A Theoretical Investigation of the Polyaddition of an AB2+A2+B4 Monomer Mixture

Hyperbranched polymers (HBPs) are widely applied nowadays as functional materials for biomedicine needs, nonlinear optics, organic semiconductors, etc. One of the effective and promising ways to synthesize HBPs is a polyaddition of AB2+A2+B4 monomers that is generated in the A2+CB2, AA′+B3, A2+B′B2, and A2+C2+B3 systems or using other approaches. It is clear that all the foundational features of HBPs that are manufactured by a polyaddition reaction are defined by the component composition of the monomer mixture. For this reason, we have designed a structural kinetic model of AB2+A2+B4 monomer mixture polyaddition which makes it possible to predict the impact of the monomer mixture’s composition on the molecular weight characteristics of hyperbranched polymers (number average (DPn) and weight average (DPw) degree of polymerization), as well as the degree of branching (DB) and gel point (pg). The suggested model also considers the possibility of a positive or negative substitution effect during polyaddition. The change in the macromolecule parameters of HBPs formed by polyaddition of AB2+A2+B4 monomers is described as an infinite system of kinetic equations. The solution for the equation system was found using the method of generating functions. The impact of both the component’s composition and the substitution effect during the polyaddition of AB2+A2+B4 monomers on structural and molecular weight HBP characteristics was investigated. The suggested model is fairly versatile; it makes it possible to describe every possible case of polyaddition with various monomer combinations, such as A2+AB2, AB2+B4, AB2, or A2+B4. The influence of each monomer type on the main characteristics of hyperbranched polymers that are obtained by the polyaddition of AB2+A2+B4 monomers has been investigated. Based on the results obtained, an empirical formula was proposed to estimate the pg = pA during the polyaddition of an AB2+A2+B4 monomer mixture: pg = pA = (−0.53([B]0/[A]0)1/2 + 0.78)υAB2 + (1/3)1/2([B]0/[A]0)1/2, where (1/3)1/2([B]0/[A]0)1/2 is the Flory equation for the A2+B4 polyaddition, [A]0 and [B]0 are the A and B group concentration from A2 and B4, respectively, and υAB2 is the mole fraction of the AB2 monomer in the mixture. The equation obtained allows us to accurately predict the pg value, with an AB2 monomer content of up to 80%.


Introduction
The synthesis and investigation of properties of hyperbranched polymers (HBPs) represents one of the most rapidly advancing areas in polymer science.They have a wide range of applications due to the number of unique features compared to the linear and cross-linked polymers, including high solubility, thermodynamic compatibility, low viscosity, high sorption capacity, and a high content of functional groups [1][2][3] are widely applied nowadays as functional materials for biomedicine needs [4,5], nonlinear optics [6,7], organic semiconductors [8,9], and flame-retardant materials [10,11], among others.One of the key ways to obtain HBPs is homo-polyaddition of AB m -type monomers [12][13][14].The primary advantage of polyaddition of AB m -type monomers is that it does not lead to gelation [15], allowing for a production of high-molecular-weight (MW) polymers with a degree of branching (DB) of 0.5 [16].However, obtaining AB m -type monomers often involves a complex organic synthesis; moreover, there are some considerable complications arising in the process of isolation and purification of these monomers containing highly reactive groups [17,18].This poses a notable barrier to the practical application of HBPs that are obtained through the aforementioned methods.For this reason, co-polyaddition of monomer mixtures of different types, for example, A 2 +B 3 , A 2 +B 4 , etc., have found wider application [19][20][21][22][23][24][25][26][27][28] (Scheme 1).
The introduction of this method has enabled a significant expansion of the range of monomers that are under use and also the carrying out of polyaddition as a single-step reaction.It is a known fact that this kind of co-polyaddition eventually results in the formation of a three-dimensional structure at a specific juncture, commonly referred to as the critical gelation conversion, or gel point (pg).To determine the pg value in these Flory systems, Equation (1) was offered [35].
where r = [A]0/[B]0, ρ is the ratio of B (or A) groups in branched units to the total number of these groups, and pA and pB are the conversions of A and B groups, respectively.In general, pg = max (pA, pB).Hereinafter, when [A]0/[B]0 > 1, pg = pB, because pA < pB in that range.Correspondingly, if [A]0/[B]0 < 1, then pg = pA, and when [A]0/[B]0 = 1, pA = pB = pg.
The introduction of this method has enabled a significant expansion of the range of monomers that are under use and also the carrying out of polyaddition as a single-step reaction.It is a known fact that this kind of co-polyaddition eventually results in the formation of a three-dimensional structure at a specific juncture, commonly referred to as the critical gelation conversion, or gel point (p g ).To determine the p g value in these Flory systems, Equation (1) was offered [35].
where r = [A] 0 /[B] 0 , ρ is the ratio of B (or A) groups in branched units to the total number of these groups, and p A and p B are the conversions of A and B groups, respectively.In general, p g = max (p A , p B ). Hereinafter, when [A] 0 /[B] 0 > 1, p g = p B , because p A < p B in that range.Correspondingly, if [A] 0 /[B] 0 < 1, then p g = p A , and when To reduce the p g value, co-polyaddition of asymmetric monomers (A 2 +CB 2 , A 2 +B B 2 , AA +B 3 , A 2 +C 2 +B 3 ) was introduced [29][30][31][32][33][34][35][36][37][38][39] (Scheme 1).These approaches made it possible to shift the gel point, since more AB 2 monomers were formed, and therefore, it was possible to obtain polymers with an increased MW.To describe the polyaddition of A 2 +CB 2 Polymers 2024, 16, 426 3 of 20 monomers, a number of simulations have been developed [40,41] to predict polydispersity index (PDI) values depending on the ratio of reactants (Equation ( 2)).
where DP w and DP n are the weight average and the number average degree of polymerization, p A , p B , p C are conversions of A, B, and C groups, and λ is the initial ratio of A 2 and CB 2 monomer concentrations.Previously, we successfully implemented an approach to obtain HBPs, using polyaddition of the AB 2 +A 2 +B 4 monomer mixture with controlled contents of each constituent [42,43] (Scheme 2).That technique can also be applied to the co-polyaddition of asymmetric monomers due to the formation of AB 2 monomers.monomers, a number of simulations have been developed [40,41] to predict polydispersity index (PDI) values depending on the ratio of reactants (Equation ( 2)).
where DPw and DPn are the weight average and the number average degree of polymerization, pA, pB, pC are conversions of A, B, and C groups, and λ is the initial ratio of A2 and CB2 monomer concentrations.Previously, we successfully implemented an approach to obtain HBPs, using polyaddition of the AB2+A2+B4 monomer mixture with controlled contents of each constituent [42,43] (Scheme 2).That technique can also be applied to the co-polyaddition of asymmetric monomers due to the formation of AB2 monomers.
Despite the fact that the AB2+A2+B4 monomer mixture can be obtained during the polyaddition of A2+CB2 monomers, there is a lack of current theories and ideas to adequately describe every possible combination of these monomers in the mixture.The methods described above prevent obtaining a complete picture of the impact of each constituent of the AB2+A2+B4 monomer mixture on HBP formation.
Moreover, positive or negative substitution effects taking place during polyaddition and described in a number of experimental papers [44][45][46][47] would significantly affect both the MW and the structural characteristics of the resulting polymers.The manifestation of a positive substitution effect, e.g., in the Friedel-Crafts aromatic substitution reaction of AB2, leads to the production of fully branched HBPs [38].The manifestation of a negative substitution effect, e.g., during the production of hyperbranched polyesters by co-polycondensation of an AB2-type monomer and B4-and B6-type polyfunctional cores, leads to a decrease in the MW of the final product [39].There is no doubt that the substitution effect will also affect the value of pg in cases where it may be less than 1.
The kinetic Monte Carlo method and molecular dynamics simulations are widely used nowadays to investigate the evolution of the structure of hyperbranched polymers and polymer networks [48][49][50].At the same time, the conventional kinetic method that has proven itself for the investigation of HBP formation currently remains of interest [51][52][53][54][55].
Despite the fact that the AB 2 +A 2 +B 4 monomer mixture can be obtained during the polyaddition of A 2 +CB 2 monomers, there is a lack of current theories and ideas to adequately describe every possible combination of these monomers in the mixture.The methods described above prevent obtaining a complete picture of the impact of each constituent of the AB 2 +A 2 +B 4 monomer mixture on HBP formation.
Moreover, positive or negative substitution effects taking place during polyaddition and described in a number of experimental papers [44][45][46][47] would significantly affect both the MW and the structural characteristics of the resulting polymers.The manifestation of a positive substitution effect, e.g., in the Friedel-Crafts aromatic substitution reaction of AB 2 , leads to the production of fully branched HBPs [38].The manifestation of a negative substitution effect, e.g., during the production of hyperbranched polyesters by copolycondensation of an AB 2 -type monomer and B 4 -and B 6 -type polyfunctional cores, leads to a decrease in the MW of the final product [39].There is no doubt that the substitution effect will also affect the value of p g in cases where it may be less than 1.
The kinetic Monte Carlo method and molecular dynamics simulations are widely used nowadays to investigate the evolution of the structure of hyperbranched polymers and polymer networks [48][49][50].At the same time, the conventional kinetic method that has proven itself for the investigation of HBP formation currently remains of interest [51][52][53][54][55].
Given all the facts above, we aim to develop a new structural kinetic model of the polyaddition of an AB 2 +A 2 +B 4 monomer mixture, taking into account the potential manifestation of the substitution effect during polyaddition.Additionally, it would enable us to determine the impact of each system constituent on the structural and molecular weight parameters of HBPs.

Design of the Kinetic-Structural Model
To describe the AB 2 +A 2 +B 4 system, it is essential to establish certain assumptions and conditions.These will provide a framework for describing various reactions and types of resulting compounds that may emerge.
The assumptions are as follows: • Flory assumption, i.e., function group reactivity is independent of the chain length; The designed model is based on the concept of homo-polyaddition of AB 2 -type monomers [55].To describe the AB 2 +A 2 +B 4 system properly, it is also necessary to add a new parameter to the ones that were employed in [55] (the number of linear (l) and terminal (t) units).That is the number of dendritic units (d) (Figure 1).Given all the facts above, we aim to develop a new structural kinetic model of the polyaddition of an AB2+A2+B4 monomer mixture, taking into account the potential manifestation of the substitution effect during polyaddition.Additionally, it would enable us to determine the impact of each system constituent on the structural and molecular weight parameters of HBPs.

Design of the Kinetic-Structural Model
To describe the AB2+A2+B4 system, it is essential to establish certain assumptions and conditions.These will provide a framework for describing various reactions and types of resulting compounds that may emerge.
The assumptions are as follows: • Flory assumption, i.e., function group reactivity is independent of the chain length; The designed model is based on the concept of homo-polyaddition of AB2-type monomers [55].To describe the AB2+A2+B4 system properly, it is also necessary to add a new parameter to the ones that were employed in [55] (the number of linear (l) and terminal (t) units).That is the number of dendritic units (d) (Figure 1).The addition of the d unit results in the introduction of a new kind of compound, An, which cannot be described accurately by t and l parameters only, since the number of A groups depends on d: The number of A groups in a macromolecule is equal to Amax when l units are formed without any t ones.In case of the formation of a t unit, the number of A groups in a macromolecule is 2 less, while the number of d units is only 1 less than in the compound An (Figure 1).So, the amount of A groups during the t unit formation equals −1×t + Amax, resulting in the following equation describing a real case of polyaddition as A = d + 2 − t.
The substitution effects, occurring when the polyaddition of AB2+A2+B4 monomer mixture takes place, are included in the structural kinetic model (Scheme 3).Scheme 3. Positive and negative substitution effects during the polyaddition of the AB2+A2+B4 monomer mixture, where ba is the product of interaction between A and B groups.The addition of the d unit results in the introduction of a new kind of compound, A n , which cannot be described accurately by t and l parameters only, since the number of A groups depends on d: The number of A groups in a macromolecule is equal to A max when l units are formed without any t ones.In case of the formation of a t unit, the number of A groups in a macromolecule is 2 less, while the number of d units is only 1 less than in the compound A n (Figure 1).So, the amount of A groups during the t unit formation equals −1 × t + A max , resulting in the following equation describing a real case of polyaddition as A = d + 2 − t.
The substitution effects, occurring when the polyaddition of AB 2 +A 2 +B 4 monomer mixture takes place, are included in the structural kinetic model (Scheme 3).Given all the facts above, we aim to develop a new structural kinetic model of the polyaddition of an AB2+A2+B4 monomer mixture, taking into account the potential manifestation of the substitution effect during polyaddition.Additionally, it would enable us to determine the impact of each system constituent on the structural and molecular weight parameters of HBPs.

Design of the Kinetic-Structural Model
To describe the AB2+A2+B4 system, it is essential to establish certain assumptions and conditions.These will provide a framework for describing various reactions and types of resulting compounds that may emerge.
The assumptions are as follows: • Flory assumption, i.e., function group reactivity is independent of the chain length; The designed model is based on the concept of homo-polyaddition of AB2-type monomers [55].To describe the AB2+A2+B4 system properly, it is also necessary to add a new parameter to the ones that were employed in [55] (the number of linear (l) and terminal (t) units).That is the number of dendritic units (d) (Figure 1).The addition of the d unit results in the introduction of a new kind of compound, An, which cannot be described accurately by t and l parameters only, since the number of A groups depends on d: The number of A groups in a macromolecule is equal to Amax when l units are formed without any t ones.In case of the formation of a t unit, the number of A groups in a macromolecule is 2 less, while the number of d units is only 1 less than in the compound An (Figure 1).So, the amount of A groups during the t unit formation equals −1×t + Amax, resulting in the following equation describing a real case of polyaddition as A = d + 2 − t.
The substitution effects, occurring when the polyaddition of AB2+A2+B4 monomer mixture takes place, are included in the structural kinetic model (Scheme 3).Scheme 3. Positive and negative substitution effects during the polyaddition of the AB2+A2+B4 monomer mixture, where ba is the product of interaction between A and B groups.
Scheme 3. Positive and negative substitution effects during the polyaddition of the AB 2 +A 2 +B 4 monomer mixture, where ba is the product of interaction between A and B groups.

of 20
The reactivity of B groups belongs to t units and can be determined by the k 1 rate constant, whereas one of the B groups from l units is included in the k 2 rate constant.B groups can be provided by either AB 2 or B 4 monomers and also by the interaction products of these monomers and with an A 2 -type monomer.Thus, in the case of k 1 /k 2 < 1, a positive substitution effect takes place, whereas in the case of k 1 /k 2 > 1, there is a negative substitution effect.
Alterations in all structural parameters during the studied reaction can be described as a set in Equation (3): R(l, t, d)+R(l , t , d ) where R(l,t,d) is a concentration of macromolecules with l-linear, t-terminal, and d-dendritic units.
The introduction of additional reactions with the A 2 -type monomer is necessary to describe the initial conditions properly.According to the set of reactions (3), the endless kinetic equation can be defined by the following Equation ( 4), with initial conditions being The solution to the systems containing a large number of differential equations can only be achieved through the convolution of these equations.One of the simplest ways to accomplish this is by employing generating functions: where s, p, and n are random variables.Equation ( 4) can then be convolved with the Φ function into a shorter one (6): Consequently, we can switch from Equation ( 6) to the moments of the generating function Φ (7): Polymers 2024, 16, 426 6 of 20 and then the set of differential equations (8) for moments of the generating function Φ can be obtained from the Equations ( 6) and ( 7): ) If R(l,t,d) is the content of macromolecules of the given composition, then the following set of equations can be defined (9): where L n , T n , and D n are equal to the values of the average content of linear, terminal, and dendritic units in a macromolecule.
We can determine the value of the average degree of polymerization (DP n ) as n + 1 amount of B groups involved in the reaction, i.e., the number of monomer units contained in a macromolecule, which is 2d + l + 1.Thus, DP n can be defined as follows (10): The mass average structural parameters can be determined by (11): where L w , T w , and D w are the weight average compositions of linear, terminal, and dendritic units in a macromolecule.The weighted average degree of polymerization (DP w ) can therefore be estimated by Equation (12): The condition of DP w → ∞, which is equivalent to PDI → ∞ (where PDI is a polydispersity index), can be considered a gelation criterion.The degree of branching is defined as the ratio of an actual number of branched units to the maximum possible number of these units in a macromolecule.Here, the branched units are dendritic, so DB can be determined by the following Equation (13) [16]: To conclude, the application of the structural kinetic model of AB 2 +A 2 +B 4 monomer mixture polyaddition enables the study of how p g and various structural and molecular weight characteristics are influenced by each reaction component, as well as by substitution effects, which were impossible to analyze in previous studies.
Nevertheless, at first, it is essential to provide the verification of the investigated model.

Verification of the Kinetic-Structural Model
The current model for AB 2 +A 2 +B 4 monomer mixture polyaddition is quite versatile, encompassing all systems based on various combinations of the studied monomers, namely, A 2 +AB 2 , AB 2 +B 4 , AB 2 , and A 2 +B 4 .This significantly expands the range of applications for the developed approach, enabling the use of well-known systems and solitary cases, such as A 2 +B 4 , AB 2 , and A 2 +CB 2 , for verification.
The A 2 +B 4 system is a subset of the A n +B m system, which was studied and described by Flory, resulting in Equation (1) [35].Comparison of the data obtained through (1) and the data calculated using the offered approach (initial conditions are N = [  As we can clearly see from Figure 2, there is a perfect correlation between data obtained through two different methods. Another method of verification lies in reviewing well-studied systems-one of them is a solitary AB2-type monomer.The variations in system characteristics calculated using our method (with initial conditions set as N = [AB2]0, T = [AB2]0, and the rest as zero) are illustrated in Figure 3.As we can clearly see from Figure 2, there is a perfect correlation between data obtained through two different methods.
Another method of verification lies in reviewing well-studied systems-one of them is a solitary AB 2 -type monomer.The variations in system characteristics calculated using  As we can clearly see from Figure 2, there is a perfect correlation between data obtained through two different methods.
Another method of verification lies in reviewing well-studied systems-one of them is a solitary AB2-type monomer.The variations in system characteristics calculated using our method (with initial conditions set as N = [AB2]0, T = [AB2]0, and the rest as zero) are illustrated in Figure 3. Figure 3 shows that the maximum value of DB is 0.5 at рB = 0.5, which corresponds to data from earlier papers [16].Along with that, the gel point (PDI → ∞ or DPw→ ∞) is achieved at pA → 1 (pB→ 0.5), which is the same as in a conventional Flory paper [15].
The validation of the comprehensive AB2+A2+B4 model, incorporating all constituents, involves comparing the results obtained with our model to those obtained from the following set of reactions: A2+CB2→AB2 (rate constant kc) and AB2+CB2→B4 (rate constant kb).For example, from [41], when kc/kb = 200, the pg value equals 0.  Figure 3 shows that the maximum value of DB is 0.5 at p B = 0.5, which corresponds to data from earlier papers [16].Along with that, the gel point (PDI → ∞ or DP w → ∞) is achieved at p A → 1 (p B → 0.5), which is the same as in a conventional Flory paper [15].
The validation of the comprehensive AB 2 +A 2 +B 4 model, incorporating all constituents, involves comparing the results obtained with our model to those obtained from the following set of reactions: A 2 +CB 2 →AB 2 (rate constant k c ) and AB 2 +CB 2 →B 4 (rate constant k b ).For example, from [41]  Experimental data confirm that the offered model describes the polyaddition of AB2+A2+B4 monomers properly.In [43], AB2+A2+B4 monomer mixtures of various compositions were synthesized, and it was determined experimentally that pg value accounts for less than 1 in the range of [AB2]0/[A2]0/[B4]0 ratios from 1/0.025/0.097 to 1/0.036/0.083.Figure 5 illustrates that the first case is characterized by a calculated pg value of ~0.99, while  Experimental data confirm that the offered model describes the polyaddition of AB 2 +A 2 +B 4 monomers properly.In [43], AB 2 +A 2 +B 4 monomer mixtures of various compositions were synthesized, and it was determined experimentally that p g value accounts for less than 1 in the range of [AB 2 ] 0 /[A 2 ] 0 /[B 4 ] 0 ratios from 1/0.025/0.097 to 1/0.036/0.083.Figure 5 illustrates that the first case is characterized by a calculated p g value of ~0.99, while for the second one, the calculated value equals p g ~0.94.Experimental data confirm that the offered model describes the polyaddition of AB2+A2+B4 monomers properly.In [43], AB2+A2+B4 monomer mixtures of various compositions were synthesized, and it was determined experimentally that pg value accounts for less than 1 in the range of [AB2]0/[A2]0/[B4]0 ratios from 1/0.025/0.097 to 1/0.036/0.083.Figure 5 illustrates that the first case is characterized by a calculated pg value of ~0.99, while for the second one, the calculated value equals pg~0.94.Thus, the data obtained from various sources and the results of calculation using our suggestions matched perfectly.Based on that, it can be concluded that our structural kinetic model of the polyaddition of AB2+A2+B4 monomer mixture provides accurate results.Thus, the data obtained from various sources and the results of calculation using our suggestions matched perfectly.Based on that, it can be concluded that our structural kinetic model of the polyaddition of AB 2 +A 2 +B 4 monomer mixture provides accurate results.

Results and Discussion
Using the proposed approach, it is possible to evaluate the effect of each constituent on both the structure and molecular weight parameters.

A 2 -Type Monomer Effects
The p g curves over the initial molar fraction of an A 2 -type monomer ] 0 ratios are shown in Figure 6.
The curves in Figure 6 reflect the conditions under which one can observe soluble systems transition to an insoluble state.Here, the condition for curve 1 is [AB 2 ] 0 = 0, indicating that it can be described by Equation (1).In other cases, [AB 2 ] 0 = 0 (Figure 6 (2-4)), and therefore, a broadening of the Flory curve can be observed.Also, there is a distinct minimum at the [A] 0 /[B] 0 = 1 ratio in all the p g vs. υA 2 graphs.When the [A] 0 /[B] 0 value tends to deviate from 1 in either direction, an increase in p g up to 1 is observed.The minimum point shifts towards lower υA 2 values when an AB 2 -type monomer is introduced into the system.At the same time, the p g value at the minimum point is almost unaffected by changes in the [AB 2 ] 0 /[B 4 ] 0 ratio and remains approximately (1/3) 1/2 .To understand the reasons for these observed patterns, it is necessary to analyze how υA 2 affects the specific number of branches per macromolecule (D/N) (Figure 7).Hereinafter, the values of p g at the corresponding values of υA 2 were used to calculate the D/N.
Using the proposed approach, it is possible to evaluate the effect of each constituent on both the structure and molecular weight parameters.

A2-Type Monomer Effects
The pg curves over the initial molar fraction of an A2-type monomer (υA2 = [A2]0/([AB2]0 + [A2]0 + [B4]0)) at different [AB2]0/[B4]0 ratios are shown in Figure 6.The curves in Figure 6 reflect the conditions under which one can observe soluble systems transition to an insoluble state.Here, the condition for curve 1 is [AB2]0 = 0, indicating that it can be described by Equation (1).In other cases, [AB2]0 ≠ 0 (Figure 6 (2-4)), and therefore, a broadening of the Flory curve can be observed.Also, there is a distinct minimum at the [A]0/[B]0 = 1 ratio in all the pg vs. υA2 graphs.When the [A]0/[B]0 value tends to deviate from 1 in either direction, an increase in pg up to 1 is observed.The minimum point shifts towards lower υA2 values when an AB2-type monomer is introduced into the system.At the same time, the pg value at the minimum point is almost unaffected by changes in the [AB2]0/[B4]0 ratio and remains approximately (1/3) 1/2 .To understand the reasons for these observed patterns, it is necessary to analyze how υA2 affects the specific number of branches per macromolecule (D/N) (Figure 7).Hereinafter, the values of pg at the corresponding values of υA2 were used to calculate the D/N.It can be observed in Figure 7 (1) that for the polyaddition of A2+B4 monomers, the specific number of branches per macromolecule increases with the growth of υA2 until it reaches 1, corresponding to a minimum of the pg vs. υA2 function (Figure 6).As expected, it then begins to decrease.Thus, the minimum pg value is reached when D/N = 1.
The Introduction of the AB2-type monomer into the system leads to an increase in the D/N growth rate over υA2.The maximum D/N value possible is 1 when [AB2]0/[B4]0 < 1 (Figure 7 (2)), whereas it exceeds 1 at [AB2]0/[B4]0 > 1 (Figure 7 (3,4)).Furthermore, the function reaches its maximum when pg ≤ 1.However, the introduction of the AB2-type monomer does not affect the condition under which pg reaches its minimum at ~(1/3) 1/2 , which is observed at D/N = 1.Thus, introducing the AB2-type monomer into the A2+B4 system results in an increase in the D/N of the homo-polyaddition of the AB2-type monomer and its interaction with the B4-type monomer.The mentioned process does not lead to the crosslinking of macromolecules and contributes only to an increase in the degree of polymerization, as indicated by the DPn vs. υA2 plots shown in Figure 8.It can be observed in Figure 7 (1) that for the polyaddition of A 2 +B 4 monomers, the specific number of branches per macromolecule increases with the growth of υA 2 until it reaches 1, corresponding to a minimum of the p g vs. υA 2 function (Figure 6).As expected, it then begins to decrease.Thus, the minimum p g value is reached when D/N = 1.
The Introduction of the AB 2 -type monomer into the system leads to an increase in the D/N growth rate over υA 2 .The maximum D/N value possible is 1 when [AB 2 ] 0 /[B 4 ] 0 < 1 (Figure 7 (2)), whereas it exceeds 1 at [AB 2 ] 0 /[B 4 ] 0 > 1 (Figure 7 (3,4)).Furthermore, the function reaches its maximum when p g ≤ 1.However, the introduction of the AB 2type monomer does not affect the condition under which p g reaches its minimum at ~(1/3) 1/2 , which is observed at D/N = 1.Thus, introducing the AB 2 -type monomer into the A 2 +B 4 system results in an increase in the D/N of the homo-polyaddition of the AB 2 -type monomer and its interaction with the B 4 -type monomer.The mentioned process does not lead to the crosslinking of macromolecules and contributes only to an increase in the degree of polymerization, as indicated by the DP n vs. υA 2 plots shown in Figure 8.
function reaches its maximum when pg ≤ 1.However, the introduction of the AB2-type monomer does not affect the condition under which pg reaches its minimum at ~(1/3) 1/2 , which is observed at D/N = 1.Thus, introducing the AB2-type monomer into the A2+B4 system results in an increase in the D/N of the homo-polyaddition of the AB2-type monomer and its interaction with the B4-type monomer.The mentioned process does not lead to the crosslinking of macromolecules and contributes only to an increase in the degree of polymerization, as indicated by the DPn vs. υA2 plots shown in Figure 8.In the case of polyaddition, the molecular weight of the product depends heavily on the ratio of the groups that are involved in the reaction, and also, the highest molecular weight polymer can only be obtained under equimolar conditions.Another factor affecting the molecular weight is the conversion of functional groups.The effect of conversion on the MW is often complex in nature.In any case, it is obvious that the degree of reaction completion is essential to obtaining a high-molecular-weight polymer.
Where the polyaddition of a binary mixture of A 2 +B 4 monomers is concerned, there is a correlation between achieving equimolar conditions, a functional group conversion, and the molecular weight of the final product.Due to this, a broad peak is present on the graph of the degree of polymerization as a function of υA 2 (Figure 8 (1)).The introduction of an AB 2 -type monomer into the system results in shifting the peak (Figure 8 (2)) towards the [A] 0 /[B] 0 < 1 area.A further increase in this part of the AB 2 -type monomer in the system causes the highest MW to be achieved only when the conversion approaches 1, thereby sharpening the peak (Figure 8 (3,4)).Thus, the increase in υAB 2 in the AB 2 +A 2 +B 4 system significantly enhanced the DP n of the final polymer from 5 to 9, with [AB 2 ] 0 /[B 4 ] changing from 0 to 4; also, υAB 2 → 1, and DP n → ∞.
As expected, a monotonic increase in DB is observed in the curves illustrating its variation over υA 2 , as depicted in Figure 9, up to p g ≤ 1.The inflection point indicates the gelation onset.Figure 9 shows that the introduction of AB 2 -type monomer facilitates the DB growth.
Generally, hyperbranched polymers exhibit a DB ≥ 0.4.This value can be reached with all the ratios used within this work.However, when [AB 2 ] 0 /[B 4 ] 0 < 4 (Figure 9 (1-3)), the DB value reaches 0.4 beyond the inflection point, that is, when p g < 1 (and when DP n reaches its highest values).On the other hand, at [AB 2 ] 0 /[B 4 ] 0 ≥ 4, fully soluble hyperbranched polymers with DB = 0.4 can be obtained (Figure 9 (4)).The highest DB that is possible for the polyaddition of an AB 2 -type monomer is 0.5.However, HBPs with DB > 0.5 can be obtained using a mixture of AB 2 +A 2 +B 4 monomers.The point is that the application of the monomer mixtures that can potentially help reach DB ≥ 0.4 results in a decrease in the molecular weight characteristics of the final product compared to the polyaddition of an AB 2 -type monomer.
As expected, a monotonic increase in DB is observed in the curves illustrating its variation over υA2, as depicted in Figure 9, up to pg ≤ 1.The inflection point indicates the gelation onset.Figure 9 shows that the introduction of AB2-type monomer facilitates the DB growth.Generally, hyperbranched polymers exhibit a DB ≥ 0.4.This value can be reached with all the ratios used within this work.However, when [AB2]0/[B4]0 < 4 (Figure 9 (1-3)), the DB value reaches 0.4 beyond the inflection point, that is, when pg < 1 (and when DPn reaches its highest values).On the other hand, at [AB2]0/[B4]0 ≥ 4, fully soluble hyperbranched polymers with DB = 0.4 can be obtained (Figure 9 (4)).The highest DB that is possible for the polyaddition of an AB2-type monomer is 0.5.However, HBPs with DB > 0.5 can be obtained using a mixture of AB2+A2+B4 monomers.The point is that the application of the monomer mixtures that can potentially help reach DB ≥ 0.4 results in a decrease in the molecular weight characteristics of the final product compared to the polyaddition of an AB2-type monomer.

B 4 -Type Monomer Effects
The next important stage involves investigating how a B 4 -type monomer affects the formation of hyperbranched polymers during the polyaddition of the AB 2 +A 2 +B 4 monomer mixture.Figure 10 shows that, as in the previous case, the curves of p g over the initial molar fraction of a B 4 -type monomer (υB 4 = [B 4 ] 0 /([AB 2 ] 0 + [A 2 ] 0 + [B 4 ] 0 )) tend to broaden when the AB 2 -type monomer is introduced into the system.Also, a distinctive minimum is observed on each curve at [A] 0 /[B] 0 = 1 for all [AB 2 ] 0 /[A 2 ] 0 ratios (Figure 10).

B4-Type Monomer Effects
The next important stage involves investigating how a B4-type monomer affects the formation of hyperbranched polymers during the polyaddition of the AB2+A2+B4 monomer mixture.Figure 10 shows that, as in the previous case, the curves of pg over the initial molar fraction of a B4-type monomer (υB4 = [B4]0/([AB2]0 + [A2]0 + [B4]0)) tend to broaden when the AB2-type monomer is introduced into the system.Also, a distinctive minimum is observed on each curve at [A]0/[B]0 = 1 for all [AB2]0/[A2]0 ratios (Figure 10).When the polyaddition of the AB2+A2+B4 monomer mixture takes place, a B4-type monomer can be introduced into a macromolecule as a linear (when two B-groups in the monomer have reacted) or tri-(when three B-groups in the monomer have reacted) or tetrafunctional (when four B-groups in the monomer have reacted) branching unit.Figure 10 demonstrates that an increase in υB4 results in a decrease in pg when [A]0/[B]0 ˃ 1.This can be explained by an excess of A groups in the system within this range.Here, a B4-type monomer is introduced to a macromolecule mainly as a polyfunctional branching unit.
Same as the A2-type monomer does, it leads to an increase in the number of branches per macromolecule.A further increase in υB4 causes a decrease in both the absolute and specific number of branches per macromolecule, which is associated with the growth of a When the polyaddition of the AB 2 +A 2 +B 4 monomer mixture takes place, a B 4 -type monomer can be introduced into a macromolecule as a linear (when two B-groups in the monomer have reacted) or tri-(when three B-groups in the monomer have reacted) or tetrafunctional (when four B-groups in the monomer have reacted) branching unit.Figure 10 demonstrates that an increase in υB 4 results in a decrease in p g when [A] 0 /[B] 0 > 1.
Polymers 2024, 16, 426 13 of 20 This can be explained by an excess of A groups in the system within this range.Here, a B 4type monomer is introduced to a macromolecule mainly as a polyfunctional branching unit.
Same as the A 2 -type monomer does, it leads to an increase in the number of branches per macromolecule.A further increase in υB 4 causes a decrease in both the absolute and specific number of branches per macromolecule, which is associated with the growth of a free B group amount.As a result, the possibility of forming a three-dimensional grid is significantly diminished.The decrease in the number of branches per macromolecule is related to a decrease in the number of reactive A groups.The latter causes an increase in the number of macromolecules, resulting in the trend for short-chain linear polymers to form.
Thus, subject to [A] 0 /[B] 0 > 1, a B 4 -type monomer is introduced to a chain mainly as a polyfunctional branching unit; in other words, it acts as a core for a macromolecule to form and grow.Meanwhile, at [A] 0 /[B] 0 < 1, the monomer is introduced primarily as a linear unit and, eventually, terminates the growing polymer chain (Figure 11).

AB2-Type Monomer Effects
The plot of pg over the initial molar fraction of the AB2-type monomer (υAB2 = [AB2]0/([AB2]0 + [A2]0 + [B4]0)) is of particular interest (Figure 12).In contrast to the two cases above, there are no distinctive points at which gelation would not be observed, when [A]0/[B]0 > 1.The pg → 1 only when υAB2 → 1, which corresponds to the data from [15].The minimum of the function is also observed at [A]0/[B]0 = 1, and shifting from equimolar conditions results in an increase in pg.The pg value decreases with an increase in υAB2 when [A]0/[B]0 > 1.The reason lies in the fact that under these conditions, an AB2-type monomer can be introduced into the chain mainly as a trifunctional unit, thereby increasing the number of these units per macromolecule and causing a decrease in pg.On the other hand, with an excess of B groups ([A]0/[B]0 < 1), an increase in the AB2 monomer content promotes an increase in the number of terminal and linear units in a macromolecule.Thus, an AB2-type monomer can be introduced in a growing polymer chain both as a trifunctional and as a linear unit.The graph of DP n and DB vs. υB 4 is illustrated in Figure 11.As with the A 2 -type monomer, the DP n curve goes through a maximum.However, for the [AB 2 ] 0 /[A 2 ] 0 = 2/3 ratio, we can see a broad peak that is related to the area where the gelation is observed.The DB decreases with an increasing υB 4 due to a decline in the number of cross-linked units.Thus, when no gelation occurs, hyperbranched polymers with B end groups can be obtained, with DB = 0.34 and DP n = 5.4.These characteristic values are not much higher compared to the polyaddition of the A 2 +B 4 monomer mixture (DB = 0.33 and DP n = 5.0).

AB 2 -Type Monomer Effects
The plot of p g over the initial molar fraction of the AB 2 -type monomer (υAB 2 = [AB 2 ] 0 /([AB 2 ] 0 + [A 2 ] 0 + [B 4 ] 0 )) is of particular interest (Figure 12).In contrast to the two cases above, there are no distinctive points at which gelation would not be observed, when [A] 0 /[B] 0 > 1.The p g → 1 only when υAB 2 → 1, which corresponds to the data from [15].The minimum of the function is also observed at [A] 0 /[B] 0 = 1, and shifting from equimolar conditions results in an increase in p g .The p g value decreases with an increase in υAB 2 when [A] 0 /[B] 0 > 1.The reason lies in the fact that under these conditions, an AB 2 -type monomer can be introduced into the chain mainly as a trifunctional unit, thereby increasing the number of these units per macromolecule and causing a decrease in p g .On the other hand, with an excess of B groups ([A] 0 /[B] 0 < 1), an increase in the AB 2 monomer content promotes an increase in the number of terminal and linear units in a macromolecule.Thus, an AB 2 -type monomer can be introduced in a growing polymer chain both as a trifunctional and as a linear unit.
The minimum of the function is also observed at [A]0/[B]0 = 1, and shifting from equimolar conditions results in an increase in pg.The pg value decreases with an increase in υAB2 when [A]0/[B]0 > 1.The reason lies in the fact that under these conditions, an AB2-type monomer can be introduced into the chain mainly as a trifunctional unit, thereby increasing the number of these units per macromolecule and causing a decrease in pg.On the other hand, with an excess of B groups ([A]0/[B]0 < 1), an increase in the AB2 monomer content promotes an increase in the number of terminal and linear units in a macromolecule.Thus, an AB2-type monomer can be introduced in a growing polymer chain both as a trifunctional and as a linear unit.The plots of DP n and DB vs. υAB 2 are shown in Figure 13.In contrast to all of the aforementioned options, a monotonic increase in DP n is observed with an increase in υAB 2 over the entire range.Moreover, the curves appear to be almost linear up to υAB 2 ~0.90 due to the contribution of each component of the AB 2 +A 2 +B 4 monomer mixture to the polyaddition process.Nevertheless, a further increase in υAB 2 leads to an exponential increase in DP n , associated with a negligible contribution of A 2 -and B 4 -type monomers compared to the AB 2 type.The DB graph reaches its lowest value and then tends to grow at υAB 2 ~0.90 for the exact same reasons.The plots of DPn and DB vs. υAB2 are shown in Figure 13.In contrast to all of the aforementioned options, a monotonic increase in DPn is observed with an increase in υAB2 over the entire range.Moreover, the curves appear to be almost linear up to υAB2 ~ 0.90 due to the contribution of each component of the AB2+A2+B4 monomer mixture to the polyaddition process.Nevertheless, a further increase in υAB2 leads to an exponential increase in DPn, associated with a negligible contribution of A2-and B4-type monomers compared to the AB2 type.The DB graph reaches its lowest value and then tends to grow at υAB2 ~ 0.90 for the exact same reasons.As indicated above, the Flory Equation (1) for pg determination is relevant solely for the polyaddition of An+Bm monomers, without taking the AB2-type monomer effect into account.To figure out how pg = pA can be influenced by the composition of the AB2+A2+B4 monomer mixture, the curves of pg = pA vs. υAB2 were plotted for a range of the [A2]0/[B4]0 ratio of 1-10 (Figure 14).As indicated above, the Flory Equation (1) for p g determination is relevant solely for the polyaddition of A n +B m monomers, without taking the AB 2 -type monomer effect into account.To figure out how p g = p A can be influenced by the composition of the AB 2 +A 2 +B 4 monomer mixture, the curves of p g = p A vs. υAB 2 were plotted for a range of the [A 2 ] 0 /[B 4 ] 0 ratio of 1-10 (Figure 14).line corresponds to the point where pg ≤ 1.
As indicated above, the Flory Equation ( 1) for pg determination is relevant solely for the polyaddition of An+Bm monomers, without taking the AB2-type monomer effect into account.To figure out how pg = pA can be influenced by the composition of the AB2+A2+B4 monomer mixture, the curves of pg = pA vs. υAB2 were plotted for a range of the [A2]0/[B4]0 ratio of 1-10 (Figure 14).  Figure 14 demonstrates that each graph here can be accurately described by the linear equation p g = p A = a × υAB 2 + b, where υAB 2 ranges between 0 and 0.8.
The constant term (b) can be determined using the Flory Equation ( 1) at υAB 2 = 0. Due to the fact that the parameters α and ρ are constants for every single case of polyaddition, the correlation between p g = p A and the parameter r = [A] 0 /[B] 0 will appear as (14).Therefore, the curve of the constant term (b) vs. ([B] 0 /[A] 0 ) 1/2 should be linear (see Figure S1a in Supporting Information).Here, [A] 0 and [B] 0 represent A and B groups from A 2 and B 4 , respectively.
The slope coefficient (a) appears to be influenced by an AB 2 -type monomer introduction; however, the relationship between a and ([B] 0 /[A] 0 ) 1/2 also exhibits linearity (see Figure S1b in Supporting Information).
Thus, we can estimate the p g = p A value during the polyaddition of the AB 2 +A 2 +B 4 monomer mixture through the following Equation ( 15 The equation allows for the accurate calculation of the p g = p A value when υAB 2 is up to 80%.One of the most significant advantages of the invented model is an opportunity to calculate structural and molecular weight properties while considering substitution effects (Scheme 3).

Substitution Effects
Let us simulate the case of a monomer mixture polyaddition when [AB 2 ] 0 /[A 2 ] 0 /[B 4 ] 0 = 0.63/0.060/0.31,based on A 2 +B B 2 and A 2 +CB 2 polyaddition cases.The impact of the k 2 /k 1 ratio on the structural and molecular weight parameters of the hyperbranched polymers that are obtained under these conditions is illustrated in Figures S2 and S3 (Supporting Information), respectively.
As we can see from Figure S2 (Supporting Information), the negative substitution effect leads to DB → 0. That is, the topological mechanism of the macromolecule formation changes drastically, resulting in the formation of weakly branched polymers with numerous side-chained B groups.It seems nearly impossible to obtain hyperbranched polymers under these conditions.On the contrary, when k 2 /k 1 > 1, the possibility of forming knots increases the same way that the ratio does, causing an increase in the DB.
As we expected, DP n is unaffected by the presence of the substitution effect (see Figure S3 in Supporting Information).It is evident that, when no gelation occurs, the k 2 /k 1 ratio has no impact on the completion of the process.We can conclude that DP n is indifferent to the unequal reactivity of groups, unlike DP w .As the k 2 /k 1 ratio grows, an increase in the possibility of generating dendritic units can be observed.Thus, there is a higher chance of obtaining high-molecular-weight macromolecules, causing DP w to increase.
The derived regularities are expected for any values of the [A 2 ] 0 /[AB 2 ] 0 /[B 4 ] 0 ratio.However, each component of the system has a different impact on the forming of hyperbranched polymers.
A joint influence of the substitution effect and υA 2 on p g , when [AB 2 ] 0 /[B 4 ] 0 = 2, is shown in Figure 15.It illustrates that the positive substitution effect, i.e., when k 2 /k 1 > 1, leads to an decrease in p g compared to the statistical polyaddition of an AB 2 +A 2 +B 4 monomer mixture.For instance, if υA 2 = 0.14 and k 2 /k 1 = 1, p g takes a value of 1, whereas it reaches 0.86 when k 2 /k 1 = 10.Both positive and negative substitution effects modify the topological mechanism of macromolecule formation.At the initial stage of the polyaddition of an AB 2 +A 2 +B 4 monomer mixture, when k 2 /k 1 > 1, macromolecules with numerous dendritic units are mainly formed.These macromolecules are characterized by an enhanced content of B groups, which act as cross-linking centers, causing them to form a threedimensional mesh.In contrast, when the negative substitution effect (k 2 /k 1 < 1) takes place, the formation of polymers with numerous linear units is primarily observed during the entire process.This is attributed to the lower reactivity of B groups within linear fragments that is characteristic of this specific case.Therefore, the cross-linked polymer is less likely to form compared to the statistical polyaddition of the AB 2 +A 2 +B 4 monomer mixture.
Polymers 2024, 16, x FOR PEER REVIEW 17 of 21 mechanism of macromolecule formation.At the initial stage of the polyaddition of an AB2+A2+B4 monomer mixture, when k2/k1 > 1, macromolecules with numerous dendritic units are mainly formed.These macromolecules are characterized by an enhanced content of B groups, which act as cross-linking centers, causing them to form a three-dimensional mesh.In contrast, when the negative substitution effect (k2/k1 < 1) takes place, the formation of polymers with numerous linear units is primarily observed during the entire process.This is attributed to the lower reactivity of B groups within linear fragments that is characteristic of this specific case.Therefore, the cross-linked polymer is less likely to form compared to the statistical polyaddition of the AB2+A2+B4 monomer mixture.When pg reaches 1, an inflection appears in the surface of the graph due to the cessation of changes in pg.Thus, we can define an area in the graph that is depicted in Figure 15, which is limited by inflection points where υA2 and k2/k1 can be adjusted freely, named

Figure 1 .
Figure 1.Structural units in the AB2+A2+B4 system, where ba/ab is the product of interaction between A and B groups.

Figure 1 .
Figure 1.Structural units in the AB2+A2+B4 system, where ba/ab is the product of interaction between A and B groups.

Figure 1 .
Figure 1.Structural units in the AB2+A2+B4 system, where ba/ab is the product of interaction between A and B groups.

Figure 2 .
Figure 2. Plot of pg as a function of [A]0/[B]0 for A2+B4 system.Solid line depicts the data obtained through Equation (1), and dots represent the data calculated by the offered approach ([AB2]0 = 0).

Figure 2 .
Figure 2. Plot of p g as a function of [A] 0 /[B] 0 for A 2 +B 4 system.Solid line depicts the data obtained through Equation (1), and dots represent the data calculated by the offered approach ([AB 2 ] 0 = 0).
with initial conditions set as N = [AB 2 ] 0 , T = [AB 2 ] 0 , and the rest as zero) are illustrated in Figure3.

Figure 2 .
Figure 2. Plot of pg as a function of [A]0/[B]0 for A2+B4 system.Solid line depicts the data obtained through Equation (1), and dots represent the data calculated by the offered approach ([AB2]0 = 0).
Figure3shows that the maximum value of DB is 0.5 at p B = 0.5, which corresponds to data from earlier papers[16].Along with that, the gel point (PDI → ∞ or DP w → ∞) is achieved at p A → 1 (p B → 0.5), which is the same as in a conventional Flory paper[15].The validation of the comprehensive AB 2 +A 2 +B 4 model, incorporating all constituents, involves comparing the results obtained with our model to those obtained from the following set of reactions: A 2 +CB 2 →AB 2 (rate constant k c ) and AB 2 +CB 2 →B 4 (rate constant k b ).For example, from[41], when k c /k b = 200, the p g value equals 0.40 for the [A 2 ] 0 /[CB 2 ] 0 = 1 ratio and 0.56 for the [A 2 ] 0 /[CB 2 ] 0 = 3/2 ratio, respectively.If [A 2 ] 0 /[CB 2 ] 0 = 1, the mixture of [AB 2 ] 0 /[A 2 ] 0 /[B 4 ] 0 at a ratio of 2/1/1 is produced, whereas it is 4/4/1 for the [A 2 ] 0 /[CB 2 ] 0 =3/2 case.The DP w values for these mixture compositions, obtained with our suggested approach, are shown in Figure 4 and are similar to the ones specified in [41].Polymers 2024, 16, x FOR PEER REVIEW 9 of 21