Constant Concentration Conditions Research Articles

Comparison of formulas derived from Fourier series solution of diffusion equation

The first Fourier series solution satisfying the boundary condition of zero concentration at the diffusion distance was derived by assuming that the solution consists of a steady-state solution and a transient solution. The first total amount which has passed through the first layer surface was derived from the first Fourier series solution. When the diffusion distance is large, the exponential term in the first total amount cannot be zero. For the sublimation diffusion of disperse dye in paste into polyethylene terephthalate film using the film-roll method, the plot of the first total amount against time was in good agreement with the quadratic regression curve. The second Fourier series solution satisfying the boundary condition of constant concentration at the diffusion distance was derived by assuming that the diffusion distance is the thickness of the first layer. The second total amount which has passed through the second layer surface was derived from the second Fourier series solution and is a linear function of time because the exponential term is zero due to the very small diffusion distance, the thickness of the first layer. The plot of the second total amount against time was in good agreement with the linear regression line. The time-lag formula was derived from the second total amount. The diffusion coefficients determined from the second total amount are much larger than those of the other three types. All four types showed excellent linearity in Arrhenius plot, but the activation energy for the time-lag formula is somewhat greater than the rest.

Analytical solution of three-dimensional particle transport in porous media considering a dual deposition mode

Based on the classical one-dimensional particle transport model, a three-dimensional particle transport model with a dual deposition mode considering sieving and adsorption effects was established. Through Laplace and Fourier transforms, general solutions for particle transport in saturated semi-infinite porous media under one-dimensional seepage and three-dimensional dispersion conditions were derived. According to the basic solution in the case of a point source injection on the surface of a semi-infinite body, the analytical expression in the case of a circular surface source injection was obtained by the integration method. The analytical solution is verified using two well-established numerical simulation solvers. The influence of parameters such as the hydrodynamic dispersion coefficient, sieving coefficient, adsorption coefficient, and desorption coefficient, on the mechanism of the particle transport process was analyzed under the conditions of constant concentration and circulating concentration injection of the circular surface source. The results showed that the hydrodynamic dispersion effect accelerated the transport of particles, shortened the breakthrough time, and increased the peak concentration. Moreover, a larger sieving coefficient and particle adsorption coefficient meant a smaller particle release coefficient, and more particles deposited on the solid matrix with a smaller peak particle concentration in the pores.

Soil moisture migration model based on flux-concentration relation under steady rainfall

Abstract Moisture content distribution in soil is of great importance for understanding rainfall-induced slope failure and roadbed settlement. This study aims to develop a moisture migration model that improves the quantification of moisture content at an arbitrary point of the soil at any time for the whole process of infiltration under steady rainfall. The model was derived from the Richards equation using the flux-concentration relation, which was validated by numerical solutions calculated by Hydrus-1D software to evaluate the performance of the model. Results showed good accuracy and high adaptability for the moisture migration simulation of a wide range of soil types, and is applicable for short-duration and long-duration steady rainfall. Moreover, it can also reflect the stratification phenomenon for soil profile wetting by infiltration. Our analysis indicates that the flux and surface volumetric moisture content together can bound the boundary conditions of rainfall infiltration, which presents a shift from constant-flux to constant-concentration during long-duration steady rainfall. The migration rate of the wetting front in the later stage of infiltration positively correlates with rainfall intensity under the constant-flux condition, while it finally stabilizes at Ks/(θs − θi) under the constant-concentration condition (i.e., Ks – saturated hydraulic conductivity, θs – saturated volumetric moisture content, θi – initial volumetric moisture content).

Transient thermo-solutal convection in a tilted porous enclosure heated from below and salted from above

The confinement of CO2 in deep geothermal reservoirs as a means of mitigation of greenhouse gas emissions is continuously motivating research on the retention capacity of these deep aquifers. An important physical containment mechanism is related with CO2 dissolution and thermo-solutal convection. In this context, numerical simulations are performed in this work to assess the effect of inclination, Rayleigh number, and buoyancy ratio on the convective transport in a rectangular porous medium. The porous enclosure is heated from below and cooled from above, whereas a solute is dissolved through the upper boundary with a constant concentration condition and no mass loss through the other boundaries. A set of governing parameters is considered in this assessment: two buoyancy ratios with dominant solute buoyant forces (10 and 100), three Rayleigh numbers (10, 50, and 80), and three inclination angles plus the horizontal case for reference (5°, 10°, and 15°). The solution to the problem is based on a Finite Volume method along with the fixed point iteration for the coupled differential equations, and a Conjugate Gradient algorithm for the algebraic system. The model is validated and tested under mesh analysis. The numerical results show that the inclination angle has a minor effect on the convective mixing properties of the porous medium in comparison with the Rayleigh number and the buoyancy ratio. Increasing the angle slightly decreases the mixing rate as a consequence of the formation of preferential flow paths associated with the inclination, these preferential flow paths make mixing less efficient and give rise to zonation of solute concentration.

Committed Warming Due to Earth’s Radiative Imbalance Using a Simple 5-Factor Energy Balance Model

The concept of committed warming has been neglected or understated in the climate change conversation, although climate scientists acknowledge that some additional global warming is “baked in” no matter how quickly the world switches to a clean-energy economy. The Earth’s climate is not equilibrated with anthropogenic climate forcing. Ignoring thermal inertia of the oceans, this study shows that stabilizing CO2 concentrations at today’s levels would still commit the earth to at least 0.5oC of additional warming in the next century. Due to past anthropogenic emissions the earth currently absorbs more heat than it emits, which induces a yet unrealized commitment. Even if the CO2 levels stopped increasing, the planet would have to warm further in order to balance solar energy absorbed with energy radiated back to space. This study incorporates a simplified, five-factor model (5FEBM) that uses only known or well-understood factors that drive global warming. These five key factors come close to accounting for the earth’s current radiative imbalance, as well as future temperatures predicted by global climate models. The factors include human greenhouse gas emissions, human generated aerosol emissions, human caused land surface alteration, water vapor feedback, and Planck radiation feedback. Published research on the topic of committed warming mostly corroborates the results of the 5FEBM model. Multiple sources predict a committed warming on the order of 1.5oC relative to pre-industrial conditions. This is particularly relevant to the IPCC’s recent call for more aggressive emission reduction targets to keep total warming below 1.5oC by the end of the 21st century. The evidence herein suggests that may not be possible. Ignoring other feedbacks that will surely worsen, the five factors listed above commit the earth to a global mean surface temperature (GMST) anomaly slightly more than 1.5oC in 2100. A numerical method is developed from the 5FEBM to estimate committed warming under constant concentration conditions. The method is validated by integrating a differential equation involving temperature and time at constant CO2 concentration. The effect of this warming commitment on the year-to-year GMST anomaly is graphed. A sensitivity analysis demonstrates the solution is robust with respect to assumed initial conditions. There is some divergence among estimates of committed warming based partly on assumptions about emissions. Most studies assume constant composition commitment (CCC), but some envision zero emissions. Others assume continued emissions from current fossil fuel investments that would persist through the end of the investments’ useful life. Another scenario admits to an ongoing emissions commitment due to certain energy technologies for which there is currently no commercially available replacement. Presumably, the rationale for estimating committed warming is to portray an idealized but still reasonable picture of the current climate crisis. Calling attention to committed warming seeks to motivate prompt action on climate change by showing both the possibilities and the urgency of mitigating its impacts. The CCC scenario balances these two objectives.

Investigation of the Flux–Concentration Relation for Horizontal Flow in Soils

The objective of the present work is to investigate the flux–concentration (F(Θ)) relation, where Θ is the normalized soil volumetric water content for the case of one-dimensional horizontal flow, subject to constant concentration conditions. More specifically, the possibility of describing F(Θ) by an equation of the form F(Θ) = 1 − (1 − Θ)p+1 is examined. Parameter p is estimated from curve-fitting of the experimentally obtained λ(Θ) data to an analytic expression of the form (1 − Θ)p where λ is the well-known Boltzmann transformation λ = xt−0.5 (x = distance, t = time). The results show that the equation of (1 − Θ)p form can satisfactorily describe the λ(Θ) relation for the four porous media tested. The proposed F(Θ) function was compared with the limiting F(Θ) function for linear and Green–Ampt soils and to the actual F(Θ) function. From the results, it was shown that the proposed F(Θ) function gave reasonably accurate results in all cases. Moreover, the analytical expression of the soil water diffusivity (D(Θ)) function, as it was obtained by using the equation for λ(Θ) of the form (1 − Θ)p, appears to be very close to the experimental D(Θ) data (root mean square error (RMSE) = 0.593 m2min−1).

Solutal convection in porous media: Comparison between boundary conditions of constant concentration and constant flux

We numerically examine solutal convection in porous media, driven by the dissolution of carbon dioxide (CO2) into water---an effective mechanism for CO2 storage in saline aquifers. Dissolution is associated with slow diffusion of free-phase CO2 into the underlying aqueous phase followed by density-driven convective mixing of CO2 throughout the water-saturated layer. We study the fluid dynamics of CO2 convection in the single aqueous-phase region. A comparison is made between two different boundary conditions in the top of the formation: (i) a constant, maximum aqueous-phase concentration of CO2, and (ii) a constant, low injection-rate of CO2, such that all CO2 dissolves instantly and the system remains in single phase. The latter model is found to involve a nonlinear evolution of CO2 composition and associated aqueous-phase density, which depend on the formation permeability. We model the full nonlinear phase behavior of water-CO2 mixtures in a confined domain, consider dissolution and fluid compressibility, and relax the common Boussinesq approximation. We discover new flow regimes and present quantitative scaling relations for global characteristics of spreading, mixing, and a dissolution flux in two- and three-dimensional media for both boundary conditions. We also revisit the scaling behavior of Sherwood number (Sh) with Rayleigh number (Ra), which has been under debate for porous-media convection. Our measurements from the solutal convection in the range 1, 500<Ra<135, 000 show that the classical linear scaling Sh ~ Ra is attained asymptotically for the constant-concentration case. Similarly linear scaling is recovered for the constant-flux model problem.

Diffusion-induced bending of viscoelastic beams

The contribution of local volumetric change due to the diffusion/migration of solute atoms to viscoelastic deformation is incorporated in the theory of linear viscoelasticity, following the elastic theory of diffusion-induced stress. Three-dimensional constitutive relationship in differential form for diffusion-induced stress in linear viscoelastic materials is proposed. Using the correspondence principle between linear viscoelasticity and linear elasticity and the results from the diffusion-induced bending of elastic beams, the radii of curvature of the centroidal plane of viscoelastic beams of single layer and bilayer with top layer being viscoelastic in the transform domain are obtained. For viscoelastic beams of single layer, closed-form solution of the radius of curvature of the centroidal plane is derived, and the radius of curvature is inversely proportional to the diffusion moment created by non-uniform distribution of solute atoms. For the condition of constant concentration on free surface, there is overshoot behavior; for the condition of constant flux on free surface, there is no overshoot behavior. For viscoelastic beams of bilayer with top layer being the Maxwell-type standard material, the numerical results show the presence of the overshoot behavior for a very compliant elastic layer under the condition of constant concentration on free surface, and there is no overshoot behavior under the condition of constant flux on free surface.

History-Dependent Depolymerization of Actin Filaments

Depolymerizing cytoskeletal filaments are involved in cell division, cell motility, and other cellular functions. Understanding the dynamics of depolymerization is as important as understanding the dynamics of polymerization. We study nonequilibrium depolymerization of actin filaments using a simple two-state model. We show that the polymerization history influences the dynamics of depolymerization as well as the length fluctuations during depolymerization. We also simulate depolymerization under different experimentally feasible conditions. Under conditions of constant concentration, we show that the depolymerization happens in two regimes. Under the conditions of mass conservation, the depolymerization can have three regimes.

Two-peak approximation in kinetic capillary electrophoresis

Kinetic capillary electrophoresis (KCE) constitutes a toolset of homogeneous kinetic affinity methods for measuring rate constants of formation (k(+)) and dissociation (k(-)) of non-covalent biomolecular complexes, C, formed from two binding partners, A and B. A parameter-based approach of extracting k(+) and k(-) from KCE electropherograms relies on a small number of experimental parameters found from the electropherograms and used in explicit expressions for k(+) and k(-) derived from approximate solutions to mass transfer equations. Deriving the explicit expressions for k(+) and k(-) is challenging but it is justified as the parameter-based approach is the simplest way of finding k(+) and k(-) from KCE electropherograms. Here, we introduce a unique approximate analytical solution of mass transfer equations in KCE termed a "two-peak approximation" and a corresponding parameter-based method for finding k(+) and k(-). The two-peak approximation is applicable to any KCE method in which: (i) A* binds B to form C* (the asterisk denotes a detectable label on A), (ii) two peaks can be identified in a KCE electropherogram and (iii) the concentration of B remains constant. The last condition holds if B is present in access to A* and C* throughout the capillary. In the two-peak approximation, the labeling of A serves only for detection of A and C and, therefore, is not required if A (and thus C) can be observed with a label-free detection technique. We studied the proposed two-peak approximation, in particular, its accuracy, by using the simulated propagation patterns built with the earlier-developed exact solution of the mass-transfer equations for A* and C*. Our results prove that the obtained approximate solution of mass transfer equations is correct. They also show that the two-peak approximation facilitates finding k(+) and k(-) with a relative error of less than 10% if two peaks can be identified on a KCE electropherogram. Importantly, the condition of constant concentration of B is always satisfied in macroscopic approach to studying kinetics at equilibrium (MASKE) whether or not B is in excess to A* and C*, and, thus, the two-peak approximation is applicable to MASKE. It completes a toolset of fitting-free methods for processing MASKE data and makes MASKE a simple practical method for finding k(+) and k(-) of "fast", "slow", and "intermediate-rate" non-covalent interactions.

Study of Sorption-Retarded U(VI) Diffusion in Hanford Silt/Clay Material

A diffusion cell method was applied to measure the effective pore diffusion coefficient (Dp) for U(VI) under strictly controlled chemical conditions in a silt/clay sediment from the U.S. Department of Energy Hanford site, WA. "Inward-flux" diffusion studies were conducted in which [U(VI)] in both aqueous and solid phases was measured as a function of distance in the diffusion cell under conditions of constant concentration at the cell boundaries. A sequential extraction method was developed to measure sorbed contaminant U(VI) in the solid phase containing extractable background U(VI). The effect of sorption kinetics on U(VI) interparticle diffusion was evaluated by comparing sorption-retarded diffusion models with sorption described either as equilibrium or intraparticle diffusion-limited processes. Both experimental and modeling results indicated that (1) a single pore diffusion coefficient can simulate the diffusion of total aqueous U(VI), and (2) the local equilibrium assumption (LEA) is appropriate for modeling sorption-retarded diffusion under the given experimental conditions. Dp of 1.6-1.7 x 10(-6) cm2/s was estimated in aqueous solution at pH 8.0 and saturated with respect to calcite, as relevant to some subsurface regions of the Hanford site.

Study of Sorption and Diffusion of 137Cs in Compacted Bentonite Saturated with Saline Water at 60°C

The effect of compaction of bentonite on the sorption behavior of 137Cs was studied for the safety assessment of the high level radioactive waste. The diffusion coefficients (effective De and apparent Da ) and the distribution coefficient for sorption Kd for 137Cs in compacted and dispersed bentonite saturated with saline water were investigated at 60°C by four different sorption and diffusion experiments: the in-diffusion, through-diffusion, reservoir-depletion and batch sorption experiments. The system of the through-diffusion experiment was carefully designed to maintain the boundary conditions of constant concentration at each end of the specimen. De and Da were found to be reproducible and showed good consistency among three of the diffusion experiments (through-diffusion, in-diffusion and reservoir depletion). Kd of 137Cs in compacted bentonite determined from the three types of diffusion experiments was in good accordance with that determined by the batch sorption experiment for dispersed bentonite.

Study of Sorption and Diffusion of 137Cs in Compacted Bentonite Saturated with Saline Water at 60.DEG.C

The effect of compaction of bentonite on the sorption behavior of 137Cs was studied for the safety assessment of the high level radioactive waste. The diffusion coefficients (effective De and apparent Da ) and the distribution coefficient for sorption Kd for 137Cs in compacted and dispersed bentonite saturated with saline water were investigated at 60°C by four different sorption and diffusion experiments: the in-diffusion, through-diffusion, reservoir-depletion and batch sorption experiments. The system of the through-diffusion experiment was carefully designed to maintain the boundary conditions of constant concentration at each end of the specimen. De and Da were found to be reproducible and showed good consistency among three of the diffusion experiments (through-diffusion, in-diffusion and reservoir depletion). Kd of 137Cs in compacted bentonite determined from the three types of diffusion experiments was in good accordance with that determined by the batch sorption experiment for dispersed bentonite.

Cytocidal effect and DNA damage of nedaplatin in vitro by simulating pharmacokinetic parameters.

The pharmacodynamic effects of cis-diammine(glycolato)platinum (nedaplatin, 254-S) in vitro have been reported, but the dosage and exposure time in vitro have not always been based on clinical observations of the drug's actions in vivo. Regardless of the actual exposure conditions used, the effect of cell-cycle nonspecific anticancer agents such as nedaplatin is believed to depend on the area under the drug concentration-time curve (AUC). In this study, we evaluated the pharmacodynamics of nedaplatin in vitro, especially in relation to its AUC dependency, in terms of cell survival and DNA crosslinking. BG-1 human ovarian cancer cells were treated with various concentrations of nedaplatin to simulate the pharmacokinetics of administration in a clinical setting. The BG-1 cells were exposed to nedaplatin dissolved in medium containing serum using constant concentration conditions, either high (maximum 7.69 mg/l) or low (average 1.33 mg/l). These concentrations were based on doses used in clinical studies. We then adjusted the exposure conditions in vitro to simulate the elimination of the drug from serum in vivo as follows: T1/2 alpha 1.20 h and T1/2 beta 2.70 h. The AUC values were set at 4, 8, 16, 25 and 40 mg.h/l for all exposure conditions. A colony-formation assay for the surviving fraction and an alkaline-elution assay for DNA crosslink measurement were done for the pharmacodynamic evaluation with comparison on the basis of the AUC value. Exposure to a low concentration for a long time was the most effective of the exposure conditions at the same AUC value. The greater the AUC value, the higher the crosslink index under all exposure conditions. This index tended to increase particularly after exposure to the low concentration. The natural logarithm of the surviving fraction (Y') was a linear function of the crosslink index regardless of the drug-exposure condition: ln(Y') = -87.2x + ln(5.79), R2 = 0.89. The threshold cytocidal effect was associated with a crosslink index of 0.02. There was a strong correlation between the cytocidal effect of nedaplatin and DNA crosslink formation. The cytocidal effect and DNA crosslinking in vitro depended on the exposure conditions used to define the AUC. Therefore, a new pharmacokinetic-pharmacodynamic model for nedaplatin must be constructed to investigate the most effective administration procedure in vivo.

PH-microscopy—theoretical and experimental investigations

Ultramicroelectrodes with a pH selective neutral carrier-based liquid membrane and a tip diameter of about 1 μm are used to investigate pH profiles above microelectrodes with different shape. The maximum local resolution of a pH-microscope with these electrodes is tested. It is in the order of the tip diameter (< 5 μm). The theory of stationary diffusion fields at microdisk electrodes is adjusted and summarized. The differences between constant current and constant concentration conditions at the surface are calculated and shown to become negligible at distances larger than two times the electrode's radius. Comparison between theory and locally resolved measurements at the microelectrode's surface shows the real behavior lies between these limits. Exact calculations are presented for pH gradients in solutions of protolytics with several protolysis steps. Excellent agreement is found between theory and experimental data. Images of pH profiles above microelectrodes electrochemically generating OH −-ions are presented.

Analytical solutions for one-dimensional colloid transport in saturated fractures

Closed-form analytical solutions for colloid transport in single rock fractures with and without colloid penetration into the rock matrix are derived for constant concentration as well as constant flux boundary conditions. A single fracture is idealized as two semi-infinite parallel plates. It is assumed that colloidal particles undergo irreversible deposition onto fracture surfaces and may penetrate into the rock matrix, and deposit irreversibly onto rock matrix solid surfaces. The solutions are obtained by taking Laplace transforms to the governing transport equations and boundary conditions with respect to time and space. For the case of no colloid penetration into the rock matrix, the solutions are expressed in terms of exponentials and complimentary error functions; whereas, for the case of colloid penetration into the rock matrix, the solutions are expressed in terms of convolution integrals and modified Bessel functions. The impact of the model parameters on colloid transport is examined. The results from several simulations indicate that liquid-phase as well as deposited colloid concentrations in the fracture are sensitive to the fracture surface deposition coefficient, the fracture aperture, and the Brownian diffusion coefficient for colloidal particles penetrating the rock matrix. Furthermore, it is shown that the differences between the two boundary conditions investigated are minimized at dominant advective transport conditions. The constant concentration condition overestimates liquid-phase colloid concentrations, whereas the constant flux condition leads to conservation of mass.

Free boundary problems with nonlinear diffusion

It is shown that whenever a nonlinear diffusion model can be solved on a semi-infinite domain, with the standard boundary conditions of constant concentration, then the same model can be adjusted to yield exact solutions to free boundary problems. This is true not only for those diffusion equations that can be solved directly, but also for the special models obtainable via Philip's inverse method. New solutions are developed for two practical free boundary problems. The first represents solidification over a mould with dissimilar nonlinear thermal properties and the second represents saturated/unsaturated absorption in the soil beneath a pond.

Radionuclide Migration in Fractured Porous Rock: Analytical Solution for a Kinetic Solubility-Limited Dissolution Model

An analytical solution based on Laplace transforms is developed for the problem of radionuclide transport along a discrete planar fracture in porous rock. The solution takes into account advective transport in the fracture, longitudinal hydrodynamic dispersion in the fracture along the fracture axis, molecular diffusion from the fracture into the rock matrix, sorption within the rock matrix, sorption onto the surface of the fracture, and radioactive decay. The longitudinal hydrodynamic dispersion in the fracture along the fracture axis, molecular diffusion from the fracture into the rock matrix, sorption within the rock matrix, sorption onto the surface of the fracture, and radioactive decay. The longitudinal-dispersion-free solution, which is of closed form, is also reported. The initial radionuclide concentrations in both the fracture and the rock matrix are assumed to be zero. A kinetic solubility-limited dissolution model is used as the inlet boundary condition. In addition to the radionuclide concentrations in both the fracture and the rock matrix, the mass flux in fracture is provided. The analytical solution is in the form of a single integral that is evaluated by a Gauss-Legendre quadrature for each point in space and time. As the dissolution rate constant approaches infinity, the inlet boundary condition ofmore » the kinetic solubility-limited dissolution model can be replaced by the boundary condition of constant concentration, as is shown by numerical illustration. Restated, the constant concentration boundary condition represents a conservative upper limit to the solubility-limited dissolution rate. Diffusion into the rock matrix enhances the dissolution rate, even though it can also enhance the retardation of solute transport in fracture.« less

Methods for the determination of transport parameters of gases in membranes

Some new methods for determination of the transport parameters of gases in membranes such as the diffusion coefficient D, the permeability coefficient P and the solubility coefficient S have been proposed. These methods were derived on the basis of more general solutions of the diffusion equations (Fick's first and second laws) for initial and boundary conditions of constant concentration on one side and non-constant initial concentration distribution in the membrane. The results make it possible to set up various independent techniques for the determination of membrane transport parameters from different measured variables such as pressure (or concentration) of penetrant and influx or effusion of penetrant from the permeation cell. These techniques may be useful for the study of membranes for gas permeation processes under different conditions (including conditions close to those encountered in industrial operations).

Lithium transport in the mouse brain

Using the stable isotopes of lithium 6Li and 7Li, and the nuclear reaction 6Li(n,α) 3H for detection, we have studied the isotopic exchange of lithium in various areas of the mouse brain and in the mouse plasma, under conditions of constant concentration of total lithium. The neutron irradiations were performed using ‘cold’ neutrons, at the European Institute Von Laue-Langevin. The nuclear reaction track densities were determined using an automatic image analyser. In the plasma, the isotopic ratios, 6Li/ 7Li, were measured using ‘Secondary Ion Mass Spectrometry’. The concentration of total lithium in the plasma was kept close to 0.28 mM. The brain concentrations of total lithium (referred to the tissue water content) ranged from more than 2 mM in the thalamus to less than 0.65 mM in the white matter of the cerebellum. The Nernst potential of lithium thus ranged from approx. −50 to approx. −20 mV, which means that lithium is probably not far from electrochemical equilibrium between brain cells and plasma. At any moment, the isotopic abundances of 6Li (ratio of 6Li to total lithium) in the different brain areas, were not significantly different from one another. The time-course of the isotopic abundance of 6Li in the brain was fitted by the composition of two exponential terms. The time-course of the isotopic abundance of 6Li in the plasma was also fitted by the composition of two exponential terms. These analytic curves (for the brain and for the plasma) were not significantly different from each other, at the precision of the measurements. This means that the isotopic equilibration of lithium between brain and plasma is almost instantaneous (i.e. accomplished in a few min at the most).