Gel point: Difference between revisions
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If one of the monomers in a copolymerization has a functionality greater than 2, a branched copolymer can be formed. It is also possible for the branches to react and create cross-links. In this way, “infinite” polymer networks called gels are made. |
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The '''Gel point''' in [[polymer chemistry]] is the point at which an infinite [[polymer network]] first appears. Assuming that it is possible to measure the extent of reaction, ''p'', defined as the fraction of [[monomer]]s that appear in [[cross-link]]s, the gel point can be determined.<ref>Paul, Hiemenz C., and Lodge P. Timothy. Polymer Chemistry. Second ed. Boca Raton: CRC P, 2007. 381-389</ref> |
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The critical extent of reaction for the gel point to be formed is given by: |
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Definitions: |
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P<sub>A</sub> = probability that A reacts |
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P<sub>B</sub> = probability that B reacts |
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N<sub>Ao</sub> = original number of A groups (@ t = 0) |
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N<sub>Bo</sub> = original number of B groups (@ t = 0) |
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r=N<sub>Ao</sub>/N<sub>Bo</sub> |
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Assuming A can only react with B: |
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N<sub>Ao</sub>P<sub>A</sub> = N<sub>Bo</sub>P<sub>B</sub> and rP<sub>A</sub> = P<sub>B</sub> |
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For gelation to occur, q, the fraction of all monomer units in the sample that form cross-links, must be greater than q<sub>c</sub>, the critical value of q: |
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q<sub>c</sub> =1/(f-1) |
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and |
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q=(P<sub>A</sub>P<sub>B</sub>p)/(1-P<sub>A</sub>P<sub>B</sub>(1-p)) |
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where f is the number of functional groups on the molecule with highest functionality. Additionally, only the highest functionality molecule reacts and causes branching, so another factor, p, must be considered. |
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p = (functionality of branched molecule * number of moles)/(sum(functionality * number of moles of all molecules of that type)) |
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For example, if 0.5 moles of trifuctional A, 1 mole of difunctional A, and 2 moles of difunctional B molecules were reacted: |
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p = (3*0.5 moles)/(2*1 mole+3*0.5 moles) = 1.5/3.5 = 0.43 |
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The limiting reactant in this situation is A because N<sub>Ao</sub> = 3.5 mol < N<sub>Bo</sub> = 4 mol. Solving for P<sub>A</sub> gives the fractional conversion of limiting reagent required to react for gelation to occur.<ref>Rudin, Alfred and Choi, Phillip. ''The Elements of Polymer Science and Engineering, 3rd Edition''. 2012. Elsevier Science. p 410. ISBN: 9780123821782</ref> |
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:<math>p_c = \frac{1}{N-1} \approx \frac{1}{N} </math> |
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For example, a polymer with N≈200 is able to reach the gel point with only 0.5% of monomers reacting. This shows the ease at which polymers are able to form infinite networks. |
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The critical extent of reaction for [[gelation]] can be determined as a function of the properties of the monomer mixture, r, ρ, and f:<ref>{{cite journal |
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| last = Pinner |
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| first = S.H. |
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| title = Functionality of non-equivalent mixtures |
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| journal = Journal of polymer science |
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| volume = XXI |
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| issue = 97 |
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| pages = 153–157 |
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| publisher = |
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| location = |
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| date = |
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| url = |
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| issn = |
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| doi = 10.1002/pol.1956.120219718 |
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| id = |
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| accessdate = }}</ref> |
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:<math>p_c = \frac{1}{(r+rp(f-2))^{0.5}} </math> |
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==References== |
==References== |
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Revision as of 20:27, 13 December 2013
If one of the monomers in a copolymerization has a functionality greater than 2, a branched copolymer can be formed. It is also possible for the branches to react and create cross-links. In this way, “infinite” polymer networks called gels are made.
Definitions:
PA = probability that A reacts
PB = probability that B reacts
NAo = original number of A groups (@ t = 0)
NBo = original number of B groups (@ t = 0)
r=NAo/NBo
Assuming A can only react with B:
NAoPA = NBoPB and rPA = PB
For gelation to occur, q, the fraction of all monomer units in the sample that form cross-links, must be greater than qc, the critical value of q:
qc =1/(f-1) and q=(PAPBp)/(1-PAPB(1-p))
where f is the number of functional groups on the molecule with highest functionality. Additionally, only the highest functionality molecule reacts and causes branching, so another factor, p, must be considered.
p = (functionality of branched molecule * number of moles)/(sum(functionality * number of moles of all molecules of that type))
For example, if 0.5 moles of trifuctional A, 1 mole of difunctional A, and 2 moles of difunctional B molecules were reacted:
p = (3*0.5 moles)/(2*1 mole+3*0.5 moles) = 1.5/3.5 = 0.43
The limiting reactant in this situation is A because NAo = 3.5 mol < NBo = 4 mol. Solving for PA gives the fractional conversion of limiting reagent required to react for gelation to occur.[1]
References
- ^ Rudin, Alfred and Choi, Phillip. The Elements of Polymer Science and Engineering, 3rd Edition. 2012. Elsevier Science. p 410. ISBN: 9780123821782