Generating Function Method for an SCVP System with Any Initial Distribution of Polymers

As stated above, the effect of the semi-batch process on an SCVP system can be presented as long as the relationship of an averaged polymeric quantity associated with any two nearest neighbor feedings is obtained. In this section, a novel GF method will be proposed to achieve such a goal. Without loss of generality, we suppose that there have preexisted some polymers at the beginning of the SCVP system under consideration. This could correspond to two nearest neighbor polymerization steps, where the preexisted polymers come from either a previous polymerization step or a feeding operation, or from both them. In this way, an explicit GF will be employed to study such an SCVP system, which enables us to find how an average polymeric quantity depends on these preexisted polymers.

To proceed, we suppose that, for the present SCVP system, the number of preexisted polymers of $m{\rm{ - mer}}\;{\rm{(}}m{\rm{ = 1,2,3,}}...{\rm{)}}$ in the initial stage is ${P_m}(x = 0)$, where the symbol $x$ denotes the conversion of vinyl groups A. According to the reaction mechanism of the SCVP,^[3,30,31] an active site B* or A* can initiate a vinyl group A of another molecule to form a new molecule and generate a new active site A*. Clearly, the consumption of an old active center (A*or B*) always leads to a new active center A*. This means that the number of active sites remains unchanged during the polymerization. It is also evident that there exist $m$ active sites and only one vinyl group A in an $m{\rm{ - mer}}$ is in accordance with the assumption of neglecting intramolecular reaction. Thus, the total number of active sites ${N_*} = \sum\nolimits_m {m{P_m}\;(x = 0)}$, while the total number of vinyl groups A is $N = \sum\nolimits_m {{P_m}\;(x = 0)}$. This is the initial condition of the present SCVP system.

Through successive reactions, molecules in the system can be linked together to form hyperbranched polymers of various sizes. As proposed by Müller, Yan and Wulkow, the evolution of the size distribution of polymers with time $t$ in such an SCVP system is governed by the following kinetic equation:^[30]

1 $\frac{{{\rm{d}}{P_m}(x)}}{{{\rm{d}}t}} = R\left[ {\frac{1}{2}\sum\limits_{n = 1}^{m - 1} {m{P_n}(x){P_{m - n}}(x) - \sum\limits_{n = 1}^\infty {(n + m){P_n}(x){P_m}(x)} } } \right]$

where ${P_m}(x)$ denotes the number of polymers of $m{\rm{ - mer}}$ at conversion $x$, and $R$ is rate constant for the polymerization. As usual, the same rate constants of A* and B* reacting with vinyl group A have been used under the assumption of equal reactivity.

Considering that the initial conversion of vinyl groups A has been arranged to be zero, then there would be $N(1 - x)$ free vinyl groups A in the system at a conversion $x$. According to the reaction mechanism, upon considering the change of free vinyl groups A during the polymerization, one can find that the variation of the conversion $x$ in time $t$ is subject to the equation:^[30]

2 $\frac{{{\rm{d}}x}}{{{\rm{d}}t}} = R\left[ {\sum\limits_{m = 1}^\infty {m{P_m}(x = 0)} } \right](1 - x) = R{N_*}(1 - x)$

where the number of active sites ${N_*}$ keeps a constent. Solving this equation yields

3 $x(t) = 1 - {\rm{exp}} ( - R{N_ * }t)$

This is an explicit relationship between the conversion $x(t)$ and time $t$. The combination of Eqs. (1) and (2) enables one to obtain the following equation:

4 $\frac{{{\rm{d}}{P_m}(x)}}{{{\rm{d}}x}} = \frac{1}{r}\left[ {\frac{m}{{2N(1 - x)}}\sum\limits_{n = 1}^{m - 1} {{P_n}} (x){P_{m - n}}(x) - \left(m + \frac{r}{{1 - x}}\right){P_m}(x)} \right]$

in which $r = {N_*}/N$, denotes the molar ratio of active sites to free vinyl groups at the beginning of the present SCVP system. As r=1 (this signifies that no polymer is preexisted), this equation is identical with that given by Müller, Yan and Wulkow.^[30]

Note that there are $N$ free vinyl groups A at the beginning of the system, therefore the total number of molecules would be $N(1 - x)$ when the conversion of vinyl groups A is $x$. This means that $N(1 - x) = \sum\nolimits_m {{P_m}(x)}$. Furthermore, ${P_m}(x)$ can be rewritten as ${P_m}(x) = N(1 - x){c_m}(x)$ since each molecule possesses only one free vinyl group A, in which ${c_m}(x)$ denotes the number faction of polymers of $m{\rm{ - mer}}$ at a conversion $x$. This results in the following identity:

5 $\sum\limits_m {{c_m}} (x) \equiv 1$

Substituting ${P_m}(x) = N(1 - x){c_m}(x)$ into Eq. (4) yields the differential equation satisfied by ${c_m}(x)$:

6 $\frac{{{\rm{d}}{c_m}(x)}}{{{\rm{d}}x}} = \frac{m}{r}\left[ {\frac{1}{2}\sum\limits_{n = 1}^{m - 1} {{c_n}} (x){c_{m - n}}(x) - {c_m}(x)} \right]$

In order to determine ${c_m}(x)$, a GF denoted by $G(x,\theta )$ can be introduced as follows:

7 $G(x,\theta ) = \sum\limits_{m = 1}^\infty {{c_m}} (x){\rm{exp}} (m\theta )$

where $\theta$ is an auxiliary variable. Such a GF signifies that ${c_m}(x)$ can be obtained by expanding $G(x,\theta )$ in the series of ${\rm{exp}} (\theta )$ provided that the GF is explicitly given. It is therefore necessary to derive an explicit expression of $G(x,\theta )$ for further investigations. To this end, through Eqs. (6) and (7), one can obtain a partial differential equation satisfied by $G(x,\theta )$, namely

8 $\left[ {1 - G(x,\theta )} \right]\frac{\partial }{{\partial \theta }}G(x,\theta ) + r\frac{\partial }{{\partial x}}G(x,\theta ) = 0$

It is obvious that through the initial conditions of the system, solving this equation will enable us to obtain $G(x,\theta )$, and hence ${c_m}(x)$ and ${P_m}(x)$.

The partial differential equation given by Eq. (8) is known as Burgers equation,^[59,60] and the corresponding characteristics of $G(x,\theta )$ through the point $(0,\phi )$ on the $x - \theta$ plane satisfies the following equation:

9 $\theta - \phi = \frac{1}{r}\left[ {1 - G(0,\phi )} \right]x$

where $\phi$ is a function of $x$ and $\theta$ by writing $\phi (x,\theta )$ to be determined. Clearly, the term $\dfrac{1}{r}\left[ {1 - G(0,\phi )} \right]$ plays the role of slope of the characteristic line, and thus we have $G(x,\theta ) =$$G\left[ {0,\phi (x,\theta )} \right]$ from Eq. (8). This signifies that $G(x,\theta )$ can be derived once the dependence of $\phi (x,\theta )$ on variables $x$ and $\theta$ is figured out. Making use of $G(x,\theta ) = G\left[ {0,\phi (x,\theta )} \right]$ together with Eqs. (7) and (9), we have

10 $G(x,\theta ) = \sum\limits_{m = 1}^\infty {{c_m}} (0){\rm{exp}} (m\theta ){\rm{exp}} \left\{ { - \frac{{mx}}{r}\left[ {1 - G(x,\theta )} \right]} \right\}$

From the definition of the GF given in Eq. (7), the explicit expressions of $G(x,\theta )$ and ${c_m}(x)$ could be directly calculated through the initial conditions given by ${c_m}(0)$.

The derivation of $G(x,\theta )$ is especially useful for the evaluation of average polymeric quantities of the system. For instance, upon introducing the k^th ($k = 0,1,2,...$) polymer moments, ${M_{k}}(x)$ defined by ${M_{k}}(x) = \sum\nolimits_{m = 1}^\infty {{m^{k}}{P_m}(x)}$, and then taking into account ${P_m}(x) = N(1 - x){c_m}(x)$ together with Eq. (7) follows that

11 ${M_k}(x) = N(1 - x)\frac{{{\partial ^{k}}}}{{\partial {\theta ^{k}}}}G(x,\theta ){|_{\theta = 0}}$

where $\dfrac{{{\partial ^{k}}}}{{\partial {\theta ^{k}}}}G(x,\theta )$ denotes the $k^{\rm{th}}$ partial derivatives of $G(x,\theta )$ with respect to the variable $\theta$, and $\dfrac{{{\partial ^{k}}}}{{\partial {\theta ^{k}}}}G(x,\theta ){|_{\theta = 0}}$ denotes its value evaluated at $\theta = 0$.

Bear in mind that the mass of an inimer has been used as the unit mass, the number-, weight-, and z-average molecular weights, and Đ can be calculated from the derived ${M_{k}}(x)$ by using

12

For the system under study, as a consequence of Eqs. (11) and (12), we have

13

where ${M_{\rm{n}}}(0)$, ${M_{\rm{w}}}(0)$, ${M_{\rm{z}}}(0)$, and Đ(0) are the number-, weight-, and z-average molecular weights, and Đ at x=0, respectively. This equation indicates that ${M_{\rm{n}}}(x)$, ${M_{\rm{w}}}(x)$, ${M_{\rm{z}}}(x)$, and Đ $(x)$ have been associated with their corresponding initial values at x=0. In other words, the effect of preexisted polymers on these average polymeric quantities has been explicitly carried out.

Now a question that arises naturally is whether the present GF method is valid. To demonstrate its validity, the present GF method is applied to the batch SCVP system of pure inimers due to Müller, Yan and Wulkow.^[30] Assume that there exist $N$ inimers in the initial stage of the system, then we have ${c_m}(0) = {\delta _{m,1}}$ (${\delta _{i,j}}$ is the Kronecker symbol, which is equal to 1 for $i = j$, and 0 otherwise.). Meanwhile, one can find that r=1 and $G(x,\theta ) = G\left[ {0,\phi (x,\theta )} \right] = {\rm{exp}} \left[ {\phi (x,\theta )} \right]$. With the help of Eqs. (7) and (10), we have

14 $G(x,\theta ){\rm{exp}} [ - xG(x,\theta )] = {\rm{exp}} (\theta - x)$

Making use of Eq. (11) and differentiating both sides of this equation with respect to $\theta$, the polymer moments ${M_0}(x) =$$N(1 - x)$, ${M_1}(x) = N$, ${M_2}(x) = \dfrac{N}{{{{(1 - x)}^2}}}$ and ${M_3}(x) = N\dfrac{{1 + 2x}}{{{{(1 - x)}^4}}}$ can be easily obtained. Furthermore, a direct calculation from Eq. (12) leads to

15

Alternatively, these results can be directly derived by substituting ${M_{\rm{n}}}(0) = {M_{\rm{w}}}(0) = {M_{\rm{z}}}(0) = 1$ into Eq. (13). This signifies that one can obtain these results in terms of the present GF instead of using the size distribution of polymers, and all we need is merely the initial conditions of the SCVP system.

More interestingly, $G(x,\theta )$ can also be employed to derive the size distribution of polymers by starting with Eq. (14) together with the theorem of Lagrange inversion (detailed calculations can be found in the electronic supplementary information, ESI).^[61] For brevity, here we only give the final expression of $G(x,\theta )$:

16 $G(x,\theta ) = \sum\limits_{m = 1} {{\omega _m}{x^{m - 1}}{\rm{exp}} ( - mx){\rm{exp}} (m\theta )}$

where ${\omega _m} = \dfrac{{{m^{m - 1}}}}{{m!}}$ is rigorously derived from the residue theorem.^[62] Such an explicit expression of $G(x,\theta )$ means that ${c_m}(x) = {\omega _m}{x^{m - 1}}{\rm{exp}} ( - mx)$, and thus the size distribution ${P_m}(x)$ can be given by

17 ${P_m}(x) = N(1 - x){\omega _m}{x^{m - 1}}{\rm{exp}} ( - mx)$

All the results presented in Eqs. (15) and (17) are identical with those obtained in the batch SCVP system,^[30] thereby validating the present GF method. Furthermore, as a direct generalization, the present GF method will be employed to discuss the effect of the semi-batch process on the average polymeric quantities of hyperbranched polymers. This can be carried out by considering the semi-batch process as a series of multi-step processes of feeding and polymerization because any two nearest neighbour steps have been essentially solved by the present GF method.

A Unified Treatment on the Semi-batch SCVP System

For a polymerization system under the semi-batch mode, all the feeding operations are usually performed according to a predetermined schedule. Therefore the whole polymerization is exactly a multi-step process. This means that, at the beginning of each polymerization step, there are always some preexisted polymers resulting either from a previous polymerization or from the present feeding, or from both of them. As done in the previous section, these discrete steps can be further investigated one by one with the aid of the present GF method. Throughout the work, each feeding operation was considered to be instantaneous such that the response of the system during such an infinitesimal time interval can be neglected.

We assume that only inimers are initially loaded into the reactor, which is labeled as the zeroth feeding, for convenience. Subsequently, the first polymerization step begins until the conversion arrives at a predetermined conversion ${x_1}$ (or equivalently, at a predetermined reaction time), and then the first feeding is performed. After the first feeding, the second polymerization begins until the conversion arrives at ${x_2}$. The rest steps are repeated in a similar manner until the last step polymerization is completed.

In this way, one can find that a polymerization and a feeding are always involved in each step. For clarity, a sequence of feedings and reactions in an SCVP system can be schematically illustrated in Fig. 1, where the semi-batch process was considered to be an $L{\rm{-step}}$ process. This indicates that the semi-batch SCVP system has been divided into an $L{\rm{-step}}$ polymerization process, and each step can be studied as we have presented in previous section.

1Full-screen View

Schematic illustration of the semi-batch SCVP system, which is thought of as an $L{\rm{-step}}$ processes, and each step is comprised of polymerization and feeding.

Now the semi-batch SCVP system would be investigated as an application of the present GF method, which enables us to treat the situation that allows polymers of various sizes to be added into the reactor in each feeding. From the results given in previous section, it has been found that these average polymeric quantities associated with two nearest neighbor steps are closely related to each other. As such, ${M_{\rm{n}}}$, ${M_{\rm{w}}}$, ${M_{\rm{z}}}$, and Đ could be readily derived as long as the feeding details are specified.

As schemed in Fig. 1, for any two nearest neighbor steps, for example, the $(i - 1)^{\rm{th}}$ and $i^{\rm{th}}$ steps, the initial conditions of the $i{\rm{-th}}$ step polymerization depend explicitly on the number of polymers coming from the $(i- 1)^{\rm{th}}$ polymerization and feeding. This indicates that the $i^{\rm{th}}$ polymerization must be affected by the polymerization and feeding in the $(i - 1)^{\rm{th}}$ step, so do the relevant average polymeric quantities.

Note that the $(i - 1)^{\rm{th}}$ feeding was performed when the conversion of vinyl groups A was ${x_{i - 1}}$, and after this feeding the $i^{\rm{th}}$ step polymerization takes place. Let $P_m^{(i)}({x_i} = 0)$ be the number of polymers of $m{\rm{ - mer}}\;{\rm{(}}m{\rm{ = 1,2,3,}}...{\rm{)}}$ at the beginning of the $i^{\rm{th}}$ step polymerization, which can be expressed as:

18 $P_m^{(i)}({x_i} = 0) = {P_m}({x_{i - 1}}) + F_m^{(i - 1)}$

where ${P_m}({x_{i - 1}})$ and $F_m^{(i - 1)}$ denote the number of polymers of $m{\rm{ - mer}}$ due to the $(i - 1)^{\rm{th}}$ polymerization and feeding, respectively. Obviously, now the total number of active sites is $\sum\nolimits_m {mP_m^{(i)}({x_i} = 0)}$, while the number of vinyl groups A is ${N^{(i)}} =$$\sum\nolimits_m {P_m^{(i)}({x_i} = 0)}$. This is the initial condition of the i^th step polymerization.

In analogy with Eqs. (9) and (10), the most important two equations resulting from the GF method can be found as:

19

where ${r_i} = \sum\nolimits_m {mP_m^{(i)}({x_i} = 0)} /{N^{(i)}}$, is the molar ratio of active sites to free vinyl groups in the $i{\rm{-th}}$ step, ${x_i}$ denotes the conversion of vinyl groups A, and $c_m^{(i)}(0)$ is the number fraction of polymers of $m{\rm{-mer}}$ at ${x_i} = 0$. In obtaining Eq. (19), the equation $G({x_i},\theta ) = G\left[ {0,\phi ({x_i},\theta )} \right]$ has been used.

Making use of Eq. (19), the $k^{\rm{th}}$ polymer moment in the $i^{\rm{th}}$ step polymerization defined by ${M_{{k}}}({x_i}) = \sum\nolimits_{m = 1}^\infty {{m^{k}}{P_m}({x_i})}$ can be given by

20 ${M_k}({x_i}) = {N^{(i)}}(1 - {x_i})\frac{{{\partial ^{k}}}}{{\partial {\theta ^{k}}}}G({x_i},\theta ){|_{\theta = 0}}$

Likewise, ${M_{\rm{n}}}({x_i})$, ${M_{\rm{w}}}({x_i})$, ${M_{\rm{z}}}({x_i})$, and Đ$({x_i})$ in the $i^{\rm{th}}$ step polymerization can be explicitly derived through the $k^{\rm{th}}$ polymer moments evaluated by this equation. More specifically, as a consequence of the present GF method, recursion formula (a detailed derivation has been left in ESI) can be obtained as follows:

21

where the quantity $f_{k,k'}^{(i)}$ is defined by

22 $f_{k,k'}^{(i)} = \frac{{\sum\nolimits_m {{m^k}F_m^{(i)}} }}{{\sum\nolimits_m {{m^{k'}}{P_m}({x_i})} }},\quad (k,k' = 0,1,2,3,...)$

which denotes the ratio of the $k^{\rm{th}}$ polymeric moment due to feeding to the $k'^{\rm{th}}$ polymeric moment due to polymerization in the $i^{\rm{th}}$ step.

It should be stressed that the quantity $f_{k,k'}^{(i)}$ depends not only on the number of polymers from the $i^{\rm{th}}$ feeding, but also on those from the $i^{\rm{th}}$ polymerization. Of course, the removal of polymers can be taken into account by stating in Eqs. (18) and (22) the degree of polymerization and the number of those removed polymers. This indicates that $f_{k,k'}^{(i)}$ is closely related to the feeding details such as species and amount of those fed polymers. For example, $f_{0,0}^{(i)}$ measures the molar ratio of the two types of polymers in the $i^{\rm{th}}$ step, while $f_{1,1}^{(i)}$ measures the corresponding weight ratio. As such, $f_{k,k'}^{(i)}$ plays a key role in evaluating average polymer quantities in the $i^{\rm{th}}$ step. In view of the relevance of two nearest neighbour steps, $f_{k,k'}^{(i)}$ becomes the most important factor that determines the average properties of resultant polymers.

Monte Carlo Simulation for a Semi-batch SCVP System

In this section, MC simulations on the semi-batch SCVP system would be performed for the case that only inimers are fed into the reactor. In particular, the simulation results for Cases I and II discussed in this work would be compared with the corresponding analytical results. For a stochastic chemical reaction, a set of master equations can be constructed on the basis of the reaction mechanism.^[63,64] In the present SCVP system, the change in size distribution of polymers is subject to the differential kinetic equations given in Eq. (1). Therefore the evolution of MC simulation would be governed by the corresponding master equation. For easy of presentation, here we only outline how to use MC method to simulate a semi-batch SCVP system.

Assume that the system consists of treelike molecules with size distribution ${P_n}(t)$ at a given time $t$, and after a time interval ${t_{\rm{w}}}$ a reaction takes place. As a consequence, the system evolves into a new state characterized by a renewed size distribution ${P_n}(t + {t_{\rm{w}}})$ of polymers at the time $t + {t_{\rm{w}}}$. The quantity ${t_{\rm{w}}}$ is usually called the waiting time since it denotes a time interval between two successive reactions. It is therefore important for an MC simulation to determine both the waiting time ${t_{\rm{w}}}$ and reaction type (what species are involved in each reaction).

According to the Eqs. (1) and (2), if ${\Omega _n}(t)$ is the possible reaction rate of forming treelike polymers of $n{\rm{ - mer}}$ at time $t$, we have

29 ${\Omega _n}(t) = \frac{R}{2}\sum\limits_{i = 1}^{n - 1} {n{P_i}(t)} \left[ {{P_{n - i}}(t) - {\delta _{i,n - i}}} \right]$

where the term with Kronecker symbol ${\delta _{i,j}}$ is to avoid repeated counts of reaction ways. The total possible reaction rate as a function of $t$ over all possible $n$ can be written as ${\Omega _n}(t) =$$\sum\nolimits_n {{\Omega _n}(t)}$. Thus, the waiting time, ${t_{\rm{w}}}$, can be determined by a random number ${r_1}$ through the relationship:^[63,64]

30 ${t_{\rm{w}}} = \Omega {(t)^{ - 1}}{\rm{ln}} r_1^{ - 1}$

This indicates that ${t_{\rm{w}}}$ is a stochastic quantity rather than a constant. Subsequently, ${q_n}(t)$, the probability of generating treelike molecules of $n{\rm{ - mer}}$ can be expressed as:

31 ${q_n}(t) = \frac{{{\Omega _n}(t)}}{{\Omega (t)}},\quad (n = 2,3,...)$

The simulation can be performed by a simple sampling over all reaction events because of the identity of $\sum\nolimits_n {{q_n}(t)} \equiv 1$. In other words, one can sample in the corresponding probability space such that the reaction type associated with a degree of polymerization (index $n$) could be determined. In such a routine process, two more random numbers would be involved, and the sampling method is available in the literature.^[63,64] In this way, the state characterized by the size distribution of polymers ${P_n}(t)$ (equivalently, ${P_n}(x)$) should be renewed after each reaction and feeding, whereby the relevant polymer quantities can be evaluated.

To test the program, we have firstly implemented 100 simulations on the batch SCVP system, where the number of inimers in each was fixed as N=10⁵. As a consequence of the simulation, ${M_{\rm{n}}}$ and ${M_{\rm{w}}}$ have been presented in Fig. 4, in which ${P_n}(x)$ at $x=0.97$ is also illustrated in the insert. Meanwhile, a comparison with the corresponding analytic results has been made. Clearly, the simulation results of ${P_n}$ against $n$, ${M_{\rm{n}}}$ and ${M_{\rm{w}}}$ against $x$ agree well with their analytical results, thereby validating the present MC simulation method. This enables us to simulate a semi-batch SCVP system.

4Full-screen View

${M_{\rm{w}}}$ (with error bar) and ${M_{\rm{n}}}$ against $x$, and the size distribution ${P_n}$($x = 0.97$) against $n$ (insert) for the batch SCVP system, where the lines and symbols denote the theoretical and simulation results, respectively.

Clearly, the key quantity obtained from the MC simulation is the number distribution of hyperbranched polymers, and hence the molecular weight distribution (MWD). This is measurable in experiments. Interestingly, the variation of MWD with time can be directly carried out provided that the reaction rate $R$ is known. In accordance with the above method, we have implemented 100 simulations on the semi-batch SCVP system for both Case I and Case II. In each simulation, the total number of inimers is fixed as N=10⁶, while the number of the fed inimers at each step is determined by the conversions specified in the Tables 1 and 2. For brevity, we only take the simulation results of ${M_{\rm{w}}}$ as an example to compare with the corresponding theoretical results, as shown in Figs. 5 and 6, respectively.

5Full-screen View

${M_{\rm{w}}}$ for the semi-batch SCVP system in the Case I, where lines and symbols denote the results from the GF method and MC simulations, respectively.

6Full-screen View

${M_{\rm{w}}}$ for the semi-batch SCVP system in the Case II, where lines and symbols denote the results from the GF method and MC simulation, respectively.

An excellent agreement between the theoretical and simulation results can be observed. At a fixed $L$, there was an obvious increase in ${M_{\rm{w}}}$ whenever a polymerization step is finished. Seen from panels (a) to (d) in Fig. 5, it can be found that ${M_{\rm{w}}}$ increase even by orders of magnitude with the increase of L, so does in Fig. 6. This is a general tendency of the present semi-batch SCVP system. Comparing Fig. 5 with Fig. 6, one can find that the increment of ${M_{\rm{w}}}$ in the Case I is always larger than that in the Case II. Such a tendency depends on the predetermined schedule given in Tables 1 and 2.

The above simulation results manifested that the present GF method can be employed to evaluate ${M_{\rm{n}}}$, ${M_{\rm{w}}}$, ${M_{\rm{z}}}$, and Đ of polymers by a semi-batch SCVP system. Experimentally, one might be interested in the variations of relevant average physical quantities with time. This can be easily realized in the MC simulation when the reaction rate $R$ is given. Instead of this, in this paper the conversion $x$ is used as the variable to describe the average physical quantities varying in the polymerization process. For a practical polymerization system, one can optimize feeding details to prepare hyperbranched polymers with desired average properties. Consequently, one can choose feeding ways, specify the number of batches, and predetermine conversions at each step. On the basis of these information, some average polymeric quantities of interest can be obtained by Eq. (21) in a straightforward manner.