First, measure the equilibrium swelling of the polymer in a good solvent, normally by allowing the polymer to swell for 1-2weeks in the dark to reach equilibrium. The swollen gel is then isolated and weighed. The weights of swelling solvent and polymer are determined after removing the solvent by vacuum-drying.

Apply the following equation:

Crosslink density, network chain per gram =

-[ln(1-Vp) + (Vp) + X(Vp)^2]/ {Dp(Vo)[(Vr)^(1/3) - (Vp)/2]}

Where

Vp=Volume fraction of polymer in the swollen polymer

X= Huggins polymer-solvent interaction constant

Dp=Density of polymer (g/cm^3)

Vo=Molar volume of solvent (cm^3/mol)

Do=Density of solvent (g/cm^3)

Here,

Vp=1/(1+Q),

Where Q=[weight of solvent in the swollen polymer x Dp]/[Weight of polymer x Do]

Normally, you need equilibrium swelling data and equilibrium mechanical modulus. Then, a quite simple equation is applied, in case you assume Gaussian chains. I do not remember the original paper where it has been derived, but I am sure you will find this equation in almost any Polymer Physics textbook, and I have applied it in my paper "Rheology and porosity control of poly(2-hydroxyethyl methacrylate) hydrogels".Please ask if you need further details, I can look in the Origin file for the raw calculations.

Generally, which type of polymer you are going to cross-link. For thermoplastics polymers the cross-linking by bi or tri functional chemical moiety will be calculated by swelling in best solvent by looking in solubility parameter and then from equilibrium swelling you can calculate the cross-link density. Finally cross-link density calculated from the values of specific volume of polymer and Molecular weight between cross-link (for more details you can contact me at: [email protected])

First, measure the equilibrium swelling of the polymer in a good solvent, normally by allowing the polymer to swell for 1-2weeks in the dark to reach equilibrium. The swollen gel is then isolated and weighed. The weights of swelling solvent and polymer are determined after removing the solvent by vacuum-drying.

Apply the following equation:

Crosslink density, network chain per gram =

-[ln(1-Vp) + (Vp) + X(Vp)^2]/ {Dp(Vo)[(Vr)^(1/3) - (Vp)/2]}

Where

Vp=Volume fraction of polymer in the swollen polymer

X= Huggins polymer-solvent interaction constant

Dp=Density of polymer (g/cm^3)

Vo=Molar volume of solvent (cm^3/mol)

Do=Density of solvent (g/cm^3)

Here,

Vp=1/(1+Q),

Where Q=[weight of solvent in the swollen polymer x Dp]/[Weight of polymer x Do]

This is a nice suggestion how to avoid mechanical testing, provided that the Higgins parameter is known. However, precise theories show that this parameter itself is concentration-dependent.

Moreover, there is another advantage of combined measurement of mechanical and swelling behavior. In the latter case it is possible to account for the physical crosslinks additionally, and even sometimes range the crosslinks according to their average lifetime!

Storage modulus (G') is directly related to the crosslink density (Vc) according to the following equation:

G'=(Vc)RT

where R is the gas constant and T is the temperature.

Slop (gradient) of storage modulus versus temperature curve gives you (Vc)R that R is constant, therefore you can measure crosslinking density.

Reference:

Jiang H, Su W, Mather PT, Bunning TJ, Rheology of highly swollen chitosan/polyacrylate hydrogels, Polymer, 1999, 40, 4593.

A. Hajighasem, K. Kabiri, Cationic highly alcohol-swellable gels: synthesis and characterization, Journal of Polymer Research, 2013, 20, 218.

Article Cationic highly alcohol-swellable gels: Synthesis and characterization

The metods described earlier are not wrong, but there are more direct methods for determining cross-linking density than those described here. In my article: Molecular structure of citric acid cross-linked starch films Carbohydrate Polymers 07/2013; 96(1):270-276. We show two metods not described here, titrating on the cross-linking agent and determine the molecular weight change upon breaking of cross-links. Whether these methods work for you of course depends on your system.

Is there any method to determine the crosslinking density for blended polymers?

Sir

Pls explain Vr in that equation.

And can you attach reference for the formula

Can Anybody help How can I measure the crosslinking density of room temperature cured phenolic resin (resol)?

Or, you could use NMR https://www.researchgate.net/publication/280575436_Probing_small_network_differences_in_sulfur-cured_rubber_compounds_by_combining_nuclear_magnetic_resonance_and_swelling_methods

Article Probing small network differences in sulfur-cured rubber com...

Can anyone suggest how to calculate the crosslinking density of Foam or hydrogel from compression test?

For characterization of network, the most often used is network density and average molecular weight of rubber segments situated between two cross-links. Both these characteristics may be determined in the same way, because dependence:

is valid. ( ν is crosslink density, ρ is rubber density, Mc is molecular weight of rubber segments between two cross-links ).

Sometimes the network of vulcanizates is characterized also with number of cross-links created in the vulcanizate volume unit. It can be calculated also from network density under assumption that cross-links are created between two rubber segments:

(n is number of cross-links in the vulcanizate volume unit, ν is its cross-link density)

Several types of methods are used for determination of these characteristics of the vulcanizate network. Chemical methods may be used only in that case, when exact curing mechanism of using cross-linking system is known and either change of the curing agent or definable products of the low molecule character may be sufficiently determined in quantitative way. Number of the cross-links n can be calculated in this case on the basis of material balance of appropriate chemical equation related to curing.

Physical methods are based on evaluation of deformation properties of appropriate vulcanizates. In case, that these evaluations are performed under low deformations (

https://www.tut.fi/ms/muo/vert/5_rubber_chemistry/2_vulcanization_crosslink.htm