#
Universal Relationships in Hyperbranched Polymer Architecture for Batch and Continuous Step Growth Polymerization of AB_{2}-Type Monomers

## Abstract

**:**

_{2}type monomer is investigated by a Monte Carlo simulation method, and the calculation was conducted for a batch and a continuous stirred-tank reactor (CSTR). In a CSTR, a highly branched core region consisting of units with large residence times is formed to give much more compact architecture, compared to batch polymerization. The universal relationships, unchanged by the conversion levels and/or the reactivity ratio, are found for the mean-square radius of gyration Rg

^{2}, and the maximum span length L

_{MS}. For batch polymerization, the g-ratio of Rg

^{2}of the HB molecule to that for a linear molecule conforms to that for the random branched polymers represented by the Zimm-Stockmayer equation. A single linear equation represents the relationship between Rg

^{2}and L

_{MS}, both for batch and CSTR. Appropriate process control in combination with the chemical control of the reactivity of the second B-group promises to produce tailor-made HB polymer architecture.

## 1. Introduction

_{2}-type monomer and self-condensing vinyl polymerization (SCVP). The former is a classical synthetic route originally considered by Flory [4], but recent development in polymer chemistry has made it possible to change the chemical reactivity of the second B group freely [5], and has established the chemical control method for the branching frequency.

_{2}type monomer, considered in this article. In the figure, T is the terminal unit with both B’s being unreacted, L is the linearly incorporated unit with one of two B’s being reacted, and D is the dendritic unit in which both B’s have reacted. The reaction rate constant, k

_{T}is for the reaction between an A group and a B group in T, while k

_{L}is for the reaction between A and B in L. The reactivity of the second B group is represented by the reactivity ratio, r defined by:

_{L}/k

_{T}

_{A}= 0.9. The figure was prepared using the unpublished data obtained in the investigation reported earlier [6]. In the figure, each red dot shows a pair of values, DB and P, for each polymer molecule generated in the MC simulation. The circular symbols with a blue line show the average DB within each intervals of P, which shows the expected DB for the given P-value, $\overline{DB}(P)$. The DB-value converges to DB

_{inf}, as the degree of polymerization P increases, i.e., for large polymers. On the other hand, the black broken line shows the magnitude of average DB of the whole reaction system. It is clearly shown that the values of DB are distributed around DB

_{inf}, rather than the average DB of the whole system, and DB

_{inf}is larger than the average DB. These characteristics hold true irrespective of the magnitude of reactivity ratio r, not only for a batch reactor [7] but also for a CSTR [8,9]. In this article, the HB architecture of large polymers, for which $\overline{DB}(P)$ has reached a constant value DB

_{inf}, is investigated in detail, both for a batch reactor and a CSTR.

_{inf}is that the magnitude of DB

_{inf}is essentially kept constant, irrespective of the conversion level for a given reactivity ratio r, both for a batch reactor [6,7] and for a CSTR [8,9]. Note that the average DB of the whole reaction system increases with conversion, but DB

_{inf}does not change. The value of DB

_{inf}can be estimated by the reactivity ratio r, as shown in Figure 4. Obviously, DB

_{inf}= 1 for the cases with r = ∞, both for batch and CSTR, but the value of DB

_{inf}for a CSTR is always larger than that for a batch reactor, as long as r is finite. Incidentally, analytic relationship between DB

_{inf}and r was established previously [6,7], and a smooth curve was drawn for batch polymerization in Figure 4. On the other hand, a general equation for a CSTR has not been reported, and five data points reported in the earlier publication [9] were plotted and connected.

^{2}by the following equation,

_{u}

_{= 1}is the value of WI when u = 1. Rg

^{2}is the value of ${\langle {s}^{2}\rangle}_{0}$ normalized by the squared monomer-unit length, and is proportional to ${\langle {s}^{2}\rangle}_{0}$, at least for large polymers, P >> 1. Note that Rg

^{2}is equal to the value of u${\langle {s}^{2}\rangle}_{0}$/l

^{2}, which is unchanged irrespective of the magnitude of u, as long as the number N of steps is large enough.

^{2}of the polymer molecule whose degree of polymerization is P for batch polymerization with the reactivity ratio, r = 1. As shown in the figure, the relationship between Rg

^{2}and P does not change with the progress of conversion, x

_{A}[6,7]. The curve moves to smaller Rg

^{2}, as the reactivity ratio r is increased [7]. However, even with r = ∞ for which DB = 1, Rg

^{2}is much larger than that for the perfect dendron [7], which is shown by the black curve in Figure 4. For large polymers, the power law Rg

^{2}~ P

^{0.5}is valid, irrespective of the magnitude of r [6,7]. The power exponent, 0.5 is the same as for the random branched polymers, represented by the Zimm-Stockmayer equation [15]. In this article, the universal relationship concerning Rg

^{2}, that is independent of r, will be reported, and the relationship with the Zimm-Stockmayer equation will be discussed quantitatively.

^{2}for large polymers is quite large. It is not perfectly clear, but the relationship between Rg

^{2}and P does not change significantly, even when the steady state conversion level is changed [8,9]. It was clearly demonstrated that a CSTR produces polymers with much smaller Rg

^{2}compared with batch polymerization, even when the value of DB

_{inf}is deliberately set to be the same for both types of reactors [8]. In this article, the reason for obtaining much more compact architecture in a CSTR is explored by considering the properties of the largest tri-branched cluster in a polymer molecule. The tri-branched clusters are shown by the group of D units surrounded by the red closed frames in Figure 2, and the largest cluster for this example consists of six units. Note that although the focal point unit is the D type, it is connected to only two other units and it is not considered as a tri-branched unit. A universal relationship concerning Rg

^{2}, independent of the steady state conversion level, will also be reported for a CSTR.

_{MS}. Here, the span length refers to the distance in the monomeric units [16], and L

_{MS}is equivalent to the longest end-to-end distance [17]. In the case of HB polymer shown in Figure 2, L

_{MS}= 12, which is the distance between the units with a star. There are two routes having L

_{MS}= 12, starting from the unit with a star to the unit with star 1 or 2. Interesting universal relationships will be reported for the magnitude of L

_{MS}.

^{2}and L

_{MS}. For Rg

^{2}and L

_{MS}, the universal relationships are sought. Note that the universal relationships reported so far for Rg

^{2}are with respect to the conversion level, and they change with the reactivity ratio r. In this article, further unification is explored.

## 2. Methods

^{4}polymer molecules with P > 50 were collected to determine statistically valid estimates. The reactivity ratios investigated were r = 0.5, 1, 2, 5, and ∞.

_{0}is the initial concentration of A group, or equivalently, the initial monomer concentration, and $\overline{t}$ is the mean residence time [3]. The HB polymer shown in Figure 2 was generated for a CSTR with the condition, r = 2 and ξ = 0.35. The conversion of A group, x

_{A}increases with ξ. To set the value of ξ corresponds to fixing the steady state conversion level, x

_{A}.

_{UL}= 0.5 for r = 1, and ξ

_{UL}= 0.25 for r = ∞. In the present investigation, the MC simulations were conducted for the cases with ξ ≤ ξ

_{UL}.

_{ij}}, where d

_{ij}is the distance in the number of monomeric units between the ith and jth unit. The WI when u = 1 is given by [13,14]:

_{MS}was determined by finding the largest value of d

_{ij}in the distance matrix.

## 3. Results and Discussion

#### 3.1. Largest Cluster of Tri-Branched Units

_{LC}belonging to the largest cluster and the number P of units in the polymer molecule (degree of polymerization). Each dot shows a set of values for each polymer molecule generated in the MC simulation. For a CSTR, the cases with ξ = ξ

_{UL}are shown in the figure, while x

_{A}= 0.95 for batch polymerization. General characteristics were the same for the other reaction conditions. For a CSTR, the largest cluster size P

_{LC}increases with P, and a very large tri-branched cluster exists in a large polymer molecule. On the other hand, the size of the largest cluster does not increase significantly for batch polymerization, except for r = ∞. With r = ∞, all units other than the peripheral T units and a focal point are tri-branched units, and the largest cluster size is essentially proportional to P with P

_{LC}≈ 0.5P. Except for r = ∞, the existence of a large cluster of tri-branched units is an important characteristic of polymers formed in a CSTR, while a large number of small-sized clusters are formed in batch polymerization. It is reasonable to consider that the dimension is smaller for the HB polymers formed in a CSTR. The cases with r = ∞ will be discussed later, and consider the properties of various types of clusters for the cases with the reactivity ratio, r ≤ 5 first.

_{LC,2}. In the present example shown in Figure 8, P

_{LC,2}= 2.

_{LC,2}and P for batch and CSTR with r = ∞. For a CSTR, ξ = ξ

_{UL}= 0.25, and x

_{A}= 0.95 for batch polymerization. In the case of a CSTR, P

_{LC,2}increases with P, and therefore, a very large tri-branched cluster of the second order exists in a large polymer molecule. On the other hand, it does not increase significantly for batch polymerization, which means a large number of smaller-sized tri-branched clusters of the second order are formed. The smaller Rg

^{2}obtained for a CSTR could be understood from the significant differences in the magnitude of P

_{LC,2}.

#### 3.2. Radius of Gyration and Maximum Span Length

^{2}defined by Equation (5) and the maximum span length L

_{MS}exemplified in Figure 2 are the characteristic factors describing the spatial size of an HB polymer. In this section, the universal relationships concerning Rg

^{2}and L

_{MS}are explored both for a batch (Section 3.2.1) and a CSTR (Section 3.2.2).

#### 3.2.1. Batch Polymerization

^{2}and P with r = 1 at x

_{A}= 0.95. Each red dot represents a set of Rg

^{2}and P, generated in the MC simulation. Note that the data were collected for P > 50 to clarify the statistical properties of large polymers, where $\overline{DB}(P)$ has reached a constant value, DB

_{inf}. Blue circular symbols show the averages within ΔP fractions, and therefore, the blue line connecting these points represents the expected Rg

^{2}-value for a given P.

^{2}-values that correspond to the blue curve in Figure 11 for various combinations of the reactivity ratio r and conversion x

_{A}. The curve for the expected Rg

^{2}-value does not change with the conversion level x

_{A}, and becomes smaller as the reactivity ratio r increases, as already reported earlier [6,7].

^{2}is the defined by Equation (5), and therefore:

_{inf}P/2. The value of DB

_{inf}is a constant for a given reactivity ratio, and the value of DB

_{if}P/2 is equal to the average number ${\overline{n}}_{\mathrm{b}}$ of branch points per molecule for large polymers. Note that DB = 2D/P for large polymers, and ${\overline{n}}_{\mathrm{b}}$= DB

_{inf}P/2. Because the Rg

^{2}-value for a given P, as well as the magnitude of DB

_{inf}, is the same at any conversion level x

_{A}, the calculated results for x

_{A}= 0.95 with various r’s are shown in Figure 13. All points fall on a single curve, showing a universal relationship, independent of x

_{A}and r.

_{inf}for a given r can be calculated analytically [6,7], and Figure 4 shows the calculated results graphically. Therefore, the value of Rg

^{2}can be determined by using Equation (11) in a straightforward manner, without conducting the MC simulation, for any combination of r and x

_{A}.

_{MS}. Figure 15 shows the MC simulation results for the relationship between L

_{MS}and P with r = 1 and x

_{A}= 0.95 for batch polymerization. Each red dot represents a set of L

_{MS}and P, generated in the MC simulation. Blue circular symbols show the averages within ΔP fractions, and the blue line connecting these points represents the expected L

_{MS}-value for a given P.

_{MS}-value for a given P, corresponding to the blue curve in Figure 15, for various combinations of the reactivity ratio r and conversion x

_{A}. The curve for the expected L

_{MS}-value does not change with the conversion level x

_{A}, and is a function of the reactivity ratio r. The qualitative tendency is quite similar to Rg

^{2}.

_{MS}/P for a given DB

_{inf}P/2. Because the L

_{MS}-value for a given P, as well as the magnitude of DB

_{inf}, is the same at any conversion level x

_{A}, the calculated results for x

_{A}= 0.95 with various r’s are shown in Figure 17. Note that the value of DB

_{inf}P/2 is equal to the average number of branch points per molecule for large polymers. All data points fall nicely on the same universal curve, as in the case of Rg

^{2}.

^{2}and L

_{MS}show a similar universal relationship, as shown in Figure 13 and Figure 17. Now, consider the relationship between Rg

^{2}and L

_{MS}.

^{2}and L

_{MS}for r = 1 and x

_{A}= 0.95, which shows a linear relationship. The blue line with circular symbols shows the expected Rg

^{2}for a given L

_{MS}.

^{2}at various conversion levels for r = 1. The plotted points shown by the blue circular symbols are the same as those in Figure 18a. The relationship is essentially unchanged by the conversion level, and the relationship fits reasonably well with:

^{2}for a given L

_{MS}, for various combinations of x

_{A}and r. A linear relationship seems to hold for any value of r. The black straight line shows the linear relationship given by Equation (12). Although a slight discrepancy is observed in the cases of r = 5 and ∞ for large polymers, the data points are well correlated with Equation (12). Equation (12) is the universal relationship between Rg

^{2}and L

_{MS}, applicable to any combination of r and x

_{A}in batch polymerization.

^{2}= 0.18 L

_{MS}+ 0.6 was reported both for a batch and a CSTR [26]. The proportional coefficient, 0.18 is the same as Equation (12), and the constant term is very close.

_{MS}is equal to P, and the following equation is valid for large polymers:

^{2}is the mean-square radius of gyration when each monomeric unit is considered as the random walk segment.

^{2}other than its own chain with P = L

_{MS}. In the HB polymers, the chains other than the largest span chain can make a contribution to increase the Rg

^{2}-value. The increase in the coefficient from 0.167 to 0.18 could be considered as showing the degree of contribution from the other chains to the magnitude of Rg

^{2}.

_{MS}-values is given by:

^{2}is still mainly determined by the maximum span chain, and the contribution of the other chains is not very significant. The exact physical meaning of the magnitude of coefficient is still an open question. However, the linear relationship found here is of great interest.

^{2}can be determined analytically without MC simulation, as discussed earlier. By using Equation (12), the magnitude of L

_{MS}can also be estimated in a straightforward manner, without relying on the MC simulation.

#### 3.2.2. CSTR

^{2}and L

_{MS}are considered, as was done in the previous section. For a CSTR, however, the variance of Rg

^{2}for large polymer is quite large [8,9], and it is difficult to determine the statistically valid expected Rg

^{2}-values for large polymers. Inspired by the universal curve shown in Figure 13, the value of g-ratio defined by Equation (9) is plotted with respect to the number of branch points in a polymer molecule, n

_{b}. Figure 21a shows the case with r = 1 at ξ = 0.5. In the figure, each red dot indicates a set of values for the polymer molecule generated in the MC simulation. With this type of plot, the variance of g for large polymers is rather small, and it is straightforward to determine the expected g-ratio for a given n

_{b}, shown by the blue curve with circular symbols.

_{b}, unchanged by ξ, is confirmed also for the other r cases, as shown in Figure S1 of Supplementary Materials.

_{UL}are shown in Figure 22. In order to magnify the small differences for smaller g-values, the logarithmic scale plot was used for Figure 22b. Slight differences among curves are observed, and the expected g-ratio is considered a very weak function of r. In particular, the change for r < 5 is not very significant.

_{MS}for the HB polymers formed in a CSTR. Figure 23a shows the relationship between the weight fraction of the maximum span chain L

_{MS}/P and n

_{b}for r = 1 with ξ = 0.5. Each red dot shows the individual data point, and the blue curve with circular symbols shows the expected value of L

_{MS}/P for a given n

_{b}. Figure 23b shows the expected L

_{MS}/P for various values of ξ, i.e., for different conversion levels at steady state. The expected values of L

_{MS}/P do not change with ξ, and another universal relationship is found. The universal relationship between L

_{MS}/P and n

_{b}, unchanged by ξ, is confirmed also for the other r cases, as shown in Figure S2 of Supplementary Materials.

_{UL}is shown for each reactivity ratio. Figure 24a is the normal scale plot, which shows the differences among the curves are rather small. To enlarge the differences for smaller L

_{MS}/P-values, the logarithmic plot is used for the y-axis of Figure 24b, and it is shown that up to r = 5, the differences are rather small, but the curve with r = ∞ shows slightly larger L

_{MS}/P-values. Similarly with the g-ratio for a CSTR, the value of L

_{MS}/P is a very weak function of the reactivity ratio r.

^{2}and L

_{MS}, as was done for batch polymerization in Figure 18 and Figure 19, which showed a universal relationship, represented by Rg

^{2}= 0.18L

_{MS}+ 1.

^{2}and L

_{MS}for HB polymers formed in a CSTR with r = 1 and ξ = 0.5, which shows a linear relationship. Figure 25b shows the expected Rg

^{2}at various ξ-values for r = 1. The relationship is essentially unchanged by the steady-state conversion level. The black line represents the relationship given by Equation (11). Although a slight deviation is observed for large values of L

_{MS}, overall agreement is satisfactory. Compared with Figure 18, the absolute values of Rg

^{2}and L

_{MS}are smaller for the case of CSTR, because of much more compact architecture formed in a CSTR.

^{2}-values for various combinations of ξ and r are shown in Figure S3 of Supplementary Materials. The universal relationship, represented by Equation (12), correlates reasonably well, irrespective of the magnitude of ξ and r, and Equation (12) could be considered as a universal relationship that holds for both batch and CSTR. Even though a CSTR leads to form much more compact HB polymers, such difference in branched structure does not affect the relationship, Rg

^{2}= 0.18L

_{MS}+ 1.

^{2}-value of the HB polymers formed with r = ∞ in a CSTR is still much larger than that for the perfect dendron [9].

_{inf}for a given r is represented graphically in Figure 4. At least approximately for large polymers, n

_{b}is estimated to be n

_{b}= DB

_{inf}P/2. Therefore, the magnitude of Rg

^{2}can be estimated from Figure 22, and the value of L

_{MS}could also be estimated by using the relationship, Rg

^{2}= 0.18L

_{MS}+ 1.

## 4. Conclusions

_{MS}/P, shows a universal relationship with the average number of branches per molecule, which is independent of conversion x

_{A}and reactivity ratio r. The g-ratio follows the relationship given by the Zimm-Stockmayer equation [15], which shows that the random branched structure is formed in batch polymerization.

_{MS}/P, follows a universal relationship with the number of branches in a polymer molecule, and the relationship is independent of ξ, but is a very weak function of r.

^{2}is linearly correlated with L

_{MS}, represented by Rg

^{2}= 0.18L

_{MS}+ 1, both for a batch and a CSTR, irrespective of the conversion level and reactivity ratio. The coefficient, 0.18 is essentially the same as for an SCVP [26], and could be considered as a general characteristic of HB polymer architecture. The coefficient is 0.167 for linear polymers, and is 0.5 for perfect dendrons. The physical meaning of the coefficient is still not clear, but the value of 0.18 is closer to that for the linear polymers, rather than the perfect dendron that cannot fit in the 3D space because of the Malthusian packing paradox.

## Supplementary Materials

_{b}branch points in a polymer for a CSTR with (a) r = 0.5, (b) r = 2, (c) r = 5, and (d) r = ∞, for various ξ-values, Figure S2: Relationship between L

_{MS}/P and n

_{b}for a CSTR; (a) r = 0.5, (b) r = 2, (c) r = 5, and (d) r = ∞, with various ξ-values, Figure S3: Universal relationship between Rg

^{2}and L

_{MS}for a CSTR with various combinations of r and ξ. (a) r = 0.5, (b) r = 2, (c) r = 5, and (d) r = ∞.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Schematic representation of a hyperbranched (HB) polymer molecule generated in the present Monte Carlo (MC) simulation for a continuous stirred-tank reactor (CSTR). The tri-branched clusters are shown by the red closed curves, and the T units with a star show the end units for the maximum span length.

**Figure 4.**Relationship between DB

_{inf}and r for batch and CSTR. Data points were taken from the earlier publication [9].

**Figure 5.**Expected Rg

^{2}-values of the polymers having P for batch polymerization with r = 1. Redrawn by using the data reported earlier [6].

**Figure 6.**Relationship between the number P

_{LC}of units belonging to the largest cluster and the degree of polymerization P. (a) r = 1; ξ = 0.5 for CSTR, and x

_{A}= 0.95 for batch. (b) r = 5; ξ = 0.306 for CSTR, and x

_{A}= 0.95 for batch. (c) r = ∞; ξ = 0.25 for CSTR, and x

_{A}= 0.95 for batch.

**Figure 7.**Average residence time of the units in the largest tri-branched cluster (red), of all units (green), and of the peripheral T units (blue) in each polymer molecule for the cases with r = 1. (

**a**) ξ = 0.4, and (

**b**) ξ = 0.5.

**Figure 8.**Example of the hyperbranched polymer architecture generated in the present MC simulation for a CSTR with r = ∞ and ξ = 0.2. The tri-branched clusters of the first order are represented by the groups enclosed by the red broken curves, while the tri-branched clusters of the second order by the blue broken curves. For this polymer, P = 43, D = 21, and DB = 1.

**Figure 9.**Relationship between the number P

_{LC,2}of units belonging to the largest cluster of the second order and the degree of polymerization P, for a batch and a CSTR with r = ∞. ξ = 0.25 for CSTR, and x

_{A}= 0.95 for batch.

**Figure 10.**Average residence time of the units in the largest tri-branched cluster of the second order (black dots), of the units in the largest tri-branched cluster of the first order (red), of all units (green), and of the peripheral T units (blue) for the cases with r = ∞. (

**a**) ξ = 0.2, and (

**b**) ξ = 0.25.

**Figure 12.**Relationship between Rg

^{2}and P with various r-values at x

_{A}= 0.95 (filled circle), 0.9 (open circle), and 0.7 (cross) for batch polymerization.

**Figure 13.**Universal relationship between g and the average number of branch points in a polymer, DB

_{inf}P/2 for batch polymerization.

**Figure 16.**Relationship between L

_{MS}and P with various r-values at x

_{A}= 0.95 (filled circle), 0.9 (open circle), and 0.7 (cross) for batch polymerization.

**Figure 17.**Universal relationship between L

_{MS}/P and the number of branch points in a polymer, DB

_{inf}P/2 for batch polymerization.

**Figure 18.**Relationship between Rg

^{2}and L

_{MS}for batch polymerization with r = 1. (

**a**) Raw data (red dots) and the averages within ΔL

_{MS}, i.e., the expected Rg

^{2}-values, at x

_{A}= 0.95. (

**b**) Expected Rg

^{2}for a given L

_{MS}with various conversion levels.

**Figure 19.**Universal relationship between Rg

^{2}and L

_{MS}for batch polymerization with various combinations of r and x

_{A}. (

**a**) r = 0.5, (

**b**) r = 2, (

**c**) r = 5, and (

**d**) r = ∞.

**Figure 20.**Relationship between Rg

^{2}and L

_{MS}for perfect dendrons, when the focal point is the L-type (red circle) and the D-type (blue cross). For both cases, the relationship is represented by Equation (14) for large polymers.

**Figure 21.**Relationship between the g-ratio and the number n

_{b}of branch points in a polymer for a CSTR with r = 1. (

**a**) Each data point (red) and the expected g-value (blue), at ξ = 0.5. (

**b**) The expected g-ratio for ξ = 0.5, 0.45, 0.4, and 0.3.

**Figure 22.**Expected g-ratio and n

_{b}for the HB polymers having n

_{b}branch points in a polymer with various reactivity ratio r for a CSTR. The plotted values are at ξ

_{UL}for each r. (

**a**) Normal scale plot, and (

**b**) logarithmic scale plot for the y-axis.

**Figure 23.**Relationship between L

_{MS}/P and n

_{b}for a CSTR with r = 1. (

**a**) Each data point (red) and the expected L

_{MS}/P (blue), at ξ = 0.5. (

**b**) The expected L

_{MS}/P for the HB polymers having n

_{b}branch points in a polymer with ξ = 0.5, 0.45, 0.4, and 0.3.

**Figure 24.**Relationship between L

_{MS}/P and n

_{b}for a CSTR with various reactivity ratios. The plotted values are at ξ

_{UL}for each r. (

**a**) Normal scale plot, and (

**b**) logarithmic scale plot.

**Figure 25.**Relationship between Rg

^{2}and L

_{MS}for a CSTR with r = 1. (

**a**) Raw data and the averages within ΔL

_{MS}, i.e., the expected Rg

^{2}-values, at x

_{A}= 0.95. (

**b**) Expected Rg

^{2}for various conversion levels.

© 2019 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.
Universal Relationships in Hyperbranched Polymer Architecture for Batch and Continuous Step Growth Polymerization of AB_{2}-Type Monomers. *Processes* **2019**, *7*, 220.
https://doi.org/10.3390/pr7040220

**AMA Style**

Tobita H.
Universal Relationships in Hyperbranched Polymer Architecture for Batch and Continuous Step Growth Polymerization of AB_{2}-Type Monomers. *Processes*. 2019; 7(4):220.
https://doi.org/10.3390/pr7040220

**Chicago/Turabian Style**

Tobita, Hidetaka.
2019. "Universal Relationships in Hyperbranched Polymer Architecture for Batch and Continuous Step Growth Polymerization of AB_{2}-Type Monomers" *Processes* 7, no. 4: 220.
https://doi.org/10.3390/pr7040220