# Effect of Chain Transfer to Polymer in Conventional and Living Emulsion Polymerization Process

## Abstract

**:**

^{2}>

_{0}. The conventional FRP shows a very broad molecular weight distribution (MWD), with the high molecular weight region conforming to the power law distribution. The MWD is much broader than the random branched polymers, having the same primary chain length distribution. The expected <s

^{2}>

_{0}for a given MW is much smaller than the random branched polymers. On the other hand, the living FRP shows a much narrower MWD compared with the corresponding random branched polymers. Interestingly, the expected <s

^{2}>

_{0}for a given MW is essentially the same as that for the random branched polymers. Emulsion polymerization process affects branched polymer architecture quite differently for the conventional and living FRP.

## 1. Introduction

## 2. Simulation Method

#### 2.1. Conventional Free-Radical Emulsion Polymerization

_{pr}(r) is given by [20]:

_{pr}(r) = (τ + C

_{P}) exp[−(τ + C

_{P})r],

_{P}are dimensionless numbers defined by:

_{fm}+ C

_{fCTA}[CTA]

_{p}/[M]

_{p}+ 1/(k

_{p}[M]

_{p}t

_{av}),

_{P}= C

_{fp}[P]

_{p}/[M]

_{p},

_{fm}, C

_{fCTA}, C

_{fp}are the chain transfer constants to the monomer, to the chain transfer agent (CTA), and to the polymer, respectively. [CTA]

_{p}and [M]

_{p}are the concentration of CTA and monomer in the polymer particle. [P]

_{p}is the polymer concentration, represented by the total number of monomeric units incorporated into polymer chains, and the ratio, [P]

_{p}/[M]

_{p}, is kept constant until the depletion of monomer droplets. In the final term of Equation (2), the zero-one kinetics is assumed, and t

_{av}represents the average time interval between radical entry to a particle. The propagation rate constant is represented by k

_{p}.

_{P}are assumed to be constant in the present MC simulation. Note that the number-average chain length of primary polymer radicals, $\overline{r}$

_{n,p}is given by:

_{b}is given by the following ratio:

_{b}= C

_{P}/(τ + C

_{P}),

_{b}is called the “branching probability”. Note that in the present simulation condition, the value of P

_{b}is constant throughout the simulated polymerization period, and the branching probability is the same for all primary chains for the conventional FRP. Note, however, because the primary chains formed earlier are subjected to branching reaction for a longer period of time, the expected branching density, i.e., the number of branch points on the primary chain divided by the chain length, is dependent on the birth time of the primary chain [5].

_{P})],

_{M}. By repeating such simulation for a large number of polymer particles to generate a significant number of polymer molecules, one can determine the statistical properties of the product polymers effectively.

#### 2.2. Living Free-Radical Emulsion Polymerization

_{deact}is given by:

_{deact}= k

_{2}[Trap][R

^{•}].

_{p}= k

_{p}[M][R

^{•}], one obtains the following dimensionless ratio, δ:

_{2}[Trap])/(k

_{p}[M]).

_{p}is approximately constant until the depletion of monomer droplets. The value of [Trap]

_{p}may change even for a constant polymer/monomer ratio period in emulsion polymerization. However, in order to simplify the discussion, the value of δ is assumed constant during the polymerization in the present MC simulation.

_{p}).

_{P}, defined by Equation (3), is constant, and therefore p is also a constant.

_{sa}(r) is given by the following modified most probable distribution [23]:

_{sa}(r) = p

^{r}(1 − p),

_{n,s}is given by:

_{bs}is given by:

_{bs}= C

_{P}/(δ + C

_{p}).

_{n}and the total number of monomeric units bound into polymer molecules in a particle, N

_{M}are set to be the same for both cases. The number of dormant species in a particle, which is equal to the number of product polymers in a particle, is given by N

_{M}/$\overline{r}$

_{n}. In the simulation for conventional FRP, a single linear polymer molecule having chain length ca. $\overline{r}$

_{n,p}exists initially. In order to have an equivalent initial condition for living FRP, the total of $\overline{r}$

_{n,p}/$\overline{r}$

_{n,s}dormant chains are generated. The chain lengths of these dormant chains are determined by generating random numbers that follow the distribution given by Equation (10), which can be done by using a uniform random number y between 0 and 1, as follows:

_{n,p}/$\overline{r}$

_{n,s}dormant chains whose lengths are generated by using Equation (13), and (N

_{M}/$\overline{r}$

_{n}− $\overline{r}$

_{n,p}/$\overline{r}$

_{n,s}) dormant species having length 0 in a polymer particle. Note in the present simulation study shown in the next section, N

_{M}/$\overline{r}$

_{n}> $\overline{r}$

_{n,p}/$\overline{r}$

_{n,s}for all conditions. After that, one chain end having a trapping agent is selected randomly to be activated, and a new segment chain, generated by using Equation (13), is connected to the activated chain end. Then, with probability (1 − P

_{bs}), the eliminated trapping agent is attached to the segment end. On the other hand, with probability P

_{bs}, chain transfer to polymer occurs, and the location of a midchain radical is determined by selecting one unit randomly from the already formed chains. In this case, next segment grows from the midchain radical to form a branch chain and the eliminated trapping agent moves to be attached to this segment end. These processes continue until the total number of polymerized monomeric units leaches N

_{M}, and the simulation for a single particle ends. By repeating such simulation for a large number of polymer particles, one can determine the statistical properties of the product living polymers effectively.

## 3. Results and Discussion

_{M}that defines the end point for the MC simulation is set to be N

_{M}= 1 × 10

^{6}both for the conventional and living FRP. Assuming that the molecular weight of monomer is 100 and that the density of polymer is 1 g/cm

^{3}, the value of N

_{M}= 1 × 10

^{6}corresponds to the diameter of the dried polymer particle being 68 nm. It is assumed that the polymer/monomer ratio in the polymer particle is kept constant until the end point of simulation, N

_{M}= 1 × 10

^{6}.

_{n}is 200 for C1 and L1, $\overline{r}$

_{n}= 500 for C2 and L2, and $\overline{r}$

_{n}= 1000 for C3, C4, L3, L4. Note that the number-average chain length, $\overline{r}$

_{n}is given by the following equation for the conventional FRP:

_{p}/[M]

_{p}is kept constant, the average branching density of the all polymers, $\overline{\rho}$ is given by the following equation for both conventional and living FRP.

_{b}, is not the same for all primary chains. However, the average branching probability, $\overline{P}$

_{b}is set to be the same for the corresponding conventional and living FRP, and $\overline{P}$

_{b}is increased from 0.2857 for C1 and L1 to 0.8333 for C4 and L4. In the case of living FRP, the number-average segment length during a single active period is set to be $\overline{r}$

_{n,s}= 2 throughout the polymerization.

#### 3.1. Weight Fraction Distribution

#### 3.1.1. Conventional Free-Radical Emulsion Polymerization

_{b}is large, a sharp high MW peak appears. In the present condition, the total number of monomeric units incorporated into polymer molecules is, N

_{M}= 1 × 10

^{6}, and therefore, a polymer molecule with log

_{10}r > 6 cannot exist in a polymer particle. The sharp high MW peak is formed due to the limitation of the small particle size. A sharp high MW peak essentially consists of the largest polymer molecule in each polymer particle [17].

_{w}is given by [16,17]:

_{n}, rather than the weight-average $\overline{r}$

_{w}.

_{n,p}and $\overline{r}$

_{w,p}, respectively. When such primary chains are connected through random branching, the number- and weight-average chain length are given, respectively, by [25]:

_{n}is the same for both the emulsion polymers and the random branched polymers, because the average branching density is the same. On the other hand, in the case of the emulsion polymers, the expected branching density of the primary chains formed in the earlier stage of polymerization is much larger than those formed in the later stage of polymerization [5]. The primary chains with larger values of branching density form hubs in the buildup process to connect a large number of primary chains in a polymer molecule, which allows the formation of large sized polymers. It is clearly shown that the weight-average chain length $\overline{r}$

_{w}is larger for the emulsion polymers, especially for the large branching probability cases.

_{1}is the modified Bessel function of the first kind and of the first order.

_{b}+ 1), as shown in the figure. With the relationship, W(r) ~ rN(r), the power exponent of the weight fraction distribution, W(r) is −1/P

_{b}, i.e., W(r) ~ r

^{−α}with α = 1/P

_{b}. Because the branching probability, P

_{b}is related with the chain transfer constant C

_{fp}through Equations (3) and (5), the value of C

_{fp}could be estimated through the present type of double logarithmic plot, as was illustrated for the emulsion-polymerized polyethylene [18].

#### 3.1.2. Living Free-Radical Emulsion Polymerization

_{n,p}and the average branching density $\overline{\rho}$, as well as the number-average chain lengths of the product polymers $\overline{r}$

_{n}, are the same for the corresponding polymerization conditions, i.e., C1 and L1, C2 and L2, and so on. Bimodal distributions of W(log

_{10}r) are shown for L1 and L2. On the other hand, in the present idealized model, a uniform particle size distribution is assumed. In a real system, the MWD is expected to be somewhat broader than the present prediction, and the second peak may be difficult to be observed. In addition, the second peak contains branches and the hydrodynamic volume becomes smaller than the corresponding linear polymers. The peak separation could be difficult in the usual GPC analysis.

_{n,s}= 2 in the present set of simulations.

_{P}, represented by Equation (3), which is kept constant throughout polymerization. When the ideal living chains whose MWD is represented by Equation (21) is subjected to a constant probability of chain transfer, the weight fraction distribution is represented by [23]:

_{n,p}for L1 to L4 are given respectively in Table 3. The weight-average chain lengths of the primary chains, $\overline{r}$

_{w,p}can be calculated from Equation (23). On the other hand, if these primary chains are combined to form random branched polymers, the number- and weight-average chain lengths of product polymers can be calculated from Equations (18) and (19). Note that Equations (18) and (19) are valid, irrespective of the primary chain length distribution, as long as the branching is random, i.e., the probability of having a branch point is the same for all units.

_{n}is the same for both types of branched polymers, because the average branching density is the same. On the other hand, in the case of living FRP, the branch chains must be formed after the formation of the backbone chain. The branch chain is allowed to grow a shorter period of time than the backbone chain, and is expected to be shorter than the backbone chain. In addition, the branch chains formed later are subjected to branching reaction for a shorter period of time, and therefore, the expected branching density is smaller. On the other hand, in the random branched polymers, the expected branching density is the same for all primary chains and any primary chain can become a branch chain. A long primary chain having large branching density could be connected as a branch chain in the random branching process. Larger polymer molecules can be formed in the random branching process, compared with the living emulsion polymerization, leading to a larger weight-average chain length $\overline{r}$

_{w}. It is clearly shown that the living emulsion polymerization gives smaller PDI values, indicating narrower MWD.

#### 3.2. Branching Density

#### 3.2.1. Conventional Free-Radical Emulsion Polymerization

_{n,p}, which seems to be a reasonable estimate of ${\rho}_{r\to \infty}$, at least for large branching probability cases, C2–C4.

_{n,p}. For all 4 cases, including C1, the curves are reduced to fall approximately on the same curve. The number-average chain length of primary chains, $\overline{r}$

_{n,p}is kept constant throughout the polymerization in the present model emulsion polymerization, and therefore, the limiting value is given by:

_{2}is a modified Bessel function of the first kind of the second order, and P

_{b}is the branching probability. For random branching, P

_{b}= ρ$\overline{r}$

_{n,p}.

_{n,p}and ζ, in the random branched polymer systems for a given branching probability, P

_{b}. On the other hand, the value of P

_{b}does not play a role in emulsion polymers, as shown in Figure 9.

_{b}must be smaller than unity, i.e., P

_{b}< 1, the limiting branching density, ${\rho}_{r\to \infty}$ is always larger for the emulsion polymerization, compared with the corresponding random branched polymers. The lines in Figure 10 show the relationship between $\overline{\rho}$(ζ)$\overline{r}$

_{n,p}and ζ for the random branching, and that for the emulsion polymerization, in particular the data for C4. Although the emulsion polymerization shows slightly slower increase, but the limiting value, ${\rho}_{r\to \infty}$ agrees with the case with P

_{b}= 1 in the random branched polymer system.

#### 3.2.2. Living Free-Radical Emulsion Polymerization

_{b}-value is largest with $\overline{P}$

_{b}= 0.833 as listed in Table 3, the curve of $\overline{\rho}$(r) shows an overshooting behavior that was not observed in the conventional FRPs.

_{b}is small for these conditions, and it is difficult to make discussions on the limiting branching density, ${\rho}_{r\to \infty}$. For L3, the limiting value seems to be almost the same, but the emulsion polymers show a larger branching density for the transient region of chain lengths. For L4, the branching density of the emulsion polymers is larger for smaller polymers, while it becomes smaller than the random branch for large chain length region. The behavior is quite complicated.

#### 3.3. Radius of Gyration

^{2}>

_{0}can be determined from the Wiener index (WI) [26]. The relationship between <s

^{2}>

_{0}and WI is given by: [27]

^{2}>

_{0}/L

^{2}= WI/N

^{2},

^{2}>

_{0}in each polymer is large. For larger polymers with N > 1000, the actual random walk process in the 3D space was conducted for 100 times to estimate the value of <s

^{2}>

_{0}.

#### 3.3.1. Conventional Free-Radical Emulsion Polymerization

^{2}>

_{0}/L

^{2}and chain length r divided by u. In the figure, each dot represents the value set of <s

^{2}>

_{0}/L

^{2}and r/u of the individual polymer molecule obtained in the MC simulation, and the blue solid curve with circular symbols shows the average value of <s

^{2}>

_{0}/L

^{2}within the intervals of Δr, which is the estimate of the average mean-square radius of gyration for the polymers having chain length r.

^{2}>

_{0}/L

^{2}for the random branched polymers whose primary polymer chain length distribution is the same as the present model emulsion polymers. For the random branched polymers whose primary chain length distribution conforms to the most probable distribution is given by [28]:

_{r}is the average number of branch points for the polymers with chain length r, which is given by:

_{b}= 0.2857), and is small for C4 (P

_{b}= 0.8333). In all cases, the expected mean-square radius of gyration is smaller for the emulsion polymers, which shows the non-randomness in the distribution of branch points formed in emulsion polymerization leads to compact 3D architecture in the conventional FRP.

#### 3.3.2. Living Free-Radical Emulsion Polymerization

^{2}>

_{0}of polymers having chain length r.

^{2}>

_{0}/L

^{2}of the random branched polymers whose primary polymer chain length distribution, as well as the average branching density, is the same as the present model emulsion polymers. The mean-square radius of gyration of the random branched polymers was estimated by the MC simulation, in which the primary chains whose distribution is shown in Figure 6 are connected randomly. Interestingly, for living emulsion polymerization, the expected mean-square radius of gyration for the polymers with chain length r is essentially the same as for the random branched polymers.

## 4. Conclusions

^{−α}with the power exponent α given by α = 1/P

_{b}, where P

_{b}is the probability that the chain end of the primary polymer molecule is connected to a backbone chain. The MWD is much broader than the corresponding random branched polymer system. When the branching probability P

_{b}is large, the second sharp high MW peak appears because of the limitation of the particle size.

_{n,p}is found. This limiting value is $1/\sqrt{{P}_{\mathrm{b}}}$−times larger for the emulsion polymerization, compared with the corresponding random branched polymers.

^{2}>

_{0}for the given chain length r is smaller than the corresponding random branched polymers. On the other hand, for the living emulsion FRP, the value of <s

^{2}>

_{0}for the given chain length r is essentially the same as for the random branched polymers.

## Conflicts of Interest

## Nomenclature

Symbols | |

[CTA]_{p} | Concentration of the chain transfer agent in the polymer particle |

C_{fCTA} | Chain transfer constant to the chain transfer agent |

C_{fm} | Chain transfer constant to monomer |

C_{fm} | Chain transfer constant to polymer |

C_{P} | Ratio of the reaction rates between chain transfer to polymer and propagation |

F[a,b;x] | Confluent hypergeometric function (Kummer’s function of the first kind) |

I_{1} | Modified Bessel function of the first kind and of the first order |

I_{2} | Modified Bessel function of the first kind and of the second order |

k | Number of branch points in the polymer molecule |

k_{p} | Propagation rate constant |

L | Random-walk segment length |

[M]_{p} | Monomer concentration in the polymer particle |

$\overline{m}$_{r} | Average number of branch points for the polymers with chain length r |

N | Number of random-walk segments in the polymer |

n | Number of primary chains in the particle |

N_{pr}(r) | Number-based chain length distribution of the primary chains |

N_{sa}(r) | Number-based chain length distribution during a single active period in living FRP |

N(r) | Number fraction distribution |

p | Probability that an active radical connects another monomer unit during the active period in living FRP |

P_{b} | Branching probability, which gives the probability that the chain end of a primary chain is connected to a backbone chain |

P_{bs} | Probability that the chain end of a segment formed during a single active period is connected to a backbone chain in living FRP |

[P]_{p} | Polymer concentration in the polymer particle, represented by the total number of monomeric units in polymer |

R_{p} | Polymerization rate |

R_{deact} | Deactivation rate in the reversible deactivation process in controlled/living radical polymerization |

r | Chain length (number of monomeric units in the polymer) |

$\overline{r}$_{n} | Number-average chain length of the product polymers |

$\overline{r}$_{n,p} | Number-average chain length of the primary chains |

${\overline{r}}_{\mathrm{n},\mathrm{p}}^{\prime}$ | Number-average chain length of the primary chains that constitute a branched polymer molecule |

$\overline{r}$_{n,s} | Number-average chain length of the sequence of chains (segments) formed during a single active period in living FRP |

$\overline{r}$_{w} | Weight-average chain length of the product polymers |

<s^{2}>_{0} | Mean-square radius of gyration of unperturbed chain |

t_{av} | Average time interval between radical entry to a polymer particle |

u | Number of monomeric units in a random-walk segment |

WI | Wiener Index |

W(r) | Weight fraction distribution |

W_{liv}(r) | Weight fraction distribution of the ideal linear living FRP |

W_{CP}(r) | Weight fraction distribution of the polymer chains formed through the ideal linear living FRP with a constant chain transfer probability |

y | Uniform random number between 0 and 1 |

z | Average number of active period for the linear living chain formation |

α | Exponent of the power-law distribution |

δ | Ratio of the reaction rates between the deactivation in living FRP and the propagation, defined by Equation (8) |

ζ | Reduced chain length, defined by r/$\overline{r}$_{n,p} |

ρ | Branching density |

$\overline{\rho}$(r) | Expected branching density of the polymer molecules having chain length r |

τ | Dimensionless parameter representing the ratio of reaction rates between various types of chain stoppage and the propagation, defined by Equation (2) |

Acronyms | |

ATRP | Atom-Transfer Radical Polymerization |

CTA | Chain Transfer Agent |

FRP | Free-Radical Polymerization |

GPC | Gel Permeation Chromatography |

MC | Monte Carlo |

MW | Molecular Weight |

MWD | Molecular Weight Distribution |

PDI | Polydispersity Index, PDI = $\overline{r}$_{w}/$\overline{r}$_{n} |

RAFT | Reversible Addition–Fragmentation Chain-Transfer |

RDRP | Reversible-Deactivation Radical Polymerization |

SRMP | Stable-Radical-Mediated Polymerization |

## References

- Odian, G. Principles of Polymerization, 4th ed.; John Wiley & Sons: Hoboken, NJ, USA, 2004; pp. 252–255. [Google Scholar]
- Plessis, C.; Arzamendi, G.; Leiza, J.R.; Schoonbrood, H.A.S.; Charmot, D.; Asua, J.M. Seeded Semibatch Emulsion Polymerization of n-Butyl Acrylate. Kinetics and Structural Properties. Macromolecules
**2000**, 33, 5041–5047. [Google Scholar] [CrossRef] - Nikitin, A.N.; Hutchinson, R.A.; Kalfas, G.A.; Richards, J.R.; Bruni, C. The effect of Intramolecular Transfer to Polymer on Stationary Free-Radical Polymerization of Alkyl Acrylates, 3—Consideration of Solution Polymerization up to High Conversions. Macromol. Theory Simul.
**2009**, 18, 247–258. [Google Scholar] [CrossRef] - Tobita, H. Molecular Weight Distribution in Free Radical Polymerization with Long-Chain Branching. J. Polym. Sci. B Polym. Phys.
**1993**, 31, 1363–1371. [Google Scholar] [CrossRef] - Tobita, H. Kinetics of Long-Chain Branching via Chain Transfer to Polymer: 1. Branched Structure. Polym. React. Eng.
**1993**, 1, 357–378. [Google Scholar] [CrossRef] - Gilbert, R.G. Emulsion Polymerization, A Mechanistic Approach; Academic Press: San Diego, CA, USA, 1995. [Google Scholar]
- Tobita, H.; Yamamoto, K. Network Formation in Emulsion Cross-Linking Copolymerization. Macromolecules
**1994**, 27, 3389–3396. [Google Scholar] [CrossRef] - Tobita, H. Scale-Free Power-Law Distribution of Emulsion Polymerized Nonlinear Polymers: Free-Radical Polymerization with Chain Transfer to Polymer. Macromolecules
**2004**, 37, 585–589. [Google Scholar] [CrossRef] - Cunningham, M.F. Controlled/living radical polymerization in aqueous dispersed systems. Prog. Polym. Sci.
**2008**, 33, 365–398. [Google Scholar] [CrossRef] - Zetterlund, P.B.; Kagawa, Y.; Okubo, M. Controlled/Living Radical Polymerization in Dispersed Systems. Chem. Rev.
**2008**, 108, 3747–3794. [Google Scholar] [CrossRef] [PubMed] - Ferguson, C.J.; Hughes, R.J.; Nguyen, D.; Pham, B.T.T.; Gilbert, R.G.; Serelis, A.K.; Such, C.H.; Hawkett, B.S. Ab Initio Emulsion Polymerization by RAFT-Controlled Self-Assembly. Macromolecules
**2005**, 38, 2191–2204. [Google Scholar] [CrossRef] - Luo, Y.; Wang, X.; Li, B.G.; Zhu, S. Toward Well-Controlled ab Initio RAFT Emulsion Polymerization of Styrene Mediated by 2-(((Dodecylsulfanyl)carbonothiol)sulfanyl)propanoic Acid. Macromolecules
**2011**, 44, 221–229. [Google Scholar] [CrossRef] - Zhu, Y.; Bi, S.; Gao, X.; Luo, Y. Comparison of RAFT Ab Initio Emulsion Polymerization of Methyl Methacrylate and Styrene Mediated by Oligo(methacrylic acid-β-methyl methacrylate) Trithiocarbonate Surfactant. Macromol. React. Eng.
**2015**, 9, 503–511. [Google Scholar] [CrossRef] - Gonzalez-Blanco, R.; Saldivar-Guerra, E.; Herrera-Ordonez, J.; Cano-Valdez, A. TEMPO Mediated Radical Emulsion Polymerization of Styrene by Stepwize and Semibatch Processes. Macromol. Symp.
**2013**, 325, 89–95. [Google Scholar] [CrossRef] - Li, X.; Wang, W.J.; Li, B.G.; Zhu, S. Branching in RAFT Miniemulsion Copolymerization of Styrene/Triethylene Glycol Dimethacrylate and Contro of Branching Density Distribution. Macromol. React. Eng.
**2015**, 9, 90–99. [Google Scholar] [CrossRef] - Tobita, H. Molecular Weight Distribution in Nonlinear Emulsion Polymerization. J. Polym. Sci. B Polym. Phys.
**1997**, 35, 1515–1532. [Google Scholar] [CrossRef] - Tobita, H. Bimodal Molecular Weight Distribution Formed in Emulsion Polymerization with Long-Chain Branching. Polym. React. Eng.
**2003**, 11, 855–868. [Google Scholar] [CrossRef] - Tobita, H. Scale-Free Power-Law Distribution of Emulsion Polymerized Branched Polymers: Power Exponent of the Molecular Weight Distribution. Macromol. Mater. Eng.
**2005**, 290, 363–371. [Google Scholar] [CrossRef] - Tobita, H. Power-law distribution of molecular weights of nonlinear emulsion polymers. e-Polymers
**2005**, 5, 684–694. [Google Scholar] [CrossRef] - Tobita, H. Polymerization Processes, 1. Fundamentals. In Ullmann’s Encyclopedia of Industrial Chemistry; Elvers, B., Ed.; Wiley: Weinheim, Germany, 2015. [Google Scholar] [CrossRef]
- Jenkins, A.D.; Jones, R.G.; Moad, G. Terminology for reversible-deactivation radical polymerization previously called “controlled” radical or “living” radical polymerization (IUPAC Recommendations 2010). Pure Appl. Chem.
**2010**, 82, 483–491. [Google Scholar] [CrossRef] - Tobita, H. Effect of Small Reaction Locus in Free-Radical Polymerization: Conventional and Reversible-Deactivation Radical Polymerization. Polymers
**2016**, 8, 155. [Google Scholar] [CrossRef] - Tobita, H. Molecular Weight Distribution of Living Radical Polymers. 1. Fundamental Distribution. Macromol. Theory Simul.
**2006**, 15, 12–22. [Google Scholar] [CrossRef] - Konkolewicz, D.; Sosnowski, S.; D’hooge, D.R.; Szymanski, R.; Reyniers, M.F.; Martin, G.B.; Matyjaszewski, K. Origin of the difference between Branching an Acrylates Polymerization under Controlled and Free Radical Conditions: A Computational Study of Competitive Processes. Macromolecules
**2011**, 44, 8361–8373. [Google Scholar] [CrossRef] - Tobita, H. Molecular weight distribution in random branching of polymer chains. Macromol. Theory Simul.
**1996**, 5, 129–144. [Google Scholar] [CrossRef] - Wiener, H. Structural Determination of Paraffin Boiling Points. J. Am. Chem. Soc.
**1947**, 69, 17–20. [Google Scholar] [CrossRef] [PubMed] - Nitta, K. A topological approach to statistics and dynamics of chain molecules. J. Chem. Phys.
**1994**, 101, 4222–4228. [Google Scholar] [CrossRef] - Zimm, G.H.; Stockmayer, W.H. The dimensions of chain molecules containing branches and rings. J. Chem. Phys.
**1949**, 17, 1301–1314. [Google Scholar] [CrossRef]

**Figure 1.**Reversible deactivation reactions in some of representative RDRP. In the figure, P

_{i}X or XP

_{i}is the dormant polymer with chain length i. ${\mathrm{R}}_{i}^{\u2022}$ is the active polymer radical with chain length i.

**Figure 2.**Weight fraction distribution determined by the MC simulation for the conditions, C1–C4. In the figure, PDI means the polydispersity index, defined by PDI = $\overline{r}$

_{w}/$\overline{r}$

_{n}. The PDIs are the theoretical values, with $\overline{r}$

_{w}being calculated from Equations (16) and (17).

**Figure 3.**Comparison of the full weight fraction distribution profiles of polymers formed through the conventional emulsion polymerization for the conditions, C1–C4 (red) and the corresponding random branching (blue).

**Figure 4.**Double logarithmic plot of the number fraction distribution of the polymers formed through the conventional free-radical emulsion polymerization, for the conditions C1–C4.

**Figure 5.**Weight fraction distribution determined by the MC simulation for the living emulsion polymerization, for the conditions, L1–L4. The PDI values in the figure are determined also from the MC simulation results.

**Figure 6.**Primary polymer chain length distribution of the present model living emulsion polymerization for the conditions L1–L4, calculated from Equation (23).

**Figure 7.**Comparison of the full weight fraction distribution profiles formed through the living emulsion polymerization for the conditions L1–L4 (red) and the corresponding random branching (blue).

**Figure 8.**Relationship between branching density and chain length in conventional free-radical emulsion polymerization, for the conditions C1–C4.

**Figure 9.**Expected branching density of the polymer whose reduced chain length is ζ (= r/$\overline{r}$

_{n,p}).

**Figure 10.**Expected branching density of the polymer whose reduced chain length is ζ (=r/$\overline{r}$

_{n,p}). The lines are for random branching with the given P

_{b}-values, and the circular symbols are for the conventional emulsion polymerization, C4.

**Figure 11.**Relationship between the branching density and the chain length in living free-radical emulsion polymerization, for the conditions L1–L4.

**Figure 12.**Expected branching density of the polymer with chain length r, for living emulsion polymerization (red) and random branching (blue).

**Figure 13.**Relationship between the mean-square radius of gyration, <s

^{2}>

_{0}and the chain length, r for the conventional emulsion polymerization conditions, C1–C4. The red dots show the property of each polymer molecule simulated, and the blue curve with circular symbols is the expected <s

^{2}>

_{0}–value for the given r. The black solid and dotted curves are for the random branched polymers, having the same primary chain length distribution. See more details in the text.

**Figure 14.**Relationship between the mean-square radius of gyration, <s

^{2}>

_{0}and the degree of polymerization, r for the living emulsion polymerization conditions, L1–L4. The red dots show the property of each polymer molecule simulated, and the blue curve with circular symbol is the expected <s

^{2}>

_{0}–value for the given r. The black dotted curve with $\times $-symbol is for the random branched polymers, having the same primary chain length distribution and the average branching density.

Run | τ | C_{P} |
---|---|---|

C1 | 0.005 | 0.002 |

C2 | 0.002 | 0.002 |

C3 | 0.001 | 0.002 |

C4 | 0.001 | 0.005 |

Run | 1/$\overline{\mathit{r}}$_{n} | C_{P} |
---|---|---|

L1 | 0.005 | 0.002 |

L2 | 0.002 | 0.002 |

L3 | 0.001 | 0.002 |

L4 | 0.001 | 0.005 |

_{n,s}= 2 for all cases, which means that p = 2/3.

Run | $\overline{\mathit{r}}$_{n} | $\overline{\mathit{\rho}}$ | $\overline{\mathit{r}}$_{n,p} ^{1)} | $\overline{\mathit{P}}$_{b} ^{2)} |
---|---|---|---|---|

C1, L1 | 200 | 0.002 | 142.9 | 0.2857 |

C2, L2 | 500 | 0.002 | 250 | 0.5 |

C3, L3 | 1000 | 0.002 | 333.3 | 0.6667 |

C4, L4 | 1000 | 0.005 | 166.7 | 0.8333 |

^{1)}The primary chains are defined as linear chains when the branch points formed by chain transfer to polymer are severed, both for the conventional and living FRP, which is given by 1/$\overline{r}$

_{n,p}= 1/$\overline{r}$

_{n}+ C

_{P}.

^{2)}Average probability for the chain end of a primary chain is connected to a backbone chain, which is given by $\overline{P}$

_{b}= $\overline{r}$

_{n}$\overline{\rho}$.

Run | Description | $\overline{\mathit{r}}$_{n} | $\overline{\mathit{r}}$_{w} |
---|---|---|---|

C1 | MC | 200.6 | 647.9 |

Theory | 200 | 649.9 | |

Error (%) | 0.3 | −0.311 | |

C2 | MC | 500.3 | 4024.5 |

Theory | 500 | 3936.8 | |

Error (%) | 0.06 | 2.23 | |

C3 | MC | 1000.4 | 21,753.9 |

Theory | 1000 | 22,238.4 | |

Error (%) | 0.04 | −2.18 | |

C4 | MC | 1001.9 | 108,989 |

Theory | 1000 | 109,270 | |

Error (%) | 0.19 | −0.257 |

Run | Description | $\overline{\mathit{r}}$_{n} | $\overline{\mathit{r}}$_{w} | PDI |
---|---|---|---|---|

C1 | Emulsion Polym. | 200 | 649.9 | 3.250 |

Random Branch | 200 | 560 | 2.8 | |

C2 | Emulsion Polym. | 500 | 3936.8 | 7.874 |

Random Branch | 500 | 2000 | 4 | |

C3 | Emulsion Polym. | 1000 | 22,238.4 | 22.24 |

Random Branch | 1000 | 6000 | 6 | |

C4 | Emulsion Polym. | 1000 | 108,989 | 109.3 |

Random Branch | 1000 | 12,000 | 12 |

Run | Description | $\overline{\mathit{r}}$_{n} | $\overline{\mathit{r}}$_{w} | PDI |
---|---|---|---|---|

L1 | Emulsion Polym. | 200 | 254.2 | 1.27 |

Random Branch | 200 | 351.1 | 1.76 | |

L2 | Emulsion Polym. | 500 | 755.8 | 1.51 |

Random Branch | 500 | 1478.9 | 2.96 | |

L3 | Emulsion Polym. | 1000 | 1782.3 | 1.78 |

Random Branch | 1000 | 5115.1 | 5.12 | |

L4 | Emulsion Polym. | 1000 | 2153.3 | 2.15 |

Random Branch | 1000 | 11,540.9 | 11.5 |

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tobita, H.
Effect of Chain Transfer to Polymer in Conventional and Living Emulsion Polymerization Process. *Processes* **2018**, *6*, 14.
https://doi.org/10.3390/pr6020014

**AMA Style**

Tobita H.
Effect of Chain Transfer to Polymer in Conventional and Living Emulsion Polymerization Process. *Processes*. 2018; 6(2):14.
https://doi.org/10.3390/pr6020014

**Chicago/Turabian Style**

Tobita, Hidetaka.
2018. "Effect of Chain Transfer to Polymer in Conventional and Living Emulsion Polymerization Process" *Processes* 6, no. 2: 14.
https://doi.org/10.3390/pr6020014