Carothers equation

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In step-growth polymerization, the Carothers equation (or Carothers' equation) gives the degree of polymerization, Xn, for a given fractional monomer conversion, p.

Contents

There are several versions of this equation, proposed by Wallace Carothers, who invented nylon in 1935.

Linear polymers: two monomers in equimolar quantities

The simplest case refers to the formation of a strictly linear polymer by the reaction (usually by condensation) of two monomers in equimolar quantities. An example is the synthesis of nylon-6,6 whose formula is [-NH-(CH2)6-NH-CO-(CH2)4-CO-]n from one mole of hexamethylenediamine, H2N(CH2)6NH2, and one mole of adipic acid, HOOC-(CH2)4-COOH. For this case [1] [2]

In this equation

  • is the number-average value of the degree of polymerization, equal to the average number of monomer units in a polymer molecule. For the example of nylon-6,6 (n diamine units and n diacid units).
  • is the extent of reaction (or conversion to polymer), defined by
  • is the number of molecules present initially as monomer
  • is the number of molecules present after time t. The total includes all degrees of polymerization: monomers, oligomers and polymers.

This equation shows that a high monomer conversion is required to achieve a high degree of polymerization. For example, a monomer conversion, p, of 98% is required for , and p = 99% is required for .

Linear polymers: one monomer in excess

If one monomer is present in stoichiometric excess, then the equation becomes [3]

  • r is the stoichiometric ratio of reactants, the excess reactant is conventionally the denominator so that r < 1. If neither monomer is in excess, then r = 1 and the equation reduces to the equimolar case above.

The effect of the excess reactant is to reduce the degree of polymerization for a given value of p. In the limit of complete conversion of the limiting reagent monomer, p → 1 and

Thus for a 1% excess of one monomer, r = 0.99 and the limiting degree of polymerization is 199, compared to infinity for the equimolar case. An excess of one reactant can be used to control the degree of polymerization.

Branched polymers: multifunctional monomers

The functionality of a monomer molecule is the number of functional groups which participate in the polymerization. Monomers with functionality greater than two will introduce branching into a polymer, and the degree of polymerization will depend on the average functionality fav per monomer unit. For a system containing N0 molecules initially and equivalent numbers of two functional groups A and B, the total number of functional groups is N0fav.

And the modified Carothers equation is [4] [5] [6]

, where p equals to

Related to the Carothers equation are the following equations (for the simplest case of linear polymers formed from two monomers in equimolar quantities):

where:

The last equation shows that the maximum value of the Đ is 2, which occurs at a monomer conversion of 100% (or p = 1). This is true for step-growth polymerization of linear polymers. For chain-growth polymerization or for branched polymers, the Đ can be much higher.

In practice the average length of the polymer chain is limited by such things as the purity of the reactants, the absence of any side reactions (i.e. high yield), and the viscosity of the medium.

Related Research Articles

In polymer chemistry, polymerization, or polymerisation, is a process of reacting monomer molecules together in a chemical reaction to form polymer chains or three-dimensional networks. There are many forms of polymerization and different systems exist to categorize them.

Stoichiometry Calculation of relative quantities of reactants and products in chemical reactions

Stoichiometry is the calculation of reactants and products in chemical reactions in chemistry.

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 time. Reaction rates can vary dramatically. For example, the oxidative rusting of iron under Earth's atmosphere is a slow reaction that can take many years, but the combustion of cellulose in a fire is a reaction that takes place in fractions of a second. For most reactions, the rate decreases as the reaction proceeds. A reaction's rate can be determined by measuring the changes in concentration over time.

In polymer chemistry, ring-opening polymerization (ROP) is a form of chain-growth polymerization, in which the terminus of a polymer chain attacks cyclic monomers to form a longer polymer. The reactive center can be radical, anionic or cationic. Some cyclic monomers such as norbornene or cyclooctadiene can be polymerized to high molecular weight polymers by using metal catalysts. ROP is a versatile method for the synthesis of biopolymers.

Dispersity

In chemistry, the dispersity is a measure of the heterogeneity of sizes of molecules or particles in a mixture. A collection of objects is called uniform if the objects have the same size, shape, or mass. A sample of objects that have an inconsistent size, shape and mass distribution is called non-uniform. The objects can be in any form of chemical dispersion, such as particles in a colloid, droplets in a cloud, crystals in a rock, or polymer macromolecules in a solution or a solid polymer mass. Polymers can be described by molecular mass distribution; a population of particles can be described by size, surface area, and/or mass distribution; and thin films can be described by film thickness distribution.

Polymer chemistry is a sub-discipline of chemistry that focuses on the chemical synthesis, structure, and chemical and physical properties of polymers and macromolecules. The principles and methods used within polymer chemistry are also applicable through a wide range of other chemistry sub-disciplines like organic chemistry, analytical chemistry, and physical chemistry. Many materials have polymeric structures, from fully inorganic metals and ceramics to DNA and other biological molecules, however, polymer chemistry is typically referred to in the context of synthetic, organic compositions. Synthetic polymers are ubiquitous in commercial materials and products in everyday use, commonly referred to as plastics, and rubbers, and are major components of composite materials. Polymer chemistry can also be included in the broader fields of polymer science or even nanotechnology, both of which can be described as encompassing polymer physics and polymer engineering.

Copolymer

A copolymer is a polymer derived from more than one species of monomer. The polymerization of monomers into copolymers is called copolymerization. Copolymers obtained by copolymerization of two monomer species are sometimes called bipolymers. Those obtained from three and four monomers are called terpolymers and quaterpolymers, respectively.

The rate law or rate equation for a chemical reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters. For many reactions, the initial rate is given by a power law such as

Chain-growth polymerization or chain-growth polymerisation is a polymerization technique where unsaturated monomer molecules add onto the active site on a growing polymer chain one at a time. There are a limited number of these active sites at any moment during the polymerization which gives this method its key characteristics.

Step-growth polymerization

Step-growth polymerization refers to a type of polymerization mechanism in which bi-functional or multifunctional monomers react to form first dimers, then trimers, longer oligomers and eventually long chain polymers. Many naturally occurring and some synthetic polymers are produced by step-growth polymerization, e.g. polyesters, polyamides, polyurethanes, etc. Due to the nature of the polymerization mechanism, a high extent of reaction is required to achieve high molecular weight. The easiest way to visualize the mechanism of a step-growth polymerization is a group of people reaching out to hold their hands to form a human chain—each person has two hands. There also is the possibility to have more than two reactive sites on a monomer: In this case branched polymers production take place.

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Free-radical polymerization (FRP) is a method of polymerization, by which a polymer forms by the successive addition of free-radical building blocks. Free radicals can be formed by a number of different mechanisms, usually involving separate initiator molecules. Following its generation, the initiating free radical adds (nonradical) monomer units, thereby growing the polymer chain.

The degree of polymerization, or DP, is the number of monomeric units in a macromolecule or polymer or oligomer molecule.

Flory–Huggins solution theory Lattice model of polymer solutions

Flory–Huggins solution theory is a lattice model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for the entropy of mixing. The result is an equation for the Gibbs free energy change for mixing a polymer with a solvent. Although it makes simplifying assumptions, it generates useful results for interpreting experiments.

The Mayo–Lewis equation or copolymer equation in polymer chemistry describes the distribution of monomers in a copolymer. It was proposed by Frank R. Mayo and Frederick M. Lewis.

In polymer chemistry the kinetic chain length of a polymer, ν, is the average number of units called monomers added to a growing chain during chain-growth polymerization. During this process, a polymer chain is formed when monomers are bonded together to form long chains known as polymers. Kinetic chain length is defined as the average number of monomers that react with an active center such as a radical from initiation to termination.

Gelation

Gelation is the formation of a gel from a system with polymers. Branched polymers can form links between the chains, which lead to progressively larger polymers. As the linking continues, larger branched polymers are obtained and at a certain extent of the reaction links between the polymer result in the formation of a single macroscopic molecule. At that point in the reaction, which is defined as gel point, the system loses fluidity and viscosity becomes very large. The onset of gelation, or gel point, is accompanied by a sudden increase in viscosity. This "infinite" sized polymer is called the gel or network, which does not dissolve in the solvent, but can swell in it.

Flory function
Synonym: viscosity function, , SI unit: mol−1

Path integrals in polymer science

A polymer is a macromolecule, composed of many similar or identical repeated subunits. Polymers are common in, but not limited to, organic media. They range from familiar synthetic plastics to natural biopolymers such as DNA and proteins. Their unique elongated molecular structure produces unique physical properties, including toughness, viscoelasticity, and a tendency to form glasses and semicrystalline structures. The modern concept of polymers as covalently bonded macromolecular structures was proposed in 1920 by Hermann Staudinger. One sub-field in the study of polymers is polymer physics. As a part of soft matter studies, Polymer physics concerns itself with the study of mechanical properties and focuses on the perspective of condensed matter physics.

Flory–Stockmayer theory is a theory governing the cross-linking and gelation of step-growth polymers. The Flory-Stockmayer theory represents an advancement from the Carothers equation, allowing for the identification of the gel point for polymer synthesis not at stoichiometric balance. The theory was initially conceptualized by Paul Flory in 1941 and then was further developed by Walter Stockmayer in 1944 to include cross-linking with an arbitrary initial size distribution. The Flory-Stockmayer theory was the first theory investigating percolation processes.

References

  1. Cowie J.M.G. "Polymers: Chemistry & Physics of Modern Materials (2nd edition, Blackie 1991), p.29
  2. Rudin Alfred "The Elements of Polymer Science and Engineering", Academic Press 1982, p.171
  3. Allcock Harry R., Lampe Frederick W. and Mark James E. "Contemporary Polymer Chemistry" (3rd ed., Pearson 2003) p.324
  4. Carothers, Wallace (1936). "Polymers and polyfunctionality". Transactions of the Faraday Society. 32: 39–49. doi:10.1039/TF9363200039.
  5. Cowie p.40
  6. Rudin p.170