The rate of free-radical polymerization (FRP), including the RDRP, is represented by: (1) R p = k p [ M ] [ R • ] where k_p is the propagation rate constant, [M] is the monomer concentration, and [R^•] is the total active radical concentration.

For the calculation of the polymerization rate, Equation (1) is convenient to use. On the other hand, because Equation (1) does not involve concentrations of characteristic components shown in Figure 1, Equation (1) is not suitable for the prediction and control of the RDRPs, based on the reaction mechanism. Unique representation of R_p for RDRP, R_p = k_p[M]K[Interm]/[Trap] can be obtained as follows [13].

In order for the pseudo-living condition to be valid, the deactivation rate R_deact must be much larger than the bimolecular termination of active radicals R_t, i.e., R_deact ≫ R_t. If not, a large amount of dead polymer chains are formed. Similarly, if the initiation reaction R_I is involved, the activation reaction in RDRP R_act must be much larger than R_I, i.e., R_act ≫ R_I.

As long as the active period is short enough, the polymerization rate is given by the product of the radical generation rate (R_RG) and the average number of monomeric units added during a single active period (L_ν).

Equation (2) is usually valid also for the conventional nonliving FRP, where R_RG = R_I and L_ν is equal to the kinetic chain length, ν. In RDRP, the active period is much shorter than the conventional nonliving FRP because of the fast deactivation reaction, and the validity of Equation (2) is guaranteed.

For RDRPs, R_RG and L_ν are represented by: (3) R RG = R 1 + R act ≅ R act (4) L v = R p R t + R deact ≅ R p R deact

For instance, R_RG = k₁[PX] and L_ν = k_p[M]/(k₂[X]) in SRMP. In general, the polymerization rate of RDRP is given by: (5) R p = k p [ M ] K [ Interm ] [ Trap ]

The terms, K, [Interm], and [Trap] used in Equation (5) are summarized for SRMP, ATRP, and RAFT in Table 1. For ATRP, Equation (5) was already shown in [2].

Validity of Equation (5) to SRMP, ATRP, and RAFT was confirmed by using various kinetic parameters [13].

Equation (5) may appear that the termination reaction does not affect the polymerization rate, but it is not so. For example, in SRMP and ATRP, a single termination event leaves two additional trapping agents, leading to the increase of [Trap], resulting in a smaller polymerization rate. For TEMPO-mediated styrene polymerization, it is known that the active radical concentration is represented by [ R • ] = R I / R t [24,25]. This equation conforms to Equation (5), provided R_I is not too small. The quantitative discussion on this problem can be found in [26,27].

The polymerization rate of the ith polymerization loci, R_{p, i} is given by: (6) R p , i = R RG , i L v , i

The overall polymerization rate is given by: (7) R p = 〈 R RG , i L v , i 〉where 〈 〉 represents the average over all i's. (8) 〈 R RG , i L v , i 〉 = 1 V ∑ i = 1 N V i R RG , i L v , i

In Equation (8), N is the total number of reaction loci, and V represents the total volume of the reaction loci, i.e., V = ∑ i = 1 N V i.

For instance, considering SRMP, R_RG is the radical generation rate through the activation reaction, and R_RG,i = k₁[PX]_i. On the other hand, L_ν,i is the average number of monomeric units added during a single active period, and therefore, given by: (9) L v , i = k p [ M ] i t − R • ( i ) where t − R • ( i ) is the average active radical period in the ith polymerization locus (particle), which is given by: (10) t − R • ( i ) = 1 k 2 [ X ] Act , i

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 [Trap]_Act,i and [Trap]_i is important in miniemulsion SRMP and ATRP. In SRMP and ATRP, [Interm] ≫ [Trap], and therefore, the number of trapping agents in a particle could be very small, and in a small reaction locus, there may exist a long time period for which no active radicals exist. When the activation reaction occurs to generate a radical, one trapping agent is formed. This difference of one molecule of trapping agent could be important in a nano-sized particle.

In general, the polymerization rate in dispersed system is given by: (11) R p = k p 〈 k i [ M ] i [ Interm ] i [ Trap ] Act , i 〉

Note for SRMP and RAFT, K_i is simply a constant, and K does not have to be included inside the brackets, 〈 〉.

In the conventional theoretical treatment, the average concentrations are used. When the average concentrations are used without accounting for the statistical variation of concentrations: (12) R p , NoSV = k p 〈 K i 〉 〈 [ M ] i 〉 〈 [ Interm ] i 〉 〈 [ Trap ] Act , i 〉where the subscript, “NoSV” is used to designate the polymerization rate that is calculated without accounting for any statistical variation effect, by using the average concentrations.

Theoretically, the average concentration approximation applies to the following two cases: (1) Negligible statistical concentration variation among particles; (2) The numerator terms, K_i, [M]_i, and [Interm]_i have statistical variation among particles, but there are no correlations with other terms.

For example, suppose the statistical variation of [Interm]_i is significant but the variations of K_i, [M]_i, and [Trap]_i are negligible, the average method given by Equation (12) still works. On the other hand, if both [M]_i and [Interm]_i have correlation and statistical variation, Equation (12) cannot be used. Another case where Equation (12) is invalid is the cases where the statistical variation of [Trap]_i that exists in the denominator is significant. This is because of a simple mathematical principle, i.e., the average of the inverse is always larger than the inverse of the average [10,13,14].

Figure 3 shows comparison of Equations (11) and (12), with the Monte Carlo (MC) simulation result for a miniemulsion polymerization of SRMP. Uniform particles are assumed for the calculation. The set of parameters is named SRPM-1, and the parameters are: k₁ = 2 × 10⁻³ s⁻¹, k₂ = 1 × 10⁸ L mol ⁻¹ s⁻¹, k_p = 2 × 10³ L mol ⁻¹ s⁻¹, k_t = 1 × 10⁸ L mol ⁻¹ s⁻¹, [M]₀ = 8 mol L ⁻¹, [PX]₀ = 0.04 mol L ⁻¹, and [X]₀ = 0. This condition is the same as that for SFRP-1 used in [13], and more detailed kinetic behavior in bulk and miniemulsion polymerization can be found therein. Note that the termination rate constant, k_t is defined by R_t = k_t[R^•]², not R_t = 2k_t[R^•]². The MC calculation method used in this article is described in [9]. When applying Equations (11) and (12), [M]_i, [PX]_i, and [X]_i are determined by the MC simulation, and the monomer conversion to polymer, x is obtained by the integration of Equations (11) and (12).

As shown in Figure 3, Equation (11) that accounts for the statistical variation agrees completely with the MC simulation results. On the other hand, when Equation (12) that does not account for the statistical variation of concentrations is used, clear deviation is observed for D_p = 50 nm, while the prediction from Equation (12) could be a reasonable approximation for D_p = 100 nm and 30 nm. Figure 3 shows there exists a particle size region for an SRMP in which the statistical variation among particles is vital. This region corresponds to the acceleration window observed for SRMP and ATRP [10,13,14].

Figure 4 shows comparison of Equations (11) and (12), with the Monte Carlo (MC) simulation results for a miniemulsion polymerization of RAFT. The kinetic parameters used here are RAFT-11 shown in Table 2. The characteristics of parameters used are discussed in Section 3.2. Again, Equation (11) that accounts for the statistical variation of the concentrations of the components among particles agrees perfectly with the MC simulation results. The average concentration method represented by Equation (12) shows clear deviation for smaller particle sizes. For D_p = 30 nm, use of the average concentrations leads to overestimate the polymerization rate very significantly.

The characteristic particle diameter below which the effect of statistical variation becomes significant will be represented by D p ( sv ), and simple equations to determine the D p ( sv ) -values will be presented later in Section 4. For SRMP and ATRP, D p ( sv ) corresponds to D_p,Fluct in the earlier publications [10,13,14]. On the other hand, however, before discussing the statistical variation effect, consider a much simpler problem, that the concentration of a single molecule may become very high in a nano-sized reaction locus.

For the cases with [Trap] ≪ [Interm], the concentration of a single trapping agent in a small particle may become larger than in the corresponding bulk polymerization. Therefore, Equation (5) states that the polymerization rate may become smaller than in bulk polymerization.

SRMP and ATRP fall into this category. In the reversible reactions shown in Figure 1, when an active radical is generated by the activation reaction, one trapping agent always coexists. If the exit of this trapping agent is neglected, at least one trapping agent always exists during the active period. On the other hand, when this trapping agent exits from the particle, the polymerization is not controlled, and usual free-radical polymerization occurs. However, in the present discussion, I consider the cases where RDRP is well controlled.

The concentration of a single molecule in a particle with diameter D_p is given by 1 / { N A ( π / 6 ) D p 3 }, where N_A is the Avogadro constant. When this concentration is equal to the trapping agent concentration in the corresponding bulk polymerization, the polymerization rate would be equal to that in bulk polymerization, given all the other conditions are the same. The particle size below which the polymerization rate becomes smaller than in the corresponding bulk polymerization, D p , Trap ( 1 ) is given by: (13) D p , Trap ( 1 ) = ( 6 π N A [ Trap ] bulk ) 1 / 3 where [Trap]_bulk is the trapping agent concentration in the corresponding bulk polymerization, which can be determined in a straightforward manner by solving a set of differential equations describing the time development of various concentrations. Note, D p , Trap ( 1 ) corresponds to D_p,SMC used in earlier publications [10,13,14].

The lower limit of the acceleration window shown in Figure 2 was determined by using Equation (13). The equation to determine the upper limit of the acceleration window, D p , Trap ( sv ) will be introduced in Section 4.1.

Figure 5 shows the calculated development of D p , Trap ( 1 ) during polymerization under the condition SRMP-1, whose conversion development was shown in Figure 3. The D p , Trap ( 1 )-value decreases because [Trap]_bulk increases with time. In Figure 5, the D p , Trap ( sv ) -value whose equation will be shown in Section 4.1 is also shown. The acceleration window at conversion, x = 0.02 is predicted to be D_p = 43–92.6 nm, and that at x = 0.1 is D_p = 32.6–70.3 nm.

Figure 6 shows the miniemulsion polymerization rate of SRMP-1 at x = 0.02 and 0.1 for various particle diameter, D_p. The polymerization rate was calculated by the MC simulation as follows. The concentrations at x = 0.02 and 0.1 in each particle are first determined by the MC simulation, and then Equation (11) is used to obtain the polymerization rate, R_p. The lower limit of the acceleration window, D p , Trap ( 1 ) agrees perfectly. Because the concentration of a single trapping agent is in inverse proportion to D p 3 , R p ∝ D p 3 for D p < D p , Trap ( 1 ).

Excellent agreement of D p , Trap ( 1 ) with the lower limit of the acceleration window for other SRMP conditions can also be found in [10,13], and for ATRP in [13,14].

For the cases with [Interm] ≪ [Trap], the concentration of a single intermediate molecule in a small particle may become larger than in the corresponding bulk polymerization. Therefore, Equation (5) states that the polymerization rate may become larger than in bulk polymerization, given all other terms are essentially the same. Many RAFT polymerization systems may fall into this category.

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 n̅. In the case of RAFT polymerization, the active radical experiences deactivation-activation cycle shown in Figure 1 repeatedly, and therefore, the sum of the numbers of R^• and PXP would be, at least approximately, equal to n̅, i.e., n̅ = n̅_R^• + n̅_PXP.

The time fraction of the active period during the deactivation-activation cycle, ϕ_A is given by: (14) ϕ A = t ¯ R • t ¯ R • + t ¯ PXP = 1 / k 2 [ XP ] 1 / k 2 [ XP ] + 1 / k 1 = K K + [ XP ] where t̅_R• and t̅_PXP are the average time length of the active and inactive period, respectively.

Using the ϕ_A-value, the average number of the intermediate molecule in a particle, n̅_PXP is represented by t̅_PXP = n̅(1 – ϕ_A). Therefore, the effective intermediate concentration in a particle with diameter D_p is given by n ¯ ( 1 − ϕ A ) / { N A ( 1 / 6 ) D p 3 }. When this PXP concentration is larger than in the corresponding bulk polymerization, the polymerization rate is larger than in bulk polymerization. The particle size below which the polymerization rate becomes larger than in the corresponding bulk polymerization, D p , Interm ( 1 ) is given by: (15) D p , Interm ( 1 ) = ( 6 n ¯ ( 1 − ϕ A ) π N A [ Interm ] bulk ) 1 / 3 where [Interm]_bulk is the trapping agent concentration in the corresponding bulk polymerization, i.e., [PXP]_bulk, which can be determined easily by solving a set of differential equations describing the time development of various concentrations. D p , Interm ( 1 ) corresponds to D_p,ZIM used in earlier publications [13,21].

Strictly, the particle size dependency of n̅ needs to be accounted for. However, because the drastic increase in the polymerization rate by decreasing the particle size is observed, essentially for the zero-one condition in the conventional nonliving FRP where the time fraction in which more than one radical exists in a particle can be neglected. To roughly estimate the D p , Interm ( 1 )-value, one can use the n̅-value of the zero-one condition for the corresponding nonliving FRP. For example, if the exit of a radical from the particle can be neglected, n̅ = 0.5 can be used. Because the acceleration normally starts to be observed slightly before the system reaches the zero-one condition, use of n̅-value of the zero-one condition underestimate the D p , Interm ( 1 )-value. However, considering the experimental errors, this estimate would provide a useful pointer for the design of RAFT miniemulsion polymerization processes.

The parameters shown in Table 2 in Section 2.2 are used in the present RAFT investigation. RAFT-11 is a typical example of the intermediate termination (IT) model that involves cross-termination between a propagating R^• and an intermediate adduct radical PXP, whose rate constant is represented by k_ct. For RAFT-12 and -13, k₁ is made smaller, and the average intermediate time t̅_PXP becomes larger. A larger frequency of cross-termination is expected for RAFT-12 and -13. On the other hand, RAFT-01 is a typical example of the slow fragmentation (SF) model, and it takes as large as t̅_PXP = 1/k₁ = 2 s for a PXP molecule to generate a radical. These parameters are used also in the earlier articles [13,21], and detailed kinetic behavior of these reaction conditions can be found therein. As shown in Figure 2 of [22], the model discrimination of RAFT-11 and RAFT-01 based on the bulk polymerization data is difficult.

Figure 7 shows the calculated D p , PXP ( 1 )-values for RAFT-11, -12, and -13. The D p , PXP ( 1 )-value reaches a nearly constant value in a very short period of time, because [PXP]_bulk reaches an equilibrium rapidly.

Figure 8 shows the conversion development of the RAFT miniemulsion polymerization with various particle sizes for RAFT-11, -12, and -13. Significant acceleration is observed for D p < D p , PXP ( 1 ). Note that the polymerization rate of D_p = 25 nm in RAFT-11 is smaller than that for D_p = 50 nm. This is due to the statistical variation effect, which will be discussed in detail in Section 4.2.

Figure 9 shows the calculated D p , PXP ( 1 ) development of RAFT-01, which is a typical example of the SF model without the cross-termination. In this case, k₁ is very small and t̅_PXP is large enough, therefore, PXP molecules are accumulated during polymerization. [PXP]_bulk is not small enough, and therefore, D p , PXP ( 1 )-value is too small for the real miniemulsion polymerization.

Figure 10 shows the conversion development of the RAFT miniemulsion polymerization with various particle sizes for RAFT-01. The polymerization rate is essentially unchanged down to D_p = 20 nm. At D_p = 10 nm on the other hand, the polymerization rate slows down rather than the acceleration. As will be discussed in Section 4.2, this deceleration is due to the statistical variation effect.

Equation (15) was developed on the basis of the intermediate concentration, using Equation (5) as the fundamental rate expression. On the other hand, if one focuses the attention to the active radical concentration, by using Equation (1) as the fundamental rate expression, one obtains the following representation of the threshold diameter, which is equivalent to Equation (15).

The derivation of Equation (16) is shown in the appendix of [21]. Equation (16) is useful especially for DTRP and conventional nonliving FRP, where the intermediate molecule is not involved.

As discussed in Section 2.2, the statistical variation of the component molecules in a particle must be accounted for, and Equation (11) needs to be used. The use of the average concentrations, i.e. Equation (12) may underestimate the polymerization rate, as shown in Figure 3(b), or overestimate as in the cases of Figure 4(b,c).

For the cases with [Trap] ≪ [Interm], the statistical variation of the trapping agent concentration in a particle may not be neglected. Because [Trap] resides in the denominator in the rate expression shown by Equation (11), the use of the average concentrations leads to underestimate the polymerization rate. This is due to a simple mathematical relationship, 1/〈[Trap]_Act,i〉 ≤ 〈1/[Trap]_Act,i〉. For the cases with [Interm]≪[Trap], the statistical variation of the intermediate concentration in a particle may not be neglected. On the other hand, however, if [Interm] is the only term to fluctuate, the average concentration still applies because [Interm] resides in the numerator of Equation (11). On the other hand, if the fluctuation of [Interm] correlates, for example, with [M], the statistical variation must be accounted for.

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 Table 4. These diameters are obtained on the basis of two important features of polymerization carried out in nano-sized reaction loci, (1) high single-molecule concentration effect, and (2) large statistical variation effect. These values can be calculated in a straightforward manner, and provide vital information for the design of miniemulsion polymerization processes.

For the cases with [Trap] ≪ [Interm], such as for SRMP and ATRP, there may be an acceleration window in which the polymerization rate is larger than that of the corresponding bulk polymerization. The lower and upper limits of the particle sizes are represented by D p , Trap ( 1 ) and D p , Trap ( sv ) respectively. For D p < D p , Trap ( 1 ) the polymerization rate decreases significantly with D p 3 by making D_p smaller, provided that the trapping agents do not exit from the particle. The n ¯ Trap ( sv )-value required for the calculation of D p , Trap ( sv ) is a function of k₂/k_t, and a smaller k₂/k_t-value leads to a larger n ¯ Trap ( sv )-value. As a rough indicator, n ¯ Trap ( sv ) -value is about 10 for k₂/k_t > 0.5, about 20 for k₂/k_t around 0.05, and larger than 30 for k₂/k_t < 0.01. Figure 12 can be used to roughly determine the n ¯ Trap ( sv ) -value. Note, however, from the theoretical point of view, the acceleration discussed here is with respect to the polymerization rate calculated by using the average concentrations, R_p,NoSV, and therefore, acceleration compared with the bulk polymerization is not always observed.

For the cases with [Interm] ≪ [Trap], such as for many RAFT systems, the significant rate increase by decreasing the particle size, as in the cases of conventional FRP, occurs for D p < D p , Interm ( 1 ) or equivalently, D p < D p , R • ( 1 ) For D p < D p , M ( sv ) the statistical variation effect becomes significant, and the polymerization rate becomes smaller than R _p,NoSV, and the average concentration cannot be used. For the cases with D p , Interm ( 1 ) < D p , M ( s v ) such as for the representative SF model, the rate increase by decreasing the particle size does not occur at all.

The conventional nonliving FRP and DTRP can be considered as a special case of RAFT with ϕ_A = 1, and the values of D p , R • ( 1 ) and D p , M ( sv ) for these systems can be determined by applying ϕ_A = 1 to the equations for RAFT.