View source for Degree of Cross-Linking Elastomers

{{Language_sel|LANG=ger|ARTIKEL=Vernetzungsgrad Elastomere}}
{{PSM_Infobox}}
<span style="font-size:1.2em;font-weight:bold;">Degree of cross-linking elastomers</span>
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==General information==

The properties of elastomers are determined by the cross-linking density and the chemical structure of the cross-links [1]. The crosslinking density depends on the crosslinking system, the concentration of the reaction partners, the vulcanization time (see [[Vulcanization | vulcanization]]) and temperature, as well as the length of the statistical chain segments and the chain cross-section [2]. Due to the chemical and physical cross-linking, a three-dimensional network is formed. The chemical cross-links are covalent and thermostable. The physical network chain bonds are formed as a result of chain entanglements (see [[Cross-Linking Elastomers | cross-linking elastomers]]). The microphysical structure of an elastomer network is characterized by the following parameter:

* Density of physical (''x''<sub>phys</sub>) and chemical (''x''<sub>chem</sub>) mesh points,
* Density of entanglements (''x''<sub>ent</sub>),
* Density of network chains (''n''<sub>phys</sub>, ''n''<sub>chem</sub>, ''n''<sub>ent</sub>),
* Density of the chain ends,
* Functionality of the network sites [1].

The characteristics of networks can be described quantitatively using the [[Quasi-static Test Methods | quasi-static]] [[Tensile Test | tensile test]], nuclear magnetic resonance spectroscopy (NMR), dynamic mechanical thermal analysis (DMTA) and the degree of swelling.

==Evaluation of the degree of swelling==

Determining the degree of swelling is helpful for a more detailed description of the network characteristics and for determining the degree of crosslinking. The equilibrium swelling test is based on the effect that the diffusion of a swelling agent is hindered with increasing cross-linking density. The swelling is influenced on the one hand by the osmotic pressure and on the other hand by the elastic restoring force of the network. These two effects cancel each other out in the swelling equilibrium. In accordance with ISO 1817 [4 ], the degree of swelling of an elastomer material can be determined in a solvent. For this purpose, the material is stored in a suitable swelling agent for 72 h at room temperature. The mass of the specimen is determined before storage. After removal from the swelling agent, the mass is determined after 5 minutes. The specimens continue to be stored in air at room temperature, or alternatively in a heating chamber at an elevated temperature. The masses are determined at different times until a constant mass is achieved. 

The degree of swelling is determined as follows:

{|
|-
|width="20px"|
|width="500px" | <math> Q = \frac {m_1 - m_2} {m_2} </math>
|(1)
|}

with
{|
|-
|''m''<sub>1</sub>
|width="15px" | 
|Mass of swollen specimen <br>
|-
|''m''<sub>2</sub>
|
|Mass of non-swollen specimen
|}

The degree of swelling determined in this way is a measure of the mass fraction of the elastomer in the swollen specimen.

==Determination of the cross-linking density==

Using the equation according to Flory and Rehner [5] and taking into account the degree of swelling, the crosslinking density of unfilled vulcanizates can be calculated as follows [6]:

{|
|-
|width="20px"|
|width="500px" | <math> v_c = - \frac {\text {ln}(1 - \varphi) + \varphi + \chi \cdot {\varphi}^2} {V_a (\varphi^{1/3} - 0.5\, \varphi)} </math>
|(2)
|}

with
{|
|-
|''v''<sub>c</sub>
|width="15px" | 
|Crosslinking in mol*cm-³  <br>
|-
|''V''<sub>A</sub>
|
|Molar volume of the swelling agent <br>
|
|-
|''&chi;''
| 
|FLORY-HUGGINS interaction parameter <br>
|-
|''&phi;''
|
|Volume fraction of polymer in swollen gel
|}

If elastomers are reinforced with fillers, additional network points occur between the filler particles and the macromolecules. Correction methods are used to take the influence of fillers (e.g. carbon black) into account. For this purpose, the quotient of the degree of swelling of a reinforced and an unreinforced elastomer specimen is related to the rubber mass of the compound formulation [2].

{|
|-
|width="20px"|
|width="500px" | <math> \frac {Q_{g,K}} {Q_{u,K}} = 0.33 + 0.685 \cdot e^{-z} </math>
|(3)
|}
with
{|
|-
|width="20px"|
|width="500px" | <math> z = \frac {\varphi_R} {\varphi_K} </math>
|(4)
|}
with
{|
|-
|''Q''<sub>g,K</sub>
|width="15px" | 
|degree of swelling of the reinforced elastomer in relation to the rubber mass <br>
|-
|''Q''<sub>u,K</sub>
| 
|degree of swelling of the unreinforced elastomer in relation to the rubber mass <br>
|-
|''z''
|
|exponent <br>
|-
|''&phi;''<sub>R</sub>
|
|mass fraction of the filler (e.g. carbon black) in the elastomer compound <br>
|}

With <math> Q_{g,K} = \left( \frac {\varphi_{ges}} {\varphi_K} + Q_g \right) </math>  for the reinforced elastomer mixtures and assuming that the mass fraction of the rubber in the unreinforced mixtures corresponds approximately to the total mass fraction <math> (\varphi_{ges} \sim \varphi_K) </math>, the corrected rubber volume fraction in the swollen state <math> \varphi_B^{*} </math> results as follows:

{|
|-
|width="20px"|
|width="500px" | <math> \varphi_B^{*} = \frac {(0.33 + 0.685 \cdot e^z) \cdot \varphi_B} {(1 - \varphi_B) + (0.33 + 0.685 \cdot e^z) \cdot \varphi_B} </math>
|(5)
|}

In addition, the molecular weight ''M''<sub>c,sw</sub> can be calculated on the basis of the FLORY-REHNER equation (see Eq. (2)). The FLORY-REHNER theory is based on the separability hypothesis of the free energies of chain stretching and mixing [8]. Investigations by Grassé et al [9] have shown that the values of the FLORY-REHNER equation using the phantom model provide better consistent data for the description of the elastic behaviour. The model allows the calculation of the interaction parameter between the swelling agent and the elastomeric network [10]. For this purpose, elastomers with different effective crosslinking functionalities f were investigated [9].

{|
|-
|width="20px"|
|width="500px" | <math> M_{c,sw} = -\frac {\rho_p \, V_S \, {\varphi_{p,el}}^{1/3}} {\text{ln} (1 - \varphi_p) + \varphi_p + \chi \, {\varphi_p}^2} \cdot \frac {f-2} {f} </math>
|(6)
|}

with
{|
|-
|''&varrho;''<sub>p</sub>
|width="20px" | 
|polymer density <br>
|-
|''V''<sub>S</sub>
| 
|molar volume of the solvent <br>
|-
|''&phi;''<sub>p</sub> = ''V''<sub>p</sub>/''V''
|
|polymer volume fraction in swelling equilibrium <br>
|-
|''&phi;''<sub>p,el</sub> = ''&phi;''<sub>p</sub> (1 - ''&omega;''<sub>def,sw</sub>)
|
|volume fraction of the elastic active polymer (determination using double-quantum nuclear magnetic  resonance spectroscopy (DQ NMR) <br>
|-
|''f'' = 4
| 
|for elastomers with long polymer chains
|}

==See also==

* [[Ageing Elastomers | Ageing elastomers]]
* [[Thermosets]]
* [[Density]]
* [[Volume Swelling Elastomers | Volume swelling elastomers]]
* [[Vulcanization]]


'''References'''

{|
|-valign="top"
|[1]
|Röthemeyer, F., Sommer, F.: Kautschuktechnologie. Carl Hanser, Munich Vienna (2006), 2nd Edition; (ISBN 978-3-446-40480-9)
|-valign="top"
|[2]
|[https://de.wikipedia.org/wiki/Gert_Heinrich Heinrich, G.]: Struktur, Eigenschaften und Praxisverhalten von Gummi: vom Polymernetzwerk zum dynamisch beanspruchten Reifen Teil 1, GAK Gummi Fasern Kunststoffe 50 (1997) 9, 687–693 
|-valign="top"
|[3]
|[[Reincke, Katrin | Reincke, K.]]; Bruchmechanische Bewertung von ungefüllten und gefüllten Elastomerwerkstoffen. Dissertation, Martin-Luther-Universität Halle-Wittenberg, Mensch & Buch Verlag Berlin (2005) (ISBN 978-3-86664-021-0; see [[AMK-Büchersammlung|AMK-Library]] under B 1-13)
|-valign="top"
|[4]
|ISO 1817 (2024-03): Rubber, Vulcanized or Thermoplastic – Determination of the Effect of Liquids 
|-valign="top"
|[5]
|Flory, P. J., Rehner, J.: Statistical Mechanics of Cross-Linked Polymer Network I. Rubberlike Elasticity, J. Chem. 11 (1943) 521–526; https://doi.org/10.1063/1.1723791
|-valign="top"
|[6]
|Kleemann, W., Weber, K.: Formeln und Tabellen für die Elastomerverarbeitung. Dr. Gupta Verlag, (1994); (ISBN 978-3-9803593-0-6)
|-valign="top"
|[7]
|Kelm, J., Tobisch, K., Leisen, J.: Vergleichende Untersuchungen zur Netzstellendichte an Elastomeren. Kaut. Gummi Kunstst. 51 (1998) 364–369 
|-valign="top"
|[8]
|Schlögl, S., Trutschel, M.-L., Chassé, W., Riess, G., Saalwächter, K.: Entaglement Effects in Elastomers: Microscopic vs Microscopic Properties. Macromolecules 47 (2014) 2759–2773 DOI: https://doi.org/10.1021/ma4026064
|-valign="top"
|[9]
|Chassé, W., Lang, M., Sommer, J.-U., Saalwächter, K.: Corrections to Cross-Link Density Estimation of PDMS Networks with Precise Consideration of Network Defects. Macromolecules 45 (2012) 2, 899–912 DOI: https://doi.org/10.1021/ACS.MACROMOL.5B00236 
|-valign="top"
|[10]
|Hild, G.: Interpretation of Equilibrium Swelling Data on Model Networks using Affine and ‘Phantom’ Networks Models. Polymer 38 (1997) 3, 3279–3293 DOI: https://doi.org/10.1016/S0032-3861(96)00878-6
|}

[[Category:Elastomers]]