gelation

Gelation is promoted by gelling agents.

Gelation can occur either by physical linking or by chemical crosslinking. While the physical gels involve physical bonds, chemical gelation involves covalent bonds. The first quantitative theories of chemical gelation were formulated in the 1940s by Flory and Stockmayer. Critical percolation theory was successfully applied to gelation in 1970s. A number of growth models (diffusion limited aggregation, cluster-cluster aggregation, kinetic gelation) were developed in the 1980s to describe the kinetic aspects of aggregation and gelation.{{Cite book|title=Polymer physics|last1=Rubinstein|first1=Michael|date=December 20, 2003|publisher=Oxford University Press|last2=Colby|first2=Ralph H.|isbn=978-0198520597|location=Oxford|oclc=50339757}}{{pn|date=June 2019}}

It is important to be able to predict the onset of gelation, since it is an irreversible process that dramatically changes the properties of the system.

According to the Carothers equation number-average degree of polymerization $DP\_n$ is given by

$DP\_n=\; \backslash frac\{2\}\{2-p.f\_\{av\}\}$

where $p$ is the extent of the reaction and $f\_\{av\}$ is the average functionality of reaction mixture. For the gel $DP\_n$ can be considered to be infinite, thus the critical extent of the reaction at the gel point is found as

$p\_c=\backslash frac\; \{2\}\{f\_\{av\}\}$

If $p$ is greater or equal to $p\_c$, gelation occurs.

{{main article| Flory–Stockmayer theory}}

Flory and Stockmayer used a statistical approach to derive an expression to predict the gel point by calculating when $DP\_n$ approaches infinite size. The statistical approach assumes that (1) the reactivity of the functional groups of the same type is the same and independent of the molecular size and (2) there are no intramolecular reactions between the functional groups on the same molecule.{{cite journal |last1=Stockmayer |first1=Walter H. |title=Theory of Molecular Size Distribution and Gel Formation in Branched-Chain Polymers |journal=The Journal of Chemical Physics |date=February 1943 |volume=11 |issue=2 |pages=45–55 |doi=10.1063/1.1723803 |bibcode=1943JChPh..11...45S }}{{cite journal |last1=Flory |first1=Paul J. |title=Molecular Size Distribution in Three Dimensional Polymers. I. Gelation |journal=Journal of the American Chemical Society |date=November 1941 |volume=63 |issue=11 |pages=3083–3090 |doi=10.1021/ja01856a061 }}

Consider the polymerization of bifunctional molecules $A-A$, $B-B$ and multifunctional $A\_f$, where $f$ is the functionality. The extends of the functional groups are $p\_A$and $p\_B$, respectively. The ratio of all A groups, both reacted and unreacted, that are part of branched units, to the total number of A groups in the mixture is defined as $\backslash rho$. This will lead to the following reaction

$A-A\; +\; B-B\; +\; A\_f\; \backslash rightarrow\; A\_\{f-1\}-(B-BA-A)\_nB-BA-A\_\{f-1\}$

The probability of obtaining the product of the reaction above is given by $p\_A[p\_B(1-\backslash rho)p\_A]^np\_B\backslash rho$, since the probability that a B group reach with a branched unit is $p\_B\backslash rho$ and the probability that a B group react with non-branched A is $p\_B(1-\backslash rho)$

.

This relation yields to an expression for the extent of reaction of A functional groups at the gel point

$p\_c=\; \backslash frac\; \{1\}\{\backslash \{r[1+\backslash rho(f-2)]\backslash \}^\{1/2\}\}$

where r is the ratio of all A groups to all B groups. If more than one type of multifunctional branch unit is present and average $f$ value is used for all monomer molecules with functionality greater than 2.

Note that the relation does not apply for reaction systems containing monofunctional reactants and/or both A and B type of branch units.

{{main article| Erdős–Rényi model}}

Gelation of polymers can be described in the framework of the Erdős–Rényi model or the Lushnikov model, which answers the question when a giant component arises.{{Cite journal|last1=Buffet|first1=E.|last2=Pulé|first2=J. V.|date=1991-07-01|title=Polymers and random graphs|journal=Journal of Statistical Physics|language=en|volume=64|issue=1|pages=87–110|doi=10.1007/BF01057869|bibcode=1991JSP....64...87B|s2cid=120859837|issn=1572-9613}}

{{main article|Random graph theory of gelation}}

The structure of a gel network can be conceptualised as a random graph. This analogy is exploited to calculate the gel point and gel fraction for monomer precursors with arbitrary types of functional groups. Random graphs can be used to derive analytical expressions for simple polymerisation mechanisms, such as step-growth polymerisation, or alternatively, they can be combined with a system of rate equations that are integrated numerically.

• Mechanics of gelation

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Category:Gels

Category:Drug delivery devices

Category:Dosage forms

Category:Colloids

Category:Organic chemistry

Category:Physical chemistry

Category:Polymer chemistry