In the physical sciences, relaxation usually means the return of a perturbed system into
equilibrium.
Each relaxation process can be categorized by a relaxation time
τ. The simplest theoretical description of relaxation as function of time ''t'' is an exponential law (
exponential decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
).
In simple linear systems
Mechanics: Damped unforced oscillator
Let the homogeneous
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
:
:
model
damped unforced
oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
s of a weight on a spring.
The displacement will then be of the form
. The constant T (
) is called the relaxation time of the system and the constant μ is the quasi-frequency.
Electronics: RC circuit
In an
RC circuit
A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ci ...
containing a charged capacitor and a resistor, the voltage decays exponentially:
:
The constant
is called the ''relaxation time'' or
RC time constant
The RC time constant, also called tau, the time constant (in seconds) of an RC circuit, is equal to the product of the circuit resistance (in ohms) and the circuit capacitance (in farads), i.e.
: \tau = RC econds
It is the time required to c ...
of the circuit. A nonlinear
oscillator
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
circuit which generates a repeating waveform by the repetitive discharge of a capacitor through a resistance is called a ''
relaxation oscillator
In electronics a relaxation oscillator is a nonlinear electronic oscillator circuit that produces a nonsinusoidal repetitive output signal, such as a triangle wave or square wave. on Peter Millet'Tubebookswebsite The circuit consists of a feedba ...
''.
In condensed matter physics
In
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
, relaxation is usually studied as a
linear response A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information ...
to a small external perturbation. Since the underlying microscopic processes are active even in the absence of external perturbations, one can also study "relaxation ''in'' equilibrium" instead of the usual "relaxation ''into'' equilibrium" (see
fluctuation-dissipation theorem
The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the th ...
).
Stress relaxation
In
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, ''
stress relaxation In materials science, stress relaxation is the observed decrease in stress in response to strain generated in the structure. This is primarily due to keeping the structure in a strained condition for some finite interval of time hence causing some ...
'' is the gradual disappearance of
stresses from a
viscoelastic
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
medium after it has been deformed.
Dielectric relaxation time
In
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
materials, the dielectric
polarization ''P'' depends on the electric field ''E''. If ''E'' changes, ''P''(''t'') reacts: the polarization ''relaxes'' towards a new equilibrium. It is important in
dielectric spectroscopy
Dielectric spectroscopy (which falls in a subcategory of impedance spectroscopy) measures the dielectric properties of a medium as a function of frequency.Kremer F., Schonhals A., Luck W. Broadband Dielectric Spectroscopy. – Springer-Verlag, 200 ...
. Very long relaxation times are responsible for
dielectric absorption Dielectric absorption is the name given to the effect by which a capacitor, that has been charged for a long time, discharges only incompletely when briefly discharged. Although an ideal capacitor would remain at zero volts after being discharged, r ...
.
The dielectric relaxation time is closely related to the
electrical conductivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allow ...
. In a
semiconductor
A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
it is a measure of how long it takes to become neutralized by conduction process. This relaxation time is small in
metal
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
s and can be large in semiconductors and
insulators.
Liquids and amorphous solids
An
amorphous solid
In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid, glassy solid) is a solid that lacks the long-range order that is characteristic of a crystal.
Etymology
The term comes from the Greek ''a'' ("wit ...
such as amorphous
indomethacin
Indometacin, also known as indomethacin, is a nonsteroidal anti-inflammatory drug (NSAID) commonly used as a prescription drug, prescription medication to reduce fever, pain, joint stiffness, stiffness, and swelling (medical), swelling from infl ...
displays a temperature dependence of molecular motion, which can be quantified as the average relaxation time for the solid in a
metastable
In chemistry and physics, metastability denotes an intermediate Energy level, energetic state within a dynamical system other than the system's ground state, state of least energy.
A ball resting in a hollow on a slope is a simple example of me ...
supercooled
Supercooling, also known as undercooling, is the process of lowering the temperature of a liquid or a gas below its melting point without it becoming a solid. It achieves this in the absence of a seed crystal or nucleus around which a crystal ...
liquid or
glass
Glass is a non-crystalline, often transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling (quenching) of ...
to approach the molecular motion characteristic of a
crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
.
Differential scanning calorimetry
Differential scanning calorimetry (DSC) is a thermoanalytical technique in which the difference in the amount of heat required to increase the temperature of a sample and reference is measured as a function of temperature. Both the sample and ref ...
can be used to quantify
enthalpy
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
change due to molecular structural relaxation.
The term "structural relaxation" was introduced in the scientific literature in 1947/48 without any explanation, applied to NMR, and meaning the same as "thermal relaxation".
Spin relaxation in NMR
In
nuclear magnetic resonance
Nuclear magnetic resonance (NMR) is a physical phenomenon in which nuclei in a strong constant magnetic field are perturbed by a weak oscillating magnetic field (in the near field) and respond by producing an electromagnetic signal with a ...
(NMR), various relaxations are the properties that it measures.
Chemical relaxation methods
In
chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...
, relaxation methods are used for the measurement of very fast
reaction rate
The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit ...
s. A system initially at equilibrium is perturbed by a rapid change in a parameter such as the temperature (most commonly), the pressure, the electric field or the
pH of the solvent. The return to equilibrium is then observed, usually by spectroscopic means, and the relaxation time measured. In combination with the chemical
equilibrium constant
The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
of the system, this enables the determination of the
rate constant In chemical kinetics a reaction rate constant or reaction rate coefficient, ''k'', quantifies the rate and direction of a chemical reaction.
For a reaction between reactants A and B to form product C
the reaction rate is often found to have the f ...
s for the forward and reverse reactions.
Monomolecular first-order reversible reaction
A monomolecular, first order reversible reaction which is close to equilibrium can be visualized by the following symbolic structure:
A <=> B
In other words, reactant A and product B are forming into one another based on reaction rate constants k and k'.
To solve for the concentration of A, recognize that the forward reaction (
A ->[] B) causes the concentration of A to decrease over time, whereas the reverse reaction (
B ->[] A) causes the concentration of A to increase over time.
Therefore,
, where brackets around A and B indicate concentrations.
If we say that at
, and applying the law of conservation of mass, we can say that at any time, the sum of the concentrations of A and B must be equal to the concentration of
, assuming the volume into which A and B are dissolved does not change:
Substituting this value for
in terms of
sub>0 and
''t'') yields
which becomes the separable differential equation
This equation can be solved by substitution to yield
In atmospheric sciences
Desaturation of clouds
Consider a supersaturated portion of a cloud. Then shut off the updrafts, entrainment, and any other vapor sources/sinks and things that would induce the growth of the particles (ice or water). Then wait for this
supersaturation
In physical chemistry, supersaturation occurs with a solution when the concentration of a solute exceeds the concentration specified by the value of solubility at equilibrium. Most commonly the term is applied to a solution of a solid in a liqu ...
to reduce and become just saturation (relative humidity = 100%), which is the equilibrium state. The time it takes for the supersaturation to dissipate is called relaxation time. It will happen as ice crystals or liquid water content grow within the cloud and will thus consume the contained moisture. The dynamics of relaxation are very important in
cloud physics
Cloud physics is the study of the physical processes that lead to the formation, growth and precipitation of atmospheric clouds. These aerosols are found in the troposphere, stratosphere, and mesosphere, which collectively make up the greatest ...
for accurate
mathematical modelling
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, b ...
.
In water clouds where the concentrations are larger (hundreds per cm
3) and the temperatures are warmer (thus allowing for much lower supersaturation rates as compared to ice clouds), the relaxation times will be very low (seconds to minutes).
In
ice cloud An ice cloud is a colloid of ice particles dispersed in air. The term has been used to refer to clouds of both water ice and carbon dioxide ice on Mars.
Such clouds can be sufficiently large and dense to cast shadows on the Martian surface.
Cirru ...
s the concentrations are lower (just a few per liter) and the temperatures are colder (very high supersaturation rates) and so the relaxation times can be as long as several hours. Relaxation time is given as
where:
*''D'' = diffusion coefficient
2/s">2/s*''N'' = concentration (of ice crystals or water droplets)
−3">−3*''R'' = mean radius of particles
*''K'' = capacitance
nitless
In astronomy
In
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, relaxation time relates to clusters of
gravitation
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
ally interacting bodies, for instance, stars in a
galaxy
A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...
. The relaxation time is a measure of the time it takes for one object in the system (the "test star") to be significantly perturbed by other objects in the system (the "field stars"). It is most commonly defined as the time for the test star's velocity to change by of order itself.
Suppose that the test star has velocity ''v''. As the star moves along its orbit, its motion will be randomly perturbed by the
gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
of nearby stars. The relaxation time can be shown to be
:
where ''ρ'' is the mean density, ''m'' is the test-star mass, ''σ'' is the 1d velocity dispersion of the field stars, and is the
Coulomb logarithm A Coulomb collision is a binary elastic collision between two charged particles interacting through their own electric field. As with any inverse-square law, the resulting trajectories of the colliding particles is a hyperbolic Keplerian orbit. Th ...
.
Various events occur on timescales relating to the relaxation time, including
core collapse, energy
equipartition, and formation of a
Bahcall-Wolf cusp around a
supermassive black hole
A supermassive black hole (SMBH or sometimes SBH) is the largest type of black hole, with its mass being on the order of hundreds of thousands, or millions to billions of times the mass of the Sun (). Black holes are a class of astronomical obj ...
.
See also
*
Relaxation oscillator
In electronics a relaxation oscillator is a nonlinear electronic oscillator circuit that produces a nonsinusoidal repetitive output signal, such as a triangle wave or square wave. on Peter Millet'Tubebookswebsite The circuit consists of a feedba ...
*
Time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...
References
{{Reflist
Time in physics
Time in astronomy
Celestial mechanics
fr:Constante de temps