A synthetic polymer is not a single molecule but a mixture of chains with different lengths, so its molar mass is described by a distribution and by several averages. This simulation shows how that distribution is defined and how it is measured by two common characterization methods: gel permeation (size-exclusion) chromatography, which separates chains by size, and dilute-solution viscometry, which gives the intrinsic viscosity and the viscosity-average molar mass. You can see how the shape and breadth of the distribution change each average and how limitations of the measurements cause the measured values to differ from the true values.
The sliders in the top row define a polymer sample: the shape of its molar-mass distribution, its number-average molar mass Mn, and its dispersity Đ = Mw/Mn. Choose "two populations" to blend a second Schulz–Zimm population into the sample.
distribution plots the number and weight distributions per decade of molar mass and marks Mn, Mv, Mw, and Mz.
chromatography GPC / SEC (GPC = gel permeation chromatography, SEC = size-exclusion chromatography) shows the refractive-index chromatogram the sample would give on a size-exclusion column, with the calibration line on the right axis. Lower the exclusion limit or increase band broadening to see how the averages calculated from the chromatogram differ from the true values.
viscometry simulates dilute-solution measurements at five concentrations. Huggins and Kraemer plots are extrapolated to zero concentration to obtain the intrinsic viscosity [η], and the Mark–Houwink equation converts [η] to the viscosity-average molar mass Mv.
For chains of molar mass \(M_i\), with \(N_i\) moles and mass \(w_i=N_iM_i\):
\[M_n=\frac{\sum N_iM_i}{\sum N_i}=\frac{\sum w_i}{\sum w_i/M_i}\] \[M_w=\frac{\sum N_iM_i^2}{\sum N_iM_i}=\frac{\sum w_iM_i}{\sum w_i}\] \[M_z=\frac{\sum N_iM_i^3}{\sum N_iM_i^2}\] \[M_v=\left(\frac{\sum N_iM_i^{1+a}}{\sum N_iM_i}\right)^{1/a}\]The dispersity is Đ = \(M_w/M_n\). For any distribution \(M_n\le M_v\le M_w\le M_z\); \(M_v=M_w\) when \(a=1\).
Schulz–Zimm (gamma) distribution of number fraction:
\[n(M)\propto M^{k-1}\exp\!\left(-\frac{kM}{M_n}\right),\qquad \frac{M_w}{M_n}=1+\frac{1}{k}\]Đ = 2 (\(k=1\)) is the Flory most-probable distribution of step-growth polymerization. Đ near 1 (large \(k\)) is typical of living anionic polymerization.
Log-normal distribution:
\[n(M)\propto\frac{1}{M}\exp\!\left[-\frac{(\ln M-\mu)^2}{2\sigma^2}\right],\qquad \sigma^2=\ln\frac{M_w}{M_n}\]The plots show the fraction per decade of \(M\). The weight distribution is the number distribution multiplied by \(M/M_n\), so it is shifted toward high \(M\).
Large coils sample less of the pore volume and elute first. Between the exclusion volume \(V_0\) and the total permeation volume \(V_t\) the calibration is linear:
\[\log_{10}M=A-B\,V_e\]The refractive-index signal \(h_i\) at each elution slice is proportional to the mass of polymer in that slice, so
\[M_n=\frac{\sum h_i}{\sum h_i/M_i},\qquad M_w=\frac{\sum h_iM_i}{\sum h_i}\]Band broadening spreads each species into a Gaussian of standard deviation σ (mL), which lowers the measured \(M_n\) and raises the measured \(M_w\). All chains above the exclusion limit elute together at \(V_0\) and are assigned the exclusion molar mass, so \(M_w\) is underestimated. The simulation assumes the calibration (normally made with narrow standards) applies to the sample.
Here \(t/t_0\) is the efflux-time ratio in a capillary viscometer. The exponent \(a\) is 0.5 in a theta solvent and about 0.7–0.8 in a good solvent. The simulation generates \(\eta_{sp}\) from the Huggins equation; the Kraemer points come from the same data, so they curve slightly and their intercept differs a little from [η]. The reported [η] is the average of the two intercepts. The Mark–Houwink constants are representative values; published values depend on the molar-mass range and method.
\(a\) = Mark–Houwink exponent (dimensionless)
\(A,\,B\) = calibration constants for size-exclusion chromatography (\(B\) in mL\(^{-1}\))
\(c\) = polymer concentration (g/dL)
Đ = dispersity, \(M_w/M_n\) (dimensionless)
GPC = gel permeation chromatography
\(h_i\) = refractive-index detector signal for elution slice \(i\)
\(k\) = Schulz–Zimm shape parameter (dimensionless)
\(k_H\) = Huggins constant (dimensionless)
\(k_K\) = Kraemer constant (dimensionless)
\(K\) = Mark–Houwink constant (mL/g)
\(M\) = molar mass of a chain (g/mol)
\(M_i\) = molar mass of species or elution slice \(i\) (g/mol)
\(M_n\) = number-average molar mass (g/mol)
\(M_v\) = viscosity-average molar mass (g/mol)
\(M_w\) = weight-average molar mass (g/mol)
\(M_z\) = z-average molar mass (g/mol)
\(n(M)\) = number fraction of chains with molar mass \(M\)
\(N_i\) = moles of chains with molar mass \(M_i\) (mol)
PMMA = poly(methyl methacrylate)
SEC = size-exclusion chromatography
\(t\) = efflux time of polymer solution (s)
\(t_0\) = efflux time of pure solvent (s)
\(V_0\) = exclusion volume (mL)
\(V_e\) = elution volume (mL)
\(V_t\) = total permeation volume (mL)
\(w_i\) = mass of chains with molar mass \(M_i\) (g)
\(\eta\) = viscosity of polymer solution (Pa·s)
\(\eta_0\) = viscosity of pure solvent (Pa·s)
\(\eta_r\) = relative viscosity (dimensionless)
\(\eta_{sp}\) = specific viscosity (dimensionless)
\(\eta_{sp}/c\) = reduced viscosity (dL/g)
\(\ln\eta_r/c\) = inherent viscosity (dL/g)
[η] = intrinsic viscosity (dL/g)
\(\mu\) = mean of \(\ln M\) for the log-normal distribution
\(\sigma\) = standard deviation of \(\ln M\) for the log-normal distribution; for chromatography, band-broadening standard deviation (mL)
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.