|
Poisson–Boltzmann Equation
The Poisson–Boltzmann equation is a useful equation in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor, or more. It aims to describe the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules in solution. The Poisson–Boltzmann equation is derived via mean-field assumptions. From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions. Origins Background and derivation The Poisson–Boltzmann equation describes a model proposed independently by Louis Georges Gouy and David Leonard Chapman in 1910 and 1913, respectively. In the Gouy-Chapman model, a charged solid comes into contact with an ionic solution, creating a layer of surface charges and counter-ions or double layer. Due to thermal motio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physiology
Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemical and physical functions in a living system. According to the classes of organisms, the field can be divided into medical physiology, animal physiology, plant physiology, cell physiology, and comparative physiology. Central to physiological functioning are biophysical and biochemical processes, homeostatic control mechanisms, and communication between cells. ''Physiological state'' is the condition of normal function. In contrast, ''pathological state'' refers to abnormal conditions, including human diseases. The Nobel Prize in Physiology or Medicine is awarded by the Royal Swedish Academy of Sciences for exceptional scientific achievements in physiology related to the field of medicine. Foundations Cells Although there are differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, and in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy. It is named after the Austrian scientist Ludwig Boltzmann. As part of the 2019 redefinition of SI base units, the Boltzmann constant is one of the seven " defining constants" that have been given exact definitions. They are used in various combinations to define the seven SI base units. The Boltzmann constant is defined to be exactly . Roles of the Boltzmann constant Macroscopically, the ideal gas law states that, for an ideal gas, the product of pressure and volume is proportional to the product of amount of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrostatics
Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber, (), was thus the source of the word 'electricity'. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. Even though electrostatically induced forces seem to be rather weak, some electrostatic forces are relatively large. The force between an electron and a proton, which together make up a hydrogen atom, is about 36 orders of magnitude stronger than the gravitational force acting between them. There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearized Vs Full Potential Forms Of The Poisson-Boltzmann Equation
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology. Linearization of a function Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function y = f(x) at any x = a based on the value and slope of the function at x = b, given that f(x) is differentiable on , b/math> (or , a/math>) and that a is close to b. In short, linearization approximates the output of a function near x = a. For example, \sqrt = 2. However, what would be a good approx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Trigonometric Identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity. Pythagorean identities The basic relationship between the sine and cosine is given by the Pythagorean identity: :\sin^2\theta + \cos^2\theta = 1, where \sin^2 \theta means (\sin \theta)^2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimensionless Quantity
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1), ISBN 978-92-822-2272-0. which is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Dimensionless quantities are distinct from quantities that have associated dimensions, such as time (measured in seconds). Dimensionless units are dimensionless values that serve as units of measurement for expressing other quantities, such as radians (rad) or steradians (sr) for plane angles and solid angles, respectively. For example, optical extent is defined as having units of metres multiplied by steradians. History Quantities having dimension one, ''dimensionless quantities'', regularly occur in sciences, and are formally treated within the field of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Debye Length
In plasmas and electrolytes, the Debye length \lambda_ (also called Debye radius), is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. With each Debye length the charges are increasingly electrically screened and the electric potential decreases in magnitude by 1/ e. A Debye sphere is a volume whose radius is the Debye length. Debye length is an important parameter in plasma physics, electrolytes, and colloids (DLVO theory). The corresponding Debye screening wave vector k_=1/\lambda_ for particles of density n, charge q at a temperature T is given by k_^2=4\pi n q^2/(k_T) in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures (T \to 0) are known as the Thomas–Fermi length and the Thomas–Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature. The Debye length is named after the Dutch-America ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DLVO Theory
The DLVO theory (named after Boris Derjaguin and Lev Landau, Evert Verwey and Theodoor Overbeek) explains the aggregation of aqueous dispersions quantitatively and describes the force between charged surfaces interacting through a liquid medium. It combines the effects of the van der Waals attraction and the electrostatic repulsion due to the so-called double layer of counterions. The electrostatic part of the DLVO interaction is computed in the mean field approximation in the limit of low surface potentials - that is when the potential energy of an elementary charge on the surface is much smaller than the thermal energy scale, k_ T. For two spheres of radius a each having a charge Z (expressed in units of the elementary charge) separated by a center-to-center distance r in a fluid of dielectric constant \epsilon_r containing a concentration n of monovalent ions, the electrostatic potential takes the form of a screened-Coulomb or Yukawa potential, : \beta U(r) = Z^2 \lambda_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colloid Chemistry
A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others extend the definition to include substances like aerosols and gels. The term colloidal suspension refers unambiguously to the overall mixture (although a narrower sense of the word ''suspension'' is distinguished from colloids by larger particle size). A colloid has a dispersed phase (the suspended particles) and a continuous phase (the medium of suspension). The dispersed phase particles have a diameter of approximately 1 nanometre to 1 micrometre. Some colloids are translucent because of the Tyndall effect, which is the scattering of light by particles in the colloid. Other colloids may be opaque or have a slight color. Colloidal suspensions are the subject of interface and colloid science. This field of study was introduced in 1845 by Ital ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
