7.2 Molecular Weight Distribution and Polydispersity

Polymers are complex molecules with varying chain lengths. Understanding their molecular weight distribution is crucial for predicting their properties and behavior. This topic dives into how we measure and describe these distributions.

Number-average and weight-average molecular weights help us quantify polymer distributions. The polydispersity index, derived from these averages, tells us how uniform the polymer chains are. This knowledge is key for tailoring polymers to specific applications.

Number-average vs Weight-average Molecular Weights

Definitions and Calculations

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• Number-average molecular weight ($M_n$) represents the arithmetic mean of the molecular weights of all the polymer chains in a sample
  • Calculated by dividing the total weight of the sample by the total number of molecules: $M_n = \frac{\sum_i N_i M_i}{\sum_i N_i}$, where $N_i$ is the number of molecules with molecular weight $M_i$
• Weight-average molecular weight ($M_w$) represents a weighted average of the molecular weights of all the polymer chains in a sample
  • Each chain's contribution to $M_w$ is proportional to its molecular weight: $M_w = \frac{\sum_i N_i M_i^2}{\sum_i N_i M_i}$
  • Gives more weight to larger molecules in the distribution

Relationship and Significance

• $M_w$ is always greater than or equal to $M_n$ due to the increased influence of larger molecules in the weight-average calculation
  • Example: For a polymer with equal numbers of chains with molecular weights of 10,000 and 20,000 g/mol, $M_n$ = 15,000 g/mol, while $M_w$ = 16,667 g/mol
• The ratio of $M_w$ to $M_n$ serves as a measure of the polydispersity of the polymer sample
  • A ratio close to 1 indicates a narrow molecular weight distribution, while a larger ratio suggests a broader distribution

Polydispersity Index: Significance and Interpretation

Definition and Calculation

• The polydispersity index (PDI) quantifies the breadth of the molecular weight distribution of a polymer sample
  • Calculated as the ratio of the weight-average molecular weight ($M_w$) to the number-average molecular weight ($M_n$): $PDI = \frac{M_w}{M_n}$
• A PDI of 1 indicates a monodisperse polymer, where all the chains have the same molecular weight
  • Example: A polymer sample with all chains having a molecular weight of 50,000 g/mol would have a PDI of 1
• A PDI greater than 1 indicates a polydisperse polymer with a distribution of molecular weights
  • Example: A polymer sample with 50% of chains having a molecular weight of 50,000 g/mol and 50% having a molecular weight of 100,000 g/mol would have a PDI of 1.5

Interpreting PDI Values

• The larger the PDI, the broader the molecular weight distribution and the greater the variation in chain lengths within the polymer sample
  • Example: A polymer with a PDI of 2.5 has a broader distribution than a polymer with a PDI of 1.2
• The PDI provides information about the uniformity of the polymer chains, which can affect various properties
  • Mechanical strength: Generally, polymers with lower PDI have better mechanical properties due to more uniform chain lengths and entanglements
  • Viscosity: Polymers with higher PDI may have lower viscosity due to the presence of shorter chains that act as plasticizers
  • Processability: Polymers with lower PDI are often easier to process due to more consistent flow properties and fewer low-molecular-weight components that can volatilize

Molecular Weight Distribution: Impact on Properties

Influence on Physical and Mechanical Properties

• Polymers with a narrow molecular weight distribution (low PDI) tend to have more uniform properties and better mechanical strength compared to those with a broad distribution (high PDI)
  • Example: High-density polyethylene (HDPE) with a narrow distribution has higher tensile strength and stiffness than HDPE with a broad distribution
• The presence of low molecular weight chains in a broad distribution can act as plasticizers, reducing the glass transition temperature ($T_g$) and lowering the viscosity of the polymer melt
  • Example: The addition of low-molecular-weight polystyrene to high-molecular-weight polystyrene can lower the $T_g$ and improve processability
• High molecular weight chains in a broad distribution can increase the melt viscosity, improve the tensile strength and toughness, but may also lead to processing difficulties
  • Example: Ultra-high-molecular-weight polyethylene (UHMWPE) has excellent wear resistance and impact strength but is difficult to process due to its high viscosity

Effect on Crystallization Behavior

• The molecular weight distribution can affect the crystallization behavior of semi-crystalline polymers
• A narrow distribution promotes faster crystallization and higher crystallinity due to more uniform chain lengths and easier packing
  • Example: Isotactic polypropylene with a narrow distribution has a higher crystallization rate and degree of crystallinity than polypropylene with a broad distribution
• A broad distribution can hinder crystallization due to the presence of short chains that disrupt the regular packing of longer chains
  • Example: Low-density polyethylene (LDPE) with a broad distribution has lower crystallinity and a more amorphous structure than HDPE with a narrow distribution

Determining Molecular Weight Distribution: Methods

Gel Permeation Chromatography (GPC)

• GPC, also known as size exclusion chromatography (SEC), is a common technique for determining the molecular weight distribution of polymers
• Principle: The polymer sample is dissolved in a solvent and passed through a column packed with porous beads; separation is based on the size of the polymer chains, with smaller chains eluting later than larger ones
• The elution time is related to the hydrodynamic volume of the polymer chains, which can be correlated to their molecular weight using calibration standards
  • Example: Polystyrene standards with known molecular weights are often used to calibrate GPC columns for the analysis of other polymers
• Advantages: GPC provides a direct measurement of the molecular weight distribution, is applicable to a wide range of polymers, and can handle high-molecular-weight samples
• Limitations: GPC requires the polymer to be soluble in the mobile phase, and the accuracy depends on the quality of the calibration standards and the similarity of their structure to the analyte

Light Scattering Techniques

• Light scattering techniques, such as static light scattering (SLS) and dynamic light scattering (DLS), can provide information about the molecular weight distribution
• Static Light Scattering (SLS):
  • Measures the intensity of scattered light at different angles to determine the weight-average molecular weight ($M_w$) and the radius of gyration ($R_g$) of the polymer chains
  • Example: SLS has been used to determine the $M_w$ and $R_g$ of polysaccharides, such as dextran and pullulan
• Dynamic Light Scattering (DLS):
  • Measures the fluctuations in scattered light intensity over time to determine the hydrodynamic radius and the diffusion coefficient of the polymer chains, which can be related to their molecular weight
  • Example: DLS has been employed to study the molecular weight distribution of proteins and polymer nanoparticles
• Advantages: Light scattering techniques are non-destructive, require small sample volumes, and can provide information about the size and shape of the polymer chains in solution
• Limitations: Light scattering techniques are sensitive to dust and impurities, require precise sample preparation, and may be affected by polymer-solvent interactions

Viscometry

• Viscometry is another method that can provide an estimate of the molecular weight distribution by measuring the intrinsic viscosity of the polymer solution
• The intrinsic viscosity $[\eta]$ is related to the molecular weight $M$ by the Mark-Houwink equation: $[\eta] = KM^a$, where $K$ and $a$ are constants specific to the polymer-solvent system
  • Example: For polystyrene in toluene at 30°C, $K = 1.23 \times 10^{-4}$ dL/g and $a = 0.71$
• By measuring the intrinsic viscosity of a polymer solution and knowing the $K$ and $a$ values, the viscosity-average molecular weight ($M_v$) can be calculated
  • $M_v$ is typically between $M_n$ and $M_w$ and provides an estimate of the average molecular weight
• Advantages: Viscometry is a simple and inexpensive technique that requires minimal sample preparation and can be used for quality control purposes
• Limitations: Viscometry provides an average molecular weight rather than a full distribution, and the accuracy depends on the validity of the Mark-Houwink constants for the specific polymer-solvent system