Mark-Houwink Equation Calculator for Polymer Science
Mark-Houwink Parameters
Custom Parameters
Polystyrene in Benzene (25°C)
Polystyrene in Cyclohexane (34.5°C)
PMMA in Acetone (25°C)
PVC in THF (25°C)
Polyethylene in Decalin (135°C)
Select a standard system or enter custom K and a values.
Viscosity-average molecular weight (g/mol or Da).
Please enter a positive molecular weight.
Unit: mL/g. Typically between 0.001 and 0.1.
Please enter a valid K constant.
Dimensionless. Typically between 0.5 (theta solvent) and 0.8 (good solvent).
Please enter a valid exponent (a).
Intrinsic Viscosity [η]
0.00mL/g
Log(M)–
Log(K)–
Specific Viscosity Estimate (c=1g/dL)–
Calculated using: [η] = K × Ma
Viscosity vs. Molecular Weight
● Calculated Curve
● Current Point
Projected Values
Molecular Weight (g/mol)
Intrinsic Viscosity (mL/g)
Relative Increase
Calculate Viscosity from Molecular Weight: A Comprehensive Guide
Understanding how to calculate viscosity from molecular weight is fundamental in polymer science and material engineering. The relationship between the size of a polymer molecule (its molecular weight) and its resistance to flow in solution (viscosity) allows scientists to characterize materials accurately. This guide explains the Mark-Houwink equation, how to use our calculator, and the factors influencing these measurements.
What is the Calculation of Viscosity from Molecular Weight?
When we talk about the "calculation of viscosity from molecular weight," we are typically referring to determining the intrinsic viscosity ($[\eta]$) of a polymer solution based on its viscosity-average molecular weight ($M_v$). This process uses an empirical relationship known as the Mark-Houwink-Sakurada equation.
This calculation is vital for:
Polymer Chemists: To determine the molecular weight of synthesized polymers.
Quality Control Engineers: To ensure consistency in batch production of plastics.
Material Scientists: To predict processing behavior like extrusion or injection molding flow.
Common Misconception: Many assume that viscosity increases linearly with molecular weight. In reality, the relationship follows a power law, meaning a doubling of molecular weight often results in a significantly larger increase in viscosity depending on the solvent quality.
Mark-Houwink Formula and Mathematical Explanation
The core formula used to calculate viscosity from molecular weight is the Mark-Houwink equation:
$[\eta] = K \times M^a$
Variable
Meaning
Typical Unit
Typical Range
$[\eta]$
Intrinsic Viscosity
mL/g or dL/g
0.1 – 10.0
$K$
Mark-Houwink Constant
mL/g
$10^{-3} – 10^{-1}$
$M$
Viscosity-Average Molecular Weight
g/mol (Daltons)
$10^3 – 10^7$
$a$
Mark-Houwink Exponent
Dimensionless
0.5 – 0.8
Step-by-Step Derivation
To use this formula practically, one often works with logarithms to linearize the data:
$$ \log([\eta]) = \log(K) + a \times \log(M) $$
If you plot $\log([\eta])$ against $\log(M)$, the slope of the line is $a$, and the y-intercept is $\log(K)$. This linear relationship allows researchers to determine the constants $K$ and $a$ for specific polymer-solvent pairs experimentally.
Practical Examples
Example 1: Polystyrene in Benzene
A researcher synthesizes Polystyrene and determines its molecular weight is 150,000 g/mol. They want to predict its intrinsic viscosity in Benzene at 25°C.
Select Polymer System: Choose a preset (e.g., Polystyrene in Benzene) if applicable. This auto-fills the $K$ and $a$ constants. Choose "Custom" if your specific polymer-solvent pair is not listed.
Enter Molecular Weight: Input the viscosity-average molecular weight ($M_v$) in g/mol. Ensure this value is positive.
Verify Constants: If using custom mode, input the $K$ value (in mL/g) and the $a$ exponent found in polymer handbooks (like the Polymer Handbook by Brandrup & Immergut).
Read Results: The primary result is the Intrinsic Viscosity. Intermediate values like Log(M) are provided for plotting reference.
Analyze the Chart: The graph visualizes how viscosity would change if the molecular weight were higher or lower, helping you target specific viscosity grades.
Key Factors That Affect Viscosity Results
Several variables influence the outcome when you calculate viscosity from molecular weight.
1. Solvent Quality (The 'a' Parameter)
The exponent $a$ reflects the polymer coil's shape in solution. In a "good" solvent, the polymer expands, leading to a higher $a$ (closer to 0.8) and higher viscosity. In a "theta" solvent (poor solvent), the coil contracts, $a$ drops to roughly 0.5, and viscosity decreases.
2. Temperature
Viscosity is highly temperature-dependent. The constants $K$ and $a$ are valid ONLY for a specific temperature. Changing the temperature changes the solvent quality, requiring new constants to accurately calculate viscosity from molecular weight.
3. Molecular Weight Distribution
Polymers are polydisperse, meaning they contain chains of different lengths. The Mark-Houwink equation technically uses the Viscosity-Average Molecular Weight ($M_v$). Using Number-Average ($M_n$) or Weight-Average ($M_w$) without correction can lead to errors if the polydispersity index is high.
4. Branching
Branched polymers have a smaller hydrodynamic volume than linear polymers of the same molecular weight. This results in a lower intrinsic viscosity. The standard Mark-Houwink parameters for linear polymers will overestimate the viscosity of branched equivalents.
5. Concentration Regimes
While this calculator focuses on intrinsic viscosity (infinite dilution limit), in practical scenarios (semi-dilute or concentrated), chain entanglements cause viscosity to skyrocket exponentially, deviating from the simple power law.
6. Shear Rate
Polymer solutions are often non-Newtonian (pseudoplastic). At high shear rates, the polymer chains align with the flow, reducing the apparent viscosity (shear thinning). Intrinsic viscosity measurements are typically performed at low shear rates (zero-shear viscosity).
Frequently Asked Questions (FAQ)
Q: Can I calculate molecular weight from viscosity?
Yes, simply rearrange the formula: $M = ([\eta] / K)^{(1/a)}$. This is a very common method called "Viscometry" used to determine molecular weight cheaply.
Q: What is the unit of Intrinsic Viscosity?
The most common units are mL/g or dL/g. Note that $1 \text{ dL/g} = 100 \text{ mL/g}$. Our calculator assumes $K$ is in mL/g, so the result is in mL/g.
Q: Why is the exponent 'a' usually between 0.5 and 0.8?
0.5 represents a random coil in a theta solvent (ideal conditions where steric forces cancel out). 0.8 represents a swollen coil in a good solvent. Values > 0.8 usually indicate rigid rod-like polymers.
Q: Does this apply to all polymers?
It applies to linear, flexible polymers. Rigid rods, proteins, or highly branched dendrimers follow different hydrodynamic models.
Q: How accurate is the Mark-Houwink equation?
It is an empirical relation. Its accuracy depends entirely on the accuracy of the $K$ and $a$ constants, which must be determined experimentally for every unique polymer-solvent-temperature combination.
Q: What is the difference between dynamic viscosity and intrinsic viscosity?
Dynamic viscosity (Pa·s) measures resistance to flow. Intrinsic viscosity is a measure of the polymer's contribution to the viscosity of the solution, normalized by concentration. It relates to the hydrodynamic volume of the molecule.
Q: Where can I find K and a values?
The standard reference is the "Polymer Handbook." Many scientific papers also publish these constants for novel polymers.
Q: Can I use this for melt viscosity?
No. This calculator is for solution viscosity (intrinsic viscosity). Melt viscosity follows a steeper power law (power of ~3.4) above the critical entanglement molecular weight.
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