In
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 a ...
, the Carothers equation (or Carothers' equation) gives the
degree of polymerization
The degree of polymerization, or DP, is the number of monomeric units in a macromolecule or polymer or oligomer molecule.
For a homopolymer, there is only one type of monomeric unit and the ''number-average'' degree of polymerization is given by ...
, , for a given fractional
monomer
In chemistry, a monomer ( ; ''mono-'', "one" + '' -mer'', "part") is a molecule that can react together with other monomer molecules to form a larger polymer chain or three-dimensional network in a process called polymerization.
Classification
Mo ...
conversion, .
There are several versions of this equation, proposed by
Wallace Carothers
Wallace Hume Carothers (; April 27, 1896 – April 29, 1937) was an American chemist, inventor and the leader of organic chemistry at DuPont, who was credited with the invention of nylon.
Carothers was a group leader at the DuPont Experiment ...
, who invented
nylon
Nylon is a generic designation for a family of synthetic polymers composed of polyamides ( repeating units linked by amide links).The polyamides may be aliphatic or semi-aromatic.
Nylon is a silk-like thermoplastic, generally made from petro ...
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
monomer
In chemistry, a monomer ( ; ''mono-'', "one" + '' -mer'', "part") is a molecule that can react together with other monomer molecules to form a larger polymer chain or three-dimensional network in a process called polymerization.
Classification
Mo ...
s in equimolar quantities. An example is the synthesis of
nylon-6,6
Nylon 66 (loosely written nylon 6-6, nylon 6/6, nylon 6,6, or nylon 6:6) is a type of polyamide or nylon. It, and nylon 6, are the two most common for textile and plastic industries. Nylon 66 is made of two monomers each containing 6 carbon atoms, ...
whose formula is
from one mole of
hexamethylenediamine
Hexamethylenediamine is the organic compound with the formula H2N(CH2)6NH2. The molecule is a diamine, consisting of a hexamethylene hydrocarbon chain terminated with amine functional groups. The colorless solid (yellowish for some commercial sam ...
, , and one mole of
adipic acid
Adipic acid or hexanedioic acid is the organic compound with the formula (CH2)4(COOH)2. From an industrial perspective, it is the most important dicarboxylic acid: about 2.5 billion kilograms of this white crystalline powder are produced annually, ...
, . For this case
:
In this equation
* is the
number-average value of the
degree of polymerization
The degree of polymerization, or DP, is the number of monomeric units in a macromolecule or polymer or oligomer molecule.
For a homopolymer, there is only one type of monomeric unit and the ''number-average'' degree of polymerization is given by ...
, equal to the average number of monomer units in a polymer molecule. For the example of nylon-6,6
( diamine units and 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 . The total includes all degrees of polymerization: monomers,
oligomer
In chemistry and biochemistry, an oligomer () is a molecule that consists of a few repeating units which could be derived, actually or conceptually, from smaller molecules, monomers.Quote: ''Oligomer molecule: A molecule of intermediate relativ ...
s and polymers.
This equation shows that a high monomer
conversion
Conversion or convert may refer to:
Arts, entertainment, and media
* "Conversion" (''Doctor Who'' audio), an episode of the audio drama ''Cyberman''
* "Conversion" (''Stargate Atlantis''), an episode of the television series
* "The Conversion" ...
is required to achieve a high degree of polymerization. For example, a monomer conversion, , of 98% is required for = 50, and = 99% is required for = 100.
Linear polymers: one monomer in excess
If one monomer is present in
stoichiometric
Stoichiometry refers to the relationship between the quantities of reactants and products before, during, and following chemical reactions.
Stoichiometry is founded on the law of conservation of mass where the total mass of the reactants equal ...
excess, then the equation becomes
:
:* ''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
The limiting reagent (or limiting reactant or limiting agent) in a chemical reaction is a reactant that is totally consumed when the chemical reaction is completed. The amount of product formed is limited by this reagent, since the reaction canno ...
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 f
av per monomer unit. For a system containing N
0 molecules initially and equivalent numbers of two functional groups A and B, the total number of functional groups is N
0f
av.
:
And the modified Carothers equation is
[Rudin p.170]
:
, where p equals to
Related equations
Related to the Carothers equation are the following equations (for the simplest case of linear polymers formed from two monomers in equimolar quantities):
:
where:
:*''X''
w is the ''weight average degree of polymerization'',
:*''M''
n is the
number average molecular weight The molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species (Ni) and the molar mass (Mi) of that species. In linear polymers, the individual polymer chains rarely have ...
,
:*''M''
w is the
weight average molecular weight The molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species (Ni) and the molar mass (Mi) of that species. In linear polymers, the individual polymer chains rarely have ...
,
:*''M''
o is the molecular weight of the repeating
monomer
In chemistry, a monomer ( ; ''mono-'', "one" + '' -mer'', "part") is a molecule that can react together with other monomer molecules to form a larger polymer chain or three-dimensional network in a process called polymerization.
Classification
Mo ...
unit,
:*''Đ'' is the
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 inconsi ...
index. (formerly known as polydispersity index, symbol PDI)
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
Chain-growth polymerization (American English, AE) or chain-growth polymerisation (British English, BE) is a polymerization technique where Unsaturated compound, unsaturated monomer molecules add onto the active site on a growing polymer chain one ...
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
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
of the medium.
References
{{reflist
Polymer chemistry
Equations