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Various types of controlled/living radical polymerizations, or using the IUPAC recommended term, reversibledeactivation radical polymerization (RDRP), conducted inside nanosized reaction loci are considered in a unified manner, based on the polymerization rate expression,
With the advent of reversibledeactivation radical polymerization (RDRP), the characteristics of living polymerization can be introduced to radical polymerization, creating novel possibilities to produce welldefined polymers, such as narrow distributed, endfunctionalized, block, star, and dendritic polymers. A significant number of papers are being published these days for various types of RDRPs. The RDRPs include stableradicalmediated polymerization (SRMP) such as nitroxidemediated polymerization, atomtransfer radical polymerization (ATRP), reversibleadditionfragmentation chaintransfer polymerization (RAFT), and degenerativetransfer radical polymerization (DTRP). The experimental and theoretical investigations conducted for these types of RDRPs in bulk [
The reversible deactivation reactions for various types of RDRPs are shown in
When the conventional nonliving freeradical polymerization is conducted in a dispersed system, typically for
In RDRPs, there are important characteristic particle diameters in miniemulsion polymerization below or above which the polymerization rate changes with the particle size. For SRMP and ATRP, theoretical calculation results have shown that a particle size region may exist in which the polymerization rate is larger than bulk polymerization [
In RAFT, it is known that the rate retardation occurs by increasing the RAFT concentration. Monteiro and Brouwer [
On the other hand, these two types of models can be discriminated in a straightforward manner by using the miniemulsion polymerization. The IT model leads to show that the polymerization rate increases significantly for
The threshold diameter below which the polymerization rate increases significantly by reducing the particle size
Another characteristic particle diameter
In this article, all of the above characteristic particle diameters for various types of RDRPs are represented by simple equations in a unified manner, based on the characteristic polymerization rate expression for the RDRPs,
The rate of freeradical polymerization (FRP), including the RDRP, is represented by:
For the calculation of the polymerization rate,
In order for the pseudoliving condition to be valid, the deactivation rate
As long as the active period is short enough, the polymerization rate is given by the product of the radical generation rate (
For RDRPs,
For instance,
The terms,
Validity of
The polymerization rate of the
The overall polymerization rate is given by:
In
For instance, considering SRMP,
Note that the chains are formed only during the active period, and therefore, the trapping agent concentration must be that during the active period. The difference of [
In general, the polymerization rate in dispersed system is given by:
Note for SRMP and RAFT,
In the conventional theoretical treatment, the average concentrations are used. When the average concentrations are used without accounting for the statistical variation of concentrations:
Theoretically, the average concentration approximation applies to the following two cases: (1) Negligible statistical concentration variation among particles; (2) The numerator terms,
For example, suppose the statistical variation of [
As shown in
The characteristic particle diameter below which the effect of statistical variation becomes significant will be represented by
Basically, even when a reaction medium is separated into small reaction loci, the concentration does not change. However, this statement requires a premise that each reaction locus contains a large number of molecules. When the number of molecules of a given component in a reaction locus is small, the statistical variation becomes significant, and the concentration is not the same for all reaction loci. When the volume of reaction locus is further decreased, some reaction loci contain only a single molecule of a component, while the other loci do not contain the component at all. The reaction loci that contain a single molecule may show unusually high concentration compared with the bulk system.
This is the reason for showing extremely high polymerization rate in the zeroone kinetics [
Conventionally, the rate increase in emulsion polymerization is explained by the segregation of radicals,
Now, look at the fundamental polymerization rate expression,
For the cases with [
SRMP and ATRP fall into this category. In the reversible reactions shown in
The concentration of a single molecule in a particle with diameter
The lower limit of the acceleration window shown in
Excellent agreement of
For the cases with [
Suppose that the average number of radicals in a particle for the conventional nonliving FRP using the same monomer but without using RAFT agent is given by
The time fraction of the active period during the deactivationactivation cycle,
Using the
Strictly, the particle size dependency of
The parameters shown in
The derivation of
As discussed in Section 2.2, the statistical variation of the component molecules in a particle must be accounted for, and
For the cases with [
In usual SRMP and ATRP, the relationship [
As was shown in
The degree of rate increase due to the fluctuation in [
Assuming about 10% increase in polymerization is considered significant (
Note the acceleration discussed in this section is in comparison with the polymerization rate without accounting for the statistical variation, represented by
On the other hand, with [
In order to develop a simple equation that can be used as a pointer to roughly determine the diameter below which the acceleration due to the statistical variation becomes significant
The upper curve shown in
In many RAFT systems, [
Specifically for RAFT,
Because [XP] is large, the difference of a single molecule concentration in XP is negligible, and
With the MCV effect, the polymerization rate becomes smaller than that estimated by using the average concentrations,
In the present discussion, however, it is worth noting that the transfer of monomer molecules among particles is neglected. Especially for particles with smaller
The question to be answered in this section is that how to determine the diameter,
The average time interval between radical entry for the condition shown in
The MCV effect is expected to be significant when the conversion increase in a particle during the time period of
The characteristic time for the conversion development,
On the other hand, the average time between radical entry,
Therefore, the characteristic particle diameter below which the MCV effect is significant,
To determine the
The miniemulsion polymerization rate is determined from the MC simulation data, using the following equation:
On the other hand, if the statistical variation effect is neglected, the polymerization rate is given by:
From
Because
Finally, let us consider the case with the slow fragmentation (SF) model, RAFT01 in
As shown in
The condition, [XP]_{0} = 0.04 mol L^{−1} leads to obtain the initial number of RAFT agent in a 10 nm particle being
Important threshold diameters below which the polymerization rate starts to deviate significantly (1) from the bulk polymerization, and (2) from the estimate using the average concentrations, for various types of RDRPs, are determined, as summarized in
For the cases with [
For the cases with [
The conventional nonliving FRP and DTRP can be considered as a special case of RAFT with
Reversible deactivation reaction scheme in each type of reversibledeactivation radical polymerization (RDRP). In the figure, P
Calculated polymerization rate at 10% conversion for a model stableradicalmediated polymerization (SRMP) miniemulsion polymerization with various particle diameter,
Calculated conversion development for a miniemulsion SRMP, whose parameters are described in the text, and are the same as SFRP1 used in [
Calculated conversion development for a miniemulsion RAFT polymerization, whose parameters are shown in
Calculated development of the acceleration window for the miniemulsion polymerization of SRMP1, whose parameters are the same as
Calculated polymerization rate at 2 and 10% conversion for the miniemulsion polymerization of SRMP1 with various particle diameter,
Calculated time development of the
Conversion development of (
Calculated time development of the
Conversion development of RAFT01, with various particle sizes.
Calculated conversion development of a miniemulsion polymerization with
Estimated ratio of polymerization rates, with and without statistical variation of trapping agents among particles, as a function of the average number of trapping agents in a polymer particle. The model was proposed in [
Conversion development in each particle for RAFT11 with
Conversion development of the conventional nonliving freeradical polymerization (FRP), whose kinetic parameters are the same as RAFT11, except that the RAFT agent is not used here.
Ratio between the true miniemulsion rate
Conversion development of the conventional FRP with various particle sizes. The numbers in the figure shows the particle diameter
Calculated
Ratio between the true miniemulsion rate
Conversion development of RAFT11 with various particle sizes. The numbers in the figure show the particle diameter
Conversion development of RAFT12 with various particle sizes. Noticeable difference is observed for
Calculated time development of
Conversion development for miniemulsion RAFT01 polymerization with
Explicit representation of
SRMP  [PX]  [X]  
ATRP  [PX]  [XY]  
RAFT  [PXP]  [XP] 
Parameters used in the present RAFT calculations (
RAFT11  1 × 10^{4}  1 × 10^{6}  1 × 10^{7}  Typical IT model  
RAFT12  1 × 10^{2}  1 × 10^{6}  1 × 10^{7} 

Lager intermediate time, 
RAFT13  1  1 × 10^{6}  1 × 10^{7}  
RAFT01  0.5  1 × 10^{6}  0  Typical SF model 
Concentration of a single molecule in a particle.
150  9.43 × 10^{−7} 
100  3.18 × 10^{−6} 
75  7.55 × 10^{−6} 
50  2.55 × 10^{−5} 
30  1.18 × 10^{−4} 
25  2.04 × 10^{−4} 
Obtained threshold particle diameters for various RDRPs and conventional FRP.
[ 


[ 


Conventional FRP, DTRP 


This work is supported by a grantinaid for Scientific Research, the Ministry of Education, Culture, Sports, Science, and Technology, Japan (grantinaid 21560890).