processes

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

Hidetaka Tobita ID Department of Materials Science and Engineering, University of Fukui, 3-9-1 Bunkyo, Fukui 910-8507, Japan; [email protected]; Tel.: +81-776-27-8767

Received: 28 December 2017; Accepted: 5 February 2018; Published: 7 February 2018 Abstract: Emulsion polymerization process provides a unique polymerization locus that has a confined tiny space with a higher polymer concentration, compared with the corresponding bulk polymerization, especially for the ab initio emulsion polymerization. Assuming the ideal polymerization kinetics and a constant polymer/monomer ratio, the effect of such a unique reaction environment is explored for both conventional and living free-radical polymerization (FRP), which involves chain transfer to the polymer, forming polymers with long-chain branches. Monte Carlo simulation is applied to investigate detailed branched polymer architecture, including 2 the mean-square radius of gyration of each polymer molecule, 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 2 polymers, having the same primary chain length distribution. The expected 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, 2 the expected 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.

Keywords: branched; controlled/living; emulsion polymerization; free-radical polymerization; molecular weight distribution; Monte Carlo; radius of gyration

1. Introduction Chain transfer to polymer during free-radical polymerization (FRP) leads to form branched polymer molecules. Branching has significant effects on the rheological and physical properties of the product polymers, and, therefore, control of branched architecture is an important consideration in the design and development of polymerization processes. For the production of branched polymers, various structural properties must be controlled properly. In the present investigation, the distributions of molecular weights, branch points, and the mean-square radii of gyration that represent the three-dimensional (3D) size in space are highlighted. Chain transfer to polymer leads to form short- and long-chain branches. Short-chain branches are formed by intramolecular reactions, called backbiting, through the occurrence of a five- or six-membered ring transition state of a chain-end radical [1]. The backbiting reaction does not change the molecular weight distribution (MWD) by itself. As long as the reaction rate of short-chain branching is much smaller than the propagation reaction, which is usually the case for FRPs, the effect of backbiting on the formed MWD, as well as the radius of gyration, would not be significant. Although it has been reported that the frequency of short-chain branching is greater than that of long-chain branching, especially for acrylic monomers [2,3], the backbiting reaction is neglected in the present theoretical investigation in which the effects of polymerization process on the MWD and the 3D size distribution are highlighted.

Processes 2018, 6, 14; doi:10.3390/pr6020014 www.mdpi.com/journal/processes Processes 2018, 6, 14 2 of 21

Long-chain branching is the intermolecular chain transfer reaction between a polymer radical and a backbone polymer chain. When a midchain radical formed by chain transfer to polymer propagates, a long-branch chain is formed. In this case, large polymer chains have a better chance of being attacked because they have a larger number of monomeric units. In addition, the polymer chains formed at earlier stages of polymerization are subjected to chain transfer reaction for a longer period of time, leading to a larger expected branching density [4]. Therefore, the long-chain branched architecture is highly dependent on the polymerization process employed, which grants considerable leeway for designing and developing polymerization processes by controlling the reaction environment during polymerization. Emulsion polymerization proceeds inside the polymer particles. In the polymerization locus, the polymer concentration is high from the beginning, and is higher than the corresponding bulk polymerization, as long as monomer droplets exist to work as a monomer feed reservoir. Higher polymer concentration enhances the rate of chain transfer to polymer, compared with the corresponding bulk polymerization in which the polymer concentration increases gradually from zero. Therefore, even for the type of monomer whose branching density is negligibly small and the chain transfer to polymer reaction can be neglected in bulk polymerization especially for the early stage of polymerization, the long-chain formation may not be neglected in emulsion polymerization [5]. Another interesting characteristic of emulsion polymerization is that the size of polymerization locus is often smaller than 100 nm. With this small volume, two radicals cannot coexist, causing almost immediate bimolecular termination between radicals. Unique emulsion polymerization kinetics, typically represented by the zero-one system [6] is valid for such small polymer particles. Not only the number of radicals, but the number of monomeric units in a particle is finite. The high molecular weight (MW) tail the MWD of nonlinear polymers could be affected by this finite size of the confined polymerization locus [7,8]. In this article, the effects of higher polymer concentration and finite tiny size of polymerization locus on the long-chain branching are considered for conventional and living FRP. In the experiment, ab initio emulsion polymerization, utilizing controlled/living radical polymerization, often encounters difficulty due to the colloidal instabilities [9,10]. However, various methods have been proposed to overcome the experimental problems [11–13]. In the present investigation, theoretical analysis is conducted, assuming an ideal ab initio emulsion polymerization both for conventional and living FRP. Emulsified systems provide the polymerization loci with huge surface area per unit volume, which enables one to conduct semibatch operation without suffered significantly by the slow diffusion of the added substances [14,15], which endows the process with additional operability. The present theoretical investigation is hoped to provide useful information, also for such advanced operation. In this article, the Monte Carlo simulation method proposed earlier for the investigation of the MWD formed in conventional FRP that involves chain transfer to polymer [8,16–19] is extended for the living FRP. The exploration is made not only for the full MWD, but also for the distributions of the branch points among various polymer molecules and the 3D polymer architecture both for conventional and living FRP. The differences of the effect of chain transfer to polymer are contrasted between these two different types of FRP mechanisms.

2. Simulation Method

2.1. Conventional Free-Radical Emulsion Polymerization A simplified model for the conventional free-radical emulsion polymerization that involves chain transfer to polymer is used for the present investigation. In emulsion polymerization, the monomer concentration inside the polymer particle is kept approximately constant as long as the monomer droplets exist. To highlight unique characteristic of emulsion polymerization, the simulation is conducted during a constant polymer/monomer ratio period. Processes 2018, 6, 14 3 of 21

The primary chain is defined as a linear chain when the branch point is severed. In the conventional FRP, a primary chain is formed instantaneously, when the growing primary polymer radical is stopped growing by the termination reaction or the chain transfer reaction. Neglecting the chain-length dependent kinetics, the number-based chain length distribution of the primary chain radicals in conventional FRP, Npr(r) is given by [20]:

Npr(r) = (τ + CP) exp[−(τ + CP)r], (1) where r is the chain length (degree of polymerization), and τ and CP are dimensionless numbers defined by: τ = Cfm + CfCTA[CTA]p/[M]p + 1/(kp[M]ptav), (2)

CP = Cfp[P]p/[M]p, (3) where Cfm, CfCTA, Cfp 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 tav represents the average time interval between radical entry to a particle. The propagation rate constant is represented by kp. To highlight the most fundamental characteristics of emulsion polymerization until the depletion of monomer droplets, both τ and CP are assumed to be constant in the present MC simulation. Note that the number-average chain length of primary polymer radicals, rn,p is given by:

rn,p = 1/(τ + CP). (4)

The length of a dead primary polymer chain formed by the chain transfer reaction is the same as that of a primary polymer radical. Assuming the zero-one kinetics is valid, a newly entered oligomeric radical terminates immediately if a polymer radical exists. Therefore, bimolecular termination by combination leads to form a dead primary polymer chain that is almost the same chain length as the original primary polymer radicals. Disproportionation termination forms a dead primary polymer chain and an oligomeric chain. When the existence of such oligomeric chains is neglected, the dead primary polymer chain length distribution is also given by Equation (1). In the present MC simulation, Equation (1) is used to represent the primary chain length distribution, whose number average is given by Equation (4). The probability that a newly formed primary chain has started growing from the midchain radical, i.e., the probability that a chain end of a primary chain is connected to a backbone chain, Pb is given by the following ratio: Pb = CP/(τ + CP), (5) which is also a constant in the present MC simulation. In this article, Pb is called the “branching probability”. Note that in the present simulation condition, the value of Pb 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]. The MC simulation proceeds as follows. The nucleation stage is neglected for simplicity, and it is assumed that the primary polymer chain length distribution follows Equation (1), starting from the very first chain. The first primary chain length is determined by generating a random number that Processes 2018, 6, 14 4 of 21 follows Equation (1), which can simply be done by using a uniform random number y between 0 and 1, as follows. Processes 2018, 6, x FOR PEER REVIEW 4 of 22 r = Ceiling[ln(1/y)/(τ + CP)], (6) wherewhere Ceiling[Ceiling[aa]] meansmeans thethe leastleast integerinteger notnot smallersmaller thanthan aa.. From thethe second second primary primary chain chain in a in particle, a particle, every every time a newtime primarya new primary polymer chainpolymer is generated, chain is thegenerated, probability the probability given by Equation given by (5) Equation is checked. (5) is If checked. the chain If endthe ischain connected, end is connected, the connected the branchconnected point branch is selected point is randomly selected randomly from the from monomeric the monomeric units already units boundalready into bound polymer into polymer chains. Thechains. simulation The simulation for a single for polymera single particlepolymer ends partic whenle ends the totalwhen number the total of polymerizednumber of polymerized monomeric unitsmonomeric reaches units a predetermined reaches a predetermined value, NM. By repeatingvalue, N suchM. By simulation repeating forsuch a large simulation number for of polymer a large particlesnumber toof generatepolymer aparticles significant to numbergenerate ofa polymersignificant molecules, number oneof polymer can determine molecules, the statistical one can propertiesdetermine ofthe the statistical product properties polymers effectively.of the product polymers effectively. The MC simulation results otherother thanthan the conditionsconditions reported inin thethe present articlearticle cancan be found inin [[8,16–19].8,16–19].

2.2. Living Free-Radical Emulsion PolymerizationPolymerization InIn FRP, the bimolecular bimolecular termination termination reactions reactions of of the the active active radicals radicals are are inevitable. inevitable. Therefore, Therefore, the theliving living polymerization polymerization in which in which the chain the chaintermination termination reactions reactions are totally are absent totally in absent a strict in sense, a strict is sense,impossible. is impossible. However, if However, a large percentage if a large of percentage polymer chains of polymer are dormant chains and are can dormant potentially and grow can potentiallyfurther, such grow FRP further,systemssuch can be FRP regarded systems as can pseudo-living be regarded polymerization. as pseudo-living By polymerization.introducing the Byreversible-deactivation introducing the reversible-deactivation process in FRP, polymers process having in FRP, a polymersnarrow distribution having a narrow can be distributionobtained in canthe linear be obtained polymerization in the linear that polymerization does not involve that branching does not and/or involve crosslinking. branching and/orIUPAC recommends crosslinking. IUPACusing the recommends term, “reversible-deactivation using the term, “reversible-deactivation radical polymerization” radical polymerization”(RDRP) [21], because (RDRP) [21the], becausereversible-deactivation the reversible-deactivation reaction is reaction the origin is the originof pseudo-livingness. of pseudo-livingness. However, However, the the effect effect of intermittentintermittent growth of livingliving chainchain withwith thethe existenceexistence of chain transfer to polymerpolymer is thethe essenceessence ofof comparisoncomparison inin thisthis article,article, thethe term,term, “living“living FRP”FRP” isis usedused interchangeably.interchangeably. Figure1 1 showsshows thethe reversible deactivation deactivation reactions reactions in in some some of ofrepresentative representative RDRPs, RDRPs, i.e., i.e.,stable-radical-mediated stable-radical-mediated polymerization polymerization (SRMP), (SRMP), atom-transfer atom-transfer radical radical polymerization polymerization (ATRP), (ATRP), and andreversible-addition-fragmentation reversible-addition-fragmentation chain-transfer chain-transfer (RAFT) (RAFT) polymerization. polymerization. In In order order to to formulate formulate the polymerizationpolymerization raterate expression expression for for various various types types of of RDRPs RDRPs in ain unified a unified manner manner [22], [22], the componentthe component that generatesthat generates an active an active radical radical is represented is represented as the radical as the generating radical generating species (RGS), species and (RGS), the component and the thatcomponent deactivates that andeactivates active radical an active is represented radical is repr as theesented trapping as the agent trapping (Trap) agent in Figure (Trap)1. in Figure 1.

Figure 1.1. ReversibleReversible deactivationdeactivation reactionsreactions in in some some of of representative representative RDRP. RDRP. In In the the figure, figure, PiX Pi orX or XP XPi isi • • theis the dormant dormant polymer polymer with with chain chain length lengthi.R i.i isR thei is active the active polymer polymer radical radical with with chain chain length lengthi. i.

InIn orderorder to to highlight highlight the the most most fundamental fundamental characteristics characteristics in living in living emulsion emulsion FRP, anFRP, ideal an livingideal radicalliving radical polymerization, polymerization, with no with termination no termination occurring occurring other than other the loss than of livingnessthe loss of through livingness the chainthrough transfer the chain to polymer transfer is considered.to polymer Usingis considered. the notations Using shown the no intations Figure 1shown, the deactivation in Figure 1, rate, the deactivation rate, Rdeact is given by: Rdeact is given by: • RdeactRdeact= =k 2k2[Trap][R [Trap][R].•]. (7)(7)

Dividing Equation (7) by the polymerization rate, Rp = kp[M][R•], one obtains the following dimensionless ratio, δ:

δ = (k2 [Trap])/(kp [M]). (8)

In emulsion polymerization, the monomer concentration in the polymer particle, [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

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• Dividing Equation (7) by the polymerization rate, Rp = kp[M][R ], one obtains the following dimensionless ratio, δ: δ = (k2 [Trap])/(kp [M]). (8)

In emulsion polymerization, the monomer concentration in the polymer particle, [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. Neglecting the bimolecular termination reactions, the probability that an active radical connects another monomer unit during the active period is given by:

p = 1/(1 + δ + Cp). (9)

Note that the rate ratio CP, defined by Equation (3), is constant, and therefore p is also a constant. The number-based chain length distribution during a single active period, Nsa(r) is given by the following modified most probable distribution [23]:

r Nsa(r) = p (1 − p), (10) which includes the possibility that the activated species fails to connect a single monomer, i.e., the cases with r = 0. In this article, the sequence of monomeric units formed during a single active period is called a “segment chain.” The number-average chain length of segment chains, rn,s is given by:

rn,s = p/(1 − p). (11)

The probability that the active period starts from a midchain radical, which is equal to the probability that the chain end of a segment is connected to a backbone chain, Pbs is given by:

Pbs = CP/(δ + Cp). (12)

Note that the “slow reactivation” of midchain dormant [24] is not considered in the present investigation, to highlight pure differences of living and nonliving FRP. In order to make “fair” comparison with the corresponding conventional (nonliving) FRP, the final number-average chain length (degree of polymerization), rn and the total number of monomeric units bound into polymer molecules in a particle, NM 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 NM/rn. In the simulation for conventional FRP, a single linear polymer molecule having chain length ca. rn,p exists initially. In order to have an equivalent initial condition for living FRP, the total of rn,p/rn,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:

r = Ceiling[ln y/ln p − 1]. (13)

The MC simulation starts from the following initial condition. There are rn,p/rn,s dormant chains whose lengths are generated by using Equation (13), and (NM/rn − rn,p/rn,s) dormant species having length 0 in a polymer particle. Note in the present simulation study shown in the next section, NM/rn > rn,p/rn,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 − Pbs), the eliminated trapping agent is attached to Processes 2018, 6, 14 6 of 21

the segment end. On the other hand, with probability Pbs, 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 NM, 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

The total number of monomeric units bound into polymer molecules in a particle, NM that defines 6 the end point for the MC simulation is set to be NM = 1 × 10 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/cm3, 6 the value of NM = 1 × 10 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 6 of simulation, NM = 1 × 10 . The parameters used for the conventional FRP are shown in Table1, and those for the living FRP are shown in Table2. The number-average chain length of the product polymers, rn is 200 for C1 and L1, rn = 500 for C2 and L2, and rn = 1000 for C3, C4, L3, L4. Note that the number-average chain length, rn is given by the following equation for the conventional FRP:

rn = 1/τ. (14)

Because the polymer/monomer ratio, [P]p/[M]p is kept constant, the average branching density of the all polymers, ρ is given by the following equation for both conventional and living FRP.

ρ = Cp, (15)

Table 1. Parameters used for conventional FRP.

Run τ CP C1 0.005 0.002 C2 0.002 0.002 C3 0.001 0.002 C4 0.001 0.005

Table 2. Parameters used for living FRP.

Run 1/rn CP L1 0.005 0.002 L2 0.002 0.002 L3 0.001 0.002 L4 0.001 0.005

rn,s = 2 for all cases, which means that p = 2/3.

Table3 shows the properties of product polymers that are set to be the same for the corresponding conventional and living FRP. For the living FRP, the probability that the chain end of a primary chain is connected, or the branching probability Pb, is not the same for all primary chains. However, the average branching probability, Pb is set to be the same for the corresponding conventional and living FRP, and Pb 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 rn,s = 2 throughout the polymerization. Processes 2018, 6, 14 7 of 21

Table 3. Properties set to be the same for the corresponding conventional and living FRP.

1) 2) Run rn ρ rn,p Pb 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 2) Processessevered, 2018 both, 6, x forFOR the PEER conventional REVIEW and living FRP, which is given by 1/rn,p = 1/rn + CP. Average probability for 7 of 22 the chain end of a primary chain is connected to a backbone chain, which is given by Pb = rnρ. 3.1. Weight Fraction Distribution 3.1. Weight Fraction Distribution 3.1.1. Conventional Free-Radical Emulsion Polymerization 3.1.1. Conventional Free-Radical Emulsion Polymerization The red curve in Figure 2 shows the MC simulation results for the weight fraction distribution. The red curve in Figure2 shows the MC simulation results for the weight fraction distribution. The independent variable is the logarithm of chain length, as is usually employed for the GPC The independent variable is the logarithm of chain length, as is usually employed for the GPC measurement. The blue curves in Figure 2 show the fractional MWDs consisting of k branch points. measurement. The blue curves in Figure2 show the fractional MWDs consisting of k branch points. The value of k is 0, 1, 2, 3 from the left to the right. For the second high MW peak shown in C2–C4, The value of k is 0, 1, 2, 3 from the left to the right. For the second high MW peak shown in C2–C4, the large k-value fractions are summed up, as shown in the figure. When the branching probability, the large k-value fractions are summed up, as shown in the figure. When the branching probability, Pb is large, a sharp high MW peak appears. In the present condition, the total number of monomeric P is large, a sharp high MW peak appears. In the present condition, the total number of monomeric b 6 units incorporated into polymer molecules is, NM = 1 × 106 , and therefore, a polymer molecule with units incorporated into polymer molecules is, NM = 1 × 10 , and therefore, a polymer molecule with log10r > 6 cannot exist in a polymer particle. The sharp high MW peak is formed due to the limitation log r > 6 cannot exist in a polymer particle. The sharp high MW peak is formed due to the limitation of10 the small particle size. A sharp high MW peak essentially consists of the largest polymer molecule of the small particle size. A sharp high MW peak essentially consists of the largest polymer molecule in each polymer particle [17]. in each polymer particle [17].

FigureFigure 2. 2.Weight Weight fraction fractiondistribution distribution determineddetermined by the MC simulation simulation for for the the conditions, conditions, C1–C4. C1–C4. In _ _ In the figure, PDI means the polydispersity index, defined by PDI = rw/rn. The PDIs are the theoretical the figure, PDI means the polydispersity index, defined by PDI = rw/rn. The PDIs are the theoretical values, with rw_ being calculated from Equations (16) and (17). values, with rw being calculated from Equations (16) and (17).

InIn the the present present model model emulsion emulsion polymerization, polymerization, the the weight-average weight-average chain chain length length (degree (degree of of polymerization), rw_ is given by [16,17]: polymerization), rw is given by [16,17]:   n 2Pb rw,p rw(n + 1) = 1 + rw(n) + (16) n + 1 n + 1 n + 1 (16) and and rw(1) = rw,p, (17)

, (17)

where n is the number of primary chains in a particle. Table 4 shows the comparison of average chain lengths between the MC simulation results and the theoretical values. The errors are rather small, and the accuracy of the present MC simulation is confirmed. Note that the present MC simulation is conducted on the number basis, and therefore, _ the statistical errors are expected to be smaller for the number-average chain length rn, rather than _ the weight-average rw.

Processes 2018, 6, 14 8 of 21 where n is the number of primary chains in a particle. Table4 shows the comparison of average chain lengths between the MC simulation results and the theoretical values. The errors are rather small, and the accuracy of the present MC simulation is confirmed. Note that the present MC simulation is conducted on the number basis, and therefore, the statistical errors are expected to be smaller for the number-average chain length rn, rather than the weight-average rw.

Table 4. Comparison of the calculated average chain lengths.

Run Description rn rw MC 200.6 647.9 C1 Theory 200 649.9 Error (%) 0.3 −0.311 MC 500.3 4024.5 C2 Theory 500 3936.8 Error (%) 0.06 2.23 MC 1000.4 21,753.9 C3 Theory 1000 22,238.4 Error (%) 0.04 −2.18 MC 1001.9 108,989 C4 Theory 1000 109,270 Error (%) 0.19 −0.257

In order to highlight the characteristics of the emulsion polymerization system, the comparison with the random branched polymer, in which the probability of having a branch point ρ is the same for monomeric units in polymer, is considered. Suppose we have the primary chains whose number- and weight-average chain lengths are rn,p and rw,p, respectively. When such primary chains are connected through random branching, the number- and weight-average chain length are given, respectively, by [25]: rn = rn,p/(1 − ρrn,p) (18) 2 rw = rw,p/(1 − ρrn,p) (19) Note that Equations (18) and (19) are valid irrespective of the primary chain length distribution. Comparison of the present model emulsion polymerization and random branching is shown in Table5. The number-average chain length rn 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 rw is larger for the emulsion polymers, especially for the large branching probability cases. In the present model emulsion polymerization, the primary chain length distribution follows the most probable distribution, given by Equation (1). When the primary chains conforming to the most probable distribution are connected through random branching, the full weight fraction distribution can be calculated from the following equation [25]: √     ! 1 − Pb I1 2(r/rn,p) Pb   W(r) = √ exp −(1 + Pb)(r/rn,p) , (20) rn,p Pb where I1 is the modified Bessel function of the first kind and of the first order. Processes 2018, 6, 14 9 of 21

Table 5. Comparison of the calculated average chain lengths for conventional FRP.

Run Description rn rw PDI Emulsion Polym. 200 649.9 3.250 C1 Random Branch 200 560 2.8 Emulsion Polym. 500 3936.8 7.874 Processes 2018, 6, x FOR PEERC2 REVIEW 9 of 22 Random Branch 500 2000 4 Emulsion Polym. 1000 22,238.4 22.24 Processes 2018, 6, x FOR PEERC3 REVIEW 9 of 22 Random Branch 1000 6000 6 Emulsion Polym. 1000 108,989 109.3 , (20) C4 Random Branch 1000 12,000 12 , (20) where I1 is the modified Bessel function of the first kind and of the first order. Figure3 showsFigure the3 shows comparison the comparison of the of the full full weight weight fractionfraction distribution distribution profiles profiles between between the the emulsion polymersemulsionwhere I1 is polymers the and modified the and random theBessel random function branched branched of the polymers. firstpolyme kindrs. and It It is ofis clearly the clearly first shown order. shown that the that MWD the is MWD much is much broader forbroader theFigure emulsion for the3 shows emulsi polymerization. theon polymerization.comparison of the full weight fraction distribution profiles between the emulsion polymers and the random branched polymers. It is clearly shown that the MWD is much broader for the emulsion polymerization.

Figure 3. ComparisonFigure 3. Comparison of the full of the weight full weight fraction fraction distribution distribution profiles profiles of of po polymerslymers formed formed through through the conventionalthe emulsion conventional polymerization emulsion polymerization for the conditions,for the conditions, C1–C4 C1–C4 (red) (red and) and the the corresponding corresponding random randomFigure 3. branching Comparison (blue). of the full weight fraction distribution profiles of polymers formed through branching (blue). the conventional emulsion polymerization for the conditions, C1–C4 (red) and the corresponding Figurerandom 4 branching shows the (blue). log-log plot of the number fraction distribution, N(r). The power law holds Figurefor4 shows a significant the log-log range of plot chain of thelengths number [8,18,19 fraction]. The power distribution, exponent Nfor(r ).the The number power fraction law holds for a significantdistribution, rangeFigure of 4 chainN shows(r) is lengths− the(1/P log-logb + 1), [8 as ,plot18 shown,19 of]. the in The numberthepower figure. fraction With exponent thedistribution, relationship, for the N(r number ).W The(r) ~ powerrN( fractionr), thelaw power holds distribution, exponentfor a significant of the weight range fractionof chain distribution, lengths [8,18,19 W(r)]. is The −1/P powerb, i.e., W exponent(r) ~ r−α with for αthe = 1/numberPb. Because fraction the N(r) is −(1/Pb + 1), as shown in the figure. With the relationship, W(r) ~rN(r), the power exponent of branchingdistribution, probability, N(r) is −(1/ PPbb is+ 1),related as shown with inthe the ch figure.ain transfer With theconstant relationship, Cfp through W(r) Equations~ rN(r), the (3) power and the weight fraction distribution, W(r) is −1/P , i.e., W(r) ~r−α with α = 1/P . Because the branching (5),exponent the value of the of Cweightfp could fraction be estimated distribution, throughb W (ther) is present −1/Pb, i.e., type W of(r )double ~ r−α with logarithmic αb = 1/Pb. plot,Because as was the probability,illustratedbranchingPb is related forprobability, the with emulsion-pol thePb is chainrelatedymerized transfer with polyethylenethe constantchain transfer [18].Cfp constantthrough Cfp Equationsthrough Equations (3) and (3) (5), and the value of Cfp could(5), be the estimated value of Cfp through could be estimated the present through type the of present double type logarithmic of double logarithmic plot, as plot, was as illustrated was for the emulsion-polymerizedillustrated for the emulsion-pol polyethyleneymerized [18]. polyethylene [18].

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 4. DoubleFigure logarithmic4. Double logarithmic plot of theplot numberof the number fraction frac distributiontion distribution of theof the polymers polymers formedformed through

the conventionalthrough free-radical the conventional emulsion free-radical polymerization, emulsion polymerization, for the conditionsfor the conditions C1–C4. C1–C4.

Processes 2018, 6, x FOR PEER REVIEW 10 of 22

3.1.2. Living Free-Radical Emulsion Polymerization Figure 5 shows the MC simulation results for the full weight fraction distribution, formed through living FRP. A notable difference from the conventional FRP, shown in Figure 2, is the _ narrowness of the MWD. Note that the number-average chain length of primary chains rn,p and the _ average branching density ρ, as well as the number-average chain lengths of the product polymers _ rn, are the same for the corresponding polymerization conditions, i.e., C1 and L1, C2 and L2, and so on. Bimodal distributions of W(log10r) are shown for L1 and L2. On the other hand, in the present Processes 2018, 6, 14 10 of 21 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 3.1.2.to be Living observed. Free-Radical In addition, Emulsion the second Polymerization peak contains branches and the hydrodynamic volume becomes smaller than the corresponding linear polymers. The peak separation could be difficult in Figure5 shows the MC simulation results for the full weight fraction distribution, formed through the usual GPC analysis. living FRP. A notable difference from the conventional FRP, shown in Figure2, is the narrowness of the The power-law relationship of MWD was found for the conventional emulsion FRP, as was MWD. Note that the number-average chain length of primary chains rn,p and the average branching discussed in the previous section. On the other hand, the power law does not hold for the living density ρ, as well as the number-average chain lengths of the product polymers rn, are the same for the emulsion polymerization. corresponding polymerization conditions, i.e., C1 and L1, C2 and L2, and so on. Bimodal distributions In L4, the PDI value is 2.15, while the corresponding conventional FRP gives the PDI value as of W(log10r) are shown for L1 and L2. On the other hand, in the present idealized model, a uniform large as 109.3. In the cases of conventional emulsion FRP, the existence of long-chain branches would particle size distribution is assumed. In a real system, the MWD is expected to be somewhat broader be predicted easily judging from a broad MWD. On the other hand, for living emulsion than the present prediction, and the second peak may be difficult to be observed. In addition, the polymerization, the existence of long-chain branches could be overlooked because of a relatively second peak contains branches and the hydrodynamic volume becomes smaller than the corresponding narrow MWD of product polymers. linear polymers. The peak separation could be difficult in the usual GPC analysis.

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

TheIn Figure power-law 5, the relationship fractional MWDs of MWD containing was found k branches for the conventionalare also shown. emulsion Compared FRP, with as was the discussedfractional in MWDs the previous for the section.conventional On the FRP other shown hand, in theFigure power 2, one law would does not notice hold that for thethe livingbranch emulsionpoints are polymerization. distributed much more equally for the living emulsion polymerization. Note that the averageIn L4, branching the PDI value density is 2.15,of the while whole the system corresponding is the same conventional for the correspondi FRP givesng conditions, the PDI value e.g., as C1 largeand asL1, 109.3. and so In on, the as cases shown of conventional in Table 3. emulsion FRP, the existence of long-chain branches would be predictedThe MWD easily of judging linear polymer from a broad fraction MWD. (k = On 0) theshows other a hand,long tail for toward living emulsion small chain polymerization, lengths. The thelong existence tail is formed of long-chain through branches chain transfer could to be polyme overlookedr, which because causes of irreversible a relatively deactivation narrow MWD of ofthe productchain-end polymers. radical. The primary chain length distribution in the present living polymerization system canIn be Figurecalculated5, the theoretically, fractional MWDs as follows. containing k branches are also shown. Compared with the fractional MWDs for the conventional FRP shown in Figure2, one would notice that the branch points are distributed much more equally for the living emulsion polymerization. Note that the average branching density of the whole system is the same for the corresponding conditions, e.g., C1 and L1, and so on, as shown in Table3. The MWD of linear polymer fraction (k = 0) shows a long tail toward small chain lengths. The long tail is formed through chain transfer to polymer, which causes irreversible deactivation of the chain-end radical. The primary chain length distribution in the present living polymerization system can be calculated theoretically, as follows. Processes 2018, 6, 14 11 of 21

In the case of ideal linear living FRP, where the segment distribution is kept constant throughout the polymerization without termination and chain transfer reactions, the full weight fraction distribution is given by [23]:

2 r−1 −z Wliv(r) = (1 − p) p re F[1 + r, 2; (1 − p)z], (21) Processes 2018, 6, x FOR PEER REVIEW 11 of 22 where F[a,b;x]In is thethe confluentcase of ideal hypergeometric linear living FRP, function where the (Kummer’s segment distribution function ofis thekept first constant kind), and z is the averagethroughout number the of activepolymerization period without for a chain terminatio thatn is and defined chain transfer by: reactions, the full weight fraction distribution is given by [23]: z = r /r . (22) n n,s , (21)

Note that wherern,s = F 2[a in,b;x the] is the present confluent set hypergeometric of simulations. function (Kummer’s function of the first kind), and z In theispresent the average model number reaction of active system, period for the a chain probability that is defined for an by: active radical to cause chain transfer _ _ reaction, resulting in the irreversible terminationz = forrn/rn,s the. primary chain is equal to CP,(22) represented by Equation (3), which_ is kept constant throughout polymerization. When the ideal living chains Note that rn,s = 2 in the present set of simulations. whose MWDIn is the represented present model by reaction Equation system, (21) the is probabilit subjectedy for to an aactive constant radical probabilityto cause chain oftransfer chain transfer, the weightreaction, fraction resulting distribution in the irreversible is represented termination by for [23 the]: primary chain is equal to CP, represented by Equation (3), which is kept constant throughout polymerization. When the ideal living chains whose Z ∞ MWD is represented by Equation (21) is subjected to a constant probability of chain transfer, the WCP(r) = Wliv(s)(CPr/s)[2 + CP(s − r)] exp(−CPr)ds + Wliv(r) exp(−CPr). (23) weight fractionr distribution is represented by [23]:

Equation (23) gives the weight fraction distribution of the primary chains. in the present(23) model living emulsionEquation polymerization. (23) gives the Figureweight 6fr actionshows distribution the calculated of the primary results chains for L1 in tothe L4. present The model characteristics of long tailsliving inthe emulsion low MW polymerization. region, including Figure a6 strangeshows the distorted calculated fractional results MWDfor L1 profileto L4. forThe k = 0 of L3 condition incharacteristics Figure5, agree of long qualitatively tails in the low with MW the region, primary including chain a strange length distorted distribution fractional shown MWD in Figure6. Note thatprofile the primary for k = 0 chains of L3 condition whose distribution in Figure 5, agree is given qualitatively by Figure with6 arethe combined,primary chain according length to the distribution shown in Figure 6. Note that the primary chains whose distribution is given by Figure 6 pertinent emulsionare combined, polymerization according to the kinetics,pertinent emulsion to form polymerization the branched kinetics, polymers to form whose the branched MWD is shown in Figure5polymers. whose MWD is shown in Figure 5.

Figure 6. FigurePrimary 6. Primary polymer polymer chain chain length length distribution of ofthe thepresent present model modelliving emulsion living emulsion polymerizationpolymerization for the for conditions the conditions L1–L4, L1–L4, calculated calculated from Equation Equation (23). (23).

_ The number-average chain lengths of the primary chains, rn,p for L1 to L4 are given respectively _ The number-averagein Table 3. The weight-average chain lengths chain of lengths the primary of the primary chains, chains,rn,p for rw,p L1 can to be L4 calculated are given from respectively in Table3.Equation The weight-average (23). On the other chainhand, if lengthsthese primar ofy the chains primary are combined chains, to formrw,p randomcan be branched calculated from Equation (23).polymers, On the the number- other hand,and weight-average if these primary chain le chainsngths of are product combined polymers to can form be randomcalculated branched polymers,from the number-Equations (18) and and weight-average (19). Note that Equations chain lengths (18) and of(19) product are valid,polymers irrespective canof the be primary calculated from chain length distribution, as long as the branching is random, i.e., the probability of having a branch Equationspoint (18) is and the (19).same for Note all units. 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. Processes 2018, 6, 14 12 of 21

Table6 shows the comparison between the emulsion polymerization and the random branching, in the case of living FRP. The number-average chain length rn 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 forProcesses a shorter 2018, 6 period, x FOR PEER of REVIEW time, and therefore, the expected branching density is smaller.12 of 22 On the other hand, in the random branched polymers, the expected branching density is the same for all Table 6 shows the comparison between the emulsion polymerization and the random primary chains and any primary chain can become a branch chain._ A long primary chain having branching, in the case of living FRP. The number-average chain length rn is the same for both types large branchingof branched density polymers, could because be connected the average branchin as a branchg density chain is the insame. the On random the other branchinghand, in the process. Larger polymercase of molecules living FRP, the can branch be formed chains must in the be randomformed after branching the formation process, of the backbone compared chain. with The the living emulsion polymerization,branch chain is allowed leading to grow to aa largershorter period weight-average of time than the chain backbone length chain,rw. and It is is clearlyexpected shown to that the living emulsionbe shorter polymerizationthan the backbone giveschain. smallerIn addition, PDI the values, branch indicatingchains formed narrower later are subjected MWD. 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 forTable all primary 6. Comparison chains and of any the primary calculated chain average can become chain a branch lengths chain. for livingA long FRP.primary chain having large branching density could be connected as a branch chain in the random branching process. LargerRun polymer molecules Description can be formed inrn the random branchingrw process,PDI compared with _ the living emulsion polymerization,Emulsion Polym. leading to a larger 200 weight-average 254.2 chain length 1.27 rw. It is clearly L1 shown that the living emulsionRandom polymerization Branch gives 200 smaller PDI 351.1values, indicating 1.76 narrower MWD.

Table 6.Emulsion Comparison Polym. of the calculated 500average chain lengths 755.8 for living FRP. 1.51 L2 Random Branch 500_ 1478.9_ 2.96

Run Description rn rw PDI Emulsion Polym. 1000 1782.3 1.78 L1L3 Emulsion Polym. 200 254.2 1.27 RandomRandom Branch Branch 1000200 5115.1351.1 5.12 1.76 L2 Emulsion Polym. 500 755.8 1.51 EmulsionRandom Branch Polym. 1000500 2153.31478.9 2.15 2.96 L4 L3 RandomEmulsion Polym. Branch 1000 1000 11,540.9 1782.3 11.5 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 The full weight fraction distribution of the random branched polymers, whose primary chain length distributionThe full is givenweight byfraction Equation distribution (23) canof the be random determined branched by polymers, using the whose MC primary simulation. chain Figure7 shows thelength comparison distribution of is weight given by fraction Equation distribution,(23) can be determined between by using random the MC branching simulation. andFigure emulsion polymerization.7 shows The the formedcomparison MWDs of weight in L1–L3 fraction are distribution, distorted between because random of the branching complicated and emulsion distribution of polymerization. The formed MWDs in L1–L3 are distorted because of the complicated distribution the primaryof chains. the primary Both chains. from Both the from PDI th valuese PDI values in Table in Table6 and 6 and the the full full MWD MWD profiles profiles in inFigure Figure 7, it 7, it can be concludedcan thatbe concluded the MWD that is the narrower MWD is fornarrower the emulsion-polymerized for the emulsion-polymerized polymers, polymers, compared compared with the correspondingwith randomthe corresponding branched random polymers. branched Remember polymers. Remember that for that the for conventional the conventional FRP, FRP, the the results are totally opposite,results i.e.,are totally the MWDs opposite, are i.e., much the MWDs broader are mu forch the broader emulsion for the polymers,emulsion polymers, compared compared with random with random branching, in the case of conventional FRP. branching, in the case of conventional FRP.

Figure 7. ComparisonFigure 7. Comparison of the full of weight the full fractionweight fraction distribution distribution profiles profiles formed formed through through the the living livingemulsion polymerizationemulsion for thepolymerization conditions for L1–L4 the conditions (red) and L1–L the4 (red) corresponding and the corresponding random random branching branching (blue). (blue).

Processes 2018, 6, x FOR PEER REVIEW 13 of 22

3.2. Branching Density

3.2.1. Conventional Free-Radical Emulsion Polymerization Figure 8 shows the relationship between the branching density ρ and chain length r (degree of Processes 2018, 6, 14 13 of 21 polymerization). In the figure, each dot represents the branching density and the chain length of each polymer molecule generated in the MC simulation. The blue curve shows the estimate of _ 3.2.average Branching branching Density density having chain length r, ρ(r). As shown in the figure, the expected branching density increases as the chain length r increases for smaller polymers, but the value soon 3.2.1. Conventional Free-Radical Emulsion Polymerization reaches a constant limiting value, . The branching densities of branched polymers are Figure8 shows the relationship between the branching density ρ and chain length r (degree of distributed around the value of , not the average branching density of the whole reaction polymerization). In the figure, each dot represents the branching density and the chain length of each _ polymersystem, ρ molecule given in generated Table 3. It inis theinteresting MC simulation. to note that The the blue high curve MW showspolymers the that estimate form ofthe average second branchingsharp high density MW peak having of chainC3 and length C4 inr, Figureρ(r). As 2 shown still possess in the figure,the branching the expected density branching approximately density increasesbeing equal as theto chain. length r increases for smaller polymers, but the value soon reaches a constant limiting value, ρr→∞. The branching densities of branched_ polymers are distributed around the value of ρr→The∞, notblack the dotted average line branching shows the density value ofof the1/rn,p whole, which reaction seems system, to be aρ reasonablegiven in Table estimate3. It isof interesting, at leas tot notefor large that thebranching high MW probability polymers cases, that formC2–C4. the second sharp high MW peak of C3 and C4 in Figure2 still possess the branching density approximately being equal to ρr→∞.

FigureFigure 8.8. RelationshipRelationship between branching densitydensity andand chainchain lengthlength inin conventionalconventional free-radicalfree-radical emulsionemulsion polymerization,polymerization, forfor thethe conditionsconditions C1–C4.C1–C4.

_ TheFigure black 9 shows dotted the line expected shows the branching value of 1/densityrn,p, which of polymers seems to having be a reasonable chain length estimate r, ρ(r of). ρInr→ the∞, atfigure, least forthe largeindependent branching variable probability is change cases,d to C2–C4. the reduced chain length defined by:

Figure9 shows the expected branching density of_ polymers having chain length r, ρ(r). In the figure, the independent variable is changed to theζ = reduced r/rn,p. chain length defined by: (24)

_ _ In addition, the y-axis in Figure 9 shows the product of ρ(ζ) and rn,p. For all 4 cases, including C1, ζ = r/rn,p. (24) the curves are reduced to fall approximately on the same curve. The number-average chain length of _ Inprimary addition, chains, the y-axisrn,p is inkept Figure constant9 shows throughout the product the polymerization of ρ(ζ) and rn,p in. For the all present 4 cases, model including emulsion C1, thepolymerization, curves are reduced and therefore, to fall approximately the limiting value on the is samegiven curve. by: The number-average chain length of primary chains, rn,p is kept constant throughout the polymerization in the present model emulsion polymerization, and therefore, the limiting value is given by: . (25)

It is now found that the limiting branching density is simply given by the inverse of the ρζ→∞ = ρr→∞ = 1/rn,p. (25) number-average chain length of the primary polymer molecules. It is now found that the limiting branching density is simply given by the inverse of the number-average chain length of the primary polymer molecules.

Processes 2018, 6, 14 14 of 21 Processes 2018, 6, x FOR PEER REVIEW 14 of 22

_ Figure 9. Expected branching density of the polymer whose reduced chain length is ζ (= r/r ). Figure 9. Expected branching density of the polymer whose reduced chain length is ζ (= r/rn,p).

In the presentpresent modelmodel FRP, FRP, the the primary primary chains chains conform conform to to the the most most probable probable distribution. distribution. When When the primarythe primary chains chains with thewith most the probablemost probable distribution distribution are branched are branched randomly, randomly, the expected the branchingexpected _ density, ρ(r) is givenρ by: [25] branching density, (r) is given by: [25] √  √  Pb I2 2r/rn,p Pb ρ(r) =  √  (26) rn,p I1 2r/rn,p Pb (26) or  √  p I2 2ζ Pb ρ(ζ) rn,p = Pb  √  , (27) I1 2ζ Pb or where I2 is a modified Bessel function of the first kind of the second order, and Pb is the branching probability. For random branching, Pb = ρrn,p. Equation (27) shows that a universal curve exists for the relationship, between ρ(ζ)rn,p and(27)ζ, in the random branched polymer systems for a given branching probability, Pb. On the other hand, the value of Pb does not play a role in emulsion polymers, as shown in Figure9. whereFrom I2 is Equationa modified (26), Bessel one canfunction obtain of the the limiting first kind branching of the second density order,ρr→∞ ,and as follows:Pb is the branching _ probability. For random branching, Pb = ρrn,p. p ρr→ = Limρ(r) = P /r ._ _ (28) ∞ → b n,p Equation (27) shows that a universal cur rve∞ exists for the relationship between ρ(ζ)rn,p and ζ, in the random branched polymer systems for a given branching probability, Pb. On the other hand, the Because the value of Pb must be smaller than unity, i.e., Pb < 1, the limiting branching density, value of Pb does not play a role in emulsion polymers, as shown in Figure 9. ρr→∞ is always larger for the emulsion polymerization, compared with the corresponding random branchedFrom polymers. Equation The(26), lines one incan Figure obtain 10 the show limiting the relationship branching betweendensity ρ(ζ)rn,p, asand follows: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, ρr→∞ agrees with(28) the case with Pb = 1 in the random branched polymer system. BecauseIn the present the value branched of Pb must architecture, be smaller only than one unity, of two i.e., chainPb < 1, ends the limiting of a primary branching chain density, can be connected. is always In alarger branched for the polymer emulsion molecule, polymeriza theretion, always compared exists with a single the corresponding primary chain random whose both chain ends are unconnected and all the other primary chains are connected_ at_ their head units. branched polymers. The lines in Figure 10 show the relationship between ρ(ζ)rn,p and ζ for the The branching density of a polymer molecule, consisting of k branch points and r monomeric units is random branching, and that for the emulsion polymerization, in particular the data for C4. Although given by: the emulsion polymerization shows slightly 0 slower
increase, but the limiting value, agrees k r/r n,p − 1 0 ρ = = = 1/r n,p − 1/r, (29) with the case with Pb = 1 in the randomr branchedr polymer system. In the0 present branched architecture, only one of two chain ends of a primary chain can be where rn,p is the number-average chain length of the primary chains that constitute the branched connected. In a branched polymer molecule, there 0always exists a single primary chain whose both polymer molecule. For large polymer, ρr→∞ = 1/rn,p. Therefore, Equation (25) means to show that chain0 ends are unconnected and all the other primary chains are connected at their head units. The rn,p = rn,p for large polymers in the conventional emulsion polymerization. For large emulsion branching density of a polymer molecule, consisting of k branch points and r monomeric units is polymers, no discrimination of the size of the primary chain exists√ in connecting branch chains. On the given by: 0 other hand, for the random branched polymers, rn,p = rn,p/ Pb > rn,p, which means that larger primary chains are connected preferentially in the large-sized polymers. , (29)

Processes 2018, 6, x FOR PEER REVIEW 15 of 22 where is the number-average chain length of the primary chains that constitute the branched polymer molecule. For large polymer, . Therefore, Equation (25) means to show that

for large polymers in the conventional emulsion polymerization. For large emulsion polymers, no discrimination of the size of the primary chain exists in connecting branch chains. On the other hand, for the random branched polymers, , which means that larger Processesprimary2018 chains, 6, 14 are connected preferentially in the large-sized polymers. 15 of 21

_ Figure 10. Expected branching density of the polymer whose reduced chain length is ζ (=r/rn,p). Figure 10. Expected branching density of the polymer whose reduced chain length is ζ (=r/rn,p). The The lines are for random branching with the given P -values, and the circular symbols are for the lines are for random branching with the given Pb-values,b and the circular symbols are for the conventional emulsion polymerization, C4.

3.2.2. Living Free-Radical Emulsion Polymerization Figure 11 shows the relationship between the branching density and chain length for living free-radical emulsionemulsion polymerization. polymerization. Each Each dot dot represents represents the branchingthe branching density density and the and chain the lengthchain lengthof each of polymer each polymer molecule molecule generated generated in the MCin the simulation. MC simulation. The blue The curve blue showscurve shows the estimate the estimate of the average branching density having chain length r, ρ(r). The_ discrete arrays of dots correspond to the ofProcesses the average 2018, 6, x branching FOR PEER REVIEW density having chain length r, ρ(r). The discrete arrays of dots correspond16 of 22 class of polymers having k = 0, 1, 2, ... branch points. Because the number of “arrays” is small for L1 to the class of polymers having k = 0, 1, 2, … branch points. Because the number of “arrays” is small andlarger L2, for the smaller curve polymers, for ρ(r) is notwhile_ smooth. it becomes It is interesting smaller than to the note random that in branch the case for of large L4, for chain which length the for L1 and L2, the curve for ρ(r) is not smooth. It is interesting to note that in the case of L4, for averageregion. ThePb-value behavior is largest is quite with complicated.Pb = 0.833 as_ listed in Table3, the curve of ρ(r) shows an_ overshooting whichbehavior the that average was notPb-value observed is largest in the with conventional Pb = 0.833 FRPs. as listed in Table 3, the curve of ρ(r) shows an overshooting behavior that was not observed in the conventional FRPs. The black dotted lines in Figure 11 show the values of for the conventional emulsion polymerization. It is difficult to judge if these values apply also for living polymers, because the _ curve of ρ(r) does not reach a constant value for L1 to L3. For L4, the limiting value for the living polymers, represented by the blue line, is clearly smaller than that for the corresponding conventional FRP given by the black dotted line. It seems that the limiting value, is close to _ the average branching density of the whole reaction system, that is ρ = 0.005 given by the blue horizontal line. _ Figure 12 shows the expected branching density, ρ(r). The red lines are for the living emulsion polymerization, and the blue lines are for the random branched polymers whose primary chain length distribution is the same as for the corresponding living emulsion polymerization. For L1 and L2, the expected branching density of the large polymers seems to be larger for the emulsion polymers. On the other hand, however, the branching probability Pb is small for these conditions, Figure 11. Relationship between the branching density and the chain length in living free-radical and itFigure is difficult 11. Relationship to make discussions between the on branching the limiting density branching and the chaindensity, length in .living For L3, free-radical the limiting emulsion polymerization, for the conditions L1–L4. valueemulsion seems to polymerization, be almost the for same, the conditions but the emulsi L1–L4.on polymers show a larger branching density for the transient region of chain lengths. For L4, the branching density of the emulsion polymers is The black dotted lines in Figure 11 show the values of ρr→∞ for the conventional emulsion polymerization. It is difficult to judge if these values apply also for living polymers, because the curve of ρ(r) does not reach a constant value for L1 to L3. For L4, the limiting value for the living polymers, represented by the blue line, is clearly smaller than that for the corresponding conventional FRP given by the black dotted line. It seems that the limiting value, ρr→∞ is close to the average branching density of the whole reaction system, that is ρ = 0.005 given by the blue horizontal line. Figure 12 shows the expected branching density, ρ(r). The red lines are for the living emulsion polymerization, and the blue lines are for the random branched polymers whose primary chain length distribution is the same as for the corresponding living emulsion polymerization. For L1 and L2,

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

3.3. Radius of Gyration In the present MC simulation method, one can observe the structure of each polymer molecule directly, and any type of detailed structural information can be extracted. The mean-square radius of gyration for the unperturbed random-flight chain of each polymer molecule, 0 can be determined from the Wiener index (WI) [26]. The relationship between 0 and WI is given by: [27]

0/L2 = WI/N2, (30) where L is the random-walk segment length, and N is the number of random-walk steps in the given polymer. Obviously, N is given by: N = r/u, (31) where u is the number of monomeric units in a random-walk segment, and u = 5 was used for the actual calculation. Obviously, the “random-walk segment” is different from the “segment” introduced for the MC simulation of the living emulsion polymerization. In the present simulation, the Wiener Index method is used for N < 1000. With the Wiener index method, the obtained mean-square radius of gyration is exact, with no statistical errors involved. The Wiener index method is an excellent method especially for smaller polymers whose standard

Processes 2018, 6, x FOR PEER REVIEW 16 of 22 larger for smaller polymers, while it becomes smaller than the random branch for large chain length region. The behavior is quite complicated.

Processes 2018, 6, 14 16 of 21 the expected branching density of the large polymers seems to be larger for the emulsion polymers. On the other hand, however, the branching probability Pb is small for these conditions, and it is difficult to make discussions on the limiting branching density, ρr→∞. 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,Figure while 11. Relationship it becomes smallerbetween thanthe branching the random density branch and for the large chain chain length length in living region. free-radical The behavior is quiteemulsion complicated. polymerization, for the conditions L1–L4.

Figure 12. ExpectedExpected branching branching density density of of the the polymer with chain length r, for living emulsionemulsion polymerization (red) and random branchingbranching (blue).(blue).

3.3. Radius of Gyration In the present MC simulation method, one can observeobserve the structure of each polymer molecule directly, andand anyany typetype of detailed structural information can be extracted. The mean-square radius of gyration for the unperturbed random-flight chain of each polymer molecule, 0 can be determined gyration for the unperturbed random-flight chain of each polymer molecule, 0 can be determined from the Wiener index (WI) [26]. The relationship between 0 and WI is given by: [27] from the Wiener index (WI) [26]. The relationship between 0 and WI is given by: [27]

0/L2 = WI/N2, (30) 2 2 2 0/L = WI/N , (30) where L is the random-walk segment length, and N is the number of random-walk steps in the given wherepolymer.L is Obviously, the random-walk N is given segment by: length, and N is the number of random-walk steps in the given N polymer. Obviously, is given by: N = r/u, (31) N = r/u, (31) where u is the number of monomeric units in a random-walk segment, and u = 5 was used for the whereactual ucalculation.is the number Obviously, of monomeric the “random-walk units in a random-walk segment” segment, is different and ufrom= 5 wasthe used“segment” for the actualintroduced calculation. for the Obviously,MC simulation the “random-walk of the living emulsion segment” polymerization. is different from the “segment” introduced for theIn MCthe present simulation simulation, of the living the Wiener emulsion Index polymerization. method is used for N < 1000. With the Wiener index method,In the the present obtained simulation, mean-square the Wiener radius Index of gyration method is is exact, used forwithN 0 in each polymer is large. For larger polymers with N > 1000, the actual random 2 walk process in the 3D space was conducted for 100 times to estimate the value of 0.

3.3.1. Conventional Free-Radical Emulsion Polymerization Figure 13 shows the relationship between the mean-square radius of gyration represented by 2 2 2 2 0/L and chain length r divided by u. In the figure, each dot represents the value set of 0/L and r/u of the individual polymer molecule obtained in the MC simulation, and the blue solid curve 2 2 with circular symbols shows the average value of 0/L within the intervals of ∆r, which is the estimate of the average mean-square radius of gyration for the polymers having chain length r. Processes 2018, 6, x FOR PEER REVIEW 17 of 22

deviation of 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 0.

3.3.1. Conventional Free-Radical Emulsion Polymerization Figure 13 shows the relationship between the mean-square radius of gyration represented by 0/L2 and chain length r divided by u. In the figure, each dot represents the value set of 0/L2 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 0/L2 within the intervals of Δr, which is the estimate Processesof the average2018, 6, 14 mean-square radius of gyration for the polymers having chain length r. 17 of 21

2 Figure 13. Relationship between the mean-square radius of gyration, and the chain length, r for Figure 13. Relationship between the mean-square radius of gyration, 0 and the chain length, r for the conventionalconventional emulsionemulsion polymerization polymerization conditions, conditions, C1–C4. C1–C4. The The red red dots dots show show the propertythe property of each of 2 polymereach polymer molecule molecule simulated, simulated, and the blueand curvethe blue with curve circular with symbols circular is the symbols expected is < sthe>0 –valueexpected for the given r. The black solid and dotted curves are for the random branched polymers, having the same 0–value for the given r. The black solid and dotted curves are for the random branched polymers, primaryhaving the chain same length primary distribution. chain length See moredistribution. details inSee the more text. details in the text.

2 2 2 2 In FigureFigure 13 13,, the the solid solid black black curve curve shows shows the value the ofvalue 0of/L 0 the/L randomfor the branchedrandom branched polymers whosepolymers primary whose polymer primary chain polymer length distributionchain length is thedistribution same as the is presentthe same model as the emulsion present polymers. model Foremulsion the random polymers. branched For the polymers random whose branched primary polymers chain whose length primary distribution chain conforms length distribution to the most probableconforms distributionto the most isprobable given by distribution [28]: is given by [28]:

− < s2 > r  m  4m  0.5 0 = 1 + r + r , (32) L2 6u 7 9π , (32)

where m_ r is the average number of branch points for the polymers with chain length r, which is givenwhere by:mr is the average number of branch points for the polymers with chain length r, which is given by: mr = rρ(r) (33) _ _ In Equation (33), ρ(r) is the average branching densitymr = rρ of(r). the polymer molecules having chain length(33)r. For a random_ branched polymer system whose primary chains conform to the most probable ρ distribution,In Equation (33),ρ(r) is( givenr) is the by average Equation branching (26), and density the black of solid the polymer curve shows molecules such calculation having chain results. length It isr. clearly shown that the emulsion polymerization leads to more compact structure, compared with For a random branched polymer system whose primary chains conform to the most probable the random branched_ polymers. ρ distribution,On the other(r) is hand, given however, by Equation the limiting(26), and branching the black solid density curveρr→ shows∞, or the such branching calculation density results. of It is clearly shown that the emulsion polymerization leads= to more compact structure, compared large polymers is√ larger for the emulsion polymers, i.e., ρr→∞ 1/rn,p for the emulsion polymerization whilewith theρr→ random∞ = branchedPb/rn,p for polymers. the random branching. Therefore, compact molecular architecture in emulsion polymerization might be resulted from larger branching density. In order to eliminate the factor of different branching density level, the expected branching density ρ(r) shown by the blue curve in Figure8 is used for Equation (33). The calculated results are shown by the black dotted curves in Figure 13. The ratio of limiting branching√ density ρr→∞ between the emulsion polymerization and corresponding random branching is 1/ Pb, and therefore, the difference between the black solid and dotted lines is large for C1 (Pb = 0.2857), and is small for C4 (Pb = 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. Processes 2018, 6, x FOR PEER REVIEW 18 of 22

On the other hand, however, the limiting branching density , or the branching density of

large polymers is larger for the emulsion polymers, i.e., for the emulsion

polymerization while for the random branching. Therefore, compact molecular architecture in emulsion polymerization might be resulted from larger branching density. In order to _ eliminate the factor of different branching density level, the expected branching density ρ(r) shown by the blue curve in Figure 8 is used for Equation (33). The calculated results are shown by the black dotted curves in Figure 13. The ratio of limiting branching density between the emulsion

Processes 2018polymerization, 6, 14 and corresponding random branching is , and therefore, the difference 18 of 21

between the black solid and dotted lines is large for C1 (Pb = 0.2857), and is small for C4 (Pb = 0.8333). In all cases, the expected mean-square radius of gyration is smaller for the emulsion polymers, In thewhich conventional shows the non-randomness emulsion FRP, in thethe expecteddistribution branchingof branch points density formed of thein emulsion primary chains formed inpolymerization the earlier stage leads to of compact polymerization 3D architecture is muchin the conventional larger than FRP. those formed in the later stage of polymerizationIn the [5 ].conventional The primary emulsion chains FRP, with the largerexpected branching branching densitydensity formof the hubsprimary to connectchains a large formed in the earlier stage of polymerization is much larger than those formed in the later stage of number of primary chains in a polymer molecule, leading to form a star-like compact architecture. polymerization [5]. The primary chains with larger branching density form hubs to connect a large number of primary chains in a polymer molecule, leading to form a star-like compact architecture. 3.3.2. Living Free-Radical Emulsion Polymerization 3.3.2. Living Free-Radical Emulsion Polymerization Figure 14 shows the relationship between the mean-square radius of gyration and chain length. Each red dot representsFigure 14 shows the the mean-square relationship between radius the ofmean-square gyration radius of the of gyration individual and chain polymer length. molecule Each red dot represents the mean-square radius of gyration of the individual polymer molecule obtained inobtained the MC in simulation,the MC simulation, and and the the blue blue solid solid curvecurve with with circular circular symbols symbols shows showsthe estimate the of estimate of 2 the expectedthe 0 of < polymerss2>0 of polymers having having chain chain length length rr. .

Figure 14. Relationship between the mean-square radius of gyration, 0 and2 the degree of Figure 14. Relationship between the mean-square radius of gyration, 0 and the degree of polymerization, r for the living emulsion polymerization conditions, L1–L4. The red dots show the 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

property ofexpected each polymer 0–value molecule for the given simulated, r. The black and thedotted blue curve curve with with × -symbol circular is symbolfor the random is the expected 2 0–valuebranched for the polymers, given havingr. The the black same dottedprimary chai curven length with distribution×-symbol and is the for average the random branching branched polymers, havingdensity. the same primary chain length distribution and the average branching density.

The black dotted curve with X-symbols shows the value of 0/L2 of the random branched 2 2 The blackpolymers dotted whose curve primary with polymer X-symbols chain length shows distribution, the value as of well as0 /theL averageof the randombranching branched polymers whosedensity, primaryis the same polymer as the present chain model length emulsion distribution, polymers. The aswell mean-square as the averageradius of gyration branching of density, the random branched polymers was estimated by the MC simulation, in which the primary chains 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 Figure6 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 Assuming an idealized emulsion polymerization condition, a Monte Carlo simulation is conducted for both conventional and living FRP, which involves chain transfer to polymer, resulting in forming polymers with long-chain branches. The molecular weight distribution (MWD) of the emulsion polymers formed in conventional FRP is very broad, and high molecular weight polymers conform to the power law, W(r) = r−α with the power exponent α given by α = 1/Pb, where Pb 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 Pb is large, the second sharp high MW peak appears because of the limitation of the particle size. The MWD of emulsion polymers formed in living FRP is much narrower than that of the conventional FRP. The MWD is so narrow that one may overlook the existence of long-chain Processes 2018, 6, 14 19 of 21 branches from the measured MWD data. The MWD is narrower than the corresponding random branched system. The branching density of large polymers reaches a constant value, ρr→∞ for both conventional and living emulsion FRPs, as well as for the random branched polymers.√ For the conventional FRP, a simple relationship, ρr→∞= 1/rn,p is found. This limiting value is 1/ Pb−times larger for the emulsion polymerization, compared with the corresponding random branched polymers. For the living emulsion FRP, the relationship with the corresponding random branched polymers concerning the value of ρr→∞ is rather complex, as was shown in Figure 12. 2 For the conventional emulsion FRP, the mean-square radii of gyration, 0 for the given chain length r is smaller than the corresponding random branched polymers. On the other hand, for the 2 living emulsion FRP, the value of 0 for the given chain length r is essentially the same as for the random branched polymers. The present model analysis promises to offer a broad perspective for designing and controlling the branched polymer architecture both in the conventional and living emulsion polymerization.

Conflicts of Interest: The author declares no conflict of interest.

Nomenclature The following symbols and acronyms are used in this manuscript: Symbols [CTA]p Concentration of the chain transfer agent in the polymer particle CfCTA Chain transfer constant to the chain transfer agent Cfm Chain transfer constant to monomer Cfm Chain transfer constant to polymer CP 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) I1 Modified Bessel function of the first kind and of the first order I2 Modified Bessel function of the first kind and of the second order k Number of branch points in the polymer molecule kp Propagation rate constant L Random-walk segment length [M]p Monomer concentration in the polymer particle mr 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 Npr(r) Number-based chain length distribution of the primary chains Nsa(r) Number-based chain length distribution during a single active period in living FRP N(r) Number fraction distribution Probability that an active radical connects another monomer unit during the active period in p living FRP Branching probability, which gives the probability that the chain end of a primary chain is P b connected to a backbone chain Probability that the chain end of a segment formed during a single active period is connected P bs to a backbone chain in living FRP Polymer concentration in the polymer particle, represented by the total number of monomeric [P] p units in polymer Rp Polymerization rate Deactivation rate in the reversible deactivation process in controlled/living R deact radical polymerization r Chain length (number of monomeric units in the polymer) rn Number-average chain length of the product polymers rn,p Number-average chain length of the primary chains Processes 2018, 6, 14 20 of 21

Number-average chain length of the primary chains that constitute a branched r0 n,p polymer molecule Number-average chain length of the sequence of chains (segments) formed during a single r n,s active period in living FRP rw Weight-average chain length of the product polymers 2 0 Mean-square radius of gyration of unperturbed chain tav 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

Wliv(r) Weight fraction distribution of the ideal linear living FRP Weight fraction distribution of the polymer chains formed through the ideal linear living FRP W (r) CP 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 Ratio of the reaction rates between the deactivation in living FRP and the propagation, defined δ by Equation (8) ζ Reduced chain length, defined by r/rn,p ρ Branching density ρ(r) Expected branching density of the polymer molecules having chain length r Dimensionless parameter representing the ratio of reaction rates between various types of τ 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 = rw/rn RAFT Reversible Addition–Fragmentation Chain-Transfer RDRP Reversible-Deactivation Radical Polymerization SRMP Stable-Radical-Mediated Polymerization

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