Intrinsic Viscosity Calculator (Mark-Houwink Equation)

An expert tool to calculate a polymer’s intrinsic viscosity from its molecular weight.

Calculated Intrinsic Viscosity [η]

---

The calculation is based on the Mark-Houwink equation: [η] = K × M^a. Intrinsic viscosity is a measure of a polymer’s contribution to the viscosity of a solution.

Viscosity vs. Molecular Weight Chart

This chart illustrates how the intrinsic viscosity changes with molecular weight, based on the current K and ‘a’ parameters. This demonstrates the core principle of the Mark-Houwink equation.

What is Intrinsic Viscosity?

When you want to calculate viscosity for a polymer solution, you often turn to a specific concept: intrinsic viscosity [η]. It’s not a measure of “thickness” in the everyday sense. Instead, it quantifies the contribution of a solute (the polymer) to the overall viscosity of a solution at infinite dilution. This value is incredibly useful in polymer science because it directly relates to the size and shape of individual polymer molecules in a specific solvent. By measuring it, scientists can infer the polymer’s molecular weight, which is critical for controlling its properties. The primary method to calculate viscosity in this context is the Mark-Houwink equation.

The Mark-Houwink Equation and Formula

The empirical relationship between intrinsic viscosity [η] and molecular weight (M) is described by the Mark-Houwink equation. It is the cornerstone for determining the molecular weight of polymers from viscosity data. The formula is:

[η] = K × M^a

This equation connects the macroscopic property of viscosity to the microscopic property of molecular weight. To accurately calculate viscosity using this formula, you need specific parameters for your system. For more on the fundamentals of viscosity, see this introduction to rheology basics.

Formula Variables

Variables used in the Mark-Houwink equation to calculate intrinsic viscosity.

Variable	Meaning	Typical Unit	Typical Range
[η]	Intrinsic Viscosity	dL/g, cm³/g, m³/kg	0.1 – 10 dL/g
K	Mark-Houwink Proportionality Constant	Unit depends on [η] and M	1 x 10^-5 to 1 x 10^-2
M	Viscosity-Average Molecular Weight	g/mol	10,000 to >1,000,000
a	Mark-Houwink Exponent	Unitless	0.5 (theta solvent) to 2.0 (rigid rod)

Practical Examples

Example 1: Polystyrene in Toluene

Let’s say we have a sample of polystyrene dissolved in toluene at 25°C. The Mark-Houwink parameters for this system are known.

• Inputs:
  • Molecular Weight (M): 250,000 g/mol
  • Parameter K: 0.00011 dL/g
  • Parameter a: 0.72
• Calculation:
  • [η] = 0.00011 × (250,000)^0.72
  • [η] ≈ 0.00011 × 14668 ≈ 1.61 dL/g
• Result: The intrinsic viscosity is approximately 1.61 dL/g.

Example 2: Poly(methyl methacrylate) in Acetone

Now consider PMMA dissolved in acetone at 20°C.

• Inputs:
  • Molecular Weight (M): 80,000 g/mol
  • Parameter K: 0.00057 dL/g
  • Parameter a: 0.72
• Calculation:
  • [η] = 0.00057 × (80,000)^0.72
  • [η] ≈ 0.00057 × 6765 ≈ 3.86 dL/g
• Result: The intrinsic viscosity is approximately 3.86 dL/g. This calculation shows how different K parameters significantly impact the final viscosity value, even with the same ‘a’ exponent. Understanding these solvent effects on viscosity is crucial.

How to Use This Intrinsic Viscosity Calculator

This tool makes it simple to calculate viscosity based on the Mark-Houwink equation. Follow these steps for an accurate result.

1. Enter Molecular Weight (M): Input the viscosity-average molecular weight of your polymer in the first field.
2. Enter Mark-Houwink Parameters (K and a): Provide the ‘K’ and ‘a’ constants for your specific polymer-solvent-temperature system. These values are typically found in polymer handbooks or scientific literature.
3. Select Viscosity Unit: Choose your desired output unit for the intrinsic viscosity ([η]) from the dropdown menu. The calculator will handle the conversion automatically.
4. Review the Results: The calculator instantly provides the primary intrinsic viscosity value, along with the intermediate `M^a` term for verification. The dynamic chart also updates to show the relationship.

For experimental measurements, consider our lab viscometry guide for best practices.

Key Factors That Affect Intrinsic Viscosity

Several factors influence the intrinsic viscosity of a polymer solution. Understanding them is key to correctly interpreting results and using this calculator effectively.

• Molecular Weight: As shown by the formula, this is the most direct factor. Higher molecular weight leads to a higher intrinsic viscosity.
• Polymer Architecture: The ‘a’ value reflects the polymer’s shape. Linear polymers have different ‘a’ values than branched or star-shaped polymers. Our tool assumes a linear polymer, a key aspect of polymer chain conformation.
• Solvent Quality: In a “good” solvent, the polymer coil expands, leading to a higher ‘a’ value (~0.8) and higher intrinsic viscosity. In a “poor” or “theta” solvent, the coil is more compact, resulting in a lower ‘a’ value (~0.5).
• Temperature: Temperature affects both solvent quality and polymer chain mobility. The K and ‘a’ parameters are only valid at the temperature at which they were determined.
• Molecular Weight Distribution: The calculator uses a single average molecular weight. However, real polymers have a distribution of chain lengths. This is described by the molecular weight distribution.
• Polymer Concentration: Intrinsic viscosity is an extrapolated value at zero concentration. At higher concentrations, polymer chains interact, and other viscosity models are needed.

Frequently Asked Questions (FAQ)

1. What is the difference between dynamic, kinematic, and intrinsic viscosity?

Dynamic viscosity measures a fluid’s internal resistance to flow (e.g., in Pa·s). Kinematic viscosity is dynamic viscosity divided by density (e.g., in cSt). Intrinsic viscosity is a polymer-specific measure of its contribution to solution viscosity, extrapolated to zero concentration (e.g., in dL/g).

2. Why does the ‘a’ parameter matter so much when I calculate viscosity?

The ‘a’ parameter reflects the polymer’s conformation. An ‘a’ of 0.5 indicates a compact coil in a theta solvent, while ‘a’ > 0.8 suggests a more extended chain in a good solvent. ‘a’ ≈ 1.8-2.0 is for rigid rod-like molecules. It drastically changes how molecular weight affects viscosity.

3. Where can I find K and ‘a’ values for my polymer?

These empirical constants are found in polymer science literature, handbooks (like the Polymer Handbook), and online databases. They must match your polymer, solvent, and temperature. You can explore our polymer database for common values.

4. Can I use this calculator for any polymer?

Yes, as long as you have the correct K and ‘a’ parameters. The Mark-Houwink equation is widely applicable to linear polymers.

5. What do the different viscosity units mean?

They are all measures of specific volume. 1 dL/g is equal to 100 cm³/g. The SI unit is m³/kg, where 1 dL/g = 0.1 m³/kg. This calculator allows you to convert between them easily.

6. How is intrinsic viscosity measured in a lab?

It’s typically measured using a capillary viscometer (like an Ubbelohde viscometer). The flow times of the pure solvent and several dilute polymer solutions are measured, and the results are extrapolated to zero concentration.

7. Does this calculator account for temperature?

Indirectly. The K and ‘a’ values you input are highly dependent on temperature. You must use constants that were determined at your experimental temperature.

8. Can I use this to find the molecular weight if I know the viscosity?

Yes, you can rearrange the formula: M = ([η] / K)^(1/a). This calculator is set up to calculate viscosity from M, but the underlying relationship works both ways. Our molar mass calculator may also be helpful.