Persistence length
Encyclopedia
The persistence length is a basic mechanical property quantifying the stiffness
of a polymer
or of a string
.
Informally, for pieces of the polymer that are shorter than the persistence length, the molecule behaves rather like a flexible elastic rod, while for pieces of the polymer that are much longer than the persistence length, the properties can only be described statistically, like a three-dimensional random walk
.
Formally, the persistence length, P, is defined as the length over which correlations in the direction of the tangent are lost. In a more chemical based manner it can also be defined as the average sum of the projections of all bonds j ≥ i on bond i in an indefinitely long chain.
Let us define the angle θ between a vector that is tangent to the polymer at position 0 (zero) and a tangent vector at a distance L away from position 0. It can be shown that the expectation value of the cosine of the angle falls off exponentially with distance,
where P is the persistence length and the angled brackets denote the average over all starting positions.
The persistence length can be also expressed using the bending stiffness
, the Young's modulus
E and knowing the section of the polymer
chain.
In the case of a rigid and uniform rod I can be expressed as:
where a is the radius.
For example a piece of uncooked spaghetti
has a persistence length on the order of
m (taking in consideration a Young modulus of
0.1 GPa and a radius of 1 mm). Double-helical DNA
has a persistence length of about 50 nanometers.
In polymer science, persistence length is one half of the Kuhn length
, the length of hypothetical segments that the chain can be considered as freely joined. The persistence length equals the average projection of the end-to-end vector
on the tangent to the chain contour at a chain end in the limit of infinite chain length.
Stiffness
Stiffness is the resistance of an elastic body to deformation by an applied force along a given degree of freedom when a set of loading points and boundary conditions are prescribed on the elastic body.-Calculations:...
of a polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
or of a string
String
Generally, string is a flexible piece of rope or twine which is used to tie, bind, or hang other objects.Examples of string use include:* For flying kites* string figures, designs formed by weaving string around one's fingers...
.
Informally, for pieces of the polymer that are shorter than the persistence length, the molecule behaves rather like a flexible elastic rod, while for pieces of the polymer that are much longer than the persistence length, the properties can only be described statistically, like a three-dimensional random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
.
Formally, the persistence length, P, is defined as the length over which correlations in the direction of the tangent are lost. In a more chemical based manner it can also be defined as the average sum of the projections of all bonds j ≥ i on bond i in an indefinitely long chain.
Let us define the angle θ between a vector that is tangent to the polymer at position 0 (zero) and a tangent vector at a distance L away from position 0. It can be shown that the expectation value of the cosine of the angle falls off exponentially with distance,
where P is the persistence length and the angled brackets denote the average over all starting positions.
The persistence length can be also expressed using the bending stiffness
, the Young's modulusYoung's modulus
Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain...
E and knowing the section of the polymer
chain.
In the case of a rigid and uniform rod I can be expressed as:
where a is the radius.
For example a piece of uncooked spaghetti
Spaghetti
Spaghetti is a long, thin, cylindrical pasta of Italian origin. Spaghetti is made of semolina or flour and water. Italian dried spaghetti is made from durum wheat semolina, but outside of Italy it may be made with other kinds of flour...
has a persistence length on the order of
m (taking in consideration a Young modulus of0.1 GPa and a radius of 1 mm). Double-helical DNA
DNA
Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...
has a persistence length of about 50 nanometers.
In polymer science, persistence length is one half of the Kuhn length
Kuhn length
The Kuhn length is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of N Kuhn segments each with a Kuhn length b. Each Kuhn segment can be thought of as if they are freely jointed with each other...
, the length of hypothetical segments that the chain can be considered as freely joined. The persistence length equals the average projection of the end-to-end vector
End-to-end vector
In the physical chemistry study of polymers, the end-to-end vector is the vector that points from one end of a polymer to the other end.The norm of the end-to-end vector is called the end-to-end distance....
on the tangent to the chain contour at a chain end in the limit of infinite chain length.
See also
- PolymerPolymerA polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
- Worm-like chainWorm-like chainThe worm-like chain model in polymer physics is used to describe the behavior of semi-flexible polymers; it is sometimes referred to as the Kratky-Porod model.- Theoretical Considerations :...
- Freely Jointed Chain
- Kuhn lengthKuhn lengthThe Kuhn length is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of N Kuhn segments each with a Kuhn length b. Each Kuhn segment can be thought of as if they are freely jointed with each other...
- Paul FloryPaul FloryPaul John Flory was an American chemist and Nobel laureate who was known for his prodigious volume of work in the field of polymers, or macromolecules...