Cage effect

In chemistry, the cage effect^[1] (also known as geminate recombination^[2]) describes how the properties of a molecule are affected by its surroundings. First introduced by James Franck and Eugene Rabinowitch^[3][4] in 1934, the cage effect suggests that instead of acting as an individual particle, molecules in solvent are more accurately described as an encapsulated particle. The encapsulated molecules or radicals are called cage pairs or geminate pairs.^[5][6] In order to interact with other molecules, the caged particle must diffuse from its solvent cage. The typical lifetime of a solvent cage is 10^-11 seconds.^[7] Many manifestations of the cage effect exist.^[8]

In free radical polymerization, radicals formed from the decomposition of an initiator molecule are surrounded by a cage consisting of solvent and/or monomer molecules.^[6] Within the cage, the free radicals undergo many collisions leading to their recombination or mutual deactivation.^[5][6][9] This can be described by the following reaction:

${\displaystyle R\!-\!R\;\;{\underset {k_{c}}{\overset {k_{1}}{\rightleftharpoons }}}\;\;{\underset {\text{cage pair}}{(R^{\,\bullet },^{\bullet }\!R)}}\;\;{\underset {k_{D}}{\overset {k_{d}}{\rightleftharpoons }}}\;\;{\underset {\text{free radicals}}{2R^{\,\bullet }}}\;\rightarrow \;{\text{Products}}}$^[9]

After recombination, free radicals can either react with monomer molecules within the cage walls or diffuse out of the cage. In polymers, the probability of a free radical pair to escape recombination in the cage is 0.1 – 0.01 and 0.3-0.8 in liquids.^[5] In unimolecular chemistry, geminate recombination has first been studied in the solution phase using iodine molecules^[10] and heme proteins.^[11][12] In the solid state, geminate recombination has been demonstrated with small molecules trapped in noble gas solid matrices^[13] and in triiodide crystalline compounds.^[14][15][16]

The cage effect can be quantitatively described as the cage recombination efficiency F_c where:

${\displaystyle F_{c}=k_{c}/(k_{c}+k_{d})}$^[9]

Here F_c is defined as the ratio of the rate constant for cage recombination (k_c) to the sum of the rate constants for all cage processes.^[9] According to mathematical models, F_c is dependent on changes on several parameters including radical size, shape, and solvent viscosity.^[9][17][18] It is reported that the cage effect will increase with an increase in radical size and a decrease in radical mass.

In free radical polymerization, the rate of initiation is dependent on how effective the initiator is.^[6] Low initiator efficiency, ƒ, is largely attributed to the cage effect. The rate of initiation is described as:

${\displaystyle R_{i}=2fk_{d}[I]}$ ^[6]

where R_i is the rate of initiation, k_d is the rate constant for initiator dissociation, [I] is the initial concentration of initiator. Initiator efficiency represents the fraction of primary radicals R·, that actually contribute to chain initiation. Due to the cage effect, free radicals can undergo mutual deactivation which produces stable products instead of initiating propagation – reducing the value of ƒ.^[6]

• Solvent effects
• Carrier generation and recombination
• Rate-determining step