Polymer Chemistry
Mark-Houwink Equation: How Intrinsic Viscosity Reveals Polymer Molecular Weight
Dissolve a pinch of polystyrene in tetrahydrofuran, time how long the solution drips through a thin glass capillary, and you can back out the polymer's molecular weight to within a few percent — often 100,000 g/mol or more — without ever weighing a single chain. The bridge between that stopwatch measurement and molecular weight is the Mark-Houwink equation, [η] = K·M^a, one of the most-used empirical relations in all of polymer science.
The Mark-Houwink equation (also called the Mark-Houwink-Sakurada or Kuhn-Mark-Houwink-Sakurada equation) is an empirical power law linking a polymer's intrinsic viscosity [η] — the limiting fractional thickening a dissolved chain causes per unit concentration — to its molecular weight M. K and a are two constants fixed for a given polymer/solvent/temperature combination, and the exponent a encodes the shape the chain adopts in solution.
- TypeEmpirical power law (polymer physics)
- Introduced1938-1940 (Mark, Houwink; Sakurada 1940s)
- Key equation[η] = K·M^a
- Exponent rangea = 0.5 (theta) to 0.8 (good solvent); up to 2.0 rigid rod
- Applies toViscosity-average molecular weight Mv
- Measured byUbbelohde capillary viscometer + Huggins/Kraemer extrapolation
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What the Mark-Houwink Equation Is and Where It Applies
The Mark-Houwink equation is the workhorse relation of dilute-solution viscometry. It states:
- [η] = K·M^a
where [η] is the intrinsic viscosity (units mL/g or dL/g), M is molecular weight (g/mol), K is a system-specific prefactor, and a is a dimensionless exponent. Because [η] is averaged in the same way viscosity weights the chains, the M you extract is specifically the viscosity-average molecular weight Mv, which lies between the number-average Mn and weight-average Mw (usually close to Mw).
The relation matters wherever a fast, cheap molecular-weight readout is needed: quality control of nylon and PET in fiber plants, monitoring PVC and polyethylene batches, characterizing cellulose and starch, sizing hyaluronic acid and DNA in biology, and — critically — calibrating size-exclusion chromatography (SEC/GPC) via universal calibration. Any lab with a thermostatted water bath and a glass viscometer can run it.
Derivation: From Einstein's Spheres to the Power Law
The equation is semi-empirical, but its exponent has a firm physical basis. Start with Einstein's viscosity law for hard spheres: η = η0(1 + 2.5·φ), where φ is the volume fraction of suspended particles. Rearranged, the intrinsic viscosity is proportional to the specific hydrodynamic volume: [η] ∝ Vh/M, where Vh is the volume the solvated coil sweeps out.
For a random-coil polymer, the coil size scales with molecular weight as <R²>^(1/2) ∝ M^ν, so the hydrodynamic volume Vh ∝ R³ ∝ M^(3ν). Combining via the Flory-Fox relation [η] = Φ·<R²>^(3/2)/M (Φ ≈ 2.5 x 10²³ mol⁻¹) gives:
- [η] ∝ M^(3ν − 1) = K·M^a, so a = 3ν − 1.
An ideal (theta) coil has ν = 1/2, giving a = 0.5; a swollen coil in a good solvent has ν ≈ 0.588, giving a ≈ 0.76; a rigid rod has ν = 1, pushing a toward 2.0. The power law thus falls straight out of chain scaling.
Key Quantities and a Worked Example
Worked example — polystyrene in THF at 25 C (K = 6.82 x 10⁻³ mL/g, a = 0.766). Suppose viscometry gives an intrinsic viscosity of [η] = 55 mL/g. Solve for M:
- M = ([η]/K)^(1/a) = (55 / 6.82 x 10⁻³)^(1/0.766)
- = (8065)^(1.305) ≈ 1.3 x 10⁵ g/mol.
A useful sanity check: because a < 1, [η] grows slower than M. Doubling M from 100k to 200k raises [η] only by 2^0.766 ≈ 1.70x, not 2x. This sub-linear sensitivity is why viscometry alone cannot resolve fine differences at very high M.
Characteristic numbers: flexible synthetic coils in good solvents cluster at a = 0.65-0.80; theta conditions give exactly a = 0.5; semi-flexible biopolymers (cellulose, xanthan) run 0.8-1.2; DNA and other stiff rods approach 1.7-2.0. K typically spans 10⁻⁴ to 10⁻² mL/g and falls as a rises for a given polymer.
How It's Measured in Practice
Intrinsic viscosity is not measured directly — it is an extrapolation to infinite dilution. The procedure:
- Prepare 4-5 dilute solutions (typically 0.1-1 g/dL) of the polymer.
- Time solvent (t0) and each solution (t) flowing through a Ubbelohde capillary viscometer in a bath held to ±0.02 C. Relative viscosity ηr = t/t0; specific viscosity ηsp = ηr − 1.
- Plot reduced viscosity ηsp/c and inherent viscosity (ln ηr)/c versus concentration c.
Two linear fits share a common intercept, the intrinsic viscosity:
- Huggins: ηsp/c = [η] + kH·[η]²·c (Huggins constant kH ≈ 0.3 good solvent, ~0.5 poor).
- Kraemer: (ln ηr)/c = [η] − kK·[η]²·c.
The intercept feeds directly into [η] = K·M^a. Measurements must be taken in the low-shear, Newtonian regime, and K and a must come from a table calibrated with the same polymer, solvent, and temperature.
Comparison to Related Concepts
The Mark-Houwink equation sits within a family of molecular-weight tools, each averaging chains differently:
- vs. light scattering / osmometry: These are absolute methods yielding Mw and Mn from first principles; Mark-Houwink is relative and needs pre-tabulated K, a — but it is far cheaper and faster.
- vs. the Flory-Fox equation: Flory-Fox, [η] = Φ·<R²>^(3/2)/M, is the theoretical parent; Mark-Houwink is its empirical, tabulated form.
- vs. SEC/GPC universal calibration: GPC separates by hydrodynamic volume [η]·M. Knowing K and a for two polymers lets you convert one calibration curve to another (the Benoit universal calibration), so a polystyrene-calibrated column can size PMMA.
- vs. Huggins/Kraemer: Those give you [η]; Mark-Houwink turns [η] into M. They are sequential steps, not competitors.
The exponent a is what makes it uniquely informative — no other single parameter so cleanly reports chain conformation.
Exceptions, Limits, and Significance
The relation is empirical, so it carries caveats. It fails outside its calibration window: using polystyrene-in-toluene constants for a branched or blocky sample gives wrong M, because branching contracts the coil and lowers [η] at fixed M. Values of a below 0.5 signal a poorer-than-theta solvent, chain aggregation, or a compact globule; values above 0.8 warn of stiff or rod-like chains where the simple coil scaling breaks down.
Molecular-weight range matters: K and a are only constant over a decade or two of M; very low oligomers and ultrahigh-M chains deviate. Polydispersity also biases Mv versus Mw.
- Named for Herman F. Mark and Roelof Houwink, who established the power law around 1938-1940; Ichiro Sakurada and the Kuhn brothers contributed independently, hence the longer names.
- The classic marker of the rigid-rod limit is tobacco mosaic virus, whose exponent approaches 2.0.
Despite these limits, the equation remains indispensable — a nearly century-old, tabulated shortcut still used daily in polymer plants and research labs worldwide.
| Polymer / solvent / T | K (mL/g) | a | Chain regime |
|---|---|---|---|
| Polystyrene / THF / 25 C | 6.82 x 10^-3 (0.0114 in some refs) | 0.766 | Good solvent, flexible coil |
| Polystyrene / cyclohexane / 34.5 C | ~8.0 x 10^-2 | 0.50 | Theta (ideal) coil |
| PMMA / toluene / 25 C | 7.1 x 10^-3 | 0.73 | Good solvent, flexible coil |
| Linear polyethylene / decalin / 135 C | 5.1 x 10^-4 (~0.62 in dL/g units) | 0.70 | Good solvent, flexible coil |
| Cellulose / cadoxen | 5.31 x 10^-4 | 0.78 | Semi-stiff chain |
| Tobacco mosaic virus (rigid rod) | - | ~1.7-2.0 | Rigid rod limit |
Frequently asked questions
What do K and a represent in the Mark-Houwink equation?
K is a system-specific prefactor (units mL/g or dL/g) set by the polymer, solvent, and temperature, while a is a dimensionless exponent that reports chain conformation in solution. Both are empirical constants tabulated for each polymer/solvent/temperature combination and cannot be transferred between systems. The exponent follows a = 3ν − 1, where ν is the coil-scaling exponent.
Why is the exponent a usually between 0.5 and 0.8?
Because random-coil polymers scale as coil size R proportional to M^ν, and hydrodynamic volume scales as R^3, so [η] scales as M^(3ν-1). An ideal theta coil (ν = 0.5) gives a = 0.5, and a swollen coil in a good solvent (ν approximately 0.588) gives a approximately 0.76. Rigid rods push a toward 2.0, while compact spheres drop it to zero.
Which molecular weight does the Mark-Houwink equation give you?
It gives the viscosity-average molecular weight, M_v. This average lies between the number-average M_n and the weight-average M_w, and for typical distributions it sits close to M_w (usually 80-90 percent of the way from M_n to M_w). It equals M_w only when a = 1 or when the sample is monodisperse.
How do you get intrinsic viscosity before applying the equation?
You measure flow times of several dilute solutions through an Ubbelohde capillary viscometer, compute reduced viscosity (η_sp/c) and inherent viscosity ((ln η_r)/c), and extrapolate both to zero concentration using the Huggins and Kraemer plots. Their common y-intercept is the intrinsic viscosity [η], which then goes into [η] = K·M^a.
What is a theta solvent and why does it give a = 0.5?
A theta solvent at the theta temperature is one where polymer-solvent and polymer-polymer interactions cancel, so the chain behaves as an ideal random walk with no excluded-volume swelling. Under these conditions ν = 1/2 exactly, giving a = 3(0.5) − 1 = 0.5. Polystyrene in cyclohexane at 34.5 C is the textbook example.
Can the Mark-Houwink equation detect polymer branching?
Indirectly, yes. Branched chains are more compact than linear chains of the same molecular weight, so they have a lower hydrodynamic volume and thus a lower intrinsic viscosity. If you apply linear-polymer K and a constants to a branched sample, you underestimate M; comparing measured [η] to the linear expectation gives a branching index g' that quantifies the contraction.