Polymer physics
Encyclopedia
Polymer physics is the field of physics
that studies polymer
s, their fluctuations, mechanical properties
, as well as the kinetics of reactions
involving degradation and polymerisation
of polymer
s and monomer
s respectively.
While it focuses on the perspective of condensed matter physics
, polymer physics is originally a branch of statistical physics
. Polymer physics and polymer chemistry
are also related with the field of polymer science
, where this is considered the applicative part of polymers.
Polymers are large molecules and thus are very complicated for solving using a deterministic method. Yet, statistical approaches can yield results and are often pertinent, since large polymers (i.e., polymers with a large number of monomer
s) are describable efficiently in the thermodynamic limit
of infinite many monomers (also the actual size is clearly finite).
Thermal fluctuations continuously affect the shape of polymers in liquid solutions, and modeling their effect requires using principles from statistical mechanics
and dynamics. As a corollary, temperature strongly affects the physical behavior of polymers in solution, causing phase transitions, melts, and so on.
The statistical approach for polymer physics is based on an analogy between a polymer and either a Brownian motion
, or other type of a random walk
, the self-avoiding walk
. The simplest possible polymer model is presented by the ideal chain
, corresponding to a simple random walk. Experimental approaches for characterizing polymers are also common, using Polymer characterization
methods, such as size exclusion chromatography
, Viscometry, Dynamic light scattering
, and Automatic Continuous Online Monitoring of Polymerization Reactions (ACOMP) for determining the chemical, physical, and material properties of polymers. These experimental methods also helped the mathematical modeling of polymers and even for a better understanding of the properties of polymers.
.
.
The freely-rotating chain improves the freely-jointed chain model by taking into account that polymer segments make a fixed angle to neighbouring units because of specific chemical bonding. Under this fixed angle the segments are still free to rotate and all torsion angles are equally likely.
The hindered rotation model assumes that the torsion angle is hindered by a potential energy. This makes the probability of each torsion angle proportional to a Boltzmann factor
:
In the rotational isomeric state model the allowed torsion angles are determined by the positions of the minima in the rotational potential energy. Bond lengths and bond angles are constant.
The Worm-like chain
is a more complex model. It takes the persistence length
into account. Polymers are not completely flexible; bending them requires energy. At the length scale below persistence length, the polymer behaves more or less like a rigid rod.
. This causes a reduction in the conformational possibilities of the chain, and leads to a self-avoiding random walk. Self-avoiding random walks have different statistics to simple random walks.
where
is the radius of gyration
of the polymer,
is the number of bond segments (equal to the degree of polymerization) of the chain.
For good solvent,
; for bad solvent, 
. Therefore polymer in good solvent has larger size and behaves like a fractal
object. In bad solvent it behaves like a solid sphere.
In the so called
solvent, 
, which is the result of simple random walk. The chain behaves as if an ideal chain.
The quality of solvent depends also on temperature. For a flexible polymer, low temperature may correspond to poor quality and high temperature makes the same solvent good. At a particular temperature called theta (θ) temperature, the solvent behaves as if an ideal chain
.
model assumes that polymer segments can overlap with each other as if the chain were a phantom chain. In reality, two segments cannot occupy the same space at the same time. This interaction between segments is called the excluded volume
interaction.
The simplest formulation of excluded volume is the self-avoiding random walk, a random walk that cannot repeat its previous path. A path of this walk of N steps in three dimensions represents a conformation of a polymer with excluded volume interaction. Because of the self-avoiding nature of this model, the number of possible conformations is significantly reduced. The radius of gyration is generally larger than that of the ideal chain.
of double-stranded DNA
is about 50 nm. Looking at length scale smaller than 50 nm (Known as the McGuinness limit), it behaves more or less like a rigid rod. At length scale much larger than 50 nm, it behaves like a flexible chain.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
that studies polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
s, their fluctuations, mechanical properties
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
, as well as the kinetics of reactions
Chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...
involving degradation and polymerisation
Polymerization
In polymer chemistry, polymerization is a process of reacting monomer molecules together in a chemical reaction to form three-dimensional networks or polymer chains...
of polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
s and monomer
Monomer
A monomer is an atom or a small molecule that may bind chemically to other monomers to form a polymer; the term "monomeric protein" may also be used to describe one of the proteins making up a multiprotein complex...
s respectively.
While it focuses on the perspective of condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
, polymer physics is originally a branch of statistical physics
Statistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...
. Polymer physics and polymer chemistry
Polymer chemistry
Polymer chemistry or macromolecular chemistry is a multidisciplinary science that deals with the chemical synthesis and chemical properties of polymers or macromolecules. According to IUPAC recommendations, macromolecules refer to the individual molecular chains and are the domain of chemistry...
are also related with the field of polymer science
Polymer science
Polymer science or macromolecular science is the subfield of materials science concerned with polymers, primarily synthetic polymers such as plastics...
, where this is considered the applicative part of polymers.
Polymers are large molecules and thus are very complicated for solving using a deterministic method. Yet, statistical approaches can yield results and are often pertinent, since large polymers (i.e., polymers with a large number of monomer
Monomer
A monomer is an atom or a small molecule that may bind chemically to other monomers to form a polymer; the term "monomeric protein" may also be used to describe one of the proteins making up a multiprotein complex...
s) are describable efficiently in the thermodynamic limit
Thermodynamic limit
In thermodynamics, particularly statistical mechanics, the thermodynamic limit is reached as the number of particles in a system, N, approaches infinity...
of infinite many monomers (also the actual size is clearly finite).
Thermal fluctuations continuously affect the shape of polymers in liquid solutions, and modeling their effect requires using principles from statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
and dynamics. As a corollary, temperature strongly affects the physical behavior of polymers in solution, causing phase transitions, melts, and so on.
The statistical approach for polymer physics is based on an analogy between a polymer and either a Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
, or other type of a 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...
, the self-avoiding walk
Self-avoiding walk
In mathematics, a self-avoiding walk is a sequence of moves on a lattice that does not visit the same point more than once. A self-avoiding polygon is a closed self-avoiding walk on a lattice...
. The simplest possible polymer model is presented by the ideal chain
Ideal chain
An ideal chain is the simplest model to describe a polymer. It only assumes a polymer as a random walk and neglects any kind of interactions among monomers...
, corresponding to a simple random walk. Experimental approaches for characterizing polymers are also common, using Polymer characterization
Polymer characterization
Polymer characterization is the analytical branch of polymer science.The discipline is concerned with the characterization of polymeric materials on a variety of levels...
methods, such as size exclusion chromatography
Size exclusion chromatography
Size-exclusion chromatography is a chromatographic method in which molecules in solution are separated by their size, and in some cases molecular weight . It is usually applied to large molecules or macromolecular complexes such as proteins and industrial polymers...
, Viscometry, Dynamic light scattering
Dynamic light scattering
thumb|right|350px|Hypothetical Dynamic light scattering of two samples: Larger particles on the top and smaller particle on the bottomDynamic light scattering is a technique in physics that can be used to determine the size distribution profile of small particles in suspension or polymers...
, and Automatic Continuous Online Monitoring of Polymerization Reactions (ACOMP) for determining the chemical, physical, and material properties of polymers. These experimental methods also helped the mathematical modeling of polymers and even for a better understanding of the properties of polymers.
- 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...
is considered the first scientist establishing the field of polymer physics. - French scientists contributed a lot since the 70s (e.g. de Gennes, J. des Cloizeaux).
- Doi and Edwars wrote a very famous book in polymer physics.
- Russian and Soviet schools of physics (I. M. Lifshitz, A. Yu. Grosberg, A.R. Khokhlov ) have been very active in the development of polymer physics
.
Models
Models of polymer chains are split into two types: "ideal" models, and "real" models. Ideal chain models assume that there are no interactions between chain monomers. This assumption is valid for certain polymeric systems, where the positive and negative interactions between the monomer effectively cancel out. Ideal chain models provide a good starting point for investigation of more complex systems and are better suited for equations with more parameters.Ideal Chains
The freely-jointed chain is the simplest model of a polymer. In this model, fixed length polymer segments are linearly connected, and all bond and torsion angles are equiprobable. The polymer can therefore be described by a simple random walk and ideal chainIdeal chain
An ideal chain is the simplest model to describe a polymer. It only assumes a polymer as a random walk and neglects any kind of interactions among monomers...
.
The freely-rotating chain improves the freely-jointed chain model by taking into account that polymer segments make a fixed angle to neighbouring units because of specific chemical bonding. Under this fixed angle the segments are still free to rotate and all torsion angles are equally likely.
The hindered rotation model assumes that the torsion angle is hindered by a potential energy. This makes the probability of each torsion angle proportional to a Boltzmann factor
Boltzmann factor
In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...
:
In the rotational isomeric state model the allowed torsion angles are determined by the positions of the minima in the rotational potential energy. Bond lengths and bond angles are constant.
The Worm-like chain
Worm-like chain
The 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 :...
is a more complex model. It takes the persistence length
Persistence length
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...
into account. Polymers are not completely flexible; bending them requires energy. At the length scale below persistence length, the polymer behaves more or less like a rigid rod.
Real Chains
Interactions between chain monomers can be modelled as excluded volumeExcluded volume
The concept of excluded volume was introduced by Werner Kuhn in 1934 and applied to polymer molecules shortly thereafter by Paul Flory.- In liquid state theory :...
. This causes a reduction in the conformational possibilities of the chain, and leads to a self-avoiding random walk. Self-avoiding random walks have different statistics to simple random walks.
Solvent and temperature effect
The statistics of a single polymer chain depends on the solvent. For good solvent the chain is more expanded while for bad solvent the chain segments stay close to each other. In the limit of a very bad solvent the polymer chain merely collapses to form a hard sphere, while in good solvent the chain swells in order to maximize the number of polymer-fluid contacts. For this case the radius of gyration is approximated using Flory's mean field approach which yields a scaling for the radius of gyration of:-
,
where
is the radius of gyrationRadius of gyration
Radius of gyration or gyradius is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the objects' parts from either its center of gravity or an axis....
of the polymer,
is the number of bond segments (equal to the degree of polymerization) of the chain.For good solvent,
; for bad solvent, . Therefore polymer in good solvent has larger size and behaves like a fractalFractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
object. In bad solvent it behaves like a solid sphere.
In the so called
solvent, , which is the result of simple random walk. The chain behaves as if an ideal chain.The quality of solvent depends also on temperature. For a flexible polymer, low temperature may correspond to poor quality and high temperature makes the same solvent good. At a particular temperature called theta (θ) temperature, the solvent behaves as if an ideal chain
Ideal chain
An ideal chain is the simplest model to describe a polymer. It only assumes a polymer as a random walk and neglects any kind of interactions among monomers...
.
Excluded volume interaction
The ideal chainIdeal chain
An ideal chain is the simplest model to describe a polymer. It only assumes a polymer as a random walk and neglects any kind of interactions among monomers...
model assumes that polymer segments can overlap with each other as if the chain were a phantom chain. In reality, two segments cannot occupy the same space at the same time. This interaction between segments is called the excluded volume
Excluded volume
The concept of excluded volume was introduced by Werner Kuhn in 1934 and applied to polymer molecules shortly thereafter by Paul Flory.- In liquid state theory :...
interaction.
The simplest formulation of excluded volume is the self-avoiding random walk, a random walk that cannot repeat its previous path. A path of this walk of N steps in three dimensions represents a conformation of a polymer with excluded volume interaction. Because of the self-avoiding nature of this model, the number of possible conformations is significantly reduced. The radius of gyration is generally larger than that of the ideal chain.
Flexibility
Whether a polymer is flexible or not depends on the scale of interest. For example, the persistence lengthPersistence length
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...
of double-stranded 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...
is about 50 nm. Looking at length scale smaller than 50 nm (Known as the McGuinness limit), it behaves more or less like a rigid rod. At length scale much larger than 50 nm, it behaves like a flexible chain.
See also
- Important publications in polymer physics.
- Polymer characterizationPolymer characterizationPolymer characterization is the analytical branch of polymer science.The discipline is concerned with the characterization of polymeric materials on a variety of levels...
- File dynamicsFile dynamicsIn chemistry, physics, mathematics and related fields, file dynamics is the diffusion of N identical Brownian hard spheres in a quasi-one-dimensional channel of length L , such that the spheres do not jump one on top of the other, and the average particle's density is approximately fixed...