Molecular Weight Averages & Polydispersity Distributions

Unlike low molecular weight organic molecules or monodisperse biological macromolecules (such as enzymes, insulin, or wild-type genomic DNA) where every molecule possesses an exact, identical chemical formula and invariant mass, synthetic polymers are inherently polydisperse.

Origins of Polydispersity

In any synthetic chemical polymerization process:

• Initiation events occur randomly throughout the reactor volume over time.
• Radical, ionic, or coordination active centers propagate with stochastic collision kinetics.
• Chain transfer, termination by combination or disproportionation, and mass-transfer mixing gradients introduce broad variations in individual chain lifetimes.

Consequently, a synthetic polymer sample consists of a complex statistical mixture of macromolecules possessing identical constitutional repeat units but spanning a broad continuum of chain lengths and molecular weights.

The Number and Weight Distribution Functions

A polydisperse polymer is described by its discrete or continuous molecular weight distribution (MWD):

• Number Distribution $N(M)$: The number of moles (or molecules) $N_i$ of species possessing molecular weight $M_i$. The mole fraction is:

• Weight Distribution $W(M)$: The total mass $w_i = N_i M_i$ of species possessing molecular weight $M_i$. The weight fraction is:

Because heavier molecules contribute proportionally more mass, the weight distribution $W(M)$ is always shifted toward substantially higher molecular weights relative to the number distribution $N(M)$.

Because no single number can fully describe a broad distribution, polymer science employs a family of statistical moments known as molecular weight averages.

1. Number-Average Molecular Weight $M_n$

The first statistical moment of the number distribution, representing the total mass of the sample divided by the total number of moles:

$M_n$ is highly sensitive to the presence of low molecular weight oligomers. It is measured experimentally by colligative properties (membrane osmometry, vapor pressure osmometry, cryoscopy) and end-group chemical titration.

2. Weight-Average Molecular Weight $M_w$

The second statistical moment of the number distribution (or the first moment of the weight distribution):

$M_w$ is heavily influenced by the presence of large, high molecular weight macromolecules. It is determined experimentally by static light scattering, small-angle neutron scattering (SANS), and sedimentation equilibrium.

3. Z-Average Molecular Weight $M_z$

The third moment of the number distribution (or second moment of the weight distribution):

$M_z$ is exceptionally sensitive to high molecular weight tails, micro-gels, and long-chain branching. It is measured by analytical ultracentrifugation (AUC sedimentation equilibrium). Higher moments ($M_{z+1}$) can be defined analogously.

4. Viscosity-Average Molecular Weight $M_v$

Derived from dilute solution viscometry through the Mark-Houwink-Sakurada relationship ($[\eta] = K' M^a$):

where $a$ is the Mark-Houwink exponent ($0.5 \le a \le 0.8$ for flexible coils).

• When $a = 1$: $M_v = M_w$.
• In a theta solvent where $a = 0.5$: $M_v = \left(\sum w_i M_i^{1/2}\right)^2$.

A cornerstone theorem of polymer mathematics states that for any polydisperse polymer sample, the molecular weight averages obey the strict hierarchical inequality:

Equality ($M_n = M_v = M_w = M_z$) holds if and only if the polymer is strictly monodisperse.

Proof that $M_w \ge M_n$ via the Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality states that for any real sequences $(a_i)$ and $(b_i)$:

Let $a_i = \sqrt{N_i}$ and $b_i = \sqrt{N_i} M_i$. Substituting these sequences:

Dividing both sides by $\left(\sum N_i\right) \left(\sum N_i M_i\right)$:

Equality holds if and only if $b_i = c \, a_i$ for all $i$, meaning $\sqrt{N_i} M_i = c \sqrt{N_i} \implies M_i = c$ (monodisperse sample). The proof that $M_w \le M_z$ follows identically by choosing $a_i = \sqrt{N_i M_i}$ and $b_i = \sqrt{N_i M_i} M_i$.

Position of the Viscosity Average $M_v$

Because the Mark-Houwink exponent $a$ satisfies $0 < a < 1$ for flexible random coils in good and theta solvents, Hölder's inequality dictates:

Typically, $M_v$ lies within $10 - 20\%$ below $M_w$, serving as a highly accessible proxy for weight-average molecular weight.

The breadth of the molecular weight distribution is quantitatively indexed by the polydispersity index (PDI, denoted by IUPAC as the dispersity $\text{Đ}$):

Variance and Standard Deviation of the MWD

The variance $\sigma_n^2$ of the number distribution is:

Notice that:

Therefore:

The standard deviation relative to the number-average molecular weight is:

This elegant identity proves that the dispersity $\text{Đ}$ is a direct metric of the normalized variance of the distribution.

Typical Dispersity Values across Polymerization Mechanisms

• Living Anionic & Group-Transfer Polymerization: $\text{Đ} \approx 1.01 - 1.05$ (nearly monodisperse Poisson distributions).
• Controlled / Living Radical Polymerization (ATRP, RAFT, NMP): $\text{Đ} \approx 1.05 - 1.20$.
• Step-Growth (Condensation) Polymerization at High Conversion ($p \to 1$): $\text{Đ} = 1 + p \to 2.0$ (Flory-Schulz most probable distribution).
• Free-Radical Polymerization:
• Disproportionation termination: $\text{Đ} = 2.0$.
• Combination termination: $\text{Đ} = 1.5$.
• With autoacceleration (Trommsdorff gel effect) and chain transfer: $\text{Đ} = 2.5 - 5.0$.
• Ziegler-Natta Multi-Site Heterogeneous Coordination Catalysis: $\text{Đ} = 5.0 - 20.0$ (superposition of multiple active catalytic sites).
• Hyperbranched Polymers & Random Crosslinking Gels: $\text{Đ} > 20 - 50$ (diverging as the critical gel point is approached).

The Flory-Schulz distribution (also termed the "most probable" distribution) governs linear step-growth polycondensation and chain-growth polymerization with constant chain transfer.

Statistical Derivation from Equal Reactivity Postulate

Consider a step-growth polymerization of bifunctional monomers ($A-B$ or stoichiometric $A-A + B-B$). Let $p$ be the extent of reaction (probability that any given functional group has reacted). The probability that a polymer molecule contains precisely $x$ repeating units requires:

• $(x - 1)$ reacted linkages (each with independent probability $p$).
• $1$ unreacted end group terminating the chain (with probability $1 - p$).

The number-fraction distribution (mole fraction of $x$-mers) is:

The total number of molecules in the system at conversion $p$ is $N = N_0 (1 - p)$, where $N_0$ is the initial number of monomers. The total number of $x$-mer chains is:

The Weight-Fraction Distribution $w_x$

The weight fraction $w_x$ of $x$-mers is the mass of $x$-mers divided by total mass $N_0 M_0$:

Molecular Weight Averages and Dispersity

Summing the statistical moments using standard geometric series identities:

1. Number-Average Degree of Polymerization $X_n$:

2. Weight-Average Degree of Polymerization $X_w$:

3. Z-Average Degree of Polymerization $X_z$:

4. Polydispersity Index $\text{Đ}$:

In the limit of high conversion ($p \to 1.0$), $X_n \to \infty$, and the dispersity approaches exactly $2.0$:

The maximum of the weight distribution curve occurs at:

Thus, the weight distribution peaks at precisely $x = X_n$.

When polymer chain initiation is instantaneous relative to propagation and termination or chain transfer is completely absent—as in ideal living anionic or living ring-opening polymerizations—the molecular weight distribution follows a Poisson distribution.

Mechanistic Derivation

Let all $N_I$ initiator molecules initiate chain growth simultaneously at $t = 0$. The rate of monomer consumption per active center is identical across all chains:

The probability that a growing chain adds precisely $x$ monomer units during reaction time $t$ obeys Poisson birth-process kinetics:

where $\nu$ is the average number of monomers consumed per initiator molecule (the kinetic chain length):

Molecular Weight Averages and Monodispersity

For the Poisson distribution:

• Number-average degree of polymerization:

• Weight-average degree of polymerization:

• Dispersity $\text{Đ}$:

For a typical polymer with $X_n = 500$:

The distribution is extraordinarily narrow, providing essentially monodisperse polymers used as universal molecular weight calibration standards.

To model empirical polymer distributions spanning from narrow living systems to broad industrial materials, the Schulz-Zimm distribution provides a continuous, highly flexible mathematical representation.

Mathematical Formulation

The weight-fraction distribution function $w(M)$ in the Schulz-Zimm model is:

where:

• $z$ is the coupling / breadth parameter ($z > 0$)
• $y$ is a scaling parameter
• $\Gamma(z + 1) = z!$ is the gamma function

The statistical moments are:

Relationship to Dispersity $\text{Đ}$

The dispersity is directly governed by the parameter $z$:

Solving for the parameter $z$:

• As $z \to \infty$: $\text{Đ} \to 1.0$ (monodisperse delta function).
• When $z = 1$: $\text{Đ} = 2.0$ (recovering the Flory-Schulz most probable distribution).
• When $z = 2$: $\text{Đ} = 1.5$ (recovering free-radical polymerization with combination termination).
• When $z < 1$: $\text{Đ} > 2.0$ (broad polydispersity distributions).

The ratio of the Z-average to weight-average is:

The Schulz-Zimm function is widely implemented in Gel Permeation Chromatography software to deconvolute overlapping chromatogram peaks.

Gel Permeation Chromatography (GPC), also termed Size Exclusion Chromatography (SEC), is the premier experimental technique for measuring the full molecular weight distribution of polymers.

Separation Mechanism: Entropy-Driven Size Exclusion

A GPC column is packed with porous, crosslinked polymer beads (typically styrene-divinylbenzene gels) with controlled pore diameter distributions ($10 - 10^5\text{ \AA}$). As a dilute polymer solution flows through the column:

• Large macromolecules with hydrodynamic volume $V_h$ exceeding the pore diameter cannot enter the pore network and are excluded, eluting rapidly at the interstitial void volume $V_0$.
• Small macromolecules permeate freely into both the interstitial spaces and the internal pores, eluting at the total liquid volume $V_t = V_0 + V_p$.
• Intermediate chains partition into a fraction $K_{\text{SEC}}$ of the pore volume based on their steric size.

The retention volume $V_e$ is:

Crucially, GPC separates molecules strictly by their hydrodynamic volume $V_h$, NOT by molecular weight!

Benoit Universal Calibration Principle

In 1967, Henri Benoit demonstrated that the hydrodynamic volume of any polymer chain in solution is directly proportional to the product of its intrinsic viscosity $[\eta]$ and its molecular weight $M$: From Einstein's viscosity law for equivalent hydrodynamic spheres of radius $R_h$:

Benoit established that plotting $\log([\eta] M)$ versus elution volume $V_e$ yields a single universal calibration curve onto which all polymers (linear, branched, polystyrene, PMMA, polyethylene) collapse, independent of chemical composition:

Molecular Weight Determination of Unknown Polymers

If a GPC column is calibrated using narrow polystyrene standards (subscript $\text{PS}$):

Substituting the Mark-Houwink-Sakurada relationship $[\eta] = K' M^a$:

Solving for the true molecular weight $M_x$ of the unknown polymer eluting at the same retention volume $V_e$:

This elegant relation permits precise determination of absolute molecular weights and distribution functions for any polymer whose Mark-Houwink constants are known.