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The Carothers equation relates degree of polymerization to average chain length or conversion
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Peer-reviewed literature and reference materials establish that the Carothers equation relates the degree of polymerization and average chain length to monomer conversion.

Evidence for · 2
2021 · cited by 8
A challenge in the field of polymer network synthesis by a step-growth mechanism is the quantification of the relative importance of inter- vs. intramolecular reactions. Here we use a matrix-based kinetic Monte Carlo (<i>k</i>MC) framework to demonstrate that the variation of the chain length distribution and its averages (e.g., number average chain length <i>x<sub>n</sub></i>), are largely affected by intramolecular reactions, as mostly ignored in theoretical studies. We showcase that a conventional approach based on equations derived by Carothers, Flory and Stockmayer, assuming constant reactivities and ignoring intramolecular reactions, is very approximate, and the use of asymptotic limits is biased. Intramolecular reactions stretch the functional group (FG) conversion range and reduce the average chain lengths. In the likely case of restricted mobilities due to diffusional limitations because of a viscosity increase during polymerization, a complex <i>x<sub>n</sub></i> profile with possible plateau formation may arise. The joint consideration of stoichiometric and non-stoichiometric conditions allows the validation of hypotheses for both the intrinsic and apparent reactivities of inter- and intramolecular reactions. The <i>k</i>MC framework is also utilized for reverse engineering purposes, aiming at the identification of advanced (pseudo-)analytical equations, dimensionless numbers and mechanistic insights. We highlight that assuming average molecules by equally distributing A and B FGs is unsuited, and the number of AB intramolecular combinations is affected by the number of monomer units in the molecules, specifically at high FG conversions. In the absence of mobility constraints, dimensionless numbers can be considered to map the time variation of the fraction of intramolecular reactions, but still, a complex solution results, making a <i>k</i>MC approach overall most elegant. We showcase that a conventional approach based on equations derived by Carothers, Flory and Stockmayer, assuming constant reactivities and ignoring intramolecular reactions, is very approximate, and the use of asymptotic limits is biased. Intramolecular reactions stretch the functional group (FG) conversion range and reduce the average chain lengths. In the likely case of restricted mobilities due to diffusional limitations because of a viscosity increase during polymerization, a complex x n profile with possible plateau formation may arise. Due to the nature of the step-growth polymerization mechanism with a gradual formation of dimers ( x = 2), trimers ( x = 3) and ultimately x -mers with x >> 1, a high extent of reaction and thus FG conversion (>>0.95) is required to achieve high average chain lengths [ 1 , 14 ]. Impurities and molar imbalances, which are often aggravated by different volatility tendencies, have a deleterious effect on the quality of the final polymer product. This equation is based on general stochastic insights and gives the degree of polymerization or number average chain length, i.e., x n , for a given (A or equivalently B) FG conversion p with r = 1 ignoring intramolecular reactions: (1) x n = 1 1 − p This equation reaches an asymptotic value at p = p * = 1 and requires, in practice, a final evaluation at p = 0.99. The monomers AA and BB (or AB) are considered having a chain length of 1, and it is assumed that potential byproduct is removed from the reaction mixture so that equilibrium settings can be ignored (so-called kinetically controlled step-growth polymerization) [ 19 ]. For branched step-growth polymerization systems, Carothers [ 18 ] modified his original equation stating that x n is dependent on the average functionality per monomer f av (Equation (6); f a v = N m o n , A , 0 f A + N m o n , B , 0 f B N m o n , A , 0 + N m o n , B , 0 with N mon , A ,0 and N mon , B ,0 being the initial number of monomer molecules based on FG A and B, respectively, and f A and f B the related functionality degrees, respectively). The equation is limited to stoichiometrically balanced reactions ( r = 1), describing the synthesis up to the asymptotic value located at p* = 2/ f av , which is denoted as the critical degree of FG conversion. Nevertheless, chain length averages can still be determined without first determining the CLD, and more recent results have been reported by Beginn et al. [ 33 ] and Cheng et al. [ 34 , 35 ]. Hillegers et al. [ 36 ] used, in turn, this approach to calculate path length—the number of chemical bonds in the path connecting two monomeric units in a molecule—distributions in an A 1 + A 2 + A 3 type polymerization. However, the technique based on Equation (12) involves abstract mathematics and, as such, is difficult to use. Figure 5 Benchmark between equations derived by Flory [ 22 ] and/or Stockmayer [ 27 , 28 ] (black) and the kinetic Monte Carlo ( k This is done by evaluating the effect of k i n t r a V N a v k i n t e r on the number average chain length x n as a function of the A FG conversion ( p A ); the Flory equation [ 22 ] (Equation (9); ρ = 1) is depicted as a grey dashed line; subplot ( b ) is a zoom of ( a ) for a small p A range with a 5% deviation from the equation by Flory, additionally indicated by a fuchsia dashed line; k intra/inter : intra/intermolecular rate coefficient; V : simulation volume; N Av : Avogadro number; k MC simulations based on Figure 3 . 3.2. Focus is on the effect of the number average chain length x n as a function of the A FG conversion p A (left axis; solid lines) for different ratios of k i n t r a V N A v k i n t e r and different r values: ( a ) k i n t r a V N A v k i n t e r = 1.0 × 10 − 2 and r = 1, ( b ) k i n t r a V N A v k i n t e r = 1.0 × 10 − 2 and r = 0.75, ( c ) k i n t r a V N A v k i n t e r = 1.0 and r = 1 and ( d ) k i n t r a V N A v k i n t e r = 1.0 and r = 0.75. Also given are the conventional results according to Equation (9) (Flory equation [ 22 ]; black line; no intramolecular reactions).
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step-growth polymerization, as per the Carothers' equation. The mass average molar mass (often loosely termed weight average molar mass) is another way of describing In polymer chemistry, 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 exactly the same degree of polymerization and molar mass, and there is always a distribution around an average value. The molar I… The number average molecular mass of a polymer can be determined by gel permeation chromatography, viscometry via the (Mark–Houwink equation), colligative methods such as vapor pressure osmometry,… where… In polymer chemistry, 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 exactly the same degree of polymerization and molar mass, and there is always a distribution around an average value. The molar mass distribution of a polymer may be modified by polymer fractionation. Number average molar mass (Mn), also loosely referred to as number average molecular weight (NAMW). Mass average molar mass (Mw), where w stands for weight; also commonly referred to as weight average or weight average molecular weight (WAMW). Z-average molar mass (Mz), where z stands for centrifugation (from German Zentrifuge). Viscosity average molar mass (Mv). Here, a is the exponent in the Mark–Houwink equation that relates the intrinsic viscosity to molar mass. The dispersity (also known as the polydispersity index) of a sample is defined as Mw divided by Mn and gives an indication just how narrow a distribution is. The most common technique for measuring molecular mass used in modern times is a variant of high-pressure liquid chromatography (HPLC) known by the interchangeable terms of size exclusion chromatography (SEC) and gel permeation chromatography (GPC). These techniques involve forcing a polymer solution through a matrix of cross-linked polymer particles at a pressure of up to several hundred bar. The limited accessibility of stationary phase pore volume for the polymer molecules results in shorter elution times for high-molecular-mass species. The use of low dispersity standards allows the user to correlate retention time with molecular mass, although the actual correlation is with the Hydrodynamic volume. If the relationship between molar mass and the hydrodynamic volume changes (i.e., the polymer is not exactly the same shape as the standard) then the calibration for mass is in error. The most common detectors used for size exclusion chromatography include online methods similar to the bench methods used above. By far the most common is the differential refractive index detector that measures the change in refractive index of the solvent. This detector is concentration-sensitive and very molecular-mass-insensitive, so it is ideal for a single-detector GPC system, as it allows the generation of mass v's molecular mass curves. Less common but more accurate and reliable is a molecular-mass-sensitive detector using multi-angle laser-light scattering - see static light scattering. These detectors directly measure the molecular mass of the polymer and are most often used in conjunction with differential refractive index detectors. A further alternative is either low-angle light scattering, which uses a single low angle to determine the molar mass, or Right-angle-light laser scattering in combination with a viscometer, although this latter technique does not give an absolute measure of molar mass but one relative to the structural model used. The molar mass distribution of a polymer sample depends on factors such as chemical kinetics and work-up procedure. Ideal step-growth polymerization gives a polymer with dispersity of 2. Ideal living polymerization results in a dispersity of 1. By dissolving a polymer an insoluble high molar mass fraction may be filtered off resulting in a large reduction in Mw and a small reduction in Mn, thus reducing dispersity. The number average molecular mass of a polymer can be determined by gel permeation chromatography, viscometry via the (Mark–Houwink equation), colligative methods such as vapor pressure osmometry, end-group determination or proton NMR. High number-average molecular mass polymers may be obtained only with a high fractional monomer conversion in the case of step-growth polymerization, as per the Carothers' equation. where Ni is the number of molecules of molecular mass Mi. The mass average molecular mass can be determined by static light scattering, small angle neutron scattering, X-ray scattering, and sedimentation velocity. The ratio of the mass average to the number average is called the dispersity or the polydispersity index. The mass-average molecular mass, Mw, is also related to the fractional monomer conversion, p, in step-growth polymerization (for the simplest case of linear polymers formed from two monomers in equimolar quantities) as per Carothers' equation: The z-average molar mass can be determined with ultracentrifugation. The melt elasticity of a polymer is dependent on Mz.
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  1. Going Beyond the Carothers, Flory and Stockmayer Equation by Including Cyclization Reactions and Mobility Constraints.peer-reviewedno side taken
  2. Molar mass distributionreferenceno side taken
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