KEVOS® MAKEOS · Materials
Polymer Solution and Blend Phase Check (Flory–Huggins χ, Spinodal, Critical Point, Cloud Temperature)
Will this polymer stay dissolved in this solvent, or mixed with this second polymer, at my concentration and temperature, and at what temperature will it start to separate?
Flory–Huggins phase check for a polymer in a solvent or a blend of two polymers: the interaction parameter χ at the temperature from the two solubility parameters and an entropic part, the spinodal χ at your polymer fraction from the weight-average chain length, the critical concentration and χ from the weight- and z-average chain lengths (Stockmayer), the binodal (cloud-point) χ for a uniform chain of the weight-average length, the matching critical-point, spinodal and cloud temperatures, and whether the mixture is one phase, metastable or unstable, with a table of the phase boundary.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Component 1 (solvent, or first polymer of a blend) | ||
| Molar mass of component 1 | g/mol | Molar mass of the solvent from its data sheet; for a blend, the weight-average molar mass of the first polymer. |
| Density of component 1 | g/cm³ | Density of the solvent or first polymer at about the working temperature, from the supplier’s data sheet. |
| Solubility parameter of component 1 | MPa½ | Hildebrand solubility parameter of the solvent or first polymer, from a handbook of solubility parameters or the supplier. |
| Component 2 (the polymer) | ||
| Weight-average molar mass of the polymer (Mw) | g/mol | Weight-average molar mass Mw of the polymer from the supplier’s certificate or a GPC (SEC) measurement. |
| Ratio of z-average to weight-average molar mass (Mz/Mw) | Mz ÷ Mw from the same GPC report; enter 1 for a polymer of one chain length. | |
| Density of the polymer | g/cm³ | Density of the amorphous polymer from the data sheet. |
| Solubility parameter of the polymer | MPa½ | Solubility parameter of the polymer from a handbook, a group-contribution estimate or a swelling test. |
| Mixture and conditions | ||
| Polymer volume fraction | % | Volume fraction of the polymer in the mixture, from the recipe (mass shares divided by densities). |
| Entropic part of the interaction parameter (χS) | Entropic part of χ: about 0.34 for many polymer–solvent pairs (Blanks and Prausnitz), usually near 0 for two polymers; better from a measured χ. | |
| Temperature of the mixture | °C | Lowest temperature the mixture sees in storage, transport or use. |
| Required margin above the cloud temperature | K | Distance you require between the working temperature and the cloud temperature, from your own experience of the system. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
Component 1 (solvent, or first polymer of a blend)
What this means: Molar mass of component 1
- What it is
- The molar mass of the solvent molecule, or for a blend the weight-average molar mass of the first polymer; with its density it sets the volume of one lattice cell, the unit in which the polymer’s chain length is counted.
- Units and format
- Enter a number in g/mol: g/mol. The form accepts 1 g/mol to 100000000 g/mol. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 92 g/mol as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger molar volume of component 1 shortens the polymer’s relative chain length and raises the enthalpic part of χ in proportion.
- Get it right
- For a blend enter the first polymer here, not a monomer: a lattice cell then holds a whole chain of component 1.
What this means: Density of component 1
- What it is
- The density of the solvent or first polymer, used with its molar mass to find its molar volume V₁ = M₁ ÷ ρ₁, the reference volume of the lattice.
- Units and format
- Enter a number in g/cm³: grams per cubic centimetre (1 g/cm³ = 1000 kg/m³). The form accepts 0.1 g/cm³ to 25 g/cm³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.87 g/cm³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the density near the working temperature; solvent densities fall noticeably as they warm.
What this means: Solubility parameter of component 1
- What it is
- The Hildebrand solubility parameter of component 1: the square root of its cohesive energy density. The difference from the polymer’s value sets the enthalpic part of χ.
- Units and format
- Enter a number in MPa½: MPa½. The form accepts 5 MPa½ to 60 MPa½. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 17 MPa½ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- χ rises with the square of the difference between the two solubility parameters, so equal values give the lowest χ.
- Get it right
- Take both solubility parameters from the same source so their difference is consistent; small errors in each make a large error in the squared difference.
Component 2 (the polymer)
What this means: Weight-average molar mass of the polymer (Mw)
- What it is
- The weight-average molar mass Mw of the polymer, which with its density gives the weight-average chain length Nw used for the spinodal and the binodal estimate.
- Units and format
- Enter a number in g/mol: g/mol. The form accepts 100 g/mol to 100000000 g/mol. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200000 g/mol as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Longer chains lower the critical fraction and the critical χ towards 0.5, so the mixture separates at a higher temperature.
- Get it right
- Use Mw, not Mn: the spinodal of a polydisperse polymer depends on the weight average.
What this means: Ratio of z-average to weight-average molar mass…
- What it is
- The ratio of the z-average to the weight-average molar mass, a measure of the high-mass tail of the distribution, which moves the critical concentration and critical χ of a polydisperse polymer.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger ratio raises the critical fraction and, slightly, the critical χ; the spinodal and the binodal estimate do not change.
- Get it right
- GPC reports Mz with Mw; enter 1 for a narrow standard of one chain length.
What this means: Density of the polymer
- What it is
- The density of the amorphous polymer, which turns its molar mass into a molar volume and so into a chain length in units of component-1 cells.
- Units and format
- Enter a number in g/cm³: grams per cubic centimetre (1 g/cm³ = 1000 kg/m³). The form accepts 0.1 g/cm³ to 25 g/cm³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.05 g/cm³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the amorphous density, not that of a semicrystalline moulding, which is higher.
What this means: Solubility parameter of the polymer
- What it is
- The Hildebrand solubility parameter of the polymer; polymers cannot be evaporated, so it is estimated from group contributions or from the solvent that swells or dissolves it best.
- Units and format
- Enter a number in MPa½: MPa½. The form accepts 5 MPa½ to 60 MPa½. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 19 MPa½ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- χ rises with the square of the difference from component 1’s value.
- Get it right
- Published values for one polymer can differ by 1 MPa½ or more; check the effect of that spread on the result.
Mixture and conditions
What this means: Polymer volume fraction
- What it is
- The share of the mixture’s volume occupied by the polymer, the composition at which the spinodal and binodal are found and the mixture is judged.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0.001% to 99%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Near the critical fraction the cloud temperature is highest; very dilute or very concentrated mixtures separate only at lower temperatures.
- Get it right
- A recipe in mass shares must be converted: divide each mass by its density before taking the share.
What this means: Entropic part of the interaction parameter (χS)
- What it is
- The part of the interaction parameter that does not change with temperature, an empirical entropic term added to the solubility-parameter estimate of the enthalpic part.
- Units and format
- Enter a plain number with no unit. The form accepts -2 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.34 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raising it raises χ at every temperature and so raises the cloud temperature.
- Get it right
- About 0.34 is typical for polymer–solvent pairs; for two polymers it is usually near zero; a measured χ at two temperatures gives both parts.
What this means: Temperature of the mixture
- What it is
- The temperature at which the mixture is judged; χ is evaluated here, and the margin to the cloud temperature is measured from it.
- Units and format
- Enter a number in °C: degrees Celsius. The form accepts -150 °C to 400 °C. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25 °C as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher temperature lowers χ and moves the mixture away from the two-phase region.
- Get it right
- Judge the coldest point the mixture reaches, such as unheated storage overnight, because cloud points appear on cooling.
What this means: Required margin above the cloud temperature
- What it is
- The smallest acceptable distance between the working temperature and the estimated cloud temperature, allowing for the uncertainty of χ and of the binodal estimate.
- Units and format
- Enter a number in K: kelvin (a difference of 1 K equals 1 °C). The form accepts 0 K to 300 K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the line between one phase and one phase with a small margin.
- Get it right
- Choose a larger margin when χ comes from solubility parameters rather than measurement.
Method and formulas
- Lattice cells are the size of component 1: V₁ = M₁ ÷ ρ₁, and the polymer occupies N = (M₂ ÷ ρ₂) ÷ V₁ cells (weight average Nw, z-average Nz = Nw × Mz/Mw).
- χ = χS + V₁(δ₁ − δ₂)² ÷ (R T), Klenin’s Eq. 3.1-71 with χS entered, so χ falls as the temperature rises (upper critical behaviour).
- Spinodal: 2χs = 1 ÷ (Nw φ) + 1 ÷ (1 − φ); critical fraction φc = 1 ÷ (1 + Nw ÷ √Nz) and 2χc = (1 + 1 ÷ √Nz)(1 + √Nz ÷ Nw) (Klenin Eqs. 3.2-9 to 3.2-11).
- Binodal: the χ at which your fraction coexists with a second phase of equal chemical potentials of both components, for a uniform chain of length Nw; each temperature is T = V₁(δ₁ − δ₂)² ÷ [R (χ − χS)].
- Example: Mw 200,000 g/mol (density 1.05) at 5 % in a liquid of 92 g/mol (density 0.87): Nw = 1,801, χ = 0.5106 at 25 °C against 0.5273 at the binodal and 0.5319 at the spinodal; the estimated cloud temperature is −1.6 °C, a margin of 26.6 K: one phase.
Limitations
- Mean-field Flory–Huggins model with χ independent of concentration; near the critical point and in dilute solution its predictions are only approximate (Klenin chapters 4 and 5).
- Only upper critical (cloud on cooling) behaviour is modelled; lower critical separation on heating, specific interactions such as hydrogen bonding and crystallisation are not.
- The binodal is computed for a uniform chain of length Mw; the cloud point of a polydisperse polymer depends on its whole molar-mass distribution.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check that a polymer stays dissolved in a coating, adhesive, casting or spinning solution at its storage and use temperature
- Estimate the temperature at which a solution turns cloudy on cooling
- Screen whether two polymers in a recycled or compounded blend can mix as one phase
- Choose a polymer concentration far enough from the phase boundary
Who it is for
- Materials and process engineers
- Coating, adhesive and membrane formulators
- Polymer compounders and recyclers
- Quality and laboratory staff
What you get
- χ at the temperature and χ at the binodal and spinodal for your fraction
- Critical polymer fraction and critical χ
- Estimated cloud and spinodal temperatures and the temperature at the critical point
- One phase, metastable or unstable
- Table of the phase boundary against polymer fraction
Interpreting the result
- A stability ratio χ ÷ χ(binodal) below 1 means one stable phase; between the binodal and spinodal values the mixture is metastable and separates by nucleation, often slowly; at or above the spinodal it separates spontaneously.
- Long chains have a critical fraction of only a few per cent and a critical χ just above 0.5, so a small change in temperature or solvent can move a concentrated solution across the boundary.
- For a polydisperse polymer the critical point lies on the side of the phase boundary, not at its top, so the boundary can reach a higher temperature than the critical point; read the table for the highest cloud temperature.
Before relying on the result
- Confirm the cloud temperature by cooling a sample slowly and watching for turbidity.
- Use a measured χ (osmometry, light scattering or swelling) where one exists.
References
- Klenin V. J., Thermodynamics of Systems Containing Flexible-Chain Polymers (Elsevier, 1999; ISBN 0-444-82373-5), §3.1 (state equations and liquid–liquid separation, Eq. 3.1-71 for χ from solubility parameters) and §3.2.1 (spinodal and critical point of a polydisperse polymer, Eqs. 3.2-9 to 3.2-11). Method reference; no tables or figures are reproduced.
- Flory P. J., Principles of Polymer Chemistry (Cornell University Press, 1953), chapters XII and XIII.
- Stockmayer W. H., J. Chem. Phys. 17, 588 (1949): spinodal and critical point of polydisperse polymer solutions, as cited by Klenin.
- Blanks R. F. and Prausnitz J. M., Ind. Eng. Chem. Fundam. 3, 1 (1964): the entropic part of χ for polymer solutions.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Method explained here · KEVOS® original book
Mixing and demixing: Flory–Huggins and the cold cement
In Polymer Chains by the Numbers
What it covers: Use Flory–Huggins theory to see why long chains leave a marginal solvent, estimate χ from solubility parameters, find cloud, spinodal and critical temperatures, and check a cement for cold storage.
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