified by Hohenberg and Halperin [94]. In the simple example of a binary fluid (model B) the concentration difference can be used as an order parameter m.. A gradient in the local chemical potential p(r) = 8F/ m(r) gives rise to a current j... 

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

If there are no reactions, the conservation of the total quantity of each species dictates that the time dependence of is given by minus the divergence of the flux ps vs), where (vs) is the drift velocity of the species s. The latter is proportional to the average force acting locally on species s, which is the thermodynamic force, equal to minus the gradient of the thermodynamic potential. In the local coupling approximation the mobility appears as a proportionality constant M. For spontaneous processes near equilibrium it is important that a noise term T] t) is retained [146]. Thus dynamic equations of the form... 

Concentration gradients for the analyte in the absence of convection, showing the time-dependent change in diffusion as a method of mass transport. 

Hence, the current (at any time) is proportional to the concentration gradient of the electroactive species. As indicated by the above equations, the dififusional flux is time dependent. Such dependence is described by Fick s second law (for linear diffusion) ... 

The presentation in this paper concentrates on the use of large-scale numerical simulation in unraveling these questions for models of two-dimensional directional solidification in an imposed temperature gradient. The simplest models for transport and interfacial physics in these processes are presented in Section 2 along with a summary of the analytical results for the onset of the cellular instability. The finite-element analyses used in the numerical calculations are described in Section 3. Steady-state and time-dependent results for shallow cell near the onset of the instability are presented in Section 4. The issue of the presence of a fundamental mechanism for wavelength selection for deep cells is discussed in Section 5 in the context of calculations with varying spatial wavelength. 

For the calculation of shear stress, the time-dependent impeller power, particle diameter dp and viscosity v according to v = K/9 with the representative shear gradient y = for the non Newtonian broth (see equation (17) [28]) were used. 

Rothe, GM, Determination of Molecular Mass, Stoke radius. Frictional Coefficient and Isomer-Type of Non-denatured Proteins by Time-Dependent Pore Gradient Gel Electrophoresis, Electrophoresis 9, 307, 1988. 

In the stochastic theory of lineshape developed by Blume [31], the spectral lines are calculated under the influence of a time-dependent Hamiltonian. The method has been successfully applied to a variety of dynamic effects in Mossbauer spectra. We consider here an adaptation due to Blume and Tjon [32, 33] for a Hamiltonian fluctuating between two states with axially symmetric electric field gradients (efg s), the orientation of which is parallel or perpendicular to each other. The present formulation is applicable for states with the same... 

A much simpler and more flexible approach, however, extends the scheme of pulsed magnetic field gradients discussed above to include more complex time dependences. To see how this can be done, we write down the general dependence of the phase shift accumulated at time t subject to a space-dependent Larmor frequency ... 

If the nuclei are displaced by a flow field, their positions on the gradient axis become time dependent ... 

That is, the phase shift depends on the initial position x0, the initial velocity vx0 and the initial acceleration ax0. Higher order terms vanish if the flow field is stationary on the time scale of the NMR experiment (i.e., time-dependent accelerations do not occur in this case). For a gradient pulse of duration t and strength Gx the total phase shift is [see Figure 2.9.4(a)]... 

E. O. Stejskal, J. E. Tanner 1965, (Spin diffusion measurements Spin echoes in the presence of a time-dependent field gradient),/. Chem. Phys. 42 (1), 288—292. 

In an NMR/MRI flow experiment, we would like to measure parameters such as velocity without regard to the starting position of the particle. Thus, mo is always set to zero. The moments m, are under the control of the experimenter in that they are manipulated by the choice of the time dependence of the gradient G. Thus, it is easy to see that m0 can be set to zero by simply making sure that the time integral of the gradient is zero. The easiest way to accomplish this is to have a bipolar gradient of equal absolute amplitude and duration. 

A wide variety of ID and wD NMR techniques are available. In many applications of ID NMR spectroscopy, the modification of the spin Hamiltonian plays an essential role. Standard techniques are double resonance for spin decoupling, multipulse techniques, pulsed-field gradients, selective pulsing, sample spinning, etc. Manipulation of the Hamiltonian requires an external perturbation of the system, which may either be time-independent or time-dependent. Time-independent... 

FA Long, D Richman. Concentration gradients for diffusion of vapors in glassy polymers and their relation to time dependent diffusion phenomena. J Am Chem Soc 82 513-522, 1960. 

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