Square root of time

The two most common temporal input profiles for dmg delivery are zero order (constant release), and half order, ie, release that decreases with the square root of time. These two profiles correspond to diffusion through a membrane and desorption from a matrix, respectively (1,2). In practice, membrane systems have a period of constant release, ie, steady-state permeation, preceded by a period of either an increasing (time lag) or decreasing (burst) flux. This initial period may affect the time of appearance of a dmg in plasma on the first dose, but may become insignificant upon multiple dosing. 

In the other common paradigm, desorption from a system or a material is studied. At t = 0, the system is loaded with a known concentration or activity of dmg and is immersed in an infinite, weU-stirred receiver solution maintained at concentration C = 0. The amount released per unit area is then measured over time. For a uniform slab of thickness / where the release of dmg is from both sides, the fraction released / Q is linear with the square root of time and is given by ... 

If De crr 1, non-Fickian behavior may appear, ie, a dependence other than linearity with the square root of time may be observed. 

A monolithic system is comprised of a polymer membrane with dmg dissolved or dispersed ia it. The dmg diffuses toward the region of lower activity causiag the release of the dmg. It is difficult to achieve constant release from a system like this because the activity of the dmg ia the polymer is constantly decreasiag as the dmg is gradually released. The cumulative amount of dmg released is proportional to the square root of time (88). Thus, the rate of dmg release constantly decreases with time. Again, the rate of dmg release is governed by the physical properties of the polymer, the physical properties of the dmg, the geometry of the device (89), and the total dmg loaded iato the device. 

If the driving pressure is taken to be the capillary pressure, 2yivCOS0/r, Eq. 23 may be integrated, assuming 9 and r] are constant to give the Washburn equation [43] which shows the penetration jCt is proportional to the square root of time t... 

At small driving forces a completely fiat interface cannot move at a constant speed. This is basically a result of the inherent scaling property of the diffusion equation, which scales lengths proportional to the square-root of time, so an advancing interface would slow down with time. 

Equation 1.116 is ohmic (/ a r for constant film thickness) the term x/2AB can be regarded as the film resistance. The equation is identical to parabolic film growth, for which the film thickens with the square root of time at constant potential. 

That is, the current decreases in proportion to the square root of time, with (nDQt)l/2 corresponding to the diffusion layer thickness. 

Maiti and Bhowmick also investigated the diffusion and sorption of methyl ethyl ketone (MEK) and tetrahydrofuran (THE) through fluoroelastomer-clay nanocomposites in the range of 30°C-60°C by swelling experiments [98]. A representative sorption-plot (i.e., mass uptake versus square root of time, at 45°C for all the nanocomposite systems is given in Figure 2.12. 

This means that the precision of the prediction decreases with the square root of time. This describes the random walk model. A drift can be easily built into such a model by the addition of some constant drift function at each successive time period. 

In vitro dissolution was virtually complete after 6-8 hr. Since the plot of cumulative drug release versus time is hyperbolic, the authors attempted to fit the data to the Higuchi matrix dissolution model (116,117), which predicts a linear correlation between cumulative drug release and the square root of time. Linearity occurred only between 20 and 70% release. 

Hydrogen chloride quantity captured by sodium of glass cullets at 823K as a function of square root of time is shown in Fig. 1. The amount of hydrogen chloride captured as sodium chloride was proportional to square root of time for most of the region, and thus the neutralization rates were controlled by diffusion. On the other hand partial pressure of hydrogen chloride did not affect the formation of sodium chloride even though its partial... 

We see from these equations that the current density decreases with the inverse square root of time. 

The concentration distributions found at different times after the start of current flow are shown in Fig. 11.4. It is a typical feature of the solution obtained that the variable parameters x and t do not appear independently but always as the ratio Like Eq. 11.15), this indicates that the diffusion front advances in proportion to the square root of time. This behavior arises because as the diffusion front advances toward the bulk solution, the concentration gradients become flatter and thus diflusion slows down. 

The driving force for the transfer process was the enhanced solubility of Br2 in DCE, ca 40 times greater than that in aqueous solution. To probe the transfer processes, Br2 was recollected in the reverse step at the tip UME, by diffusion-limited reduction to Br . The transfer process was found to be controlled exclusively by diffusion in the aqueous phase, but by employing short switching times, tswitch down to 10 ms, it was possible to put a lower limit on the effective interfacial transfer rate constant of 0.5 cm s . Figure 25 shows typical forward and reverse transients from this set of experiments, presented as current (normalized with respect to the steady-state diffusion-limited current, i(oo), for the oxidation of Br ) versus the inverse square-root of time. 

Fig. 2.3.9 A time series of profiles showing the is shown by the lowest trace. The inset shows ingress from right (stratum corneum) to left the advance of the glycerine front against the (viable epidermis) of glycerine into human skin square root of time from which Fickian in vitro. The skin before application of glycerine diffusion is inferred.

The penetration front was measured from the images in Figure 3.4.9 and plotted against the square root of time in Figure 3.4.10. The plot indicates that this relationship is linear and its slope is a measure of the sorptivity [28]. This type of experiment, coupled with gravimetric measurements, allow for the modeling of... 

Fig. 3.4.10 Sorptivity determined from Figure 3.4.9 [27]. There is a linear relationship between penetration depth and the square root of time.

It is evident from this expression that the variation of C(t) will decrease in proportion to the inverse of the square root of time. Since C(l) changes rapidly during the early part of the sampling period, there is an expected increase in variability in C(l) which will, in turn, affect the variability of calculated values for D. Therefore, this simple but useful error analysis shows that the expected variation in calculated values for the diffusion coefficient will be diminished if the concentrations are measured later during the relevant sampling period. 

A moving front is usually observed in swelling glassy polymers. A diffusion-controlled front will advance with the square root of time, and a case II front will advance linearly with time. Deviations from this simple time dependence of the fronts may be seen in non-slab geometries due to the decrease in the area of the fronts as they advance toward the center [135,140], Similarly, the values of the transport exponents described above for sheets will be slightly different for spherical and cylindrical geometries [141],... 

This solution is valid for the initially linear portion of the sorption (or desorption) curve when MtIM is plotted against the square root of time. These equations also demonstrate that for Fickian processes the sorption time scales with the square of the dimension. Thus, to confirm Fickian diffusion rigorously, a plot of MJM vs. Vt/T should be made for samples of different thicknesses a single master curve should be obtained. If the data for samples of different thicknesses do not overlap despite transport exponents of 0.5, the transport is designated pseudo-Fickian. ... 

The result shown in Eq. (54) is a square root of time relationship for moisture uptake. Mulski [20] demonstrated that for sodium glycinate in a hydrophobic porous matrix, moisture sorption follows Eq. (54). 

In contrast to normal diffusion, Ar2n does not grow linearly but with the square root of time. This may be considered the result of superimposing two random walks. The segment executes a random walk on the random walk given by the chain conformation. For the translational diffusion coefficient DR = kBT/ is obtained DR is inversely proportional to the number of friction-performing segments. 

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